The 11th International Congress on Mathematical Education, Monterrey, Mexico, July 6 - 13, 2008
DG 9: Promoting creativity for all students in mathematics education
INTUITIVE – CREATIVE – GIFTED – LOGICAL
An Analysis for the Discussion Group DG9 at ICME 11
Hartwig Meissner
Abstract: What does it mean to be ‘creative’ or to be ‘gifted’? How can we distinguish? Analyzing the mental processes when solving mathematical problems we discover two types of mental
activities or behavior. On the one hand we distinguish a logical and conscious mode of action and
on the other an ‘intuitive’ and mainly unconscious kind of behavior. They sometimes complement
one another but sometimes also conflict with each other. In this paper we will reflect the interaction
of “creative – gifted” on the one hand and of “intuitive – logical” on the other hand.
Key words: Creativity, Giftedness, Intuitive, Logical.
1. CREATIVITY
What does creativity mean? Many experts from different disciplines give various descriptions,
but there are no standardized answers. A broad variety of views already is given in this DG9 volume. The answer also is different when mathematicians regard mathematical creativity and invention in their own work or when mathematics educators analyze observations of mental processes of
problem solvers in the mathematics classroom (see for example Liljedahl 2008 on the one hand and
Meissner 2003, 2005 on the other1).
Creativity is a highly complex phenomenon. To develop and to further creativity in mathematics education teachers and students need more than a correct and solid mathematical knowledge.
Specific environments are necessary and a complex net of properties and relations2. In our research
group in Muenster we try to concentrate on three aspects, on individual and social components to
solve challenging problems to develop important abilities. The students must learn to explore and
to structure a problem, to invent own or to modify given techniques, to listen and argue, to define
goals, and to cooperate in teams. These are demanding abilities and not simple skills. They rely and
depend on a complex system of cognitive processes.
2. GIFTEDNESS
Similar to ‘creativity’ there are many different descriptions or definitions of what ‘giftedness’
might mean, see for example Sternberg e. a. (2005), Heller e. a. (1993), Kiesswetter e. a. (1988), or
Kaepnick (1998)3. Probably the most common model of giftedness is the Three-Ring Model from
Renzulli, see Appendix 3. To describe giftedness Renzulli concentrates on the three components
G1
above average ability (upper 15 – 20%)
G2
creativity (original thinking, ingenuity, divergent thinking, ...)
G3
task commitment and intrinsic motivation (energy and ‘perspiration’ for a successful
problem solving and for mastering specific performance areas, enjoying challenges, ...)
Putting these three interlocking clusters of traits together, giftedness then appears as the interaction among these three clusters. In other words, creativity is one of the necessary conditions for
giftedness. But we also can see that creativity is not necessarily a subset, neither of ‘giftedness’, nor
of ‘above average abilities’ nor of ‘task commitment & motivation’: Also non gifted children can be
creative. We therefore will analyze the related cognitive processes.
1
see also http://wwwmath1.uni-muenster.de/didaktik/u/meissne/WWW/creativity.htm
A long list of items related to creativity is given in Appendix 1
3
A long list of indicators for mathematical giftedness is given in Appendix 2 (Kaepnick 1998)
2
The 11th International Congress on Mathematical Education, Monterrey, Mexico, July 6 - 13, 2008
DG 9: Promoting creativity for all students in mathematics education
3. MENTAL PROCESSES
For a successful problem solving we need a polarity in thinking4. Referring to Dual Process
Theories our cognition operates in two quite different modes which we will call here System 1 and
System 2 (for more details see Kahneman/Frederick 2005 and Leron/Hazzan 2006). Working on a
mathematical problem will happen in parallel where a spontaneous or intuitive thinking (System S1)
may interfere with an analytical or reflective thinking (System S2). S1 processes are fast and automatic and need not much working memory, but they are very resistant against changes. To transform or to coordinate S1 experiences into appropriate and more flexible S2 experiences the processes must become conscious. Discussions are an important tool to bring unconscious processes
into consciousness5.
Most teachers or students or even researchers in mathematics often are unaware of their spontaneous and intuitive thinking. In the mathematics education class room often we more or less do
not realize or even ignore or suppress intuitive or spontaneous ideas. The traditional mathematics
education does not emphasize unconsciously produced feelings or reactions. In mathematics education often there is no space for informal pre-reflections, for an only general or global or overall
view, or for uncontrolled spontaneous activities. Guess and test or trial and error are not considered
to be a valuable mathematical behavior in the class room. But all these components are necessary to
develop spontaneous or common-sense ideas.
When we discuss creativity and giftedness especially these S1 components are necessary.
Creative or gifted children use their common-sense knowledge very intensively. They very often
react unconsciously, they spontaneously develop new ideas, and often we can detect a ‘cognitive
jump’. These observations allow an additional interpretation of the Three-Ring Model from Renzulli, see Appendix 6: The subsets A and AT mainly represent the analytical, logical S2 components
while the subsets C and CT can be interpreted as the area of spontaneous, intuitive and often unconscious S1 components. Thus we can summarize:
 Creativity basically determines the intuitive part of giftedness, while
 Giftedness needs an effective interplay of intuition and conscious knowledge.
REFERENCES
Velikova, E.A. (2002). Stimulating mathematical creativity in 9th – 12th grade students. The Government Specialized Scientific Council, Sofia, Bulgaria.
Velikova, E.A. & Georgieva, M. (2001). Diagnostic of mathematical abilities. Proceedings of the
Union of Scientists – Rousse, Ser.5, Mathematics, Informatics and Physics, 1, 114-120.
……………………………………………………………………………………………..……….…
ABOUT THE AUTHOR
Dr. Hartwig Meissner
Prof. em. in Mathematics Education
Westf. Wilhelms-Univ. Muenster
Einsteinstr. 62
D-48149 Muenster – Germany
4
Already Vygotzki talks about spontaneous and scientific concepts, Ginsburg (1977) compares informal work and
written work, or Strauss (1982) discusses a common sense knowledge vs. a cultural knowledge. Strauss especially has
pointed out that these two types of knowledge are quite different by nature, that they develop quite differently, and that
sometimes they interfere and conflict (“U-shaped” behavior).
5
For some more aspects see Appendix 4
The 11th International Congress on Mathematical Education, Monterrey, Mexico, July 6 - 13, 2008
DG 9: Promoting creativity for all students in mathematics education
Email: meissne@uni-muenster.de
The 11th International Congress on Mathematical Education, Monterrey, Mexico, July 6 - 13, 2008
DG 9: Promoting creativity for all students in mathematics education
References
Ginsburg, H. (1977). Children’s Arithmetic. New York: Van Nostrand.
Heller, K. A., Moenks, F. J., Passow, H. A. (1993). International Handbook of Research and Development of Giftedness and Talent. Oxford.
Kahneman, D., Frederick, S. (2005). A Model of Heuristic Judgement. In Holyoak, K.J., Morrison,
R.J. (Eds.), The Cambridge Handbook of Thinking and Reasoning, pp. 267 - 293. UK:
Cambridge University Press.
Kaepnick, F. (1998). Mathematisch begabte Kinder: Modelle, empirische Studien und Foerderprojekte fuer das Grundschulalter. Peter Lang, Frankfurt am Main.
Kiesswetter, K. (1988, Ed.). Das Hamburger Modell zur Identifizierung und Foerderung von mathematisch besonders befaehigten Schuelern. Berichte aus der Forschung Bd. 2. Fachbereich Erziehungswissenschaft, Universität Hamburg.
Leron, U., Hazzan, O. (2006). The Rationality Debate: Application of Cognitive Psychology to
Mathematics Education. Educational Studies in Mathematics, Vol. 62/2, pp. 105 – 126.
Meissner, H. (2002). Einstellung, Vorstellung, and Darstellung. Proceedings of the 26th Conference
of the International Group for the Psychology of Mathematics Education, Vol. 1, pp. 156 –
161. University of East Anglia, Norwich UK.
Meissner, H. (2003). Stimulating Creativity. Proceedings of the Third International Conference
"Creativity in Mathematics Education and the Education of Gifted Students", pp. 30 - 36.
Rousse, Bulgaria.
Meissner, H. (2005). Challenges to Provoke Creativity. Proceedings of the 3rd East Asia Regional
Conference on Mathematics Education (EARCOME 3), Symposium on Creativity and
Mathematics Education. Shanghai, China.
Renzulli, J. S. (1978). What makes giftedness? Reexamining a definition. Phi Delta Kappa, 60, pp.
180-184, 261.
Renzulli, J. S. (1998). The Three-Ring Conception of Giftedness. In: Baum, S. M., Reis, S. M.,
Maxfield, L. R. (Eds.). Nurturing the gifts and talents of primary grade students. Mansfield Center, CT: Creative Learning Press.
The 11th International Congress on Mathematical Education, Monterrey, Mexico, July 6 - 13, 2008
DG 9: Promoting creativity for all students in mathematics education
Sternberg, R. J., Davidson, J. E. (2005, Eds.). Conceptions of Giftedness - Second Edition; Cambridge University Press, Cambridge, pp. 246 – 279.
Strauss, S. (1982, Ed.). U-shaped Behavioral Growth. Academic Press, New York.
Vygotsky, L. S. (1978). Mind in Society: The Development of Higher Psychological Processes.
Cambridge, MA: Harvard University Press.
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Web page: http://wwwmath.uni-muenster.de/didaktik/u/meissne/WWW/mei148.doc
Liljedahhl 2008 Haifa
Meissner 2008 Haifa
The 11th International Congress on Mathematical Education, Monterrey, Mexico, July 6 - 13, 2008
DG 9: Promoting creativity for all students in mathematics education
Appendix 1:
not only
Creativity in Mathematics Teaching
"delivering" mathematics
"teaching" creativity as a topic
abilities like . . .
. . . invent new / important ideas,
. . . discover new relationships,
. . . imagination (visual / spatial abilities),
. . . flexibility,
. . . modify given techniques,
. . . connect ... fields of experiences, . . .
social aspects like . . .
. . . communicate,
. . . cooperate,
. . . team work,
. . . convince / argue,
. . . motivate,
. . . engagement,
. . . competitive atmosphere, . . .
"human" aspects like . . .
. . . humour and curiosity
. . . identification with,
. . . acceptance of oneself and others,
. . . success and happiness,
. . . fascination and satisfaction,
. . . self-confidence
. . . interest areas,
. . . daily life experiences, . . .
or
but
The 11th International Congress on Mathematical Education, Monterrey, Mexico, July 6 - 13, 2008
DG 9: Promoting creativity for all students in mathematics education
Appendix 2:
Indicators for math. Giftedness6
mathematics related abilities like . . .
. . . being mathematically sensitive
(for numbers, figures, operations, structures,
esthetical aspects),
. . . being original and having fantasy in
mathematical activities,
. . . remembering mathematical facts,
. . . ability to structure mathematical facts,
. . . ability to switch levels of representation,
. . . reversible thinking and transfer,
. . . visual / spatial thinking.
general human aspects like . . .
...
...
...
...
...
...
...
...
6
being highly mentally active,
being intellectually inquisitive,
task commitment and motivation,
enjoying problem solving,
ability to work with concentration,
perseverance,
independence,
ability to cooperate.
Kaepnick, F.: Mathematisch begabte Kinder: Modelle, empirische Studien und Foerderprojekte
fuer das Grundschulalter. Peter Lang, Frankfurt am Main, Germany, 1998
The 11th International Congress on Mathematical Education, Monterrey, Mexico, July 6 - 13, 2008
DG 9: Promoting creativity for all students in mathematics education
Appendix 3
above
average ability
(top 15-20%)
A
AC
creativity
ACT
C
GIFTED
NESS
AT
CT
T
task commitment
& motivation
Three-Ring Model of Giftedness (Renzulli)
The 11th International Congress on Mathematical Education, Monterrey, Mexico, July 6 - 13, 2008
DG 9: Promoting creativity for all students in mathematics education
Appendix 4:
2 Types of Thinking and Working …
„analytic-logical“
deterministic
analytic
logical
reflective
conscious
algebraic transformations,
formulae, algorithms, …
top 15-20%
„intuitive - common sense“
instinctive
trial and error
naïve
spontaneous
often unconscious
trying, guess and test, experiencing, discovering, …
creativity
AC
A
C
GIFTEDNESS
AT
ACT
CT
T
task commitment
& motivation
… in the Three-Ring Model from Renzulli
The 11th International Congress on Mathematical Education, Monterrey, Mexico, July 6 - 13, 2008
DG 9: Promoting creativity for all students in mathematics education