INTERNATIONAL ORGANIZING/PROGRAMME COMMITTEE OF TSG4
CO-CHAIRS:
- EDWARD BARBEAU, Department of Mathematics, University of Toronto, Toronto, Canada
- HYUNYONG SHIN, Department of Mathematics Education, Korea National University of Education, Korea
MEMBERS: - EMILIYA
VELIKOVA, Department of Algebra and Geometry, Centre of Applied
Mathematics and Informatics, Faculty of Education, University of
Rousse, Rousse, Bulgaria
- ALEX FRIEDLANDER, Weizmann Institute of Science, Rehovot, Israel
- SHAILESH SHIRALI, Rishi Valley School, Rishi Valley, India
ASSOCIATED MEMBER: - AGNIS ANDŽĀNS, University of Latvia, Riga, Latvia
A DDRESS TO THE PARTICIPANTS
It is my great privilege and honor to greet the participants of the 10^{th }International
Congress on Mathematical Education. Gifted students of mathematics
today are not simply the gift of Nature neither unpredictable
prodigies. They are trainees of concrete scholars. They are coming
together from big and not so big states however always they are coming
from regions with developed schools of science tradition and math
education.
As rector of the University of Latvia I am
proud that scholars and students of our country are able to present our
local culture and especially that of math education in this forum. In
Latvia the Extramural School of mathematics of the University as well
the School of Professor “Cipher” is well known for students starting
from the age of 12, and has the history of more than 30 years. In the
Open Contest of math 2004 e.g., they were able to collect more than
4000 participants. It is great, that today always running between too
railway stations, two TV serials or even two sandwiches we are
reasoning how to stimulate ability of not traditional thinking or
thinking at all. Scholars and teachers must come together for shearing
experience of best approaches how to select the talent and how to
stimulate its growth.
Today we are not allowed to left teachers
alone and scholars of the university must be in close relationship with
them. We must work together and as close as possible.
On behalf of the University of Latvia, Riga I wish you productive meeting and number of difficult math problems for tomorrow!
Rector of the University of Latvia:
Professor Ivars Lācis, PhD
A DDRESS TO THE PARTICIPANTS
As the Rector of the University of Rousse it is my honor to congratulate the participants of the 10^{th}
International Congress on Mathematical Education, attending the
activities of Topic Study Group 4: Activities and Programs for Gifted
Students.
The University is currently offering
32 different bachelor courses in the field of engineering, management,
law, teaching and informatics. In the Faculty of Education are running
four of them. Master courses devoted to preparing high level teacher
trainers are included in the teaching program of the Faculty of
Education as well.
Throughout recorded history and
undoubtedly even before records were kept, people have always been
interested in persons who have displayed superior abilities.
In nowadays the encroachment of new
information technologies and the continuous integration of mathematics
with other sciences have a great need of:
- gifted mathematicians able to discover and summarize diverse conclusions within the information flow and to generate new ideas;
- personalities
able to develop their creative potential, enrich both their knowledge
and experience and apply them in socially useful areas and activities.
Those
factors and the conditions of contemporary global problems define that
the society needs new models of educational and upbringing activities.
Based on the positive experience they ensure opportunities for
developing gifted and talented students not as consumers or destroyers
but as creators of knowledge and future builders of a peaceful world.
From this point of view the TSG4 and
many of the Congress activities are aimed to dive further enlightment
of these important problems.
At the University of Rousse work
many holders of awards of national and International mathematics
competitions and some of them are the creators in Rousse of one of the
earliest schools in Bulgaria for training of gifted and talented
students in mathematics and informatics.
I sincerely wish you successful and
beneficial participation at the Topic Study Group 4: Activities and
Programs for Gifted Students!
Rector of the University of Rousse:
Assoc. Prof. Eng. Marko Todorov, Ph.D.
HIGH-ABILITY STUDENTS IN REGULAR HETEROGENEOUS CLASSES
Alex Friedlander
Abstract:
I would like to assert here that regular mathematics classes, with a
heterogeneous student population, can provide a suitable learning
environment for higher-ability students.I will give several examples of
tasks, and responses provided by higher-ability students in the context
of two projects. Next, I will discuss some characteristics of these
tasks and their potential for advancing the learning of mathematics by
these students.
BACKGROUND
Mathematics educators have presented
various models designed to cater to the needs of high-ability students
in mathematics. Most of these models (for example, enrichment lessons,
extracurricular math clubs or teams, accelerated or advanced math
classes or special schools,) are based on a homogeneous population of
high-ability students. However, for a wide variety of reasons, most of
the relevant student population are not reached and hence, do not
participate in any of these learning environments.
I would like to assert here that
regular mathematics classes, with a heterogeneous population, can also
provide a suitable, and frequently, even an effective learning
environment for higher-ability students. I will attempt to support this
claim by presenting and discussing several episodes from two learning
projects, basically aimed at a general student population.
The Compu-Math
project created a technologically based learning environment that
systematically covers the entire mathematics syllabus for grades 7-9.
As described by Hershkowitz and her colleagues (2002), the project is
based on the following principles:
- Investigation of open problem situations;
- Work in small heterogeneous groups, where the problem is investigated and discussed;
- Consolidation of the mathematical concepts and processes that arose in group work, and were discussed by the wholeclass;
- Investigations
that utilize computerized tools to facilitate operations within and
between various mathematical representations, to reduce the load of
formal algorithmic work, to enable the construction of mathematical
concepts and processes, to provide feedback regarding hypotheses and
solution strategies, and to resolve a real need to explain important
processes and products;
- Interactions
between students in a group or in the class as a whole, between
students and computerized tools, and between students and the teacher;
- Reflections on the learning process;
- Development of students’ mathematical language and their ability to provide convincing arguments.
The second project – Discoveries, presents
a collection of activities for grades 1-6. The activities were
originally designed for higher-ability math students, but the classroom
implementation demonstrated that the activities provide good learning
opportunities for a wider range of student abilities.
First, I will give several examples of tasks
presented in the context of these two projects and the responses to
these tasks provided by high-ability students. Next, I will discuss
some characteristics of these tasks and their potential for advancing
the learning of mathematics by high-ability students.
TASKS AND RESPONSES
Example 1. The sum of a polygon’s exterior angles (Grade 9).
The issue of exterior angles was
raised by the teacher after investigating of the sum of an n-sided
polygon’s interior angles. Two (higher-ability) students presented the
following arguments:
sum for the (n+1)-sided polygon is (n+1)· 180°
- so that this transition adds 180° to the sum. We can see, however,
that all these additional 180° were “taken up” by the interior angles –
and that “leaves us nothing to add” to the exterior angles. Thus, the
sum of the exterior angles does not increase and remains the same
through the whole process of increasing the number of sides. At the
initial stage of this process, we have the triangle, and the sum of its
exterior angles is 3 · 180° (the sum of the interior and exterior angles) minus 180° (the sum of the interior angles) – that is 360°.
Example 2. Calendar patterns (Grade 7).
In an introductory course to
algebra, the class found patterns in the dates of a monthly calendar
and gave reasons for their generality. Next, the teacher asked her
students to find and justify patterns in squares of dates (see Fig. 3).
The teacher was aware that an algebraic
argumentation for the “diagonal products” required that two binomials
be multiplied [(x + 1)(x + 7)], and therefore this was not within the
reach of her students at this stage of the course. However, she asked
whether somebody could justify this pattern. The students made various
attempts to prove the pattern. Some provided numerical examples, and
others gave incomplete algebraic explanations. The same student that
raised the issue provided the following argument:
Let us start with the product x· (x + 8) =
If we increase the first factor by 1, the product will grow by (x + 8).
Thus (x + 1)· (x + 8) = + x + 8.
Now, if we take away 1 from the second factor, the product will decrease by (x + 1).
Thus (x + 1)· (x + 7) = + x + 8 – x – 1.
So we see that this product is larger by 7 than the first one.
Example 3. Groups of matches (Grade 3)
The students had to put 21 matches
into two boxes, so that the numbers of matches relate to each other in
various ways. One of the conditions required that the number of matches
in one box equals half of the matches in the second. Besides the trial
and error method employed by many, two of the more talented students
offered the following solutions:
Ben:
"I took all twenty-one matches in my hand and put two on one side… and
after each two on this side, I put one on the other side… two on one
side and one on the other side…and so I continued and put on one side
fourteen, and on the other seven."
Tomer: "I divided the 21 by 3 and got 3 times 7 … added up two sevens and got 14 and 7… and then checked."
DISCUSSION
I would like to discuss some common characteristics of the situations described above.
- The
tasks were taken from a longer sequence of questions that focused on
the same problem situation. The tasks that belong to the same sequence
were of various levels of cognitive difficulty, and were usually
presented in an increasing order. The seventh graders, for example,
were first required to look for other, simpler patterns in monthly
calendar sheets in general, and in squares in particular.
The sequencing of tasks allows all
students to contribute to the solution process. Our observations show
that sequenced tasks provide effective learning opportunities for
higher-ability students as well, and enable them to realize their
potential. Their work in heterogeneous groups provides higher-level
solutions to the more basic questions and has a leading role at the
more advanced stages of the problem. Frequently, they are required to
provide explanations to other group members, and this activity has the
additional benefit of improving their communication skills and raising
their own level of understanding.- The
tasks were presented before, or at an early stage of learning a
mathematical concept. The lack of the necessary tools did not allow for
an algorithmic solution, or for the employment of previously
encountered methods or formulas. The same task, however, can be solved
routinely at later stages of learning. In our case, for example, the
third graders were not familiar with the method of dividing a quantity
into two parts, according to a given ratio, and the seventh graders did
not learn how to multiply two binomials in order to prove the
discovered pattern.
A
lack of possibilities in applying an algorithmic or a routine solution
raises the awareness of the need for learning a concept or a solution
method, encourages students to employ higher-level thinking skills, to
make connections to other domains and to be creative. A learning
environment that does not nurture
these skills is particularly disadvantageous for higher-ability
students, and consequently creates a need to separate these students
from the general student population. - Each
task can be solved in various ways and at various mathematical levels.
I presented here only a few solutions that were provided by
higher-ability students. Many other solutions (some of them incomplete
and others incorrect) were given for the same tasks. I presented here
only those solutions that were particularly original or required an
unusually high level of thinking for that particular age group.
Frequently, the use of a numerical example as a prototype for a more
general case or working by trial and error prove to be successful for
solutions or explanations, but we consider them to be at a lower
mathematical level.
My
main purpose in presenting the three episodes, was to show that
learning in heterogeneous classes can provide opportunities to employ
higher-level thinking, and hence can be favourable for higher-ability
students as well.
To conclude, I would like to emphasize
that the tasks and the students described here constitute segments of a
larger model. Moreover, the problems are a part of a cyclic learning
model, where each cycle includes the following components:
- an
open problem situation, which is solved by informal methods, and
increases the need for learning a particular concept or method,
- consolidation of mathematical concepts and processes,
- additional investigations of the problem or other applications,
- exercises and tasks that involve the learned concepts and processes in a mathematical context.
Our
classroom observations show that this model provides effective learning
opportunities for students of various mathematical abilities. In the
team investigation of open and unknown problems, there were authentic
interactions among students, and these do not create unreasonably large
gaps among students' contributions to the solution process. On the
other hand, the examples described above indicate that in this learning
environment, higher-ability students have the opportunity to realize
their potential and to provide qualitatively better solutions.
Moreover, these students raise the general level of their group's work
by playing a larger role in the solution process of the more difficult
parts of a problem, and by ensuring that all team members understand
the solution of the task at hand.
REFERENCE
1. Hershkowitz, R., Dreyfus, T., Ben-Zvi, D., Friedlander, A., Hadas, N., Resnick, T. & Tabach, M.
Mathematics curriculum development for computerized environments: A
designer-researcher-teacher-learner activity. In L.English (Ed.).
Handbook of International Research in Mathematics Education (pp.
657-694). Mahwah, NJ: Lawrence Erlbaum, 2002.
ABOUT THE AUTHOR
Alex Friedlander
The Weizmann Institute of Science
Rehovot
ISRAEL
E-Mail: Ntfried@wisemail.weizmann.ac.il
ONE BEAUTIFUL OLYMPIAD PROBLEM: CHESS 7X7
Alexander Soifer
Abstract
New Olympiad problems occur to us in
mysterious ways. This problem came to me one Summer morning of 2003 as
I was reading a never published 1980s manuscript of a Ramsey Theory
monograph, while sitting by a mountain lake in Bavarian Alps. It all
started with my finding a hole in the manuscript, which prompted a
construction of a counterexample (part b of the present problem). I do
not think that this Olympiad problem would have been born without the
manuscript’s authors allowing this minor mistake!
Chess 7x7 (21^{st} Colorado Mathematical Olympiad, April 16, 2004, A.Soifer).
a. Each
member of two 7-member chess teams is to play once against each member
of the opposing team. Prove that as soon as 22 games have been played,
we can choose 4 players and seat them at a round table so that each
pair of neighbors has already played.
b. Prove that 22 is the best possible; i.e., after 21 games the result of (a) cannot be guaranteed.
I found three truly marvelous solutions,
which are too long to fit in the marginal time allowed me in this
section of the ICME-10 Congress. The good news is, you have thus an
opportunity to hunt for a solution on your own. Good hunting!
ABOUT THE AUTHOR
Alexander Soifer
Princeton University, Mathematics, Fine Hall, Princeton, NJ 08544, USA
DIMACS, Rutgers University, Piscataway NJ, USA &
University of Colorado at Colorado Springs, USA
USA
asoifer@princeton.edu
asoifer@uccs.edu
http://www.uccs.edu/~asoifer
ALGORITHMS AND SYMBOL-GRAPHIC LANGUAGE IN MATHEMATICS EDUCATION AND
USING OF LAST IN THE INTERNET TECHNOLOGIES
Alexandr Chumak, Vladimir Chumak
Abstract
Solutions of main mathematical problems are
based on using of algorithms. Contemporary computer technologies give
the possibilities to create interactive algorithms for mathematics
education.
We present basic and algorithmic
didactic materials based on using the symbol- graphic language. They
are compact, visual and free of the international barriers. The
symbol-graphic language is developed for creation of these didactic
materials.
The Word service abilities allow to make them interactive for using in
the Internet technology. Basic didactic materials (BDM) contain
symbol-graphic formulations of basic definitions, theorems, formulas
from the instruction materials, which have been chosen for learning.
BDM are used for algorithmic didactic materials creation. They contain
algorithms of proofs of basic theorems and solutions of key problems.
We suggest to develop three forms of these algorithms: Speed up,
Linear, Forked algorithms.
Samples of such algorithms you may see on the Mathematical Education web site:
http://www.mycgiserver.com/~chumak/index.jsp (Main page/demo Testing/Algorithms).
ABOUT THE AUTHORS
Alexandr Chumak, Assoc. Prof., Ph.D.
Department of Mathematics
Kharkov National University of Radio Electronics
Lenin Avenue 14, Kharkov, 61166, UKRAINE
Tel.: +38 057 702 13 72
Fax : +38 057 702 10 13
E-mail: chumak@kture.kharkov.ua
Vladimir Chumak, Second year student
Artificial Intelligence Chair
Computer Science Faculty
Department of Mathematics
Kharkov National University of Radio Electronics
Lenin Avenue 14, Kharkov, 61166, UKRAINE
Tel.: +38 057 702 13 72
Fax : +38 057 702 10 13
E-mail: vchumak@ukr.ne
THE ADVANCED EDUCATION AND SCIENCE CENTRE
OF THE M.V. LOMONOSOV MOSCOW STATE UNIVERSITY – THE KOLMOGOROV COLLEGE
Anatolii Chasovskikh, Yury Shestopalov
Abstract: The
contribution outlines the history of creation, activities, and main
objectives of one of the leading educational enterprises in Russia for
gifted students in the field of natural sciences (mathematics, physics,
computer science, chemistry, and biology), The Advanced Education and
Science Centre of the M.V. Lomonosov Moscow State University---The
Kolmogorov college. The Centre was established in 1963 on Academician
Andrei N. Kolmogorov initiative and from the very beginning was set up
by the special Governmental decree as a boarding school. In 1988 the
school was incorporated into the structure of the Moscow State
University. More than 6000 pupils finished the College. Many students
of the Kolmogorov college won international competitions in
mathematics, physics and computer science. The College with enrollment
of about 350 students and a two-year educational cycle has elaborated a
unique system of testing and selecting gifted high-school pupils
through all over the country.
HISTORY
On the 23^{rd}
of August 1963 the USSR Council of Ministers adopted a resolution “On
establishing specialized boarding schools in the field of physics,
mathematics, chemistry and biology”. According to this resolution four
boarding schools were opened in Moscow, Novosibirsk, Leningrad and
Kiev. The Moscow school was set up under Moscow University. Many
outstanding Soviet scientists, physicists and mathematicians like
Andrei N. Kolmogorov, Isaac K. Kikoin, Ivan. G. Petrovsky took an
active part in its establishing.
In October 1988 by a Governmental
decree the Advanced Education and Science Centre of the Moscow State
University was set up. It consists of Moscow boarding school, Research
Department and the teaching staff departments.
TODAY
Nowadays AESC is one of the leading
secondary educational establishments in Russia. Its main goal is to
select and educate senior pupils from different regions of Russia,
those who show special interest in the field of mathematics, physics,
computer science, chemistry and biology. The educational process is
conducted by the professors and teachers of the five departments:
Mathematics, Physics, Computer Science, Chemistry, and Humanities. Now
the problem of establishing a department of biology is being
considered, the initiative was introduced by the faculty of
bioinformatics and bioengineering.
STRUCTURE
According to the AESC
Regulations the principal body that governs the Center is its Academic
Council. The Head of the Council now is Moscow University Vice-Rector,
professor Alexander V. Sidorovich. The Council consists of five Deans
of Moscow University faculties, three members of Russian Academy of
Sciences, corresponding members of Russian Academy of Sciences,
professors and seven University faculties administration
representatives.
EDUCATIONSYSTEM
The system of education at
the Center is based upon lectures and seminars, it is very close to the
system of Russian universities. When studying the Center students
attend practical work classes and take credit tests and exams in the
end of each term. For that reason the Center graduates can adapt to the
student life quite easily. The curricula and education programmes are
composed in such a way that they not only provide the necessary
preparation for the future study at University but also (and that is
the main thing) promote the development of student’s creativity in the
future.
ADMITTANCE SYSTEM
The influx of senior pupils who show
disposition to studying science subjects is provided by the unique
admittance system introduced by Kolmogorov school founders. The system
still exists and is very effective. Every spring more than 40 Russia
regional centers receive the Center Admittance Board representatives.
ACTIVITIES
Besides the compulsory courses the
pupils attend some special courses and take part in the work of special
seminars. The Center’s students win the prizes of different science
Olympiads of a very high level and deliver the reports at science
conferences. Every May beginning from 2001 the Center organizes a
Pupils science conference “Kolmogorov Readings”. The conference show
the results of the different Russia regions pupils’ creative work in
the field of science. III Kolmogorov Readings (2003) were devoted to
the centenary of Andrei N. Kolmogorov. An intellectual competition of
pupils, the young physicists tournament was founded at the Center. Now
it’s a prestigious international competition, its 2003 final was held
in Sweden, 22 best teams of all the world took part in its work. Some
preparatory work is assumed for the tournament participation. This work
is supposed to be connected with the solving of research problems,
learning how to conduct a scientific discussion, learning English, etc.
All that turns the pupils participation in the tournament into a
continuous creative enthusiasm.
TEACHERS AND TEACHING
Among school teachers
there are both very experienced teachers and very young scientists,
Philosophy Doctors and Full Professors. Many of them also teach at
Moscow University science faculties. The Center teachers are the
authors of manuals, books, original methodical findings. For more than
ten years the Center has been carrying out the research work in the
field of the Russian language under the supervision of professor A. N.
Kachalkin. Beginning from 2003 this direction of research has a
financial support within the Federal Programme “The Russian Language”.
Department of computer
science has worked out a new educational course “Mathematical
background of computer science”. It is of an interdisciplinary
character and is oriented for the pupils major in physics and
mathematics of a secondary school. The course’s authors Associate
professors E. Andreeva and I. Falina got a grant of the National
foundation for the scientific personnel preparation. The grant was
received on the competitive basis within the Programme of Russian
Government.
The Center teachers have
also contributed to the creation of multimedia manuals. One of the most
popular of them is a book of problems in computer science. The use of
this book allows to increase intensity of the educational process, to
rise the teachers’ professional return and provides the pupils’
motivation as to the studying of the computer science problems. The
inculcation of new computer science teaching methods at the Center and
the usage of the scientific results give the pupils the opportunity to
successfully take part in different competitions in this field of
science. Thus in 2003 our pupils won 6 first prizes out of ten, among
the victors were A. Lakhno and M. Ivanov. At All-Russia Computer
Science Olympiad the Center’s pupils got one diploma of the first
degree, four diplomas of the second degree and one diploma of the third
degree. In 2003/2004 academic year the Center team has become Russia’s
champion on programming.
INTERNATIONAL COOPERATION
Recently, special
programs of cooperation and exchange were initiated on international
level, in particular the agreement with the school for gifted children
established in Pusan, in the Republic of Korea.
REFERENCES
1. Kolmogorov v vospominanijah (Kolmogorov Memoires), A.N. Shirjaev, Ed., Moscow, Fizmatlit, 1993.
2. Javlenie cherezvychajnoe. Kniga o Kolmogorove (A book about Kolmogorov), Moscow, FAZIS, MIROS, 1999.
3. Est’ FMSh …: Sbornik fol’klornyh
proizvedenij (There is a college: A collection of traditionals and
lyrics), Moscow, Moscow University Press, 1995.
4. Sbornik statej ko dnju rozhdenija
A.N. Kolmogorova (A collection of papers devoted to A.N. Kolmogorov’s
birthday), Moscow, Nauchno-technicheskij centr “Universitetskij”, 2003.
ABOUT THE AUTHORS
Anatolii Chasovskikh
Yury Shestopalov
youri.shestopalov@kau.se
WORK WITH GIFTED STUDENTS IN THE INVESTIGATIONS OF POLYFORMS
Andrejs Cibulis, Ilze France
Abstract: The
paper deals with students` achievements in the investigation of
polyforms, including the compatibility problem for polyominoes and
polyiamonds as well as problems for tetratans. Attention is focused on
a surprising result in constructing convex shapes from tetratans.
Key words: Polyomino, Polyiamond, Compatibility, Polytans, Tetratans
INTRODUCTION
The problem of compatibility of
polyforms is attractive, however, very difficult in general and it has
been solved only in a few cases. This problem is a very good theme for
the gifted pupils and students to carry out research. The author’s
first findings on compatibility of pentominoes (a registered trademark
of Solomon W. Golomb) were announced in the third congress of WFNMC
(China, Zhong Shan, 1998). Having joined the efforts of several authors
the work Polyomino number theory has appeared in three parts, see [1-3]. The classic reference book on polyominoes is [4].
Notions.Polyominoes are connected plane figures formed of joining unit squares edge to edge. A polyominoA is said to divide another figure B if B may be assembled from copies of A. We also say that A is divisor of B, B is divisible by A, and B is multiple of A . If two figures have a common multiple, they are said to becompatible. Aleast common multiple
of two compatible figures is a common multiple with minimum area.
Analogously when a square (being the generating element for
polyominoes) is replaced by the equilateral triangle or regular hexagon
we obtain polyiamonds and polyhexes respectively. As far as it is
known, the first findings of hexiamonds’ compatibility were published
in Rodolfo Kurchan’s “Puzzle Fun” [5].
PUPILS` RESEARCHES
They are briefly described in [6]. Some new information has been added in the further text.
Valdemārs Plociņš and Margarita Lukjanska (Latvia) investigated compatibility of hexominoes and n-iamonds (when n <
6) respectively in their papers for young scientists’ contests (2002,
2003). In 2004 the contest paper of Alīna Česnovicka was dedicated to
compatibility of n-hexes (when n <
5). She has stated compatibility of 379 pairs of polyhexes by
constructing common multiples of a small size. The largest ones consist
only of 6 copies of pentaxehes. M. Lukjanska`s contest work on
compatibility of polyiamonds was highly appreciated in the 15^{th} European Union Contest for the Young Scientists, Budapest, Hungary in 2003. She was awarded Honorary Prize and was selected to represent the European Union Contest for the Young Scientists at the 46^{th}
London Internat ional Youth Science Forum, 28 July - 11 August, 2004.
In 2003 V. Plociņš succeeded in finding a 360-mino being the common
multiply for P- and A-hexominoes shown in Fig. 1.
Tetratans
(superTangrams) are the polyforms obtained by combining four unit
isoscel es right triangles snugly in every possible way. [8]. Several
convex t-shapes
(i. e. assemblable from different tetratans) can be found on the
Internet. The perfect contest paper “Analysis of t-polytans” for young
scientists was elaborated by Jurijs Bedratijs in 2002.
Hexagon containing all tetratans is
shown in Fig. 2. Heptagon shown in Fig. 3 is assembled from 13
tetratans. It has a very large number of solutions – 16821.
One can observe origins of tetratans in the
well-known puzzle “Tangram” containing two tetratans among its seven
pieces. There are two sources stimulating investigation of tetratans:
the manuscript of A. Liu [7] and the webpage of Henry Picciotto [8].
There are only 8 convex t-shapes assemblable from all 14 tetratans.
This result (including the number of assembles) coincides with that one
given on the Internet [9]. According to J. Bedratijs there are 297
convex t-shapes (including 5 tetratans). He also found 14 t-shapes with
4 axes of symmetry. J. Bedratijs came to an unexpected conclusion that
later became a theorem on an inseparable pair. Moreover, such an
inseparable pair is unique. He had analysed a large amount of solutions
(of convex shapes) obtained by the computer programme elaborated by A.
Blumbergs.
Colouring and parity
Colouring principles and parity are
of a great importance in solving many olympiad problems. Often they
help us also to solve more difficult problems. Generally speaking
colouring and parity are a powerful combinatorial toosl.
To investigate polytans we pay
attention to two types of colouring, namely A-colouring and
B-colouring, shown in Fig. 4 and Fig. 5 respectively.
There are four colours used in these
two colourings of the plane. Each tetratan covers 8 triangles. We
represent 8 as the sum of four numbers (c_{1 }, c_{2}, c_{3}, c_{4} ), where c_{k} is the number of triangles of k colour. We define a tetratan to beeven with respect to A-colouring (or B-colouring) if all the differences c _{1}- c_{2}, c_{2} - c_{3 }, c_{3} - c_{4}
are even. Analogously if all these differences are odd tetratan is said
to be odd. In the same fashion we can count the number of triangles
covered by an arbitrarily polytan and define its parity.
Let us note that each tetratan keeps
its parity (independently of its position) for A-colouring. The same
refers to B-colouring. Moreover, only two tetratans, namely the ones
shown in Fig. 6 and Fig. 7, change their parity when we pass from A- to
B-colouring. The tetratan T_{1 }(Fig.
6) is even with respect to A-colouring. It covers the following number
of red, grey, yellow and blue triangles: (2, 2, 2, 2). This tetratan is
odd with respect to B-colouring. The number of the covered triangles is
as follows: (1, 4, 1, 2), or (2, 3, 0, 3). These numbers may vary only
in the cyclic order that keeps parity. In its turn, tetratan T_{2 }(Fig.
7 changes its parity from odd to even with respect to A- and B-
colouring. These preliminary statements allow us to obtain the main
result.
Theorem (on the inseparable pair).
Each convex t-shape contains either two tetratans or none shown in Fig. 6 and Fig. 7.
Proof . The key to the proof is the following lemma.
Lemma . A convex t-shape keeps parity in both the colourings.
Let us first observe that the unit
square is of the same parity (actually even parity) in the both the
colourings. This immediately implies Lemma for rectangles, moreover,
for shapes consisting of unit squares. Now let us prove the Lemma for
convex t-shapes.
Remark
. The even number of unit triangles is not a sufficient condition to
keep the parity of convex polytan simultaneously in both the
colourings, e. g. the trapezoid in Fig. 8 is odd for A- and even for
B-colouring.
As tetratan consists of four unit
triangles the number of unit triangles of any t-shape is multiple of 4.
The same refers to the number of triangles touching the boundary of
t-shape. There are four positions (see Fig. 9) for triangles touching
the boundary of polytan. Irrespective of position each pair of
triangles shown in Fig. 8 keeps parity. Let us suppose that we have
excluded all such pairs. Then the boundary of convex t-shape may
contain four unit triangles shown in Fig. 9. As a set of these four
triangles keeps parity Lemma has been proved.
Some problems for research - Hexomino
P (see Fig. 1) is compatible with each of the remaining 34 ones.
Determine all others hexominoes sharing this property. At present 7
such hexominoes are known.
- Investigate the similar problem for hexiamonds.
- Prove or disprove a compatibility of the pentominoes (X, Z) and (X, W).
- Prove or disprove a compatibility of the hexiamonds shown in Fig. 11.
- Is the polytan shown in Fig. 3 the one having the maximum number of solutions?
CURRICULUM POSSIBILITIES FOR WORK WITH GIFTED STUDENTS
In Latvia the learning content of
mathematics is determined by the subject standard. The transition to
new primary education standards is taking place. It is planned that
they will be in force as of 2005. In the new standard both mathematical
and problem solving skills are equally important in developing
students` research skills in everyday learning process.
Until now a greater attention was
paid to the acquisition of certain knowledge of mathematics and
algorithms. It is welcome to give an inspiration for the gifted
students already in everyday learning process so that students would be
able to carry out serious research, for example in the field of
polyforms, when finishing a primary or secondary school. Pupils of
Latvia acquire two separate subjects – algebra and geometry starting
from Form 7. It gives the possibility to carry out different research
projects from simple tasks in a primary school to scientific research
in a secondary school. The aims of geometry are to develop
understanding of figures, to investigate and to classify them, etc.,
therefore just geometry develops the skills needed for various
research. The standard prescribes to acquire the skills to investigate
and analyse, for instance, the tetragons, but in the learning and
teaching process such figures as pentominoes and others are also dealt
with.
The education process orientated to pupils` research can be broken down into three levels:
The first level is work during the lesson.
A lesson is a basis for creating
interest in mathematics. It is very important to provide pupils with
tasks of different levels that can be found in the textbooks. For
instance, the following tasks are given in the textbook of geometry
[10] meant for Form 8:
1. A
rectangle 6 x 10 consists of equal squares. Cut it along square lines
into 12 figures so that there are no equal ones among them.
2. Solve a similar task for rectangles consisting of 3 x 20, 4 x 15, 5 x 12 equal squares.
3. A square consists
of 8 x 8 squares. One of them is cut out. Is it possible to cut the
rest of the square into isosceles rectangular triangles each consisting
of halves of two squares (see Fig. 12)?
The solutions of these tasks
require creativity and give the opportunity for pupils to work with
‘untraditional’ geometrical figures and solution methods.
Pupils get an idea about figure variety
and their common features by solving such tasks already in lessons, and
thus pupils` interest in research is developed.
The second level is
mathematics Olympiads. The preparation work is done usually after
lessons and the content is not included in the standard of the subject.
Pupils get additional mathematical knowledge and skills helping them to
develop various solution approaches.
The third level
is scientific research carried out by pupils of Forms 9-12. School
determines who will carry out this research - either all pupils or
those who want it themselves. Pupils choose a subject and his/her own
topic or a topic suggested by a teacher. As distinct from olympiads
this independent research sometimes yield more profound mathematical
results. Such a research is carried out within one year or in a longer
period of time.
ACKNOWLEDGMENT
This paper was partially supported by a state–investment project “Latvian Education Informatization System”.
REFERENCES
1. Cibulis A., Liu A. and Wainwright B. Polyomino number theory (I ), Crux Mathematicorum, v. 28, No. 3, April 2002, 147-150.
2. Barbans U., Cibulis A., Lee G., Liu A., and Wainwright B. Polyomino Number Theory (II
), in the book "Mathematical Properties of Sequences and other
Combinatorial Structures" published by Kluwer Academic, 2003, 93-100
pp.
3. Barbans U., Cibulis A., Lee G., Liu A., and Wainwright B. Polyomino Number Theory (III ), Gathering for Gardner, Atlanta, April, 2002.
4. Golomb S. W. Polyominoes: Puzzles, Patterns, Problems and Packings , Princeton University Press, NJ, 1994. (First edition: Polyominoes, New York: Scribners Sons, 1965)
5. Puzzle Fun by Rodolfo Marcelo Kurchan (Parana 960 5”A” (1017) Buenos Aires, Argentina, N6, August 1995.
6. Cibulis A. Common Multiples of Polyominoes and Polyiamonds: Theoretical, Practical, Learning and Teaching Aspects
, Proceedings of the Third International Conference “Creativity in
Mathematics Education and the Education of Gifted Students”, Bulgaria,
Rousse, August, 2003, 223-227.
7. Liu A.Super-Tangram , Andy Liu’s Exchange Puzzle, 20th International Puzzle Party, Los Angeles – August 2000.8. http://www.picciotto.org/math-ed/puzzles/index.html9. http://alpha.ujep.cz/~vicher/puzzle/polyform/tan/tan.htm
10. Andžāns A., Falkenšteine E., Grava A. “Geometry for Forms 7-9, Part V. Squares”, Riga, Zvaigzne ABC, 1997, pp. 82. (in Latvian)
ABOUT THE AUTHORS
Andrejs Cibulis, Ph.D., Assoc. Prof.
University of Latvia
29 Rainis boulevard, Riga, LV-1459
LATVIA
Phone: ++371 7211421
E-mail: Andrejs.Cibulis@mii.lu.lv
Ilze France, Mg.Math.
Ministry of Education and Science of the Republic of Latvia,
Centre for Curriculum Development and Examinations
2 Vaļņu Street
Riga, LV-1050
LATVIA
Phone: ++371 7814354
E-mail: ilze.france@isec.gov.lv
HOW DO GIFTED STUDENTS BECOME SUCCESSFUL?
A STUDY IN LEARNING STYLES
Bettina Dahl
Abstract:
The purpose of this paper is to argue that gifted students need special
programmes to avoid, for instance, psychological disturbances and/or
being turned of school and furthermore the paper argues that successful
students learn in a qualitative differently way from less successful
students and that even among the successful there are differences in
how they learn.
Keywords: Gifted, successful, analysing, generalising, problem-solving procedures, memory, learning theories, learning strategies.
INTRODUCTION
In this paper I discuss if gifted
students can take care of themselves, how successful students learn
mathematics compared to lower-achieving students, and if there are
similarities or differences in how successful high school students
learn mathematics.
1. CAN GIFTED STUDENTS TAKE CARE OF THEMSELVES?
There are many synonymous for
‘gifted’ such as ‘talented’, ‘able’, ‘successful’, ‘capable’
‘high-achieving’ etc. Basically these terms fall into two groups. Some
describe the state of actually being “good” (yet another expression)
namely: ‘successful’ and ‘high-achieving’ while others more describe a
person who has the potential
for being ‘successful’ or ‘high-achieving’. These terms are: ‘gifted’,
‘talented’, ‘able’, and ‘capable’. By introducing this distinction I at
the same time argue that there is not necessarily a direct link between
having the potential to become successful and actually being or becoming
successful in either daily life or the classroom. Some might argue that
gifted students can take care of themselves and helping gifted students
is perceived as taking resources from weaker students. I will argue
that it is a question of equality of opportunity to provide for the
needs of the gifted. The focus of the education system should be on
meeting every student where he is and help him to reach his full
potential. Special emphasis on the needs of the gifted is for instance
seen in the United Kingdom where Ofsted (Office for Standards in
Education) considers the needs of able students as part of equality of
opportunity. The DfEE (Department for Education and Employment) has
furthermore in two Circulars (14/94 & 15/94) in 1994 recommended
that in primary and secondary schools all School Prospectus should
include details of arrangements to identify and provide for
exceptionally able students ([2], pp. 16-17). One can argue as follows:
If we accept that it is the duty
of society ... to provide educational opportunities for all children
appropriate to their individual abilities and aptitudes, and if one
further accepts that some children are exceptional … then the issue is
settled. … For children to receive specialized educational treatment in
such circumstances is not for them to get more than their fair share;
they are simply receiving what, in their individual circumstances, is
appropriate. ([6], p. 4)
UNESCO’s Salamanca Statement, 1994,
declares that “The guiding principle that informs this Framework is
that schools should accommodate all children regardless of their
physical, intellectual, social, emotional, linguistic or other
condition. This should include disabled and gifted children, street and
working children, children from remote or nomadic populations, children
from linguistic, ethnic or cultural minorities and children from other
disadvantaged or marginalized areas or groups“([10], p. 6) and further:
“every child has unique characteristics, interests, abilities and
learning needs; education systems should be designed and educational
programmes implemented to take into account the wide diversity of these
characteristics and needs ([10], p. viii
). Hence, it is not “un-just” to help successful and gifted students;
they too have a right to receive what fits them. Furthermore: “all
children are born as unique individuals, each different from the other,
and in developing them we need to make them more equal by overcoming
whatever inabilities they may have and more different from one another
by developing their abilities and propensities” ([12], p. 31). In that
sense, special education (for both weak and strong students) both
improves inabilities and develops the person’s talent(s).
No student can progress towards the
limit of his capacity unless he has an opportunity to learn: “Mozart
might have had an extraordinary aptitude for music, but this could
hardly have been realized unless his parents possessed a piano. It is
at best inefficient to rely on nature or chance to develop talents,
while for potentially gifted children in homes with limited cultural
horizons it borders on neglect” ([6], p. 5). Studies have furthermore
shown that some gifted students are underachieving and sometimes suffer
psychological disturbances including poor concentration, exaggerated
conformity, excessively inhibited behaviour, anxiety, social isolation
and aggressiveness, or the opposite such as extreme passivity ([6], p.
6). Other studies have shown that if gifted students are held back or
bored in school, some of them will be ‘turned off’ by school, achieve
far below the level of which they are capable, drop out, fail, or even
become delinquent ([6], p. 14). Another study showed that children who
could read before beginning in school do not develop new competencies
if they are just being taught what they already know, and many of the
early readers later loose interest in reading. These students therefore
need special attention and need to be challenged ([5], p. 6). It is
further stated that some gifted students deliberately hold themselves
back:
Some able students receive a
shock when they move on to university. The leisurely study habits which
had ensured reasonable grades in the mixed ability classes in secondary
schools prove to be inadequate for the more intellectually demanding
environment of the university. … there are too many students of high
ability who wastefully drop out. … it is very probable that many gifted
children ‘learn to be average’ or deliberately hold themselves back in
order to have a quiet life in school: this is the phenomena of ‘faking
bad’. ([6], pp. 14-15)
Gifted students therefore need
adequate stimulation. Studies suggest that association with other
students of high ability raises a student’s level of performance. One
study showed that the “overall intellectual level within a group had an
effect on the development of the level of individuals within the group
- contact with clever people tended to raise the level of ability of
the less clever” ([6], p. 13). Another study showed that “down to an IQ
of about 65, mentally retarded students taught with normal peers
achieved better than those who were taught in self-contained classes”
([6], p. 13). And further “that students of high ability
were penalized academically by being taught with students of lesser
ability” ([6], pp. 13-14). Hence, it might seem as a Catch-22
situation: when each student seems to do better when taught together
with more gifted students, and suffer from being with less gifted
students, there will always be a “looser” in the “game”. However, it
does not have to be this way if it is the teachers’ duty to stimulate
the students according to their abilities, which is also what is argued
below:
Refusing to make special
provision for the unusually able, on the grounds that they are
necessary for the optimal development of the other children, means that
adults shrug off the task of promoting the development of less gifted
youngsters onto the shoulders of clever children. Naturally, educators
should be looking at the needs of the less gifted, but not at the
expense of the gifted and talented. ([6], p. 14)
2. GIFTED STUDENTS COMPARED WITH OTHER STUDENTS
Analysing
When gifted students work on a mathematical problem they perceive the mathematics of it analytically
, which means that they isolate and assess the different elements in
its structure, systematise them, and determine their ‘hierarchy’. At
the same time they perceive the mathematical material synthetically
, and here combine the elements into complexes and investigate the
mathematical relationships ([4], pp. 227-228; [9], p. 15). Gifted
students perceive problems as a composite whole, while average students
see a problem in its separate mathematical elements. It is only through
analysing the problem that the average students are able to find the
connections of the mathematical elements. Lower-achieving students have
great difficulties in establishing these connections, even when they
achieved help. The speed of the analytical-synthetic process in the
gifted student is so fast that they see its ‘skeleton’ at once. It is
often impossible to trace the process. The fast grasping of a problem’s
structure has been observed to be the result of exercises, but gifted
students need only a minimal number of exercises to make the
analytical-synthetical perception arise ‘on the spot’ ([4], pp.
228-232).
Example 1 ([4], p. 230)
A 6^{th }grade class gets the following problem:
A jar of honey weighs 500 g, and the same jar, filled with kerosene, weighs 350 g.
How much does the empty jar weigh?
A gifted 3^{rd} grader (V.L.) answers (E is the experimenter):
V.L.: | And then? |
E: | That’s the whole problem. |
V.L.: | No, that isn’t all. I still must know how much heavier honey is than kerosene. |
E: | Why? |
V.L.: | Without
that, there could be many solutions. There are two unequal quantities,
connected by the fact that some of their parts are equal. There could
be very many of these parts. To limit their number, we must introduce
one more quantity, characterizing the ’remainder’. |
A less gifted 6^{th} grader was not able to solve this problem, even when he got the hint: ”honey is twice as heavy as kerosene”.
Generalising
The ability to ‘grasp’ structural
relationships in a generalised form is a central feature for the
productive thinking ([4], p. 234). The gifted students do this on the
spot whereas lower-achieving students need a lot of practice and
exercises covering all possible cases and levels before an elementary
level of generalization is possible ([4], pp. 240-242). Gifted students
can analyse one phenomenon and generalise from this by separating the
essential features from inessential. Their method is to infer “the
features’ generality from their essentiality. … to be essential means
to be necessary and, consequently, it should be
common to a number of phenomena of this type, that is, it should
inevitably be repeated” ([4], p. 259). Lower-achieving students
perceive the generality of features by contrast.
Example 2 ([4], p. 241)
A gifted student, O.V., had previously solved just a single example using the formula of the square of a sum: (a+b) = a + b + 2ab. Then he got the problem: (C+D+E) × (E+C+D).
(E is the experimenter.)
Procedures for problems-solving
The trials for
problem-solving for lower-achieving students are blind, unmotivated,
and unsystematic. On the contrary gifted students have an organised
plan of searching ([4], p. 292). Gifted students switch easily from one
mental operation and method to another, they have great flexibility and
mobility in their mental processes in solving mathematical problems,
and it is therefore easy for them to reconstruct established thought
patterns. For average students it is much harder to switch to a new
method of problem-solving. Lower-achieving students experience even
greater difficulties in that ([4], pp. 278-282). For the gifted
students the trials are a way to thoroughly investigate the problem
through extracting information from each trial. Without having finished
the trial, gifted students seem to know if they are on the right track.
This is owing to the existence of an acceptor, which is a psychological
control-appraisal mechanism, where ‘line-of-communication’ is received
from each mathematical operation. Under this acceptor lies a
generalised and concentrated system of past mathematical experience
([4], p. 293). The gifted students thoroughly investigate the problem,
which may suggest that they enjoy working with mathematics. The
emotional factor is seen in that they often try to solve the problem in
a more simple way or improve the solution and they show satisfaction
when the solution was economical, rational, and elegant ([4], p. 285),
which is seen in the example below.
Example 3 ([4], p. 279)
Memory
Gifted students do not
have a “better” memory than lower-achievers, but gifted students
usually remember the general character of a problem-solving operation
and not the problem’s specific data. On the contrary, lower-achieving
students usually only remember the problem’s specific facts. The
mathematical memory of gifted students is selective and only keeps the
mathematical information that represents generalised and curtailed
structures. This means that the brain is not loaded with extra
information which makes it possible to retain the information longer
and use it more easy ([4], pp. 299-300).
Example 4 ([4], pp. 298-299)
A lower-achieving student, I.G., solved the problem: 113 – 112 with
the experimenter’s help. After one week she had forgotten the
mathematical relationship (difference of squares) but remembered that
the problem had used the numbers 112 and 113.
The figure below ([4], p
297) shows the forgetting-curve of generalized relations, concrete
data, and unnecessary data for gifted students.
3. SIMILARITIES AND DIFFERENCES IN HOW SUCCESSFUL HIGH SCHOOL STUDENTS LEARN MATHEMATICS
I have previously done research
([1]) in how ten successful high school students (aged 17-20) explain
how do they come to understand a mathematical concept new for them.
Four students were Danish (Z, Æ, Ø, Å) and six were English (A, B, C,
D, E, F). They were interviewed in pairs and fours: Z-Æ-Ø-Å, A-C, D-E,
B-F. All studied mathematics at the highest level possible in each of
their school system and their teachers selected them as successful. The
study rests on the assumption that successful students have a
metacognition, which means that they have knowledge about and
regulation of their cognition. Knowledge of own learning means that one
has relatively stable information about own learning processes. This
knowledge develops with age and there is a positive correlation between
the degree of one’s insight into own learning and one’s performances on
many tasks. Regulation of own learning is the planning before one
begins to solve a problem and the ongoing evaluation and control while
one learns something new or solves a problem ([8], pp. 138-141). I
asked general explorative questions to not be leading. In their own
words, the students, among other things, describe the relationship
between visualization and verbalization and the individual and the
social side of learning. I used the learning theories of Ernest, von
Glasersfeld, Hadamard, Krutetskii, Mason, Piaget, Polya, Sfard, Skemp,
and Vygotsky in the analysis. The ten students fall in different groups
regarding their preference for learning style. For language reasons the
examples below are from the English interviews.
Visualization and verbalization
Regarding verbalization, Student A,
C, Z, and Æ tell that an oral explanation helps the one that is
talking. For instance Student Æ tells that very often if she tries to
explain the mathematics to a person, then when she is explaining it,
she understands it herself. Student C adds that verbalization (saying
things out load) helps the visualization:
C: | If
you just read it in your head, I just read it and I don’t understand.
If you come across a phrase which sounds really awkward, like the one
here: “in the plane whose interior intersects the diagram in one of the
configurations involved”, you just read it over, but you don’t think
about it, but if you read it out, then you think about the disc, and
then you visualise the disc, and then you visualise the plane. |
If one focuses on
visualization, Student Z describes that she does not need to see
pictures but to see things in writing. Another group consists of the
students being moderately positive to visualization (A, D, E, F, Æ, Å).
They describe that pictures sometimes makes it easier. The two examples
below are from two interviews, but both students talk about some knot
theory that I had given them to work on, to initiate a discussion:
F: | The
first thing to do [when explaining to classmates] would be to draw some
knots and then show the respective oriented diagrams whatever they are
called, these graphs. |
E: | Actually
the graphics are a big help, I know I was relieved when I got to this
first graphic [D & E laugh], it showed, it kind of showed what they
were. |
A third group (C, Ø)
perceive visualization as something that is very important for
learning. Student Ø says for instance that visualisation can be a great
advantage for instance in terms of cosine equation. Student A, C, and Æ
argue that the relationship between visualisation and verbalisation
depends on the mathematics. For instance Student A:
A: | Sometimes
if you just READ a sentence like the first here “A link is a finite
collection of mutually disjoint disjointed simple closed curves”, if
you just take that as the words it is difficult for you to seeany other
way that it can work. If you start to visualize what it is talking
about then you can SEE all the different ways in which it can happen. |
I: | So
is it important that the words come first and then you can see what the
words are about, or do you want to see it first and then be explained
and then get the words? |
A: | Well.
I don’t know really. It probably depends on the sort of problem. If it
is a very visual problem where you have to think it through maybe in 3
dimensions, I think it is probably better to have pictures first and
maybe dealing with graphs as well. If you are doing vectors it may be
better to have the picture first, and then the words explaining how it
works. But if it is more a linear methodological process it might be
better to have the words first and then pictures to help you
understand, cause it is the words you are trying to understand. But
where there is something visual like this or graphs or vectors it’s the
pictures that you are trying to understandso the one supports the other
depending on what it is you are trying to learn, I think. |
The students group as below. They do not all express something in this connection.
1. Oral explanation helps the one that is talking: Student: A, C, Z, Æ
2. Mainly verbal: Student Z
3. Relatively visual: Student A, D, E, F, Æ, Å
4. Mainly visual: Student C, Ø
5. Depends on circumstances, for instance the type of mathematics: Student A, C, Æ |
Individual or social learning
It seems that the students argue
that learning has both a social and an individual side. The value of
the social side is mainly when the students experience problems with
self-learning. After input from the outside, they can continue on their
own. Particularly Student C, D, E, F, Z, Æ, Ø, and Å argue this.
Conversely, Student A and B express that they learn more through the
discussions than through self-activity. I will now investigate some of
these students further. First an example from the interview with
Student D and E who both emphasize the individual side:
E: | I
know people do er differently but it’s all very individual even if you
work on something together er and you’re both aiming to solve the
problem, I mean you’ll do it completely differently from someone else
and quite oftenly I find I don’t like other people’s styles, you know,
you always get your own. |
D: | Yea,
but that’s the whole point in learning why you do something rather than
how, because a lot of the times there are some things I do my own
little way, I don’t necessarily follow the textbook example and it,
that doesn’t matter cause I understand what I’m doing, what I want to
achieve by doing it, and I can get the right answer by doing it a
different way to the textbook, but, you know, it just works for me. |
Student Z tells that basically the
teacher cannot help her learn mathematics. She has to work for herself,
she has to take the responsibility for her learning.
The social side seems to be most
important to Student A and B. They tell that they learn more through
the discussions that by sitting for themselves. Student A tells that it
is better to work for oneself through looking at examples than being
told “this is wrong, try again”. This might sound like support of
individually aimed theories such as Piaget’s ([7], pp. 16-19), who
argued that the students through own activities in interaction with the
surrounding world construct their knowledge. In that case, it would be
a contrast to what Student A also tells, namely that he learns best
through the discussions. However, what Student A actually says is that
it is better to “work for himself” instead of “being told”. To “be
told” is not necessarily the same as a discussion, perhaps rather the
opposite. Student A therefore priorities as follows: 1. discussion, 2.
work for himself, 3. be told. Student A is therefore probably more a
“Vygotsky-student”. Vygotsky ([11], pp. 56-57) argued that
internalisation happens through activity and communication in social
interaction. Student B supports this (I is the interviewer):
B: | I think it certainly helps if you can discuss it with someone else. Two brains are better than one. |
I: | Why? |
B: | Er,
one person can have one idea which should trigger another idea in the
other person’s head which the first person wouldn’t have had, and then
the second person having said that thing, and then one thing leads to
another if you got two people to think. |
Student C and Æ do on own initiative
use words such as “combination” or “two-way thing” to describe the
relationship between the individual and the social side of learning,
however still with emphasis on the individual side. An example:
C: | It
all boils down to the teaching method and the teacher. It’s a two-way
thing you see, it’s more about you learning, you being able, no, you
learning as well you being taught properly. If you are taught in a way
that you can fit in, you know, then it is good. |
Student Æ and Z tell that
one can learn through discussions with oneself, perhaps even better
than in groups. The students group as follows:
1. Individual: Student C, D, E, F, Z, Æ, Ø, Å
2. Social: Student A, B
3. Combination (emphasis on the individual side): C, Æ
4. Learn from discussing with oneself: Student Æ, Z |
Relating to the discussion
above about if it is best to work with students on the same level or
not, Students A, C, and Z argue that if they have problems learning, it
is best to discuss with someone who also has not understood. The reason
is that they do not want to be bullied. Student Å says that discussion
with “equals” is good in any case, and Student B argues that it is more
helpful to discuss with someone who knows.
The students’ learning styles
It seems that the students
divide themselves into different groups that either support individual
or social learning, or support visualization or verbalization with a
preference for one, or describe a kind of combination between these
factors. The preferences are independently of nationality or gender (D,
Z, Æ, Ø, Å are girls; A, B, C, E, F are boys). The students therefore
learn in different ways. These students constitute a rather homogeneous
group of gifted students, which might suggest that if one looks at the
whole spectrum of students, even more learning strategies or
preferences might emerge. It is therefore vital with variation in the
teaching. Teaching after a one-sided pedagogical theory or idea will
lead to that some students are lost.
Furthermore Student A says
that the learning strategies one uses are connected with the ways one
has been taught to do things. Other students explain the following in
two of the interviews (I is the interviewer):
D: | When
I first came here [to the new school], the first couples of weeks I
found math very difficult because it is kind of hard to adapt to a
different teaching style. |
I: | I
noticed when you talked about presenting it to the class you wanted to
give them examples and you also mention, while you were talking, that
it would be nice with examples. Why? |
B: | That’s
because the way that we’ve always been taught is using examples
thoroughly to explain, so that’s the way we think the people in our
class will understand it easiest, explain through examples. |
This phenomena might be explained by that the teaching methods must be part of, what I would express as a zone of proximal teaching (ZPT)
, inspired by Vygotsky’s ZPD. According to Vygotsky, the potential for
learning is limited to the “zone of proximal development (ZPD)”, where
ZPD is the area between the tasks a student can do without assistance,
and those, which require help ([11], p. 86). Hence, if a (new) teacher
uses teaching methods that are too “far away” from teaching styles the
students are used to, the students might have difficulties in learning.
4. CONCLUSIONS
Gifted students need special
programmes not just to make sure that they, for instance, avoid
psychological disturbances, but there are also major differences in how
successful and less successful students learn mathematics and even
among the successful students there are some differences in learning
styles. Also Hadamard wrote about different mathematical minds and
“even among men who are born mathematicians, important mental
differences may exist” ([3], p. 11). Successful students furthermore
experience problems when facing a new teaching style but they do
nevertheless seem to overcome these difficulties. This is to some
extent supported by [4]’s investigations that showed that successful
students switch easily from one mental operation and method to another.
REFERENCES
1. Dahl, B.
(2004). 'Analysing Cognitive Learning Processes Through Group
Interviews Of Successful High School Pupils: Development And Use Of A
Model'. Educational Studies In Mathematics. (Forthcoming)
2. Eyre, D. (1995). School Governors & More Able Children (The National Association for Able Children in Education (NACE)).
3. Hadamard, J. (1945). An Essay on The Psychology of Invention in the Mathmematical Field (New York, Dover).
4. Krutetskii, V. A. (1976). The Psychology of Mathematical Abilities in Schoolchildren (Chicago, The University of Chicago).
5. Kyed, P. (ed.) (2002). Undervisning Af Elever Med Særlige Forudsætninger [Teaching Of Gifted Students]. (Vejle Denmark, Krogh).
6. McLeod, J. & Cropley, A. (1989). Fostering Academic Excellence (Oxford, Pergamon).
7. Piaget, J. (1970). Genetic Epistemology. (New York, Columbia University).
8. Schoenfeld, A. (1985). Mathematical Problem Solving. (London, Academic).
9. Tall, D. (Ed.) (1991). Advanced Mathematical Thinking (Dordrecht, Kluwer).
10. UNESCO (1994). The Salamanca Statement and Framework for Action on Special Needs Education. Adopted, Salamanca, Spain, 7-10 June 1994.
11. Vygotsky, L. S. (1978). Mind in society: The Development Of Higher Psychological Processes. (Cambridge Massachusetts, Harvard University).
12. Young, P. & Tyre, C. (1992). Gifted or Able? Realizing Children's Potential (Buckingham, Open University).
ABOUT THE AUTHOR
Bettina Dahl (Soendergaard), Ph.D.
Norwegian Centre for Mathematics Education
Norwegian University of Science and Technology (NTNU)
Realfagbygget A4, 7491 Trondheim
NORWAY
E-mail: bdahls@math.ntnu.no
DIFFERENTIATING MATHEMATICS VIA USE
OF NOVEL COMBINATORIAL PROBLEM SOLVING SITUATIONS:
A MODEL FOR HETEROGENEOUS MATHEMATICS CLASSROOMS
Bharath Sriraman
Abstract: It
is a well-established fact that the curriculum at most public schools
do not provide any explicit enrichment for gifted students in the
mathematics classroom. In the United States, researchers in the
curricular division of gifted education have suggested parallel
curricula, curriculum compacting and curriculum differentiation [9] as
a mean to provide enrichment to the regular school curriculum. The
shortcoming of these approaches is that they require considerable
professional development of teachers and subsequent support from the
school administration for successful implementation in the classroom.
In fact numerous studies have shown that this approach is only
implemented in a small fraction of the K-8 classrooms and is a
statistical rarity in the 9-12 classrooms. In this paper I present a
radically different approach to differentiation, one that is
specifically content driven and allows for creative mathematics to
manifest in the classroom. In particular I describe the structure of
two teaching experiments, their implementation and the fascinating
outcomes with 9th grade public school students at a rural American high
school. The paper concludes with an argument for the use of novel
problem-solving situations with an underlying structure as a means of
both differentiating and enriching the regular mathematics curriculum.
Key words: Abstraction;Curriculum
Differentiation; Combinatorics; Generalization;
Heterogeneous-Classrooms; Pedagogy; Mathematical giftedness; Problem
Solving.
INTRODUCTION
Today, in the U.S., in
spite of the availability of alternative research-based curricula, the
traditional mathematics sequence of
Algebra-Geometry-PreCalculus-Calculus dominates the curricular
landscape. Calculus is the utopian dream and thought of as a capstone
class by most college bound students. As a full time teacher in a rural
Midwestern U.S. public school, I was responsible for teaching various
sections of " traditional" Algebra to 9th grade students (13-14 year
old). These classes consisted of students of varying mathematical
abilities and included some mathematically gifted students, who had
received glowing recommendations from their previous middle school
teachers and counselors as having the potential to do "accelerated"
work. The 9th grade Algebra curriculum focused on the study of the
properties of the real number system with tangential excursions to
analytic geometry. Research indicates that most gifted students have
already mastered up to 60% of the mathematics content encountered in
high school [8]. This posed the conundrum of potentially boring these
students in my classroom and perhaps even robbing them of utilizing
their mathematical capabilities. The solution to this conundrum was
found in an often quoted survey article [4] which called for increased
emphasis in enumerative or non-continuous mathematics because of its
independence from Calculus.
Each teaching experiment
was approximately 3 months long and was structured as follows. The
students in my Algebra classrooms had to maintain a mathematics journal
in which they solved problems from Combinatorics and Number Theory,
topics rarely covered in the traditional high school curriculum.
Students were given cues that asked them to reformulate the problem in
their own words, devise an explicit strategy prior to solving the
problem, and reflecting on the solution. The pedagogical goal was to
pose problem situations that had an underlying structure or a principle
which could be potentially discovered by the gifted students in the
classroom, but simple enough for all students to have the opportunity
to create representations, think abstractly and create generalizations
at varying levels.
EXPERIMENT 1
The first experiment consisted of
assigning 8 problems over the course of 3 months. The solutions of four
of the eight posed problems were characterized by the construction of
Steiner Triple Systems (STS), a useful combinatorial structure. For
instance, the following STS problems (Fig. 1) were posed un-sequentially in
the order 1-3-6. The five mathematically gifted students in the class
were able to uncover this sequence and gain an insight into the
structure of Steiner Triple Systems. They were successful in devising
an efficient strategy for counting triples in these problems and making
a substantial attempt at tackling more general questions.
Gifted students who discovered
the underlying structure of these three problems were given the
following additional challenging questions.
1. In
an arrangement of “n” objects in triplets, how many ways can each pair
of objects appear in a triplet once and only once? How many pairs are
possible?
2. How can you create problems like the ones above, which are always solvable (or unsolvable)?
EXPERIMENT 2
In the second experiment, five problems
which represented diverse situations but whosesolutions were
characterized by the application of the pigeonhole principle were
assigned sequentially over
the course of 3 months. In this experiment, the four students (out of
the 9) who were identified as mathematically gifted were able to
discover the pigeonhole principle by abstracting the structural
similarities from the assigned problems.
DISCUSSION, CONCLUSIONS AND FUTURE WORK
Based on these experiments of using
combinatorial problem-solving situations as means of differentiating
the mathematics content the following general commendations are
provided. As reported here carefully chosen classes of problems which
involve the discovery of a combinatorial principle [5] or a
combinatorial structure [6] in a heterogeneous setting along with the
use of journals as a means of establishing communication with students,
led to the positive outcomes of gifted students abstracting structural
similarities, conceptually linking related problems, utilizing their
creativity pursuing general solutions and creating theoretical
generalizations [2]. Typically, empirical generalizations are the
result of being unable to discern the structural features of a problem and are based on the superficial aspects of a task or problem. A theoretical generalization on
the other hand is characterized by the discernment of the essential
elements of a task or a problem [1], [2]. The distinction between
empirical and theoretical generalizations is illustrated by using the
notion of ‘roundness’ [1]. That is, "roundness" can be empirically
abstracted from a dish, a wheel, etc., but this empirical, everyday
notion of circularity does not reveal the underlying theoretical
construct of circularity as a locus of points at a constant distance
from a fixed point which is not readily apparent in the mere appearance
of roundness [1]. This does not mean that empirical generalizations do
not have any pedagogical value. The teacher can use the students’
empirical generalizations and hopefully facilitate the discovery of the
theoretical generalization.
In conclusion, I highly recommend problems
that result in the discovery of a combinatorial structure or principle.
One research implication would be construct a "metric" that sorts a
class of related combinatorial problems into increasing levels of
difficulty, which can then be given to teachers to use in the regular
classroom. The novelty of this approach of using combinatorial
problem-solving situations in the heterogeneous classroom is that they
are easy to understand which allows weaker students to somehow model
such problems by trial and error strategies thereby resulting in the
positive outcomes of problem-solving, systematizing thinking, and
empirical generalizations [2]. The use of a class of contextually
different but structurally isomorphic problems can be used in the
heterogeneous classroom to foster independent thinking as students
initially tackle the problem. The teacher can also encourage
flexibility and sharing of solutions as students begin to explore
simpler contexts and alternative representations. Ultimately it is
hoped that teachers will focus on problem structure as students
generalize across dimensions, and further present problem-posing
opportunities as a culminating activity [7].
REFERENCES
1. Davydov, V.V. (1988). The concept of theoretical generalization and problems of educational psychology. Studies in Soviet thought, 36, 169-202.
2. Davydov.V.V. (1990). Type of generalization in instruction: Logical and psychological problems in the structuring of school curricula.” In Soviet Studies in Mathematics Education (Vol. 2), edited by Jeremy Kilpatrick, National Council of Teachers of Mathematics, Reston: VA.
3. Gardner, M. (1997). The Last Recreations. New York: Springer-Verlag.
4. Kapur, J. (1970). Combinatorial analysis and school mathematics. Educational Studies in Mathematics, 3, 111-127.
5. Sriraman, B. (2003) . Mathematical giftedness, problem solving, and the ability to formulate generalizations. TheJournal of Secondary Gifted Education . 14(3): 151-165.
6. Sriraman,B (2004) . Discovering Steiner Triple Systems via Problem solving. The Mathematics Teacher, 97 (5), 320-326.
7. Sriraman, B. & English, L.D. (2004). Combinatorial mathematics: Research into practice. To appear in the Connecting Research into Teaching section of The Mathematics Teacher.
8. U.S. Department of Education, office of Educational Research and Improvement. (1993).National Excellence: A case for developing America's talent. Washington, DC: U.S. Government Printing Office.
9. Van Tassel-Baska (1998)
. Curriculum for the gifted: Theory research and practice. In J. Van
Tassel-Baska, J. Feldhusen, K.Seeley, G. Wheatley, L. Silverman and W.
Foster (Eds.), Comprehensive curriculum for gifted learners , (pp.1-19). Boston, MA: Allyn and Bacon.
ABOUT THE AUTHOR’
Bharath Sriraman, Ph.D., Asst. Professor of Math & Math-Education
Department of Mathematical Sciences
The University of Montana, Missoula, MT 59812
USA
E-mail: sriramanb@mso.umt.edu
RESULTING EFFECT OF CONSECUTIVE ACTIVITIES
Borislav Lazarov
Abstract:
The concept of cognitive activeness of students could be accepted as a
criterion for the success of different strategies for students dealing
with mathematicsbeyond the ordinary curriculum. A qualitative
description of the change of student’s activeness could help to
understand the impact of a significant activity on the learning
process. Under consideration are several models of change of activeness
and an idea for operationalization of the qualitative concept of
activeness.
INTRODUCTION
The concept of motivation is
crucial in understanding the psychology of a significant student
studying mathematics, participating in competitions and setting as a
personal goal a high level of math knowledge. Different points of view
focused on students’ motivation to study mathematics are highlighted in
large number of investigations. But even a simple analysis of student’s
motivation is a difficult task for the experts and an impossible
mission for an ordinary teacher. More sophisticated is the case dealing
with group of students, where individual motivating factors should be
overviewed and generalized.
But the behaviour of a group of
students could be studied, predicted and changed during the process of
education without deep analysis of motivation factors. An
operationalization of this concept could be the category students’ activeness. The concept of students’ activeness (some authors use the notion creative activeness
[1]) was in focus mainly of Russian and Bulgarian scientists for a long
period. Egorov [2] pointed some sources considering students’
activeness from the late 19^{th}
century. A descriptive definition of students’ activeness is given by
Aristova [3], p 32. Further we will introduce a qualitative variable
and represent it in a quantitative expression for measuring students’
activeness and its change caused by different activities.
MAIN CONCEPTS
Following Lazarov [4] we introduce the variable situative activeness (SAC) of students as a synthetic characteristic of students’ behaviour
during a short period in the process of math education including the
attitude of a student to mathematics, student’s motivation to deal with
mathematics alone and to take part in extracurricular activities. The
SAC can be factorized in the following directions:
- student’s interest in mathematics during the lessons;
- time that a student spends for a significant homework;
- problem solving as a part of everyday preparation for school;
- problem solving as a preparation for an extracurricular activity;
- usage of math books extra school textbook, journals etc.
The
above list should be considered as a flexible frame and could be
extended according to the specifics of age, type of education or
purpose of the math study.
It is convenient to introduce also the notion integral activeness
(IAC) for the average value of the SAC for a longer period. Since SAC
is approximately a day-long characteristic, IAC is a reflection of
student’s behaviour during a couple of weeks or even during a term.
The changes in SAC are provoked by activators (ACF). An ACF can be a test, an exam, a competition etc.
QUALITATIVE DESCRIPTION OF SAC CAUSED BY SIGNIFICANT ACF
The hypothetic graph of SAC is given below.
The period in which a significant ACF acts
could be separated into three phases. Phase 1 includes the preparation;
phase 2 is the period of performance; phase 3 is the period of fade.
The amplitude of the graph corresponds to the change in SAC. It depends
on different parameters such as initial and final values of SAC, the
priority range of ACF, i.e. how important for the student is a good
performance at this event etc. The same ACF has different amplitude for
different groups of students: as a rule extracurricular activities
cause bigger amplitude for more able students; classroom tests generate
bigger amplitude for students with no special interest in mathematics.
The change in SAC depends also on student‘s self-estimation for good
performance. It is smaller for students with too high or too low level
of self-estimation. On the contrary the amplitude of the graph is
bigger for the students that are challenged by the ACF.
ACF INTERACTION
The graph of SAC caused by
consecutive ACFs could be different. The shape of the graph depends
mainly on priority range of each ACF and their accommodation on the
time-axes. Let us first consider the hypothetic graph of SAC caused by
two ACF: A1 and A2 and let the earlier be A1. We will show by graphs
some effects that are possible.
Superposition
The effect of superposition appears
when the first phase of A2 overlaps the third phase of A1 but the
second phases of A1 and A2 are separated by a small interval. In such a
case the resulting SAC is higher than the SAC caused by any of ACFs
taken one by one.
Shadow
The effect of shadow appears
when the second phase of A1 touches the second phase of A2. In such a
case the ACF of higher priority range dominates. On the picture A1 has
higher priority range and the resulting SAC is higher than that of A1
but is lower than the SAC caused by A2 if A2 acts independently.
Depression
The effect of depression appears
when the second phase of A2 overlaps the second phase of A1. In such a
case the resulting SAC is lower than the SAC caused by any of ACFs
taken one by one.
QUANTITATIVE DESCRIPTION
The SAC can be registered and measured by the quantitative variable spectrum of activeness . To define the spectrum of activeness we first declare a list of indicators which characterize the SAC of a student.
1. Interest during the lesson to the currently considered topics.
2. Taking part in discussions during the lesson.
3. Time which the student spends dealing with mathematics outside school.
4. Solving problems as a part of everyday preparation for school.
5. Solving problems for itself or as a preparation for an extracurricular activity.
6. Interest to math magazines, journals, web-sites etc.
7. Interest to math books.
8. Discussing math topics outside classroom.
9. Willingness for attending extracurricular activities.
10. Level of self-estimation about own problem solving abilities.
The above indicators are specific for more
able students at the secondary school. It is clear that some of them
correlate, e.g. both 6 and 7 point to similar features of student’s
behaviour but there is some difference between book-readers and
magazine-readers. Some indicators overlap but none of them contains
another one. A scale Ai assigned to the indicator i ( i=1,2,…,10) and a parameter ai, taking its values in Ai, gives us the oportunity to calculate the value of a numeric variable S=a 1+a 2+...+ a10, which yelds a quantitative expression of SAC .
The chart below shows statistics from a
case study held in the 119th Secondary school in Sofia with students of
10th grade. There was a CAF on April 2 and an EAF on April 5. The
average value of S for
the group of students with high SAC reaches its maximum on March 31, bu
t the group of students with medium SAC attains its maximum on April 2.
HOW TO KEEP SAC HIGH
The above effects could be
implemented to rise the SAC and to keep it on a high level for a long
time. The key factor in designing a strategy for teaching more able
students is the proper accommodation of ACF during the scholastic year.
Since there are compulsory ACF such as tests and term examinations
(call them CAF), it is crucial to place the extracurricular ACF (call
them EAF) in a way to avoid the negative effects as depression and
particularly shadow and to gain benefit from the positive effects as
superposition. The next picture shows an exemplary accommodation of
CAFs and EAFs during a term.
PSYCHOLOGICAL BACKGROUND
It is easy to see that the above model refers to the McClelland’s Trichotomy of Needs [5]:
Need for achievement is a need to
accomplish and demonstrate competence or mastery; a person who
continuously asks for and masters increasingly difficult tasks
demonstrates a need for achievement.
CONCLUSIONS
A math teacher can apply different
strategies in teaching only in a class where the level of activeness of
the major part of students is at least medium. In such an environment
the teacher can explain more details during the lesson and leave
students a part of routine exercises; teacher can recommend students
some books or journals for additional preparation, or to state topics
for discussions. But if the level of activeness is law the first task
of the teacher should be rising of students’ activeness in an
appropriate level. Extracurricular activities are very reliable tool
for such a purpose. For example in Bulgaria more than 30 math
competitions are held during the scholastic year and any teacher can
include the most appropriate of them in students time schedule to keep
their SAC high.
But it is delicate task to do such a
selection. An easy competition does not change the SAC of more able
students and a too hard competition may have even negative effect on
students with low preparation.
OPEN QUESTIONS
The list below contains some problems which the author considers as important, but has not find satisfactory solution of them.
- How to establish benchmarks of indicators to distinguish low from medium and medium from high SAC?
- How to recognize real from pseudo SAC?
- Is it useful sometimes to consider an EAF as a classroom test, i.e. to change an EAF to CAF?
- How to explain the phenomenon: the students with the best scores from math competitions do not have the highest SAC?
ACNOLEDGMENTS
The author thanks Australian Mathematics Trust and Prof. P.J. Taylor for the support that made this presentation possible.
REFERENCES
1. Shchedrovickij,G. Psychological features of creative activeness of students. Moscow, 1962 (in Russian)
2. Egorov, S. The problem of activeness and self-dependence of students at thedidactics of 19 ^{th} century and the beginning of 20^{th} century. Moscow, 1968 (in Russian)
3. Aristova, L. Activeness in students’ learning. Moscow, 1968 (in Russian)
4. Lazarov, B.
Student’s activeness as basic criterion for effectiveness of
extracurricular activities. RIK-I-S, Sofia, 2003. (in Bulgarian)
5. McClelland, D. The Achieving Society, NJ, Princeton,1961.
ABOUT THE AUTHOR
Borislav Lazarov
Dept. of Mathematics
Higher School of Transport
158 Geo Milev Str.
1574 Sofia, BULGARIA
National Institute of Education
E-mail: byl@abv.bg
ADDRESSING MATHEMATICAL PROMISE IN THE
NEW ZEALAND CONTEXT
Brenda Bicknell
Abstract:
This paper describes the state of play in New Zealand for the
identification and provision of gifted and talented students. The
results from a national survey on the extent, nature and effectiveness
of planned approaches for identifying and providingfor gifted and
talented students specific to the intellectual/academic ability area of
giftedness are presented.The purpose of this paper isto combine recent
national data on gifted education and policy development with current
practice in mathematics classrooms.
Key words :Gifted and Talented,Policy, Curriculum Initiatives, Identification, Provision
INTRODUCTION
In New Zealand the National
Education Guidelines establish a common direction for state education
and include National Administration Guidelines, policy statements, and
national curriculum statements. Schools are managed by elected boards
that have responsibility for meeting both general policy objectives for
all schools and specific policy objectives applying to that school.
Until very recently there was no national statement that made explicit
the obligations for Boards to address the needs of gifted and talented
students in the school system.
However, in 2002 a commitment was
made by the government to support the achievement of gifted and
talented students by commissioning research, developing an online
learning centre, appointing school advisers and funding education
programmes targeted at gifted and talented learners. To further
strengthen the position for gifted and talented learners an amendment
to policy was announced in March 2004. This was a change to one of the
National Administration Guidelines so that “from Term 1 2005, schools
will be required to identify their gifted and talented learners, and
develop and implement teaching and learning strategies to address their
needs” [1]. The Minister in announcing this change acknowledged that
gifted and talented learners are found within any group in society and
that the initiative “explicitly recognises that schools are powerful
catalysts for the development of talent” [1].
IDENTIFICATION AND PROVISION
In 2003, a national investigation of
current identification and provisions for gifted and talented students
was conducted by a team of researchers led by Dr Tracy Riley from
Massey University [2]. It was premised on the acknowledgement of a
“somewhat limited research” base [3]and driven by the need to identify
strengths and gaps in provision so that future directions in gifted and
talented education may be informed by both theory and practice. It was
recognised that “giftedness and talent can mean different things to
different communities and cultures … and there is a range of
appropriate approaches towards meeting the needs of all such students”
([4], p. 2).
The questionnaire probed schools’
identification and provisions for gifted and talented students. All
schools in New Zealand were invited to participate in the survey; there
was a 48% response rate (n=1285). An analysis of the respondent
schools’ demographics indicates that the sample is representative.
Sixty per cent of the schools reported that gifted and talented
students had been formally identified over the last 12 months. The
areas of giftedness included intellectual/academic, visual/performing
arts, creativity, physical and sport, social/leadership, and
culture-specific abilities.Of the schools reporting formal
identification (n=768), the most frequently identified area of
giftedness was in the domain of intellectual/academic giftedness
(96.9%). The intellectual/academic area of giftedness was described as
referring to students with exceptional abilities in one or more of the
essential learning areas (i.e., mathematics, language and languages,
technology, health and physical education, social sciences, science,
and the arts).
Although the survey did not
specifically address mathematics, the findings under the
intellectual/academic ability area of giftedness can be used to give
insight into formal identification procedures, and provisions for
mathematically gifted students. These findings can be linked to the
literature in terms of effective practice and outcomes for students.
Identification
This is one of the critical issues
in gifted education and most publications related to identification
recommend the use of multiple methods of identification. It is not
surprising therefore that only 4.1% of the schools formally identifying
intellectual and academic abilities reported a reliance on one method.
Almost half the respondent schools (49.1%) indicated use of between two
and four methods of identification. It is acknowledged that the
multi-method approach is more likely to identify those with
mathematical promise [5] and a multi-method approach is also more
likely to be inclusive rather than exclusive so that there is
representation of students who may otherwise be overlooked in the
identification process [6].
The most often utilised methods for
formal identification by the 768 schools that
identifiedintellectual/academic giftedness was teacher observation
(94.1%) and achievement tests (89.7%). Teacher observation may have
been the most strongly supported method of identification, but issues
are raised about potential ineffectiveness. Effectiveness is known to
be variable and is attributed to the level of formality of the
identification process and the teacher’s professional knowledge and
expertise [6]. Teachers may not have proficient knowledge about the
mathematically gifted and may be focusing on a narrow set of skills
such as computational ability. Teacher bias and stereotyping may also
contribute to ineffectiveness.
In New Zealand teachers commonly use
results from a nationally standardised mathematics test, the
Progressive Achievement Test (PAT) (published by the New Zealand
Council for Educational Research) for identifying students as gifted
and talented when they score above the 90^{th}
percentile. These tests assess a student’s recall, computational
skills,aspects of understanding, and application of various
mathematical operations and concepts. One of the suggested applications
of the test is to assist teachers in selecting able students requiring
special mathematics programmes [7]. However the tests are not designed
specifically for gifted and talented students; they are age and class
norm referenced. It is highly likely that gifted students will reach a
ceiling and this effect is not taken into account.A study that used
Receiver Operating CharacteristicAnalysis showed that, independent of
any chosen percentile, the PAT was 78% accurate in identifying
mathematically gifted and included errors of commission and omission
[8].Multiple choice tests such as the PAT are limited in that the
results can only be considered summatively. It is important in
mathematics for teachers to be able to gain insight into aspects such
as the student’s problem solving abilities [9], communication of
mathematical ideas, attitude and interest towards mathematics, and
application of mathematical understandings [10].
Other methods reported as being used
(in order of preference from highest to lowest) include teacher-made
tests, teacher rating scales, portfolios, parent nominations,
self-nomination, IQ tests, peer nomination and whanau (extended
family) nominations. Teacher-made tests provide the opportunity for
more open-ended and divergent questions so that a student can reveal
creative problem solving approaches or elegant solution paths. Student
portfolios can provide a rich variety of samples of student work
showing evidence of levels of achievement, interest and perseverance.
They can provide useful information not evidenced in test results.
Examples include reports from a statistics or measurement investigation
and geometric models. Self-nomination is recommended by the Ministry of
Education as part of the multi-method approach [11]. Individual student
behaviours and characteristics, including culture, age and self
perceptions are likely to influence the potential effectiveness of self
nomination. Only when a class climate or school culture permits
students to acknowledge openly that they are gifted are they likely to
nominate themselves for gifted programmes [12]. The Ministry of
Education recommends parents and whanau as valuable sources of
information in the identification process [11]. Parents can provide
information that may not be available from classroom observations and
tests. They may also recognise mathematical talent at a young age and
advocate for their children to be challenged mathematically by the
classroom programme [13].
Provision
The survey probed schools’
provisions for gifted and talented students and included their
preference for enrichment and acceleration, school-based provisions,
and provisions within classrooms. The majority of schools (61.4%)
reported a preference for a combination of acceleration and enrichment
approaches to provision for their gifted and talented students. Of
those schools not preferring a combined approach, enrichment was more
favourably viewed(35%) than acceleration. The most frequently reported
provision was in the area of intellectual/academic abilities. Of the765
respondents that provided school-based programmes for intellectually
and academically gifted and talented students the most frequently
reported strategy was withdrawal or pull-out programmes (67.6%). Over
half the schools reported the use of cross-age grouping (52.7%),
competitions (54.4%), and external examinations (50.7%). Of all the
reported school-based provisions, early entry was the least frequently
provided strategy.
Classroom-based provisions were
reported by the majority of schools (82%) in the survey.These
classroom-based provisions included ability grouping, independent
study, teacher planning, learning centres, individualised education
plans, curriculum compacting/diagnostic-prescriptive teaching and the
use of a consulting specialist teacher. Of the 1049 schools reporting
classroom-based provisions 87% indicated that they used ability
grouping. There is certainly confusion in the literature about the use
and interpretation of the term ‘ability grouping’ and debate continues
on whethergifted and talented students should be homogeneously grouped
(by ability) or heterogeneously grouped (mixed ability).Given that
grouping is merely an organisational strategy it is not necessarily the
type of grouping that should be of concern rather the quality of the
programme. Research shows that grouping students in a class by ability
especially where the curriculum is accelerated as well as enriched is
an effective strategy [14]. Curriculum
compacting/diagnostic-prescriptive teaching was only used by 30% of the
schools providing for gifted and talented students within the regular
class. Curriculum compacting makes sense for teaching mathematically
gifted and talented students in that material that students have
already mastered can be identified and replacement strategies can be
provided that allow for more meaningful and productive use of time.
Although this study enabled us to document the type and frequency of
classroom provision it gives no indication of the quality or
effectiveness of the classroom-based programmes. This is an area
recommended for further research.
In the learning domain of
mathematics what are the options in New Zealand for providing for the
mathematically gifted?The national curriculum statement‘Mathematics in
the New Zealand Curriculum’ document [15] aims to support teachers in
developing enrichment programmes for the more able students. This
curriculum statement is organised in to strands (mathematical
processes, number, measurement, geometry, algebra, and statistics) and
levels (1-8) to cater for the range of students (ages 5-18) in the
school system. The expectation is that students work at a level that
best suits their needs and abilities. “It is not expected that all
students of the same age will be achieving at the same level at the
same time, nor that an individual student will necessarily be achieving
at the same level in all strands of the mathematics curriculum” ([15],
p. 17). The document provides teachers with additional learning
experiences labelled as Development Band activities. This feature was
introduced to encourage teachers to extend and enrich the programmes of
the more able mathematics students and to support their efforts in that
direction.
The intention of the development
band is to encourage teachers to offer broader, richer and more
challenging mathematical experiences to faster students. Work from the
development band should allow better students to investigate whole new
topics which would not otherwise be studied and to work at a higher
conceptual level. ([15], p. 19)
Throughout the curriculum document
for each strand and level there are suggested topics and activities for
development band work. These include number bases, history of
mathematics, cryptography, modulo arithmetic, and topology. To date
these are presented as topics with limited support provided for
teachers. For our primary teachers this represents a real challenge as
many do not have the content knowledge to be able to take a topic and
present it in a suitably challenging way for the gifted student. These
teachers usually need to access additional support from mathematics
specialist teachers.
The New Zealand Association of
Mathematics Teachers (NZAMT) provides a Development Band Certificate
Course for students from Years 4 to 12. The material is presented in
modules and based on themes incorporating mathematics from a variety of
related content areas. The modules are based on an investigative
approach with some structure for the students but also pose questions
that give opportunity for flexible pathways and more open-ended
investigations. It is this type of investigation that helps maintaina
student’s interest and stimulates intellectual development [16].
Competitions are used by 54% of the
schools and are available for students at a variety of levels. These
include local, national and international competitions, team and
individual entries. Competitions give gifted and talented students an
opportunity to put their talents to the test, a chance to showcase
their special abilities and to receive recognition and acknowledgement
[17]. Competitions are available for students from about 8 years of
age. These include team problem solving competitions at the local
level, and national and Australasian competitions for individuals.
Competitions are also provided through the NZAMT website, and Otago
University and Auckland University. Some of these competitions are
completed online and engage students in problem solving skills through
more interactive ways.
One of the fifteen Ministry funded
Talent Development Initiatives includes the New Zealand Mathematical
Olympiad. A series of activities involving many students culminates in
the selection of the team for the International Mathematical Olympiad.
However, leading up to this all potential students are invited to
complete a set of problems, the solutions are submitted for marking and
from this 24 students are chosen to attend a training camp. Students
can access further problems from a website especially set up to provide
practice problems and support.
Other provision options in New
Zealand schools for students gifted and talented in mathematics include
early entry to intermediate, secondary and tertiary study (early entry
to primary school is not a legal choice), dual enrolment with the New
Zealand Correspondence School, grade skipping, cluster grouping,
pull-out programmes, virtual schools and mentoring. Eleven per cent of
the schools in the survey utilised mentoring which is one of the
strategies recommended by the Ministry of Education [11]. The
mentorship involves a partnership with a more experienced older student
or adult who can provide guidance and help develop expertise as an
appropriate role model [18]. Mentoring is recommended as a
culturally-appropriate strategy for our gifted and talented Maori
students [19].Mentoring builds upon the Maori tradition where a tohunga, a master works with a child to nurture his or her gifts and talents.
CONCLUSIONS AND IMPLICATIONS
The national survey demonstrated a
growing awareness of the need to provide for gifted and talented
students. As the Minster of Education stated; “it is an exciting time
for the education of gifted and talented students” ([8], p.1). The
government is committed to supporting the achievement of gifted and
talented students and has made amendments to national requirements so
that these students will have to be explicitly identified as a group
and schools will need to make appropriate provisions. As a consequence
of the Ministry’s mandatory requirements forgifted and talented
students, it is more likely that the mathematically gifted will be
identified and provided with an education matched to their individual
learning needs.
Despite this growing awareness of
the need to provide these students with an individualised and
appropriate education,schools reported that the implementation of
suitable programmes was hindered by a lack of teacher professional
development in gifted education.Professional development was cited as a
critical factor in developing effective practices. With a reported
preference for implementing a combined approach of enrichment and
acceleration the implementation of this approachneeds to be supported
by teacher professional development to ensure that programmes are
informed by a sound understanding. Future studies could consider the
evaluation of the effectiveness of teacher professional development and
subsequent identification and provisions for mathematically promising
students. Other factors identified as barriers to effective
identification and provisions includedteacher confidence and
competence, funding, resources, and time.
Gifted and talented learners are
found in every group within society yet our Maori students are
under-represented; they are not being identified and culturally
appropriate provisions are not being planned, implemented or evaluated.
It means that Maori perspective and values (and those of other ethnic
minority groups) must be embodied in all aspects of the education of
gifted students. This provides a real challenge for those teaching in
bicultural and multicultural schools.
The school is a powerful catalyst
for the development of talent and a responsive school will consider a
range of potentially effective approaches that are differentiated and
individualised. In New Zealand there is a lack of reported evaluation
of the effectiveness of provisions. Likewise the international
literature abounds with advocacy and descriptive articles but with
limited empirical studies outside of the United States. Programmes
themselves for students gifted and talented in mathematics should be
based upon sound research. The curriculum should be rich in depth and
breadth and at a pace commensurate with their abilities. It should not
only meet academic needs but address social and emotional needs.
There are valuable implications to
be drawn from the research. The extensive literature review revealed a
paucity of national and also international research validating
recommended approaches. The questionnaire results further indicated
that even when approaches have been validated in the literature they
were not being implemented and conversely some practices unsupported by
research were being implemented. The message has to be that practices
in both the identification and provision should be grounded in
reputable theory and research.
The survey reported in this
discussion paper raises as many questions as it provides answers. The
questionnaire did not allow for any indication of the quality of
reported practices and so the effectiveness of the school practices
remains an unknown. Effectiveness is contingent upon how programmes are
designed and delivered, hence the need for ongoing evaluation and for
the evaluation to be used to inform changes in practice.
The research demonstrates that New
Zealand schools are making an effort to identify and provide for the
education of gifted and talented students but as one respondent
wrote;“It’s along journey and we ain’t there yet!”
REFERENCES
1. Mallard, T ., 2004, Research to help schools support gifted students . Retrieved April 12 2004 from http://www.beehive.govt.nz
2. Riley, T. L., Bevan-Brown, B., Bicknell, B., Carroll-Lind, J., & Kearney, A., 2004, The extent, nature and effectiveness of planned approaches in New Zealand schools for providing for gifted and talented students . Wellington: Ministry of Education.
3. Ministry of Education, 2001, Working Party on Gifted Education. Report to the Minister of Education. Retrieved December 12 2003 from http://www.executive.govt.nz/minister/mallard/gifted_education/index.html
4. Office of the Minister of Education , 2002, Initiatives for gifted and talented learners. Wellington: Ministry of Education.
5. Wertheimer, R ., 1999, Definition and identification of mathematical promise. In L. J. Sheffield (Ed.), Developing mathematically promising students (pp. 9-26). Reston, VA: National Council of Teachers of Mathematics.
6. Davis, G. A., & Rimm, S. B. (1998). Education of the gifted and talented (4^{th} edition). Needham Heights, MA: Allyn & Bacon.
7. Reid, N. A., 1993, Progressive achievement test of mathematics , Wellington: New Zealand Council for Educational Research.
8. Neiderer, K., Irwin, R. J., Irwin, K. J., & Reilly, I. J., 2003, Identification of mathematically gifted children in New Zealand. High Abilities Studies, 14(1) 71-84.
9. Niederer, K., & Irwin, K. C., 2001, Using problem solving to identify mathematically gifted students.
In M. van den Heuvel-Panhuizen, (Ed.), Proceedings of the 25th
Conference of the International Group for the Psychology of Mathematics
Education (pp. 431-438). Utrecht, The Netherlands: Freudenthal
Institute.
10. Wieczerkowski, W., Cropley, A. J., & Prado, T. M. , 2000, Nurturing Talents/Gifts in Mathematics. In K. A. Heller, F. J. Monks, R. J. Sternberg, & R. F. Subotnik (Eds.), International handbook of giftedness and talent (pp. 413-425). Oxford: Elsevier.
11. Ministry of Education, 2000, Gifted and talented students: Meeting their needs in New Zealand schools . Wellington: Learning Media.
12. Gross, M. U. M., & Sleap, B., 2001,Literature review on the education of gifted and talented students . Retrieved April 14 2004 from: www.aph.gov.au/senate/committee/eet_ctte/gifted/submissions/sub032.doc
13. Lupkowski-Shoplik, A. E., & Assouline, S. G., 1994, Evidence of extreme mathematical precocity: Case studies of talented youths. Roeper Review, 16(3), 144-151.
14. Kulik, C. C., & Kulik, J. A., 1992, Meta-analytic findings on grouping programs. Gifted Child Quarterly, 72 - 76.
15. Ministry of Education , 1992,Mathematics in the New Zealand Curriculum. Wellington: Ministry of Education.
16. Eddins, S., & House, P., 1994, Flexible pathways: Guiding the development of talented students. In C.A.Thornton & N.S. Bley (Eds.), Windows of opportunity: Mathematics for students with special needs (pp. 309-322). Reston, VA: National Council of Teachers of Mathematics.
17. Riley, T. L. & Karnes, F. A ., 1999, Forming partnerships with communities via competitions. Journal of Secondary Gifted Education, 10(3), 129-134.
18. Casey, K. M., & Shore, B. M., 2000, Mentors’ contributions to gifted adolescents affective, social and vocational development. Roeper Review, 22(4), 227-231.
19. Bevan-BrownJ., 1996, Special Abilities. A Maori Perspective. In D. McAlpine & R. Moltzen (Eds.), Gifted and talented: New Zealand perspectives (pp. 91-110). Palmerston North: ERDC Press.
ABOUT THE AUTHOR
Brenda Bicknell
NEW ZEALAND
CHALLENGE AND CONNECTEDNESS
IN THE MATHEMATICS CLASSROOM:USING LATERAL STRATEGIES WITH GIFTED ELEMENTARY STUDENTS
Carmel M. Diezmann, James J. Watters
Abstract:
Generating interest in mathematics for gifted students starts in the
early years of schooling. In this paper we summarise our research
undertaken over a number of years, which has adopted a lateral approach
to challenging the mathematically gifted. The paper argues that
teachers can provide rich mathematical learning opportunities within
the regular classroom through strategies that capitalise on the
essentials of the regular curriculum but which provide that extra
challenge needed by gifted students. These strategies enhance both
mathematical knowledge and motivation towards mathematics.
Keywords: mathematically gifted, elementary students, instructional strategies, regular classroom.
INTRODUCTION
In this proposal, we focus on
instructional strategies to support mathematically gifted elementary
students in the regular classroom. Mathematically gifted students are
distinguished from their non-gifted peers by their mathematical
reasoning, their capacity for learning, and their mathematical
orientation [10]. Although strategies such as acceleration, enrichment
programs, curriculum compacting, competitions can support gifted
elementary students, they may be inaccessible to some students, or fail
to cater for particular types of gifted students (e.g., underachievers,
minority groups). Additionally, students participating in specialist
gifted programs usually spend some time in their regular classroom.
Therefore, of particular importance is ways to develop rich
mathematical learning opportunities within the regular classroom. Many
teachers have limited knowledge, resources and time to provide
challenging experiences for mathematically gifted students. However, in
less than optimal environments, gifted students are “at risk” and may
demonstrate boredom, loss of interest in or commitment to mathematics,
limited metacognition, and poor behaviour [3], [13]. It is not
surprising that teachers often attempt to cater for mathematically
gifted through higher-level work in addition to normal classroom work.
However it is fallacious to assume that mathematical giftedness relates
solely to speed, accuracy and proficiency in set tasks. Our research
has focussed on developing instructional strategies for general
classroom use that differentiate the curricula in ways that enrich
learning experiences laterally
rather than sequentially. These approaches undertaken in the regular
classroom enhance gifted students’ reasoning, accommodate their
capacity for learning, and foster their interests in accordance with
best practice in gifted education [12].
LATERAL INSTRUCTIONAL STRATEGIES
Lateral strategies are designed to
be challenging but interconnected to the regular curriculum.
Underpinning these strategies is the view that tasks should be a
mechanism for empowering students as mathematicians rather than
discrete activities to be completed. Examples of four lateral
strategies drawn from a range of studies with variously aged elementary
students follow. Common to these strategies is the nexus between what
is advocated in research on gifted students and contemporary approaches
to teaching mathematics.
1. The level of challenge in regular classroom tasks can be increased be problematising mathematical tasks
[8], [6]. Problematisation includes inserting obstacles to the
solution, removing some problem information, or requiring students to
use particular representations or develop generalisations. For example,
after easily calculating the sum of the numbers from 1 to 10, a young
gifted student used a generalisation to calculate the numbers from 1 to
100 [4]. This task was appropriately challenging for the gifted student
but still connected to the regular curriculum focus of summing
sequential numbers.
2. Implementing mathematical investigations
requires students to apply and create mathematical knowledge in posing
and solving novel problems. Investigations are central to the reforms
advocated internationally to develop children’s mathematical power [1].
For example, after some guided investigations with Smarties [sweets],
young children formulated investigable questions about Smarties and
explored these: “How many regular Smarties weigh the same as one giant
Smartie?” (See Figure 1) and “What is the most popular coloured
Smartie?” [9]. The open-ended nature of this task enabled gifted
students to explore their mathematical interests and to develop their
capabilities.
3. Extending manipulative use capitalises
on visual-spatial or kinaesthetic representations to support
higher-level thinking. For example, young gifted students’ construction
of a number-line to represent the distances of the ten brightest stars
required the application of knowledge of large numbers, relative
magnitude and scale [2]. This application goes beyond the more typical
use of a prepared number line to represent a simple set of numbers.
4. Modifying educational games
can provide rich mathematical and social learning opportunities for the
gifted. For example, in a simple place value card game, students are
dealt three cards in order to represent a three-digit number. They then
take turns to select one card at a time from the pile of remaining
cards with the goal of making their number larger. For example, if
Student A was dealt “345”, and picked up an “8” they could make “845”
(See Figure 2). This game becomes more challenging if students select
the card to be replaced before they see their new card (See Figure 3).
As shown by Students B and C’s responses, the selection of the card to
be replaced results in substantially different outcomes (See Figure 3).
This game proved to be a particularly useful thought-revealing activity
[11] because it uncovered gifted students’ propensity for calculated
risk taking, their erroneous reasoning, and indicators of their
metacognitive processes.
CONCLUSION
The adoption of more lateral
instructional strategies expands gifted students’ mathematical
knowledge through challenging experiences that are connected to the
regular curriculum. These lateral strategies have six particular
advantages. Firstly, lateral strategies are not add-ons or extensions
but take the existing curriculum and problematise, adapt and enrich the
experiences for gifted students. Secondly, these approaches lend
themselves to both collaborative and independent learning [5]. Thirdly,
through strong linkages to the regular curriculum, lateral strategies
provide underachieving gifted students with opportunities to oscillate
between regular activities and more challenging activities according to
their capability, confidence and motivation. Fourthly, lateral
strategies address the mathematics reform agenda, for example through
attention to learning through problem solving [7]. Fifthly, because
lateral approaches capitalise on elementary teachers’ pedagogical
knowledge, for example in using games, teachers have opportunities to
develop their confidence and competence in working with gifted
students. Finally, lateral strategies are thought-revealing activities
[11] that provide opportunities for the identification and development
of gifted students’ mathematical ability.
REFERENCES
1. Baroody, A., & Coslick, R. T. Fostering children’s mathematical power: An investigative approach in K-8 mathematics instruction. Mahwah, NJ: Lawrence Erlbaum, (1998).
2. Diezmann, C. M., & English, L. D. Developing young children’s multi-digit number sense. Roeper Review, 24(1), (2001), pp. 11-13.
3. Diezmann, C. M., & Watters, J. J. Bright but bored: Optimising the environment for gifted children. Australian Journal of Early Childhood, 22 (2), (1997), pp. 17-21.
4. Diezmann, C. M., & Watters, J. J. Catering for mathematically gifted elementary students: Learning from challenging tasks. Gifted Child Today, 23(4), (2000), pp. 14-19.
5. Diezmann, C. M., & Watters, J. J. The collaboration of mathematically gifted students on challenging tasks. Journal for the Education of the Gifted, 25(1), (2001), pp. 7-31.
6. Diezmann, C. M., & Watters, J. J. The importance of challenging tasks for mathematically gifted students. Gifted and Talented International, 17 (2), (2002), pp. 76-84.
7. Diezmann C, Thornton C., & Watters J. Meeting special needs. In F. Lester & R. Charles (Eds.), Teaching mathematics through problem solving, Reston, VA: NCTM.(2003), pp. 169-182.
8. Diezmann C. M, Watters J. J., & English, L. The needs of mathematically gifted Learners: Raising the challenge of academic tasks. Paper presented at the International Congress of Mathematics Education, Tokyo, Japan, 31 July - 6 August 2000.
9. Diezmann, C. M., Watters, J. J., & English, L. D. Difficulties confronting young children undertaking investigations. Proceedings of the 26th Annual Conference of the International Group for the Psychology of Mathematics Education.Utrecht, Holland: PME. (2001), pp. 353-360
10. House, P. (Ed.) Providing opportunities for the mathematically gifted K-12. Reston, VA: NCTM. (1987).
11. Lesh, R. Hoover,
M., Hole, B., Kelly, A., & Post, T. Principles for developing
thought revealing activities for students and teachers. In A. Kelly
& R. Lesh (Eds.), Handbook of research design in mathematics and science education Mahwah, NJ: Lawrence Erlbaum. (2000), pp. 591-646.
12. National Association for Gifted Children Pre-K–Grade 12 gifted program standards. Washington, DC: NAGC. (1998).
13. Sheffield, L. J. Developing mathematically promising students. Reston, VA: NCTM. (1999).
ABOUT THE AUTHORS
Dr Carmel Diezmann
Centre for Mathematics and Science Education, Faculty of Education
Queensland University of Technology
Victoria Park Road, Kelvin Grove,
Brisbane Q4059
AUSTRALIA
+61 7 3864 3803 (Phone)
+61 7 3864 3989 (Fax)
e-mail c.diezmann@qut.edu.au
Dr James J Watters
Centre for Mathematics and Science Education, Faculty of Education
Queensland University of Technology
Victoria Park Road, Kelvin Grove
Brisbane Q4059
AUSTRALIA
+61 7 3864 3639 (Phone)
+61 7 3864 3643 (Fax)
e-mail j.watters@qut.edu.au
url: http://education.qut.edu.au/~watters/
GENERAL METHODS IN JUNIOR CONTESTS:
SUCCESSES AND CHALLENGES
Dace Bonka, Agnis Andžāns
Abstract:
the uses of the methods of mathematical induction, invariants, extremal
element and interpretation in problem solving contests for 5-8 Grade
students are dis cussed. Examples of corresponding problems from
Latvian contests are provided.
Key words: Math Contests for Junior Students, Mathematical Induction, Invariants, Extremal Element, Mean Value, Interpretation.
INTRODUCTION
Latvia is a small country with app.
2 300 000 inhabitants in North Eastern Europe. Only 10 years ago it
gets rid of Soviet occupation which has enormously damaged Latvian
nation, economy and culture.
Latvia has no rich natural resources
as oil, iron, diamonds etc. So almost only its resource to restore the
country is well-educated people. With such a small population each
talent is of great importance. Therefore serious and constant efforts
are made to develop the abilities of each child in a best way and to
help the teachers in this work.
Advanced mathematical education starting from 3-4 ^{th} Grades is considered as one of the most essential tools for at least two reasons:
A. There
are four main roads of inquiry discovered in the world: rational,
empiric, emotional and modeling. The main representatives of theseroads
are respectively mathematics, natural sciences, literature and
informatics. Thus mathematics represents one of these roads and is an
important tool for development of skills for two others – empiric and
modeling.
B.
A mathematical way of thinking – the creation and analysis of
amathematical model, the inner need to fortify own judgment
andconclusions, the creation of a strong deductive view of the world is
an important element of forming the personality.
Mathematics serves the pupil as an example of truth, objective and independent from daily needs and individual desires.
SCHOOL CURRICULA AND GENERAL MATHEMATICAL METHODS
Since late seventies the educational
system of Latvia is being reformed almost in a non-stop way. The recent
changes are aimed to switch from acquiring large amount of knowledge to
acquiring large amount of skills, including information search and
evaluation skills. So the general methods of reasoning should be
considered as having great educational value. Mathematics provide
examples of such general methods. Among most important of them are
mathematical induction (MIM), invariant method (IM), extremal element
method (EEM), mean value method (MVM) and the method of interpretation
(MI). They are even not only purely mathematical methods, but rather
general principles discovered by the mankind in a long period of time.
Accordingly to the new standards of
education [1] the acquaintance with these methods should be welcome for
all middle school students, adding new uses of them in high school.
Nevertheless, though they are common in various contests and a lot of
teaching aids are prepared and published (see, e.g., [2]), little
effort is made to incorporate them into the official school curricula
(see, e.g., [3]).
VERSIONS OF GENERAL METHODS MOST SUITABLE FOR MIDDLE GRADE STUDENTS
The abovementioned methods are
described in a number of papers and monographs (see, e.g., [4]-[6]).
Nevertheless, our opinion shows that only part of their uses is
suitable for middle grade students. Namely, the classes of problems
best for introductory examples are following:
- for MIM – inductive constructions, inductive algorithms, the recurrent relations;
- for MVM – the simple form of Dirichlet principle, the uses of it in finding the extremal values;
- for IM – the invariance of the result of counting, the idea of parity and its uses;
- for EEM – the greatest and the largest element of a set of numbers, the convex hull;
- for MI – uses of geometry in algebra, physical interpretations.
At
the beginning of studying these methods it is important that the
general idea of the method must not be hidden in the formal
manipulations connected with writing down the solution "for thegeneral
case". Therefore the so called "general special cases" are ofgreatest
importance; the analysis is done only for some values ofparameters
which contain all the essential features of the general case.Such a
concept has been derived from the concept of "full system ofexamples"
in program testing.
The studying of the abovementioned methods has alsofollowing positive general pedagogical effects:
- the demonstration of the unity of mathematics,
- aesthetically considerations,
- the possibility to use the underlying ideas of the methodsoutside mathematics,
- the broadening of the concept of proof.
SYSTEM OF MATH CONTESTS FOR JUNIOR STUDENTS IN LATVIA
There are two main classes of competitions, mainly in problem solving.
A. Mathematical Olympiads
They are organized at three levels:
- school Olympiads, often supported by universities; they are usually held in November,
- regional Olympiads held in 39 different places in Latvia each year in February,
- Open math Olympiad held each year in April. This competition is a very large one; morethen 3000 participants arrive in Riga.
All these competitions are open to everybody who wants to participate.
Other present-way competitions are organised at schools, at summer camps etc.
B. Corresponding contests
There are many students who need
more than some 4-5 hours (usually allowed during math Olympiads) to go
deep enough into the problem. For such children a system of
correspondence contests has been developed:
- “Club of Professor Littledigit” (CPL) for students up to the 9^{th}
Grade. There are 6 rounds each year, each containing 6 relatively easy
and 6 harder problems. Problems are published in the newspaper “Lat
vijas Avīze” (having the largest circulation in Latvia), and on the
INTERNET.
- “Contest of young mathematicians” for students up to 7^{th}
Grade, originally developed for weaker students than the participants
of CPL, especially in Latgale, the eastern region o f Latvia. The
problems are published in regional newspapers and on the INTERNET, and
today it has become popular all over Latvia.
SAMPLE PROBLEMS DEMONSTRATING THE APPLICATIONS OF GENERAL COMBINATORICAL METHODS TO CONTEST PROBLEMS
General principles established in Latvia for constructing a contest problem set are described in [7].
From the previous it is clear that
math contests should cover broad spectrum of mathematics, as more as
better. It is particularly important alsobecause olympiad and contest
problems from previous years arebroadly used afterwards in everyday
teaching practice. Main criteria accordingly to which the set should
bewell – balanced follow:
- it
should cover main areas of school mathematics: algebra, geometry,number
theory and combinatorics. Analysis is included into algebra here,and
combinatorics is understood in a broad sense including not onlycounting
but also existence and non-existence of combinatorial
objects.Particularly, the general combinatorial methods (mathematical
induction,invariants, mean value, extremal element, interpretation)
must be reflected;
- it should contain both problems of deductive nature and problems ofalgorithmical nature;
- there should be problems of "prove it!" type along with problems in which the answer must be found by the solver;
- "discrete" mathematics and "continuous" mathematics both are to be represented.
It
is clear that all these desires hardly can be implemented in a small
set of five or six problems. Indeed, there are very few sets of
problems which can be calledsatisfactory from all mentioned points of
view. Good balance can and must beachieved during the whole school year.There
are few problems for middle grades that use the full version of it.
Usually inductive constructions consisting of 5-6 steps are a good
choice.
Example 1. Does there exist a 10-digit number consisting only of digits 2 and 3 and dividing by 1024?
The construction is done by adding digits one by one to the end of the number and ensuring that n-digit number is dividing by 2^{n}.
- Method of invariants
- in geometry problems.
Correspondence
contests are organized mainly for those grades where only basic
concepts of geometry are studied. Therefore the main part of geometry
problems there are of combinatorical nature, e.g., problems of
dissection.
Example 2. A floor in a room of dimensions 6 m ´ 10 m must be covered by tiles of two types each consisting of 4 squares (see fig.1).
Can it be done if we have 5 tiles of type a) and 10 tiles of type b)?
Appropriate colouring and the use of
the fact that the number of elements in a finite set doesn’t change if
the counting order is changed solve the matter.
- in algebra and arithmetic’s problems.
Invariants connected with a divisibility, mainly, parity, are the most common tools here.
Example 3.
There are 6 cats and 7 dogs drawn on the blackboard. You can erase one
dog, or you can erase two cats and draw a dog instead of them. Prove
that the last animal on the blackboard will be a dog.
Replace each cat by number 1 and each dog by number 2, and investigate how does the sum of all numbers on the blackboard change.
The main challenge here is to realize what kind of extremality should be considered.
Example 4.
Does there exist a rectangle consisting of whole equal quadratic cells
which can be dissected into “squares” and “crosses” (see fig.2) so that
at least one part of each type is obtained?
Suppose it is possible, and consider the “cross” being situated no lower than any other one.- Method of interpretations
In
solving problems by means of this method we translate a problem into an
“appropriate language” in which its solution is much easier or even
trivial. It can also be said that we build a model of the problem,
solve the corresponding problem for the model and then translate the
solution back to the original language. The classical example is
analytical geometry. Clearly the effectiveness of this method increase
together with the knowledge of various branches of mathematics. So it
is not very appropriate for younger students. The main interpretations
here are those using graphs or modelling one game by another.
Example 5.Let A_{1}A_{2}...A_{100} be a regular 100-gon. Each two vertices are connected with a straight road. There are two players at A_{1} and A_{3}
correspondingly. The move consists ofgoing from one vertice to another
using only one road. It is forbidden to go to the vertice where there
is another player at the moment; it is forbidden to go along the road
that has been already used. Who has made the last move is the winner.
Let's pay the attention to the fact that the game with initial positions of players at A_{1} and A_{2} is an easy exercise, but the game on the 101-gon is an unsolved problem.
The algorithm for the game of example 5can be developed from the isomorphism similar to that between the hexagons in Fig. 3.
The pigeonhole principle (Dirichlet principle) is the undoubtfull leader here for all grades.
Example 6.
What is the largest number of natural numbers between 2 and 120
inclusively such that no one of them is a prime but each two of them
are coprime?
The answer “4” is obtained by
considering the least prime divisor of each of selected numbers. Note
that together with pigeonhole the extremal element method is also
presented in this solution.
A large number of sorted problems with solutions can be found in teaching aids published in [2].
ACKNOWLEDGMENT
This paper was prepared partially with the support of the state – investment project “Latvian Education Informatization System”.
REFERENCES
1. The Regulations of the State Standard of Basic Education. The Cabinet of Ministers of Latvia, Riga, 2000 (in Latvian).2. http://www.liis.lv
3. I.France, L.Ra
māna. The Strategies of Content and Teaching of Mathematics: Necessity
of Changes. – Proc. Int. Conf., Liepāja: LPA, 2002, pp. 67-70 (in
Latvian).
4. A.Engel. Problem-Solving Strategies. Springer, 1998.
5. J.Tabov, P.Taylor. Methods of Problem Solving. AMT, 1996, 2003.
6. T.Andreescu, R.Gelca. Mathematical Olympiad Challenges. Birkhauser, 2000.
7. A.Andžāns,
L.Ramāna. What Problem Set Should be Called Good for a Mathematical
Olympiad. – Matematika ir matematikos destymas – 2002. Kaunas,
Technologija, 2002, pp. 5-8
ABOUT THE AUTHORS
Dace Bonka, Mg.Math.
University of Latvia
19 Rainis Boulev. Riga, LV-1586
LATVIA
Phone: +371 7034498
E-mail: dace.bonka@lu.lv
Agnis Andžāns, DSc., Prof.
University of Latvia
19 Rainis Boulev. Riga, LV-1586
LATVIA
Phone: +371 7034498
E-mail: agnis@lanet.lv
IS COGNITIVE STYLE RELATED TO LINK BETWEEN
PROCEDURAL AND CONCEPTUAL
MATHEMATICAL KNOWLEDGE?
Djordje Kadijevich, Zora Krnjaic
Abstract: This
study examined the relation between cognitive style and link between
procedural and conceptual mathematical knowledge. It used a sample of
34 mathematically talented eleventh-grade students. A significant
positive correlation was found between the students’ achievements on
the administered Embedded Figures Tests (where
“field-dependence-independence” cognitive style has a very specific
perceptual connotation) and the measures of link between their scores
on procedural and conceptual mathematical knowledge. The same relation
was again found in a group of particularly talented students who
participated in mathematical competitions (N = 16), but not in the
control group comprising other talented students (N = 18).
Key words: Cognitive
Style, Procedural Knowledge, Conceptual Knowledge, Linking Procedural
and Conceptual Knowledge, Talented Students, Mathematics Education,
Upper Secondary Education.
INTRODUCTION
As regards cognitive style,
academically gifted students may be more field independent than their
counterparts involved in the regular education program [1]. A previous
analysis of the structure of test achievements (including EFT test [2])
shows that highly gifted students in mathematics and technical
sciences, who are scholarship holder candidates in Serbia, are
characterized by a form of general fluid intelligence contained in
figurative tests [3].
Linking procedural and conceptual
mathematical knowledge is an important yet neglected goal of
mathematics education, the attainment of which is a complex but
achievable enterprise [4]. When the effects of developing procedural
and conceptual knowledge and establishing links between them are
examined, cognitive style should be included since some students,
because of their less flexible (say more field dependent) cognitive
style, may demonstrate unbalanced gains in these knowledge types
resulting in missing or poor links between them [5].
Having in mind the presented
research context, the objective of this study was to examine the
relation between cognitive style and link between procedural and
conceptual mathematical knowledge. The rest of this paper presents how
these variables were operationalized, what were the main results, and
why such results might be obtained.
METHOD
The study used a sample of 34 mathematically talented students who came from two eleventh-grade classes of Matematička Gimnazija - the specialized high school in Belgrade for mathematically talented students.
The study had a correlative design.
The variables were: class (1 – control group comprising self-financed
students, 2 – target group comprising students financed by the state;
all these students passed the school entrance examination, but free
education is reserved for those who achieved better total scores at
that examination), cognitive style, procedural knowledge, conceptual
knowledge, and link between procedural and conceptual knowledge
(hereafter denoted by P-C link).
Cognitive style was measured by the
last (perhaps the hardest) 16-item subtest of Bukvi ć’s modification of
Embedded Figures Tests [2] standardized for Yugoslav population. This
instrument was administered under a group setting (one class at a time;
both classes within 45 minutes) in exactly 10 minutes by a psychologist
(the second author of this report). The subject’s cognitive style was
represented by the first principal component factor score obtained from
the subjects’ answers. The factor score reliability
(Lord-Kaiser-Caffrey) was .83.
Procedural and conceptual knowledge
were measured by scores given to different solutions of the following
task taken from [6]: “In the square below, M and N are midpoints of the
corresponding sides. Determine the numerical value of sin a.”
Answers to this task were written in
a questionnaire administered under a group setting (one class at a
time; both classes within 45 minutes) in exactly 20 minutes by a
mathematician (the first author of this report) who also precisely
scored the students’ answers. For each correct solution, student
received 1 point for conceptual knowledge and 1 point for procedural
knowledge. Partial credit was given when: (1) student wrote a solution
plan (how the task can be solved) that was partially or fully correct
(for conceptual knowledge), and (2) some of the required calculations
(plan implementation) were performed correctly (for procedural
knowledge).
P-C link was measured by formula 2PC/(P^{2 }+ C^{2}
) introduced in [7], where P and C denote total scores on procedural
and conceptual knowledge, respectively. For those students where PC
equaled 0 (when one or both types of knowledge was (were) not
demonstrated), P-C links were equal to 0. Note that such a defined link
takes values from interval [0, 1], where a bigger number indicates a
stronger P-C link.
The data collection was realized at
the end of the fall semester in January 2004 during regular school
lessons. The authors told the subjects that this study would examine
their problem solving performance and the subjects willingly provided
the requested data.
RESULTS
The correlations among
procedural knowledge, conceptual knowledge, P-C link, cognitive style,
and class are presented in Table 1.
The correlations between cognitive style and P-C link for the two classes are given in Table 2.
DISCUSSION
Two important findings emerged from this study.
First, there was a significant positive correlation between cognitive style and P-C link.
Second, while this
relation also held true for the target group of particularly talented
students participating in mathematical competitions, this was not the
case for the subjects of the control group.
As Table 1 evidences,
there were no significant differences between the two classes with
respect to cognitive style, procedural knowledge and conceptual
knowledge, which is acceptable (not expectable) as all these students
belong to the same highly selected student sample (two students solved
the problem in 3 ways and two in two ways; three students pursued a
specific way of solving the problem ¾ not listed among 16 different solutions summarized in [6] ¾
and one of them succeeded; an easier variant of this solution is given
in the appendix). But, while particularly talented students were
competition oriented (even in 20 minutes, two students solved the
problem in 3 ways), most students in the control group were not so
directed. So, the competitors, who obtained more unbalanced scores on
procedural and conceptual knowledge than those in the control class
(recall that correlation was -.38), may in general, compared to other
talented students, be more prone to procedural errors or calculation
ignorance when conceptual knowledge is correct, which was evidenced by
the student’s answers (P-C link for 8 students, a half of the group,
was 0; in the control group just 3 students had such a link). However,
the more competitor’s cognitive style was field independent, the
stronger P-C link he/she established, which is an important finding
that, to our reading, has not been reported so far. This was, however,
not the case for those in the control group, which, among others, might
be caused by a less competitive approach to day-to-day learning
requiring a more relaxed cognitive processing. Another reason was
suggested by the examined data: while no significant differences
between the classes were found for the variance of the link measure
(.20 vs. .11; Levene's Test: F = 2.90, p = .098), the variance of the cognitive style measure was higher in the competitor class (1.43 vs. .61; Levene's Test: F = 6.81, p
= .014), which was a less homogenous sub-sample. Despite such plausible
explanations, further investigations are still needed, which should
also include ordinary (not mathematically talented students) from
gymnasium or vocational school.
ACKNOWLEDGEMENT
The authors wish to
express many thanks to the students for their successful participation
and the school staff for their professional support and assistance.
APPENDIX ¾ A SPECIFIC WAY OF SOLVING THE PROBLEM
As the hypotenuses of the shaded
right-angled triangles (Fig. 2) are perpendicular (one triangle can be
rotated into the other by angle of 90?), AD is an altitude of triangle
ABM. If AB = 2, the area of triangle ABM (or BMA) is 2, and, since BM =
, AD is equal to . As = , the numerical value of is (obtained from the relation ).
The above-mentioned successful
solver utilized the fact that MD is an altitude of triangle ANM, whose
area, compared to that of triangle ABM, cannot be obtained at a glance.
However, when the length of MD is known, the numerical value of can easily be found (from the relation = ).
REFERENCES
1. Terrell, S. R . The Use of Cognitive Style as a Predictor of Membership in Middle and High School Programs for the Academically Gifted
. Paper Presented at the Annual Meeting of the American Educational
Research Association, April 1-5, 2002, New Orleans, Louisiana.
Available at: http://www.gifted.uconn.edu/siegle/aera/NewOrleans/SteveTerrell.pdf.
2. Witkin, H. A. Oltman, P. K., Raskin, E., & Karp, S. A. A Manual for the Embedded Figures Tests. Consulting Psychologists Press, Palo Alto, 1971.
3. Krnjaic, Z.
Intellectual Giftedness in Young People (in the Serbian language).
Institute of Psychology, Faculty of Philosophy, Belgrade, 2002.
4. Kadijevich, Dj. & Haapasalo, L. Linking procedural and conceptual mathematical knowledge through CAL. Journal of Computer Assisted Learning, 17, 2, 156-165, 2001.
5. Kadijevich, Dj., Maksich, S. & Kordonis, I.
Procedural and conceptual mathematical knowledge: comparing
mathematically talented with other students. In Velikova, E. (Ed.), Proceedings of the Third International Conference Creativity in Mathematics Education and the Education of Gifted Students (pp. 103-108). V-publications, Athens, 2003.
6. Barry, D. An Abundance of Solutions. Mathematics Teacher, 85, 5, 384-387, 1992.
7. Kadijevich, Dj.
Impact of mathematical-self concept on linking procedural and
conceptual mathematical knowledge (in the Serbian language). Presented
at 8^{th} Scientific Meeting “Empirical Research in Psychology”. Abstracts (p. 21). Institute of Psychology & Laboratory of Experimental Psychology, Faculty of Philosophy, Belgrade, 2002.
ABOUT THE AUTHORS
Djordje Kadijevich, Ph.D.
Graduate School of Geoeconomics
Megatrend University of Applied Sciences
Makedonska 21
11000 Belgrade
SERBIA & MONTENEGRO
Cell phone: +381 11 3373 796
E-mail: djkadijevic@megatrend.edu.yu
Zora Krnjaic, M.S.
Institute of Psychology, Faculty of Philosophy
University of Belgrade
Cika Ljubina 18-20
11000 Belgrade
SERBIA & MONTENEGRO
Cell phone: +381 11 639 724
E-mail:
zkrnjaic@f.bg.ac.yu
TEACHING CAPABLE STUDENTS IN
DEVELOPMENTAL MATHEMATICS COURSES
Elena Koublanova
Abstract:
In Developmental Mathematics courses, the variation in the abilities of
students can be recognized by comparing them in a learning environment
such as work in study groups. Capable individuals soon become group
leaders, whose role is not only to solve a problem but also to explain
a solution to other students. Working in study groups is particularly
effective in solving word problems. Ancient word problems that
implement elements of games serve well for in-group studies. Other
problems, which are useful for collaborative learning, come from
introductory topics in Logic. Examples of word problems appropriate for
in-group studies are presented.
Key words : Capable students, collaborative learning, study group, old word problems.
INTRODUCTION
In two-year colleges, a number of
students need introductory and developmental courses to refresh their
basic knowledge in Mathematics. Most of such students are adults who
took Mathematics years ago. Success in studying Mathematics is
particularly important for adult students because it increases their
confidence and self-esteem and creates necessary conditions for further
education. A population of students in Developmental Math classes is
non-homogeneous and includes individuals that either failed or never
could complete a school program as well as capable students who studied
Mathematics a long time ago. While chances are high that an instructor
would concentrate on weaker students in order to bring them to an
average level, students with higher potentials can also benefit from
such classes. Learning specific topics in study groups could be
effective for both average and strong students. This approach is
particularly productive in solving word problems. Ancient word problems
that contain elements of puzzles serve successfully for in-group
learning.
SOLVING ALGEBRAIC AND LOGIC WORD PROBLEMS IN STUDY GROUPS
The abilities of students in
introductory mathematics courses can often be recognized by comparing
students in the learning environment. When working in study groups,
capable students become group leaders, whose role is not only to solve
a problem but also to explain a solution to others. Such students would
demonstrate an interest in the subject, creativity in problem solving,
and capacity to properly present material. Apt students usually enjoy
“teaching” and tend to use an adequate mathematical language and
logical reasoning. Leadership in study groups gives capable students
satisfaction and confidence and results in better understanding the
subject.
Even a simple word problem, if it
requires writing a linear equation in one variable or a system of
linear equations, can be difficult for average students in
developmental courses. Ancient word problems, which describe a problem
with humour and implement an element of a game, help students to relax
and overcome a mathematical anxiety. Many ancient word problems can be
found in literature, and it is always possible to find a problem
suitable for a given audience. Some problems can be solved by
reasoning, whereas others require writing equations or systems of
equations. In this paper, we present examples of old Greek, Arabian,
Hindu, and Russian problems. These problems are suitable for
collaborative work in Algebra classes or can be used as “warm-up”
exercises in any Math courses.
WORD PROBLEMS INVOLVING LINEAR EQUATIONS IN ONE VARIABLE
Sinbad and Hinbad
[5; p. 12]. Sinbad and Hinbad each owned the same number of horses. How
many should Sinbad give Hinbad so that Hinbad has six more than Sinbad?
This problem can be solved more easily by reasoning than by writing an
equation. Problems of this type, although rather simple, may be
challenging and serve as an indicator of a student’s ability for
logical thinking. After such simple puzzles, problems that require
writing and solving linear equations are offered. Many ancient
problems, including the well-known “Diophantus Tomb Problem” [2; p.
25], involve linear equations. Writing an equation is the most
important step in solving word problems. Isaac Newton, in his book “The
Universal Arithmetic”, wrote that, in order to answer a question
related to quantities, one just needs to translate a problem from a
native language to the language of Algebra. He gave examples of how
such a translation is performed. Not much has changed from Newton’s
time, and contemporary textbooks in Beginning and Intermediate Algebra
use exactly the same language to describe steps in solving word
problems. Some problems that require writing a linear equation are
given below.
Ancient Greek Puzzle [5;
p. 26]. Demochares has lived one fourth of his life as a boy, one fifth
as a youth, one third as a householder, and has spent thirteen years
beyond that. How old is he?
Apples Thief [5;
p. 33]. While three watchmen were guarding an orchard, a thief slipped
in and stole some apples. On his way out he met the three watchmen one
after the other, and to each in turn he gave a half of apples he then
had, and two besides. Thus he managed to escape with one apple. How
many had he stolen originally?
Old French Problem [2;
p. 22]. A man spent one third of his money and lost two- thirds of the
remainder. He then had 12 coins. How much money did he have at first?
Father and Son [4;
p. 54]. A father was 32 years old, and his son was 5 years old. In how
many years will the father be ten times as old as the son?
Even though this problem is a simple one in terms of writing equation,
it has a negative solution that needs an explanation such as “it was
two years ago”.
WORD PROBLEMS INVOLVING SYSTEMS OF LINEAR EQUATIONS
Horse and Mule [4;
p. 35]. A horse and a mule each carried a heavy load, and the horse
complained about its weighty pack. The mule said to the horse: “Why are
you complaining? If I take one bag from you, my burden would be twice
as heavy as yours. But if you take one bag from my back, your load
would be equal to mine.” How many bags did each of them carry?
Starlings and Trees [3;
p. 12]. Flying starlings saw some trees on their way. When the
starlings sat one on a tree one starling remained without a tree. When
two starlings sat on each tree one tree remained without a starling.
How many starlings and how many trees were there?
Hindu Age Problem [2;
p. 42]. I am twice the age that you were when I was your age. When you
get to be my age our ages will total 63 years. How old are we?
LOGIC PROBLEMS AND PUZZLES
Another set of problems, in which
gifted students can benefit from in-group studies, comes from Logic.
Maurice Kraitchik, who gathered a wonderful collection of ancient
curious problems and puzzles, wrote in the introduction to his book
“Mathematical Recreations” [2; p. 13]: “…there are problems…whose
solutions are obtained by the direct exercise of the powers of
reasoning, without the intervention of formulas and computations. For
mathematics is applied logic in its simplest and purest form”. Although
such a characterization of Mathematics looks quite simplified, many
mathematicians as well as students in Math classes can appreciate the
charm of ancient logic problems. Logic problems and puzzles may be
offered to start collaborative work in any Math course. They would
provide an opportunity for group members to better recognize each
other, without the pressure of “serious” mathematical problems, and
allow gifted students to show their potential in Mathematics and
logical reasoning.
Two Medieval Problems [1; p. 27, 28].
1. A man goes to a well of water
with two jars, of which one holds exactly three pints and the other
exactly five pints. How can he bring back exactly four pints of water?
2. Three men robbed a gentleman
of a vase, containing 24 ounces of balsam. While running away they met
a glass-seller, from whom they purchased three vessels. On reaching a
place of safety they wished to divide the booty, but found that their
vessels contained 5, 11, and 13 ounces respectively. How could they
divide the balsam into equal portions?
What are Their Ages ? [5; p. 72]. Iskandar once asked his friend Kamar the ages in years of his three children. The following conversation ensued:
K: The product of their ages is thirty six.
I: That does not tell me their ages.
K: Well, by coincidence, the sum of their ages is your own age.
I: (after several minutes of thought) I still don’t have enough information.
K: Well, if this will help, my son is more than a year older than both his sisters.
I: Oh good! Now I know their ages.
What were their ages?
The Problem of the Pandects [2;
p. 28]. A hungry hunter came upon two shepherds, one of whom had three
small loaves of bread, and the other five, all of the same size. The
loaves were divided equally among the three, and the hunter paid eight
cents for his share. How should the shepherds divide the money? Versions of this problem can be found nearly in every collection of old problems.
Three Greek Philosophers [2;
p. 15]. Wearied by their disputations and by the summer heat, three
Greek philosophers lay down for a little nap under a tree of the
Academy. As they slept a practical joker smeared their faces with black
paint. Presently they all awoke at once and each began to laugh at the
other. Suddenly one of them stopped laughing, for he realized that his
own face was painted. What was his reasoning?
Robbery [5;
p. 45]. A man was being tried for robbery. Three witnesses came forth
and made the following statements. First witness: “The defendant has
committed over a dozen robberies in the past!” Second witness: “That is
not true!” Third witness: “He certainly committed at least one
robbery!” As it turned out, only one of the witnesses had told the
truth. Is the defendant guilty of robbery or is he innocent?
In general, it is easier to solve a
logic problem or puzzle than to explain the solution to others. For
students, who solved a problem by intuition, the following step of
justification and reasoning provides a good possibility for learning
logic rules of inferences. The knowledge of such rules is important for
studying Math courses in which not many proofs are presented.
CONCLUSION
Capable students can benefit from
collaborative learning in Math courses of all levels. In Developmental
Math courses, working in study groups is particularly useful for
solving word problems. Ancient word problems can help average and
capable students to overcome mathematical anxiety and enjoy learning
Mathematics. As leaders of study groups, students with higher
potentials can enhance their abilities in Mathematics, improve their
skills in logic and critical reasoning.
In the Middle Ages, competitions in
solving word problems were carried out, and participants had to present
solutions in a verbal form. A similar approach is applicable to the
work in study groups. Using the instructor’s advice, apt students can
thoroughly work up all the steps of translating verbal expressions into
the language of Algebra and explain solutions to other students. Our
experience shows that solving word problems by an instructor is less
effective, no matter how many examples are given, than students’
collaborative work and independent discussion.
REFERENCES
1. Ball, W.W.R. Mathematical Recreations & Essays, The Macmillan Co., NY, 1963.
2. Kraitchik, M. Mathematical Recreations, Dover Publications Inc., NY, 1942.
3. Olechnik, S.N., Nesterenko, U.V., Potapov, M.K. Old Recreational Problems, Nauka, Moscow, 1988 (In Russian).
4. Perelman, J. Recreational Algebra, Nauka, Moscow, 1976 (In Russian).
5. Smullyan, R. The Riddle of Scheherazade, Harcourt Brace & Co., NY, 1998.
ABOUT THE AUTHOR
Elena Koublanova, Ph.D., Associate Professor
Department of Mathematics, Community College of Philadelphia
1700 Spring Garden Street, Philadelphia, PA 19130,
USA
Office phone: 1-215-751-8928, E-mail: ekoublanova@ccp.edu
EXTRACURRICULAR WORK WITH CREATIVE-PRODUCTIVE GIFTED STUDENTS – PROGRAM AND ACTIVITIES
Emiliya Velikova
Abstract:The paper presents: - a
new Model of Joint and Independent Creative Work (MJICW) between a
leading teacher (or a team) and a“creative-productive gifted” student;
- original
authors methods (a system of mathematical problems,a system of
transformations, idea for creating new transformations and problems,
teaching methods), corresponding to the MJICW and also authors
experience with gifted students in Bulgaria, England, Greece.
Key words :Creative work, Transformations, Methods of Instructions, Students’ problems
BACKGROUND
The encroachment of new information
technologies and the continuous integration of mathematics with other
sciences in the contemporary world call for:
- gifted
mathematicians, able to discover and summarize diverse conclusions
within the information flow, and to generate new ideas;
- personalities,
able to develop their creative potential, enrich their knowledge and
experience and apply them to socially useful areas and activities.
Those
factors define the search for new models of educational and upbringing
activities, which, based on the positive experience, ensure
opportunities for developing
thestudent not only as a person who studies his/her lessons or a
consumer, but also as a creator of knowledge, a person who uses the
methods of creative mathematical activity for acquiring creative
application of information and mental processes in an integrated and
inductive way, directed to real-life problems, in conformity with
his/her interests and abilities.
ACTIVE MODEL FOR JOINT AND INDEPENDENT CREATIVE WORK WITH CREATIVE-PRODUCTIVE GIFTED STUDENTS
The organized combination
of activities, directed towards developing the productive potential of
students and creating students’ products, is viewed as joint and
independent creative work between a leading teacher (or a team) and
a“creative-productive gifted” student. There are three inter-related
parts to it[1], [9], [13], [14]:
- independent research and creative activity of the teacher himself;
- joint
work between the leading teacher and the gifted student with the
purpose of preparing the latter adequately for independent creative
activity;
- independent
creative activity of the gifted student, through which he satisfies his
needs for achievements and their expression, by creating, formulating
and presenting the product of his creation in front of an appropriate
audience.
The
model (MJICW) is principally based on the Schoolwide Enrichment Model
of J.Renzulli and S.Reis [6], [7] but also includes original authors
methods (a system of mathematical problems that includes geometrical
inequalities [13], a system of transformations [2], mathematical
methods for creating new transformations and new problems [13],
teaching methods [1], research methods [13])and also authors
experience[10], [14].
The main principles, set in the model, are:
- the volunteer principle – a strong desire of both student and teacher to participate inMJICW;
- principle
of creative applicability – application of abilities, knowledge and
skills for generating a creative product, meant for a specific
audience.
The main goals are - stimulating students for mathematical creativity in a specific area of mathematics;
- providing conditions for realization of creative activity.
Activities of Type I (general preparation, general exploratory activities).
The main goals are:
- forming a strong interest in students towards a specific mathematical area;
- developing dedication to the task set, i.e. converting the interests into internal motives for achievement.
Contents of activities Type I.
Through those activities students get the opportunity to enter the
world of science, according to their knowledge and interests, to
experiment with their ideas without being evaluated. Those activities
are: intensive group or independent discussion of scientific questions
and problems, which can be more easily solved through transformations,
lectures and meetings with experts in the respective area, etc.The
activities of this type strongly influence the natural curiosity and
activity of students, bring satisfaction, develop their interest,
initiative and independent thinking. The leading teacher is responsible
for planning and implementing those activities.
Activities Type II
(preparation for creative work, group training activities). They reveal
to students the charm of research andprepare them for self-creative
work.
The main goals are:
- developing
the creative abilities, producing research skills, acquiring knowledge
to a degree of applicability in the field of the student’s interests ;
- acquaintance with the methods of creative work in the area;
- achieving an optimal motivation level of creative work.
Contents of activities Type II.
Acquainting with the methods of creative mathematical work in the area
of transformations – the leading teacher and the mathematician present
to the students their experience in creating new problems and
transformations, generating new dependencies,generalized relationships,
etc.A particular attention is given to investigation, analysis and
decision-making, acquisition of new knowledge, formation and
presentation of results. The example set by the members of the leading
team is extremely important.
Activities of Type III
(creative work) comprise of independent choice and realization of
creative mathematical work in and through which the student expresses
himself/herself as a creator.
The main goals are to
provide conditions for mathematical creativity and presentation of
creative products to an appropriate audience.
Contents of the activities of Type III.
The student follows his/her own ideas for as long as he/she needs,
tries different ways of solving the problem chosen by him/her for
achieving the goal set.The methodological help of the teacher in the
collective planning of activities,determining the appropriate reference
and target audience, discussing the ideas, additional training if
necessary, proofreading of the paper, forming and presentation of the
results, etc. The leading teacher organizes contacts with experts,
provides access to scientific reference and technologically new
information sources, audience for the presentation, etc.
MJICW is a
completepsychological and pedagogical system, which displays the
characteristic elements of pedagogical and research, as well as
creative activity in the specific area of mathematics [13].
PROGRAM FOR WORKING WITH CREATIVE-PRODUCTIVE GIFTED STUDENTS
The program correspond to the MJICWincludes [2], [13]:
- Developing a system of mathematical problems that includes or can be solved by geometrical inequalities;
- Developing a system of transformations;
- Developing methods for creating new transformations and problems;
- Developing teaching methods.
- Creative training of students.
The method of instruction corresponding to the activities of Type I has been divided into three topics: - Examples of interesting problems that include geometrical inequalities.
- The Best Low and Upper Bound of Function.
- Specific Inequalities.
The topics have been divided into several parts in respect to problem solving methods such as: - Geometry helps Inequalities;
- Geometry helps Algebra;
- Vectors help, and others.
The problems have been chosen according to the following criteria: - to include mathematical methods, both known to the students, and nonstandard;
- to be diverse and to meet the requirements at a competitive level;
- to have been applied to similar purposes at home or abroad;
- to serve as grounds for student mathematical creative process.
Example:
Problem 1. If x, y, z are positive real numbers, prove the inequality
(1) [5].
Solution. Weconstruct (Fig. 1). Thenbyapplying the cosine theorem for the triangles ADB, BDC ? ADC respectively, we get
(2) .
The inequality (1) follows from the existing of , i.e. from the inequality , which we apply in (2).
The equality occurs in (1) when the point .
Then
Problem for self work
For any prove the inequalities:
(6) [4],
(7) [3].
When do the equalities occur?
Thetransformationofalgebraicproblemintogeometric oneisapplied into the topic “The best low and upper bound of a function”.
Problem. Find the best upper bound of the function
(8) ,
where .
The problems on view are developed, the areas of work or the given conditions are changed, and new problems are created.
New problem. Prove the inequality
(9) ,
where and ? > 0.
Instructions:
- Individual work on the basis problem with known methods (one month before).
- Discussions on students solutions.
- Showing very attractive solutions by the method of transformations.
- Solution of closely problems.
- Creating new problems by enrichment, analogies, etc.
The teaching method, corresponding to the activities of Type II is developed in the next three topics: - Method of Transformations. Application of Transformations for Solving Geometrical Inequalities.
- Dual Geometric and Algebraic Inequalities;
- GeometricandMixed Inequalities for a Triangle;
- Discovering New Transformations and Their Application for Creating Original Hard Problems.
Example of mathematical transformation:
Theorem 1.Let ABC be a triangle with the usual elementsa, b, c, F, , R, r, . Then there exists a triangle ABC_{1}with elements (Fig.2) [2]:
(10) , , , , , , , , , , , , .
Consequence. For any true triangle inequality (equality)
(11) ,
there exists another true inequality (equality) in the form
(12) ,
where the elements are given by (10), i.e.
(13) .
Definition. Activityduringwhichfromtheinequality(11) weget intoinequality (11) (12), i.e. we carry out a transfer (13), is called “Parallelogram Transformation” in respect to the side c of the triangle .
The creation of a transformation
requires from a given triangle ? to construct a new one ?' and to find
mathematical relations in the form (10) between the elements of these
triangles. It is possible, in specified conditions, to construct many
new triangles ?' and to create and prove many new transformations.
The transformation has a lot of applications:
- Creating of new asymmetric inequalities:
- Creating of “Generators” of inequalities:
(16) , ,
where an equality occurs when and only when .
For every in (16) different new inequalities are created or well-known inequalities, such as (15) if are proved.
- Creating of combination of transformations.
- Others.
Instructions. - Examining
a transformation, whose formulae are closer to students’ knowledge;
applying those transformations for the solution of competition problems
and creation of new ones.
- Show
the methods of creative activity of mathematicians – the paths for the
creation of some transformations, their summaries, combinations and
scientific applications, for the creation of new problems.
Theprocessofsolvingeveryproblemincludesthethree important stages of the general empiric model of the Method of Transformations: - transformationof
theproblemintoan easyone - transformation (for example, algebraic
problem is transformed into easier geometric problem);
- solving the dual problem by known methods (for example, geometric methods);
- transformationof
the results in respect to the given problem (for example, into
algebraic solution) – inverse transformation (if it exists).
Problem for self work. Prove the inequality
(17) ,
wherex, y, z are positive real numbers. The condition for x, y, z is given. When does an equality occur in (17)?
New Problem.Prove the inequality
(18) ,
wherea, b, c aresidesofanytriangle. The condition for a, b, c is given. When does an equality occur in (18)?
New Problems. Prove the inequalities:
whereA, B, C are angles of an arbitrary triangle. When do equalities occur?
The teaching method, corresponding to
the activities of Type III depend on the students’ interest, ideas,
speed of work, etc. It can include:
- meeting of students and creator of some transformations proper for high school education of mathematics;
- presenting some transformations andcombinationoftransformationsand the creative methods of the authors for their creation;
- finding two suitable triangles using geometry or algebra methods, depending on the interests and abilities of the students;
- determining the transformational formulae in a “beautiful” appearance;
- choosing prominent inequalities from the scientific sources in the field;
- application of transformations;
- choice of “beautiful” inequalities;
- designing of problems;
- presentation of results in front of a proper auditorium;
- assessment of one’s ownpresentation.
RESEARCH ON THE APPLICATION OF MJICW
One of the main tasks is to research the
role of the application of MJICW for creative character of the
students’mathematical problems.
Students from 9^{th}-12 ^{th} grade are the object of the research.
The subject of the research isthe creative character of the problems created by the students.
For example, through the
comparative analysis of the creative character of students’ products
(Fig.3÷5, tables 2, 3) it was found that:
- the
problems of the students from the experimental group have the
properties of creative products in the field of geometrical
inequalities;
- the theoretical and application instruments based on the method of transformations, form the foundation for student creativity.
From the survey conducted and the observation it was concluded that in order to achieve the goals of MJICW it is important to: - acquaint parents and students with the experiment;
- have a high-achieving leading teacher;
- plan activities in advance;
- develop study materials in advance;
- supply enough reference material;
- have a well-organized teaching team with good communication;
- have a specialist in transformations participate in the experiment.
In
conclusion, the comparative analysis, factor analysis, correlation
analysis and regression analysis of the experiment results showed that
the application of MJICW as a set of stimuli, provides conditions for
the development of the productive potential of students and for making
creative products.
SOME CREATIVE STUDENTS PROBLEMS
Problem of A.Velikov [12]. Prove the asymmetric inequality
(22) ,
wherea, b, care sides of a triangle with perimeter 2.
Problem of T.Petrov [11]. Find one or more solution of the equation
(23) .
ProblemofJ. Nikolov.Prove the inequality
(24) ,
where , and ?, ?, ?, r, R the usual elements of an arbitrary triangle
Problem of K. Ivanova. Prove the inequality
(25) ,
where a , b, g , r, s, R are elements of an acute triangle.
Problem of N.Velikova. Prove the inequality
(26) ,
where a, b, c, s are elements of a triangle and .
Problem of D.Lorenov. Prove the inequality
(27) ,
where a, b, c, R, ma, mb, mc are the usual elements of an arbitrary triangle.
Problem of M.Penchev. Prove the inequalities:
(28) ,
where s, r, R are the usual elements of an arbitrary triangle.
CONCLUSIONS
MJICW is specified for
extracurricular work only in mathematics for students in grades IX-XII
and requires the participation of a leading teacher and a
mathematician, who have some achievements in the area specified. The
model includes:
- activities which can be applied to extracurricular or regular studies in each area of mathematics;
- specific activities, which can be applied only to the field of transformations.
MJICW can be applied to:
a) extracurricular forms of education in mathematics for “gifted” students belonging to different age groups, as follows:
- determining of an appropriate topic or area in mathematics and complementing MJICW with new activities;
- development of appropriate lessons and aids;
- completing
the methods of education in accordance with the requirements of the
working area and the age features of the students;
b) regular education of students in grades IX – XII, which should: - create a strong interest in students towards different problems and areas of mathematics;
- plan where to apply the products created by students;
c) training of future teachers of mathematics by: - carrying out of theoretical analysis of the teacher’s psychology and his/her willingness to work with gifted students;
- testing of problems created and lessons developed, as well as their presentation by the future teachers.
REFERENCES 1. Bilchev,
S.J.&E.A.Velikova Jointly and Independently Creative Work Of
Talented Students and There Tutors I: The Process Of Learning Of
Mathematics, Mathematics and Education in Mathematics,
Proceedings of the Twenty Fourth Spring Conference of the Union of
Bulgarian Mathematicians, Svishtov, April 4 - 7, 95, Sofia, Bulgarian
Academy of Sciences, (1995), p. 88 - 108, plenary report (Bulgarian).
2. Bilchev, S.J.
& E.A. Velikova On an Generation of Parallelogram Transformation,
Mathematics and Education in Mathematics, Proceedings of the Fifteenth
Spring Conference of the Union of Bulgarian Mathematicians, Sunny
Beach, April 2-6, 1986, Sofia, Bulgarian Academy of Sciences, 1986,
pp.525-531.
3. Klamkin, M. S. Problem 805. CRUX Math., 1, 1983.?.22.
4. Klamkin, M. S. Problem 1394. CRUX Math., 10, 1988.?.302.
5. Problem ? 1090 ?vant, 2, 1988.p.26
6. Renzulli, J.S.
& S.M.Reis The Schoolwide Enrichment Model: A Comprehensive Plan
for the Development of Creative Productivity. In: Hanbook of Gifted
Education, N. Colangelo (Ed.), Boston, MA: Allyn & Bacon, 1991.
pp.111-141.
7. Renzulli, J.S.
& S.M.Reis The Schoolwide Enrichment Model: A How-to Guide for
Educational Excellence. (2nd ed.), Mansfield Center,CT: Creative
Learning Press, 1997.p.423.
8. Renzulli, J.S.
& S.M.Reis. Research Related to the Schoolwide Enrichment Traid
Model. Gifted Child Quarterly, 38, No1, Winter, 1994. pp.7-20.
9. Velikova, E.A.
Models and Programs for Diagnostics and Training of Gifted Students in
Mathematics, Proceedings of the Union of Scientists - Rousse, Bulgaria,
Ser.5, Mathematics, Informatics and Physics, Vol.2, 2002, pp.15-21.
10. Velikova, E. A.,
P.Vlamos Creative Education on Mathematics for Gifted Students,
Proceedings of the Second International Conferences “Creativity in
Mathematics Education and the Education of Gifted Students”, 15-19 July
2002, The University of Latvia, Riga, Latvia, 2002.pp.106-107.
11. Velikova, E. A.,
S.J.Bilchev & P.Vlamos Student’s Hard Geometry Problems, Geometry
& Mathematics Competitions, A. Rejali and F. Sajadi (Eds.), Isfahan
University of Technology & Isfahan Mathematics House, Iran, Paper
Contributed to Topic area 1, 4^{th} WFNMC Conference (4-11 Audust, 2002), Melbourne, Australia, 2002.pp.227-234.
12. Velikova, E. A., S.J.Bilchev & P.Vlamos Hard Problems Created by Gifted Students, Digital Journal “Revista Escolar De La Olimpiada Iberoamericana De Matematicas” , Spain, June, 2002. www.oei-campus.org/oim/revistaoim
13. Velikova, E.A.
“Stimulating the Mathematical Creativity of Students (9-12 Classes)“,
Dissertation thesis, State Specialized Scientific Council of the High
Attestation Commission of the Republic of Bulgaria, Sofia, 2002, 369
pages
14. Velikova, E. A.
One Bulgarian Experience in Identifying and Developing Mathematically
Talented Students,In:Developing Mathematically Promising Students,L.
Sheffield (Ed.), Published by the National Council of Teachers of
Mathematics, USA, 1999, pp 203-206.
ABOUT THE AUTHOR
Emiliya Velikova , Ph.D., Faculty Socrates and International Relations Co-ordinator
Department of Algebra and Geometry, Centre of Applied Mathematics and Informatics,
Faculty of Education, University of Rousse
8 Studentska str., 7017 Rousse
BULGARIA
Cell phone: +359/889 625 222 , Fax: +359/82/845 708
E-mail:
emily@ami.ru.acad.bg, emivelikova@yahoo.com
www.cmeegs3.rousse.bg
- The Third International Conference “Creativity
in Mathematics Education and the Education of Gifted Students”, August
3-9, 2003, Rousse, Bulgaria
www.icme-10.com
– TSG4: Activities and Programs for Gifted Students, ICME-10, July 4-11, 2004, Copenhagen, Denmark
IDENTIFYING OF CREATIVE-PRODUCTIVE GIFTED STUDENTS
IN MATHEMATICS
Emiliya Velikova, Svetoslav Bilchev, Marga Georgieva
Abstract:
The paper presents a diagnostic technology for finding
“creative-productive gifted” students based on thecombined theoretical
and experimental research on the problem from the aspect of the
sciences: psychology, pedagogy, methods of teaching in mathematics,
statistics.
Key words : Giftedness, Characteristics of Gifted Students, Mathematical abilities, Diagnostic technology
BACKGROUND
Every society is interested in
gifted persons who are able to develop their creative potential, enrich
their knowledge and experience and apply them to socially useful areas
and activities.
The formation of future creators is
a long and hard process, which starts at school. Hence, the teachers
play the most important role in the identifying process.
The emphasis on clarifying the
notion “creative giftedness” is placed on the basic characteristics of
the gifted and talented students in the contemporary theories and
research and J.Renzulli’s three-ring conception of “giftedness”.
The “schoolhouse (academically)
gifted” students quickly adapt to the school environment, have a high
IQ, numerous manifestations of it, etc. The “creative-productive
gifted” students possess or can develop a cross-section of three
clusters of characteristics: above-average general and specific
abilities, ahigh level of creativity and a high level of task commitment
[4]. They have a high creative quotient (CQ),often take a risk, etc. In
J.Renzulli’s conception “creative-productive giftedness” is looked upon
not as a “magical gift”, a genetically determined fact, because the
qualities from the three classes are encountered in all normal people,
but as a chance of developing a combination of qualities in an
environment close to the real creative work [4], [5]. G.Piryov notes
that those qualities get a higher level and more specific development;
they are more successfully combined in creative personalities’
characters [3]. In this sense J.Renzulli postulates: “our orientation
must be redirected towards developing “gifted behaviours” in certain
students (not all students) at certain times (not all the time) and
under certain circumstances” [4]. Giftedness is a condition that can be
developed [4].
J.Renzulli emphasize [4]:
- Both types are important.
- There is usually interaction between the two types.
- Special
programs should take appropriate provisions for encouraging both types
of giftedness as well as the numerous occasions when the two types
interact with each other.
CHARACTERISTICS OF GIFTED STUDENTS
The comparisons of general and
typological characteristics in Mathematical work (Table 1, Table 2) of
these two types of giftedness express many research results on
creative-productive people that although no single criteria can be used
to determinate giftedness,
Persons who have achieved
recognition because of the unique accomplishments and creative
contributions process a relatively well-defined set of three
interlocking clusters [4], [2]:
1) Well above average abilities (general and specific)
- general
abilities: high level of abstract thinking, verbal and numerical
reasoning, spatial relations, memory, and word fluency; adaptation to
the shaping of novel situations encountered in the external
environment; actualization of information processing, rapid accurate,
and selective retrieval of information.
- mathematical abilities:
2) High level of creativity
3) High level of task commitment
DIAGNOSTIC TECHNOLOGY
The main aim of the diagnostic
technology is choosing “creative-productive gifted” students for joint
and independent creative work with a leading teacher (or a team) in
Mathematics (Fig. 1) [1], [8], [9]. The main goal of the work with
gifted students is stimulating students for mathematical creativity in
the specific area of mathematics.
The first step of the diagnostic technology: identifying creative-productive gifted students:
1. Measuring
above average general abilities by IQ. As the students’ intelligence is
growing up it is necessary to apply adapted IQ test [6]. IQ has to be
more than 80%.
2. Measuring creativity by standard CQ test (Mensa). CQ has to be more than 70%.
3. Measuring above average mathematical abilities by [7]:- teachers’ assessment - Y1 – teacher in regular mathematics training;
- teachers’
assessment - Y2 – teacher in extracurricular work. Teacher training for
assessment the expression of students’ mathematical abilities is
necessary. The teachers observe: activity character and results; verbal
expression of thinking process; explanations and answers in the process
of problem solving.
- experts’ assessment using Mathematical test (CMA).
The sum of Y1, Y2 and CMA has to be more than 50%.
The second step of the diagnostic technology: defining the students’ and their parents’ longing.
RESULTS OF APPLYING THE DIAGNOSTIC TECHNOLOGY
The diagnostic technology was
applied as a part of an experiment in finding creative-productive
gifted students and stimulating their mathematical creativity for real
science work [8]. 69 students (9-12 classes) that showed interest in
Mathematics took part in it. The analysis of the results shows that the
research instruments have a high level of reliability. Some of the
results are present: Fig.2 ÷ Fig. 6, Table 3.
Remark: Every index in Table 3 [8] stands for a mathematical ability [2]
CONCLUSION
Diagnostic technology can be applied
to newly developed research on the extracurricular work in many areas
of mathematics, or developed for other age groups as well
REFERENCE
1. Bilchev, S.J.&E.A.Velikova Jointly and Independently Creative Work Of Talented Students and There Tutors In TheProcess Of Learning Of Mathematics, Mathematics and Education in Mathematics,
Proceedings of the Twenty Fourth Spring Conference of the Union of
Bulgarian Mathematicians, Svishtov, April 4 - 7, 95, Sofia, Bulgarian
Academy of Sciences, (1995), p. 88 - 108, plenary report (Bulgarian).
2. ?rutetskii, V. A. Psychology of mathematical abilities of Students., M., Prosveshtenie, 1968 (Russian).
3. Piryov, G. Psychology in Your Life. Horizonts, S., Partizdat, 1975. p.301 (Bulgarian).
4. Renzulli, J.S.
The Three-Ring Conceptions of Giftedness: A Deve lopmental Model for
Creative Productivity. In Conceptions of Giftedness, R.J.Sternberg
& J.E.Davidson (Eds.), Cambridge, London, New York, New Rochelle,
Melbourne, Sydney, Cambridge University Press, 1986. pp.53-92.
5. Renzulli, J.S. & S.M.Reis
The Schoolwide Enrichment Model: A Comprehensive Plan for the
Development of Creative Productivity. In Hanbook of Gifted Education,
N. Colangelo (Ed.), Boston, MA: Allyn & Bacon, 1991. pp.111-141.
6. Serebryakov, V & S.Langer Check The Intelligence of Your Child by IQ Tests for Children From 7 Till 17 Years Old, S., LIK, 1999. p.124 (Bulgarian).
7. Velikova, E.A. & M.Y.Georgieva, Diagnostic
of Mathematical Abilities,Proceedings of the Union of the Scientists –
Rousse, vol.1, R., Bulgaria, 2001. pp.67-73 (Bulgarian).
8. Velikova, E.A. “Stimulating
the Mathematical Creativity of Students (9-12 Classes)“, Dissertation
thesis, Government Specialized Scientific Council of the High
Attestation Commission of the Republic of Bulgaria, Sofia, 2002, 369
pages
9. Velikova, E. A. One
Bulgarian Experience in Indentifying and Developing Mathematically
Talented Students,In DEVELOPING MATHEMATICALLY PROMISING STUDENTS,L.
Sheffield (Ed.), Published by the National Council of Teachers of
Mathematics, USA, 1999, pp 203-206.
ABOUT THE AUTHORS
Emiliya Velikova, Ph.D., Faculty Socrates and International Relations Coordinator
Department of Algebra and Geometry, Centre of Applied Mathematics and Informatics,
Faculty of Education, University of Rousse
8 Studentska str., 7017 Rousse
BULGARIA
Cell phone: +359/889 625 222 , Fax: +359/82/845 708
E-mail:
emily@ami.ru.acad.bg, emivelikova@yahoo.com
www.cmeegs3.rousse.bg
- The Third International Conference “Creativity
in Mathematics Education and the Education of Gifted Students”, August
3-9, 2003, Rousse, Bulgaria
www.icme-10.com
– TSG4: Activities and Programs for Gifted Students, ICME-10, July 4-11, 2004, Copenhagen, Denmark
Svetoslav Bilchev, Ph.D., Associate Professor, Dean
Department of Algebra and Geometry
Centre of Applied Mathematics and Informatics,
Faculty of Education, University of Rousse
8 Studentska str., 7017 Rousse
BULGARIA
Cell phone: +359/886 735 536, Fax: +359/82/845 708
E-mail:
slavy@ami.ru.acad.bg
Marga Georgieva, Ph.D., Professor, Vice-Dean
Faculty of Education
“St. Ciril and St. Methodius” University of Veliko Turnovo
2 T. Tarnovski str., 5000 Veliko Turnovo,
BULGARIA
Cell phone: +359/888 409 044, Phone: +359/62/ 649 832, Fax: +359/62/649 834
E-mail: myg@uni-vt.bg , tyg@vali.bg
EDUCATIONAL NETWORK COMMUNICATING HEURISTIC AND SOPHISTICATED MENTAL MODELS OF
MATHEMATICAL KNOWLEDGE –
DEVELOPING OF PEDAGOGICAL REASONING TO SUPPORT GIFTED/TALENTED STUDENTS IN GREECE
Eugenia Meletea
Abstract: The presentation includes: a. The issue of the education of Gifted/Talented students in Greece,
b. The results
from 2002, since the Scientific Association – Non Profit Organization “
?PLOUN”, (Development of Pedagogical Reasoning to Support
Gifted/Talented Students) appeared,
c. “IASONAS”: Educational Networks, Programs, Activities and Studies for Gifted/Talented Students,
d. “APLOUN” and
“IASONAS”, as an “Arch”, beginning and introducing a Specific Issue (as
the Education of Gifted/Talented Students in Greece), and the use of
Models of Mathematical Knowledge to develop:
e. Educational Networks,
f. Proposals of Holistic and Specific Strategies,
g. Quality Developing Activities,
h. Heuristic of Sophisticated Mental Models of Mathematical Knowledge.
For the first time in Greece, a
non-profit organization (APLOUN), and the “IASONAS”, Educational Open
Networks, Programs, Activities and Studies, deal with the issue of the
Educational Needs of Gifted/Talented students. Participation in the
above and my experience as an independent Researcher during the last 10
years, looking for teaching quality indicators, appropriate education
materials and methods is considered.
A. THE EDUCATION OF GIFTED/TALENTED STUDENTS IN GREECE
In recent years, an important
research initiative into matters of special education has been underway
in Greece. However, while measures have been taken by the Greek state
for the education of individuals belonging to specific special
categories, such as children with physical, mental or learning problems
et al., there is no such welfare concerning the special Gifted/Talented
children/young adults.
Several of these Gifted/Talented
children/young adults are not discovered and properly treated from the
pre-school age; they demonstrate poor performance at school,
subsequently being referred to for control of learning difficulties or
problems in behavioral and social adaptability.
In Greece, during the last 10 years
schools start to accept and support students with Talent in Music (20
Secondary Public Schools). For other category of Talent there do not
exist schools to support and educate students. Only in Big Cities there
are supporting classes for Athletes and underway is the program for the
schools to support Talents in Fine Arts.
Many of competitions are taking
place and are organized by the Greek Ministry of Education, several
Public and Scientific Associations, for students in the Secondary
Education. The Greek Mathematical Association (EME) organizes several
competitions in Mathematics: there are National (Panellenic) and
International Competitions and Olympiads, see [3]. Other competitions
also are taking place (Chemistry, Computers, etc).
In the Greek Public elementary
schools, there is no any Talent support program. Only in some Private
Schools they use basic programs supporting Talent, and there is the
philosophy of the Enrichment Model. Greek Schools are not allowed to
apply Acceleration programs; there is no way for Greek Students to
Grade Skipping, if this is necessary for them.
From multi-source research it has
been estimated that about 50% Gifted/Talented children abandon their
studies by their adolescence. The absence of welfare and special
planning to meet the special needs of these children results in a large
part of them, though possessing outstanding abilities, demonstrate poor
academic performance and have difficulties in adapting to their
environment. Furthermore, insufficient or misconceived pedagogical
approaches serve to undermine the fragile emotional world of these
children/young adults, due to the suppression or obstruction of the
development of their high mental abilities, with direct consequences to
their regular emotional development.
B. THE GREEK SCIENTIFIC ASSOCIATION “ APLOUN ”
The Greek Scientific Association –
Non Profit Organization called “APLOUN”, consists of people from
various fields, who, out of their sensitivity to the particular issue,
have set the proper concern for these children as their goal.
The foundation of “APLOUN” aims to
the development of a framework to support Gifted/Talented
children/young adults, hereby setting as its basic objectives the
following [5]:
- Informing the general public about the special needs facing Gifted/Talented pupils/students.
- Timely
distinction and recognition of those children who possess outstanding
abilities, so that they are properly treated, initially by family, and
later by their broader educational and social environment.
- The
initiation of research regarding the education of Gifted/Talented
pupils/students in our country, aiming to locate those factors that
contribute to their regular mental as well as emotional development.
- The
investigation of the concept of differentiability and also of the
special needs facing Gifted/Talented children/young adults.
- The
encouragement of collaborative learning within an environment providing
‘different’ individuals with equal opportunities for educational group
effort, with specialized material based on recent educational methods
and means of teaching.
- The
production of specialized educational material based on models already
in application in other countries, to meet the special needs facing
Gifted/Talented pupils/students, namely the Heuristic Model and the
School-wide Enrichment Model, which may also be taken advantage of by
pupils of standard classes. [2].
- Special training and reinforcement of educational staff in new methods and productive procedures.
- The
development of a communication network aiming to inform the general
public, the educators, the specialized scientists and parents.
- The connection of the Society with international organizations involved in similar endeavors.
- The participation and reinforcement of parents in their effort to support and meet the needs of Gifted/Talented pupils/students.
B. 1. SUPPORT TO “APLOUN” IS GIVEN BY:- The
Laboratory of new Technologies in Communication, Culture, Education and
Media Studies of the National & Capodistrian University of Athens,
Dpt. of Communication & Media Studies, Athens, Greece.
- The Medical Office for Developmental Paediatrics at the 1^{st} University Clinic of the University of Athens (Juvenile Hospital “Aghia Sophia”); it is appointed as President.
- The Department of Physical Education and Athletics of the University of Athens.
- Professors from several Departments and Universities in Greece and Oversees.
B. 2. “APLOUN”’S ORGANIZATION STRUCTURE
In an initial effort aiming to the
internal organization of the Association so as to achieve the
fulfillment of its goals, was unanimously decided, following
discussion, the formation of twelve (12) committees, that is, the
creation of twelve (12) fields of activity, which will operate
autonomously but will also communicate among each other. Association
members will have the opportunity to declare, in accordance with their
personal interests, in which of these fields they would be interested
to contribute. There will be no remuneration at the present phase for
the members’ involvement in the Association’s activities, owing to the
Association’s financial standing, which does not allow for that.
More specifically, the
administrative board unanimously decides to form the following twelve
(12) Committees, while also defining the responsibilities there of and
appointing the President of each Committee:
Committee A.: The central Communication Network of the National Association “APLOUN”.
Committee B.: Centre for timely distinction and recognition of those children Deemed to have outstanding abilities.
Committee C.: Centre for the Psychological Support of the Gifted/Talented children/young adults.
Committee D.: Center for Assessment of Motor Behavior and Athletic Performance of Gifted/Talented Children/teenagers.
Committee E.:
Education and Teachers Training - Collaboration of Teachers for the
Production of educational material aimed to educate Gifted/Talented
pupils.
Committee F.: Educational Programs – Parents/Family Counselling.
Committee G.: Development of Communicative, Cultural, Educational and Scientific Activities.
Committee H.: Able Underachievers.
Committee I.: Global Curriculum Development, Extra Curriculum Activities – Preschool Education.
Committee J.: Global Curriculum Development, Extra Curriculum Activities – Primary Education.
Committee K.: Global Curriculum Development, Extra Curriculum Activities – Secondary Education.
Committee L.: Fine Arts: Gifted/Talented Pupils/Students
COMMITTEE FORMATION AND DEFINITION OF THEIR RESPONSIBILITIES
Committee A.: The central Communication Network of the National Association “APLOUN”.
The responsibilities of this committee are:
a. The
creation and development of a communication network for the information
of the general public, educational staff, specialized scientists as
well as parents/families
b. The development of
a communication network among the members of the administrative board
and the presidents of the Association’s committees, the trustees, the
representatives and the society members
c. The information of the general public about the special educational needs of Gifted/Talented pupils/students
d. The downloading of material on the Internet and the promotion of the Association through it
e. The connection of
the National Organisation with other International Organisations
involved in similar endeavours, with international research centres,
relevant Associations and Bodies abroad, and
f. The creation of a
database, which will emerge out of the collection of all the material
from the Committees’ activities, as well as those of the trustees and
the Association members, e.g. Communicational and/or educational
material, relevant bibliography, research and programs.
Committee B.: Centre for timely distinction and recognition of those children deemed to have outstanding abilities
The responsibilities of this committee
are the timely distinction and recognition of those children deemed to
have outstanding abilities, so that they are properly treated, at first
on the part of their family and then, by their educational and broader
social environment.
Committee C.: Centre for the Psychological Support of the Gifted/Talented children/young adults
The responsibilities of this committee are the following:
a. Psychological Support of the Gifted/Talented children/students, and
b. Research regarding
Gifted/Talented children/students, involving the location of factors,
which contribute to their regular mental and emotional development.
Committee D.: Centre for Assessment of Motor Behavior and Athletic Performance of Gifted/Talented Students
The responsibilities of this Committee are the following:
a. The
timely distinction and recognition of those children/young adults
deemed to have outstanding abilities, so that they are properly
treated, at first on the part of their family and then, by their
educational and broader social environment.
b. The Sports training of Gifted/Talented pupils/students.
c. The educational support and social integration of gifted and talented students, by means of sporting activities, and
d. The support for research into the above-mentioned issues
Committee E.: Education
and Teachers Training - Collaboration of Teachers for the Production of
educational material aimed to educate Gifted/Talented Pupils, D.3)
Research on teacher professional development [4].
Committee F.: Educational Programs – Parents/Family Counselling
The responsibilities of this Committee are the following:
a. The recording of educational programmes, for Gifted/Talented pupils/students, which have already been developed in Greece.
b. The participation
in, and reinforcement of, parents and family in the active support of
Gifted/Talented children/young adults, and the reinforcement of
cooperation with specialists and educators.
Committee G.: Development of Communicative, Cultural, Educational and Scientific Activities
The responsibilities of this Committee are the following:
a. Special
categories of Gifted/Talented pupils/students (in Maths, Music,
Sciences, Technology) investigation of the concept of differentiability
and the special needs facing Gifted/Talented pupils/students, to
develop their special Association skills and abilities.
Classroom Strategies in Primary and Secondary Education.b. Development
of Communicative, Cultural, Educational and Scientific activities,
aiming to fulfill the Association’s goals by means of the following:
organisation of seminars, lectures, conferences, programmes in Mass
Media, publication of printed matter such as magazines and newspapers
or posters and leaflets.
Committee H.: Able Underachievers
Committee J.: Global Curriculum Development, Extra Curriculum Activities - Primary Education
The responsibilities of this Committee are the following:
a. Global Curriculum Development to Support Gifted/Talented Students.
b. Extra Curriculum Activities in Secondary Education.
Committee K.: Global Curriculum Development, Extra Curriculum Activities – Secondary Education
The responsibilities of this Committee are the following:
a) Global Curriculum Development to Support Gifted/Talented Students.
b) Extra Curriculum Activities in Secondary Education.
Committee L.: Support Gifted/Talented Students: Fine Arts
The responsibilities of this Committee are the following:
a. Special
Educational activities, to discover children Special Talents in Fine
Arts, liberate imagination, ability to perform and harmonic development
b. Multicultural Music-Theater activities
B.3. SOME OF THE ACTIVITIES OF "APLOUN"- OnedayConference
on the subject: “Intellectual and Talented Students: Pedagogical
Approach”, that tookplace on February 2003, in the Central Hall of
University of Athens, under the aegis of Ministry of Education.
- In
a meeting of ours with the Minister of Education, we submitted our
proposals and were given his promise that the issue of Gifted/Talented
students will be dealt with in Greece as well. The appropriate
legislative regulation is already underway.
C. “IASONAS” - Educational Open Networks, Programs, Activities and Studies (
www.iasonas.org)
It provides Access and Services in
Open Network for Advanced Studies of Developing Pedagogical Reasoning
Goals, helping all students to growth, support, educate and develop
their personal interests and talents.
In order to cover the
needs that usually some students have and to motivate them to discover,
create, develop and perform their personal talents and needs the
Educational Open Networks, Programs, Activities and Studies “IASONAS”
decide the following activities:
1. Framework: “Plaisio Dialogue Workshops on Network - Framework – Databank
Development of Pedagogical reasoning in order to Support HarmonicDevelopment of Gifted/Talented Students.
Questions, Opinions and Dialog are
the following issues for Gifted/Talented students’ Support and
Education needs. The dialogue is necessary at the time in Greece
because the society needs to accept that it is a very important mistake
not to respond to the educational needs of Gifted/Talentd children, and
not to give them the possibility to improve their personal
characteristics.
The areas for discussion are organized to create an interactive network.
2. Production of
Educational Materials that identify, discover, create end develop
Students Personal Talents, in cooperation with educators, writers,
publishers and computer specialists using Multimedia Managers and
Content Management systems. The following databases are under
consideration:
Creative & Interactive Educational Network
With issues from the Greek Classical Literature, History, Philosophy and Ancient Greek Mathematics
Interactive Databank with Software and Multimedia applications to developand cultivate Mathematics
3. Development of Multicultural and Multidisciplinary Educational Activities
Program “ELLAS” - Expanding Learning Laboratory Activities and Studies.
Library on Network in Cooperation with Organizations and People.
Collected Materials, Programs, Events, and Activities.
a. Development of Communicative, Cultural, Educational and Scientific Activities
b. Use of the Information and Communication Technologies in the teaching and learning process.
c. Connecting Schools with the Local Communities and Industry
d. Connecting Schools between themselves
e. Multicultural Communication between Schools
f. Enrichment of the School Curriculum
g. Development of the interactive School Materials
h. Development of the Global Curriculum
i. Extra Curriculum Activities
j. Active Learning
k. Cooperation between Students
l. Harmonic Human Development Patterns
4. Music – Theater Group “Spheres Harmony” to discover, create, develop performance students’ personal talents and needs.
5. Program “Aristotle’s” Teachers Training Programs and Seminars
6. Group “Odyssea and Odysseus”: Teachers/Parents of Gifted/Talented Students
Activities and seminars to Support Parents/Family
7. Villages “Orpheas”, “Apollon” etc:
Summer Schools for Gifted/Talented Students
Supporting Programs for Students, Educators, Specialists and Parents
Activities Performance Center
Congress Center
D.1. WORKSHOPS ON NETWORK - FRAMEWORK – DATABANK
Questions, Opinions and Dialog in
the following issues for Gifted/Talented Students Support and Education
are needed [4]. The dialogue is necessary because the society needs to
accept that it is a very serious mistake not to give to the gifted
children the ability to improve the Educational possibilities they have.
The following areas for discussion are organized to create an interactive network:
Workshop A: Identification of Gifted/Talented Children/Students
- Definitions
- Timely recognition of children with outstanding ability.
- Gifted/Talented Children/Students: School performance - Academic achievement - Special or Specific Learning Difficulties
Workshop B: Opinions and understanding about Gifted/Talented Students Support and Special Education needs - Public
- Parents/Family
- Educators
- Pupils/Students
Workshop C: Educational programmes and methods for Gifted/Talented Students - Use of the educational and communicational technology
- Development of educational, communicational and cultural activities
- Extracurricular activities
Workshop D: Global Curriculum development for Gifted/Talented Students Special Education needs - Educational Materials
- Production of appropriate educational material
Workshop E: The educator of Gifted/Talented Children/Students - Professional education
- Training programs
- Support
Workshop F: Psychological support of Gifted/Talented Children/Students
Workshop G: Support Parents/Family of Gifted/Talented Children/Students
D. 2. USING TECHNOLOGY TO PRODUCE EDUCATIONAL MATERIALS AND NETWORKS:
To develop multicultural
communication networks, we need universal communicating language, like
music [4]. The emotions and the multidimensional discipline Music
creates are educational, cultural and multicultural tools. Mathematics
is next to the music, the second universal language. The concepts and
skills that belong to mathematics can be a multicultural communication
language. Mathematics is embedded into knowledge communication systems
and organizing information that expresses decision making processes,
explicit encoding of knowledge, qualitative understanding. Starting
from Basic Concepts, Machine learning techniques and the Artificial Intelligence , we design reflective educational materials expanding mathematics.
Selected issues from Classical Literature also appear in this project.
The interactive simulation
educational network to communicate heuristic and reflective mental
models for the special educational needs of Gifted/Talented Students
needs:
1. Pedagogical
principles for coaching, individual differences in learning, personal
learning plans and self-improving Experts/Tutors Systems.
2. Developing Framework for pedagogically organized Global Curriculum
3. Using new
Technologies and advanced Technology to develop a Technology – based
learning environment, using also the Distance Learning.
4. Sophisticated
mental models to teach Basic Concepts, expanding their meaning to the
level that encourages the Gifted/Talented Students to the success.
D. 3. RESEARCH ON TEACHER PROFESSIONAL DEVELOPMENT AND TEACHERS TRAINING PROGRAMS
In state to prepare teachers to perform
in a professional and creative way in teaching the gifted and talented
pupils/students we scheduled the following activities [6]:
a. The support for research, employing the Action Research method,
related to the investigation of the competencies of Teachers concerning
the efficiency of their educational work by means of the creation of a
communication network to reinforce the collaboration and interaction
among teachers and other educational staff.
b. Collaboration for the production of educational material aimed to educate Gifted/Talented pupils.
c. The education and
training of teachers, organisation and realisation of educational
seminars to promote teachers to use new educational methods as well as
productive procedures.
d. The research in teaching quality indicators and the impact on the students’ reaction [4].
e. The production of
specialized educational material based on models already in application
in other countries, to meet the special needs facing gifted and
talented students, namely the Heuristic Model and the School-wide
Enrichment Model, which may also be taken advantage of by
pupils/students of standard classes [2].
f. The encouragement
of collaborative learning within an environment providing “different”
individuals with equal opportunities for educational group effort, with
specialized material based on recent educational methods and means of
teaching.
D. 4. EDUCATIONAL PROGRAMS AND ACTIVITIES
Taking under consideration the relation
and influence between Education & Culture, it is obvious that
Family, School and the broader Community have a common responsibility
for developing young people's discipline and skills. In order to creat
inderdiciplinary materials and applications in teaching & learning,
we develop cooperation of scientists and performers [6]. Holistic
educational programs to support gifted, with (as Plato also point)
Music, Mathematics, Gymnastic, Art History, Archaeology, Geography and
other subjects included, will be developed.
In cooperation with Professors
from several Departments of the University, and also in cooperation
with the Scientific Associations, like the Greek Mathematics
Association, we assume that we will improve pedagogical reasoning,
creativity and development of teachers’ and students’ personal talents,
developing a human world.
REFERENCES
1. Plato, Republic, book VII.
2. Renzulli, J. & Reis, S. (1986) The enrichment triad/revolving door model: A schoowide plan for the development of creative productivity. J. Renzulli, Systems and models for developing programs for the gifted and talented. Mansfield Center, CT: Creative Learning Press.
3. Dimakos, G. (2003) The Greek Mathematical Competitions and the Mathematical Olympiads, Proceedings of 1^{st} APLOUN Conference, 09.02.04
4. Meletea E. (2003) (Paper on the 1^{st} APLOUN one Day Conference 09.02.04)
5. Thomaidou L., Meletea E. (2003) Research and Action for the Education of the Gifted/Talented students in Greece: Vol. 25, Dec-Mar 2003, Patakis Editions, Athens.
6. Meletea E. (2004) (Paper on the 1^{st} APLOUN’ s Congress 13 – 16 May)
ABOUT THE AUTHOR
Eugenia T. Meletea
Mathematician – System Analyst
President of “? P L O U N ”
The Greek Scientific Association –
Non Profit Organization - Development of Pedagogical Reasoning to
Support Gifted/Talented Students
Director of “ I A S O N A S ” -
Educational Open Networks, Programs, Activities and Studies for Gifted/ Ta lented Students
17, Panagi Kyriakou,
Amaroussion, 15124, Athens
GREECE
Tel. Fax: (30) 210 80 68 563
(30) 210 96 74 019
e-mail: emel@tellas.gr
THE QUALITY OF THE REASONING IN
PROBLEM SOLVING PROCESSES
GeorgeGOTOH
Abstract: This
paper takes into consideration the state of instruction for making
students master a high level mathematical thinking through mathematical
problem solving. The attention is paid to the quality of the reasoning
in problem solving processes by looking at the problem that can be
solved in different ways with different mathematical concepts in each
as an example. Based on some levels being assumed, the design for the
lesson and unit for acquiring the new concept creatively is shown.
Key words: Cognitive entity, Creativity, Mathematical structure,Problem solving.
INTRODUCTION
Problem solving and
creativity have been one of the most important themes since the 1980s.
The relationship between these two is not so clear, especially when it
comes to the effective way of enhancing students’ creativity. Because
of the complexity and variety of the concept of the word “creativity”,
this term has become associated with various aspects of creative
behaviour and mental functioning that range along a cognitive-emotive
continuum.
In this paper, the focus
is upon the process of creativity, not the definition of creativity
with a great source of apprehension. First, the three-stage model of
development of mathematical thinkingwith regard to cognitive entity is
shown for the basis of the following.Second, an example that can be
solved in different ways is considered.Then the conformity of the model
with the example and the assignment of further research is considered.
1. The stages of development of mathematical thinking
In mathematics education, the
concern about problem solving has not decreased after the 1980s. And
one of the most important themes in problem solving research is how to
make students master higher-order-thinking ability, which brings them
flexible and creative ways of reasoning towards non-routine problem.
By taking a somewhat closer look at
the aspects of different kinds of problem solving as a heuristic, there
seems to be three stages that can be hypothesized as follows:
- Stage1: The empirical (informal) activity
- Stage2: The algorithmic (formal) activity
- Stage3: The constructive (creative) activity
Stage1: The empirical (informal) activity
In this stage, some kind of technical or
practical application of mathematical rules and procedures are used to
solve problems without a certain kind of awareness.
Stage2: The algorithmic (formal) activity
In this stage, mathematical
techniques are used explicitly for carrying out mathematical
operations, calculating, manipulating and solving.
Stage3: The constructive (creative) activity
In this stage, a non-algorithmic
decision making is performed to solve non-routine problem such as a
problem of finding and constructingsome rule.
It has been hypothesized that the
learning activities are to be developed in this order. That is to say,
the context of problem solving is prepared by previous experiences and
set by a preparatory stage in which mathematical procedures become
interiorized through action before they can be the objects of
mathematical thought. However, it is quite clear that there is a
qualitatively great difference between Stage2 and Stage3. Especially,
little research on developing, teaching and evaluating problems which
require the reasoning with various conceptual entities and the insight
of the underlying problem structure has been done.
2. Course materials
Taking three stages described above
into consideration, it can be said that the ultimate goal of
mathematical problem solving is to acquire not only knowledge and
techniques but also the attitude that enables students to think about
the main structure of the problem without adhering to things they have
learned before.
Therefore, an example that contains many elements of approach corresponding to each stage described above is considered.
Example
Stage1: The empirical (informal) activity
Maybe almost all the students who
haven’t met a similar problem will have difficulties with this one. For
them, there are no rules or techniques available except “try and error”
method as all line segments are measured. In that case some students
will be able to find the property;
However, this will happen by chance, without the awareness of its’ theoretical (mathematical) foundation.
Stage2: The algorithmic (formal) activity
We can use the assumption given by the problem and write down formulas like
Then we’ll get the formula;
Stage3: The constructive (creative) activity
Here is a little different way of manipulating formulas;
Then
“What does this mean?”
“Let’s draw!” (Figures are omitted)
- Add the point F under the conditions of
- Make parallelogram BCEF (add the point E)
- Connect B to E
- Make parallelogram MNGH (show )
- Connect A to E
“Pay attention to triangle ABE”
Then
,
“Show it”
“Can you find any relation to the givenproblem?”
Now we can invent the following solution (Figures are omitted).
- Construct parallelogram ABF’D (F’: an internal point of the quadrangle ABCD)
- Construct parallelogram BCE’D (BC//F’E’ and BC=F’E’)
- Connect B to E’, A to E’ respectively (construct triangle ABE’)
Then
AM=BM, AN=E’N
3. Discussion
The quality of problem solving
approach is categorized into approximately three stages in accordance
with the way of using mathematical knowledge. And the final stage
(Stage3) requires a higher-order-mathematical thinking that is more
difficult to be mastered.
Based upon rationales and examples,
it can be said that one of the most effective ways of reaching Stage3
is to emphasize the relation between Stage1 and Stage2 because both
deductive and empirical reasoning are essential to discover some
mathematical rule of nature.
Therefore, it is important to
reacknowledge the educational value of naive manipulation anddevelop
the way of teaching and evaluating problems that can be solved in
different ways with different mathematical concepts.
REFERENCES
1. Gotoh, G. Lesson
Design for Reconstructing Mathematical Knowledge. In Konno, Y(Ed)., The
Reconstruction of School Knowledge. Chap.3, pp.204-217. Gakubunsha,
2002. (In Japanese)
2. Tall, D. Advanced Mathematical Thinking. Kluwer Academic Publishers. 1991.
3. N. Mashiko. The
difference of problem solving processes between higher order thinking
tasks and standard tasks in introductory algebra. Naruto University of
Education. Vol.9, 1994, pp.151-166 (In Japanese)
4. N. Mashiko. Some
problem characteristics of nonroutine problems in school mathematics.
Naruto University of Education. Vol.11, 1996, pp.175-185 (In Japanese)
ABOUT THE AUTHOR
George GOTOH
Media Network Center
Waseda University
1-104 Totsuka-machi
Shinjuku-ku
169-8050Tokyo
JAPAN
E-mail: gotojoji@mnc.waseda.ac.jp
EUROPEAN PROJECT: MATHEU
IDENTIFICATION, MOTIVATION AND SUPPORT OF MATHEMATICAL TALENTS IN EUROPEAN SCHOOLS
Gregory Makrides,Emiliya Velikovaand partners
Abstract:
In many European schools the mathematics curriculum is designed to
serve the average and special needs students without identifying and
supporting potentially talented/competent students in mathematics. The
aim of this project is to develop methods and educational tools, which
will help the educators to identify and motivate talented students in
mathematics as well as to support their development within the European
Community without any discrimination. The project intends to merge
forces and establish a network through the Mathematical Societiesand
universities in the European area to support the aims of the project as
well as to use new technologies in the support, dissemination and
sustainability of the developed structure of cooperation.
MATHEU is a new approved
project under Socrates-Comenius 2.1 action with characteristic
activities that are expected to enhance the learning of mathematics in
the European region as described above. The presentation will discuss
the philosophy, aims and objectives and the work plan of the project
and will invite interested individuals and organisations to support the
aims of the project.
INTRODUCTION
ThedecisionoftheEuropeanUnion,
COM/2001/678,says, «Inasocietyofknowledge, Democracyrequires the
citizens to have scientific and technological knowledge as part of the
basic competence».
The future objective aims
of the European Educational Systems, which were agreed on 12 February
2001 from the Education Council in Stockholm, identify Mathematics as
one of the major priority subjects. The basic objective is the increase
of interest in mathematics from early age and the impulsion of youth to
follow careers in these subjects, more specifically in the research in
these fields.
The types of students who
will be able to contribute in the research of these fields are more
likely to be students who are talented in these fields and more
specifically in mathematics.
Certain activities towards
this objective are already taking place in some countries. The aim of
MATHEU is to bring together experts from the partner countries and to
exchange ideas, background knowledge and experience and to develop
together a system that will work for the whole of Europe.
Talented students in
mathematics have to be discovered in early stages and in a systematic
way. The usual method for identifying such students is through
competitions but it is generally acceptable that many talented students
in mathematics are never discovered simply because they do not
participate in competitions or simply because they were not among the
top ten during the competition process, or they are talented students
who cannot perform under strict time limits.
European countries have to
find ways to keep their talents and brains in Europe. In order to
accomplish this, mathematicians, academicians and educators have to
work together in a European dimension and to design a programme, which
will change attitudes of governments, universities and foundations in
favour of supporting the gain of mathematical talents in Europe and
decreasing the brain drain outside the European Community. Talented
students need attention, love, support, training, recognition,
identification. MATHEU promises to offer solutions to all these for the
development of European talented students through their teachers,
educational administrators and other
bodies-organisations-institutions-government, as well as through the
direct links via the Internet.
AIMS AND OBJECTIVES
The aim of MATHEU is the
development of methods and supporting material for the identification
of talented students in mathematics in European Schools and their
development and support. The project aims to establish a network of
sustainable support through universities, mathematical societies and
foundations of the partner countries at first and later throughout the
European community at large.
Teachers in Europe need to
be trained with methods and to be provided with material in order to be
able to identify and support such talented students.
The main objective is to
help Europe to gain the maximum contribution to and from these
students, who will become the backbone for the scientific and
technological knowledge necessary to make Europe the major technology
developer and economic power in the years to come.
These aims and objectives will be accomplished by:
- Analysis
of the flexibility of existing mathematics curricula in European
Schools with emphasis in the partner countries focusing on the aspect
of talented students
- Analysis
of methods and tools used in European countries for the identification,
motivation and support of talented students in mathematics
- Design
methods and tools for identifying potentially talented students in both
primary and secondary education levels and for training teachers so
that they can bring the students to express their 'talent' in
mathematics (talent as ability to face and solve problematic situation
and to appreciate the role of theoretical thought)
- Design
special pedagogical methods and subject material for the development
and promotion of talented students in European schools
- Develop
methods/solutions and a programme for changing attitudes within
government , universities and foundations in providing fellowships and
support in order to keep mathematical brains in Europe
- Design a special Web-site devoted to this purpose which will enable the sustainability of the project aims
To
achieve the above a team of European mathematicians, experts in either
Didactics or Subject or Technology has been assembled to address the
problem in an integrated and coherent manner.
Developing a unified European
training programme that offers tools for identifying, motivating and
supporting talented students in mathematics is a new approach, which
can be used complementary to the existing curriculum systems in the
European community without any discrimination. In addition, using new
technologies to provide sustainable support for a group of students
with high competence in a particular topic is a recent development.
Keeping mathematical talents in Europe through a “brain-gain effect” is
also a new philosophy and approach. Also, part of the work will
investigate the problem of mathematics talented students with learning
disabilities, which currently lacks investigation.
We see mathematics
educators as educators able to practice the teaching in either a mixed
ability environment or in a selective environment. Regardless of the
environment, mathematics educators need to have the necessary tools and
skills in order to evaluate the competence of their students and to be
able to encourage, motivate and support the development of those who
appear to be strong in subject areas such as mathematics. Educators
with such developed skills will achieve higher methodological
standards. The project is also investigating the problem of “talented
students with learning disabilities”.
Specifically, MATHEU seeks to develop a programme emphasizing the role of the teachers in:
- Identifying
mathematical talent through a range of measures that go beyond
traditional standardized tests. Measures should include observations,
student interviews, open-ended questions, portfolios, and teacher-,
parent-, peer- and self-nomination. Recognition should be made of the
fact that mathematical talents can be developed; they are not just
something with which some students were born.
- Presenting interesting tasks that engage students and encourage them to develop their mathematical talents.
- Improving
opportunities for mathematics learning and a much more challenging,
nonrepetitive, integrated curriculum which is needed to help students
develop mathematical talents. Students must be challenged to create
questions, to explore, and to develop mathematics that is new to them.
They need outlets where they can share their discoveries with others.
- Encompassing
a variety of methods including differentiated assignments, a core
curriculum, pull-out programmes, in-class programmes, magnet schools,
and extracurricular activities such as after-school or Saturday
programmes, mentorship programmes, summer programmes, and competitions.
- Improving the
ways in which students learn mathematics. Teachers must become
facilitators of learning to encourage students to construct new,
complex mathematical concepts. Students must be challenged to reach for
ever-increasing levels of mathematical understanding.
TARGET GROUPS
All mathematics educators at
all educational levels (primary and secondary) associated with the
partner institutions will directly benefit. Educators in other areas
may also benefit as the methods and tools for identifying mathematical
talented students could also be used to identify talented students in
other subjects. Through the dissemination process individual educators
from all countries in the European Community and beyond as well as
educator staff of educational authorities will benefit from the
outcomes of MATHEU through their participation in the training course
to be offered under Comenius Action 2.2. Finally, the largest group to
benefit is the European Community’s potentially talented/competent
students in mathematics.
Target group 1: Teachers/Teacher trainers/Teacher trainees
MATHEU will allow the
individual non-gender biased, to develop knowledge on pedagogical
methods, use of tools appropriate to support different levels of
students.
Target group 2: Educational administrators/Inspectors
In addition to the above
listed impacts this target group will become better curriculum
developers, will raise the quality of their teacher support, improve
their own background in the needs of the topic and will gain an
educational tool.
Target group 3: Educational Psychologists/Counsellors
Nowadays, educational
establishments involve educational psychologists and counsellors and
therefore MATHEU has to provide for these educators as they play a very
important role in the educational development of all types of pupils
and students. Counsellors could become the catalysts for the
“brain-gain effect” in Europe through their counselling to the students.
OUTPUTS OF MATHEU
The outputs of the MATHEU project are listed below:
- A
Tool that identifies talented/competent students in mathematics at two
different age levels. Methods/Activities for motivating potentially
talented/competent students in mathematics
- A
European Manual and CD-Rom, which will contain the tool above together
with material needed to support the development of such students. The
Manual will be initially translated in seven languages, English,
German, Greek, Italian, Bulgarian, Czech and Romanian.
- A
Course Design in English for teacher trainees and teacher trainers for
primary and secondary levels for the target age levels using the tool,
methods and the manual mentioned above
- An
Information and Dissemination Symposium for administrators/government
decision makers, for university enrolment
managers/deans/representatives, for presidents/ representatives of
Foundations and Societies
- The
MATHEU Web-site, designed to provide sustainable communication between
the partners and to provide support to talented students in mathematics
as well as to mathematics educators of different levels. The site will
initially support the languages of the partner countries.
GENERAL REMARKS
The recent developments of the
project suggest that Identification, Motivation and Support (IMS) are
divided in two age groups, the 9-14 age group and the 15-18 age group.
For each age group a number of topics were agreed as a basis for IMS
development. It was agreed to use the idea of a curriculum and level of
difficulty ladder for each topic. A group of experts is now developing
these ladders which will then be evaluated and will constitute the
basis for Identification. Motivation elements have been discussed and
are circulated among the partners of the project to evaluate in their
own countries and to try to finalize them by the end of 2004. Support
materials have been collected for different topics and will be
developed further in connection with the final form of the ladder in
each topic and at each age level.
REFERENCES
Electronic articles
1. John F. Feldhusen, Talent Development in Gifted Education,ERIC Digest E610, 2001. http://searcheric.org/digests/ed455657.html
2. Dana T. Johnson, Teaching Mathematics to Gifted Students in a Mixed-Ability Classroom , ERIC Digest E594, 2000. http://ericec.org/digests/e594.html
3. Richard C. Miller, Discovering Mathematical Talent, ERIC Digest E482, 1990. http://ericec.org/digests/e482.html
4. Joan Franklin Smutny, Teaching Young Gifted Children in the Regular Classroom , ERIC Digest E595, 2000. http://searcheric.org/digests/ed445422.html
General Websites
5. www.nfer-nelson.co.uk
6. www.math.bas.bg/bcmi/
7. www.excalibur.math.ust.hk
8. www.unl.edu/amc
9. www.math.scu.edu/putnam/intex.html
10. www.mathleague.com
11. www.olympiads.win.tue.nl/imo
12. www.olemiss.edu/mathed/problem.htm
13. www.mathforum.com/library
14. www.geom.umn.edu
15. www.problems.math.umr.edu
16. www.math.fau.edu/MathematicsCompetitions
17. www.schoolnet.ca
18. www.mathpropress.com/mathCener.htm
19. www.enc.org/topics/inquiry/ideas
Specific Websites
20. MATHEU website: http://www.matheu.org
21. The National Research Center on the Gifted and Talented (NRC/GT) http://www.gifted.uconn.edu/nrcgt.html
22. Johns Hopkins University: The Center for Talented Youth (CTY) http://cty.jhu.edu/
23. Northwestern University’s Center for Talent Development (CTD) http://www.ctd.northwestern.edu/
24. The Education of Gifted and Talented Students in Western Australia http://www.eddept.wa.edu.au/gifttal/gifttoc.htm
ABOUT THE AUTHORS
Gregory Makrides , Ph.D., Project Coordinator
Dean and Associate Professor of Mathematics
INTERCOLLEGE-Cyprus
President of the Cyprus Mathematical Society
E-mail: makrides.g@intercollege.ac.cy
www.matheu.org
Emiliya Velikova, Ph.D., Faculty Socrates Coordinator
Department of Algebra and Geometry
Centre of Applied Mathematics and Informatics
Faculty of Education
University of Rousse,
8 Studentska str., 7017 Rousse
BULGARIA
Cell phone: +359/889 625 222, Fax: +359/82/845 708,
E-mail: emily@ami.ru.acad.bg emivelikova@yahoo.com
www.cmeegs3.rousse.bg
www.icme-10.com
DESIGNING A MICROWORLD:
ACTIVITIES AND PROGRAMS FOR GIFTED STUDENTS AND ENHANCING MATHEMATICAL CREATIVITY
Hanhyuk Cho, Hyuk Han, Manyoung Jin, Hwakyung Kim,
Minho Song
Abstract: The
purpose of this paper is to introduce a microworld designed for
mathematical creativity and gifted education. We have developed a
microworld named JavaMAL by combining LOGO and DGS microworlds. Using
JavaMAL, students can make and manipulate semi-dynamic objects “tiles”
to explore mathematics. We have tested JavaMAL for creative mathematics
program and gifted education, and we have confirmed that there are
positive educational results from the teaching experiments.
Key words: Tile, Semi Dynamic, LOGO, JavaMAL, Microworld, DGS, Creative environment, Gifted education
1. INTRODUCTION
In his Enrichment Triad
Model Theory, Renzulli[10]considers three different levels of learning
for mathematically gifted students: (1) surveying and understanding the
given problems, (2) developing their strategic skills in solving the
problems, (3) producing the final solutions and products through
relevant project activities. In this sense, it is important to
investigate and design educational environment for the gifted learners
to explore problems and make creative products. Accordingly, this paper
is on the design of a mircoworld [4], a playground
for creative mathematical education. In brief, microworlds are
primarily exploratory learning environments where the learners can
manipulate or create objects and test their effects through discovering
spaces and constraining simulation of real-world phenomena [8].
The LOGO [1], [5], [2] the
representative microworld, is an environment where students can create
and produce mathematical figures using the most basic commands fd and rt
. The Fig. 1 shows the textual command and the figure drawn by the
turtle executing the command. As a result of this textual command in
the LOGO environment, the learners would be able to produce a more
visually and creatively oriented configurations. This is the most
salient feature of the LOGO mircroworld. Here, the basic commands fd and rt
are easily understood, and by appropriate arrangement and operation of
these two commands, the learners are able to make more creative and
diverse commands to draw figures.
The previous researches on
LOGO opened the gates of new possibilities in creative mathematics
education. As new computer techniques and technology are being
developed, however, the LOGO environment needs to be revised. Based on
its fundamental philosophy, constructionism, we have improved the LOGO
environment to be more dynamic and diverse for creative education.
Meanwhile, there have been
researches on the Dynamic Geometry System (DGS) along with Cabri and
GSP and its application to overcome the static aspect of “usual” school
geometry. The DGS is an experimental environment in which the learners
hypothesize, justify, and confirm geometric and algebraic properties
through mouse exploration, and the DGS help students overcome the
static features of the diagrams and mathematical figures in the
textbook. In short, the DGS system plays an important role by providing
an excellent environment for conjecturing, testing and confirming.
Sherin [12] proposed yet
anotherup-graded DGS by adding textual commands to the system. We
consider Sherin's attempt educationally desirable and have tried to
combine the LOGO and DGS microworlds, keeping the features of both
microworlds but adding new educational functions.In this environment,
named JavaMAL microworld, “making” is emphasized and the “making” comes
firstfollowed by experimenting and confirming. For this attempt, we
intoroduce textual command system using the keyboard as well as the
mouse-clicking sequence command system in addition to the already
existing pull-down menu system. In sum, this paper proposes the
designing of new functions that can connect the LOGO with the DGS using
newly invented command systems, and it will demonstrate some examples
of creative mathematical education programs that are possible in this
new environment.
2. TILE: SEMI-DYNAMIC OBJECT
In the previous section,
we surveyed the strengths of the LOGO and the dynamic function of the
DGS. In our effort to connect the two, we have provided dynamic feature
to the LOGO on one hand and have created a function that makes the DGS
more static on the other hand.
Such a complementary is
needed in mathematics education for the following reasons: First, let’s
observe the following procedure of making creative products in the LOGO
environment. The left hand side of t he Fig. 2 shows a command ? for drawing basic figure using the primitive commands fd and rt . The command ? in
the middle of Fig. 2 shows that the previous process has been
“internalized” and “condensed”, and the right side of Fig. 2 shows a
function command ? that can be used to draw more complicated figures when one encapsulates the process of drawing as an basic object.
The three processes in the
Fig. 2 are similar to those of Sfard’s [11] concept formation scheme:
interiorization, condensation and re ification. In fact, the Fig. 2 can
be regarded as the prime representaion in the LOGO environment that
shows the necessary steps of concept formation development. Following
the Circulation Theory in cognitive development, we design a “tile”
command so that a semi-dynamic object “tile”is created by the commands fd and rt , and the object can be manipulated again by the commands fd and rt This is the basic idea of the tile command.
Thefollowing Fig. 3 shows a sequence of making tiles. That is, a student draws a basic figure using the Logo commands fd and rt, and
makes tiles using a tile command. Then the student makes another tile
by copying and reducing the tile, and the student moves and rotates the
tiles, again using fd and rt
tile commands. Hence, the student can control not only the turtle’s
movements of the turtle, but also those of the tiles constructed by the
turtle using the “fd tile” and the “rt tile” commands. Again, as shown in the Fig. 3, a tile is an object made by fd and rt commands, and it is a basic object moved freely by mouse dragging or by fd and rt commands.
Laborde [9] showed that
the DGS contains a rich learning context for proof since the “dragging”
sequence in the invariant geometric properties stimulate students’
thinking. The DGS helps the learners to experiment with moving figures
on the screen, which is an impossible task for the generation whose
tools were consisted of pen-and-paper. Be that as it may, even though
it is possible for students to conjecture certain geometric pr operties
by dragging his or her mouse in the DGS, these properties need to be
justified. In the following Fig. 4, we can see that the tile ACE is
made by mouse-clicking sequences, and is being rotated to confirmed the
conjecture D ACE ? D DCB.
The pur pose of the DGS
environment is to explore the invariant geometric properties by mouse
dragging when the situation is changed. The DGS is good for providing
an appropriate environment for exploration and experiment of various
properties. The reasoning of properties begins, however, with a fixed
instance of the dynamic situations. That is, mouse dragging dynamic
representations assure the invariability of the figure, but the
deductive reasoning takes a step-by-step processbetween the two static
situations. For this purpose, we introduce a “tile command” to take
snapshots of the dynamic movements, and then the tiles are manipulated
to check the figure’s properties such as congruence and similarity
properties. In part, the tile is expected to be a medium for deductive
reasoning. In the Fig. 4 one is able to assume that the triangles are
congruent and check one’s assumption bymanipulating the tile. The Fig.5
shows the tile’s geometric operationto check the congruency and
similarity of the figures. By allowing the blue triangle semi-dynamic,
that is movable but keeping the same shape and size, and by using the
tile commands, one can transform it in many ways (translation,
reflection. rotation. and dilation).
As a student moves the
tile with the LOGO commands on the keyboard, student can also
manipulate it with the mouse. Student can drag a tile, just as student
drag a point. Unlike a point, however, student can reflect and rotate
the tile, which is semi-dynamic with a shape and size. There are two
ways to move a tile: using textual commandson the keyboard and the
mouse-clicking sequences to connect the LOGO and the DGS.
From the Fig. 6, we can see that the Pythagoras trees are given not only by the
LOGO's turtle command, but
by the DGS construction command. We can also see that the tiles are
created by a turtle command and manipulated by mouse dragging.
The JavaMAL microworld is
designed not only to combine the LOGO and the DGS, but also to
introduce bridges between the two such as “tiles”. From our teaching
experience, we find that the tiles in JavaMAL can provide a rich
environment such that students can make their own creative mathematical
products and explore mathematical properties from the given
mathematical premises. Since JavaMAL is written in
JAVA language, it can be used on the internet ( http://edunet4u.snu.ac.kr).
Communications between the students and the teachers in
cyber space are also available on the internet, the unified environment
of democratic access [7] and internet board. It is accessible to
anyone, anytime and anyplace. Now some new examples of creativity
education will be given in the next two chapters that the JavaMAL
microworld can provide.
3. EXAMPLE: GROUP ACTIVITY BY MAKING MODULES
In creativity education, a
small group as well as an individul project is encouraged. Social
constructivism emphasizes the social construction of knowledge. Group
activities can stimulate the group members so that they can produce
more creative results after they share ideas through communication.
Group activity in cyber space as well as in classrooms should be
emphasized,and theJavaMAL microworld is designed for this purpose. The
JavaMAL microworld is available on the internet, and the group activity
can be realized through the internet board using its textual command
system. The Fig. 7 shows an example of how a small group members work
together not only in the classroom but also in the web. The group
members draw a sketch of “The Garden,” after exchanging their ideas in
the classroom. The garden is divided into the ‘sun’, ‘tree’ and
‘butterfly,’ and each module is made by each members for movements,
which, in turn, comes together and make the whole project complete. The
feedback from the group members for an individual module can be
exchanged for further discussion not only in the classroom but also in
the cyber space.
The function of tile here
is, therefore, to add the semi-dynamic features for the modules in the
sketch and initiate them to move. It is the tile that makes the ‘sun’
blazing, the ‘tree’ growing, and the ‘butterfly’ flying. Also, each
tile comes together to construct a creative product as a whole. This
kind of module based group activity is a good example of creativity
education in a microworld environmentsince it foster both an individual
and a small group projects.
4. EXAMPLE: TESSELLATION AND TANGRAM BY MAKING TILES
So far, we have talked
about the function of tile as a snapshot in the dynamic DGS. The tile
provides certain functions such as testing the congruence and
similarity (of figures), and confirming and verifyingtheconjectures.
The tile is expected to contribute to the creativity education in many
ways. For example, imagine a space filled with many pieces of tile, or
a semi-dynamic tile expecting to play a role as teaching aids,
tessellation and tangram. It is common to use ready made tiles in most
activities such as LEGO activities and tangram puzzles. Meanwhile, in
the JavaMAL environment the learners are able to “make” tilesby
themselves, and design and solve the puzzle with the tiles of their own
making. In an advanced level of manipulation of the given objects, the
learners may able to make the objects themselves, and then manipulate
them to solve the puzzle. The Fig. 8 shows tessellation and tangram in
which a student can ‘make’a basic piece by mouse-clicking sequence,
then by using the tile piece, he or she copies it and fills the space.
While tessellation and tangram
focus on puzzle-solving,JavaMAL also focuses on making a piece itself.
In JavaMAL environment, making is emphasized prior to manipulation, and
making is possible thanks to tile.
5. CONCLUSIONS AND FUTURE WORK
We surveyed the JavaMAL
microworld as a playground for creativity education. In order to
connect the static LOGO with the dynamic DGS, we introduced a new
function, “tile”.We mentioned a couple of examples for creativity
education to apply what we studied. From our pilot experiments, we
confirmed that the JavaMAL microworld is well-designed for mathematical
creativity and education for the gifted. This year, we plan to do
well-structured teaching experiments using well-organized curriculums
and instructional designs based on Renzulli's Enrichment Triad Model
Theory. We will modify the JavaMAL microworld by monitoring the
reactions from students, and we will try to obtain desirable
educational outcomes by designing not only the JavaMal microworld but
instructional strategies to provide an enriched learning environments
based on the constructivism philosophy.
In the future, we plan to
up-grade the JavaMAL environment so that it can be used to represent
three-dimensional geometric objects. As we were able to fill the space
with the basic pieces, we’llattempt to design the environment where the
pieces will make a polyhedron. We are going to add some new functions,
that is, drawing the folding net of a polyhedron using the LOGO
commands, folding it to make a polyhedron, and manipulating it. Here
the tile acts as a medium that can make the folding net to move. The
folding net of a polyhedron is drawn by the basic LOGO command, fd and rt
. The folding net is, in turn, folded and becomes three-dimentional,
and a couple of polyhedrons merge together to make some creative
product. The Fig. 9 shows the process of the students’ activity to make
decahedron. First, the students figure out what the folding net might
be look like. Second, they draw the folding net using fd and rt
. Then, they make the folding net to move and to fold using the
functions of tile. Finally, they can translate or rotate the
complicated polyhedron. Furthermore, the complicated polyhedrons can be
changed and connected to each other, to make another creative product
such as a doll and a train.
This paper is primarily
based on Constructionism that emphasizes the active construction of the
external artifact. We studied on the creativity education in different
microworlds, and considered its educational values for creativity and
education for the gifted. We propose the necessity of the tile
functions and proposed some programs using the tile for creativity
education. The designing of a microworld for creativity education as
well as its application in the given situation will be an important
issue for the gifted and creativity education.
REFERENCES
1. Abelson, H., & diSessa, A. A. (1980). Turtle Geometry: Computation as a Medium for Exploring Mathematics, Cambridge, Mass : MIT Press.
2. Clements, D. H., & Battista M. T. (2001). Logo and Geometry, Reston, Virginia: NCTM.
3. diSessa, A. A. (2000). Changing Minds, Cambridge, MA: MIT Press.
4. Edwards, L. D. (1995). Microworlds as Representation. In A. A. diSessa, C. Hoyles, R. Noss, & L. Edwards (Eds.), Computers and Exploratory Learning , Berlin: Springer.
5. Hoyles, C., & Sutherland, R. (1999). Logo mathematics in the classroom, London and New York: Routledge.
6. Kafai, Y. & Resnick, M. (Eds.) (1996). Constructionism in Practice: Designing, Thinking, and Learning in a Digital World, Lawrence Erlbaum Associates, Publishers.
7. Kaput, J.
(1999). The Mathematics of change and Variation from a Millennial
Perspective, In C. Hoyles, C. Morgan, &G. Woodhouse (Eds.), Rethinking the Mathematics Curriculum, Falmer Press, 155-170.
8. Jonassen, H. D. (1996). Computers as Mindtools for Schools, Prentice Hall.
9. Laborde, C. (2000).Dynamic Geometry Environments as a Source of Rich Learning Contexts for the Complex Activity of Proving, Educational Studies in Mathematics, 44, 151-161.
10. Renzulli, J. S. & Reis, S. M. (1991).The
Schoolwide Enrichment Model: A comprehensive plan for the development
of creative productivity. In N. Colangelo, & G. A. Davis (Eds.) Handbook of Gifted Education , Needham Heights, MA: Allyn and Bacon, 111-141.
11. Sfard, A.
(1991). On the dual nature of mathematical conceptions: reflections on
processes and objects as different sides of the same coin, Educational Studies in Mathematics,22, 1-36.
12. Sherin, B. (2002). Representing geometric constructions as programs: A brief exploration ,International Journal of Computers for Mathematical Learning 7(1), 101-115.
13. Wilensky, U. J. (1993). Connected Mathematics- Building Concrete Relationship with Mathematical Knowledge , Thesis of doctor of philosophy at the Massachusetts Institute of Technology.
ABOUT THE AUTHORS
Cho, Hanhyuk; Han, Hyuk; Jin, Manyoung; Kim, Hwakyung; Song, Minho
Department of Mathematics Education
College of Education, Seoul National University
Shinlim-dong, Kwanak-gu
Seoul 151-748, Korea
?-mail addresses:
hancho@snu.ac.kr
hyuk0916@snu.ac.kr
jin4489@snu.ac.kr
indices2@snu.ac.kr
neodmino@hanmail.net
Supported by Korea Research Foundation KRF-2003-015-C00011
ANALYSIS OF THE MATHEMATICAL DISPOSITION
OF THE MATHEMATICALLY GIFTED STUDENTS
IN THE MIDDLE SCHOOL OF KOREA'
Hye Sook Park, Kyoo-Hong Park
Abstract:
We study on the mathematical disposition of mathematically gifted
students in the middle of KOREA. For this purpose, we use the tools of
psychological test of disposition of math disliking which was developed
by Kim et al.(2001) to analyze the mathematical disposition of gifted
students and investigate the characteristic of it.
Key words: Mathematical Disposition, Gifted Students, Math Disliking Factor
1. AIM AND NECESSITY OF THE STUDY
The education in the school must
help each student to show their ability of latent faculties. But there
are many places of bad educational surroundings in Korea, such as the
entrance examination, poor facilities of the school, parents' tendency
of distrust to the school education and enervation of teachers on the
educational guidance in the daily life. These are the causes of the
stumbling block to the education which bring up student's ability of
latent faculties in Korea.
Recently, the government of Korea
gave emphasis to the education of gifted students in the school. The
law of promotion of the gifted students' education was established by
presidential law and ordinance at April, 2002 in Korea. By this law and
ordinance, the education centres of gifted students were established in
the 15 universities all over the country. And some of science high
schools carry out the program of gifted students. Also each offices of
educational district established gifted students’ class and training
the teachers who take part in the education of gifted students.
In this research, we study the
mathematical disposition of mathematically gifted students in the
middle school before we study the teaching methods of mathematically
gifted students. For this study, we use the tools of psychological test
of disposition of math disliking which was developed by Kim and 7
others(2001) to analyze the mathematical disposition of mathematically
gifted students and investigate the characteristic of it.
2. THE CONTENTS OF THE STUDY
The mathematical disposition is a
mind of good feeling and interest in the mathematics and it is the same
as mathematical cast of mind which was defined by Krutetskii (NCTM
1987). We can see the self confidence, good feeling and the tenacity on
the homework for mathematics in the mathematical disposition.
The disposition of math disliking
which was developed by Kim et al(2001) has several domains of factors
such as psychological factor, environmental factor, academic trait of
mathematics factor. In the prestudy, we studied the disposition of math
disliking of underachievers(Park and 4 others 2004). And we use the
result of prestudy to analyze the mathematical disposition of
mathematically gifted students.
The factors causing math disliking are divided into 11 groups as follows:
1. Psychological
and environmental domain contains 4 factor groups, i.e., cognitive
factor(mt1), mental ability factor(mt2), teacher related factor(en1)
and math perception related factors(en2).
2. Academic trait of
mathematics domain contains 7 factor groups, i.e., comprehension
related factor(com), hierarchy factor(rl1), connection related
factor(rl2), operation related factor(rl3), analysis reasoning related
factor(rl4), basic application factor(ap1) and composite application
factor(ap2).
When we evaluate the cumulative percentage
for each factor of math disliking, the increasing appearance of the
point of disliking factors means the factor of math disliking is weak.
On the contrary, the decreasing appearance of the point of disliking
factors means the factor of math disliking is strong. So the factor of
math disliking point excess more than 70% is defined the mathematical
disposition and the factor of math disliking point shortage less than
30% is defined the mathematical disposition of math disliking.
By using the definitions of mathematical
disposition, we investigate the relations between each factor and the
mathematical disposition of mathematically gifted students. From this
result, we will find the desirable teaching methods for the
mathematically gifted students.
We already have analyzed the properties of math disliking disposition of underachievers of middle school(Park et al., 2004).
The following (Table 1) shows the
means of the cumulative percentage of the math disliking factors for
each level of the achievement and the table 2 shows the number and
percentages of math disliking students-who have more than or equal to 3
math disliking factors- of each math disliking factor for each level of
achievement.
From the above tables, we can find that
mathematical underachievers show remarkable difference in the math
disliking factors of cognitive factor(mt1), teacher related factor(en1)
hierarchy factor(rl1), connection related factor(rl2), operation
related factor(rl3) and basic application factor(ap1).
The above result is the analysis of
disposition of math disliking focused on the underachievers(Park et al,
2004). It may be quite different from those of gifted students. So, it
is needed to survey the difference between the mathematically gifted
students and the others. For this purpose, we made about 100
mathematically gifted students of middle school an object of
investigation by questionnaire, and we are analyzing the data presently.
3. THE EXPECTED EFFECT FOR THE STUDY
In this study, we expect to develop
of the textbook for mathematical gifted students, to develop the
program for teaching the mathematical gifted students after analyzing
their mathematical disposition.
REFERENCES
1. Gagne, F. Constructs and Models Pertaining to Exceptional Human Abilities, International Handbook of Research and Development of Giftedness and Talent , edited by Heller, K. A. etc., Oxford: Pergamon Press, 1993, pp.69-87.
2. House, P. A.(Ed.) Providing Opportunities for the Mathematically Gifted, K-12 , Reston, Virginia: NCTM, 1987
3. Kirk, S. A. Educating Exceptional Children , 2nd ed., Rev., Boston: Houghton Mifflin Co.. 1972
4. Kim et al.
Studies on Exploring Math Disliking Factors and Devising Tools to
Analyze Students’ Disliking Trends about School Mathematics, J. Korea
Soc. Math. Ed. Ser. A : The Mathematical Education, Vol. 40 No. 2, 2001. pp.217-239.
5. Renzulli, J. S. & Reis, S. M. The Schoolwide Enrichment Model: A Comprehensive Plan for Educational Excellence, Creative Learning Press, INC, 1985.
ABOUT THE AUTHORS
Prof. Hye Sook Park, DSc.
Dept. of Mathematics Education
Seowon University
Cheongju, Chungbuk 361-742,KOREA
E-mail: hyespark@seowon.ac.kr
Prof. Kyoo-Hong Park, Ph.D.
Dept. of Mathematics Education
Seowon University,
Cheongju, Chungbuk 361-742, KOREA
E-mail: parkkh@seowon.ac.kr
ACTIVITIES IN NEW CURRICULUM FOR GIFTED STUDENTS-TRIALS IN SUPER SCIENCE HIGH SCHOOLS IN JAPAN-
Kyoko Kakihana, Suteo Kimura
Abstract: The
Super Science High school (SSH) project was set up by the Ministry of
Education, Cultural, Sports, Science and Technology (MEXT) in 2002 to
develop a high school science and mathematics curriculum for gifted
students. Every year twenty-six high schools are selected to take part
in the project for a period of three years. In this article, we will
explain what the Super Science High school project is as well as
categorize the new curricular activities involved.
Key words: Curriculum, Motivation, gifted students, Creative Process,
BACKGROUND
These days everything
changes so fast that things learnt in school often become outdated in a
very short time. In Japan, despite a policy over the last 50 years of a
standardized education for all, in the last 10 years, we have seen a
widening gap in achievement between top-level and low-level.Many
students complained that they did not like mathematics or science. As
the result, the Ministry of Education, Cultural, Sports, Science and
Technology (MEXT) set the guidelines for mathematics education at the
below average level in an attempt to reduce the burden for students.
However, many university professors complain that the achievement level
of students in mathematics classes is decreasing, even to the level
where they cannot compute basic calculations such as fractions (Okabe,
T. and others, 1999). On the other hand, rapid development in the
science and technology make it essential to produce creative students
with a high-level ability in science and mathematics. Therefore, MEXT
recognized the need to promote comprehensive research and development
in order to accomplish the highest creative achievements in worldwide
comparisons by nurturing the merits of science, technology, and
scientific research, and by seeking harmony and balance between science
and technology on the one hand and scientific research on the other
(Japan's Science and Technology Policy, 2001). Along the formulation of
the "Science and Technology Basic Plan" at the end of March 2001, MEXT
set a plan to develop students based on the creativity of science and
technology, including curriculum development for gifted students in
science and mathematics. Then, the Super Science High School (SSH)
project started in 2002.
THE SUPER SCIENCE HIGH SCHOOL PROJECT
The purpose of this
project is to develop a curriculum in mathematics, physics, biology,
chemistry, and earth science for gifted and mature students. The
three-year project was inaugurated in 2002 and each high school
participates for 3 years. MEXT selects 26 high schools every year.
Seventy-seven schools applied for the project in the first year. The
budget for the project was 727,000,000 yen (about 6,000,000 dollars)
for 26 schools in 2002, and 1,186,000,000yen (about 9,000,000 dollars)
for 52 schools in 2003. Selected high schools are expected to develop
curricula based on science and mathematics in cooperation with
universities or research institutes. Specialists in each field and
researchers in education for each subject make up the research group
that will examine and analyze their activities and curricula.
RESULTS
We referred to reports
from the SSH and the data from the websites of these high schools to
categorize the curricular topics and their activities.
(1) Lectures by university professors to introduce new areas in mathematics
All schools planned
lectures by university professors. In scientific topics (physics,
biology, chemistry, and earth science), there were many lectures by
people from companies, but few lectures on mathematical topics.
Nineteen mathematical topics were lectured by eleven high schools. They
were beyond the government guidelines in mathematics, such as, “Fractal
science and the Logalism,” “A Polyhedron and Geometry,” “The Forefront
of Modern Mathematics: Chaos Theory.” These topics matured students
about new areas in mathematics. In two high schools, students
investigated the lecture topic before the lecture and continued to
study it after. For example, Benoit Mandelbrot gave a lecture on
fractals at a high school attached to the Education Department of Kyoto
University. This is a very new topic in high school curricula. To
prepare for listening to the lecture, students studied affine geometry,
sequences, logalism, and such beforehand. After the lecture, they
investigated fractals on the computer and reported what they learned.
Students in Nagaoka High
School in Niigata prefecture took lectures “Chaos Theory” to introduce
modern mathematics by the Professor Kawamura. Results of a
questionnaire showed 43% of students were practically able to take an
image what chaos theory is and 55% of students were interested in the
lecture. Forty-eight percent of students became motivated to learn
mathematics as a result of the lectures.
(2) Constructing New Curricula that Combine Mathematics With Other Subjects
Six schools tried to
combine mathematics with other subjects. For example, Honjyo High
School which is attached to Waseda University, constructed a curriculum
that combines mathematics with physics and chemistry.At first, students
learn what a vector is, the reason why you need a vector, and
calculation with vectors in their mathematics class. Then they learn
about projection from a slanting direction, relative speed, and the
composition of forces in their physics class. As table 1 shows students
take the mathematics class and physics class in turn, and the classes
are called “corroboration classes”.
Students in Daiichi Girls
High School in Miyagi prefecture took a lecture “The Roles of
Mathematics in Economics”. In their impression of the lecture, some
students wrote that they understood from the lecture the importance of
mathematics even for students who are not majoring in science,
engineering or mathematics and other students were interested the
relation between mathematics and the economy
Students in Chiba High
School took lectures entitled “Fractal Science”. The goal of these
lectures was to connect mathematics to physics and biology. After the
lectures students visited a university laboratory and observed by
microscope fractals in the pattern of bacterial development and
crystallization. It was written in students’ visiting reports that they
understood fractals more clearly after visiting the laboratory.
(3) Trial of New Curricula and New topics for High School Mathematics
Five schools held lectures
about fractals by university professors. They held extra classes to
understand the topic, and some of the classes used computers.
Other schools tried new
materials such as “the structure of ciphers,” “learning mathematics and
science in English,” “statistics.”. In Japan, text books which are used
in school are usually published by MEXT. In this project, six schools
produced their own text book. Some of them changed the order of
teaching. Some of them added new topics connected with other subjects
like modeling, more advanced materials like epsilon-delta logic or
introduction of symbolic logic.
(4) Using Technology
Five schools conducted
classes with technology. Three of them used Mathematica, and two of
them used a programming language. In the high school attached to the
Education Department of Kyoto University, students made a program to
draw Sierpinski triangles and explored the characteristics of these
triangles. Two of them planned to use a graphical calculator. At Kaiho
High School in Okinawa, students used the calculator to explore the
foci of quadratic curves and the functions of sounds. A software for
learning functions, Grapes, produced by a high school teacher,
Katsuhisa Tomoda, and offered in his website as a freeware ( http://okumedia.cc.osaka-kyoiku.ac.jp/~tomodak/grapes/volume. html) is widely used in high schools.
(5) Increasing Mathematics Classes and Enhancing the Topics in Standards.
Most of the SSH schools
increased mathematic classes by two or three per week and went beyond
government guidelines by teaching more advanced and challenging
material. In other subjects, they tried to make scientific experiments
beyond the government guideline material. Some of them made experiments
at the university laboratory. Experiments in other subjects motivated
students in learning mathematics. Increasing mathematics classes must
be effective for all science. One student comments, "it was too
difficult to understand the university professor’s lecture in a physic
class, without basic mathematics knowledge. So, I should study
mathematics more" in his report.
CONCLUSION
New trials of mathematics
classes lead to motivating students to learn mathematics more
effectively. Moreover experiences in other subjects also motivated them
to learn mathematics. In this research, systematization of these trails
in the SSH for a curriculum for gifted students is left.
ACKNOWLEDGEMENTS
We deeply appreciate the
twenty-six SSH 2001 schools for allowing us to view their reports and
for cooperating with our research.This research is supported by the
science research fund of MEXT 14022101.
REFERENCES
1. Japan's Science and Technology Policy (2003) http://www.mext.go.jp/english/org/ science/07a.htm
2. Okabe, T., Tose, N. and Nishimura, K. (1999), University students who are not able to calculate fractions, Toyo Keizai Shinpou
ABOUT THE AUTHORS
Kyoko Kakihana, Ph.D., Professor
Department of Computer Science
Tokyo Kasei Gakuin Tsukuba Women’s University
3-1 Azuma Tsukuba
Ibaraki 305-0031
JAPAN
Cell phone: +81 29 858 6292
E-mail: kakihana@cs.kasei.ac.jp
Suteo Kimura, Ph.D.
University of Meijyo
JAPAN
DEVELOPMENT OF ENRICHMENT PROGRAMS
FOR THE MATHEMATICALLY GIFTED:
FOCUSED ON THE CONIC SECTION
Kang Sup Lee, Dong Jou Hwang, Woo Shik Lee
Abstract: In
this study, we developed teaching and learning contents of an
enrichment program for the mathematically gifted high school students
focused on the conic section. Class activities in the contents were
selected and organized based on topic-centered, activity oriented,
open, interdisciplinary and student-selected approaches, and also
designed according to the Triad Enrichment Model developed by Renzulli.
I. INTRODUCTION
The gifted education should be
composed of various educational contents and methods by considering the
characteristics of the gifted. In Korea, since 1999 which is the origin
year of the Gifted Education Act, the gifted education program can be
characterized by variety of approaches: special purposed high schools
for the gifted and talented, after-school enrichment programs in
elementary and junior high schools, acceleration and enrichment
programs provided by gifted education centers affiliated with school
boards and universities, and cyber education systems for the gifted.
The Korean Education Research
Institutes (KEDI) developed the general curriculum for the gifted and
talented [7], [4] and the curriculum for the mathematically gifted and
talented [6]. Based on the curriculums mentioned above, practical
programs are in progress of development. For the area of mathematical
enrichment program, after-school activities in elementary and junior
high school were studied [16], [8], [9], [3], [13], [14]. However, the
mostly developed programs were for the gifted in middle school. We have
the only material ‘Mathematics III’, the textbook of science high
schools, which is developed from KAIST 1999 (revised 2003). In their
study, Shin, etc [24] pointed out that the current system for the
enrichment programs of special purposed high schools does not
efficiently provide know-how on how to guide the mathematically gifted
students to become creative knowledge producers. For these reasons,
teaching and learning materials for the science high school and general
high school’s mathematically gifted class students are needed.
The other hand, exploration of ERIC
from 1981 to 2003 showed 233 documents related to the mathematically
gifted/talented program and of those found there were 131 (56%)
acceleration program documents, 89 (38%) enrichment program documents
and 16 (6%) acceleration/enrichment program documents. As the result,
mathematically gifted/talented program should be implemented for
intensive learning through acceleration and enrichment [12].
The purpose of this study is carried
out to develop an enrichment program with conic section for the
mathematically gifted students who are identified by and participate in
the enrichment programs provided by the science high schools in Korea.
This study is based on the characteristics of the mathematically
gifted, nature of enrichment programs for the gifted, principles of
developing program for the gifted.
II. PLAN OF DEVELOPMENT ON ENRICHMENT PROGRAM
1. Development Trends on Enrichment Program for Mathematical Gifted
The National Council of Teachers of
Mathematics (NCTM) [17] recommends that all mathematically talented and
gifted students should be enrolled in a program that provides broad and
enriched view of mathematics, which holds them to higher expectations.
NCTM endorses use of Renzulli’s Traid Enrichment Model in providing
opportunities for the mathematically gifted K-12. NCTM, however,
contents that in almost all cases, gifted students benefit more from
enrichment than acceleration. For a limited number of extremely
talented and productive students, NCTM supports accelerated programs
enhanced by enrichment. Any program for mathematically talented
students should be expected to measure up on the following essential
components [11]. First, the mathematical content of the program must be
of a high quality. These differences should be reflected in the
difficulty, cognitive level, breadth, and depth of the curriculum.
Second, programs for the mathematically gifted and talented must
nurture high-level thinking processes. Third, efforts must be made to
include applications of mathematics to real-world situations as well as
the examination of standard topics in greater depth. Fourth, the
ability to communicate is essential in learning mathematics. Fifth,
mathematics, with its unique content characteristics, provides an
effective vehicle for developing study skills and work habits. Sixth,
the program must provide opportunities for students to explore
mathematical ideas in a creative fashion. Seventh, gifted students need
frequent and imaginative use of manipulative materials and other
instructional aids. Mathematics should be related to other content
areas of school program.
As we plan programs for these top
students, we first examine the purposes behind our programs. There are
many reasons why we might want to help students develop their
mathematical abilities [23]. These include: First, Helping students
become deep mathematical thinkers. Second, Developing an informed
citizenry. Third, allowing students to experience the joy and the
beauty of mathematics. Forth, enabling students to be competitive at
the university level and beyond. Fifth, developing world leaders in our
increasingly technological world.
Other developments that had their
origins in special programs are currently being examined for general
practice, These developments include: a focus on concept rather than
skill learning, the use of interdisciplinary curriculum and theme-based
studies, student portfolios, performance assessment, cross-grade
grouping, alternative scheduling patterns, and perhaps most important,
opportunities for students to exchange traditional roles as
lesson-learners and doers-of-exercises for more challenging and
demanding roles that require hands-on learning, first-hand
investigations, and the application of knowledge and thinking skills to
complex problems [20].
Gu, etc [6] and Hwang, etc [12] have
picked out some points on concerning the methods of teaching and
learning for the mathematically gifted in order to maximize the
mathematically gifted considering their properties. Their suggestions
are as follows:
First, the curriculum is plotted to
develop not only the intellectual area but also the emotional and
social area of the children and emphasizes interdisciplinary approach.
Second, the curriculum should be
composed of various educational content and method by considering the
characteristics of the high schools gifted.
Third, the curriculum emphasizes the
enhancement of creative thinking and higher-order thinking through
enrichment than acceleration.
Fourth, the curriculum should be
composed of various modern mathematics content, which will lead future
society as well as education program, which enables the children to
prepare for the future information and creativity society.
Fifth, the curriculum is an
educational program that enables student to perform an expertise study
on the area where they are interested in and specialized in, inclusive
of individual project or study project to reinforce the communication
function in the mathematical thinking.
Sixth, the curriculum provides an
education program, which has good connectivity to the elementary
school, middle school, university and other educational agencies.
KEDI [14], Kim [15] and Han [9] have
suggested teaching-learning materials as well as a direction in
development of the teaching-learning material for the gifted students.
On their research, when developing materials, followings should be
considered; creative problem solving ability in concerned subject,
developing high-order thinking power, interest in the concerned subject
content and activity, concentration on assignment, developing
self-confidence, developing self-directed learning attitude, focusing
more on reinforcing than speed renovating study, individualized
education, diverse degree of difficulty and presenting reinforcing
activity assignment, including content from various area, connecting
with problem situation which can occur in actual life, composing of
sub-subject, interdisciplinary subject, including content or activity,
organizing various group, various teaching and learning material,
application on location, emphasizing on creative products.
2. Contents of Enrichment Program Focused on the Conic Section
In this study, all activities were
organized according to the Triad Enrichment Model developed by Renzulli
[19], which consists of Type I, Type II and Type III enrichment
activities. Type I activities are planning and implementing process.
Type II activities training consists of process that should be taught
in connection with a Type III activity. Type III activities are
individual and small group investigations of real problems.
Type I activities are mainly
organized to strengthen motivation of students to study on a topic.
Each activity devised and organized to increase motivation and required
students to exercise the variables conic section activities and
mathematical thinking and to produce products at the professional
level. It also required providing free and rich educational
environments for the mathematically gifted. The goal of Type II
activities is to help students to have deep understanding and master
skills required to carry out mathematical investigation studies. Each
activity was organized to exercise the property of the conic section
and to find the cryptosystem on elliptic curves. The goal of Type III
activities is to provide students with opportunities of carrying out
research projects on the topic being studied.
The developed teaching and learning
material is composed to the first step as making plan, the second step
as acquiring the knowledge and function, the third step as performing
them and the fourth step as presenting, evaluating and reviewing. Each
step contains various activities. The teaching and learning materials
of this study is what reinforced the unit of ‘quadratic curve’ in the
‘Mathematics III’ of high school textbook. This teaching and learning
material is developed for gifted high school students in mathematics as
well as for the general high school students who are advanced and
strongly interested in mathematics. Sixteen class activities are
composed as project performing procedure in four steps so it can be
utilized efficiently when required a group instruction for the
mathematically talented students or a focused teaching.
First step: It calls interests and
active participation of students into the subject through the various
activities; finding the conic section used in real life, paper folding
of conic section using rectangle, expressing the activity of paper
folding of conic section mathematically, proving conic section using
the Belgian mathematician Dandelin (G. P. Dandelin)’s sphere and etc.
Second step: This is to provide the
students sufficient knowledge concerning conic section through
activities; studying on the property of focus reflection of the conic
section, making inequality in the shape of parabola, finding the
relationship between the unit complex number and trigonometric function
using the polar form of the complex number, finding the relationship
between orthogonal equation and polar equitation, finding the property
of conic section. Also it enables students to study on the discrete
logarithm problem which is basic of RSA cryptosystem using the conic
section property and algorithms for simple signature scheme.
Third step: This enables students to
express the conic curve as a polar equation based on the knowledge and
function acquired from the performance, and to study on cryptosystem
and crypto experiment on the elliptic curve as well as the algorithms.
And it instructs them to compose a subject report based on the
properties of elliptic curve.
Fourth step: In this presentation
and evaluation/reflection step, students are to present, discuss on the
their subject report and evaluate it according to the objective
followed by the overall review and modification on project procedure
and products (subject report or idea suggestion) based on the
evaluation.
3. Teaching and Learning Plan
The topics, main contents and activities in the proposed program are shown as in the following Table 1.
III. DISCUSSION AND CONCLUSION
We showed the new directions for development the enrichment program in the high schools for the mathematically gifted.
We could think about mathematics
students along a continuum or hierarchy as shown in the following
diagram. Sheffield [22] showed that illiterates ? doers ? computers ?
consumers ? problem solvers ? problem posers ? creators. The notions of
fluency, flexibility and novelty were adapted and applied in the domain
of mathematics by Balka [1], who asked subjects to pose mathemati cal
problems that could be answered on the basis of information provided in
a set of stories about real world situations. Problem posing, along
with problem solving, is central to the discipline of mathematics and
the nature of mathematical thinking [25]. When mathematicians engage in
the intellectual work of the discipline, it can be argued that the
self-directed posing of problems to be solved is an important
characteristic [18].
Also, Runco and Chand [21] explains
that problem finding is the starting point and key to producing
creative products. Given the "creating a problem" ; characteristic of
problem posing and the "bring into being" nature of creativity one
might see problem posing as a kind of creativity. In fact, problem
finding has sometimes been considered as a creative process in itself
[5]. Studies in mathematical creativity were reviewed [10] and one may
see problem posing ability as a creative ability.
We assume the teacher's belief about
mathematics and a role of textbooks are connected to fostering
mathematical creativity, and role of textbooks is important. "Problem
finding and problem posing" can be never seen in Korean high schools
textbooks. To make textbooks contain problem finding and problem posing
we have a proposal of Three-Stage Mathematical Creativity Enrichment
Model for mathematically gifted and regular students.
REFERENCES
1. Balka, D. S. (1974). Creativity ability in mathematics. Arithmetic Teacher, 21(7). 633-636.
2. Bang, S., Hong, J., & Hwang, D. (2001). Development of Curriculum for the Gifted in the High Schools. Journal of the Korea Society of Mathematical Education Series F: Studies in Mathematical Education,6. pp.223-245.
3. Bang, S., Lee, S., & Lee, W. (2002). Development of Enrichment Programs for the Mathematically Gifted: Focused on the 9th grade. Journal of the Korea Society of Mathematical Education Series F: Studies in Mathematical Education,7. pp.103-119.
4. Cho, S., Kim, H., Kim S., Bang., S. & Hwang., D. (2000). Development of Curriculum for the Scientifically Gifted . Seoul: Korean Education Development Institute.
5. Dillon, J. T. (1988). Levels of problem finding vs problem solving. In: questioning exchange 2(2), 105-115.
6. Gu, J., Cho, S., Kim, H., Soe, H., Jang, Y. Lim, H., Bang, S., & Hwang, D. (2000). Development of Curriculum for the Gifted - Fundamental Study of development the curriculum for the gifted in the high schools . Seoul: Korean Education Development Institute.
7. Gu, J., Cho, S., Kim, H., Soe, H., Jang, Y., Hwang, D., & Lim, H. (1999). Development
of Curriculum for the Gifted - Fundamental Study of development the
curriculum for the gifted in elementary and the middle schools. CR 99-20. Seoul: Korean Education Development Institute.
8. Han, I. (2000). A study of development the Enrichment Program Materials: Focused on a problem for construction. Journal of the Korea Society of Mathematical Education Series F: Studies in Mathematical Education,5. pp.221-232.
9. Han, I. (2001). The direction for development the program in middle and the high schools for the mathematical gifted. Proceeding of the Korean Society for the Gifted.
10. Haylock, D. W. (1987). A framework for assessing mathematical creativity in schoolchildren in Educational Studies in Mathematics 18, pp. 59-74.
11. House, P. (1987). (Ed.) Providing Opportunities for the Mathematically Gifted, K-12. Reston, VA: National Council of Teachers of Mathematics.
12. Hwang, D., Hong, J., & Seo, J. (2002). An Analytic Study of Mathematics Gifted/Talented Education Program of U.S.A. by ERIC Search. Journal of the Korea Society of Mathematical Education Series F: Studies in Mathematical Education, 7. pp. 121-131.
13. KEDI (2002). Development of Instructional Materials for the Gifted Students in the Regular Schools. Seoul: Korean Education Development Institute.
14. KEDI (2003). Development of Instructional Materials for the Gifted Students in the Regular Schools. Seoul: Korean Education Development Institute.
15. Kim, S. (2001). The direction for development the program in elementary schools for the mathematical gifted. Proceeding of the Korean Society for the Gifted.
16. Lee, S., & Han, I. (2000). A study of development the Geometrical Enrichment Program Materials: Focused on a analogy. Journal of the Korea Society of Mathematical Education Series F: Studies in Mathematical Education,5. pp.165-174.
17. NCTM (2000). Principles and Standards In Mathematics. Reston, VA: National Council of Teachers of Mathematics.
18. Pólya, G. (1954). Mathematics and plausible reasoning. Princeton, NJ: Princeton University Press.
19. Renzulli, J. S. (1978). The enrichment triad. Mansfield Center, CT: Creative Learning Press.
20. Renzulli, J. S., & Reis, S. M. (2000). The Schoolwide Enrichment Model.
In: Heller, K. A., Monks, F. J., Sternberg, R. J., & Subotnik, R.
F. (Eds.), International Handbook of Giftedness and Talent (2nd Ed.).
Oxford: Pergamon Press.
21. Runco, M. A., & Chand, I. (1995). Cognition and creativity. Educational Psychology Review , 7, 243-267.
22. Sheffield, L. J. (Ed.) (1994). The Development of Gifted and Talented Mathematics Students and the National Council of Teachers of Mathematics Standards. Research-Based Decision Making Series. Mathematics. The National Research Center on the Gifted and Talented.
23. Sheffield, L. J. (1999). Developing Mathematically Promising Students , Reston, VA: National Council of Teachers of Mathematics.
24. Shin, H., Lu, G., & Han, I. (2000). A study on the Gifted Education for the Mathematical Special Class in the Special High Schools. Journal of the Korea Society of Mathematical Education Series F: Studies in Mathematical Education,5. pp.125-140.
25. Silver, E. A. (1994). On mathematical problem posing. For the learning of mathematics , Vol14(1). 19-28.
ABOUT THE AUTHORS
Kang Sup Lee
Professor and Dean of College of Education, Dankook University and
President of Korea Society of Mathematics Education
Dept. of Mathematics Education
Dankook University
Hannam-dong, Youngsan-Ku, Seoul 140-714
KOREA
E-mail: leeks@dankook.ac.kr
Dong Jou Hwang
Graduate School of Dankook University and
Division of Electronic & Info-Communication
Yeungjin Junior College
218 Bokhyun-dong, Buk-ku, Daegu 702-721
KOREA
E-mail: hdj0719@chol.com
Woo Shik Lee
Dept. of Mathematics
Gyeongbuk Science High School
418-1, Yongheung-dong, Buk-ku, Pohang City
Gyeongsangbuk-do 791-170
KOREA
E-mail: lws54@chol.com
PROJECT M^{3 }: MENTORING MATHEMATICAL MINDS
Kathy Gavin, Linda Sheffield
Abstract: The
following proposal is designed to address questions of interest to the
participants in Topic Study Group 4 (TSG4): Activities and Programs for
Gifted Students by presenting findings from Project M^{3} , Mentoring Mathematical Minds that focus on:
a) What activities and programs are
useful for identifying gifted students and assessing their
potential?What are the cognitive processes of gifted students? What are
characteristic features of talent in mathematics?
b) What are special learning
environments for gifted students, and how does one work with them
there? What happens when they interact with peers?
PROGRAM OVERVIEW
In 1980, the National Council of
Teachers of Mathematics (NCTM) made a bold statement, "The student most
neglected in terms of realizing full potential, is the gifted student
of mathematics." As test scores indicate, progress since that time for
gifted students in the United States has been slow or nonexistent in
this area. This is especially true for underrepresented students from
economically disadvantaged backgrounds.
Project M^{3}
, Mentoring Mathematical Minds, is a five-year collaborative research
effort of faculty at the University of Connecticut, the University of
Northern Kentucky, and Boston University and teachers, administrators,
and third through fifth grade students in ten schools of varying
socioeconomic levels in Connecticut and Kentucky.
As part of Project M^{ 3},
a team of national experts in the fields of mathematics, mathematics
education, and gifted education are creating a total of 12 curriculum
units of advanced mathematics (four units per grade level) accompanied
by professional development modules. A mathematics talent pool of
students is identified in each of the ten schools (total N = 800) and
the units are being implemented in a variety of settings. We will also
modify some of these units to use with all students across ability
levels and backgrounds in differentiated classroom settings. Pre and
post achievement and attitude data are being gathered using
standardized and criterion referenced tests. To enhance the
effectiveness of these units, we provide extensive professional
development for a total of 40 teachers, including yearly summer
institutes, school year in-service, and an Internet portal for
continuous communication and dissemination of resources.
Research questions focus on three
items: measuring the changes in mathematics achievement and attitudes
for talent pool students after exposure to the intervention model;
measuring the difference in mathematics achievement and attitudes
between the experimental and comparison groups; and measuring the
changes in mathematics achievement and attitudes of students exposed to
modified units in differentiated classroom settings. The research
questions will contribute to the summative evaluation component. The
formative evaluation will include an annual assessment of the delivery
of training using classroom observations, teacher interviews and
surveys, and student focus groups.
IDENTIFICATION OF GIFTED STUDENTS AND ASSESSMENT OF POTENTIAL
In order to identify, create and
serve students with mathematical promise, especially those in
economically disadvantaged areas, a variety of measures are being used.
These include traditional measures such as achievement tests and
teacher recommendations as well as other instruments such as a
nonverbal ability test and measures of creativity. Students in the
program have widely divergent scores on a variety of measures. Data
indicate that no single measure is sufficient to identify the majority
of students from diverse backgrounds with mathematical promise.
CURRICULUM AND LEARNING ENVIRONMENTS
The initial group of students selected for Project M^{3}
: Mentoring Mathematical Minds are third graders who began the program
in Fall 2003. One of the most critical aspects of the program is the
use of a student-centered inquiry approach that encourages students to
think like mathematicians, asking questions that enable them to make
sense of mathematics. Students study four units per year that were
developed to add depth and complexity to the typical elementary
mathematics curriculum following recommendations from the National
Council of Teachers of Mathematics Principles and Standards for School
Mathematics and based on best practices in gifted education. Each
lesson has “Think Deeply” questions and a Mathematician’s Journal that
students use to develop and organize their mathematical reasoning.
Students who are ready for more challenge are presented with “Think
Beyond” questions that encourage them to delve more deeply into the
mathematics. “Hint Cards” are available for students who need more
information to get started on an investigation. Students frequently
work with a partner and in small groups that provide stimulating and
necessary dialogue to foster conceptual understanding. This is often
followed by whole class discourse giving students an opportunity to
further develop and consolidate their own mathematical reasoning and
questioning skills as they work with classmates to develop and analyze
complex skills and concepts.
Additional information on Project M^{3} can be found on our website at http://www.projectm3.org.
ABOUT THE AUTHORS
Dr. Kathy Gavin , Project Director
E-mail: kathy.gavin@uconn.edu
Dr. Linda Sheffield, Project Co-director
E-mail: sheffield@nku.edu
PROCESS OF TRAINING AND ADMISSIONTO A MOFET SCIENCE CLASS
Elena Levit, Larisa Marcu, Orna Schneiderman
Abstract: Mofet Science classes start in 7^{th} grade. All students who want to study in a science class have to take a preparation course [1] (“Mechina”).
CONCEPT AND OUTLINE
The transition of a child from elementary
school to middle school occurs simultaneously with the transition from
childhood to adolescence.
According to Piaget, at ages 11-12 children
are able to understand the world surrounding them, and its application
in their lives [2]. An adolescent understands the role of independence
and the importance of taking responsibility.
The preparatory course offers the adolescent
an opportunity to develop skills in decision-making and accepting
responsibility, similar to the way an adult would [3].
Moreover, the preparatory course helps
develop the adolescent’s process of self-evaluation, and attempts to
further independent identity [1], [3].^{ }
According to the educational/psychological
literature, the assessment of individual intellectual capability is
very important in utilizing the creative processes to forecast, plan
and set up targets and make decisions [4], [5], [6].
The course is aimed to reduce as far as
possible, the conflict between the adolescent’s inner values and those
of the society in which he or she lives.The balance between these two
sets of values inspires the development of the new skills and
capabilities of the adolescent to create new relationships and take
responsibility for his or her decisions [1], [3].
Middle schools accept students based on their
grades from elementary school, and other tests. Mofet accepts children
based on theirexpected ability to improve their grades by teaching them
new knowledge, and establishing learning habits and motivation in this
transition to middle school.
The demands of the preparatory course are
very strict. It is an elite course and the students are proud to
graduate. Indeed, the course eases the transition from elementary to
middle school. The student studies in the course with his or her peers,
which is helpful and diminishes pressure and anxiety. At the end of the
course, we obtain high quality students, with outstanding learning
skills, who believe in their ability to succeed academically.
The preparatory course has ameliorated the
performance of students with average achievements, and their
integration into the science classes.
Knowing each student and his unique needs
enables the teacher and the staff to cope with his problems
efficiently. This is based on a diagnostic process aimed to reveal the
potential in every individual. In this process, both teacher and
student obtain a tool; a direct personal relationship is necessary for
building common trust [7].
The development of intellectual processes is
dependent upon the nature of instruction and the overall abilities of
the student [7].
1. CHARACTERISTICS OF THE INTENDED PARTICIPANTS
1. Students interested in science.
2. Students without any exceptional behavior problems.
3. Students
motivated to study.Motivation, according to Mofet’s method, is the key
to success. It is important to note that motivation is the engine of
development of any child. Mofet believes that with motivation, event
the child who is considered “weak” at the onset of study will be able
to attain significant results.
2. THE OBJECTIVES OF THE PREPARATORY COURSE 1. Learning the work patterns in Mofet science classes.
2. Acquiring effective learning methods and skills.
3. Filling gaps in knowledge.
4. Exploiting natural curiosity of children as a stimulating and propelling factor in learning.
5. Increasing self-confidence and establishing the ability to cope with exam anxiety because of past failures.
6. Creating a supportive learning environment [1], [7], [4], [5].
7. Expanding in depth the material studied in elementary school.
8. Developing mathematical, logical and analytical skills [1], [6], [7], [8].
3. COURSE STRUCTURE 1. Subjects studied: Mathematics, Physics and English, total of 60 hours.
2. The preparatory course is given in 20 sessions of 3 academic hours each or 15 sessions of 4 hours each.
3. The
sessions are partly based on frontal teaching and partly on group work,
giving emphasis and attention to the unique problems of each student.
4. The special structure of the course gives the students optimal training for the beginning of their studies in the Mofet class: 1. Learning different approaches to problem solving.
2. Learning models of elimination and educated guessing.
3. Learning methods to solve problems with the aid of evaluation and appraisal.
4. Repetition of basic subjects in mathematics in a deeper and broader view.
5. Teaching English with a special method, fulfilling the potential of each student and assuring better ability in middle school.
6. Quizzes are used to monitor the personal progress of each student, and are used for feedback to each student personally.
7. A workshop is conducted, aiming at: 1. Linking the students who come from different schools.
2. Differentiation among the terms leadership, leader, and public. 1. The importance of job definition, authority delegation and responsibility within a work team.
2. The workshop includes the following activities:
3. Acquaintance exercise.
4. Correlation of expectations and an agreement.
5. Determining leadership.
6. Cooperation versus competition.
7. Organizing a team.
4. PREPARATORY COURSE ADMISSION 1. The
first stage of this process is a day of interviews with the student and
his/her parents, the school principal, the school advisor, the class
coordinator of the school and a representative from Mofet.
2. During
the interview we can tentatively assess the interaction of the child
within the family and comment on family interaction in the presence of
the child and his parents.
3. Examples from the student’s questionnaire completed in the interview:
4. What do you know about Mofet classes? 1. Why do you think you are a suitable candidate for a Mofet class?
2. What are your expectations from a Mofet class, academically and socially?
3. How do you intend to fit into the Mofet class? What ideas can you bring up for the bonding of the class?
4. Do
your parents want you to study in Mofet class? Yes/no (Afternoon
activities: sports (hours per week) Arts (hours per week), any other,
hobbies.)
5. Parents and other family
members are very significant factors, whose involvement or lack of it
will influence the educational team’s success. The acquaintance of the
educational team with the parents and the family members will lead to
constructing common work patterns, and to an intensification of their
participation in their child’s education process (Fig. 1).
Mofet, as an external body, has no commitment to
accept a child to a Mofet class because of parental pressure. The talks
with the students are most significant: a student who does not find the
program suitable can consider other alternatives.
The home has an extensive influence on the
commitment to study; therefore all children and their parents will
sign, at the beginning of the course, a binding agreement in the
following areas:
- Academic demands (preparing homework, successfully complying with work in class)
- Complying
with disciplinary regulations (one breach, two instances of tardiness
or missing a course twice will result in immediate suspension).
5. TEAMWORK
In a modern school, the teacher
should know how to plan the intellectual development of each and every
student [6], [8]. Therefore, the studies in Mofet are a means for
developing the intellectual capability of each student, i.e. the
importance of studying Math as a tool for developing mathematical
thinking [8], as well as a tool for developing creative potential,
reason and general learning skill [2], [4], [5].
1. Designating
a suitable person to coordinate the whole project of the preparatory
course, from the stage of planning until the end.
2. An excellent teachers team, professional and experienced.
3. Availability of teaching material, teaching aids and equipment needed for the course.
4. Regular meetings of the team
5. Building
cooperation with the parents, and establishing a mutual channel for
dealing with all the problems (discipline, studies, motivation,
anxiety).
6. The teachers’
multidisciplinary teamwork is based on the principle that the team
should work in interaction, focusing on the student they undertook to
promote. The educational team learns to work in cooperation and takes
total responsibility for the education program. The core of teamwork is
mutual fertilization, and taking a professional stance towards the
student as an individual. Feedback from the student enables each team
member to deal accurately with each of the students’ needs. In this way
an encouraging atmosphere is created, supporting initiative,
creativity, and the search for new ways and approaches in teaching.
Mofet's teachers are creative enough to use a
beneficial variety of activities in the classroom, to be able to adapt
a course book to meet the needs of the learners, and to design
materials and activities tailored to specific classes.
Our approach to the role of the teacher in the learning process is represented in the Fig. 2.
6. INITIAL OPERATIONS
1. A
first quiz aimed to assess the initial level of each student, and that
of all the group of students, in order to select suitable teaching
material for any gaps that have been identified.
2. Forming small learning groups, which enables the teacher to identify problems and deal with them immediately.
3. Establishing
learning habits: each student will receive a booklet with the
curriculum, explanations, working exercises and reading material, so
that he or she can plan towards defined targets.
4. Establishing
work practices of a high-school level, starting with an orderly copying
from the blackboard, taking notes during lectures, and preparing
homework.
5. A daily check of the student’s output, locating problems and immediate reaction.
6. Planning free time: studying, social activity and hobbies.
7. Lesson Structure: 1. Frontal
lessons, independent learning and group teaching, simulations,
competitions, viewing and discussing educational films, research and
discovery activities, activities for enhancement and elaboration on
subjects from the studies, construction of mathematical and physical
models (demonstrations)
2. Mathematics and Physics
- presentation of data, analysis, solving exercises and problems in
math, geometry, logic, comprehension of math through physics.
8. English – grammar, reading texts, vocabulary, composition, understanding.
9. After
several lessons, the work is done in groups or in couples, when the
teacher knows the students’ level quite well, and can match them in
learning groups.
10. During the lessons the focus is on:- Emphasizing multiple possibilities – there is no one right answer.
- An
atmosphere of love (trust, understanding, confidence, tolerance towards
the other) and support, enabling each individual to access in-depth
knowledge, to locate platforms to express his/her creativity, and to be
able to take risks.
- An atmosphere
that supports the strength of effort. When there is no room for
mistakes or failures, one does not tend to take risks, and a large part
of one’s creativity is not expressed.
- A
transition to a research method of study in which the student defines
what interests him/her and in what method, what time and place, he/she
chooses to study. The teachers will take into consideration the
interpersonal aspects between the students.
The following illustrates a sample lesson for mathematics students:
What digit is located in a particular location?
1. Students make a table (searching for the last digit in each column).
2. What can be deduced from the table?
3. Question 1 : Prove that 1 + 2 + 3 + …+ n cannot end with the digits 2, 4, 7, 9.
Question 2 : Prove that no three whole numbers (x, y, z) exist whereby
x^{4 }+ y^{4 }=3 z^{4}.
4. Explain: Filling in the table leads the students to draw the following conclusions:- the square of a whole number cannot end with the digits 2, 3, 7, 8.
- the fourth power of whole numbers and the doubled squares of four can end in 1, 5, 6.
- any power of numbers ending in 1, 5, 6 will end in 1, 5, 6, respectively.
- the third power of any whole number can end in any digit.
- the multiple of two consecutive natural numbers n (n+1) = n^{2} +n can end in the digits 0, 2, 6 only.
Solution to Question 1:
According to the table, n^{2 }+n can end in 0,2,6, therefore can end in 1, 2, 3, 5 only.
Solution to Question 2:
According to the table, x^{4} + y^{4 }can end in 0, 1, 2, 6, 7 only. Therefore 3z^{4} can end in 3, 5, 8 only.
Further expansion
Question 3: Prove that there are no whole numbers, a, b, c, whereby the sum of
a^{ 8} + b^{8} + c^{8} will not end in the digit 9.
Solution to Question 3:
According to the table, each of the variables in the equation can end
in only 1, 5, or 6. The sum of the three variables can end in 1, 2, 3,
5, 6, 7, or 8.
Question 4 : Determine the last digit of the number 2^{12} +24^{3} and check whether the next to last digit is even or odd.
Solution to Question 4: 2^{12} + 24^{3} + 4^{6} + 4^{3} � 6^{3 }+ 4^{3} (4^{3} +6^{3}) + 64 � …0 = …0
Summary :
these are activities designed to enrich the student’s knowledge of the
subject. Students learn to ask and answer questions, as well as to draw
conclusions from the activity for the purpose of solving new problems.
7. ASSESSMENT OF STUDENTS’ ACHIEVEMENTS
Students are assessed in the following procedures:
1. Observation and analysis of events in real time
2. Pedagogic monitoring
3. Exams and quizzes
4. In each lesson, an exercise or a quiz or a dictation is given, to assess the student’s relative progress.
5. A feedback form
is circulated mid-course to receive student reaction to course
progress. The students are requested to answer the form fully. On this
basis the pedagogic team can learn and draw conclusions how to improve
the second half of the course.
Examples of questions from the feedback form:
- Is the material you study in the course difficult to understand or easy?
- Is the material interesting?
- Are the lessons presented in an interesting way?
- My feeling in general in the course (organization, teachers, attitude)
The
following is a sample of the type of questions used to gain
understanding of the student's attitudes toward the learning process,
in this case their views on homework:
6. At the end of the course each student receives a self-assessment sheet , to sum up his or her participation in the course.
Some questions are:
- How do you think you will cope with material you have not yet mastered (that will demand a lot of effort to understand)?
- How did you feel about your struggle with homework?
- Are you interested in joining the Mofet class now that you have experienced the course, and to what extent?
- Evaluate the extent to which the Mofet plan is suitable for you?
- In
doing the homework - did you use your personal knowledge, or you did
you need additional knowledge from parents/friends/books/Internet?
- Are the students you met in the Mofet course appropriate classmates?
- Does your decision to study in Mofet depend on whether or not one of your classmates will join? If yes, what is his name?
7.
A personal assessment is given to each student, consisting of
discipline, ability to cope with difficult targets, teamwork,
independent work, curiosity, motivation to study and scientific
orientation.
8. At the end of the course each student receives an achievement diploma written as follows:
”In the short period of the Mofet course
you have shown high motivation and wonderful diligence; you’ve
succeeded in learning new terminology and coping with difficulty,
finding new friends and getting to know the teachers. We wish you
success wherever you go.”
9. At the end of the course, the Decisions Committee
– a professional steering team of the middle school, whose members are
the principal, teachers from the course, the school advisor and Mofet’s
national representative – holds a discussion and decides who will be
admitted to the Mofet classes.
8. DIAGNOSTIC PROCEDURES AND THE POTENTIAL FOR PREDICTING SUCCESS
The following flowchart (Fig. 3) summarizes
the various stages in the process of screening students in advance of
their acceptance to the Mofet program.
PROVEN ACHIEVEMENT LEVELS
Our experience has shown the following level of achievement by those students who complete the Mofet program.
- 40%
of students completing the Mofet program sit for 5-point matriculation
exam in mathematics, physics or chemistry as well as participate in
academic courses concurrent with their matriculation studies in high
school.
- 20% of
students participate in academic courses while in high school, earning
credits toward their academic degrees, concomitant to their
matriculation studies in high school.
- 15% of
students complete the 5-point matriculation exam in mathematics,
together with an additional 4 points in physics or chemistry.
- 15% of
students complete the 4-point matriculation exam in mathematics, as
well as the physics or chemistry exam, enabling their admission to
academic institutions.
- 10% of students complete their matriculation without an emphasis on mathematics and science subjects.
REFERENCES 1. Schneiderman O., Levit E., Marcu L., Zakharova A. From Mediocrity to Excellence: The MOFET Group for the Advancement of Teaching//
Proceedings of the Third International Conference on CREATIVITY IN
MATHEMATICS EDUCATION & THE EDUCATION OF GIFTED STUDENTS, ICCME
& EGS’03, Rousse, 2003. pp 144.
2. Piaget J. Psychology of Intelligence // Selected psychological works, Moscow: Prosveshenie (in Russian) 1969.
3. Tzukerman G.A., Masterov B.M. Psychology of Self-development. Moscow: Interpraks (in Russian) 1995.
4. Davydov V.V. Theory of Developing Education. Moscow: Intor. 1996.
5. Vigotsky L. Psychology of Children . Collected works, V.4, Moscow: Pedagogy (in Russian) 1984.
6. Kholodnaya M.A. Psychology of Intelligence: Paradoxes of research . Moscow: Bars (in Russian) 1997.
7. Kalmykova Z.I.Productive Thinking as a Basis of Ability to Training . Moscow: Pedagogy (in Russian) 1981.
8. Krutetzky V.A. Psychology of Mathematical Abilities. Moscow: Prosveshenie (in Russian) 1968.
9. Sukhomlinsky V.A. About Intellectual Education. Kiev: Radyans’ka shkola (in Russian) 1983.
10. PiagetJ. Les Structures Mathématiques et les Structures Opératoires de l`intelligence: L'enseignement des mathématiques. Paris (in French)1960.
ABOUT THE AUTHORS
Elena Levit, Ph.D., lecturer & scientist
Mofet
ISRAEL
Larisa Marcu, Pedagogic Coordinator
Mofet
ISRAEL
Orna Schneiderman, Director
Mofet
Hakfar Hayarok, 47800
ISRAEL
Tel: 03-6440493
E-mail: mofet1@bezeqint.net
THE UNITY OF MATHEMATICS EDUCATION
The creative teacher can make important
contributions on many levels of education. We have gathered here mostly
to talk about creativity and the education of the gifted. This paper
explores some ways in which we can learn about teaching average
students from our work with the gifted. It will also give examples of
how we can find out more about working with gifted students by
examining our work with average students. Finally, it will give some
examples of how we can construct problem sets which start with very
simple ideas and build towards rather significant mathematical results.
ABOUT THE AUTHOR
Mark Saul, Director
National Science Foundation
USA
E-mail:
msaul@nsf.gov
UNLOCKING INTERLOCKING MATHEMATICAL STRUCTURES – AN EXPERIMENT AT THE KIDMATIKA MATH CLUB
Miriam Amit,AlexeiBelov
Abstract:
Working with young students is both challenging and frustrating.
Exposing them to innovative concepts, higher mathematics, complex ideas
and sophisticated proofs is challenging, but it can also be frustrating
because they lack the requisite knowledge and skills. Usually,
innovative ideas in mathematics rely heavily on previous knowledge and
on skills such as algebraic techniques, argumentation methods,
abstraction and generalizations. We sought to solve this problem in
part by creating a special program that circumvents the obstacle of
prerequisites and taps the ability of the young pupils. This program
was successfully tried out at the Kidmatika Math Club at Ben-Gurion
University in Beersheva.
It is well known in psychology that
there are logical and geometrical ways of thinking, corresponding to
the development of the left and right hemispheres of the brain,
respectively. Geometrical thinking involves dealing with both plane or
two-dimensional (2D) geometry and with spatial or three-dimensional
(3D) geometry. Although there are many similarities between 2D and 3D
geometry, there are also substantial differences, not only in subject
matter but also in the skills and abilities needed to cope with these
two domains. Even students talented for mathematics may lack 3D visual
ability, both because 3D geometry is neglected in the schools and
because they have had very little experience in handling 3D problems or
developing spatial skills.
The purpose of the experiment
described below was to combine the presentation of new mathematical
ideas with the development of 3D visual ability in young students,
thereby overcoming the barrier of prerequisite mathematical knowledge.
The topic we chose to present was the mathematical idea of interlocking
structures. Suppose we have several convex figures on a plane. It is
always possible to find one figure that can be shifted without
moving/shifting/touching the others. Hence any arrangement of convex
figures can be disassembled one by one. Is this also true for 3D? The
natural answer would seem to be Yes, backed up by everyday experience
with blocks or spherical figures; however, in 3D space the answer is
No! Arrangements of convex bodies have recently been found such that
they lock each other and none can be removed. This kind of arrangement
is called an interlocking structure. The theory of block arrangements
is currently at an early stage of development, but it has already
produced unexpected results and found important applications, such as
in the construction of large roofs without cement.
SETTING
The Kidmatika Math Club for young
people is an after-school program operating twice a week on the
university campus. It has 400 members aged 10-15 who are chosen on the
basis of talent and motivation. Not only have they passed a battery of
examinations, they have also proved to be persistent and committed to
the math club. The guiding principle is a belief in talent cultivated
from a young age, accompanied by hard work and constant practice, as in
sport or music. The pupils work in small groups guided by 14 highly
qualified teachers – mathematicians with special expertise in working
with gifted children. These teachers have
developed a unique program that combines exposure to subject areas not
included in the school curriculum and the development of strategies for
problem solving and proofs. The subject areas comprise logic, number
theory, combinatorics, 2D and 3D geometry, optimization, non routine
problems, algebraic laboratory, algorithms, creative innovation, math
and culture, and more... The students participate in all the national
competitions and Olympiads and have captured major prizes all over the
country. The club was founded and is still directed by the first
author, and the two author conduct some of the lessons.
THE EXPERIMENT
In the experiment reported here, we
started with the planar case and asked students to show and prove that
in any arrangement of convex figures, any single figure can be shifted
away. This is a simple well-known experience, and we were expecting an
intuitive, non formal proof. There was a heated discussion with the
students, who offered different approaches justified by experiences (we
had real figures in the classroom) and drawings. We also discussed with
them the problematic transition from an intuitive/feeling approach to
formal mathematical expression. At the next stage, we asked students if
the same conclusions were true for 3D. The immediate answer was Yes. We
asked them to convince us, and of course they found 3D structures
capable of assimilating every single component. However, our students
knew two things: a) experience is not proof; and b) the very fact that
we asked them about 3D indicated a possible trap. After further
discussions, they began to rethink the problem. We split the group into
teams, provided them with small wooden cubes and boxes full of sand
into which the cubes could be placed, and asked them to build a
structure from which no figure could be removed. To our considerable
surprise, after about 40 minutes two teams succeeded (one completely
and the other partially) in creating an interlocking structure.
At the last stage, together with the
pupils we developed a semi-formal proof based on the fact that, in an
interlocking structure, movement of any body in any direction will be
stopped by opposite vectors. We did not use the formal symbols for
vectors, confining ourselves to drawings of arrows and to verbal
argumentation. Based on the discussion that followed and the group's
reactions, we assume that the proof was understood by most of the
students. We finished by showing pictures of interlocking tetrahedral
structures.
CLOSING REMARKS.
The students in the group ranged in
age between 12 and 14, and none had had previous experience in or
knowledge of the domain. Yet they performed very well in the session,
understood the complicated idea of interlocking structures, and applied
this understanding to a concrete model as well as using an abstract
approach. We do not claim this experience can be repeated with any
class, but it is tangible proof
that with the right pedagogy and a carefully orchestrated didactic
strategy, even powerful mathematics such as interlocking structures can
be grasped and unlocked by simple means.
The whole session was videotaped and
documented. Our presentation will include a detailed outline of the
experiment, including a video and concrete materials.
ABOUT THE AUTHORS
Belov Alexei
kanel@dnttm.ru
RUSSIA
Dr. Miriam Amit
Head, Center for Science and Technology Education
Institute for Applied Research
Ben-Gurion University of the Negev
Beer Sheva
ISRAEL, 84105
CHARACTERISTICS OF THE BULGARIAN
MATHEMATICAL EDUCATION
ANALYSIS OFTHE SUBSTANCES OF TEXTBOOKS
IN SECONDARY EDUCATION
Nobuaki Kawasaki
1. PURPOSE
Bulgarian students have participated in
mathematical Olympics every year.So I am interested in the substances
of Bulgarian mathematical education. Last summer, I went to Rousse in
Bulgaria.At that time, I bought some textbooks inmathematics. This
time, I would like to focus on theteaching materialsof secondary
education. I tried to find the characteristics of Bulgarian
mathematical education by comparing both countries' textbooks in
mathematics.
2. MAIN SUBSTANCES OF MATHEMATICS IN BULGARIA AND JAPAN
I want to introduce some examples in
order to find characteristics of the teaching materials of mathematics
inBulgaria and Japan, and to find some differences between the
curricula.
When we want to compare curricula of
both countries, there are some points of view. This time, I want to
research each country'scurriculum of mathematics from two points of
view. One point is that what substances are considered during each
school year, and the other point is how to teach them.
Chart 1 in the next page has
shownthe comparison between Bulgarian curriculum and Japanese one about
some kinds of teaching materials of mathematics in secondary education.
The grade is from 9^{th} to 10^{th} one.I made Chart 1 with the textbooks of both countries as a reference.
According to the Chart 1, Bulgarian
curriculum has put emphasis on algebraand geometry. Especially,
equation and inequalitysolving is substantial,and all students study
them. On the other hand, Japanese curriculum has put emphasis on
analysis, but some students do notstudy them, because Japanese
curriculumallows thechoice.There are some remarkable examples in the
Chart 1.
Except for the teaching material in
the 12th grade, all teaching materials in the Bulgarian row of Chart 1
have been included in the required subject. However, in Japan, only
trigonometric ratio has been included in required subject and other
teaching materials have been included in selective subjects.
The arrows in the solid line have
displayed the comparison of the grade where eachcurriculum of
mathematics has adopted teaching materialsof algebra and geometry. In
addition, the arrows in the dotted line have displayed the comparison
of the grade where each curriculum of mathematics has adopted teaching
materials of analysis. The arrows in the solid line haveexpressed that
Bulgarian curriculum of mathematics has adopted teaching materials of
algebra earlier than Japanese curriculum of mathematics.
The arrows in the dotted line
haveexpressed that Japanese curriculum of mathematics has adopted
teaching materials of analysis earlier than Bulgariancurriculum of
mathematics.
3. THE DIFFERENCE OF THE METHOD OF INSTRUCTION
Each country has adopted the same
sorts of many teaching materials of mathematics. However, the method of
instruction is not the same. I introduce some typical examples in this
paper.The examples are equation, trigonometric ratio,differentiation,
integration, recurrence formula, and rearrangement of data.
(1) EQUATION
Bulgarian curriculum of mathematics contains complete substances of equation. The reason is as follows:
(i) The Time When Equation is Studied
Bulgarian students study many kinds
of equationsearlier than the Japanese students do. The Chart 1has shown
that all Bulgarian students study almost all kinds of the equation in
9th or 10th grade. On the other hand, Japanese students study them in
11th or 12th grade, and some students do not study them because of the
choice system.
According to the Bulgarian textbooks
of mathematics, substances of exponential law and logarithm law are in
the 10th grade's textbook, and substances of exponential function and
logarithm function are in the textbook of 11th grade. The substances of
exponential law and logarithm law are considered as algebra.
On the other hand, in Japanese, there are all these substances in the textbook of 11th grade.
(ii)The Number of Exercises in the Textbooks
In comparison with the Japanese
textbook, there are many exercises in the Bulgarian textbooks. I think
that Bulgarian students have high ability of calculation. In case of
Japan, mathematical thinking or many kinds of problem solving is more
important than the ability of calculation. There are not so many
exercises in the Japanese textbooks of mathematics in comparison with
the Bulgarian ones.
(iii)Difference Method of Solution
In case of Japan, students often use
graph for solving equation. But Bulgarian students solve it as problem
of algebra, not using graph.
The example is thus equation :
The solution of the Bulgarian students is and , then
As ,
then .
The solution of the Japanese students is:
The graphs of and are given on Fig. 1.
It is possible to calculate the coordinate ofthe intersection in two graphs.
(The calculation is the same as Bulgarian
one.)
The value of coordinate is ,then this value is the solution of theequation.The value is
not the solution of the equation. According to the graph, this value is
the coordinate of the intersection of another two graphs that are and .
If students must verify whether the
value of solution is correct, Bulgarian students use the character of
root sign, and Japanese students use the character of intersection of
two graphs.Of course, students of both countries know these two types
of answers.
Iwould like to say that the
Bulgarian students are accustomed with algebraic thinking, and Japanese
students are accustomed with analytic thinking.
(2) TRIGONOMETRIC RATIO
Students of both countries learn
Pythagorean Theorem in 10th grade. However, Bulgarian students learn
the basic of trigonometric ratio as the application of Pythagorean
Theorem. In Bulgarian textbook, Pythagorean Theorem and trigonometric
ratio are in the same chapter. Japanese students have learned
Pythagorean Theorem without the substances of trigonometric ratio in
10th grade, and they have learned the basic of trigonometric ratio as
the application of the characters of triangle in 11th grade.
Bulgarian textbooksof mathematics
have an interesting character. The character is that there are many
theorems or formulas about the characters of triangle or circle. In
Bulgarian textbooksof mathematics about geometry, there are many
formulas that Japanese students cannot use as such. I want to introduce
two examples. The first example is the Parallelogram Theorem, and the
second example is the length of bisector.
Parallelogram Theorem
There is Parallelogram Theorem in Japanese textbook of mathematics. The formula is as follows:
(1)
This is Parallelogram Theorem, and students remember the formula as theorem.
However, another expression of this theorem is possible. Bulgarian textbook defines the length of segments as follows (Fig. 2):
AB= , BC= , CA= , AM=
The formula (2) is derived by using
trigonometric ratio, and this is another expression of Parallelogram
Theorem.If we use the formula(2), we can calculate the length of the
median line directly in a short time.
(2)
Each formula is usedwhen we need the length
of the median line, and the formula (2) is more convenient than the
formula (1).Then the formula (2) is more practical than the formula
(1).
The Length of Bisector
There is a triangle ABC, and the segment AL is a bisector of (Fig. 3).In Bulgarian textbook, there is the method how we can calculate the length of bisector directly.
Let AB= , AC= , AL= , then the length of AL is as follows:
(3)
In case of Japanese, the formula (4) is introduced in
the textbook:
AL^{2}=AB ×AC-BL ×LC (4)
Of course,
Japanese teacher can prove the formula (3) or(4).
However,they do not remember the formula(3)or (4).
Japanese teachers or students do not
remember so many formulas as there are in the Bulgarian textbooks of
mathematics. As Bulgarian students can use these many formulas, I think
that they can solve geometric problemsfaster than Japanese students can.
(3) DIFFERENTIATION
In Japanese, there are teaching
materials of differentiation in the elective subject's textbook of 11th
grade. The sort of function is integral one. Other sorts of functions
are in the elective subject's textbook of 12th grade.
On the other hand, teaching
materials of differentiation are only in the elective subject's
textbook of 12th grade in Bulgaria, and the students study
differentiation of almost all sorts of functions.The method of curve
tracing is one of the applications of differentiation.
(4) INTEGRATION
In Japanese, the substances of
integration are the teaching materials of elective subject. The
students can learn them if they elect the subject named "math II" or
"math III". However, I could not find these substances in Bulgarian
textbooks.There are substances of the integration of integral function
in math II, and there are ones of the integration of other functions in
math III. (Other functions mean transcendental function, fractional
function, irrational function, and so on.) The method of quadrature is
one of the applications of integration.
There are substances of quadrature
in the Bulgarian textbook of mathematics, but the contents is the
method of using the Lemma of CAVALIERI.
(5) RECURRENCE FORMULA
There are substances of recurrence formula in each country's textbook of mathematics, but the method of teaching is not same.
In Bulgaria, there are substances of
recurrence formulain textbook of mathematics for understanding some
features of arithmetic sequence or geometric one. The examples are as
follows:
There are these substances in Japanese
textbook too. However, in case of Japanese, there is certainly next
kind of recurrence formula in the textbook.To find the general term is
one of the purposes of considering recurrence formula .
As we can change this formula to , the sequence{ } is a geometric one, and we can find the formula of general term.
In Bulgaria, recurrence formula is
the way for finding the characters of arithmetic sequence or geometric
one. However, Japanese students learn the substances of recurrence
formula to grasp therecursive thinking.
(6) REARRANGEMENT OF DATA
In Bulgaria, the students certainly
can study some parts of statistic's field. In the 11th grade, all
students study rearrangement of data. The substances are how to make
frequency distribution, histogram, and how to calculate the mean value
of the data.However, in Japan,substancesof statistics are included in
selective subjects. Soonly few students study statistics.
4. ABOUTEACH COUNTRY'S MATHEMATICAL EDUCATION
In case of Bulgaria, there are many
theorems and their proofs in the textbooks of mathematics, and students
use them for solving mathematical problems positively. The textbooks
are similar to some technical books of mathematics that are used in
Japanese universities. For example, there are many topics of expansion
of theorem. Therefore, if the teaching instruction is desirable,
students may be able to realize the pleasure of making mathematics.
Moreover, if students have experience to use theorems or their
expansions, they will be able to solve problems of mathematics in short
time. Bulgarian curriculum of mathematics has adopted many substances
of algebra, geometry, and statistics.
In case of Japanese curriculum of
mathematics, our teachers think that abilities of problem solving are
more important than those of calculating are. So there are not so many
exercises of calculation but there are many substances of thinking.
If we could mix the mathematical
educations of both countries, I think it may be possible to find the
method of new mathematical education. Then our teachers should become
able to teach or train not only gifted students but also many students
desirably.
ABOUT THE AUTHOR
Nobuaki Kawasaki
Senior High School, Otsuka University of Tsukuba
MUSIC AND MATHEMATICS: RELATIONSHIPS BETWEEN INTERVALS AND RATIOS IN MATHEMATICS EDUCATION
Oscar Joao Abdounur
Abstract
In this communication I shall consider educational
aspects of the development of ratio and proportion, focusing on the
arithmetization undergone by these concepts in the light of the
relations between mathematics and music.Since such relations, even if
confined to the context of ratio and proportion, are
fairlywide-reaching and also that the process ofarithmetization is
quite complex, we shall concentrate mainly on the instructional aspects
of a structural peculiarity presented in such a fascinating dynamics.
This peculiarity is the so-called compounding ratios,
a curious feature present in the structure of ratio since the Classical
Period whose irregular transformation into the operator multiplication
is quite representative of the importance of theoretical music in the
arithmetization of ratios. As a consequence we shallalso point out
features of the differences between identity and proportion , which are capable of being didactically explored with a mathematic-musical approach.
The reason for choosing music for the present
approach is not only historical, but more specifically didactic insofar
as the subtle semanticdifferences between compounding and multiplication and also between identity and proportion
are clearer if one thinks of ratios as musical intervals when looking
at such constructs. Grattan-Guinness argues that the well-known
difficulties in teaching fractions can be alleviated by converting the
latterintoratios, andthususing a musical approach. These considerations
corroborate the need to explore didactically specific contexts in which
differences between given constructs manifest themselves moreclearly.
In order to fulfill the aforementioned aim we
shall first of all introduce some historical aspects of ratio in
mathematical-musical contexts as well as of the corresponding structure
in which compounding makes sense, and then follow these with examples
of the practice of compounding on the monochord and by the
didactic-epistemological aspects that underlie such a practice.
The present musical approach widens our
comprehension of ratio and proportion in mathematics not only because
of its historical-cultural contextualization and the interdisciplinary
aspect which underlies it, but also, and most importantly, because of
the role that analogical thought plays in the construction of meaning,
in this case, that of ratio and proportion. If we wanted to extend
Kieren's argument about rational numbers to ratios, we could claim that
to understand the ideas of ratios ,
one must have adequate experience with their many interpretations.
Throughout the history of mathematics and theoretical music, ratio and
proportions assumed different meanings with discrete or continuous
natures in regard to geometry, music and/or arithmetic. Among such
meanings, ratio can be seen as a tool of comparison by means of
proportions, a musical interval, a fraction, a number, an invariant
with respect to proportion, a common thread between distinct contexts
with regard to proportions whereas proportion can be seen as a vehicle
to compare ratios, an equality, a relation, a function etc. The
aforementioned device not only provides a fertile ground for the
understanding of the subtle differences and structural similarities
underlying the diversity of interpretations associated with ratio and
proportions but also contributes to constructing and to experiencing in
a broader way their associated meanings.
In a general sense, discovering common
schemes and archetypes is an efficient way of constructing concepts
that concern in principle different areas. An analogy or metaphor used
in a sensible and discerning way may re-configure a student‘s thought
in a problematic situation of learning, enabling a better understanding
of matters that escape immediate intuition, or that seem too abstract
to him/her, such as the many interpretations associated with ratio and
proportions as well as with the wide variety of structures historically
associated with them.
ABOUT THE AUTHOR
Oscar Joao ABDOUNUR
University of Sao Paulo,
Rua do Matao, 1010 – Cidade Universitaria – Cep 05508 900 – Sao Paulo – SP –
BRAZIL
E-mails: abdounur@mpiwg-berlin.mpg.de
abdounur@ime.usp.br
SELF MADE MATHEMATICS
Key words : General education, mathematical research and education, researchers and high school students.
The author has been chairing the Junior
Mathematical Congress'96 Miskolc, Hungary, one of the official
satellite meetings of the 2nd European Congress of Mathematics, and a
chain of similar meetings almost each year.
The meetings were aimed at bringing
together the future mathematicians of Europe, especially those aged
between 13-19 years. Apart from attending lectures given by invited
scholars and meeting famous European mathematicians, the young
participants themselves gave talks and exhibited posters.
The activity in between the
conferences and congresses in sustained by a smaller group of talented
young people, secondary school students of two local grammar schools,
the Földes Ferenc Gimn?zium and Herman Ottó Gimn?zium.
During the year the students perfom
their activity in a special group, we call it the SELF MADE MATHEMATICS
group. They learn a bit of Mathematics out of the normal school
material, and they present their „findings” for the whole group in the
form of a student presentation. We hope that what they do is to convert
the idea of the learning by doing method to the deeper understanding by
explaining.
The efficiency of the method is due to the facts:
- While preparing oneself for a presentation the material must be understood much deeper.
- One must be prepared to defend the statements.
- While
explaining, the knowledge is restructured in new manner, instead of
simple reception (passive), this is the new dimension of transmission
(active).
The best
papers of the actual year have been published in the online edition of
the Hungarian language journal Pi edited in Miskolc, see:
http://www.uni-miskolc.hu/~matpi/
ABOUT THE AUTHOR
Péter Körtesi
Institute of Mathematics
University of Miskolc
Miskolc, P.O.Box 10., H 3515
HUNGARY
Phone: + 36-46-565111-1795
Facsimile: + 36-46-565146
E-mail: matkp@gold.uni-miskolc.hu
THE ROLE OF EDUCATIONAL METHOD IN TEACHING OFGIFTED AND TALENTED STUDENTS
Risto Malcevski, Valentina Gogovska
Abstract:
During the process of learning mathematics, students are in similar
position as scientists. It means that they “discover” mathematical
truth alone or with teacher’s help. Because of that scientific methods
used in mathematical research are at the same time educational teaching
methods and they have significant meaning for development of gifted and
talented students. Basically without qualitative knowledge of
scientific methods, especially of their use, it’s almost impossible for
students to make structural knowledge, which, of course,is not
necessary for everyone, but at the same time has special importance for
gifted and talented students, starting from theirbeginnings.
In this work short
explanation of scientific methods used in mathematics is given and few
examples according to which we can introduce some scientific methods on
higher level are considered. This is especially important, because
individual further research is main tool for development of gifted and
talented students.
Key words :
Scientific methods, Observation and experiment, Comparison, Analysis
and synthesis, Generalization, Systematization and abstraction.
INTRODUCTION
A scientific method
represents a way of noticing some fact, which will allow to the
examiner to discover basic characteristic of overviewedobjects or
phenomena. The most commonly used scientific methods in mathematical
research are:
Observation and experiment
Comparison
Analysis and synthesis
Generalization, systematization and abstraction
Observation is a scientific method which is used according to an a priori
developed plan, with the aim to discover, to establish and to
investigate some properties of some objects and phenomena or connection
between other objects or phenomena. During it the objects and phenomena
are commonly investigated in their natural environment.
Experiment
is a scientific method of learning about objects and phenomena where
the examiner intervenes in their natural condition and development
creating artificial conditions, dividing them into parts or combining
with other objects and phenomena. Experiment and Observation are
closely connected, because during the experiment every object is
observed.
Comparison
is a thoughtful operation during which a thoughtful discovery of
similaritiesor differences between investigatedobjects and phenomena is
made. It’s necessaryto respectthe following principles:
comparison should have real meaning, theobjects compared should be connected,
comparison should be realizedaccordingly to a plan,
comparison should be realized entirely.
Analysis
decomposes the given object or phenomena intocharacteristic elements,
with the main idea to investigate them individually, remembering that
they are composed parts. As athoughtful operation analysis starts from
consequences and goes on to causes.
Synthesis
is the merging of given parts or properties into unique composition. As
a thoughtful operation synthesis starts from causes and goes on to
consequences.
Analysis and synthesis
are the most important psychological characteristics of thinking,
because in the process of thinking we first analyse and then
synthesizeon the basis of results of analysis.At the end we come to
generalization, systematization and abstraction which are the results
of analysis and synthesis.We should mention that there is not a strict
boundary between analysis and synthesis, because they are strongly
connected. Because of that analysis and synthesis like scientific
methods are always connected forming unique analytic – synthetic method.
Generalization
is a result of joining separate basic properties, which are essential
for given class of subjects and phenomena. The generalization allows to
work with larger set of objects instead of the given set .
A thinking process reverse to generalization is specialization
. In this process we separate some properties from the set of
properties of investigatedobjects or phenomena. We can say that
specialization is the investigation of a subset N instead of the whole
set .
Systematization is
a thoughtful operation during which the investigatedobjects are
organized into system according to some principle or properties.
Before systematization we
apply: analysis, synthesis, generalization and comparison.Their results
are used and realized within systematization. Final results are systems
of phenomena. The most important type of systematization is classification , which is the constructing of groups of objects, based on similarities and differences between them.
Thinking operation during which we isolate unessential and accent essential properties of given subject or source is abstraction
. Abstraction can be sensual or thoughtful. All mathematical ideas are
formed with thoughtful abstraction. Abstraction and generalization are
strongly connected.
Concretization is
a thinking process reverse to abstraction. It discovers the content of
scientific abstractions, including concrete facts or relations.
LAYOUT
1. HOW THE SCIENTIFIC METHODS SHOULD BE ACCEPTED BY STUDENTS?
In the process of
scientific exploration, but also in the education the scientific
methods are connected and because of that their independent learning
has a meaning only in the process of theoretical teaching of scientific
methods, but not in the practice. From the other side for student the
early mastering of them is necessary because without good knowing of
scientific methods it is impossible to get structural knowledge which
is necessary for successful development of gifted and talented
students. Practice shows that realization of mathematics during the
lessons and commonly accepted forms of working almost don’t allow
accepting scientific methods by the students. It’s a big cause for
concern because this question is marginal: nobody takes care of it,
neither the creator of educational systems nor the teacher. They use
different excuses for their behavior like:
shortage of teaching time,
nonappropriate connection between psychophysical ability of students and abstractness of scientific methods
These aren’t real excuses,
because with skilled planning of teaching process we can pass the
problem of time shortage,and we don’t accept excuse about
nonappropriate connection because we think it doesn’t really exist.It's
more about not perceiving enough about the problem and missing of
preparation and ability of teachers and program creators. We should
mention that acceptance of scientific methods by primary and secondary
school students should be achieved only if:
solving of heavier
problems can be separated into parts in which we can use different
scientific methods, like exercises in geometrical constructions
(analysis, construction, proof and discussion)
for specific purpose we
introduce specific topics not only allowing the students acquire new
knowledge, but also effectively introducingthem to scientific methods
without strict definitions.
We really think that this
kind of introducing will allow qualitativeapplications of scientific
methods by gifted and talented students. We will give some examples of
improving the level of mastering scientific methods by the students in
high school.
Example 1. Great German mathematician Gauss (1777-1855), being a ten yearold child, quickly solved the following task: Find the sum of all natural numbers from 1 to 100.
After waiting long enough
for students’ answer about their idea, we go on following “fertile”
idea, and if we don’t get it we tell them that probably Gauss used the
following table of two progressions each consisting of numbers: 1, 2,
..., 100 during solving this task:
The sum of the numbers in can
be found in two ways. First way consists of finding the sums of the
numbers in each column(1+100, 2+99, 3+98, ..., 98+3, 99+2, 100+1) and
findingthe sum of these sums. So the result is
(1)
Second way of finding the sum of all numbers in is finding the sum in every row:
1+2+3+...+98+99+100 and 100+99+98+...+3+2+1.
Than we add these two sums and get
(2)
Finally, from (1) and (2) we get the result:
.
In the next step we can tell the students the next task to introduce the method of generalization.
Find the formula for the sum .
Of course the students
should solve the task individually, but during that it’s useful for
teacher to mention that this is a generalization of the idea of Gauss.
Further we give the next task, which is similar to the previous one.
Task 1: Find the sum of squares of first n natural numbers
.
The way of finding this sum is not obvious, but some students will use Gauss idea and will write
But in this example the sums of numbers in columns of table are different, for example
etc.,
so students will discover
quickly that this way is not appropriate for the task. In this case we
get infertile idea, from which students go away quickly. It’s important
that students (individually or with teacher’s little help)go to the
formula
which can be written as
.
Cleary, last formula is true for every natural number p, and if we put we get
Students will easily find out further that they should add the equalities and get
from which we get
(3)
After some manipulations we get
(4)
The action of counting the sum of squares of first n natural numbers is different from the action of counting the sum of first n natural numbers. It’s logical that students will use the previous procedure for finding the sum of cubes of first n natural numbers, which we denote with . Following analogy they will probably write
from this, using analogy, they will get
After adding they will get
from where
. (5)
In the last equality everything is already known, except the sum , which can be found from (5):
. (6)
In the next steps students will try to find the sum of fourth powers of first n
natural numbers, then that of the fifth ones, etc. But this is an
already known procedure, so it doesn’t add new quality to the learning
of scientific methods. We should try to reach a new quality, which can
be done by insignificant intervention in which we will introduce the
method of specialization.
Let’s set
and we write
Adding these equalities we get
(7)
from where . Further, using (3) and (5) we get
(8)
(9)
In the next step students
should notice the connection between the formulas (7)-(9) and the role
of Newton’s Binomial formula to reformulate the results:
,
,
,
where , are binomial coefficients. After that students should come to the general identity for :
(10)
We should expect that students could prove (10) using analogy.We should expect that they will write down the equalities
........ ...............................................................................
and add them to get (10).
The obtaining of previous
equalities allowed us to demonstrate scientific methods:
generalization, specialization, abstraction, and introducing the
recurrence. For the last one students should notice that we can use
(10) to find S_{k}if we know already S_{k}_{-1}, S_{k}_{-2}, …, S_{1}, and S_{0 }, or that we can find , consequently one after another, because we know that .
Using of scientific
methods in teaching mathematics is a priority. During this teacher
should take special care of gifted and talented students.
From the history we know
about attempts to discover universal scientific methods connected with
the universal scientific procedure of preparation young generation for
scientific research. Because this did not happen educational systems
are paying more attention to partial implementation of scientific
methods. This is a huge mistake for mathematics, especially forgifted
and talented students. We could not find any mathematical discovery
based on only one mathematical method in the history. The most
important thing working with gifted and talented students is using
different methods interacting each with other. It’s essential if we
consider a problem, which not only uses different mathematical methods
but also leads to open question in the science. The next task is such
anexample.
Example 2. We will consider Ramsey type colourings.We will use scientific methods to investigate this combinatorial topic.
Step 1. Students are introducedto the subject.
Let be a given set. Every map is calledcolouring of the set in colors, and map is called function of colouring.
For a given function of colouring we define the relation with if and only if . Clearly, is an equivalence and because of that set is the union of disjunctive classes of equivalence. A set , is monochromatic, if is a subset of some class of equivalence, the same as is a constant function.
Ramsey theory is a very
important part of combinatorics, which started its development in 1930
with work of English mathematician F.P.Ramsey. The basic aim of it is
to establish the conditions under which every “irregular” coloring of
some structure (points of the line, surface, elements of a set etc.)
provides a “regular” monochromatic substructure only if the given
structure is “big” enough. A classical example of Ramsey-type problem
is: “Prove that in every colouring of sides and diagonals of a regular
hexagonin two colors a monochromatic triangle can be found”.Its
solution can be obtained by elementary use of Dirichlet’s principle.
A heavier problem of the same type:”Find the smallest n such that for each colouring of the sides and diagonals of a regularn -gon ink colours a monochromatic triangle can be found” is still unsolved. It is well known that for , and for ,but even for only inequality is
known. We must mention that this kind of problems is a relatively
simple one in Ramsey theory, which is very complicated and offers
numerous unsolved combinatorial problems.
Step 2. Discussion with students about the necessaryprior knowledge.
It’s necessary to learn
about Dirichlet’s principle and basic combinatorial principles before
we start to solve colouring problems.We give some definitions further.
Let n be a natural number and . Set is finite, if a bijection existsfor some natural n. Then hasn elements and . We will consider finite sets only.
Principle of equality . If a bijection between the sets and exists, then .
Principle of sum.
a) If , then .
b) If is a family of sets, , for which holds if , then
.
Principle of product. , for .
Dirichlet’s Principle. Let objects, ,be put into boxes. Then at least one box has at least objects in it.
Step3 . Involving into research (starting problems)
It’s useful to give one or
two elementary problems which will be generalized further.So the
principles of generalization will be learned automatically.
Problem 1. The line is coloured in two colours. Does there exist a segment whose endpoints and midpoint are monochromatic?
Naturally students will
try to find the number of possibilities of colouring three points in
two colours. They will discover that three points can be coloured in
two coloursin different
ways, and that among these three points two monochromaticpoints exist.
This result implicatesDirichilet’s principle, having nine points among
which two are monochromatic.
Further, students will
experiment and will notice the “rule of distance” between the points in
the individual triples. It’s necessary, with teacher’s help, to
conclude that they should take nine triples of points of form (for different ), etc.
After that they can go to the next problem.
Problem 2. A plane is coloured in two colours. Does there exist an isosceles right-angled triangle whosevertices are monochromatic?
The idea of solution is
similar to that of the previous problem using also its result,so this
problem is appropriate for acquiring analytic-synthetic scientific
method. Let’s start with three monochromatic points A, B, C on a line
such that AB=BC. Consider isosceles right-angled triangles AMB, BNC,
AKC, such that M, N, K are on the same side of the line AC; then
discuss the possible colourings of M, N, K. Cleary the answer is
positive.
Step 4. Acquiring the method of generalization
Problem3. A plane is coloured in two colours. Does there exist a trianglewith monochromatic vertices similar to the given triangle?
This problem and its solution is a generalization of the previous problem (all isosceles right-angled triangles are similar).
Step 5. Acquiring themethod of specialization
Problem 4.
Prove that the results of the problems 1 – 3 still hold if the whole
plane is replaced by an appropriate finite set of points (these sets
are different for each problem.)
Step 6. Further deverbing of the method of generalization
During solving the next
tasks it’s essential to work individually and to discuss the methods
used. It means that students should individually recognize and comment
their actions. For this it’s importantto know the scientific methods
and their differences.
In further we will assume that instead of the whole planeonly points with integer coordinates are coloured.
Tasks:
Problem 5. A plane is coloured in two colours. Prove the existence of an isosceles right-angled triangle with monochromaticvertices.
Problem 6. A plane is coloured in colours. Prove the existence of a rectangle with monochromatic vertices.
Problem 7. A plane is coloured in two colours. Prove the existence of a square with monochromatic vertices.
Problem 8. A plane is coloured in colours and is an arbitrary finite set of integer points. Prove that there exists a monochromatic set similar to the given set .
Problem 9. A natural number is placed in each integer point of the plane (the numbers can be different). Prove that for every there existsa square with sides parallel to the coordinate axis, such that the sum of the numbers inside the square dividesby .
Remark. In this placea question can be set about the connection between the previous problem and the next task “Each set consisting of natural numbers has a subset whose sum of elements dividesby .”
CONCLUSIONS
The educators should
achieve that the scientific methods are acquired by students in primary
and secondary school.They can achieve it solving heavier problems that
can be separated into parts to which different scientific methods can
be applied(an example:analysis, constructions, proof and discussion in
geometrical exercises)or maybe introducing specific topics allowing the
students to get new knowledge without strict definitions, but
introducing scientific methods. The practical application of this work
will show whether this is a successful attempt to stimulate the
mathematical creativity of gifted students.
Learning and using the
scientific methods develops such qualities of thinking as activeness,
depth and width as well as the criticism of thinking.
REFERENCE
1. Ganchev, I.; Portev, L.; Sidorov, ?.et al. : Methodic of Teaching Mathematics, 1chast, Modul, Sofia, 1996
2. Malceski, R.: Methodic of Mathematics Lessons, Prosvetno delo, Skopje, 2003
3. Malceski, R.; Gogovska, V.: Using Inequality Between Means for Structural Mathematical Knowledge
4. Polya, D.: Mathematical Discovery, Nauka, Moskva, 1976
5. Celakoski, N.: Didactic of Mathematics, Numerus, Skopje, 1993
ABOUT THE AUTHORS
Risto Malcevski, Prof.
Faculty of Natural Science
Institute of Mathematics-Skopje,
FORMER YUGOSLAVIAN REPUBLIC OF MACEDONIA
Valentina Gogovska
Faculty of Natural Science
Instituteof Mathematics-Skopje,
FORMER YUGOSLAVIAN REPUBLIC OF MACEDONIA
E-mail: valet@iunona.pmf.ukim.edu.mk
ACTIVITY OF A GIFTED STUDENT WHO FOUND
LINEAR ALGEBRAIC SOLUTION OF BLACKOUT PUZZLE
Sang-Gu LEE
Abstract:
The purpose of this paper is to introduce an activity of student who
found purely linear algebraic solution of the Blackout puzzle. It shows
how we can help and work with gifted students. It deals
withalgorithm,mathematical modeling, optimal solution and software.
Key words: Motivation, Blackout puzzle, linear algebra, basis, algorithm, mod 2 arithmetic.
INTRODUCTION
From the conditions of real life
problems, gifted students see some aspects that others don’t see.
Blackoutgame, which was introduced in the official homepage of popular
movie “Beautiful Mind”,is a one-person strategy game that has recently
gained popularity as a diversion on handheld computing devices. An
animated Macromedia Flash version of the puzzle can be found from the
official website for the 2001 movie ‘A Beautiful Mind’.
We will show what was the question
and answer. We will introduce student’s answer, which is very
elementary and intuitive. We also made a software based on his
algorithm. This process only used the basic knowledge of linear algebra
and can be extended to the fullsize Go board problem and teach how we
work with gifted students.
LAYOUT: L inear Algebraic Solution of Blackout Puzzle
I. Background of Blackout Puzzle
II. Main Questions
III. Our Solution of the Blackout Puzzle
IV. Conclusion
I. Background of Blackout Puzzle
In my recent linear algebra
class we were talking about the movie "A Beautiful Mind", starring
Russell Crowe as Nobel Laureate John F. Nash, Jr. (2001), where Nash
was playing Go game with his friend. Some of my students told me that
they have played "the Blackout puzzle" from the Korean official website
of the movie. ( http://www.cjent.co.kr/beutifulmind/ ).
One of my students asked
me "Can we find an optimal solution for the game?” and further "Is
there any possibility that we can not win the game if the given setting
is fixed?"
I gave a chance to think about it for one of my students who liked to do so.
He and I met a couple of times personally and made a Mathematical model of the game,
and he brought me the right answer.What he found was that we can always
win the game. The model is fully based on the basic knowledge of linear
algebra.
We made a search of the
game at that time but we did not find any good reference, so we did it
in our own way. Later we found the following web site
http://home.sc.rr.com/jacobsfam/jared/blackout.html , so we went further.
(1) Introduction of Blackout puzzle
Blackout is a one-person
strategy game that has recently gained popularity as a diversion on
handheld computing devices. An animated Macromedia Flash version of the
puzzle can be found from the official website for the 2001 movie “A
Beautiful Mind”.
(2) How to Play
The Blackout board is a
grid of any size. Each square takes on one of two colors. (The diagram
above used blue and red.) The player takes a turn by choosing any
square. The selected square and all squares that share an edge with it
change their colors.
The object of the game is to get all squares on the grid (tile) to be the same color (Black or White).
When you click on a tile
the highlighted tile icons will change or " flip" from their current
state to the opposite state. Remember, the goal is to change all of the
tile icons to black (or white).
(3) How to Solve Any 3×3 Game
The diagram below illustrates the shortest sequence of moves for resolving possible scenarios on a 3x3 board.
II. Main Questions
Q 1. "Is there any possibility that we can not win the game if the given setting is fixed?"
Q 2. "Can we always find an optimal solution for the game?”
Q 3. "Can we make a program to give us an optimal solution?"
III Our Solution of the Blackout puzzle
There are patterns of blackout grid.
Among these 512 patterns, there are patterns
such that we can win the game with only one more click as following.
(Twice of the following basic 9 patterns as we can change all initial
colors.)
We checked several
examples and had enough trial and error to convince us to answer the
first question with any given initial condition.
(Example) Assume the following initial condition
This setting is not one of the above = 18 patterns, but the following 3 clicks make it all white.
Our first step to find a winning strategy was to recognize these 18 patterns.
Now we try to make a Mathematical Model of this game.
Only actions that we can perform are 9 clicks because we only have 9 stones on the board. We assume "the white stone 1 and black stone 0". Then we can classify effects of each action as an addition of one vector
(or 3x3 matrix). Any series of our actions is a linear combination of
these. Now we use modular 2 arithmetic to make the Zero vector or all
1's vector (or matrix, resp.) to finish the game
So we now have the above 9 vectors (in fact, twice of them) to consider which will end the game with just one more click.
[Modelling example] We have 5 black(blue) stones and 4 white(red) stones in board as below.
Then the given matrix is
and we can click some of 9 positions to take action on it. This can be represented by
So, our problem is to find some a, b, c, d, e, ¦ , g, h and i such that
We can use any computational tool and obtain
But we only need integer vector x, so
and
We only need 0 and 1 because clicking 2n + 1 times of one stone is same as clicking once, and clicking’s of one stone is same as doing nothing.
So, our answer is
This shows that if we click on positions (1, 1), (1, 3), (2, 1), (3, 2), we will get all white stones on the board with only 4 clicks.
In the following Fig. 9, the
command "(Wizard) " tells us "1 3 4 8" that indicates which 4 stones we
have to click to win. The number "4" shows we won with 4 clicks(MOVE).
We can run the program from http://matrix.skku.ac.kr/sglee/blackout_win.exe
This works always. Why does this happen? So our next question is
Q 2. "Can we always find an optimal solution for the game?”
[ Proof ] From the 9 matrices
make nine column vectors with the above, and make a symmetric matrix (because of the symmetry in the board) whose columns are these vectors.
Then we have a linear system of equations to find x.
where is a given (condition) matrix.
Then RREF(A)=I_{9} and . So the columns (rows) are linearly independent, and the system has a unique solution.
(Furthermore all this process can be done in Modular 2 arithmetic.)
For the example considered earlier we have
Let
given (1,1) (1,2) (1,3) (2,1) (2,2)
or
(2,3) (3,1) (3,2) (3,3) Goal 1 Goal 2
We want to show that such , exist for any B.
Remember that we can represent 3x3 matrices by vectors and let
Given (1,1) (1,2) (1,3) (2,1)
Recall: Ax = - B is consistent iff rank (A) = rank[A |-B]
Let 0 , j
Case 1. If is a zero matrix, it’s OK because a, b, c, d, e, ¦ , g, h, i (mod 2) give a solution.
Case 2. If is not a zero matrix, then it is clear that
and
In any case, ( or ) is consistent.
So, for given , x, 0, j, and b,
where the vector b=[b _{1},b_{2},b_{3},b_{4},b_{5},b_{6 },b_{7},b_{8},b_{9}] ^{t} comes from the given (0,1)-matrix ,
the system Ax + b = 0 (or j ) has a solution.
Then x is obtained as
x = 0 - b (or x = j - b )
where
Let x' x (mod 2). Then x' is a real optimal winning strategy vector (matrix) which can be deduced from x.
Now entries of x' are all 0 or
1 as is in real game situation and we can always find a (0,1) matrix as
a real optimal winning strategy vector(matrix).
With this idea, one of my student made a computer program in C++ based on this algorithm
which tells us an optimal strategy to win.
We can download it and run from http://matrix.skku.ac.kr/sglee/blackout_win.exe .
This software also verified our conjecture, and showed the proof was valid.
CONCLUSIONS AND FUTURE WORK
This practical approachto a real
problem of gifted students and leading teacher gave a stimulating
mathematical creativity for both. This process can be adapted to
resolve other real world problems with basic mathematical knowledge.
REFERENCES
1. Park, H.-S., Go Game With Heuristic Function , Kyongpook Nat. Univ. Elec. Tech Jour. V15, No.2 pp. 35-43, 1994
2. Park, J.-B.(2003), Software http://matrix.skku.ac.kr/MT-04/blackout_win.exe
3. Uhl, J. & Davis, W. , Is the mathematics we do the mathematics we teach?, Contemporary issues in mathematics education , Volume 36, pp. 67-74, Berkeley: MSRI Publications,1999
4. JAVA program by CJ entertainment Inc. , Movie: The Beautiful Mind, Blackout puzzle (2002), http://www.cjent.co.kr/beautifulmind/
ABOUT THE AUTHOR
Prof.Sang-Gu LEE, Ph.D.
Sungkyunkwan University
Suwon, South KOREA
Phone: +82-31-290-7025
E-mail:
sglee@skku.edu
MATHEMATICAL GIFTEDNESS IN EARLY GRADES:
CHALLENGING SITUATION APPROACH
Viktor Freiman
Abstract: The
purpose of this paper isto analyze how the model of a teaching based on
challenging situation approach would help to engage young children in a
meaningful mathematical activity and thus contribute to the
identification and nurturing of mathematical talents in primary school
through: - the questioning, search for patterns, inquiry about mathematical relationships
- the elaboration of efficient strategies and methods and creating of new tools of problem solving
- the discussion and communication: thinking about results and reflection on methods
Key words: Mathematical
giftedness, Challenging environment, Challenging situation, Early
Grades, Didactical tools, Re-organization and (re-)construction of
mathematical knowledge
INTRODUCTION
The biographers of famous
mathematicians often refer to the evidence of particular nature of
their talent which can be detected already at a very young age. One can
ask where this deep insight in mathematics comes from. How can teachers
discover their talent and nurture it? And, as a result of this
discovery, what kind of classroom environment would be advantageous for
these children?
Many educational systems
are now implementing new reforms of mathematics curriculum that leave
the classroom door open to the innovative methods of teaching that meet
the interests of all the students. The question is how we as teachers
are prepared to handle this multifaceted educational task. Our
particular interest is directed to the identification and fostering
mathematically gifted students in the elementary mathematics classroom.
Based on our experience of
teaching a challenging mathematics curriculum in Grades K-6 we
constructed a developmental ‘recursive chain’ of identification and
fostering mathematical giftedness: the challenging situation requires
rupture with old knowledge and construction of (new) abilities, thus at
the same time revealing the obstacles and giving the teacher an
opportunity to address them to all students (not only the gifted ones);
reflection on the situations and the shortcomings of thinking about it
leads to new questions and indeed creates a new challenging situation.
STUDIES IN MATHEMATICAL GIFTEDNESS: IDENTIFICATION AND FOSTERING
What tools could help to identify
mathematical giftedness in young children? Analysis of various studies
in mathematics education shows a large spectrum of theories and
practices. Baroody and Ginsburg [3] remark that even among children
just beginning school, there is a wide range of individual differences.
Kindergarteners and first-graders are far from uniform in their
informal mathematical knowledge and readiness to master formal
mathematics. With each grade, individual differences increase. Although
there seems to be no agreement on terminology nor on procedures of
identification of the gifted, all researchersagree that school marks
donot reflect mathematical abilities. School success in mathematics
does not imply the presence of mathematical ability. Conversely,
also,children who donot succeed in school mathematics are not
necessarily mathematically unable.
Many researchers indicate that the process of identification
of gifted children requires many different steps and techniques
including classroom observation, surveys, interviewsand a variety of
tests. According to Kulm [12], Young& Tyre [18] and Johnson [9] an
appropriate assessment should be aimed at revealing the extent,
complexity, and functional characteristics of mathematical thinking
rather than focusing simply on final, well formed ability or
performance. It worth to find out what children can do and how well and
quickly they can learn to do more. Finally, the characteristics that
separates a gifted from non-gifted child in mathematics is the quality of the child's thinking.
Unfortunately, as it was mentioned
by Greenes [8], the bulk of our mathematics program is devoted to the
development of computational skills and we tend to assess students'
ability or capability based on successful performance of these
computational algorithms (so called "good exercise doers") and have
little opportunity to observe students’ ’high order reasoning skills.
Many authors point at the teacher's
role in the process of identification of mathematically able children
as a crucial element of the process of identification of very able
students. Fore example, Kennard [10] affirms that the provision of
challenging material and forms of teacher-pupil interaction is a
revealing key to mathematical abilities voting for interactive and
continuous model for providing identification through challenge.
An identification of mathematically
gifted children is only the first step in a long termwork with them,
which is usually followed by various methods of fostering and nurturing.
Such instructional program,
according to Sheffield [16], helps students develop their mathematical
abilities to the fullest. It may thus allow them to move faster than
others in the class to avoid deadly repetition of material that they
have already mastered. She opts for a program that ”introduces students
to the joys and frustrations of thinking deeply about a wide range of
original, open-ended, or complex problems that encourage them to
respond creatively in ways that are original, fluent, flexible and
elegant [16:46]”
In the Enrichment-Triad Model
developed by Renzulli, one can find the following three types of
activities which are important for nurturing mathematical talents [14:
218]:
- General exploratory activities to stimulate interest in specific subject areas
- Group
training activities to enable students to deal more effectively with
content through the power of mind using critical thinking, problem
solving, reflective thinking, inquiry training, divergent thinking,
sensitivity training, awareness development, and creative or productive
thinking
- Individual
and small-group investigation of real problems in which giftedness
manifests as a result of student's willingness to engage with more
complex, self-initiated investigative activities.
As pedagogical task of fostering the development of mathematical thinking, Baroody [2], Ernst [5] and Fishbein [6] stress the use of a problem-solving approach which
focuses on the processes of mathematical inquiry: problem solving,
reasoning, and communicating approach in which a student plays an
active role. Thus, the didactical transition of mathematical process
from the application of facts, skills and concepts, to the full range
of problem-solving strategies including problem-posing happens when the
classroom teaching becomes more open and challenging. Facing a
challenging task child might not be able to find solutions
spontaneously. Rather, he gets engaged in a constructional process
combining various conditions producing a method to work on the problem
systematically. This aspect of finding a method consciously is
fundamental for the development of mathematical reasoning. A teacher's
task becomes to create an environment that would require a mathematical
attitude, mathematical concepts, and mathematical solutions.
The literature survey shows a variety of
models and methods of identification and fostering mathematical
giftedness. In our study, we focus on those which can be applied to the
everyday teaching practice. We base our study on the assumption that
mathematical giftedness appears already in the early school age.
Therefore, we as primary school teachers cannot wait till
mathematically gifted children are detected in the middle grades and
transferred into special programs. We have to takecare of these
children as early as we can and create favourable conditions for their
development on the everyday basis. Questions of teaching and learning
organization become thus crucial for identification and fostering
mathematically gifted children.
PARADOXES OF TEACHING
Our challenging situation teaching
model is a didactical response to the teaching paradoxes described by
Brousseau [4], Shchedrovitskii [15] and Sierpinska [17]:
- Everything
the teacher undertakes in order to make the student produce the
behaviours that, she expects, tend to deprive this student of the
necessary conditions for the understanding and the learning of target
notion (Brousseau)
- When we
as educators want our children to master some kind of action, we often
tend to teach it directly by giving children tasks which are identical
to this action. But classroom practice shows that the children not only
do not learn actions that go beyond the tasks, they do not even learn
the actions we teach them within the tasks (Shchedrovitskii)
- In
order to access a higher level of knowledge or understanding, a person
has to be able to proceed at once with an integration and
re-organization (of previous knowledge). But we can not tell the
students ‘how to re-organize’ their previous understanding, we can not
tell them what to change and how to make shifts in focus or generality
because we would have to do this in terms of a knowledge they have not
acquire yet (Sierpinska, referring to Piaget and Bachelard)
If
the teacher does not pay attention to these paradoxes, children would
not be able to show (nor to develop) their abilities to formalize, to
generalize, to curtail, to demonstrate flexible thinking, to evaluate
critically their thinking strategies. But these abilities form a
specific mental structure that Krutetskii [11] calls a ‘mathematical
cast of mind’ and their development gives an opportunity to identify
and to nurture mathematical giftedness.
CREATION OF CHALLENGING ENVIRONMENT FOR GIFTED CHILDREN IN A REGULAR CLASSROOM
Our study of mathematical giftedness
[7] shows that, at a very young age, children are eager to learn
mathematics, they enjoy it, and teachers should use every opportunity
to nurture their fresh minds. Thus, a special environment has to be
created in order to maintain their genuine interest. We shall call this
environment challenging, as it is composed of a variety of situations
that provoke mathematical questioning, investigations, and use of
different strategies, reasoning about problems and reasoning about
reasoning.
We propose the use of teaching
approaches based on challenging situations in order to engage all
students into meaningful learning through:
- Early beginning of work on challenging mathematical tasks : 3-5 year old (fostering precocious mind)
- Stimulating questioning (fostering critical / reflective mind) :
- Why?
- What if not?
- Is there a different way?
- Does it always hold? etc.
- Encouraging search for new original ideas by means of open-ended tasks (fostering creative / investigative mind
- Promote full and correct explanations (fostering logical / systematic mind)
- Introduce children to the complexity and variety of mathematical concepts and methods (fostering looking at the world with mathematical eyes)
- Provide children with tasks that require complex data organisations and reorganisations (fostering selective / reversible / analytical / structural mind ).
Our
8 year-long teaching experiment has been conducted at Académie
Marie-Claire, a Canadian private bilingual elementary school with
French and English both taught as a first language. The school is
located in the west suburban area of Montreal, in Quebec. Along with a
strong linguistic program (with a third language, Spanish or Italian),
the school insists on offering enriched programs in all subjects
including mathematics to all its students independently of their
abilities and academic performance. The school thus promotes education
as a fundamental value by instilling the will to learn while developing
such intellectual aptitudes as being able to analyse and synthesize, critical thinking, and art of learning.
The mathematics curriculum is composed
of a solid basic course whose level is almost a year ahead in
comparison to the program of the Quebec's Ministry of Education
(Programme de formation de l'école québécoise, 2001) and an enrichment
(deeper exploration of difficult concepts and topics: logic, fractions,
geometry, numbers as well as a strong emphasis on problem solving
strategies). The active and intensive use of "Challenging mathematics"
; text-books [13]along with carefully chosen additional materials helps
us create a learning environment in which the students participate in
decisions about their learning in order to grow and progress at their
own pace. Each child competes with himself (herself) and is encouraged
to surpass himself (herself). Since the school doesn't do any selection
of students for the enriched mathematics courses, all children of
Académie Marie-Claire participated in the experiment. With some of
them, this author started to work at their age of 3-5, as a computer
teacher.
In the challenging situations used in our own teaching, we favour open-ended problems
which are situated in a conceptual domain familiar enough to the child
who appropriates the situation as his/her own and engages in an
interplay of trials and conjectures, examples and counter-examples,
organisations and reorganisations [1].
In each situation, we observe various elements of the child's mathematical behaviour:
- How
the child enters into the situation (introductory stage,
pre-organisation) and how different ways of presenting the problem
affect children's actons;
- How
the child constructs his/her process of problem solving (choice of
strategy, use of manipulative, systematic search, autonomy,
self-control, mathematical components);
- How
the child acts in case of an error (destroys his/her previous work and
starts from scratch or tries to modify/correct certain actions);
- How the child modifies his/her strategy when the conditions are slightly / completely changed;
- How
the child presents his/her results (orally or in writing, clearly or
not, communicating or not with other participants (children or adults),
symbolism used by the child, organisation of results (on paper)).
Since
a real challenge is possible only when the situation is new for the
learner, the challenging situation must contain the rupture with what
the student has previously learned, provoking the student to reflect on
the insufficiency of the past knowledge and construct new means, new
mechanisms of action adapted to the new conditions, activating her full
intellectual potential. A challenging situation often presents the
child with a problem, which goes above or beyond the average level of
difficulty. The child is encouraged to surpass what is normally
expected from children of her age, thus demonstrating her precocity,
which is a sign of mathematical giftedness.
A challenging situation could also
provide the student with an opportunity to face an obstacle of a pure
mathematical nature, the so-called epistemological obstacle. In order
to overcome it, the student will have to re-organize her mathematical
knowledge, create new links, new structures following laws of logical
inference.
A challenging situation helps to
create a friendly environment in which a child compete with herself
sharing her discoveries with other children and learning from others.
Thus it allows mathematically gifted children who are not high
achievers to participate actively in class and to succeed. Altogether,
this approach helps to reveal mathematical giftedness while fostering
further progress in children.
ANALYSIS OF EXAMPLES
In this section, we will analyze
several examples of children’s work in a challenging environment. These
examples illustrate our approach and show how it enables young
children’s genuine mathematical thinking and helps to identify and
foster mathematical giftedness.
Construction of new means by gifted students
The meaning of multiplication and
division is not fully developed in 8-9 year old children. They
constantly meet situations in which they need to evaluate critically
their means, adapt them to new situation, or construct the new ones.
This process can be seen once children go through the problems with the
calendar (How many days there are in 3 weeks, in 3 weeks and 4 days?
How many weeks there are in 21 days, in 27 days?).These problems can be
solved by simple counting and grouping (using manipulatives or
pictures). However, more advanced children would use more sophisticated
strategies like addition or multiplication (division would be hardly
expected).
Now, suppose, we ask children to
calculate a number of weeks in the year (in order, for example, to know
the number of issues of a weekly published magazine in one year). Even
if they know the fact that a year has 365 (or 366) days and a week has
7 days, they will not be able to divide 365 by 7. Once applying a
strategy of drawing and counting, they will face the difficulty to
represent this big number. So, they are forced to modify their
strategy.
Amelie (8) started searching for a
different way of counting. She constructed a chart which represents a
year divided into 12 small periods (month) (Fig 1.). She invented
symbols representing a week (‘X’) and a day (‘C’). Thus she obtained a
structure suitable for the use of grouping (days into weeks) and
counting (weeks).
She has thus developed a new
mechanism which included old means (counting by 7), old objects (days,
weeks, year), old process (grouping), and old products (number of days
in a month) as well as a reflective action on the problem as a whole
(what does not work in my direct approach), that pushed her towards the
creation of new means (symbols and procedures). But beside all this, we
can see the influence of the combination of abilities and
meta-abilities (to work on a problem and to work on the work), which is
an important characteristic of a mathematically gifted child.
Logical thinking in mathematically gifted children
According to our model, mathematical
reasoning is an important part of fostering theoretical thinking in all
children and especially in those who are identified as mathematically
gifted. Since logical thinking is a foundation of mathematical
reasoning, we can expect that presenting our students with challenging
situations that stimulate the growth of logical thinking, we help them
reason at a higher theoretical level. That is why logic is an important
part of our challenging elementary curriculum.
On the one hand, our children learn
to solve different logical puzzles, use logical operations (negation,
implication, class inclusion, etc.), play different strategy games
(like chess). On the other hand, they are constantly invited to think
logically in various mathematical situations (like working with
definitions, looking for logical explanations, proofs, using examples,
non-examples and counter examples). We challenge them constantly with
'little questions' provoking logically grounded mental actions. For
example, the following problem is very difficult for an 8 year old
child:
When it's sunny, Tim always puts
on his white hat. When he puts his hat he never puts on his blue shoes.
Yesterday, he was wearing his blue shoes during a whole day. What was
the weather that day?
It requires a deep understanding of
negation and ability to apply it in a complex situation. Sarah's
solution (Fig.2) is particular in many ways. First, she uses two key
words describing her solution: proof (‘preuve’) and inverse
(reasons about the reasoning), develops an efficient strategy (making
necessary links between data: sun means wearing hat, hat means no
shoes, so sun means no shoes), uses schemas to communicate her
thinking, gives a short explanation (shoes means no sun, so it rains
(‘alors, c’est inverse’; ‘Reponse: il pleut’).
It comes as no surprise that, facing
challenging tasks, gifted children use their abilities to the full in
making logical inferences and thus demonstrate their potential of
proving. In their 'proofs' they use schemas, symbols (letters),
relationships between data and generalisations. Let’s look for example
at Alice’s work (10 year old, Fig. 3, Fig. 4) on the following problem:
Alex has no brother and is 2
years younger than Peter. Boys are older than girls. Melissa is not the
oldest of the group. The sum of ages of Peter and Elisabeth is the same
as the sum of ages of Alex and Melissa. Put the ages of four children
in the order, from the youngest to the oldest. Give the age of persons
that we know exactly. Find the difference of girls’ ages.
Here is a transcript of child’s work (translated from French):
Alex – 13 year old,
Elisabeth - ? Melissa - ? Peter – 2 years older than Alex – 15 year
old. If the sum of ages of Peter and Elisabeth = 20*, than the age of
Elisabeth = 5 and of Melissa = 7, because the sum of ages of Peter and
Elisabeth is the same as the sum of ages of Alex and Melissa and Alex is 13 .
As we see, the child took concrete
values of the variable (assuming that the sum of ages of Peter and
Elisabeth equals 20). That gave her a possibility to establish an order
between all ages. But, the key point of her reasoning is the following
footnote:
Here is a transcript of it (translated from French):
* this number can be always
different but the difference between two ages (Elisabeth’s and
Melissa’s) will be always two and Melissa will be always older.
We can see that this child shows abilities to
- look for relationships ("Si
la somme des ages de Peter et d'Elisabeth = 20, alors l'?ge d'Elisabeth
= 5 et celui de Melissa = 7, car la somme de son ?ge et celui d'Alex
est égale ? celle de Peter et d'Elisabeth et d'?ge d'Alex est 13 ans."),
- think about relationships making generalisations ("
ce nombre (20) peut toujours changer, mais la différence d'?ge entre
celui d'Elisabeth et celui de Mélissa sera toujours 2 et Mélissa sera
toujours plus ?gée")
- appropriate
use of prepositional phrases in mathematical explanations (therefore
(‘alors’), because (‘car’), always (‘toujours’)
Switching from one representation form to another during investigation process
Our next example presents the work of an
8 year old child on the problem. We started our investigation with
grade 3 children with the question of what number of lines could relate
2 points. Than we started to ask the same question about 3 points, 4
points. Then, the students had to find a way to do it for 10 points and
then for 101 points. Children had to construct a whole process of
investigation working systematically, looking for patterns and making
generalizations. So, Matthew started to look geometrically into small
number of points. Then he discovered that we have to add natural
numbers starting with 1 and finishing one less than our number of
points. So he came out with calculation of the some of first 100
numbers. This work he does also systematically (making only two
calculation mistakes). He didn’t succeed here to discover a faster way
but for the child of 8 is calculation of big numbers is still
meaningful and interesting task. But what is particularly interesting
in terms of mathematical giftedness is to see this ability to switch
from one representation (geometric) to another (arithmetic). We can
also see the effect of an open ended problem that gives a chance to a
child to conduct her own investigation.
Illustration of Matthew’s work: geometric representation (small numbers, search for patterns)
Illustration of Matthew’s work: arithmetic representation (big numbers, systematic work, perseverance)
CONCLUSIONS AND FUTURE WORK
A challenging situation approach is
a system of teaching based on a challenging curriculum as a whole. It
presents thus an efficient didactical tool of constructing a learning
environment in which every child would be able to demonstrate her
highest level of ability.
Challenging situations cannot be
used only on exceptional occasions in a teaching approach. Some of them
must, of course, be carefully prepared, but, for the approach to work,
it must become a style pervading all teaching all the time at all
levels of education. The teacher must be ready to use any opportunity
that presents itself in class (e.g. a puzzling question posed by a
students, an interesting error or unusual solution) to interrupt the
routine and engage in reflective and investigative activities on the
spot, or suggest that students think about the problem at home. Thus,
in fact, what is needed is not occasional challenging situations, but a
‘challenging learning environment’.
This is why, using a challenging
situation model, we are not only able to get gifted children involved
in genuine mathematical activity but also help all children to increase
their intellectual potential.
A challenging situation approach has
another opening for gifted children: they can always go further, go
beyond situations, ask new questions, initiate their own
investigations, and be more creative in their mathematical work.
This gives us as teachers a chance
to understand better what it is that makes mathematical talent appear
and grow and thus leads to the creation of more efficient didactical
tools that would help to keep their interest in learning mathematics.
REFERENCES
[1] Arsac, G., Germain, G., & Mante, M. (1988). Probl?me ouvert et situation-probl?me. Lyon: Université Claude Benaud.
[2] Baroody, A. (1993). Problem solving, reasoning, and communication, K-8: helping children think mathematically. Macmillan Publishing Company.
[3] Baroody, A., & Ginsburg, H. (1990). Children's Learning: A Cognitive View. In: R.Davis, C.Maher, & N. Noddings(Eds.), Constructivist Views on the teaching and Learning of Mathematics (pp 51-64). Reston, Va: NCTM..
[4] Brousseau, G. (1997). Theory of didactical situations in mathematics . Dordrecht: Kluwer Academic Publishers.
[5] Ernst, P.(1998). Recent development in mathematical thinking. In: R. Burden, & M. Williams (Eds.), Thinking through the curriculum (pp.113-134). London, New York: Routledge.
[6] Fishbein, E. (1990). Introduction. In: P.Nesher, & J.Kilpatrick (Eds.), Mathematics and cognition: A research synthesis by the International Group for the Psychology of Mathematics Education.Cambridge: University Press.
[7] Freiman, V.
(2003). Identification and Fostering of Mathematically Gifted Children,
A Thesis In The Department Of Mathematics and Statistics, Concordia
University, Montreal, Canada
[8] Greenes, C. (1981, February). Identifying the Gifted Student in Mathematics. Arithmetic Teacher, 14-17.
[9] Johnson, M. (1983, January). Identifying and Teaching Mathematically Gifted Elementary School Children. Arithmetic Teacher, 25-26; 55-56.
[10] Kennard, R. (1998). Providing for mathematically able children in ordinary classrooms”, Gifted Education International, Vol. 13 , No 1, 28-33.
[11]Krutetskii V.A. (1976). The psychology of mathematical abilities in school children . Chicago: The University of Chicago Press.
[12]Kulm, G. (1990). New Directions for Mathematics Assessment. In: G. Kulm, Assessing higher order thinking in mathematics . Washington, DC: American Association for the Advancement of Science.
[13] Lyons, M., & Lyons, R. (2001-2003). Défi mathématique. Cahiers de l'él?ve. 1-6. Montreal: Cheneli?re McGraw-Hill.
[14]Ridge, L., & Renzulli, J. (1981). Teaching Mathematics to the Talented and Gifted. In: V.Glennon (Ed.), The Mathematical Education of Exceptional Children and Youth, An Interdisciplinary Approach (pp. 191-266). NCTM.
[15] Shchedrovitskii, G. (1968) Pedagogika i logika. Unedited version (in Russian).
[16]Sheffield, L. (1999) Serving the Needs of the Mathematically Promising. In: L. Sheffield (Ed.), Developing mathematically promising students (pp. 43-56). NCTM.
[17] Sierpinska, A.(1994). Understanding in mathematics, London: The Falmer Press.
[18] Young P., & Tyre C.(1992). Gifted or able?: realising children's potential. Open University Press.
ABOUT THE AUTHOR
Viktor Freiman, M.T.M., Ed.Dr., Professeur agrégé
Département d’enseignement au primaire et de psychologie éducationnelle
Faculté des sciences de l’éducation, Université de Moncton
Pavillon Jeanne-de-Valois, Moncton, NB
Canada E1A 3E9
Phone: 1 506 858 44 37
Fax: 1 506 858 43 17
E-mail: freimanv@umoncton.ca
www.umoncton.ca/cami
“ALEF EFES“: STUDENTS CREATE AND PUBLISH
AMATHEMATICAL QUARTERLY AND AN INTERACTIVE SITE
Ziva Deutsch, Akiva Kadari, Thierry Dana-Picard
Abstract:
We present a recreational Mathematics Magazine produced by the gifted
students from the Math Department at Michlalah - Jerusalem College. The
magazine is published in two versions: printed and on-line
(interactive), and is targeted to a broad audience. It also includes
components for gifted high-school students.By that way we introduce
gifted population to an innovative creative process.
Key words: Creative Process, Creative Environment, Interactivity, Gifted Students.
I. INTRODUCTION
The need for specific
activities in Mathematics for gifted students has been researched in
detail. The positive influence of computerised teaching and
computerised self-learning has also been investigated. References to
such studies are given in the report by Ravaglia[5], [6]. It isof great
importance that gifted high-school students be given the opportunity to
learn more advanced courses than their peers. In some curricula,
specific courses are actually built for this gifted audience, at
various levels[5], [6], but the syllabus skeleton usually remains the
same, the main difference being the course’s depth and pace.
In this paper, we describe a project
which is based on a very different approach.The world of Mathematics is
a very broad field. Gifted students are able to discover mathematical
regions beyond that which is taught in their usual course of study. So,
why not inspire them to create their own mathematical material, either
educational or recreational? The result will be recreational pursuit
with a clear educational value.
We emphasise the following: not all
member students of “Alef Efes“’s board, are gifted teacher trainees.
Additionally, the audience does not only consist of endowed high-school
students. Nevertheless, working with gifted people produces finer
materials which can inspire a more general audience to discover new
horizons and new trends in mathematics[8].
“Alef Efes” is an exclusive
quarterly mathematical magazine in Hebrew, appearing in two
non-identical forms: a printed version for subscribers and a free
on-line interactive version. It is prepared by a group of gifted
teacher traineesguidance from the departmental staff primarily created
for the gifted high-school student. It publishes recreational
Mathematics together with classical mathematical topics in a unique
way. The magazine consists of original articles on popular subjects,
riddles on various levels, mathematical games, aspects of the history
of mathematics, and more. Non-mathematical topics which
containmathematical features are also included. We describe these
features in the next section.
II. COMPOUND PROJECT
1. The Magazine’s Goals
The audience of “Alef Efes” is comprised of different groups, in varying proportions:
a. Gifted high-school students and their teachers;
b. High-school students who are less drawn to math(not those who have a
profound lack of interest for Mathematics);c. Undergraduate college students;
d. Teachers at all levels not necessarily involved in gifted youth programs.
Because of the non-homogeneous audience,
there are various levels of difficulty within the activities and the
proposed challenges. We will describe this point later.
In particular, high school students are
faced with unfamiliar mathematical material. Thanks to “Alef Efes” they
have an opportunity to discover mathematical topicswhich are not
included in their regular curriculum, and to discover mathematical
applications in everyday life. All this is done in an interesting and
attractive manner, displaying and adding mathematical flavour to
ancient problems. This process develops curiosity for new topics as
well.
2. The Magazine’s Double Format
Each of the magazine's dual formats
provides a unique advantage. The journal's purpose is to make
Mathematics enjoyable and to encourage mathematical education within a
wide variety of populations. This goal becomes attainable if we are
able to shatter the myths and fears that surround this subject. For
that purpose, the graphic features of the printed journal comprise a
significant issue, and much attention is devoted to their development.
Theinteractive dynamic on-line
version, uses the advantages of the Internet, and allows the visitor to
try and solve problems on his own, to check the solutions, or make use
of on-line hints. The mathematical games, which appear on the site,
take advantage of new methods (such as use of Java, Java script and
Flash), allowing interactive educational activitiesand illuminating the
underlying mathematical principles in depth. The interactivity is a
central feature of the project, in particular to decrease the
“mathematical fear level" of a general audience.
The original web site, written in
Hebrew contains hundreds of riddles, paradoxes, mathematical games and
articles. For the convenience of the readers of this paper, we
translated a few examples into English and included them in a mini
English version of the site.
3. Content
A recurrent myth claims that a
gifted population is homogeneous. This claim is not necessarily true,
and even a group of very bright students of the same age often show
different interests and abilities in Mathematics. Therefore, " Alef
Efes" must contain articles and activities of a wide range of
mathematical topics.
Such activities can be compared to
enriching activities in class teaching. Nevertheless Alef Efes does not
intend to replace the teacher, but rather contributes strongly to
self-teaching, within the borderlines of the official syllabi (and
often beyond them).
Let us give some examples:
a. Expository Articles
The site contains a rich collection of articles of various subjects:
- Sundry examples from geometry and their applications such as the 4-colour problem, platonic solids.
- Number theory discussions as in magic squares (with a reference to Albrecht D ?rer’s Melancholia, Fibonacci sequences, etc).
- Recursive processes like Hanoi towers, Pascal Triangle etc.
b. Math Riddles
Motivating questions, quizzes and
challenges, are proposed at various levels, including those designed
for an audience with out high mathematical abilities. The different
levels of difficulty are indicated by specific icons, as displayed next
to every puzzle both in the printed version and at the site.
Regarding this, [8] quotes [3]:
“learning takes place when students’ abilities and interest are
stimulated by the appropriate level of challenge”.
Examples of a few riddles appear in the mini site at the following URL [2]:
http://alefefes.macam.ac.il/english/riddles/home.asp?miun=1
.
c. Mathematical Games
Recreational activities are a very
important part of a “low-fear” approach to Mathematics. As noted
previously, gifted students do not form a homogeneous population, and
being gifted does not mean being strongly self-confident. The online
games proposed by the web-version of “Alef Efes” contribute to lower
the reluctance level before mathematical situation. Of course, the
mostly graphical nature of such games contributes to developing
geometrical intuition, both two-dimensional and three-dimensional.
Critics are sometimes formulated against Mathematical Education’s
classical way, such as “Until the age of 6, children have a good
three-dimensional perception of the world; thenwe flatten them to plane
geometry”. Alef Efes brings its two cents to enhancing a better
three-dimensional view.
Two examples of our math games appear in the mini English site at the following URL[2]:
http://alefefes.macam.ac.il/english/games/home.asp
d. Paradoxes
One of the most attractive sections
of both the printed and the on-line versions is the paradox section.
Well known and less famous paradoxes amaze the readers. The dynamic
part of the on-line version enables lively illustrations of the
paradoxes. For example look at the so-called Galileo paradoxes in
Hebrew[1]:
http://alefefes.macam.ac.il/paradox/paradox.asp?n=3
e. History of Mathematics and Ethnomathematics
Mathematics is a constantly
developing field made up and discovered by human beings, therefore the
journal emphasizes historical topics dedicated either to biographies of
famous mathematicians or to episodes from the history of Mathematics,
such as the bridges of Koenigsberg, the life of Archimedes and the
famous palimpsest of one of Archimedes’s lost works that was recently
found and sold.
Special attention is paid to
Ethnomathematics (African, Arabic, Indian etc.) Moreover, the journal
includes many articles from the Jewish mathematical sources, and Hebrew
manuscripts.
f. Special Topics
Besides the regular topics appearing
in every issue of the printed version, every new issue includes updated
new items, such as mathematical problems on a chessboard, ancient
mathematical problems, the “Olympic Corner”, etc. These are intended to
either broaden the mathematical landscapes discovered by the readers or
to develop deeper mathematicalinsight. In particular, the challenges
appearing in the “Olympic Corner” present activities from other
countriestargeted to endowed students. This enables the readers to feel
as part of a worldwide community.
4. Who Is Involved in the Edition?
This project enrols gifted students
of the Department of Mathematics at Michlalah-Jerusalem College,
Israel, in creating, formatting, and editing the material for both the
hardcopy and the on-line version of the magazine.
The creative process is planned
around a slightly modified version of Renzulli-Reis Triad model [7].
That was initiated through an on-line course.
In that course the students deal
with interactive tools, gain experience, they discover and sort
unprocessed printed material and material from the internet, or other
sources (Type One activities), they develop skills to write original
material for the magazine, to translate it into Hebrew and to edit it
for the Israeli students.
Subsequently, they organize the
output according to the level of mathematical understanding of the
targeted audience (Type Two activities), and process it into a finished
product appropriate for either the print or for electronic
publications. For the level of Type Three activities, the department
students are less autonomous; they are directed by decisions of the
editorial board of the journal.
We should emphasise the fact that
one of the main goals of the "Alef Efes" project is to develop gifted
teacher trainee’s creativity, and to encourage autonomous research.
These are not frequently developed at the undergraduate level
throughout the world, as the regular classroom work is generally
repetitive and imitates the educator’s expository work. Additionally
the teacher trainees developbroadened graphical features (programming
animations, applets, etc.). The importance of animations and applets is
obvious in quite every course in Mathematics; for an example see[4].
5. How Is the Work Done?
The students work in groups, both in
a synchronous framework and in asynchronous interactions, via a
specific web-forum. This allows for a continuous, uninterrupted flow of
work throughout the week, with constant feedback and open discussions.
Of course, the department's staff is present at each step, but wide
autonomy is left to students.
III. MORE ABOUT THE AUDIENCE
As noted by J. Stepanek [8], elitism is not
perceived as a relevant issue, and in many countries, classes are
highly non-homogeneous, mirroring the general population. Aproject
aimed exclusively at a high-ability population is therefore irrelevant.
On that basis, the editorial board of
"Alef Efes" designed the publication to address an additional issue. A
large part of the high-school students and a certain percentage of
their teachers feel, at the very least, uncomfortable when dealing with
mathematical topics, whether they come from the official syllabus or
appear as a by-product of another activity. In the worst case, they
don't deal with such a material.
The presentation of “serious”
material in a recreational way, using attractive illustrations and high
standard designfor the printed version, and all the graphical updated
animations for the on-line version, assist in decreasing the level of
fear, and increase curiosity about mathematics.
2. Interactions
The connection with a wide an
inhomogeneous audience implies the necessity of a high level of
interactivity. The online journal fulfils that need in various ways:
- A
few forums are active in the site and supply constant connection in
between the editorial board and the audience. The forums are dedicated
to discussions about interesting open problems in Mathematics and
reader’s suggestions of new puzzles.
- “The
Problem of the Month” is a very popular section offering a new
challenging puzzle at the beginning of every month. The winner receives
free subscription for the printed journal.
- A very active guest book enables the readers to give their opinion and send remarks.
3. Statistics
As mentioned previously the audience of
“Alef Efes” comprises varied populations according to
mathematicallevel, professional occupation or age group. There is a
difference between the printed and the online version of the periodical:
- The
printed version is published in 3000 copies (An impressive number,
considering the language and the country size). About 50% of the
subscribers are institutional subscribers (schools, libraries, colleges
and universities). The other 50% comprise teachers, students and math
enthusiasts.
- The on-line
free version is very popular (about30000 visitors per month).The
following table displays the audience distribution along the day; two
maxima appear, one in the morning corresponding to classroom
activities, the other one in the afternoon, showing home activity
mainly of young students.
- The following diagram shows the relative popularity of the various sections of the on-line site:
IV. CONCLUSION
The double magazine (hardcopy and
on-line interactive) is an integral part of the students’ teacher
training at the Mathematics Department of Michlalah-Jerusalem College,
enabling the student to participate in a form of cooperative learning
within a very creative environment, in conjunction with the faculty. A
by-product of this cooperation is “learn how I work, it’s more
important than what I teach”.
"Alef Efes" is a special experiment
in which the undergraduate students are not only exposed to existing
materials, but also participate in original creation of new material
and most importantly, have fun with Mathematics.
REFERENCES
1. URL: http://alefefes.macam.ac.il
2. Mini site in English: http://alefefes.macam.ac.il/english
3. Caine, R.N. &G. Caine
, Making Connections : Teaching And The Human Brain, Alexandria, VA,
Association for Supervision and Curriculum Development. 1991
4. Kidron,I. &N. Zehavi, The Role of Animation in Teaching the Limit Concept , The International Journal of Computer Algebra in Mathematics Education , 9 (3), 2003
5. Ravaglia, R., Suppes, P., Stillinger, C., & Alper, T.M., Computer-Based MathematicsAnd Physics For Gifted Students, Gifted Child Quarterly, 39: 7-13. 1995, Available: http://www-epgy.stanford.edu/research/mset.pdf
6. Ravaglia, R., Sommer, R., Sanders, M., Oas, G. & DeLeone, C., Computer-Based Mathematics And Physics For Gifted Remote Students,Available: http://www-epgy.stanford.edu/research/mset.pdf
7. Renzulli, J.S. and Reis , S.M.,
The Enrichment Triad/Revolving Door Model: A School-Wide Plan For The
Development Of Creative Productivity, Systems and Model For Developing
Programs For The Gifted And Talented, in J.S. Renzulli (Eds), Mansfield
Center, CT, Creative Learning Press, 216-266. 1986
8. Stepanek, J.,
Meeting The Needs Of Gifted Students: Differentiating Mathematics And
Science Instruction, Northwest Regional Educational Laboratory,
1999Available: http://www.nwrel.org/msec/images/resources/justgood/12.99.pdf
9. Tyler-Wood, T.,M.V. Perez-Cereijo &T.Holcomb, Technology Skills Among Gifted Students, Journal of Computing in Teacher Education, 18 (2), 57-60.
ABOUT THE AUTHORS
Ziva Deutsch, Ph.D.
Department of Mathematics
Michlalah - Jerusalem College
Bayit Vegan
Jerusalem
ISRAEL
e-mail: zivad@macam.ac.il
Akiva Kadari, M.A
Department of Mathematics
Michlalah - Jerusalem College
Bayit Vegan
Jerusalem
ISRAEL
e-mail: akiva@macam.ac.il
Thierry Dana-Picard*, Ph. D.
Department of Applied Mathematics
Jerusalem College of Technology
Havaad Haleumi Street 21
Jerusalem 91160
ISRAEL
e-mail: dana@mail.jct.ac.il
WEB BASED MATHEMATICAL PROBLEM SOLVING DATABASEFOR GIFTED STUDENTS
Dimitris V. Papanagiotakis, Panayiotis M. Vlamos
Abstract
Key words: Problem
Database, Problem Solving, Collaboration, Creativity in Problem
Solving, Web, Creative Process, Web Site, Step by Step Approach,
Solution, MathML, XML, JSP.
Themainaspectofthispaperistopresentan
Integrated Web Based Step by Step Mathematical Problem Solving System.
The system consists of a Database where the problems and the solutions
are stored using HTML and MathML format. The mathematical problems
database is accessible to registered members who are interestedin
solving the problems. The members can be students from all over the
world.
The main objective of the system is
the collaboration among gifted students on solving complicated
mathematical problems. The key concept of the system is the Step by
Step approach to the solution of the problem by each group of students
using different paths. Each path may lead to the solution or may
not.Another innovation of the system is that each step is recorded by
the system so at the end the students can have a full tree structure
image of the different paths that lead to the solution.
As a result, of the Step by Step
process and the collaboration we have the increase of the students’
creativity and imagination. The system was developed using state of the
art web technologies Tomcat Web Server, mySQL, JSP 2.0, HTML,
MathML,XML.
ABOUT THE AUTHORS
Dimitris V. Papanagiotakis
Defacto Research Center for Culture and Innovation
Greece
Panayiotis M. Vlamos, Ph.D., Assoc. Prof., President
Defacto Research Center for Culture and Innovation
Greece
MATHEMATICAL SCHOOLS, COMPETITIONS AND PUBLICATIONSFOR PRIMARY AND
SECONDARY SCHOOL STUDENTS INMACEDONIA
Donco Dimovski
Abstract:
The aim of this short note is to give an overall description of most of
the mathematical schools, competitions and publications for primary and
secondary school students in Macedonia. The primary school in Macedonia
is an 8 year school, and the secondary one is a 4 year school. Usually
the students start to attend the primary school when they are six and a
half years old.
Key words: Competitions.
I. MATHEMATICAL SCHOOLS
There are several different types of
mathematical schools organized in Macedonia. The mathematical contests
in almost all of them are not included in the curricula, they are
organized usually by the Union of Mathematicians of Macedonia (SMM) and
some of its sections, and usually the students that participate at
these schools are the ones which like mathematics and show big interest
in it. Below, some of these schools will be described in more detail.
I.1. Mathematical schools for high school students
The Mathematical school for high
school students in Macedonia started in 1976/77 school year. At the
beginning it was organized by the Mathematical Institute at the
University of Skopje. Later, the organizers of this school were the
Institute of Mathematics and Computer Science at the Faculty of
Mathematics and Natural Sciences in Skopje, SMM and the section for
high schools at the Ministry of Education of Macedonia. Today the
organization of this School is solely based at the Institute of
Mathematics, at the Faculty of Mathematics and Natural Sciences in
Skopje and SMI, but because of the shortage of money its activity is
very low.
The aim of this school is to broaden
the knowledge of talented high school students in mathematics and other
disciplines close to mathematics. With its activity, the school gives
the opportunity to high school students to see different aspects of
mathematical sciences and helps them in the choice of their
professional carrier. Most of the participants of the school in the
past years have chosen to study mathematics, informatics and technical
sciences, and a lot of them today are mathematicians and technical
engineers, but some of them study medicine, law, economics.
In 1967, the school started as a
Summer school held in Ohrid, with the topics: Elements of vector
algebra, and Elements of number theory. The next 1967/68 year, an
Evening school was organized for students in Skopje, working in the
Fall and Spring 2 hours per week. Since 1968, the Summer school had two
stages: Seminar in June held in Skopje, and Summer school held in
Ohrid.
Some of the topics of the school in
the years 1967-1971 were the following: Basics of computer sciences
(1967); Three classical geometric problems (1968); Programming on
IBM-1130 (1968); Methods for solving geometric construction problems
(1968); Divisibility of numbers (1968); Approximate calculations
(1968); Sets and maps (1969); Elements of probability theory (1969);
Inversion (1969); Matrix theory 1970); Fortran (1970); Metric spaces
(1971).
From 1972 till 1994 the form of the School was as follows.
A topic and a main lecturer for the
school was chosen. The main lecturer prepared written material,
separately for all of the three stages. For the first stage classes
were formed in high schools in Macedonia, working in the spring, 2
hours per week, using the prepared written material. The number of
students attending the first stage ranged from about 350 to 1400. At
the end of the first stage all the students that attended the school
took the same exam prepared by the main lecturer. About 90 students
that have shown the best results on the exam, participated in the
second stage, held in Skopje in June. At this stage the students
continued to work on the topic, using the written material for this
stage, for about 3-4 days (12 hours). Again, at the end the students
who attended the second stage took an exam prepared by the main
lecturer. About 35 studentswho had shown the best results on this exam,
participated at the third stage, the Summer school, usually held in a
tourist resort on Ohrid lake. Again, at this stage the students
continued to work on the topic, using the written material for this
stage, for about 10 to 15 days (30 to 40 hours). At the end of this
stage the students took an exam on the material of the three stages.
Sometimes the prepared material was published as a book in the edition
"Library Mathematical school".
Some of the topics of the school in
this period were the following: Post and Turing machines and normal
algorithms (1972); Fortran programming (1972); Mathematical logic
(1973); Basic of linear programming (1974); Basic of graph theory
(1975); Set theory (1976); Algorithms, computers and their applications
(1977); Semigroups and finite automata (1978); Project planning (1979);
Logic and computers (1980); Probability and statistics (1981); Basic
programming (1982); Finite automata (1984); Artificial intelligence
(1985); Mathematical Olympiad problems (1987); Computer programming
(1988); Inversion (1989); Combinatorics (1991); Dynamical systems
(1992); Topology via logic (1993); Difference equations (1994).
I.2. Mathematical schools for primary school students
There are several mathematical
schools for primary school students held in Macedonia, and most of them
have local character. Two of them will be described in more details.
The Society of Mathematicians of
Stip organizes a mathematical school for primary school students
(grades 5,6,7 and 8) since 1987. It has two stages: Winter school and
Summer school. The winter school is held in January for about 7 days,
and the summer school is held in June for about 10 days. The students
work on a topic (different topics for different grades) using a written
material. The number of students from almost all parts in Macedonia
attending these schools ranges from 100 to 350.
The Society of Mathematicians of
Skopje organizes a mathematical school for the students in Skopje,
grade 7 and 8. This school works during the school year, in classes
formed of students from almost all primary schools in Skopje, 2 hours
per week. A written material, prepared usually by a group of
mathematicians, is given to the students who participate at the school,
and they attend lectures given usually by primary school mathematics
teachers. Similar schools were organized also in other towns in
Macedonia.
II. MATHEMATICAL COMPETITIONS
The mathematical competitions for
high school students of Macedonia have been organized in 1955, 1957.
Later, since 1959, such competitions are organized each year. Since
1978, a Regional mathematical competition is held each year at the
beginning of February, for 2 and a half hours, each year in a different
town in 6 regions of Macedonia. At these competitions students usually
work on 4 problems, different problems for different grades, prepared
by a national problem committee. Usually about 800 to 1000 students
take part at the Regional mathematical competition. About 140 students
(about 35 from each grade) with the highest number of points from the
Regional competition take part at the Republic mathematical
competition, usually held at the end of March, for 3 and a half hours.
The Republic competition was held in Skopje till 1979, and ever since
it is held in different towns in Macedonia.
A Macedonian Mathematical Olympiad
is organized each year since 1994, in the end of April. About 30 to 40
students with highest score from the Republic competition take part at
the Olympiad. All of them work on 5 problems for 4 and a half hours.
A school competition for primary
school students, grade 8, 7, 6 and 5 (and since last year grade 4), is
held each year. Students work on 4 problems, two of them prepared by
the National problem committee, and two of them prepared by school
teachers of mathematics. After the school competition, the students,
grade 8, 7 and 6 (and since last year grade 5), with highest score
participate at the Municipal competition, usually held in March. All of
the students work on same problems prepared by the National problem
committee. A Republic mathematical competition for primary school
students, grade 7 and 8 (and since last year grade 6) is held usually
in May. About 140 students (about 70 from each grade, and since last
year 60 + 60 + 20), with the highest scores from the Municipal
competition take part at the Republic mathematical competition, usually
held at the beginning of May.
A Macedonian Junior Mathematical
Olympiad is organized each year since 1997, in June. About 30 to 40
students with highest score from the Republic competition take part at
the Olympiad. All of them work on 5 problems for 4 and a half hours.
During the year several types of
preparations for primary and secondary school students are organized in
several places in Macedonia.
Since 1995, the European Kangaroo
Competition is held in Macedonia, with 7000 to 25000 participants in
different years. In cooperation with the Union of Mathematicians of
Bulgaria, about 100 to 300 students participate at the tournament
Cernorizec Hrabar each year. Also, for some years, students from
Macedonia participated at the Tournament of Towns.
III. MATHEMATICAL PUBLICATIONS
Usually, the system of national
competition is supported by publishing the appropriate literature for
the participants of these competitions. The same is practiced by the
SMM, the main organizer of the national mathematical competitions, via
the appropriate publishing activities.
The mentioned literature for the
students of the primary schools consists of the publication "NUMERUS",
published since 1975. It is published in four books per school year. In
this publication, except the popular literature for the young readers,
there is a material for preparation of the students for the National
mathematical competitions. This publication contains also complete
information about the National Mathematical competitions in Macedonia
and the JBMO (the Junior Balkan Mathematical Olympiad). Also, there is
a special edition "Library Numerus", which for the time being consists
of 9 books.
Similar situation is with the
publishing activities for the high school participants in the National
mathematical competitions. Namely, for the needs of these competitions,
SMM is publishing the publication "SIGMA", three times per school year,
since 1979. This publication contains: papers for extending the
knowledge in mathematics of high school students; complete information
about the National mathematical competitions in Macedonia, the BMO (the
Balkan Mathematical Olympiad), the International Mathematical Olympiad,
and the Tournament of Towns. Also, there is a special edition "Library
SIGMA"with 10 books.
The Institute of Mathematics at the
Faculty of Mathematics and Natural Sciences in Skopje publishes the
edition "Library Mathematical School" with 13 books, 4 of which are
problem books with problems and solutions of the Republic, Regional,
and Yugoslav Federation Mathematical Competitions.
For the purposes of Winter and
Summer camps and Mathematical Schools, for primary and secondary school
students, several internal working materials are published.
ABOUT THE AUTHOR
Donco Dimovski, Ph.D., Prof.
St. Cyril and Methodius University
Faculty of Natural Sciences and Mathematics
Institute of Mathematics
Gazibaba b.b. P.O.Box 162, 1000 Skopje
FYR of Macedonia
E-mail: donco@iunona.pmf.ukim.edu.mk