فصلنامه علمی

نوع مقاله: مقاله پژوهشی

نویسندگان

1 دانشکده ریاضی،دانشگاه علم و صنعت ایران،تهران،ایران

2 دانشکده ریاضی دانشگاه علم و صنعت ایران، عضو هیأت علمی دانشگاه تربیت دبیر شهید رجائی،تهران،ایران

3 دفتر مطالعات بین المللی تیمز/ پرلز، پژوهشگاه مطالعات آموزش و پرورش، سازمان پژوهش وزارت آموزش و پرورش

چکیده

خلاقیت ریاضی اغلب یک پدیده ناشناخته محسوب می‌شود. به نظر می‌رسد بیشتر دانشمندان علاقه‌مند نیستند فرآیندهای تفکر خود را که موجب خلق و ابداع می‌شود توصیف و تحلیل نمایند.  یکی از اهداف مقاله حاضر، که متکی بر یافته­های تحقیقی معاصر می­باشد، آن است که با اشاره به بعضی از تعاریف و ویژگی‌های خلاقیت ریاضی، به توصیف و تحلیل فرآیندهای تفکر ریاضیدانان در حین خلق ریاضیات، بپردازد. در این راستا، یک مدل چهار مرحله­ای شامل  مراحل آماده سازی، کمون ، جرقه ذهنی و تأیید را مورد بررسی قرار می­دهد. با مروری بر ادبیات، معلوم می­شود که یک تعریف خاص و قراردادی برای خلاقیت ریاضی وجود ندارد. بعضی از تعاریف تأکید می­کنند که یک عمل خلاقانه در ریاضیات می­تواند شامل خلق یک مفهوم مفید یا کشف یک رابطه شناخته نشده و یا تغییر در سازماندهی ساختار یک نظریه ریاضی باشد. چالش­ها در معرفی و توسعه خلاقیت ریاضی به علت تنوع زیاد تعاریف و ویژگی­های آن است. درک، شهود، بصیرت و توانایی تعمیم دادن از جمله نیرو محرکه­های خلاقیت ریاضی هستند. خطاپذیری یکی از ویژگی‌های فعالیت‌های خلاقانه ریاضی می­باشد که باید مورد تقدیر قرار گیرد؛ زیرا همین وجود خطا است که می­تواند باعث پیشروی‌های اساسی ‌شود. لذا در آموزش ریاضی باید با توجه به این ویژگی­ها ، شرایط و بستری فراهم نمود تا یادگیری معنا دار (درک و بصیرت) از طریق برقراری ارتباطات و اتصالات بین مفاهیم و روش­ها ایجاد شود و فرصت پرورش مهارت تعمیم دادن از این طریق فراهم گردد.

کلیدواژه‌ها

موضوعات

عنوان مقاله [English]

The Perspective of Creativity in the Process of Learning Mathematics

نویسندگان [English]

  • M. Nadjafikhah 1
  • N. Yaftian 2
  • S. Bakhshalizadeh 3

1 Faculty of Mathematics, Iran University of Science and Technology, Tehran, Iran

2 Faculty of Mathematics, Iran University of Science and Technology, Faculty Member, Tarbiat Dabir Shahid Rajaei University, Tehran, Iran

3 Research Officer, Timss/ PIRLS National Study Center, Research Institute for Education, OERP, Ministry of Education

چکیده [English]

Mathematical creativity is often considered as a mysterious phenomenon. Most mathematicians seem to be not interested in analyzing their own thinking processes and do not describe how they work or conceive their theories. One of the goals of this paper, which is based on research findings from contemporary literature, presents some definitions and characteristics of mathematical creativity and also describes and analyzes mathematicians’ thinking processes during creating mathematics. For this purpose, a four-stage model is considered consisting of: preparation, incubation, illumination and verification. Referring to literature, it is evident that there is not a specific conventional definition of mathematical creativity. According to some of definitions, a creative act in mathematics could consist of: creating a new fruitful mathematical concept; discovering an unknown relation; and reorganizing the structure of a mathematical theory. The challenges in the identification and development of mathematical creativity are due to the large variety in definitions and characteristics of mathematical creativity. Understanding, intuition, insight and generalization are some of moving powers of mathematical creativity. Fallibility is one of the characteristics of creative mathematical activities which should be appreciated since this existing chance of fallibility could lead to major success achievements of human. Hence, considering all above mentioned, learning-teaching mathematics should be so that it provides environment that fosters ability to make connections among concepts and processes and also provides opportunities to generalize.

کلیدواژه‌ها [English]

  • Creativity
  • Mathematical Creativity
  • Creative Mathematical Activities

[1] Hadamard J., the Psychology of Invention in the Mathematical Field, Princeton University Press, 1945. ]2 ]اعتماد شاپور، تحقیق و خالقیت، نشرریاضی، سال دو، شماره 2 ،1332. [3] Bryson B., A Short History of Nearly Everything, Doubleday, Australia, 2003. [4] Ervynck G., Mathematical creativity, in :Tall D., Advanced mathematical thinking, Kluwer Academic Publishers, New York, 1991, pp.42- 52. [5] Tall D., The psychology of advanced mathematical thinking, in: Tall D., Advanced mathematical thinking, Kluwer Academic Publishers New York, 1991, pp.3-21. [6] Mann E.L., Creativity: The essence of mathematics, Journal for the Education of the Gifted, Vol.30, No.2, 2006, pp.236-260. [7] Sriraman B., The characteristics of mathematical creativity, The International Journal on Mathematics Education [ZDM], Vol.41, 2009, pp.13-27. [8] Mann E.L., Mathematical Creativity and School Mathematics: Indicators of Mathematical Creativity in Middle School Students, University of Connecticut, 2005. [9] Poincaré H., Science and method, New York, Dover, 1948. [10] Poincaré H., Mathematical Creation, in: Newman J.R., (ed.), the world of mathematics, New York, Simon and Schuster, Vol.4, 1956, pp.2041-2050. [11] Sriraman B., Are giftedness & creativity synonyms in mathematics? An analysis of constructs within the professional and school realms, The Journal of Secondary Gifted Education, Vol.17, 2005, pp.20–36. [12] Noraini I. and Norjoharuddeen M.N., Mathematical Creativity, Usage of Technology, Procedia- Social and behavioral Sciences, (ISI/SCOPUS Cited Publication). 2010. [13] Shriki A., Working like real mathematicians: Developing prospective teachers’ awareness of mathematical creativity through generating new concepts, Educational Studies in Mathematics, 2010. [14] Liljedahl P. and Sriraman B., Musings on mathematical creativity, For the Learning of Mathematics, Vol.26, No.1, 2006, pp.20–23. [15] Leikin R., Exploring mathematical creativity using multiple solution tasks, in: Leikin R., Berman A. and Koichu B., (Eds.), Creativity in mathematics and the education of gifted students, Rotterdam, the Netherlands, Sense Publisher,Vol.9, 2009, pp.129-145. [16] Rickart Ch., Structuralism and Mathematical Thinking, in: Sternberg R.J. and Ben-Zeev T., (Eds.), the Nature of Mathematical Thinking, Mahwah, NJ, Lawrence Erlbaum, 1996, pp. 285-300. ]11 ]اردشیر محمد ، ففرمن و فلسفه ریاضی الکاتوش، فرهنگ و اندیشه ریاضی، شماره 27 ،1321،صفحه 0 الی 09. [18] Lakatos I.M., Proofs and refutations: The logic of mathematical discovery, Cambridge University Press, 1976. [19] Sriraman B., Mathematical giftedness, problem solving and the ability to formulate generalizations, The Journal of Secondary Gifted Education, Vol.14, No.3, 2003, pp.151-165. [20] Polya G., Mathematical discovery: on understanding, learning and teaching problem solving, New York, Wiley, 1962. [21] Wallas G., The art of thought, New York, Harcourt Brace, 1926. ]22 ]گویا زهرا، رشد آموزش ریاضی، یادداشت سردبیر، .1321 ،86 شماره [23] Polya G., How to Solve It, Princeton University Press, Princeton, 1945. [24] Sio U.N. and Ormerod T.C., Does Incubation Enhance Problem Solving? A Meta-Analytic Review, Psychological Bulletin, Vol.135, No. 1, 2009, pp.94–120. [25] Ellwood S., Pallier G., Snyder A. and Gallate J., the Incubation Effect: Hatching a Solution? Creativity Research Journal, Vol.21, No.1, 2009, pp.6–14. [26] Vul E., and Pashler H., Incubation benefits only after people have been misdirected, Memory and Cognition, Vol.35, No.4, 2007, pp.701–710. [27] Kaufman J.C., Sternberg R.J., (Eds.), the international handbook of creativity, Cambridge, Cambridge University Press, 2006. [28] Chamorro-Premuzic T., Creative Process, in: Kerr B., (Ed.), Encyclopedia of Giftedness, Creativity and Talent, Sage Publications, 2009, pp.188-191. مهدی نجفی خواه و همکاران 294 نشریه علمی پژوهشی فناوری آموزش، سال پنجم، جلد 5 ،شماره 4 ،تابستان 0931 [29] Christensen B.T., Creative Cognition: Analogy and Incubation, Department of Psychology, University of Aarhus, Denmark, 2005. [30] Liljedahl P., the AHA! Experience: mathematical contexts, pedagogical implications, unpublished doctoral dissertation, Simon Fraser University, Burnaby, British Columbia, Canada, 2004. [31] Kim K.H., Creative Problem Solving, in: Kerr B.,(Ed.), Encyclopedia of Giftedness, Creativity and Talent, Sage Publications, 2009, pp.188-191. [32] Zhong C., Dijksterhuis A., and Galinsky A.D., The merits of unconscious thought in creativity, Psychological Science, Vol.19, 2008, pp.912-918. [33] Seabrook R. and Dienes Z., Incubation in problem solving as a context effect, in: Alterman R. and Kirsh D., (Eds.), Proceedings of the 25th annual meeting of the Cognitive Science Society, Austin, TX: Cognitive Science Society, 2003, pp.1065–1069. [34] Davidson J.E., the Suddenness of Insight, in: Sternberg R.J. and Davidson J.E., (Eds.), the nature of insight, Cambridge, MA: MIT Press, 1995, pp.125–155. [35] National Council of Teachers of Mathematics, Principles and standards for school mathematics, Reston, VA: Author, 2000. [36] Silver E.A., Fostering Creativity through Instruction Rich in Mathematical Problem Solving and Problem Posing, in: ZDM, Zentralblatt für Didaktik der Mathematik, Vol. 29, No.3, 1995, pp.75–80. [37] Dubinsky E., Reflective abstraction in advanced mathematical thinking, in: Tall D., Advanced mathematical thinking, Kluwer Academic Publishers NewYork, 1991, pp.95- 123. [38] Thurston W.P., Three dimensional manifolds, Kleinian groups, and hyperbolic geometry, Bull Amer Math, Vol.6, No.3, 1982, pp.357-381.