The 12th INTERNATIONAL CONGRESS ON
MATHEMATICAL EDUCATION
Proceedings of Discussion Group DG 2
CREATIVITY IN MATHEMATICS EDUCATION
July 8-15, 2012, Coex, Seoul, Korea
http://icme12.org/
THE USE OF A DIGITAL ENVIRONMENT FOR
DEVELOPING THE CREATIVITY OF
MATHEMATICALLY GIFTED HIGH SCHOOL
STUDENTS
LILLA KORENOVA
Abstract: Accelerated cognitive growth is typical for gifted
students, which can be developed under sufficient motivation
and creativity using a proper didactic environment. Other
typical characteristics of gifted students involve curiosity, joy
from discovering, asking uncommon questions, and a good
ability to argue. Therefore a constructivist approach in teaching
is very appropriate for developing the creativity of
mathematically gifted students. In this contribution we present
several examples of teaching mathematics in a digital
environment (the workshop method, etc.).
Key words: Creativity, ICT, constructivism
INTRODUCTION
What is creativity?
Nowadays there exist a few hundred definitions of creativity. From all the
definitions which describe creativity, probably Torrance’s definition is the bestknown. He describes it as a creation of something new, as a process of forming
thoughts or hypotheses, testing hypotheses and communicating the results. The
ICME 12 - DG 2: Creativity in Mathematics Education,
July 8-15, 2012, Coex, Seoul, Korea
The use of digital environment for developing the creativity of mathematically
gifted high school students
creative process then leads into something new, until then never created.
Based on the world’s leading experts, Turek describes a few axioms of
creativity. From those we choose an important axiom for us: Creativity, creative
abilities and creative thinking can all be developed, formed, trained, i.e., they
can be improved. (Turek1999)
Determining the level of creativity plays a key role in the topic of creativity. The
topic of creativity is extraordinarily complex and insufficiently researched.
Some of the basic factors of creativity are fluency, flexibility, originality,
sensitivity, redefinition and elaboration. They are not of the same importance
and are connected with divergent thinking, which is characterized by creating
alternatives, hypotheses, considering possibilities, making up new approaches,
and problem solving. Another type of thinking is convergent thinking, which is
logical thinking using analysis, synthesis, induction, deduction, conclusion and
generalization. (Turek, 1999)
What is the difference between intelligence and creativity?
The difference between creativity and intelligence is that intelligence is an
intellectual ability based on convergent thinking, which signals a quality of
orientation in problematic situations by way of a choice of an appropriate
solution, and is a measure of flexibility and ease in change of orientation. On the
other hand, creativity is a process based also on the gaps in knowledge, a
sensibility towards missing fragments, the process of discovering problems, and
searching new original approaches, creating, verifying and judging hypotheses.
Creativity tests are intended to determine intellectual abilities. The best-known
tests of creative thinking are Guilford’s and Torrance’s tests. Torrance presents
these characteristics of a creative person: curiosity, a tendency to ask,
independent thinking and acting, humor, fantasy, playfulness with thoughts,
radicalism, self-confidence, assertiveness, a good ability to observe, sensibility,
a smaller binding to reality, rejecting authority.
Intelligence, as well as creativity, is all about intellectual activity. However, they
are not the same. In general high intelligence can be connected with lower
creativity, although a high level of creative thinking does not occur with a very
low level of intelligence.
What is talent (mathematical)?
With the term of creative abilities are connected the terms “gift” and “talent.”
An extraordinary ability of an individual expressed in a specific field of human
creativity is called a gift or a talent towards performing a specific action. Mainly
psychologists have been researching the structure of talent - looking for the
factors which themselves create and affect it. They found out that talent is not
only a set of high abilities, but also other personal characteristics. So was
created the Renzulli’s Three-Ring Conception of Giftedness, which shows talent
ICME 12 - DG 2: Creativity in Mathematics Education,
July 8-15, 2012, Coex, Seoul, Korea
Lilla Korenova
as an intersection between high abilities, creativity and motivation.
(Renzulli,1977) Other psychologists associated this model with environmental
factors (family, school) and other factors (e.g., lucky circumstances). Renzulli
derived his model based on the analysis of implications resulting from that in
practice, a decisive criterion in assessing talent was the IQ. He set apart three
basic groups of personal characteristics and understands talent as an interaction
between them. (Picture 1.)
Picture 1
From Picture 1, a teacher can realize that he can influence the creativity and
involvement of the students by motivation, thus helping to develop the students’
talents.
So we do not know what creativity is, but we know it can be developed! We know
that by increasing motivation, students’ involvement is increased and that helps
to develop their talents. These statements hold valid for mathematically gifted
students.
How do we motivate more creative activities?
The digital world for the students is very natural. All of them work with the
Internet, use cell phones, computers, notebooks, tablets, various software; they
live in the ICT world. A digital environment in teaching (also) mathematics is
very motivating for the students.
Why a constructivist approach towards teaching?
There exist many concepts of teaching focused on the development of creativity.
The two best-known are the problem method and the project method. Both can
include teaching methods such as the heuristic method, research method, method
of controlled exploring and others. In the heuristic method, students are actively
participating in discovering new information and methods of work. However,
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ICME 12 - DG 2: Creativity in Mathematics Education,
July 8-15, 2012, Coex, Seoul, Korea
The use of digital environment for developing the creativity of mathematically
gifted high school students
they do not solve entire problems on their own, only specific parts and stages of
solving. The research method includes such activity of students, which gets near
the creative work of a scientist. Amongst all these methods, pedagogic
constructivism is dominating.
The theory of constructivist learning, which was developed by the Swiss
psychologist Jean Piaget, is based on the assumption that a student, while
actively interacting with his environment, gradually constructs an inner system
of understanding. The whole process of learning should be done in a stimulating
educational environment, which inspires students to explore. (Lukáč, S.a kol,
2010)
By such an environment we mean a digital environment, taking advantage of
ICT, computers, tablets, interactive whiteboards, software, etc. This kind of
environment is close to the students, it is their workaday reality.
Constructivism in pedagogy requires that the following should be included into
teaching: real-life (authentic) problem solving, creative thinking, group work,
object manipulation, visual teaching aids, e.g., interactive computer software.
(Turek, 2010)
Constructivism has many strands, however, they are all focused on learning with
understanding. A student creates his understanding by himself, by considering
new information and comparing it with previous information, experience, and
schemes. So basically it is the problem method of teaching. (Turek, 2010)
Amongst several constructivist approaches, we would find the method of
controlled exploring, or the ERR (evocation - realizing of the meaning reflection). An extremely effective form of the controlled exploring method is
the workshop method, which is based on educating in groups, actively
recognizing the student, immediate check of work, and a proper rate of progress.
(Lukac, S.a kol. 210)
An excellent stimulating digital educational environment for all the methods of
constructivist approach is the open-source software GeoGebra.
In this paper we present GeoGebra as a tool for a stimulating digital
environment. In such a digital environment, and by using constructivist
approach in teaching, we can develop the creativity of students. Students have
the possibility to create hypotheses, verify, and solve new problems. They can
extend their curiosity, tendency to ask, independent thinking, and ability to
observe. During the presentation of their results, their self-confidence and
ICME 12 - DG 2: Creativity in Mathematics Education,
July 8-15, 2012, Coex, Seoul, Korea
Lilla Korenova
assertiveness also grows. All of that can help and improve the students’
creativity.
We present a few concepts for controlled exploration in high schools in the
GeoGebra environment. The concepts are created as a part of the preparation of
future teachers in the field of the didactics of the digital teaching of mathematics
(in the course “Didactic software in teaching mathematics”), as well as a part of
a national project, “Modernization of the educational process in high schools”,
where we proposed and tested a few concepts of constructivist learning in the
GeoGebra environment.
REFERENCES
Lukáč, S.a kol. (2010). Využitie informačných a komunikačných technológií v
predmete Matematika pre stredné školy. Košice : elfa s. r. o., 2010. ISBN
978-80-8086-149-0.
Partová, E. (2011). Vyučovanie matematiky pomocou moderných technológií.
Bratislava : Univerzita Komenského v Bratislave, 2011. ISBN 978-80223-3144-9.
Renzulli, J. (1977). The enrichment triad model: A guide for developing
defensible program for the gifted and talented. Mansfield Center :
Creative Learning Press, 1977. ISBN 0-936386-01-0.
Turek, I. (2010). Didaktika. Bratislava : Iura Edition spol s. r. o., 2010. ISBN
978-80-8078-322-8.
Turek, I. (1999). Tvorivé riešenie problémov. Bratislava : Metodické centrum v
Bratislave, 1999. ISBN 80-8052-054-2.
Velikova, E. (2011). Geo-Gebra: Technology can
mathematical activities in the classroom.
http://www.igmcg.org/files/newsletter_1.pdf.
advance creative
[Online] 2011.
Žilková, K.( 2009). Školská matematika v prostredí IKT. Bratislava : Univerzita
Komenského v Bratislave, 2009. ISBN 978-80-223-2555-4.
ABOUT THE AUTHOR
PaedDr. Lilla Korenova, PhD.
Comenius University in Bratislava
Faculty of Mathematics, Physics and Informatics
Slovakia
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ICME 12 - DG 2: Creativity in Mathematics Education,
July 8-15, 2012, Coex, Seoul, Korea
The use of digital environment for developing the creativity of mathematically
gifted high school students
Е-mail: korenova@fmph.uniba.sk
ICME 12 - DG 2: Creativity in Mathematics Education,
July 8-15, 2012, Coex, Seoul, Korea