ࡱ> ~ @7bjbj >p؝؝1*&="    !!!8!tp"l T"":8#8#8#***SSSSSSS$URXS 3b)*33S  8#8#*S<<<3 8# 8#S<3S<<O  Q8#" uXeV!o6ePRS0TyPX;lX(Q      QX Q,*,<.L0***SS  !}<j  ! INTUITIVE CREATIVE GIFTED LOGICAL AN ANALYSIS FOR THE DISCUSSION GROUP DG 9 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 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 other). 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 relations. 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. 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). 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. MENTAL PROCESSES For a successful problem solving we need a polarity in thinking. 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 consciousness. 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 J.Renzulli, see Appendix 4: 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 Ginsburg, H. (1977). Childrens 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, Universitaet 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. Liljedahl, P. (2008). Mathematical Creativity: In the Words of the Creators. Proceedings of the 5th International Conference "Creativity in Mathematics Education and the Education of Gifted Students", pp. 153 - 159. Haifa, Israel. 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. Meissner, H. (2008). Creativity or Giftedness? Proceedings of the 5th International Conference "Creativity in Mathematics Education and the Education of Gifted Students", pp. 165 - 172. Haifa, Israel. 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. 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. ABOUT THE AUTHOR Dr. Hartwig Meissner Prof. em. in Mathematics Education Westf. Wilhelms-Univ. Muenster Einsteinstr. 62 D-48149 Muenster Germany E-mail:  HYPERLINK "mailto:meissne@uni-muenster.de" meissne@uni-muenster.de Appendix 1: Creativity in Mathematics Teaching not only "delivering" mathematics or "teaching" creativity as a topic but 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, . . . Appendix 2: Indicators for math. Giftedness 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 . . . . . . being highly mentally active, . . . being intellectually inquisitive, . . . task commitment and motivation, . . . enjoying problem solving, . . . ability to work with concentration, . . . perseverance, . . . independence, . . . ability to cooperate. Appendix 3  above average ability AC creativity (top 15-20%)  A ACT C GIFTED NESS AT CT T task commitment & motivation Three-Ring Model of Giftedness (J.Renzulli) Appendix 4: 2 Types of Thinking and Working analytic-logicalintuitive - common sensedeterministic analytic logical reflective consciousinstinctive trial and error nave spontaneous often unconsciousalgebraic transformations, formulae, algorithms, trying, guess and test, experiencing, discovering,   AC top 15-20% creativity A C GIFTEDNESS AT ACT CT T task commitment & motivation in the Three-Ring Model from J.Renzulli  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  A long list of indicators for mathematical giftedness is given in Appendix 2  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).  Kaepnick, F.: Mathematisch begabte Kinder: Modelle, empirische Studien und Foerderprojekte fuer das Grundschulalter. Peter Lang, Frankfurt am Main, Germany, 1998     Intuitive Creative Gifted Logical An Analysis for the Discussion Group DG 9 at ICME 11 Hartwig Meissner PAGE 86 DG 9: Promoting Creativity for All Students in Mathematics Education, Section 1 PAGE 83 ICME 11, Mexico, 2008 Proceedings of the Discussing Group 9 : Promoting Creativity for All Students in Mathematics Education The 11th International Congress on Mathematical Education Monterrey, Mexico, July 6-13, 2008 PAGE 80 DG 9: Promoting Creativity for All Students in Mathematics Education, Section 1 [\^_no    ݾ{k{k{[I=h1F]h1F]CJ\aJ"h1F]h1F]5CJ\aJmH sH h1F]h1F]6CJaJmH sH h1F]h1F]6CJNH]aJh1F]h1F]6CJ]aJh1F]heK6CJaJh1F]heK56CJaJh1F]heKCJaJh1F]h1F]5CJ aJ mH sH h1F]heK5CJaJmH sH h~Z5CJ$\aJ$mH sH "h1F]h1F]5CJ$\aJ$mH sH  h1F]heKOJQJaJmH sH 0\no   t$ & Fd\$^`a$gd`$ $d\$a$gd1F]$$ & Fd@&\$^`a$gd`$ d\$`gd1F]$d\$]^a$gd1F] $d\$a$gd1F]d\$gd1F]12W57   bpqt/0q356;@H68Avh1F]h1F]6CJNHaJh1F]h1F]6CJNH]aJh1F]h1F]6CJ]aJh1F]h1F]5CJ\aJh1F]h1F]CJaJmH sH h1F]h1F]5CJaJh1F]h1F]6CJaJh1F]h1F]CJNHaJ!jh1F]h1F]0JCJUaJh1F]h1F]CJaJ.tqn#rtM$0^`0a$gdd)p $d\$a$gd1F]d\$gd1F]$ & Fd\$a$gd1F]$ & Fd\$^`a$gd`$$d\$^a$gd1F] $d\$a$gd1F]AabCDJUYemn!&,@MBC#.prt~wiwh1F]h1F]6CJ]aJh1F]h1F]CJ\aJh1F]h1F]5CJaJh1F]h1F]5CJaJmH sH h1F]h1F]CJaJmH sH "h1F]h1F]5CJ\aJmH sH h`$CJaJhL\CJaJh1F]h1F]CJ]aJh1F]h1F]6CJaJh1F]h1F]CJNHaJh1F]h1F]CJaJ( KMp12B !!6!R!ʺyiyWyiyWyiyGyh1F]h1F]6CJaJmHsH"h1F]h1F]6CJ]aJmHsHh1F]h1F]CJ\aJmHsHh1F]h1F]CJaJmHsH"h1F]h1F]6CJ]aJmH sH  h1F]h1F]CJNHaJmH sH h1F]h1F]CJaJmH sH h1F]h1F]CJ\aJmH sH h1F]h1F]CJaJh1F]h1F]0JCJNHaJh1F]h1F]0JCJaJh1F]h1F]0JCJ\aJ R!;"##$%%&&c'd'u''''''G(gd1F] ^`gd1F]$d\$^`a$gd1F]d\$^`gd1F]$0^`0a$gd&U$0^`0a$gdd)pR!h!!!";"O"h""##<####### $$$$$%%/%Y%%%%)&*&&&&&&&&8'd'̽oh1F]h1F]CJ\aJmHsHh1F]h1F]CJaJmHsHh1F]h1F]6CJ]aJ h1F]h1F]CJNHaJmH sH h1F]h1F]6CJaJmH sH h1F]h1F]CJaJmH sH h1F]h1F]CJ\aJmH sH h1F]h1F]6CJaJh1F]h1F]CJaJh1F]h1F]CJ\aJ(d'q'r't'''''((-(.(E(F(G(H(T(U(W(z({(|((())p**+++㹫㘌}m}a}O}O}O}"h1F]h1F]56CJaJnHtHh1F]CJaJnHtHh1F]h1F]5CJaJnHtHh1F]h1F]CJaJnHtHhqh1F]5CJaJ$h1F]h1F]CJaJmH nHsH tHh1F]h1F]0J5CJaJ jh1F]h1F]5CJUaJhL\5CJaJh1F]h1F]5CJaJmHsHh1F]h1F]5CJaJh1F]h1F]5CJaJmH sH G({(|(((()&)W)k))))))*(*:*L*p***** + x`gd1F]x^`gd1F]xgd1F]d\$gd1F] +0+G+^+++++,8,N,{,,,,-G-i----.*.Y.u. x^gd1F]x^`gd1F] x`gd1F]xgd1F]++++++++i---- ..2.5.d.g...........们vm[QhqPJnH tH #jhqPJUmHnH sHtH hq5CJ(aJ(hqhq5CJaJh1F]h1F]5CJaJh?CJaJnHtH"h1F]h1F]56CJaJnHtHh1F]h1F]CJaJnHtH(jh1F]h1F]CJH*UaJnHtH&h1F]h1F]5CJNH\aJnHtH"h1F]h1F]5CJ\aJnHtHhq5CJaJu............ /-///@/I/S/\/]/$p^p`a$gdq$@ ^@ `a$gdq$^`a$gdq$a$gdqgdqxgd1F]...-/.///4/5/8/;/>/?/@/S/U/Y/[/\/]/a/c////////Ǽ㼭ǼǃtbPb#hqhq5CJ PJaJ nH tH #hqh1F]5CJ PJaJ nH tH hq6CJ(PJaJ(nH tH hq6PJnH tH hqCJPJaJnH tH  hq5CJPJ\aJnH tH hq5CJ(PJaJ(nH tH hq5PJnH tH hqPJnH tH #jhqPJUmHnH sHtH hq5CJPJaJnH tH hqCJ(PJaJ(nH tH ]/c/v//////////// 0%0 $$Ifa$gd6_ $Ifgd6_$a$gdqgdqxgd1F] $xa$gd1F]$a$gdq$a$gdq$p^p`a$gdq$^`a$gdq/////// 0 0%0&0Z00000011111±zzjzR>&jhqCJUmHnHsHtHu/jhqB* CJUmHnHphpsHtHuhqB* CJ(NHnHphptHhqB* CJ(nHphptHhqCJ(nHtH!hq5B* CJ(\nHphptHhqB* CJ(nHphtH!hq5B* CJ(\nHphtHhqnHtHhqhq5CJaJnHtHhqhqhqCJaJhqhq5CJaJh1F]h1F]CJaJ%0&040=0E0P0Z0f0v0|000xxxxxxxxxx $$Ifa$gd6_zkd$$IfF05& t0&644 la 0001ww $If`gd6_zkd$$IfF05& t0&644 la1111 1 11A1E1Z1[1rbbb L gdq  ^` gd'gdqzkd$$$IfF05& t0&644 la 1 111111)1*191:1@1A1H1Z1[1g1l11111111rgXLhqCJ aJ nHtHhq5CJ \aJ nHtHhq5\nHtHhq5CJ(\nHtHhqnHtHhqCJ nHtHh'B* CJ(NHnHphptHh'B* CJ(nHphptH!h'h'B* CJ(nHphptHh'B* CJ(nHphtHh'CJ(nHtHhqCJnHtHhqCJ(nHtH&jhqCJUmHnHsHtHu[1g1h1y1z1{111111111021222344V5W5gd1F] $d\$a$gd1F]$a$gdq$@&gdq L gdqgdq$@&gdq11 222/20212222222223333333324344445 5)5ƻvjj`XNXh#NHmHsHh#mHsHjh#0JUh#CJNHPJaJh#CJPJaJmH sH h#CJPJaJh#CJaJmH sH h#CJaJjh#0JCJUaJh1F]hma CJaJh1F]h1F]CJaJh1F]h1F]CJaJnHtHhQ5CJ aJ nHtHhqhq5CJ aJ nHtHhq5CJ(aJ(nHtH)595:5V5W5X5Z5[5]5^5`5a5c55555555555555555ִ֬{j{W{j%h?0JCJOJQJaJmHnHu hAoIh#0JCJOJQJaJ)jhAoIh#0JCJOJQJUaJh#5mH sH $h1F]h#CJOJQJaJmH sH h#OJQJ!h#CJOJQJ\aJmH sH !hL\CJOJQJ\aJmH sH 'h1F]h#CJOJQJ\aJmH sH jh:qUh:q h#NHh#W5Y5Z5\5]5_5`5b5c555555555D6E6Q6g6$a$gd+$',&#$+D,a$gd+$a$gdd)py,&#$+D,gd*$$a$gd1F]$a$gd1F]$a$$a$gd1F]55586C6E6F6L6M6O6P6Q6Z6f6h666/717278797;7<7=7>7B7F7ʵtfXfʵF#h;h#0J6CJOJQJaJh-Ch#56CJaJh-Ch#56CJ aJ h;h#CJOJQJaJh#0JCJOJQJaJ%h?0JCJOJQJaJmHnHu hAoIh#0JCJOJQJaJ)jhAoIh#0JCJOJQJUaJh#h#6CJOJQJaJ'h;h#6CJOJQJaJmHsHh;h#6CJOJQJaJg6h6666 70717=7>77777 $d\$a$gd1F]$a$gd+$ 9r  n!&#$+Dq]a$gd+y,&#$+D,gd+$ 9r  ]a$gdN$  ]a$gdN F7H777777ƻh1F]hma CJaJh:qh#h#6CJOJQJaJh;h#6CJOJQJaJ'h;h#6CJOJQJaJmHsH< 00&P 1hPP:pL\. 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