Creativity of an Aha! Moment and Mathematics Education 900444744X, 9789004447448

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Creativity of an Aha! Moment and Mathematics Education

Creativity of an Aha! Moment and Mathematics Education Edited by

Bronislaw Czarnocha and William Baker

‫ءؘؗ؜ؘ؟‬ᄩؕ‫ءآائآ‬

Cover illustration: Artwork by Jose Garcia, Art Director, Hostos ѹѹ All chapters in this book have undergone peer review. Library of Congress Cataloging-in-Publication Data ƞƸƣƾᄘDžƞƽƹƺơƩƞᄕƽƺƹƫƾƶƞǂᄕƣƢƫƿƺƽᄙᄩƞƴƣƽᄕƫƶƶƫƞƸᄕƣƢƫƿƺƽᄙ Title: Creativity of an aha! moment and mathematics education / edited by Bronislaw Czarnocha and William Baker. ƣƾơƽƫƻƿƫƺƹᄘƣƫƢƣƹᄖƺƾƿƺƹᄘƽƫƶƶᄧƣƹƾƣᄕᄴᇾᇼᇾᇽᄵᄩ ƹơƶǀƢƣƾ bibliographical references and index.

ƢƣƹƿƫƤƫƣƽƾᄘ҂ѹѹ҄ᇾᇼᇾᇽᇼᇼህሂᇾᇿᄬƻƽƫƹƿᄭᄩ҂ѹѹ҄ᇾᇼᇾᇽᇼᇼህሂᇾሀᄬƣƟƺƺƴᄭᄩѿ҉Ѹ҄ ህሃሄህᇼᇼሀሀሀሃሀሀሄᄬƻƞƻƣƽƟƞơƴᄭᄩѿ҉Ѹ҄ህሃሄህᇼᇼሀᇿሃሂᇾᇾህᄬƩƞƽƢƟƞơƴᄭᄩѿ҉Ѹ҄ ህሃሄህᇼᇼሀሀሀሂሀᇿሀᄬƣƟƺƺƴᄭ Subjects: ҂ѹ҉Ѿ: Mathematics--Study and teaching. Classification: ҂ѹѹ҇ѷᇽሂᄙሃሀᇾᇼᇾᇽᄬƻƽƫƹƿᄭᄩ҂ѹѹ҇ѷᇽሂᄬƣƟƺƺƴᄭᄩѺѺѹ ሁᇽᇼᄙሃᇽᅟᅟƢơᇾᇿ ҂ѹƽƣơƺƽƢƞǁƞƫƶƞƟƶƣƞƿƩƿƿƻƾᄘᄧᄧƶơơƹᄙƶƺơᄙƨƺǁᄧᇾᇼᇾᇽᇼᇼህሂᇾᇿ ҂ѹƣƟƺƺƴƽƣơƺƽƢƞǁƞƫƶƞƟƶƣƞƿƩƿƿƻƾᄘᄧᄧƶơơƹᄙƶƺơᄙƨƺǁᄧᇾᇼᇾᇽᇼᇼህሂᇾሀ

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Contents ƽƣƤƞơƣᏻ‫ث؜‬ ơƴƹƺǂƶƣƢƨƣƸƣƹƿƾᏻ‫ث‬ ƫƾƿƺƤ ƫƨǀƽƣƾƞƹƢƞƟƶƣƾᏻ‫؜ث‬ ƺƿƣƾƺƹƺƹƿƽƫƟǀƿƺƽƾᏻ‫ة؜ث‬   ƹƿƽƺƢǀơƿƫƺƹᏻᇳ Bronislaw Czarnocha ᇳ  ƽƿƩǀƽƺƣƾƿƶƣƽᅷƾƫƾƺơƫƞƿƫƺƹƩƣƺƽDŽᏻᇵᇳ Bronislaw Czarnocha

҆ѷ҈Ҋማ Bisociation in the Classroom 2

Teaching-Research Analysis: The Constructivist Teaching Experiment as ƞƣƿƩƺƢƺƶƺƨDŽƺƤƣƞơƩƫƹƨᏻᇷᇻ Bronislaw Czarnocha

ᇵ  ƶƞƾƾƽƺƺƸ ƞơƫƶƫƿƞƿƫƺƹƺƤƩƞᄛƺƸƣƹƿ ƹƾƫƨƩƿƾᏻᇺᇲ Bronislaw Czarnocha and William Baker ᇶ  ƾƾƣƾƾƸƣƹƿƺƤƿƩƣƣƻƿƩƺƤƹƺǂƶƣƢƨƣơƼǀƫƽƣƢƢǀƽƫƹƨƞƹƩƞᄛ ƺƸƣƹƿ ƹƾƫƨƩƿᏻᇳᇳᇲ Bronislaw Czarnocha ᇷ  ƩƣƺƶƣƺƤƿƩƣƣƞơƩƣƽƫƹ ƞơƫƶƫƿƞƿƫƹƨƿƩƣƩƞᄛƺƸƣƹƿᏻᇳᇵᇻ William Baker 6

The Work of the Teaching-Research Team of the Bronx: ƽƣƞƿƫǁƫƿDŽᏻᇳᇸᇶ William Baker, Olen Dias, Edme Soho, Hector Soto and Lauren Wolf

‫؜ة‬

ѹ‫ئاءؘاءآ‬

҆ѷ҈Ҋሜ The Aha! Moment and Affect 7

Creativity in the Eyes of Students: Espoused and Enacted Beliefs in  ƞƿƩƣƸƞƿƫơƞƶƽƺưƣơƿƾᏻᇴᇲᇷ  Hannes Stoppel and Benjamin Rott

ᇺ  ǀƫƶƢƫƹƨƺƹƨᅟƣƽƸƣƞƹƫƹƨƫƹƞƿƩƣƸƞƿƫơƞƶƩƫƹƴƫƹƨᄘƩƞᄛƞƹƢ ƩᅟƩǀƩᄛᏻᇴᇴᇸ David Tall ᇻ  ƺƹƞƿƫǁƣƣƽƾƻƣơƿƫǁƣƺƹƩƞᄛƺƸƣƹƿƾᏻᇴᇸᇲ Gerald A. Goldin ᇳᇲ  ƶƶǀƸƫƹƞƿƫƹƨƩƞᄛƺƸƣƹƿƾƿƩƽƺǀƨƩƿƩƣƣƶƞƿƫƺƹƾƩƫƻƾƟƣƿǂƣƣƹ ƺƨƹƫƿƫƺƹᄕƤƤƣơƿᄕƞƹƢƺƹƞƿƫƺƹᏻᇴᇹᇳ Bronislaw Czarnocha and Peter Liljedahl

҆ѷ҈Ҋም Bisociation and Theories of Learning ᇳᇳ  ƫƾƺơƫƞƿƫƺƹᄕƽƣƞƿƫǁƫƿDŽᄕƞƹƢ ƹƿƣƽƫƺƽƫDžƞƿƫƺƹᏻᇵᇲᇳ William Baker ᇳᇴ  ǂƺƿƞƨƣƩƞƹƨƣƾƫƹƹƿƫơƫƻƞƿƫƺƹᄘƺƨƹƫƿƫǁƣƺǀƽơƣƾƺƤƩƞᄛ ƺƸƣƹƿƾᏻᇵᇴᇳ Ron Tzur ᇳᇵ ƩƞᄛƺƸƣƹƿƾᄕƫƾƺơƫƞƿƫƺƹᄕƞƹƢǀƶƿƫƤƺơƞƶƿƿƣƹƿƫƺƹᏻᇵᇶᇷ John Mason and Bronislaw Czarnocha

҆ѷ҈Ҋሞ Bisociativity from Without ᇳᇶ ƩƣƩƞᄛƺƸƣƹƿƞƿƿƩƣƣǃǀƾƺƤƫƹƢƞƹƢƽƞƫƹᏻᇵᇸᇷ Stephen R. Campbell ᇳᇷ  ƫƾƺơƫƞƿƫǁƣƿƽǀơƿǀƽƣƾᏻᇵᇻᇺ Hannes Stoppel and Bronislaw Czarnocha

ѹ‫ ئاءؘاءآ‬

ᇳᇸ ƺƹơƶǀƾƫƺƹƾᏻᇶᇴᇷ Bronislaw Czarnocha ᇳᇹ  ƺƶƶƣơƿƫƺƹƺƤƩƞᄛƺƸƣƹƿƾᏻᇶᇶᇳ Bronislaw Czarnocha ƶƺƾƾƞƽDŽᏻᇶᇸᇳ

ƹƢƣǃᏻᇶᇸᇸ

‫؜؜ة‬

Preface The presented volume introduces a new approach to mathematical creativity via the theory of bisociation formulated in the Act of Creation by Arthur ƺƣƾƿƶƣƽᄬᇳᇻᇸᇶᄭᄙƩƣƿƩƣƺƽDŽƺƤƟƫƾƺơƫƞƿƫƺƹƢƣƽƫǁƣƾƤƽƺƸƺƣƾƿƶƣƽᅷƾƞƹƞƶDŽƾƫƾƺƤ the insight phenomenon called Aha! moment or eureka experience conducted in the domain of humor, scientific discovery, and art. The wide familiarity of the phenomenon to the general student population as that sudden moment when everything becomes clear, the definition of creativity through the Aha! moment gives us the basis for the concept of creativity of and for all. Ʃƣ ǁƺƶǀƸƣ Ʃƞƾ ƿǂƺ ƺƟưƣơƿƫǁƣƾᄘ ᄬᇳᄭ ƿƺ ƫƹǁƣƾƿƫƨƞƿƣ ơƶƞƾƾƽƺƺƸ Ƥƞơƫƶƫƿƞƿƫƺƹᄕ emergence, and depth of knowledge assessment of an Aha! moment insight, ƞƹƢᄬᇴᄭƿƺƫƹǁƣƾƿƫƨƞƿƣƽƣƶƞƿƫƺƹƾƩƫƻƾƟƣƿǂƣƣƹƟƫƾƺơƫƞƿƫƺƹƿƩƣƺƽDŽǂƫƿƩƿƩƣƤƫƣƶƢ of creativity and learning theories. The summary of both investigations is presented in the Conclusions together with new research questions generated by the work of different authors in different chapters of the book. Two of the reƾǀƶƿƾƤƽƺƸƿƩƣƺƹơƶǀƾƫƺƹƾƾƿƞƹƢƺǀƿᄘᄬᇳᄭƤƺƽƸǀƶƞƿƫƺƹƺƤƿƩƣƟƞƾƫƾƤƺƽƩƞᄛƻƣƢagogy, whose aim is to introduce facilitation of bisociative creativity at every ƶƣǁƣƶƺƤƸƞƿƩƣƸƞƿƫơƞƶƢƣǁƣƶƺƻƸƣƹƿᄕƞƹƢᄬᇴᄭƤƺƽƸǀƶƞƿƫƺƹƺƤƿƩƣƟƞƾƫƾƤƺƽƿƩƣ theory of learning-through-creativity of Aha! moments.

ƿƫƾƾƿƽƫƴƫƹƨƿƺƹƺƿƣƿƩƣǂƣƞƶƿƩƺƤơƺƹƹƣơƿƫƺƹƾƟƫƾƺơƫƞƿƫƺƹƸƞƴƣƾǂƫƿƩƢƫƤferent domains and theories of learning presented in the volume: from the ƸƞƿƩƣƸƞƿƫơƾơƶƞƾƾƽƺƺƸᅷƾƩƞᄛƸƺƸƣƹƿƾƿƺƿƩƣƹƣǀƽƞƶƹƣƿǂƺƽƴƾƺƤƞƽƿƫƤƫơƫƞƶƫƹtelligence, through neuroscience to computer creativity and bisociative search engines, from the psychology of affect to the psychology of conation. Bisociation easily networks with constructivist theories of learning, which leads us to conjecture the possibility of creativity-based learning and teaching of mathematics. The editors of the volume hope to facilitate new moments of useful reflection among readers upon the fundamental role of creativity in learning and teaching mathematics in our classrooms.

Acknowledgements The editors of the book want to express deep appreciation to the whole writing team of the book for the commitment to deeply investigate different aspects of the Aha! moment and its role in the development of mathematical creativity by mathematics students. Thank you, Stephen Campbell, Olen Dias, Gerald Goldin, Peter Liljedahl, John Mason, Benjamin Rott, Edme Soho, Hector Soto, Hannes Stoppel, David Tall, Ron Tzur and Lauren Wolf. We want to express our gratitude to Mr. Jose Garcia, the Art Director at Hostos ѹѹ, who designed the cover, the copy editors Pamela and Ariela Fuchs, and Ed Hatton; to students and Peer Leaders of the mathematics remedial intermeƢƫƞƿƣƞƶƨƣƟƽƞơƶƞƾƾƺƤƻƽƫƹƨᇴᇲᇳᇹǂƩƺƩƞǁƣƻƞƽƿƫơƫƻƞƿƣƢƫƹƿƩƣƩǀƹƿƤƺƽƩƞᄛ Moments Teaching-Experiment and experienced five Aha! insights discussed in the book. ƣǂƞƹƿƿƺƺƤƤƣƽƿƩƞƹƴƾƿƺƽᄙƽǀƹƢƞƽƞƟƩǀᄬᇳᇻᇸᇳᅬᇴᇲᇳᇵᄭǂƩƺᄕƞƾƿƩƣƤƫƽƾƿ among us, had grasped the role of bisociativity in her experimental classroom of the teaching experiment Problem Solving in Remedial Arithmetic—A JumpƾƿƞƽƿƿƺƣƤƺƽƸᄬᇴᇲᇳᇲᄭᄙƣƞƽƣƤƺƶƶƺǂƫƹƨƿƩƣƻƞƿƩǂƞDŽơƽƣƞƿƣƢƟDŽƩƣƽƢƫƾơƺǁƣƽDŽᄙ

Figures and Tables Figures ѿᄙᇳ ѿᄙᇴ ᇳᄙᇳ ᇳᄙᇴ ᇳᄙᇵ ᇵᄙᇳ ᇶᄙᇳ ᇶᄙᇴ ᇶᄙᇵ 4.4 ᇹᄙᇳ ᇹᄙᇴ ᇹᄙᇵ ᇹᄙᇶ ᇺᄙᇳ ᇺᄙᇴ ᇺᄙᇵ ᇺᄙᇶ ᇺᄙᇷ ᇺᄙᇸ ᇺᄙᇹ ᇺᄙᇺ ᇺᄙᇻ ᇺᄙᇳᇲ ᇳᇲᄙᇳ ᇳᇴᄙᇳ ᇳᇴᄙᇴ ᇳᇵᄙᇳ

ƿƞƹƢƞƽƢƿƣƞơƩƫƹƨᅟƽƣƾƣƞƽơƩơDŽơƶƣᄬƺǀƿƣƽᄭƞƹƢơƽƣƞƿƫǁƣƿƣƞơƩƫƹƨᅟƽƣƾƣƞƽơƩơDŽơƶƣ ᄬƫƹƹƣƽᄭᄙᏻᇻ ƽƫƻƿDŽơƩᅟƟƞƾƣƢƾƿǀƢƣƹƿƞƾƾƫƨƹƸƣƹƿᄙᏻᇳᇻ ƺƣƾƿƶƣƽƿƽƫƻƿDŽơƩᄬƤƽƺƸƺƣƾƿƶƣƽᄕᇳᇻᇸᇶᄕƫƹƾƫƢƣơƺǁƣƽᄭᄙᏻᇵᇴ ƹƣǁƣƹƿƟƫƾƺơƫƞƿƣƢǂƫƿƩƿǂƺƻƶƞƹƣƾƺƤƿƩƫƹƴƫƹƨᄬƤƽƺƸƺƣƾƿƶƣƽᄕᇳᇻᇸᇶᄕƻᄙᇵᇷᄭᄙᏻᇶᇲ ƫƹƨƞƹƢDŽƞƹƨƾDŽƸƟƺƶᄙᏻᇶᇸ ƣƿƤƺƽơƺƹƤƣơƿƫƺƹƞƽDŽƟƺǃᄬƤƽƺƸƞƽƹƣƾᄕᇴᇲᇲᇲᄕƻᄙᇵᇷᄭᄙᏻᇺᇻ ƣǁƣƶƾƺƤƿƩƫƹƴƫƹƨƫƹƶƺƺƸƿƞǃƺƹƺƸDŽƞƹƢƣƟƟᅷƾƢƣƻƿƩƺƤƴƹƺǂƶƣƢƨƣᄬơƽƣƞƿƣƢ ƟDŽƣƟƟƫƣƣƽƴƫƹƾᄕᇴᇲᇲᇺᄭᄙᏻᇳᇳᇸ ƫƸᅷƾƻƽƺƟƶƣƸᄙᏻᇳᇴᇶ ƿƞƨƣƾƫƹƿƩƣƤƫƽƿƽƣƣƢƣǁƣƶƺƻƸƣƹƿᄙᏻᇳᇴᇸ Algebra tile model for 2xሕናᇵǃናᇳᄙᏻᇳᇵᇵ ƽƺơƣƣƢƫƹƨƺƤƿƩƣơƺǀƽƾƣƾᄙᏻᇴᇲᇻ ǃơƣƽƻƿƺƤƻƽƺưƣơƿƣƶƞƟƺƽƞƿƫƺƹƺƤᄴᇳᄵᄬƞᄭƺƤƿDŽƻƣᄬᇳᄭƞƹƢᄴᇴᄵᄬƟᄭƺƤƿDŽƻƣ ᄬᇴᄭᄙᏻᇴᇳᇳ ƽƫƢƨƫƹƨƨƽƞƻƩƾƟDŽơƺƹơƣƻƿƾᄬƤƽƺƸƣƽƿƩƺƶƢᄕᇴᇲᇳᇴᄕƻᄙᇶᄭᄙᏻᇴᇳᇴ ƩƞƹƨƣƾƫƹƿƩƣƾƿƞƿǀƾᄬᄕᄭƤƽƺƸ ƹƿƣƽǁƫƣǂƿƺ ƹƿƣƽǁƫƣǂᄬƾƣƣƞƟƶƣᇹᄙᇴƤƺƽ the frequencies at both t቞ and tበᄭᄙᏻᇴᇳᇻ ƩƣƟƽƞƫƹƤƽƺƸƞƟƺǁƣᄙᏻᇴᇴᇺ ơƽƺƾƾᅟƾƣơƿƫƺƹƞƶǁƫƣǂƺƤƿƩƣƟƽƞƫƹƤƽƺƸƿƩƣƶƣƤƿƾƫƢƣᄙᏻᇴᇴᇻ ƶƺƿƿƫƹƨƿƩƣƾƶƺƻƣƤǀƹơƿƫƺƹƾƤƺƽƾƫƹᄬxᄭƞƹƢơƺƾᄬxᄭᄙᏻᇴᇵᇴ ǀƶƿǀƽƞƶƿƽƞƹƾƫƿƫƺƹƟƣƿǂƣƣƹƢƫƤƤƣƽƣƹƿơƺƸƸǀƹƫƿƫƣƾᄬƤƽƺƸƞƶƶᄕᇴᇲᇳᇻƞᄭᄙᏻᇴᇵᇷ ƣƽƾƺƹƞƶƿƽƞƹƾƫƿƫƺƹƟƣƿǂƣƣƹƢƫƤƤƣƽƣƹƿƸƞƿƩƣƸƞƿƫơƞƶơƺƹƿƣǃƿƾᄬƤƽƺƸƞƶƶᄕ ᇴᇲᇳᇻƞᄭᄙᏻᇴᇵᇸ ƸƺƿƫƺƹƞƶƽƣƞơƿƫƺƹƾƿƺƨƺƞƶƾƞƹƢƞƹƿƫᅟƨƺƞƶƾᄬƤƽƺƸƞƶƶᄕᇴᇲᇳᇵᄕƻᄙᇳᇴᇲᄭᄙᏻᇴᇵᇺ ƞƶơǀƶƞƿƫƹƨᇵƿƫƸƣƾᇴᇵᄕǁƫƾǀƞƶƶDŽƞƹƢƾDŽƸƟƺƶƫơƞƶƶDŽᄙᏻᇴᇵᇻ ƩƣƿƩƽƣƣǂƺƽƶƢƾƤƽƞƸƣǂƺƽƴᄙᏻᇴᇶᇷ

ƹƤƫƹƫƿƣƸƞƨƹƫƤƫơƞƿƫƺƹƺƤƞƶƺơƞƶƶDŽƾƿƽƞƫƨƩƿƨƽƞƻƩᄙᏻᇴᇶᇹ ƺƽƣƾƺƻƩƫƾƿƫơƞƿƣƢƣƸƟƺƢƫƸƣƹƿƞƹƢƾDŽƸƟƺƶƫƾƸᄙᏻᇴᇶᇺ ƫƸᅷƾƻƽƺƟƶƣƸƿƺƾƺƶǁƣᄙᏻᇴᇹᇳ ƞƽᄬƤƫǁƣƣƼǀƞƶƻƞƽƿƾᄭᄕƞƽᄬƨƽƞDŽƻƫƣơƣᄭᄕƞƹƢƞƽᄬƨƽƞDŽƻƫƣơƣƫƿƣƽƞƿƣƢƿƩƽƣƣ ƿƫƸƣƾᄭᄙᏻᇵᇴᇵ ƩƫƾƫƾᇵᄧᇷƺƤƞơƩƺơƺƶƞƿƣƟƞƽᄙƽƺƢǀơƣƿƩƣǂƩƺƶƣƟƞƽƞƹƢƣǃƻƶƞƫƹDŽƺǀƽ ƞƹƾǂƣƽᄙᏻᇵᇴᇶ ƣƞƽơƩƫƹƨƤƺƽƞƽƣƶƞƿƫƺƹƾƩƫƻᄙᏻᇵᇷᇲ

‫؜؜ث‬

Ѽ‫ؗءؔئؘإبؚ؜‬Ҋؔؕ‫ئؘ؟‬

ᇳᇶᄙᇳ ƞƨƫƿƿƞƶᄬƶƣƤƿᄭƞƹƢƸƫƢƾƞƨƫƿƿƞƶᄬƽƫƨƩƿᄭǁƫƣǂƾƺƤƞƢƫƾƾƣơƿƣƢƩǀƸƞƹƟƽƞƫƹᄙᏻᇵᇸᇸ ᇳᇶᄙᇴ ƺƽƿƫơƞƶơƺƶǀƸƹƾơƺƹƾƫƾƿƫƹƨƺƤƻDŽƽƞƸƫƢƞƶơƣƶƶƾᄬƤƽƺƸƽƞDŽᄕᇳᇻᇳᇺᄕƻᄙᇺᇶᇸᄭᄙᏻᇵᇸᇹ ᇳᇶᄙᇵ ƞƾƾƫǁƣƸDŽƣƶƫƹƞƿƣƢƞǃƺƹƫƹƿƣƽơƺƹƹƣơƿƫǁƫƿDŽƟƣƿǂƣƣƹƢƫƤƤƣƽƣƹƿƟƽƞƫƹ ƽƣƨƫƺƹƾᄙᏻᇵᇸᇹ ᇳᇶᄙᇶ DŽƣƶƫƹƞƿƣƢƞǃƺƹƿƽƞơƿƾƽƣǁƣƞƶƣƢǀƾƫƹƨƢƫƤƤǀƾƫƺƹƿƣƹƾƺƽƫƸƞƨƫƹƨᄬơƺǀƽƿƣƾDŽ ƺƤƺƽƢƺƹƫƹƢƶƸƞƹƹƞƿƿƩƣơƫƣƹƿƫƤƫơƺƸƻǀƿƫƹƨƞƹƢ Ƹƞƨƫƹƨ ƹƾƿƫƿǀƿƣᄕ ƹƫǁƣƽƾƫƿDŽƺƤƿƞƩᄕƞƹƢƹƢƽƣǂƶƣǃƞƹƢƣƽᄕᄙᄙƣơƴƞƟƺƽƞƿƺƽDŽ Ƥƺƽ ǀƹơƿƫƺƹƞƶƽƞƫƹ ƸƞƨƫƹƨƞƹƢƣƩƞǁƫƺƽᄕƹƫǁƣƽƾƫƿDŽƺƤƫƾơƺƹƾƫƹᅟ ƞƢƫƾƺƹᄭᄙᏻᇵᇸᇺ ᇳᇶᄙᇷ ƽƺƢƸƞƹƹƞƽƣƞƾƟƞƾƣƢƺƹơDŽƿƺƞƽơƩƫƿƣơƿǀƽƞƶƢƫƤƤƣƽƣƹơƣƾƺƤƹƣǀƽƺƹƞƶ ƞƾƾƣƸƟƶƞƨƣƾᄬƤƽƺƸƞƹƾƺƹᄕᇳᇻᇴᇲᄕƻᄙᇴᇺᇺᄭᄙᏻᇵᇸᇺ ᇳᇶᄙᇸ ƫǁƣƾƿƞƨƣƾƺƤƿƩƣơƽƣƞƿƫǁƣƻƽƺơƣƾƾᄬƤƽƺƸƞƢƶƣƽᅟƸƫƿƩᄕᇴᇲᇳᇷᄕƻᄙᇵᇶᇸᄭᄙᏻᇵᇹᇴ ᇳᇶᄙᇹ ƩƣơƽƣƞƿƫǁƣƻƽƺơƣƾƾƞƾᅸƨƽƞƢƣƾƺƤơƺƹƾơƫƺǀƾƹƣƾƾᅺᄬƤƽƺƸƞƢƶƣƽᅟƸƫƿƩᄕᇴᇲᇳᇷᄕ ƻᄙᇵᇶᇺᄭᄙᏻᇵᇹᇴ ᇳᇶᄙᇺ ǀƸƞƹѻѻѽƤƽƣƼǀƣƹơDŽƟƞƹƢƾᄙᏻᇵᇹᇶ ᇳᇶᄙᇻ ƢƢƟƞƶƶƾƿƫƸǀƶǀƾƾƶƫƢƣƾᄬƤƽƺƸƣƩƞƣƹƣƣƿƞƶᄙᄕᇴᇲᇲᇸᄭᄙᏻᇵᇺᇳ ᇳᇶᄙᇳᇲ ƞƻƿǀƽƫƹƨƞƹƩƞᄛƸƺƸƣƹƿǁƫƞƿƩƣƫƹƿƣƨƽƞƿƫƺƹƞƹƢƿƫƸƣᅟƾDŽƹơƩƽƺƹƫDžƞƿƫƺƹƺƤ ƻƩDŽƾƫƺƶƺƨƫơƞƶƞƹƢƟƣƩƞǁƫƺƽƞƶƺƟƾƣƽǁƞƿƫƺƹƾᄙᏻᇵᇺᇴ ᇳᇶᄙᇳᇳ ƣƿƞƫƶƣƢƣDŽƣƿƽƞơƴƫƹƨƺƤƿƩƣƩƞᄛƸƺƸƣƹƿᄙᏻᇵᇺᇴ ᇳᇶᄙᇳᇴ ƫƸƣƾDŽƹơƩƽƺƹƫDžƫƹƨƻƩDŽƾƫƺƶƺƨƫơƞƶƞƹƢƟƣƩƞǁƫƺƽƞƶƢƞƿƞƾƣƿƾᄙᏻᇵᇺᇶ ᇳᇶᄙᇳᇵ ƣƤƿᄘѻѻѽƫƹƾƫƨƩƿƣƤƤƣơƿᄬƤƽƺƸǀƹƨᅟƣƣƸƞƹƣƿƞƶᄙᄕᇴᇲᇲᇶᄕƻᄙᇷᇲᇷᄭᄙƫƨƩƿᄘ An independent analysis component of our ѻѻѽ data corresponds to a phenomenon they have identified with the ѻѻѽƫƹƾƫƨƩƿƣƤƤƣơƿᄬƤƽƺƸƺǀƹƫƺƾѵ ƣƣƸƞƹᄕᇴᇲᇲᇻᄕƻᄙᇴᇳᇳᄭᄙᏻᇵᇺᇷ ᇳᇶᄙᇳᇶ  ƹƢƣƻƣƹƢƣƹƿơƺƸƻƺƹƣƹƿƾƺƤƟƽƞƫƹƞơƿƫǁƫƿDŽƞƾƾƺơƫƞƿƣƢǂƫƿƩƽƣƞƾƺƹƫƹƨᄕ ơƺƸƻƽƣƩƣƹƾƫƺƹᄕƞƹƢƫƹƾƫƨƩƿᄙᏻᇵᇺᇷ ᇳᇶᄙᇳᇷ ƽƞƫƹƞơƿƫǁƫƿDŽơƺǀƻƶƣƢǂƫƿƩƣDŽƣƿƽƞơƴƫƹƨƿƣƶƶƾƿƩƣƾƿƺƽDŽᄬƾƣƣ ƫƨǀƽƣƾᇳᇶᄙᇳᇳᅬ ᇳᇶᄙᇳᇶᄭᄙᏻᇵᇺᇹ ᇳᇷᄙᇳ ᅸƞƶƴƫƹƨƞƽƺǀƹƢƿƩƣƸƞƿƽƫǃᅺᄙᏻᇶᇲᇴ ᇳᇷᄙᇴ ƹƣǁƣƹƿƟƫƾƺơƫƞƿƣƢǂƫƿƩƿǂƺƻƶƞƹƣƾƺƤƿƩƫƹƴƫƹƨᄬƤƽƺƸƺƣƾƿƶƣƽᄕᇳᇻᇸᇶᄕƻᄙᇵᇷᄭᄙᏻᇶᇲᇶ ᇳᇷᄙᇵ ƩǀƸƺƽƺǀƾƹƞƽƽƞƿƫǁƣƺƾơƫƶƶƞƿƫƹƨƟƣƿǂƣƣƹƿǂƺƻƶƞƹƣƾƺƤƽƣƤƣƽƣƹơƣᄬƤƽƺƸ ƺƣƾƿƶƣƽᄕᇳᇻᇸᇶᄕƻᄙᇵᇹᄭᄙᏻᇶᇲᇷ ᇳᇷᄙᇶ ƺƸƟƫƹƞƿƫƺƹƺƤƸƞƿƽƫơƣƾᄙᏻᇶᇲᇸ ᇳᇷᄙᇷ ƺƹƾƿƽǀơƿƫƺƹƺƤƞƸƞƿƽƫǃᄙᏻᇶᇳᇲ ᇳᇷᄙᇸ ƺƹƾƿƽǀơƿƫƺƹƫƹƞǀƹƫƺƹƺƤƸƞƿƽƫơƣƾᄙᏻᇶᇳᇳ ᇳᇷᄙᇹ ƽƫƢƨƫƹƨƟDŽƞƾƫƹƨƶƣơƺƹơƣƻƿᄬƤƽƺƸʢƿƿƣƽѵƣƽƿƩƺƶƢᄕᇴᇲᇳᇴᄕƻᄙᇶᇷᄭᄙᏻᇶᇳᇷ ᇳᇷᄙᇺ ƶƨƣƟƽƞƫơơƺƹưǀƨƞƿƫƺƹƞƾƿƩƣƾƫƹƨƶƣƟƽƫƢƨƫƹƨơƺƹơƣƻƿᄙᏻᇶᇳᇸ ᇳᇷᄙᇻ ƽƫƢƨƫƹƨƨƽƞƻƩƾƟDŽơƺƹơƣƻƿƾᄬƤƽƺƸƣƽƿƩƺƶƢᄕᇴᇲᇳᇴᄕƻᄙᇶᄭᄙᏻᇶᇳᇹ ᇳᇷᄙᇳᇲ ƽƫƢƨƫƹƨƨƽƞƻƩƺƤƿƩƽƣƣƞƶƨƣƟƽƞƫơƢƺƸƞƫƹƾᄙᏻᇶᇳᇺ ᇳᇷᄙᇳᇳ ƽƫƢƨƫƹƨƨƽƞƻƩƾƟDŽƾƿƽǀơƿǀƽƞƶƾƫƸƫƶƞƽƫƿDŽᄬƤƽƺƸƣƽƿƩƺƶƢᄕᇴᇲᇳᇴᄕƻᄙᇷᄭᄙᏻᇶᇳᇻ ᇳᇷᄙᇳᇴ ƿƽǀơƿǀƽƞƶƾƫƸƫƶƞƽƫƿDŽƺƤƣƶƣơƿƽƺƸƞƨƹƣƿƫƾƸƞƹƢƨƣƹƣƽƞƶƽƣƶƞƿƫǁƫƿDŽᄙᏻᇶᇴᇳ

Ѽ‫ؗءؔئؘإبؚ؜‬Ҋؔؕ‫ ئؘ؟‬ ᇳᇹᄙᇳ ᇳᇹᄙᇴ ᇳᇹᄙᇵ ᇳᇹᄙᇶ ᇳᇹᄙᇷ

‫؜؜؜ث‬

ƿƞƨƣƾƺƤƿƩƣ ƫƽƽƣƣƞƾƾƫƨƹƸƣƹƿᄙᏻᇶᇶᇹ ƶƨƣƟƽƞƿƫƶƣƸƺƢƣƶƤƺƽᇴǃሿናᇵǃናᇳᄙᏻᇶᇷᇷ ƹƣƶƶᅷƾƞǂᄘƹaƾƫƹ஘a = nbƾƫƹ஘bᄙᏻᇶᇷᇹ ஘aኙ஘bና஘cᄙᏻᇶᇷᇹ ஘bኙᄧᇴᄙᏻᇶᇷᇺ

Tables ѿᄙᇳ ᇶᄙᇳ ᇹᄙᇳ ᇹᄙᇴ ᇳᇲᄙᇳ ᇳᇴᄙᇳ ᇳᇵᄙᇳ ᇳᇷᄙᇳ ᇳᇷᄙᇴ ᇳᇷᄙᇵ ᇳᇹᄙᇳ

ƽƞƹƾƶƞƿƫƺƹƺƤƿƣƽƸƾƟƣƿǂƣƣƹƟƫƾƺơƫƞƿƫƺƹƿƩƣƺƽDŽƞƹƢƫƞƨƣƿƫƞƹƿƩƣƺƽƫƣƾᄙᏻᇴᇴ ǀƸƣƽƫơƞƶƽƣƶƞƿƫƺƹƾƩƫƻƾƢƫƾơƺǁƣƽƣƢƞƿƣƞơƩƾƿƞƨƣƺƤƿƩƣ ƫƽƽƣƣƩƞᄛ ƸƺƸƣƹƿᄙᏻᇳᇴᇹ  ƹƿƣƽƻƽƣƿƞƿƫƺƹƺƤƿƩƣƾƿƞƿǀƾƣƾᄬᄕᄭƺƤƣƹƞơƿƣƢơƽƣƞƿƫǁƫƿDŽᄬᄭƞƹƢƣƾƻƺǀƾƣƢ ơƽƣƞƿƫǁƫƿDŽᄬᄭᄙᏻᇴᇳᇸ ƟƾƺƶǀƿƣƤƽƣƼǀƣƹơƫƣƾƺƤƾƿƞƿǀƾᄙᏻᇴᇳᇹ ƹƞƶDŽƾƫƾƺƤƫƸᅷƾƩƞᄛƸƺƸƣƹƿᄙᏻᇴᇹᇸ ƞƿƣƨƺƽƫƣƾƺƤƩƞᄛƸƺƸƣƹƿƾƟƞƾƣƢƺƹƣƤᅚ ᄙᏻᇵᇵᇸ ƩƣƿƞƟƶƣƺƤƹƞƹƢᄬƹᄭᄙᏻᇵᇶᇹ ƺƸƻƞƽƫƾƺƹƺƤƞƾƾƺơƫƞƿƫƺƹƞƹƢƟƫƾƺơƫƞƿƫƺƹᄙᏻᇶᇲᇴ ƾƾƺơƫƞƿƫƺƹƾƟƣƿǂƣƣƹƿƩƣơƶƞƾƾƫƤƫơƞƿƫƺƹƾƺƤƺƢƣƹƞƹƢƺƣƾƿƶƣƽᄙᏻᇶᇳᇴ ƾƾƺơƫƞƿƫƺƹƾƟƣƿǂƣƣƹƾƿǀƢƣƹƿƾᅷƻƣƽơƣƻƿƫƺƹƺƤơƽƣƞƿƫǁƫƿDŽᄕƿƩƣƣǃƿƣƹƾƫƺƹƞƹƢ ơƺƸƟƫƹƞƿƫƺƹƺƤƸƞƿƽƫơƣƾᄕƞƹƢƿƩƣơƶƞƾƾƫƤƫơƞƿƫƺƹƾƺƤƺƢƣƹƞƹƢƺƣƾƿƶƣƽᄙᏻᇶᇳᇵ ǀƸƣƽƫơƞƶƽƣƶƞƿƫƺƹƾƩƫƻƾƢƫƾơƺǁƣƽƣƢƞƿƣƞơƩƾƿƞƨƣƺƤƿƩƣ ƫƽƽƣƣƩƞᄛ ƸƺƸƣƹƿᄙᏻᇶᇶᇹ

Notes on Contributors William Baker ƺƤ ƺƾƿƺƾ ƺƸƸǀƹƫƿDŽ ƺƶƶƣƨƣ ᄬƫƿDŽ ƹƫǁƣƽƾƫƿDŽ ƺƤ ƣǂ ƺƽƴᄭ ƿƣƞơƩƣƾ ƞ Ƥǀƶƶ range of mathematics course, from developmental mathematics through to calculus sequences. As a member of the teaching-research team, he is interested in how educational research can inspire creativity in his teaching methodology and his students. He believes that a guided-discovery method should be used flexibly to encourage participation and to give students a feel for the beauty of mathematics. As a teacher he struggles to balance the need to cover the curricula yet leave enough time and space for students to reason with and discover mathematics. Stephen R. Campbell is an associate professor in the Faculty of Education at Simon Fraser University, British Columbia. His scholarly focus is on the historical and psychological development of mathematical thinking from an embodied perspective informed ƟDŽ ƞƹƿᄕ ǀƾƾƣƽƶᄕ ƞƹƢ ƣƽƶƣƞǀᅟƺƹƿDŽᄙ ƫƾ ƽƣƾƣƞƽơƩ ƫƹơƺƽƻƺƽƞƿƣƾ ƸƣƿƩƺƢƾ ƺƤ psychophysics and cognitive neuroscience as a means for operationalizing affective and cognitive models of math anxiety and concept formation. Bronislaw Czarnocha ƺƤ ƺƾƿƺƾƺƸƸǀƹƫƿDŽƺƶƶƣƨƣᄬƫƿDŽƹƫǁƣƽƾƫƿDŽƺƤƣǂƺƽƴᄭƫƾƞƿƣƞơƩƣƽᅟƽƣsearcher and a quantum physicist turned mathematics educator. His present ƫƹƿƣƽƣƾƿƫƾƫƹƢƣǁƣƶƺƻƫƹƨƿƣƞơƩƫƹƨᅟƽƣƾƣƞƽơƩᄬҊ҈ᄭƫƹƿƺƞƤǀƶƶᅟƤƶƣƢƨƣƢƫƹƾƿƽǀơƿƫƺƹal methodology in the context of Aha! pedagogy as well as in understanding and expanding the application of bisociative creativity to various domains of intellectual and artistic endeavors. He was co-recipient of a grant from the Naƿƫƺƹƞƶơƫƣƹơƣ ƺǀƹƢƞƿƫƺƹᄬ҄҉ѼᄭƿƺƢƣǁƣƶƺƻƹƣǂƞƻƻƽƺƞơƩƣƾƿƺƫƹơƺƽƻƺƽƞƿƫƹƨ indivisibles into calculus instruction. He was also awarded a Socrates grant ᄬƹƺǂơƞƶƶƣƢ ƽƞƾƸǀƾᄭƤǀƹƢƣƢƟDŽƿƩƣ ǀƽƺƻƣƞƹƹƫƺƹƤƺƽƫƹƿƣƽƹƞƿƫƺƹƞƶƻƽƺƤƣƾsional development of teacher-researchers. Olen Dias is a professor of mathematics at Hostos Community College of the City UniǁƣƽƾƫƿDŽƺƤƣǂƺƽƴᄙƩƣƻƞƽƿƫơƫƻƞƿƣƢƫƹƿǂƺѹҋ҄ҏ-wide developmental mathematic projects and likes to teach from developmental mathematics through to calculus sequence. She was a co-founder of supplemental instruction at the college and is a member of Teaching Research Team of the Bronx. She is involved in creating co-req. model course, flipped classes, and development of

҄‫ءآئؘاآ‬ѹ‫ ئإآابؕ؜إاءآ‬

‫ةث‬

ƻƣƹ Ƣǀơƞƿƫƺƹƞƶƣƾƺǀƽơƣƾᄬ҅ѻ҈ᄭƸƞƿƣƽƫƞƶƤƺƽƻƽƣᅟơƞƶơǀƶǀƾᄙƽᄙƫƞƾƫƹơƺƽporates different pedagogical approaches in her classes for improvement of teaching and learning. Gerald A. Goldin is a distinguished professor of mathematics education, mathematics, and ƻƩDŽƾƫơƾƞƿǀƿƨƣƽƾƹƫǁƣƽƾƫƿDŽᄕƣǂƣƽƾƣDŽᄙ ƫƾƽƣƾƣƞƽơƩƫƹơƶǀƢƣƾƺǁƣƽᇴᇲᇲƻǀƟlications embracing these three fields. He directed several major ҉Ҋѻ҃ educaƿƫƺƹƻƽƺưƣơƿƾᄕƫƹơƶǀƢƫƹƨƣǂƣƽƾƣDŽᅷƾƿƞƿƣǂƫƢƣDŽƾƿƣƸƫơ ƹƫƿƫƞƿƫǁƣƞƹƢƣƿƽƺƞƿƩᄘƩƣƣƹƿƣƽƤƺƽƞƿƩƣƸƞƿƫơƾƫƹƸƣƽƫơƞᅷƾƫƿƫƣƾᄙ ƣƫƾƞƽƣơƫƻƫƣƹƿƺƤƿƩƣ Humboldt Research Prize for work in quantum physics. His current educational research focuses on affect and engagement in mathematical learning and ƻƽƺƟƶƣƸᅟƾƺƶǁƫƹƨᄙ ƣƞƶƾƺơƺᅟƞǀƿƩƺƽƣƢᄬǂƫƿƩƣƹƹƫƤƣƽᄙƺƩƣƽƿDŽᄭƿǂƺƫƶƶǀƾƿƽƞƿed storybooks for young children, The Mouse of GoldᄬᇴᇲᇲᇸᄭƞƹƢThe Fierce and Gentle WolfᄬᇴᇲᇳᇳᄭᄕƻǀƟƶƫƾƩƣƢƫƹơƺƿƶƞƹƢƟDŽƣƽƞƤƫƹƞƽƣƾƾᄙ Peter Liljedahl is a professor of mathematics education in the Faculty of Education at Simon

ƽƞƾƣƽ ƹƫǁƣƽƾƫƿDŽᄕ ƽƫƿƫƾƩ ƺƶǀƸƟƫƞᄙ ƣ ƫƾ ƿƩƣ ƤƺƽƸƣƽ ƻƽƣƾƫƢƣƹƿ ƺƤ ƿƩƣ ƹƿƣƽƹƞƿƫƺƹƞƶ ƽƺǀƻ Ƥƺƽ ƿƩƣ ƾDŽơƩƺƶƺƨDŽ ƺƤ ƞƿƩƣƸƞƿƫơƾ Ƣǀơƞƿƫƺƹ ᄬ҆҃ѻᄭ ƞƹƢ ƫƾ the current president of the Canadian Mathematics Education Study Group ᄬѹ҃ѻ҉ѽᄭᄕƞƾǂƣƶƶƞƾƞƾƣƹƫƺƽƣƢƫƿƺƽƤƺƽƿƩƣInternational Journal of Science and Mathematics Education ᄬѿҀ҉҃ѻᄭᄙ ƣƿƣƽ ƫƾ ƞ ƤƺƽƸƣƽ ƩƫƨƩ ƾơƩƺƺƶ ƸƞƿƩƣƸƞƿƫơƾ teacher who has kept his research interest and activities close to the classroom. He consults regularly with teachers, schools, school districts, and ministries of education on issues of teaching and learning, assessment, and numeracy. John Mason is professor emeritus at the Open University and Honorary Research Fellow ƞƿƿƩƣƹƫǁƣƽƾƫƿDŽƺƤǃƤƺƽƢᄙ ƣƾƻƣƹƿᇶᇲDŽƣƞƽƾƞƿƿƩƣƻƣƹƹƫǁƣƽƾƫƿDŽǂƽƫƿƫƹƨ distance-learning courses in mathematics and in mathematics education. He ƫƾƿƩƣƞǀƿƩƺƽƺƤƸƺƽƣƿƩƞƹᇴᇷƟƺƺƴƾƞƹƢƹǀƸƣƽƺǀƾƻƞƻƣƽƾƺƹƿƣƞơƩƫƹƨƞƹƢ learning mathematics. He engages in mathematical thinking with an eye to locating and refining useful tasks for bringing to the surface the human powers that underpin mathematical thinking, and the ubiquitous themes of mathematics. He is particularly interested in the role played in learning mathematics by mental imagery and attention. Benjamin Rott ƢƫƢƩƫƾƩƫƹᇴᇲᇳᇴƫƹ ƞƹƺǁƣƽᄕƣƽƸƞƹDŽᄕǂƞƾƞƻƺƾƿƢƺơƞƿƿƩƣƹƫǁƣƽƾƫƿDŽƺƤ Ƣǀơƞƿƫƺƹ ƽƣƫƟǀƽƨǀƹƿƫƶᇴᇲᇳᇶƞƹƢƞƹƞƾƾƫƾƿƞƹƿƻƽƺƤƣƾƾƺƽƞƿƿƩƣƹƫǁƣƽƾƫƿDŽƺƤ

‫؜ةث‬

҄‫ءآئؘاآ‬ѹ‫ئإآابؕ؜إاءآ‬

ǀƫƾƟǀƽƨᅟ ƾƾƣƹǀƹƿƫƶᇴᇲᇳᇹᄙ ƣƫƾƹƺǂƞƤǀƶƶƻƽƺƤƣƾƾƺƽƤƺƽƸƞƿƩƣƸƞƿƫơƾƣƢǀơƞtion at the University of Cologne, Germany. His research interests are matheƸƞƿƫơƞƶƻƽƺƟƶƣƸᅟƾƺƶǁƫƹƨᄕƟƣƶƫƣƤƾᄕƞƹƢƨƫƤƿƣƢƹƣƾƾᄬƫƹơƶǀƢƫƹƨơƽƣƞƿƫǁƫƿDŽᄭᄙ Edme Soho is an assistant professor in the Department of Mathematics at Hostos ComƸǀƹƫƿDŽƺƶƶƣƨƣᄬƫƿDŽƹƫǁƣƽƾƫƿDŽƺƤƣǂƺƽƴᄭᄙ ƣƫƾƞƹƞƻƻƶƫƣƢƸƞƿƩƣƸƞƿƫơƫƞƹ with experience using and instructing others on the use of mathematical statistical and computational modeling tools in multiple fields. He enjoys collaborative, interdisciplinary research with professionals of diverse backgrounds. His primary research interests lie in mathematical modeling, dynamical systems, dynamics of infectious diseases, population dynamics, epidemiology, and immunology. Hector Soto of the Public Policy and Law Unit of the Behavioral & Social Sciences DepartƸƣƹƿƞƿ ƺƾƿƺƾƺƸƸǀƹƫƿDŽƺƶƶƣƨƣᄬƫƿDŽƹƫǁƣƽƾƫƿDŽƺƤƣǂƺƽƴᄭƩƞƾƟƣƣƹ referred to by many community activists as a true educational warrior because of his lifetime commitment to the service of communities of color and progressives. He is a founder of the Center for Neighborhood Leadership and Community Learning Partnership, a national organization that promotes the development of community change degree programs. Prof. Soto has worked to expand the debate in intellectual circles over civic engagement. He is also a member of the Hostos Community College research team, studying creativity in education to improve academic interest and motivation among Hispanic, minority and female students in the criminal justice system and how it affects their lives. Hannes Stoppel ƢƫƢƩƫƾƩƫƹᇴᇲᇳᇻƫƹ˦ƹƾƿƣƽᄕƣƽƸƞƹDŽᄙ ƽƺƸᇴᇲᇲᇲƿƺᇴᇲᇳᇵƩƣǂƞƾƞƿƣƞơƩƣƽƺƤ mathematics, physics, and computer science at the Max-Planck-Gymnasium ƫƹƣƶƾƣƹƴƫƽơƩƣƹᄙ ƽƺƸᇴᇲᇳᇵƿƺᇴᇲᇳᇺƩƣǂƞƾƞƿƣƞơƩƣƽƞƿƿƩƣƫƹƾƿƫƿǀƿƣƤƺƽƸƞƿƩƣƸƞƿƫơƾƞƹƢƫƿƾƣƢǀơƞƿƫƺƹƞƿƿƩƣƹƫǁƣƽƾƫƿDŽƺƤ˦ƹƾƿƣƽƞƹƢƫƾƹƺǂƿƣƞơƩƣƽƺƤ mathematics, physics and computer science at the Max-Planck-Gymnasium in Gelsenkirchen. He attends to the provision of scholarships of outstanding students. His research interests are beliefs and creativity. David Tall is emeritus professor of Mathematical Thinking at the University of Warwick in Coventry and visiting professor at Loughborough University London. His publications include How Humans Learn to Think Mathematicallyᄬѹҋ҆ᄕᇴᇲᇳᇵᄭᄕ

҄‫ءآئؘاآ‬ѹ‫ ئإآابؕ؜إاءآ‬

‫؜؜ةث‬

Advanced Mathematical ThinkingᄬƶǀǂƣƽᄕᇳᇻᇻᇳᄭᄕƿƩƽƣƣơƶƞƾƾƫơƿƣǃƿƟƺƺƴƾǂƫƿƩ

ƞƹ ƿƣǂƞƽƿ ƺƹ Foundations ᄬᇴƹƢ ƣƢᄙᄕ ҅ҋ҆ᄕ ᇴᇲᇳᇶᄭᄕ Algebraic Number Theory ᄬᇶƿƩƣƢᄙᄕѹ҈ѹƽƣƾƾᄕᇴᇲᇳᇷᄭƞƹƢComplex AnalysisᄬᇴƹƢƣƢᄙᄕѹҋ҆ᄕᇴᇲᇳᇺᄭᄕƾƺƤƿǂƞƽƣ ƤƺƽƽƞƻƩƫơƞƶơǀƶǀƾƞƹƢƺǁƣƽᇴᇷᇲƻƞƻƣƽƾƫƹƸƞƿƩƣƸƞƿƫơƾƞƹƢƸƞƿƩƣƸƞƿƫơƾ education. His recent interests are in developing a comprehensive theory of long-term growth of mathematical thinking including cognitive, affective, historical, cultural, and technological aspects. Ron Tzur ƫƾƞƻƽƺƤƣƾƾƺƽƺƤƸƞƿƩƣƸƞƿƫơƾƣƢǀơƞƿƫƺƹƞƿƿƩƣƹƫǁƣƽƾƫƿDŽƺƤƺƶƺƽƞƢƺƣƹǁƣƽᅷƾ School of Education and Human Development. He completed his PhD at the ƹƫǁƣƽƾƫƿDŽƺƤƣƺƽƨƫƞᄬƿƩƣƹƾᄭᄙ ƫƾƽƣƾƣƞƽơƩƤƺơǀƾƣƾƺƹơƩƫƶƢƽƣƹᅷƾơƺƹƾƿƽǀơƿƫƺƹ ƺƤƣƞƽƶDŽƹǀƸƟƣƽƞƹƢƤƽƞơƿƫƺƹƞƶƴƹƺǂƶƣƢƨƣᄕƺƹƿƣƞơƩƣƽƾᅷƻƽƺƤƣƾƾƫƺƹƞƶƢƣǁƣƶƺƻment, and on linking mathematical thinking/learning with brain processes. ƣ Ʃƞƾ Ɵƣƣƹ ƾƣƽǁƫƹƨ ƞƾ ƞ ƻƽƫƹơƫƻƞƶ ƫƹǁƣƾƿƫƨƞƿƺƽ ƺƹ ƿƩƣ ƤƺǀƽᅟDŽƣƞƽᄕ ᇣᇵ Ƹƫƶƶƫƺƹᄕ ҄҉Ѽ-funded project, “Student-Adaptive Pedagogy for Elementary Teachers.” This project implements and studies a professional development intervention ƢƣƾƫƨƹƣƢƿƺƾƩƫƤƿǀƻƻƣƽƣƶƣƸƣƹƿƞƽDŽƿƣƞơƩƣƽƾᅷƸƞƿƩƣƸƞƿƫơƾƿƣƞơƩƫƹƨƿƺǂƞƽƢƞ ơƺƹƾƿƽǀơƿƫǁƫƾƿƞƻƻƽƺƞơƩƞƹƢƸƣƞƾǀƽƣƫƸƻƞơƿƺƹƾƿǀƢƣƹƿƾᅷƶƣƞƽƹƫƹƨᄧƺǀƿơƺƸƣƾᄙ Lauren Wolf ƫƾƞƹƞƾƾƫƾƿƞƹƿƻƽƺƤƣƾƾƺƽƺƤƸƞƿƩƣƸƞƿƫơƾƞƿ ƺƾƿƺƾƺƸƸǀƹƫƿDŽƺƶƶƣƨƣᄬƫƿDŽ ƹƫǁƣƽƾƫƿDŽƺƤƣǂƺƽƴᄭᄙƽᄙƺƶƤƫƾƾƺǀƨƩƿƞƤƿƣƽƤƺƽƩƣƽǀƹƫƼǀƣƿƣƞơƩƫƹƨƾƿDŽƶƣ and activist approach. She teaches everything from developmental algebra to graduate-level mathematics. She taught in the prison for seven years and has experience in many different teaching environments but found her place in the South Bronx. Her research interest includes teaching math to prison inmates as well as understanding how her students in the South Bronx relate to mathematics and encouraging them into undergraduate ҉Ҋѻ҃ research.

Introduction Bronislaw Czarnocha

The book we offer readers for their consideration, enjoyment, and intellectual stimulation focuses attention on that sudden moment of insight many of us experience when, suddenly and unexpectedly, our mind is enlightened with the solution to or a new understanding of a problem that had been bothering us for some time. Everything becomes clear, and pleasurably clear. This is an Aha! moment, sometimes called a Eureka experience, and it was named by ƽƿƩǀƽƺƣƾƿƶƣƽᄬᇳᇻᇸᇶᄭƞƾƿƩƣᅸƞơƿƺƤơƽƣƞƿƫƺƹᄙᅺ Our focus of attention on this manifestation of human creativity brings forth ƫƿƾơƣƹƿƽƞƶơƺƹơƣƻƿƺƤƺƣƾƿƶƣƽᅷƾƿƩƣƺƽDŽᄕƿƩƣơƺƹơƣƻƿƺƤƟƫƾƺơƫƞƿƫƺƹƞƾƿƩƣơƺƨƹƫtive/affective mechanism underlying the dynamics of the act of creation. Bisociation, distinct from association, describes the situation when a new entity, a new concept, or a new joke arises out of dialectical interaction between two ƻƽƣǁƫƺǀƾƶDŽƾƣƻƞƽƞƿƣǂƞDŽƾƺƤƿƩƫƹƴƫƹƨᄕƤƽƞƸƣƾƺƤƽƣƤƣƽƣƹơƣᄕƺƽᄙƞƾƺƣƾƿƶƣƽơƞƶƶƾ ƿƩƣƸᄕᅸƸƞƿƽƫơƣƾƺƤƣǃƻƣƽƫƣƹơƣᅺᄬƻᄙᇶᇷᄭᄙƺƽƣƻƽƣơƫƾƣƶDŽᄕƿƩƽƺǀƨƩƺǀƿƿƩƣƟƺƺƴǂƣ are guided by the following definition: The bisociation act is the spontaneous leap of insight that connects previously unconnected matrices of experienceሾ through the discovery of a “hidden analogy.” ƹƢ ƞƶƿƩƺǀƨƩ ƺƣƾƿƶƣƽ ƫƾᄕ ƿƩƽƺǀƨƩƺǀƿ Ʃƫƾ ƿƽƣƞƿƫƾƣᄕ ƾƴƫƶƶƤǀƶƶDŽ ƻƺƫƹƿƫƹƨ ƿƺ ƿƩƣ hidden analogies in a different example of big C creativityሿ showing the bisociative nature of the discovery, nonetheless he understood quite well that the creative act goes beyond the recognition of hidden similarity or hidden likeness. “But where does the hidden likeness hide, and how is it found?” he asks. ᄴ ᄵƹƸƺƾƿƿƽǀƶDŽƺƽƫƨƫƹƞƶƞơƿƾƺƤƢƫƾơƺǁƣƽDŽƿƩƣᅸƾƣƣƫƹƨᅺƫƾƫƹƤƞơƿƫƸƞƨƫƹƫƹƨᄖƫƿƫƾƢƺƹƣƫƹƿƩƣƸƫƹƢᅷƾᄕƞƹƢƸƺƾƿƶDŽƿƩƣǀƹơƺƹƾơƫƺǀƾƸƫƹƢᅷƾƣDŽƣᄙƩƣ analogy between the life of one kind of microbe inside a cow and another ƴƫƹƢƺƤƸƫơƽƺƟƣƫƹƞƤƺƽƨƺƿƿƣƹơǀƶƿǀƽƣƿǀƟƣᄴƺƣƾƿƶƣƽƽƣƤƣƽƾƩƣƽƣƿƺƞƾƿƣǀƽᅷƾ Ƣƫƾơƺǁƣƽƫƣƾᄵ ǂƞƾ ƹƺƿ ᅸƩƫƢƢƣƹᅺ ƞƹDŽǂƩƣƽƣᄖ ƫƿ ǂƞƾ ᅸơƽƣƞƿƣƢᅺ ƟDŽ ƿƩƣ imagination; and once an analogy has been created, it is of course there Ƥƺƽƞƶƶƿƺƾƣƣᄙᄬƻᄙᇴᇲᇲᄭ

ᇙ ‫ةء؟؟؜إؘؕ؞؝؜؟؞ء؜ءآ؞‬ᄕ‫ؘ҄ؗ؜ؘ؟‬ᄕሦሤሦሥᏺᄩᏺѺ҅ѿᄘሥሤᄙሥሥሪሧᄧርራሬርሤሤረረረሪረሧረᇇሤሤሥ

ሦ

ѹ‫ؔ؛ؖآءإؔح‬

ƩƞƿᅷƾǂƩDŽƿƩƣƢƫƾơƺǁƣƽDŽƺƤƿƩƣᅸƩƫƢƢƣƹƞƹƞƶƺƨDŽᅺᄬƻᄙᇵᇴᇲᄭƫƾƿƩƣơƽƣƞƿƫǁƣƻƽƺcess, the creative act, which brings the creative product for everyone to see: creation of the bridge between two separated matrices of experience. Thus, ƺƣƾƿƶƣƽᅷƾƢƣƤƫƹƫƿƫƺƹƺƤƟƫƾƺơƫƞƿƫǁƣơƽƣƞƿƫǁƫƿDŽơƺƹƤƶƞƿƣƾᄕƞƾƺƢƣƹᄬᇴᇲᇲᇶᄭƾƞDŽƾᄕ the process aspect of creativity, that is, the act of making a connection—with its novel aspect, the connection—which gives us a unified view of the creative ƞơƿᄙƩƞƿᅷƾǂƩƣƽƣƫƿƾƽƫơƩƹƣƾƾƽƣƾƫƢƣƾᄕƟǀƿƫƿƫƾƞƶƾƺƞƿƽƞƻƿƩƞƿƺƤƿƣƹƶƣƞƢƾƿƺƿƩƣ separation of its dynamic and structural aspects. Despite the seeming precision of the definition, there are several ways of interpreting it. One way is by considering bisociation as the spontaneous moment of insight together with the connected structure it brings about, as most of the examples of insight inform. However, there is also a consistent interpretation that focuses the meaning of bisociation solely on the outcome. ƣƽƿƩƺƶƢᄬᇴᇲᇳᇴᄭƞƾƾƣƽƿƾƿƩƞƿƫƿƫƹƤƺƽƸƞƶƶDŽơƞƹƟƣƾƣƣƹƞƾᅸᄬƾƣƿƾƺƤᄭơƺƹơƣƻƿƾƿƩƞƿ ƟƽƫƢƨƣƿǂƺƺƿƩƣƽǂƫƾƣƹƺƿᅭƺƽƺƹƶDŽǁƣƽDŽƾƻƞƽƾƣƶDŽᅭơƺƹƹƣơƿƣƢƢƺƸƞƫƹƾᅺᄬƻᄙᇴᄭᄙ ƺƸƻƞƽƫƹƨƿƩƫƾǂƫƿƩƺƣƾƿƶƣƽᅷƾƢƣƤƫƹƫƿƫƺƹᄕǂƣƹƺƿƣƿƩƣƞƟƾƣƹơƣƺƤƿƩƣƻƩƽƞƾƣ “spontaneous leap of insight.”

ƹƾƿƣƞƢ ƺƤ ƫƹƾƫƨƩƿ ǂƣ Ʃƞǁƣ ƫƿƾ ƻƽƺƢǀơƿᄘ ƿƩƣ ƾƣƿ ƺƤ ơƺƹơƣƻƿƾ ƿƩƞƿ ơƺƹƹƣơƿƾ or bridges previously unconnected matrices/domains. We have also lost the ƾƻƺƹƿƞƹƣƺǀƾƶƣƞƻƺƤƺƣƾƿƶƣƽᅷƾƢƣƤƫƹƫƿƫƺƹᄕƾƺƿƩƞƿƟƫƾƺơƫƞƿƫǁƣƴƹƺǂƶƣƢƨƣƢƫƾcovery employs only the mechanical, that is, no spontaneous aspects inherent in connecting two different structures. This method is the basis of the artificial ƫƹƿƣƶƶƫƨƣƹơƣ ᄬѷѿᄭ ǀƹƢƣƽƾƿƞƹƢƫƹƨ ƺƤ ƟƫƾƺơƫƞƿƫƺƹᅭơƺƸƻǀƿƣƽƾ Ƣƺ ƹƺƿ ƾƣƣƸ ƿƺ ƩƞǁƣƩƞᄛƸƺƸƣƹƿƾᄙƺƽƣƢƣƿƞƫƶƣƢƢƫƾơǀƾƾƫƺƹƾơƞƹƟƣƨƶƞƹơƣƢƞƿƫƹƩƞƻƿƣƽᇳᇷᄙ ƺƽƣƺǁƣƽᄕơƺƹƾƫƢƣƽƞƿƫƺƹƾƺƤƞƴƣƽᄬƩƞƻƿƣƽᇳᇳᄭƾǀƨƨƣƾƿƞƾƿƫƶƶƢƫƤƤƣƽƣƹƿǂƞDŽƿƺ interpret bisociation, not through connecting two matrices or skills but rather through disconnecting them and through that disconnection creating a new ƞƟƾƿƽƞơƿƣƢơƺƹơƣƻƿᄬƾƣƣƢƫƾơǀƾƾƫƺƹƾƺƤƞƟƾƿƽƞơƿƫƺƹƫƹƩƞƻƿƣƽƾᇳᇳƞƹƢᇳᇵᄭᄙ This interpretation of the bisociative frame introduces its inverse, one starting from, for example, the two components tightly connected within the reaƾƺƹƫƹƨƺƤƢƫǁƫƾƫƺƹƺƤƤƽƞơƿƫƺƹƾǂƫƿƩƿƩƣƾƞƸƣƢƣƹƺƸƫƹƞƿƺƽƾᄬƩƞƻƿƣƽᇳᇵᄭᄙƾ time progresses, the new, not yet fully abstracted, connection, the short-curtailed algorithm, emerges and separates itself from the original method, thereby creating the beginning of the bisociative frame. The continuation of the process takes place within that emerging bisociative frame until the reasoning between the two is unconsciously forgotten and consciously disregarded, creating the new relation of abstraction. This interpretation suggests abstraction as the inverse bisociative process, in some contexts.

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Two Objectives of the Writing Project

The volume of inquiries into the creativity of an Aha! moment was initiated by ƸƣƸƟƣƽƾƺƤƿƩƣƣƞơƩƫƹƨᅟƣƾƣƞƽơƩƣƞƸᄬҊ҈ƣƞƸᄭƺƤƿƩƣƽƺƹǃᄕƞƨƽƺǀƻƺƤ mathematics faculty from ѹҋ҄ҏ community colleges in the Bronx borough of ƣǂƺƽƴƫƿDŽᄙ The Ҋ҈ Team has been systematically studying classroom mathematical insights since conducting the collaborative teaching experiment ProbƶƣƸᅟƺƶǁƫƹƨƫƹƣƸƣƢƫƞƶƽƫƿƩƸƣƿƫơᄘǀƸƻƾƿƞƽƿƿƺƣƤƺƽƸƫƹᇴᇲᇳᇲƾǀƻƻƺƽƿƣƢƟDŽ a ѹҋ҄ҏ ѹቀѿ҈ѽᇹƨƽƞƹƿᄬƺƶƶƞƟƺƽƞƿƫǁƣƺƸƸǀƹƫƿDŽƺƶƶƣƨƣ ƹơƣƹƿƫǁƣƣƾƣƞƽơƩ ƽƞƹƿᇱᇹᄭᄙƽǀƹƢƞƽƞƟƩǀᄕቀ a member of the Ҋ҈ Team of the Bronx, observed many Aha! moments in her remedial algebra experimental classroom that ƾƩƣơƺƺƽƢƫƹƞƿƣƢǂƫƿƩƺƣƾƿƶƣƽᅷƾƿƩƣƺƽDŽᄕƿƞƴƫƹƨƿƩƣƤƫƽƾƿƾƿƣƻƾƺƹƿƩƣƻƞƿƩƿƩƞƿ took us to our present multifaceted inquiry into the creativity of mathematical insight. The central skeleton of that pathway has been the integration of classroom practice with adequate theories of learning. The design of the volume reflects the integrative process. The initial part, Bisociation in the Classroom, takes us into a discussion of teaching practice, the results of classroom facilitation, and ƿƩƣƫƹǁƣƾƿƫƨƞƿƫƺƹƺƤƞƹƩƞᄛƸƺƸƣƹƿƫƹƾƫƨƩƿƨǀƫƢƣƢƟDŽƺƣƾƿƶƣƽᅷƾƟƫƾƺơƫƞƿƫƺƹᄕ primarily from the cognitive point of view. That part is followed immediately by the second part, The Aha! Moment and Affect, where several chapters delve into the depth of the relationship between cognition and affect in the context ƺƤƿƩƣƟƫƾƺơƫƞƿƫƺƹᄬDžƞƽƹƺơƩƞᄕᇴᇲᇳᇶᄭᄙ ƩƣƿƩƫƽƢƻƞƽƿᄕƫƾƺơƫƞƿƫƺƹƞƹƢƩƣƺƽƫƣƾƺƤƣƞƽƹƫƹƨᄕƢƣǁƣƶƺƻƾƞƴƣƽᅷƾᄬᇴᇲᇳᇸᄭ conjecture that bisociative creativity can be identified and found within many theories of learning, particularly in the constructivist theories. The fourth part, Bisociativity from Without, offers several views from the outside of the strictly educational domain, from neuroscience and from artificial intelligence, which has recently created a new subdomain of computer creativity. Hostos Community College, the present site of the Ҋ҈ Team of the Bronx, ǂƞƾƣƾƿƞƟƶƫƾƩƣƢƫƹᇳᇻᇸᇺƫƹƽƣƾƻƺƹƾƣƿƺƢƣƸƞƹƢƾƸƞƢƣƟDŽƿƩƣơƺƸƸǀƹƫƿDŽƺƤƿƩƣ South Bronx—at the time, and for many years after,ቁ the poorest congressional district in the United States. The college population has traditionally been predominately Latino and female, and the college serves a multicultural South ƽƺƹǃƻƺƻǀƶƞƿƫƺƹƺƤǂƺƽƴƫƹƨᅟơƶƞƾƾƺƽƫƨƫƹƾᄙƻƿƺᇺᇷነƺƤƤƽƣƾƩƸƣƹƹƣƣƢƿƺƿƞƴƣ ƟƞƾƫơƸƞƿƩᄕƺƤƿƣƹƾƿƞƽƿƫƹƨƞƿƿƩƣƶƣǁƣƶƺƤƞƽƫƿƩƸƣƿƫơᄖƺƹƶDŽᇴᇹነƺƤƾƿǀƢƣƹƿƾơƺƸplete their degree.

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The dire conditions of education in the community impelled concerned faculty to focus their research and teaching on the improvement of learning among students of the college and more broadly among urban students, referred to in this volume as rank-and-file or underserved and underrepresented. That is why one of the central aims is to introduce facilitation, understanding, and assessment of creative efforts at the lowest levels of learning—the level at which many students in our study find themselves in basic math classes. The Creativity of the Aha! Moment and Mathematics Education has two objectives. 1.1 Objective 1 The first objective is to formulate an approach to creativity in mathematics learning which addresses itself to all students: typically achieving, rankand-file students of any mathematics classroom, high-achieving, and gifted ƾƿǀƢƣƹƿƾᄙ ƿƾƞƫƸƫƾƿƺƢƣǁƣƶƺƻƞƹƩƞᄛƻƣƢƞƨƺƨDŽƺƤƸƞƿƩƣƸƞƿƫơƾƟƞƾƣƢƺƹƢƫƤferent methods of designing creative learning environments in the classroom and facilitating Aha! moments. Several researchers have pointed to the dearth of educational research on creativity within the general population. ƽƫƽƞƸƞƹƣƿƞƶᄙᄬᇴᇲᇳᇳᄭƾƿƞƿƣƾƿƩƞƿᅸᄴƿᄵƩƣƽƺƶƣƺƤơƽƣƞƿƫǁƫƿDŽǂƫƿƩƫƹƸƞƿƩƣƸƞƿƫơƾ education with students who do not consider themselves gifted is essentially ƹƺƹᅟƣǃƫƾƿƣƹƿᅺᄬƻᄙᇳᇴᇲᄭᄙƩƞƸƟƣƽƶƫƹᄬᇴᇲᇳᇵᄭƞƾƾƣƽƿƾƿƩƞƿ Missing is information on what initiatives are in place to develop and facilitate mathematical creativity in underserved and under-identified populations. This type of discussion would be informative to the field of gifted education and counter the criticism that the field is not inclusive. ᄬƻᄙᇺᇷᇸᄭ

ƹƿƩƫƾǁƺƶǀƸƣᄕǂƣƻƽƺƻƺƾƣƟƫƾƺơƫƞƿƫƺƹᄕƿƩƣƿƩƣƺƽDŽƺƤƿƩƣƩƞᄛƸƺƸƣƹƿᄕƞƾƿƩƣ approach for the creativity of all students. The reason is that the Aha! moment is a phenomenon known and experienced by the general population—a pheƹƺƸƣƹƺƹƿƩƞƿǀƹƿƫƶƹƺǂƩƞƾƹᅷƿƽƣơƣƫǁƣƢƞƤǀƶƶᄕƞƻƻƽƺƻƽƫƞƿƣƿƽƣƞƿƸƣƹƿƫƹƶƫƿƣƽƞture on creativity in mathematics. We derive the theoretical support and justification for this proposal from the following: ᅬ ƺƣƾƿƶƣƽᅷƾ ƺƟƾƣƽǁƞƿƫƺƹ ƿƩƞƿ ᅸƞƹDŽ ƾƣƶƤᅟƿƞǀƨƩƿ ƹƺǁƣƶƿDŽ ƫƾ ƞ Ƹƫƹƺƽ Ɵƫƾƺơƫƞƿƫǁƣ ƞơƿᅺᄬƻᄙᇸᇷᇹᄭƾƺƿƩƞƿᅸƸƫƹƺƽᄕƾǀƟưƣơƿƫǁƣƟƫƾƺơƫƞƿƫƺƹƻƽƺơƣƾƾƣƾᄚƞƽƣƿƩƣƸƞƫƹ ǁƣƩƫơƶƣ ƺƤ ǀƹƿǀƿƺƽƣƢ ƶƣƞƽƹƫƹƨᅺ ᄬƻᄙ ᇸᇷᇺᄭᄙ ƹƿǀƿƺƽƣƢ ƶƣƞƽƹƫƹƨ ƫƾ ƶƣƞƽƹƫƹƨ ƟDŽ oneself, a self-taught process essentially different from the process of assimilation of information given by a tutor. Guided-discovery methods provide

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the methodology which can well approximate “untutored learning” in classroom conditions. ᅬ ƞƢƞƸƞƽƢᅷƾᄬᇳᇻᇶᇷᄭǁƫƣǂƿƩƞƿ “ᄴƟᄵƣƿǂƣƣƹƿƩƣǂƺƽƴƺƤƿƩƣƾƿǀƢƣƹƿǂƩƺƿƽƫƣƾ to solve a problem in geometry or algebra and a work of invention, one can say that there is only the difference of degree, the difference of a level, both works being of similar natureᅺᄬƻᄙᇳᇲᇶᄕƣƸƻƩƞƾƫƾƞƢƢƣƢᄭᄙ ᅬ ǃƻƣƽƫƣƹơƣƢ ƸƞƿƩƣƸƞƿƫơƾ ƿƣƞơƩƣƽƾ ƻƻƶƣƟƞǀƸ ƞƹƢ ƞǀƶ ᄬᇴᇲᇲᇻᄭ ƞƾƾƣƽƿ that “remedial mathematics students can exhibit creativity as often and as clearly as an advanced calculus student,” confirming their own experience teaching basic mathematics. ᅬ The developments in artificial intelligence suggests, according to Boden ᄬᇴᇲᇲᇶᄭᄕƿƩƞƿƿƩƣƹƞƿǀƽƣƺƤơƽƣƞƿƫǁƣƫƹƾƫƨƩƿƫƾƞƨƣƹƣƽƞƶơƩƞƽƞơƿƣƽƫƾƿƫơƺƤƿƩƣ human brain and mind. These observations suggest that from the cognitive point of view, there is no difference between mathematical invention of a general character and solving a mathematical problem by a student of any level. Bisociation is the common ƸƣơƩƞƹƫƾƸᄙƩƣƢƫƤƤƣƽƣƹơƣƫƾƫƹƿƩƣƫƹƿƣƹƾƫƿDŽƞƹƢƢƣƻƿƩƺƤƴƹƺǂƶƣƢƨƣᄬƺᄭ reached independently, whether the originality of the insight has an objective, social significance, or solely a subjective one, for the learner. We take note here of the distinction between big C, or historical, creativity, the act of creation that impacted the population and society at large, and little C, or personal, creativity, the creativity whose product is new and original for the learner, even if known to others.ቂ Cognitive unity between the creative efforts of a mathematician and the creative efforts of the student solving mathƣƸƞƿƫơƞƶƻƽƺƟƶƣƸƾƫƾƫƸƻƺƽƿƞƹƿᄙ ƿƾǀƨƨƣƾƿƾƞơƺƸƸƺƹƞƻƻƽƺƞơƩƿƺƞƶƶƾƿǀƢƣƹƿƾᄕ rank-and-file students, and gifted ones. The complexity and the nature of the hints, questions, and problems asked will of course be different; however, the nature of the cognitive act and effort is similar. The role of creativity, especially of the creativity of an Aha! moment, among students who come from families or a social strata that have not had contact with higher education, is fundamental because of its connection with affect. We can interest and motivate students toward mathematics through positive affect while working with the subject. ƩƞƻƿƣƽᇻƢƫƾơǀƾƾƣƾƢƣƣƻƣƽƾƿƽǀơƿǀƽƣƾƺƤƿƩƣƟƺƹƢƫƹƨƻƽƺơƣƾƾǂƫƿƩƸƞƿƩƣmatics through the concept of conation, that is, by connecting the practice of ƸƞƿƩƣƸƞƿƫơƾǂƫƿƩƿƩƣƟƞƾƫơƩǀƸƞƹƹƣƣƢƾƺƤƞƾƿǀƢƣƹƿᄙƩƞƻƿƣƽᇳᇲơƺƹƤƫƽƸƾƿƺ us that students do bond with mathematics by becoming pleasurably aware, through the distinct pleasure of the moment of insight, of their own potential and thus are motivated to pursue the subject within the horizon of success.

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The experience of Aha! moments creates the emotional basis for the pursuit of mathematics in classrooms and beyond. As several researchers have observed above, our knowledge and understanding of the creativity of every person is lacking. The research methods are adapted to the creativity of the gifted, which has been given a lot of attention in mathematics education. ƽƞƟƩǀ ƞƹƢ DžƞƽƹƺơƩƞ ᄬᇴᇲᇳᇶᄭ ƻƽƺƻƺƾƣƢ ƺƣƾƿƶƣƽᅷƾ Ɵƫƾƺơƫƞƿƫƺƹ ƞƾ ƿƩƣ ƹƣǂ definition of creativity in our field, which allows for the process of its democratization. An Aha! moment is a common experience in the general population including both rank-and-file members as well their gifted and highly able counterparts. Bisociation theory offers a new cognitive tool and instrument that make possible exploration of the creativity of everyone. 1.2 Objective 2 The second objective of this book is to investigate the relationship of bisociation with the field of creativity and learning research. Certain components of the relationship are known: bisociation takes place ƞƶƺƹƨƿƩƣƻƞƿƩǂƞDŽƤƽƺƸƿƩƣƫƹơǀƟƞƿƫƺƹƿƺƿƩƣƫƶƶǀƸƫƹƞƿƫƺƹƾƿƞƨƣƺƤƞƶƶƞƾᅷƾ ᄬᇳᇻᇴᇸᄭƣƾƿƞƶƿᅟƟƞƾƣƢƿƩƣƺƽDŽᄙƩƣƣƸƻƩƞƾƫƾƺƹƺƽƫƨƫƹƞƶƫƿDŽơƺƹƹƣơƿƾƫƿǂƫƿƩƿƩƣ Guilford and the Torrance approaches. One of the main differences is the connection of bisociativity with affect, which forms the cognitive/emotional dualƫƿDŽƺƤƿƩƣƩƞᄛƸƺƸƣƹƿᄬDžƞƽƹƺơƩƞᄕᇴᇲᇳᇶᄭᄙ The central focus of the second objective is the relationship of bisociation with different theories of learning with the aim to investigate the conjecture that the creativity of the Aha! moment is present at every conceptual act of learning. ƞƴƣƽᄬᇴᇲᇳᇸᄖƩƞƻƿƣƽᇳᇳᄭƩƞƾƻƺƾƫƿƫƺƹƣƢƿƩƣƟƫƾƺơƫƞƿƫǁƣƤƽƞƸƣƺƤƿǂƺǀƹơƺƹƹƣơƿƣƢ ƻƶƞƹƣƾƺƤƽƣƤƣƽƣƹơƣǂƫƿƩƫƹƿƩƣƻƽƺưƣơƿƫƺƹƾƿƞƨƣƺƤƽƣƤƶƣơƿƫǁƣƞƟƾƿƽƞơƿƫƺƹᄬƫƞƨƣƿѵ ƞƽơƫƞᄕᇳᇻᇺᇻᄭᄙƹƿƩƞƿƟƞƾƫƾᄕƣǃƻƶƺƽƣƢƸƺƽƣƢƣƣƻƶDŽƫƹƩƞƻƿƣƽᇳᇳᄕǂƣơƺƹưƣơƿǀƽƣ the presence here of two parallel intertwined dynamic processes through which the concept development takes place, reflective abstraction and bisociation. The two theories translate here one into another but with one significant difference: the dynamic of reflective abstraction is incremental gradual moveƸƣƹƿᄬƽƹƺƹƣƿƞƶᄙᄕᇴᇲᇳᇶᄕƻᄙᇳᇸᇴᄭǂƩƫƶƣƿƩƣƢDŽƹƞƸƫơƺƤƟƫƾƺơƫƞƿƫƺƹƫƾƢƫƾơƽƣƣƿᄕ ƿƩƽƺǀƨƩƶƣƞƻƾᄙƩƞƻƿƣƽᇳᇳƣǃƿƣƹƢƾƿƩƫƾƢƫƾơǀƾƾƫƺƹƿƺƞƶƶƶƣǁƣƶƾƺƤƶƣƞƽƹƫƹƨᄕƫƹǁƣƾƿƫƨƞƿƫƹƨƿƩƣƹƞƿǀƽƣƺƤƿƩƣơƺƹƹƣơƿƫƺƹƟƣƿǂƣƣƹƿƩƣƿǂƺᄙ ƤƫƹƢƣƣƢơƽƣƞƿƫǁƫƿDŽƫƾƾƺ ơƶƺƾƣƶDŽƽƣƶƞƿƣƢƿƺƶƣƞƽƹƫƹƨƞƾƩƞƻƿƣƽᇳᇳƾǀƨƨƣƾƿƾᄕǂƣƩƞǁƣƞƸǀơƩƾƿƽƺƹƨƣƽƽƣƞson to demand creativity—in particular, the creativity of an Aha! moment—as the basis for the design of the mathematics curriculum. Objective 2 of the book leads to the investigation of how the integrative ƤƺơǀƾƺƤƿƩƣƹƣǂƢƣƤƫƹƫƿƫƺƹƩƣƶƻƾǀƾƿƺǀƹƢƣƽƾƿƞƹƢᄬᇳᄭƿƩƣƸƣơƩƞƹƫƾƸƺƤơƽƣƞƿƫǁƫƿDŽᅸƫƹǁƫǁƺᅺƺƤƿƩƣƾƻƺƹƿƞƹƣƺǀƾƶƣƞƻƞƹƢᄬᇴᄭƫƿƾƽƺƶƣǂƫƿƩƫƹƺƿƩƣƽƿƩƣƺƽƫƣƾ of creativity and learning discussed in different chapters.

ѿ‫ءآ؜اؖبؗآإاء‬

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The new definition of creativity has a specific feature of focusing attention ƺƹƿƩƣƻƽƺơƣƾƾƺƤơƺƹƹƣơƿƫƹƨƢƫƤƤƣƽƣƹƿᄕƾƣƻƞƽƞƿƣơƺƸƻƺƹƣƹƿƾƺƤƞƿƩƫƹƴƣƽᅷƾƺƽ ƞƹƞƽƿƫƾƿᅷƾƴƹƺǂƶƣƢƨƣᄙ ƿƾƣƞƽơƩƣƾƤƺƽƿƩƣƾDŽƹƿƩƣƾƫƾƺƤǀƹƫƤƫơƞƿƫƺƹƽƞƿƩƣƽƿƩƞƹ analytical separation. ƩƞƻƿƣƽᇳƫƹƿƽƺƢǀơƣƾƟƫƾƺơƫƞƿƫƺƹƞƹƢƢƫƾơǀƾƾƣƾƫƿƾƾơƺƻƣᄙ 1.3 Chapter 1, Arthur Koestler’s Bisociation Theory The chapter introduces bisociativity along several avenues of thought. First, ƫƿƨƽƺǀƹƢƾƿƩƣƢƣƤƫƹƫƿƫƺƹƺƤƿƩƣơƺƹơƣƻƿƿƩƽƺǀƨƩƿƩƣƺƣƾƿƶƣƽƿƽƫƻƿDŽơƩᄕǂƩƫơƩ shows the changing nature of creativity by the changes in the emotional climate while moving the attention from humor to scientific discovery and to art. At the same time the logical structure of creativity that is bisociation stays the same throughout the passage along the different domains. Second, the chapter places bisociation in the context of two major approaches to creativity, that of the structural Gestalt approach and the Guilford/Torrance phenomenological view originating in the Guilford understanding of creativity through divergent and convergent thinking. ƩƫƽƢᄕ Ʃƞƻƿƣƽ ᇳ Ƣƫƾơǀƾƾƣƾ Ʃƺǂ ƺƣƾƿƶƣƽᅷƾ ƣƸƻƩƞƾƫƾ ƺƹ ƿƩƣ ơƺƹƹƣơƿƫǁƫƿDŽ of unconnected frames of reference as the spontaneous nature of the Aha! moment begins the third general approach to understanding creativity leading through the JanusianቃƿƩƫƹƴƫƹƨƺƤƺƿƩƣƹƟƣƽƨᄬᇳᇻᇹᇻᄭƿƺƿƩƣơƺƹƹƣơƿƫǁƫƿDŽ of neural networks, which constitute the basis of the ѷѿ approach to computer ơƽƣƞƿƫǁƫƿDŽᄬƺƢƣƹᄕᇴᇲᇲᇶᄭᄙ The discussion of several creative moments known from intellectual history provides the background for the discussion of the nature of bisociative thinking, whose characteristic feature is the presence of the bisociative frame—one of the central concepts throughout the volume. The discussion of the relationship between bisociation, association, and a habit, which starts in this chapter, ƤƫƹƢƾ ƫƿƾ ƣơƩƺ ƫƹ ƾƣǁƣƽƞƶ ƺƤ ƿƩƣ Ƥƺƶƶƺǂƫƹƨ ơƩƞƻƿƣƽƾᄙ Ʃƞƻƿƣƽ ᇳ ơƶƺƾƣƾ ǂƫƿƩ ƞƹ extensive discussion of bisociation as the third pathway to creativity.

2

Teaching-Research Methodology

Ʃƣ ƿƣƞơƩƫƹƨᅟƽƣƾƣƞƽơƩ ᄬҊ҈ᄭ ƸƣƿƩƺƢƺƶƺƨDŽ ơƞƶƶƣƢ ƿƩƣƣƞơƩƫƹƨᅟƣƾƣƞƽơƩ ƣǂ ƺƽƴ ƫƿDŽ ᄬҊ҈ᄧƫƿDŽᄭ ƸƺƢƣƶ ᄬDžƞƽƹƺơƩƞᄕ ᇴᇲᇳᇸᄭ ǀƹƢƣƽƶƫƣƾ ƿƩƣ Ƣƣƾƫƨƹ ƺƤ ƿƩƣ ǁƺƶǀƸƣƻƽƣƾƣƹƿƣƢƩƣƽƣᄙƩƣƸƣƿƩƺƢƺƶƺƨDŽƫƾƟƞƾƣƢǀƻƺƹƿǂƺƻƽƫƹơƫƻƶƣƾᄘᄬᇳᄭƿƩƣ ƟƞƶƞƹơƣƺƤƿƣƞơƩƫƹƨƻƽƞơƿƫơƣƞƹƢƣƢǀơƞƿƫƺƹƞƶƽƣƾƣƞƽơƩƞƹƢᄬᇴᄭƿƩƣƨƺƞƶƺƤƿƩƣ improvement of learning. ƶƞƾƾƽƺƺƸ ƿƣƞơƩƫƹƨ ƣǃƻƣƽƫƸƣƹƿƾ ᄬDžƞƽƹƺơƩƞᄕ ᇳᇻᇻᇻᄖ DžƞƽƹƺơƩƞ ѵ ƽƞƟƩǀᄕ ᇴᇲᇲᇸᄖDžƞƽƹƺơƩƞѵƞưᄕᇴᇲᇲᇺᄭƺƤƤƣƽƣƢƞǀƹƫƼǀƣƿƺƺƶᄕǂƫƿƩƫƹǂƩƫơƩƿƣƞơƩƫƹƨƞƹƢ

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research found mutually nourishing ground and led to the formulation of the Ҋ҈ᄧƫƿDŽƸƺƢƣƶᄙƣƞơƩƫƹƨᅟƽƣƾƣƞƽơƩƺƤƤƣƽƣƢƞơƺƹƾƫƾƿƣƹƿƸƣƿƩƺƢƺƤơƺƹƹƣơƿing and integrating teaching and research in the search of maximally effective methods for the improvement of learning. Ҋ҈ᄧƫƿDŽƺƽƨƞƹƫDžƣƢƫƿƾơƶƞƾƾƽƺƺƸƫƹǁƣƾƿƫƨƞƿƫƺƹǂƺƽƴƞƶƺƹƨƞƿƣƞơƩƫƹƨᅟƽƣƾƣƞƽơƩ ᄬҊ҈ᄭ ơDŽơƶƣ ƸƣƿƩƺƢƺƶƺƨDŽ ᄬDžƞƽƹƺơƩƞ ƣƿ ƞƶᄙᄕ ᇴᇲᇳᇸᄭ ǂƫƿƩ Ƥƺǀƽ ƹƺƢƣƾ ƺƤ diagnosis: design of intervention, implementation with the collection of data, ƞƹƢ ƿƣƞơƩƫƹƨᅟƽƣƾƣƞƽơƩ ƞƹƞƶDŽƾƫƾ ƺƤ ƿƩƣ Ƣƞƿƞ ǂƩƫơƩ ơƺƹƾƫƢƣƽƾ ƿƣƞơƩƣƽƾᅷ ơƽƞƤƿ knowledge, and the relevant research results. The aim of the analysis was to establish the degree of effectiveness of the intervention and design leading to its improved version for the next cycle. The by-product of the cycle or of the iterative series of cycles are the research observations and results. Thus, the teaching-research cycle produces two outcomes, new teaching methods and new research results. New teaching methods are the focus of the second, upper-right node, where the intervention design process is adapted to the state of knowledge of the class. The new research results are the focus of the third, a left-lower node. This ƹƺƢƣƫƾơƣƹƿƽƞƶƿƺƿƩƣơDŽơƶƣƸƣƿƩƺƢƺƶƺƨDŽƟƣơƞǀƾƣƫƿƺƤƤƣƽƾƞƟƫƾƺơƫƞƿƫǁƣᄬƩƞƻƿƣƽᇳᄭƻƺƾƾƫƟƫƶƫƿDŽƺƤơƽƣƞƿƫǁƫƿDŽƤƺƽƿƩƣƫƹƾƿƽǀơƿƺƽƾǂƩƺƻƽƞơƿƫơƣƫƿᄙ ƿƫƾƿƩƣƹƺƢƣ where teaching practice meets research knowledge, often producing new, ǀƹƣǃƻƣơƿƣƢ ƽƣƾƣƞƽơƩ ƽƣƾǀƶƿƾ ƺƽ ƿƣƞơƩƫƹƨ ƸƣƿƩƺƢƾᄙ ƹ ƿƩƫƾ ƹƺƢƣᄕ ƿƩƣ Ɵƫƾƺơƫƞtive frames grounded in two components, teaching practice and research, are being formed, underlying the creative Aha! moments of teacher-researchers. Due to their separation, the interface between research and practice is at presƣƹƿƻƽƣƨƹƞƹƿǂƫƿƩƿƩƣƻƺƾƾƫƟƫƶƫƿƫƣƾƺƤƩƞᄛƸƺƸƣƹƿƾᅷơƽƣƞƿƫǁƣƫƹƾƫƨƩƿƾƿƩƽƺǀƨƩ which both can be joined. ƩƣƻƽƺơƣƾƾƺƤƫƹƿƣƽƞơƿƫƺƹƺƤƽƣƾƣƞƽơƩǂƫƿƩƿƣƞơƩƫƹƨƻƽƞơƿƫơƣƫƾƿǂƺƤƺƶƢᄘᄬᇳᄭ

ƿƻƽƺơƣƣƢƾƤƽƺƸƻƽƞơƿƫơƣᄕƿƩƞƿƫƾᄕƤƽƺƸƿƩƣƢƣƾƫƨƹƟƞƾƣƢƺƹƿƩƣƺƟƾƣƽǁƞƿƫƺƹƾƺƤ ƻƽƞơƿƫơƣᄙ ƿƣƹơƺǀƹƿƣƽƾƽƣƾƣƞƽơƩƞƹƢƶƣƞƽƹƫƹƨƿƩƣƺƽƫƣƾƞƶƺƹƨƿƩƣƿƩƫƽƢƹƺƢƣᄙ fundamental question at that stage is which learning theory or research results ƤƫƿƫƹƿƺƿƩƣƿƣƞơƩƣƽᅷƾơƶƞƾƾƽƺƺƸƺƟƾƣƽǁƞƿƫƺƹƞƹƢƞƹƞƶDŽƾƫƾƺƤƢƞƿƞᄙᄬᇴᄭ ƿƻƽƺơƣƣƢƾ from research. Here the question is how to adapt the learning environment and teaching methodology to the chosen theory of learning. Both decisions can facilitate pedagogical creativity, and both are based upon familiarity with the process of practice/theory coordination and integration. The generalization of that process—the theory/theory coordination and ƫƹƿƣƨƽƞƿƫƺƹᄬƺƽƹƣƿǂƺƽƴƫƹƨƺƤƿƩƣƺƽƫƣƾᄭǀƹƢƣƽƶƫƣƾƞƹƢơƺƹƾƿƫƿǀƿƣƾƿƩƣƿƩƣƺƽƣƿƫơƞƶƤƽƞƸƣǂƺƽƴƤƺƽƸƞƹDŽơƺƹƾƫƢƣƽƞƿƫƺƹƾƺƤƿƩƣƻƽƣƾƣƹƿƣƢǁƺƶǀƸƣᄬƾƣƣƟƣƶƺǂᄭᄙ Ʃƣ ƶƣƹƨƿƩ ƺƤ ƞ ƿƣƞơƩƫƹƨᅟƽƣƾƣƞƽơƩ ơDŽơƶƣ ǁƞƽƫƣƾᄙ ƿ ơƞƹ Ɵƣ ƞ ƻƞƽƿ ƺƤ ƞ ƾƫƹgle class time, several consecutive classes of a course, or the full semester or

‫ ؘإبؚ؜ؙ‬ѿᄙᇳᏻƿƞƹƢƞƽƢƿƣƞơƩƫƹƨᅟƽƣƾƣƞƽơƩơDŽơƶƣᄬƺǀƿƣƽᄭƞƹƢơƽƣƞƿƫǁƣƿƣƞơƩƫƹƨᅟƽƣƾƣƞƽơƩơDŽơƶƣᄬƫƹƹƣƽᄭ

ѿ‫ءآ؜اؖبؗآإاء‬

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ƞơƞƢƣƸƫơDŽƣƞƽᄙ ƿƣƽƞƿƫƺƹƺƤƞƹƫƹƿƣƽǁƣƹƿƫƺƹƿƩƽƺǀƨƩƾƣǁƣƽƞƶƾǀơƩơDŽơƶƣƾƞƶƶƺǂƾ for a high level of its teaching-research refinement. During our nine-year-long work on the creativity of an Aha! moment, the notion of the teaching-research cycle generalized to multiyear cycles, which focused its teaching-research attention on classroom creativity of students and ƿƣƞơƩƣƽƾᄙ ƞơƩƾǀơƩơƽƣƞƿƫǁƣƿƣƞơƩƫƹƨᅟƽƣƾƣƞƽơƩᄬѹҊ҈ᄭơDŽơƶƣƫƹơƶǀƢƣƾᄘᄬᇳᄭƞƿƶƣƞƾƿ ƺƹƣƾƣƸƣƾƿƣƽƺƽƞDŽƣƞƽᅟƶƺƹƨơƶƞƾƾƽƺƺƸƿƣƞơƩƫƹƨƣǃƻƣƽƫƸƣƹƿᄖᄬᇴᄭƞƾƣƽƫƣƾƺƤҊ҈ classroom investigations of the theme similar to that presented in Chapter 6, the work of the Ҋ҈ƣƞƸᄘơƽƣƞƿƫǁƫƿDŽᄖᄬᇵᄭƞƾƣƽƫƣƾƺƤƽƣƶƞƿƣƢƻƽƣƾƣƹƿƞƿƫƺƹƾƞƹƢ ƻǀƟƶƫơƞƿƫƺƹƾᄖƞƹƢᄬᇶᄭƿƩƣƻƽƺơƣƾƾƺƤƟƺƺƴǂƽƫƿƫƹƨƞƟƺǀƿƿƩƣƫƹǁƣƾƿƫƨƞƿƣƢƿƩƣƸƣᄙ The book-writing process plays the role of the third node in the classical Ҋ҈ cycle discussed above, and it is the manifestation of the long-term search for ƸƞƿƩƣƸƞƿƫơƞƶƸƣƞƹƫƹƨƢƫƾơǀƾƾƣƢƟDŽƞǁƫƢƞƶƶƫƹƩƞƻƿƣƽᇺᄙ The book-writing process initiation time cannot be precisely planned within the ѹҊ҈ cycle: it takes place at the moment the members of the Ҋ҈ Team recognize that the scope of the results and ideas developed needs a longer time for reflection focused on restructuring and connecting the emerging themes of classroom Ҋ҈ƫƹǁƣƾƿƫƨƞƿƫƺƹƾᄙ ƿƫƾƿƩƣƿƫƸƣǂƩƣƹƟƺƺƴᅟǂƽƫƿƫƹƨơƺƹƿƽƫƟǀƿƺƽƾᄕ teachers, teacher-researchers, and researchers reach a new understanding, making the new connections within the theme of creativity, often with the ƩƣƶƻƺƤƩƞᄛƸƺƸƣƹƿƾᄬƩƞƻƿƣƽᇳᇴᄭᄙ The first such ѹҊ҈ơDŽơƶƣơƺƸƸƣƹơƣƢƫƹᇴᇲᇳᇲƫƹƿƩƣơƺƹƿƣǃƿƺƤƿƩƣƿƣƞơƩƫƹƨ experiment supported by a ѹҋ҄ҏ ѹቀѿѽ҈7 grant,ቄ when the idea of bisociation appeared in the classroom of V. Prabhu. Our realization that the Ҋ҈ᄧƫƿDŽ model constitutes a bisociative frame in the presence of the contemporary ƢƫǁƫƢƣ Ɵƣƿǂƣƣƹ ƿƣƞơƩƫƹƨ ƞƹƢ ƽƣƾƣƞƽơƩ ᄬƤƺƽ ƣǃƞƸƻƶƣᄕ ƫƣƽƞƹ ƣƿ ƞƶᄙᄕ ᇴᇲᇳᇵᄭ ǂƞƾ ƿƩƣ Ƹƞƫƹ ƿƩƣƸƣ ƺƤ DžƞƽƹƺơƩƞ ƣƿ ƞƶᄙ ᄬᇴᇲᇳᇸᄭᄕ ǂƩƣƽƣ ƺǀƽ ƽƣƤƶƣơƿƫƺƹƾ ǀƻƺƹ ƾƿǀƢƣƹƿƾᅷơƽƣƞƿƫǁƫƿDŽƞƹƢƺǀƽƽƺƶƣƫƹƫƿƾƤƞơƫƶƫƿƞƿƫƺƹơƺƞƶƣƾơƣƢƫƹƿƺƿƩƣƟƺƺƴƟDŽƿƩƣ Ҋ҈ Team, The Creative Enterprise of Mathematics Teaching Research: Elements of Methodology and Practice—From Teachers to Teachers. The process of writing the book has been analogous to the left-bottom node of the Ҋ҈ cycle, where the continuous teaching experience encounters intense research reflection leading to the design, writing of the book, and births of new points of view and approaches to be introduced into practice of the next cycle. The first ѹҊ҈ cycle provided us with a new view upon our own Ҋ҈/ ƫƿDŽƸƣƿƩƺƢƺƶƺƨDŽƫƹƿƩƞƿǂƣƽƣƞƶƫDžƣƢƿƩƞƿƺǀƽƿƣƞơƩƫƹƨᅟƽƣƾƣƞƽơƩƞƻƻƽƺƞơƩ had resulted in so many new ideas to write and to think about and to transform the practice, because it is inherently bisociative joining two very disconnected matrices of experience: mathematics teaching and research in mathematics education.

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That observation focused our attention on the design of teaching-research investigations, particularly on the nature of the creative learning environƸƣƹƿᄬƽƞƟƩǀᄕᇴᇲᇳᇸᄭᄙƣƣƸƟƞƽƴƣƢǀƻƺƹƞƹƫƹǁƣƾƿƫƨƞƿƫƺƹƺƤƿƩƣƽƣƶƞƿƫƺƹƾƩƫƻ between bisociation and Piagetian theories of learning in the conviction that ơƽƣƞƿƫǁƫƿDŽƣƹƿƣƽƾƣǁƣƽDŽƾƿƣƻƺƤơƺƹơƣƻƿǀƞƶƢƣǁƣƶƺƻƸƣƹƿᄬƞƴƣƽᄕᇴᇲᇳᇸᄭᄙ The second ѹҊ҈ cycle, whose left-bottom node is the design and writing of the present volume, was initiated toward the final years of the previous cycle. The moment when the conclusions of the previous cycle could be applied, verified, and developed in the teaching practice is the beginning of the second ѹҊ҈ cycle, which has become focused on a more direct investigation of Aha! ƸƺƸƣƹƿƾƫƹƺǀƽơƶƞƾƾƽƺƺƸᄕǀƾƫƹƨƿƩƣƶƣƹƾƣƾƺƤƺƣƾƿƶƣƽᄙ The second creative cycle has included an extensive process of facilitating and collecting cognitively well-described Aha! moments. Three different ƞơƿƫƺƹƾƩƞǁƣƟƣƣƹƫƹƫƿƫƞƿƣƢᄘᄬᇳᄭƞƻƫƶƺƿơƶƞƾƾƽƺƺƸƣǃƻƣƽƫƸƣƹƿǂƫƿƩƾƿǀƢƣƹƿƾᄧ ƻƣƣƽƶƣƞƢƣƽƾƞƾƾƿǀƢƣƹƿᅟƽƣƾƣƞƽơƩƣƽƾƫƹƿƩƣơƶƞƾƾƽƺƺƸƾƺƤƽƣƸƣƢƫƞƶƞƶƨƣƟƽƞᄖᄬᇴᄭ the hunt for Aha! moments organized by the publication Mathematics Teaching-Research Journal OnlineᄖƞƹƢᄬᇵᄭƞѹҋ҄ҏ-wide conference called “Creativity in ҉Ҋѻ҃ᄕᅺǂƩƣƽƣƺƹƣƺƤƿƩƣƩƞᄛƸƺƸƣƹƿƾƺƤƿƩƣơƺƶƶƣơƿƫƺƹƫƹƩƞƻƿƣƽᇳᇹᄕ Ʃƣƽƣ ƾƞƣơƿƺƽᄞᄕǂƞƾƣƹơƺǀƹƿƣƽƣƢᄙƞƿǀƽƞƶƶDŽᄕƞƶƶƿƩƣƞơƿƫƺƹƾǂƣƽƣƾǀƻƻƺƽƿƣƢ by professional presentations and publications. The effort to formulate principles of Aha! pedagogy to be introduced into teaching practice at the next Ҋ҈ cycle of work will complete the second ѹҊ҈ cycle in agreement with the Ҋ҈ƸƣƿƩƺƢƺƶƺƨDŽᄙƞƽƿᇳᄕƫƾƺơƫƞƿƫƺƹƫƹƿƩƣƶƞƾƾroom, introduces readers to some of the results of the second ѹҊ҈ cycle. Part 1: Bisociation in the Classroom Chapter 2: Teaching-Research Analysis: The Constructivist Teaching Experiment as a Methodology of Teaching This chapter investigates the contradiction between the well-grounded conƾƿƽǀơƿƫǁƫƾƿƞƻƻƽƺƞơƩƿƺƶƣƞƽƹƫƹƨƞƹƢƫƿƾƤƞƫƶƫƹƨơƶƞƾƾƽƺƺƸƫƸƻƶƣƸƣƹƿƞƿƫƺƹᄙ ƿ shows the bisociative nature of teaching-research in the context of a constructive teaching experiment methodology, which can be seen both as the research instrument and as the constructivist methodology of teaching. We identify the source of its failing implementation as the lack of recognition of the inherently constructivist teaching methodology within the frame of the constructivist approach. We situate the absence of recognition within the hierarchical social/institutional divide between academia and the teaching profession. The chapter critiques scripted lesson plans that instruct teachers

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what questions to ask and tells them what students might respond and how to react as contradictory with constructivist principles. The scripted lesson plans are liable to introduce a degree of automation ƫƹ ƿƣƞơƩƣƽƾᅷ ƿƩƫƹƴƫƹƨ ƿƩƞƿ ơƺƹƿƽƞƢƫơƿƾ ƿƩƣ ơƺƹƾƿƽǀơƿƫǁƫƾƿ ƞƻƻƽƺƞơƩᄕ ǂƩƫơƩ ƫƾ geared to independent thinking and problem-posing and problem-solving. Our research suggests that teaching-research based on the goal of improvement of classroom learning is fully consistent with the constructivist approach, and it re-establishes the missing methodology of constructive teaching. Chapter 3: Classroom Facilitation of Aha! Moment Insights ƩƞƻƿƣƽᇵƻƺƫƹƿƾƿƺƿƩƣƢƫƤƤƣƽƣƹƿƸƣƿƩƺƢƺƶƺƨƫƣƾƫƹƾƿǀƢDŽƫƹƨƿƩƣơƽƣƞƿƫǁƫƿDŽƺƤ gifted and rank-and-file students. The chapter discusses the role of the social classroom creative environment in promoting student collaboration for facilƫƿƞƿƫƹƨƩƞᄛƸƺƸƣƹƿƫƹƾƫƨƩƿƾᄙƣơƿƫƺƹƾᇵᄙᇵƞƹƢᇵᄙᇶƢƣƶǁƣƫƹƿƺƿƩƣƞƹƞƶDŽƾƫƾƺƤ ƤƞơƫƶƫƿƞƿƫƺƹƸƣƿƩƺƢƾƣƹơƺǀƹƿƣƽƣƢƫƹƩƞƻƿƣƽᇳᇹᄬᅸƺƶƶƣơƿƫƺƹƺƤƩƞᄛƺƸƣƹƿƾᅺᄭ and in several papers describing others. The two essentially different cases are a classroom collaborative complex problem-solving environment and one-onone teacher/mentor scaffolding student understanding methods. Different roles of the teacher within each of the methods are thoroughly ƢƫƾơǀƾƾƣƢᄙƩƣ ƞƹƞƶDŽƾƫƾ ƺƤ ơƶƞƾƾƽƺƺƸ ƫƹƿƣƽƞơƿƫƺƹ ƢƣƾơƽƫƟƣƢ ƫƹ ƞƽƹƣƾ ᄬᇴᇲᇲᇲᄭ reveals the relationship between internalization and interiorization; the analysis of different pathways leading to Aha! moment insight reveals that some of them do not take place within problem-solving activity, but result out of a drive for conceptual understanding. Those are the instances of conceptual ƶƣƞƽƹƫƹƨᄕƞơƺƹơƣƻƿƫƹƿƽƺƢǀơƣƢƟDŽƫƸƺƹƣƿƞƶᄙᄬᇴᇲᇲᇶᄭᄙ Chapter 4: Assessment of the Depth of Knowledge Acquired during an Aha! Moment Insight This chapter addresses institutional challenges that exclude underrepresented and underserved students from the development of their creativity. The definition of this category of students appears in Section 4.2, while the analysis of ƫƹƾƿƫƿǀƿƫƺƹƞƶơƩƞƶƶƣƹƨƣƾƫƾƫƹƣơƿƫƺƹᇶᄙᇵᄙƩƣƢƫƾơǀƾƾƫƺƹƻƺƫƹƿƾƿƺƿƩƣƹƣƣƢƿƺ formulate alternative approaches and methods to foster creativity in regular students, and Section 4.4 proposes such a method of analysis, leading to the assessment of the nature of the connections made in Aha! moment creative insights. ƩƣƸƣƿƩƺƢƫƾƞƻƻƶƫƣƢƿƺƩƞƻƿƣƽᇳᇹᄕᅸƺƶƶƣơƿƫƺƹƺƤƩƞᄛƺƸƣƹƿƾᄕᅺǂƩƫơƩƞƽƣ ơƶƞƾƾƫƤƫƣƢƫƹƿƺƿƩƽƣƣơƞƿƣƨƺƽƫƣƾᄕƸƫƶƢᄕƹƺƽƸƞƶᄕƞƹƢƾƿƽƺƹƨᄙ ƿƸƣƞƾǀƽƣƾƿƩƣƢƣƨƽƣƣ

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of progress in understanding reached during the insight. While developed originally with classroom mathematical insights, some of the big C creativity Aha! moments of the collection fit neatly into the scheme. The new methodology of assessment is geared to the creativity of rank-and-file students, who by their low mathematical skill level have been excluded from being considered creative. Chapter 5: The Role of the Teacher in Facilitating the Aha! Moment The role of the teacher involves acting as a mediator between student knowledge, the actions that are viable for them, and the curriculum. The most universal aspect of this role is managing a social learning environment: classroom ƢƫƾơƺǀƽƾƣᄙƺƾǀƻƻƺƽƿƞơƽƣƞƿƫǁƣƶƣƞƽƹƫƹƨƣƹǁƫƽƺƹƸƣƹƿᄕƿƩƣƿƣƞơƩƣƽᅷƾƽƺƶƣƫƾƿƺ guide students to discover their own meaning for mathematics. This involves a focus on soliciting student insight and responses and building upon these insights to promote shared knowledge with a focus on reasoning and meaning, not simply procedural rules and algorithms.

ƹ ƿƩƫƾ ƾƣƹƾƣᄕ ƿƩƣ ƿƣƞơƩƣƽᅷƾ ƸƣƢƫƞƿƫƺƹ ƫƹǁƺƶǁƣƾ Ɵƞƶƞƹơƫƹƨ ƺƽƫƣƹƿƞƿƫƺƹ ƿƺ ensure students are exposed to and internalize the curriculum with the mindset that ultimately students must create or construct meaning themselves. This chapter examines the role of the teacher in managing discourse in constructivist pedagogy, creativity research, and teaching research within the social setting to examine the role of the teacher in promoting a culture of inquiry. Central to this chapter is the translation of constructivist teaching experiment methodology based upon guided discovery into the classroom. This effort, which can be viewed as one of the basis of Aha! pedagogy, is contrasted with the view that the reasoning of students along trajectories or pathways can be scripted for classroom use. Chapter 6: The Work of the Teaching-Research Team of the Bronx: Creativity ƺƣƾƿƶƣƽ ᄬᇳᇻᇸᇶᄭ ƻƺƫƹƿƾ ƺǀƿ ƿƩƞƿ ƿƩƣ ƶƞơƴ ƺƤ Ƣƫƽƣơƿ ƣǃƻƣƽƫƣƹơƣ ǂƫƿƩ ƿƩƣ ơƽƣative-discovery process within mathematics and science has led to a robotic, ƞƶƸƺƾƿƢƣƩǀƸƞƹƫDžƫƹƨƣƢǀơƞƿƫƺƹƞƶƣǃƻƣƽƫƣƹơƣᄙ ƹƿƩƫƾƶƫƨƩƿᄕƫƿƫƾƺƹƶDŽƿƩƣƣǃƻƣrience of discovering math for themselves that will ultimately motivate students who view mathematics as the memorization and application of rules to appreciate the beauty involved in interpreting and resolving a problem sitǀƞƿƫƺƹᄙ ƹƿƩƫƾƾƣơƿƫƺƹᄕǂƣƽƣǁƫƣǂƶƫƿƣƽƞƿǀƽƣƺƹơƽƣƞƿƫǁƫƿDŽǂƫƿƩƫƹƸƞƿƩƣƸƞƿƫơƾ as it relates to classroom situations involving students who have not necessarily been labeled gifted. More specifically, we review the notion of creative

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ѹ‫ؔ؛ؖآءإؔح‬

ƽƣƞƾƺƹƫƹƨ ƣƽƾƩƴƺǂƫƿDžƣƿƞƶᄙᄬᇴᇲᇳᇹᄭǀƾƣƢƿƺơƶƞƾƾƫƤDŽƾƿǀƢƣƹƿƫƹƾƫƨƩƿǂƫƿƩƫƹƿƩƣ social discourse of the classroom. We then present examples of student insight within the classrooms of our team of teacher researchers and analyze them within our bisociative frame ᄬƩƞƻƿƣƽ ᇳᄭᄙƩƣ ƞƹƞƶDŽƾƫƾ ƺƤ ƿƩƣƾƣ ƸƺƸƣƹƿƾ ƺƤ ƾƿǀƢƣƹƿ ơƽƣƞƿƫǁƫƿDŽ Ƥƺơǀƾƣƾ ƺƹ ƿƩƣƿƩƽƣƣơƽƫƿƣƽƫƞƤƺƽơƽƣƞƿƫǁƫƿDŽƢƣǁƣƶƺƻƣƢƫƹƩƞƻƿƣƽᇳᇳᄙ ƫƽƾƿᄕƿƩƣƾƣƞƽơƩƻƽƺcess leading up to the moment of insight; second, the connection made by the student during the moment of insight; and third, the creative reasoning that underlies the novel activity that results from such a connection. Part 2: The Aha! Moment and Affect ƩƣƫƸƻƞơƿƺƤƞƹƩƞᄛƸƺƸƣƹƿƺƹƞƶƣƞƽƹƣƽᅷƾƻƺƾƫƿƫǁƣƞƤƤƣơƿƫƾƢƫƾơǀƾƾƣƢƫƹ ƸƞƹDŽơƶƞƾƾƽƺƺƸƢƣƾơƽƫƻƿƫƺƹƾƺƤƿƩƣƻƩƣƹƺƸƣƹƺƹᄬƤƺƽƣǃƞƸƻƶƣᄕƞƽƹƣƾᄕᇴᇲᇲᇲᄖ ƫƶưƣƢƞƩƶᄕᇴᇲᇳᇵᄭᄙƩƞƿᅷƾƸƺƽƣᄕƿƩƣƽƺƶƣƺƤƾƺᅟƨƣƹƣƽƞƿƣƢƻƺƾƫƿƫǁƣƞƤƤƣơƿǀƻƺƹƾƿǀƢƣƹƿƾᅷƞƿƿƫƿǀƢƣƿƺƸƞƿƩƣƸƞƿƫơƾƩƞƾƟƣƣƹƺƟƾƣƽǁƣƢƟDŽƽƞƟƩǀᄬᇴᇲᇳᇸᄭƫƹƩƣƽƺƽƫƨƫnal teaching experiment. Therefore, the following idea suggests itself: to design a pedagogy based on the facilitation of an Aha! moment in order not only to provide students with the real creative experience of insight and discovery, but also to change their generally negative attitude to mathematics through the ƻƽƺơƣƾƾƺƤƟƺƹƢƫƹƨǂƫƿƩƫƿᄬƣƣƶƶƫƾѵƺƶƢƫƹᄕᇴᇲᇲᇸᄭᄙƩǀƾᄕƿƩƣƿƞƾƴƺƤƞƽƿᇴᄘ The Aha! Moment and Affect is to investigate the processes through which a ƶƣƞƽƹƣƽᅷƾƟƺƹƢƫƹƨǂƫƿƩƸƞƿƩƣƸƞƿƫơƾƿƞƴƣƾƻƶƞơƣᄙ A couple of words about the relationship between cognition and positive ƞƤƤƣơƿƞƽƣƫƹƻƶƞơƣᄙƺǀƹƫƺƾƞƹƢƣƣƸƞƹᄬᇴᇲᇳᇷᄭᅭƿƩƣƿǂƺƽƣƾƣƞƽơƩƣƽƾǂƩƺƾƣ teams designed very effective methods of neuroscientific investigation of Aha! ƸƺƸƣƹƿ ƫƹƾƫƨƩƿ ᄬƩƞƻƿƣƽ ᇳᇶᄭᅭƫƹƤƺƽƸ ǀƾ ƞƟƺǀƿ ƿǂƺ Ƹƞưƺƽ ƽƣƾƣƞƽơƩ ƣƤƤƺƽƿƾ establishing the fact that a good mood increases the frequency of insight: ƾƩƟDŽƣƿƞƶᄙᄬᇳᇻᇻᇻᄭƞƹƢƸƞƟƫƶƣƣƿƞƶᄙᄬᇴᇲᇲᇷᄭᄙƩǀƾᄕǂƣƾƣƣƞǁƣƽDŽƫƹƿƣƽƣƾƿƫƹƨ dialectical relationship: on the one hand, the positive mood increases the frequency of insight, and, on the other hand, the experience of insight creates a positive mood. We see that mainstream neuroscience seems to be primarily interested in the impact of a good mood upon insight, whereas classroom teaching requires deep understanding of the inverse process and how insight produces a good ƸƺƺƢᄕǂƩƫƶƣơƺƹƿƽƫƟǀƿƫƹƨƿƺƟƺƹƢƫƹƨǂƫƿƩƸƞƿƩƣƸƞƿƫơƾᄙƺǀƹƫƺƾƞƹƢƣƣƸƞƹ ᄬᇴᇲᇳᇷᄕƻᄙᇳᇳᇻᄭƻƽƺǁƫƢƣƞƹƺƿƩƣƽƫƹƿƣƽƣƾƿƫƹƨƺƟƾƣƽǁƞƿƫƺƹᄘᅸƩƺƾƣƻƣƺƻƶƣƫƹƞƻƺƾitive mood base their judgments on the overall pattern; those in a negative mood base their judgments on the parts.” These observations agree with our ơƶƞƾƾƽƺƺƸƾƫƿǀƞƿƫƺƹᄘǂƣơƞƹᅷƿǁƣƽDŽƸǀơƩơƺƹƿƽƺƶƿƩƣƸƺƺƢƺƤƺǀƽƾƿǀƢƣƹƿƾᄕƺƹ

ѿ‫ءآ؜اؖبؗآإاء‬

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one hand, and on the other, most rank-and-file students base their judgments in classes of mathematics on the scattered parts, which suggests they are not, ƨƣƹƣƽƞƶƶDŽᄕƫƹƞƨƺƺƢƸƺƺƢᄙƩƞƿᅷƾǂƩDŽƿƩƣƢǀƞƶƽƺƶƣƺƤƿƩƣƩƞᄛƸƺƸƣƹƿƫƾƾƺ ƫƸƻƺƽƿƞƹƿ ᄬDžƞƽƹƺơƩƞᄕ ᇴᇲᇳᇶᄭᄙ ƿ ưƺƫƹƾ ƾƣƻƞƽƞƿƣ ƻƞƽƿƾ ƫƹƿƺ ƺƹƣ ƾƿƽǀơƿǀƽƣᄕ ƞƹƢ ǂƩƫƶƣƢƺƫƹƨƫƿƶƫƤƿƾƞƶƣƞƽƹƣƽᅷƾƸƺƺƢᄙƿǀƢƣƹƿƾᅷǁƣƽƟƞƶƣǃƻƽƣƾƾƫƺƹƾƺƤƿƩƣƫƽƞƤƤƣơƿ give us information about their basic human needs as they manifest while doing mathematics, and that allows for the redesign of the instruction to make it closer in agreement with those needs. The structure of the part is interesting in that it starts from the impact of the ƺǀƿƾƫƢƣǂƺƽƶƢǀƻƺƹƾƿǀƢƣƹƿƾᅷǀƹƢƣƽƾƿƞƹƢƫƹƨƺƤƿƩƣƫƽƺǂƹơƽƣƞƿƫǁƫƿDŽᄬƩƞƻƿƣƽ ᇹᄭᄙƣǃƿᄕƩƞƻƿƣƽᇺƣǃƻƞƹƢƾƿƩƞƿƢƫƾơǀƾƾƫƺƹƿƺƣƸƻƩƞƾƫDžƣƿƩƣƽƺƶƣƺƤƿƩƣƻƺƾitive Aha! and the negative Uh-huh! impact on the development of personal ơƽƣƞƿƫƺƹ ƺƤ Ƹƣƞƹƫƹƨ ƫƹ ƸƞƿƩƣƸƞƿƫơƾᄙ Ʃƞƻƿƣƽ ᇻ ƻƽƣƾƣƹƿƾ ƞ ƹƣǂ ƿƩƣƺƽƣƿƫơƞƶ framework for the discussion of affect in mathematics education involving the ơƺƹơƣƻƿƺƤơƺƹƞƿƫƺƹᄕǂƩƫƶƣƩƞƻƿƣƽᇳᇲƞƢƢƽƣƾƾƣƾơƺƨƹƫƿƫƺƹᄕƞƤƤƣơƿᄕƞƹƢơƺƹƞtion and develops the details of the relationship between cognition and affect, ƻƺƫƹƿƫƹƨƿƺǂƫƢƣƣǁƫƢƣƹơƣƤƺƽƿƩƣƢƣǁƣƶƺƻƸƣƹƿƺƤƿƩƣƟƺƹƢƫƹƨƻƽƺơƣƾƾᄬƣƣƶƶƫƾѵƺƶƢƫƹᄕᇴᇲᇲᇸᄭᄙƣƤƫƹƢƩƣƽƣƞƾǀƽƻƽƫƾƫƹƨƞƨƽƣƣƸƣƹƿƟƣƿǂƣƣƹƿƩƣƿƩƣƺƽƣƿƫơƞƶƾǀƨƨƣƾƿƫƺƹƾƺƤƺƶƢƫƹᄬƩƞƻƿƣƽᇻᄭƞƹƢƿƩƣƣƸƻƫƽƫơƞƶƽƣƾǀƶƿƾƺƤƫƶưƣƢƞƩƶ ᄬᇴᇲᇲᇶᄭƻƽƣƾƣƹƿƣƢƫƹƿƩƣơƩƞƻƿƣƽᄙ Chapter 7: Creativity in the Eyes of Students: Espoused and Enacted Beliefs in Mathematical Projects Ʃƣ ơƩƞƻƿƣƽ ƫƹǁƣƾƿƫƨƞƿƣƾ ƾƿǀƢƣƹƿƾᅷ ƾƣƶƤᅟƞƾƾƣƾƾƸƣƹƿ ƺƤ ƿƩƣƫƽ ƺǂƹ ơƽƣƞƿƫǁƫƿDŽ ƫƹ the context of a project-based learning high school course focused on coding and cryptography. The central question of the teaching experiment was how students understand their own creativity. The authors point out that within the knowledge of creativity in the educational profession, there is no information about how students view and assess their own creativity: solely “outside” views of teachers and researchers are considered. They emphasize that if teachers or lecturers expect creativity from their students, they need to clarify their own understanding of creativity first. ConƾƣƼǀƣƹƿƶDŽᄕƿƩƣƽƣƫƾƞƹƣƣƢƿƺƢƫƾƿƫƹƨǀƫƾƩƟƣƿǂƣƣƹƽƣƾƣƞƽơƩƣƽƾᅷƞƾƾƣƾƾƸƣƹƿƺƤ ƾƿǀƢƣƹƿơƽƣƞƿƫǁƫƿDŽƞƹƢƾƿǀƢƣƹƿƾᅷƺǂƹƞƾƾƣƾƾƸƣƹƿᄙƫƾƺơƫƞƿƫǁƫƿDŽƿƩƣƺƽDŽƞƾƾƣƣƹ ƟDŽƺƣƾƿƶƣƽᄬᇳᇻᇸᇶᄭƞƹƢƺƢƣƹᄬᇴᇲᇲᇶᄭơƞƹƟƣƩƣƶƻƤǀƶƫƹƢƣƶƫƹƣƞƿƫƹƨƿƩƣƹƞƿǀƽƣƺƤ ƿƩƣƢƫƤƤƣƽƣƹơƣƾƟƣƿǂƣƣƹƾƿǀƢƣƹƿƾᅷƞƹƢƿƩƣƫƽƿƣƞơƩƣƽƾᅷƻƣƽơƣƻƿƫƺƹƾᄙ ƞƾƣƢ ƺƹ ƾƿǀƢƣƹƿƾᅷ ƽƣƾƣƞƽơƩ Ƣƫƞƽƫƣƾᄕ ƿƩƣ ƞǀƿƩƺƽƾ ƢƫƾƿƫƹƨǀƫƾƩ Ɵƣƿǂƣƣƹ ƾƿǀdent-espoused and student-enacted creativity and look for the relationship ƟƣƿǂƣƣƹƿƩƣƸƿƩƽƺǀƨƩƺǀƿƿƩƣƻƽƺưƣơƿᄙƩƣƫƽƽƣƾƣƞƽơƩƼǀƣƾƿƫƺƹƾƞƽƣᄬᇳᄭƩƞƿ

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ѹ‫ؔ؛ؖآءإؔح‬

ƞƽƣƾƿǀƢƣƹƿƾᅷƻƣƽơƣƻƿƫƺƹƾƺƤơƽƣƞƿƫǁƫƿDŽᅭƿƩƣƾƺᅟơƞƶƶƣƢƣƾƻƺǀƾƣƢơƽƣƞƿƫǁƫƿDŽᄞƞƹƢ ᄬᇴᄭƩƫƶƣǂƺƽƴƫƹƨƺƹƸƞƿƩƣƸƞƿƫơƞƶƻƽƺưƣơƿƾᄕƢƺƾƿǀƢƣƹƿƾƞơƿƫƹƞơơƺƽƢƞƹơƣ with their perception of creativity? This is the so-called enacted creativity. Ʃƣ ơƩƞƻƿƣƽ ƿƽƞơƣƾ ơƩƞƹƨƣƾ ƫƹ ƾƿǀƢƣƹƿƾᅷ ƞƾƾƣƾƾƸƣƹƿ ƺƤ ơƽƣƞƿƫǁƫƿDŽ ǂƩƫƶƣ ǂƺƽƴƫƹƨƺƹƿƩƣƻƽƺưƣơƿᄕƤƫƹƢƫƹƨƺǀƿƿƩƞƿᇳᇶƺǀƿƺƤᇴᇶƾƿǀƢƣƹƿƾƻƞƽƿƫơƫƻƞƿƫƹƨƫƹ the study changed their views about their own creativity. The authors point out that student self-regulation, a problem-solving orientation, and their own perception of creativity affect each other. Chapter 8: Building Long-term Meaning in Mathematical Thinking: Aha! and Uh-huh! This chapter focuses on how we humans make sense of mathematics in the long term, which involves not only grasping the positive side that gives us insight into understanding and creating more sophisticated mathematical ideas, but also understanding the negative side that impedes our progress. The chapter seeks to promote the positive development of mathematical thinking, but this cannot be done while ignoring the negative aspects that arise from personal difficulties with mathematics and emotions such as fear and anxiety. The chapter begins with a study of the structure and operation of the human brain sufficient to understand the relationship between mathematical thinking and personal aspects of emotional reactions and attitudes toward mathematics. Then we consider the cultural development of mathematical thinking to gain insight into how communities can hold very different views of the nature of mathematics. The author emphasizes that the connection of previously unlinked frameworks in mathematics operates not only in the creation of original ideas but also in the teaching and learning of mathematics, where learners are faced with the need to make new connections in their own minds. This involves not only the linking of different mathematical ideas, but also ƞƽƞƹƨƣƺƤƞƿƿƫƿǀƢƣƾƞƹƢƣƸƺƿƫƺƹƾƻƣƽƾƺƹƞƶƿƺƣƞơƩƫƹƢƫǁƫƢǀƞƶᄙ ƹƿƩƣƤƫƹƞƶƾƣơtions, the author presents his own Aha! moments of insight in the domain of mathematics education: the personal Aha!s that have offered him insight into ƿƩƣƢƣǁƣƶƺƻƸƣƹƿƺƤƸƞƿƩƣƸƞƿƫơƞƶƿƩƫƹƴƫƹƨᅭƾǀơƩƞƾơƺƹơƣƻƿƫƸƞƨƣᄬƞƶƶѵ ƫƹƹƣƽᄕᇳᇻᇺᇳᄭᄕƿƩƣƹƺƿƫƺƹƺƤᅸƶƺơƞƶƾƿƽƞƫƨƩƿƹƣƾƾᅺƿƺƨƫǁƣƣƸƟƺƢƫƣƢƸƣƞƹƫƹƨƿƺ ƿƩƣƢƣƽƫǁƞƿƫǁƣᄬƞƶƶᄕᇳᇻᇺᇷᄭᄕƸƞƿƩƣƸƞƿƫơƞƶƾDŽƸƟƺƶƾƫƹƿƣƽƻƽƣƿƣƢƤƶƣǃƫƟƶDŽƞƾƻƽƺơƣƾƾƞƹƢơƺƹơƣƻƿᄬƻƽƺơƣƻƿᄭᄬƽƞDŽѵƞƶƶᄕᇳᇻᇻᇶᄭᄕƿƩƽƣƣǂƺƽƶƢƾƺƤƸƞƿƩƣƸƞƿƫơƾ ᄬƞƶƶᄕ ᇴᇲᇲᇶᄭᄕ ƾƿƽǀơƿǀƽƣ ƿƩƣƺƽƣƸƾ ƿƺ ƨƫǁƣ ƣƸƟƺƢƫƣƢ ƞƹƢ ƾDŽƸƟƺƶƫơ Ƹƣƞƹƫƹƨ ƿƺ ƤƺƽƸƞƶƾƿƽǀơƿǀƽƣƾᄬƞƶƶᄕᇴᇲᇳᇵᄭᄕƞƹƢƸƞƴƫƹƨƾƣƹƾƣƺƤƸƞƿƩƣƸƞƿƫơƞƶƣǃƻƽƣƾƾƫƺƹƾ ƿƩƽƺǀƨƩƾƻƺƴƣƹƞƽƿƫơǀƶƞƿƫƺƹᄬƞƶƶƣƿƞƶᄙᄕᇴᇲᇳᇹᄭᄙ

Ƹƻƺƽƿƞƹƿ ƾǀƨƨƣƾƿƫƺƹƾ Ƥƺƽ Ʃƞᄛ ƻƣƢƞƨƺƨDŽ ƢƣƽƫǁƣƢ ƤƽƺƸ ƿƩƣ ơƩƞƻƿƣƽᅷƾ ơƺƹtent are as follows: Approaches that extensively incorporate facilitation of Aha!

ѿ‫ءآ؜اؖبؗآإاء‬

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moments cannot use Aha! moments to form the whole substance of lessons, as one sees in Japanese Lesson Study. A specific Aha! moment needs to be a highlight in the teaching and learning which focuses on an important idea that gives substantial insight. For this to happen, the teacher must be aware of the mathematical meaning, placed in a longer-term plan, with appropriate preparation, time for quiet contemplation, and dedicated effort for consolidation of the ideas. Chapter 9: A Conative Perspective on Aha! Moments ƩƞƻƿƣƽᇻƻƽƣƻƞƽƣƾƿƩƣƨƽƺǀƹƢƤƺƽƿƩƣƢƫƾơǀƾƾƫƺƹƾƺƤơƺƨƹƫƿƫƺƹƞƹƢƞƤƤƣơƿƫƹƿƩƣ follow-up chapter by including conation—a concept less well-known in the ƣƢǀơƞƿƫƺƹƻƽƺƤƣƾƾƫƺƹᄙ ƿƻƣƽƿƞƫƹƾƿƺƿƩƣƹƣƣƢƾᄕƻǀƽƻƺƾƣƾᄕƫƹƿƣƹƿƫƺƹƾᄕƺƽǂƫƶƶƺƤ ƿƩƣƫƹƢƫǁƫƢǀƞƶᅭƿƩƞƿƫƾᄕƫƿƣƹơƺƸƻƞƾƾƣƾƿƩƣǂƩDŽƺƤƩǀƸƞƹƞơƿƫǁƫƿDŽᄬƤƺƽƣǃƞƸƻƶƣᄕƹƺǂƣƿƞƶᄙᄕᇳᇻᇻᇸᄭᄙ ƣƽƣǂƣƴƣƣƻƫƹƸƫƹƢƿƩƽƣƣƼǀƣƾƿƫƺƹƾᄘᄬᇳᄭƩƞƿƫƾƫƿƿƩƞƿ drivesƞƹƫƹƢƫǁƫƢǀƞƶƿƺǂƞƽƢƿƩƣƸƺƸƣƹƿƺƤƩƞᄛᄞᄬᇴᄭWhy are such moments ƞơơƺƸƻƞƹƫƣƢƟDŽƣƸƺƿƫƺƹƾƺƤƫƹƿƣƹƾƣƻƺƾƫƿƫǁƣǁƞƶƣƹơƣᄞᄬᇵᄭƩƞƿƸƫƨƩƿƞơƺƹƞtive perspective suggest for rethinking prevailing approaches to mathematics education and for the role of Aha! moments in teaching? The discussion of conation is anchored in consideration of basic human needs.

ƿƾƩƺǂƾƩƺǂƿƩƽƺǀƨƩƿƩƣƞƹƞƶDŽƾƫƾƺƤƿƣƽƸƾƢƣƾơƽƫƟƫƹƨƾƿǀƢƣƹƿƞƤƤƣơƿǂƣơƞƹơƺƹclude about the personal needs or desires that drive student engagement, especially in-the-moment engagement. The chapter provides a model for describing student mathematical engagement; it points to further development of the bondƫƹƨƞƾƻƣơƿᄬƣƣƶƶƫƾѵƺƶƢƫƹᄕᇴᇲᇲᇸᄭᄕǂƩƫơƩǂƞƾƞƿƿƽƫƟǀƿƣƢƺƽƫƨƫƹƞƶƶDŽƿƺƿƩƣƫƹƿƣƽaction between positive outcome emotions and mathematical cognition. Here, outcome emotions of the Aha! moment, serving an essential repreƾƣƹƿƞƿƫƺƹƞƶƤǀƹơƿƫƺƹᄕƞƽƣƫƹƾƺƸƣǂƞDŽƣƹơƺƢƫƹƨƿƩƣƤǀƶƤƫƶƶƸƣƹƿƺƤƿƩƣƻƣƽƾƺƹᅷƾ ƤǀƹƢƞƸƣƹƿƞƶƹƣƣƢƾᄙ ƿƫƾƫƹƿƩƫƾơƺƹƞƿƫǁƣƢƫƸƣƹƾƫƺƹƿƩƞƿƟƺƹƢƫƹƨǂƫƿƩƸƞƿƩƣmatics occurs. From the point of view of Aha! pedagogy, mathematics educators should be aware of the many ways in which mathematics can meet fundamental needs—from satisfying curiosity to gaining in power, from achieving recognition to participating fully in a community. And, ideally, we should provide a variety of contexts offering opportunities to fulfill such at-the-moment needs through mathematics. Chapter 10: Illuminating Aha! Moments through the Relationships between Cognition, Affect, and Conation The chapter focuses attention on two essential components of Aha! moments: ơƺƨƹƫƿƫǁƣƫƹƾƫƨƩƿƞƹƢƞƤƤƣơƿƫǁƣƫƸƻƞơƿᄙ ƿƶƫƾƿƾƢƫƤƤƣƽƣƹƿƢƣƤƫƹƫƿƫƺƹƾƺƤƿƩƫƾƻƩƣƹƺƸƣƹƺƹᄕǂƩƫơƩƽƣƤƶƣơƿƿƩƣƫƸƻƺƽƿƞƹơƣƺƤƫƿƾƢƫƤƤƣƽƣƹƿƞƾƻƣơƿƾᄙ ƹƨƣƹƣƽƞƶᄕƿƩƣ

ሥሬ

ѹ‫ؔ؛ؖآءإؔح‬

definitions reflect the cognitive bent of the insight; however, observations from the classrooms suggest the need to incorporate affect into its very definition. ƩƣơƩƞƻƿƣƽƾƿƞƽƿƾǂƫƿƩǂƩƞƿᅷƾơƞƶƶƣƢƞƻƞƽƞƢƫƨƸƞƿƫơƣǃƞƸƻƶƣƺƤƿƩƣƫƹƾƫƨƩƿ ᄬƾƣƣƿƩƣƫƸƩƞᄛƸƺƸƣƹƿƫƹƩƞƻƿƣƽᇳᇹᄭǂƩƫơƩơƶƣƞƽƶDŽƾƩƺǂƾƿƩƣƫƹƿƣƽƞơƿƫƺƹ between both cognitive and emotional components. ƣơƿƫƺƹ ᇳᇲᄙᇴ ƺƹ ơƺƨƹƫƿƫƺƹ ƽƣǁƫƣǂƾ ƿƩƣ ơƺƨƹƫƿƫǁƣ ƞƾƻƣơƿƾ ƺƤ ƿƩƣ ƫƹƾƫƨƩƿᄕ ƻƺƫƹƿƫƹƨ ƿƺ ƢƫƤƤƣƽƣƹƿ ƶƣǁƣƶƾ ƺƤ ơƺƨƹƫƿƫǁƣ ơƺƸƻƶƣǃƫƿDŽ ƻƽƣƾƣƹƿ ƫƹ ƾƿǀƢƣƹƿƾᅷ ƞƹƢ ƽƣƾƣƞƽơƩƣƽƾᅷ Ƣƣƾơƽƫƻƿƫƺƹƾᄙ ƿ ƣƸƻƩƞƾƫDžƣƾ ƿƩƣ ƻƺƾƾƫƟƫƶƫƿDŽ ƺƤ ƨƣƹƣƽƞƶƫDžƞƿƫƺƹ ƺƤ ƟƫƾƺơƫƞƿƫƺƹƿƺƸǀƶƿƫƾƺơƫƞƿƫƺƹƫƸƻƶƫơƫƿƫƹ ƫƹƾƿƣƫƹᅷƾƢƣƾơƽƫƻƿƫƺƹƺƤƿƩƣƿƩƺǀƨƩƿ ƻƽƺơƣƾƾᄬ ƫƹƾƿƣƫƹᄕᇳᇻᇶᇻᄭᄙƩƣơƩƞƻƿƣƽƢƫƽƣơƿƾƺǀƽƞƿƿƣƹƿƫƺƹƿƺƿƩƣƤƞơƿƿƩƞƿƿƩƣ act of connecting unconnected matrices of thought suggests that creativity of Aha! moments is the process of building the schema of thinking: the network of relationships between relevant concepts. The ability to create such connected networks and to move along them means the ability to think. ƣơƿƫƺƹᇳᇲᄙᇵƺƹƞƤƤƣơƿƞƹƢơƺƨƹƫƿƫƺƹƻƽƺǁƫƢƣƾƢƞƿƞƤƽƺƸƿƩƣƿƣƞơƩƫƹƨƣǃƻƣƽiment showing the large scope and richness of the impact of cognitive work ƫƹƻƽƺƟƶƣƸᅟƾƺƶǁƫƹƨƺƹƾƿǀƢƣƹƿƣƸƺƿƫƺƹƾᄬƫƶưƣƢƞƩƶᄕᇴᇲᇳᇵᄭᄙƾƿƩƣƢƞƿƞƽƣơƺƽƢƾ ƾƿǀƢƣƹƿƾᅷƣƸƺƿƫƺƹƾƞƤƿƣƽƿƩƣƣǃƻƣƽƫƣƹơƣƺƤƿƩƣƫƽƫƹƾƫƨƩƿᄕƸƺƾƿƽƣƾƻƺƹƾƣƾƫƹƢƫcate the presence of positive emotions, although the presence of negative emotions is noted, primarily in the early stages before the insight takes place. The description of student affect is followed by its descriptions by research mathematicians. Here their affect is more of a transcendental nature in their descriptions of aesthetic and philosophical reflections upon the nature of insight. The chapter coalesces in the third section on the relationship between affect and emotion, pointing to the high possibility of development of mathematical ƫƹƿƫƸƞơDŽƞƹƢƾƿǀƢƣƹƿƟƺƹƢƫƹƨǂƫƿƩƸƞƿƩƣƸƞƿƫơƾᄬƣƣƶƶƫƾѵƺƶƢƫƹᄕᇴᇲᇲᇸᄭᄙ

3

Theoretical Framework: Networking of Theories

The natural theoretical framework for the work in the volume is the recently ƤƺƽƸǀƶƞƿƣƢ ᄬƞƹƢ ƾƿƫƶƶ ƫƹ statu nascendiᄭ ƹƣƿǂƺƽƴƫƹƨ ƿƩƣƺƽƫƣƾ ƞƻƻƽƺƞơƩᄙ ƫƴƹƣƽᅟƩƾƟƞƩƾ ƣƿ ƞƶᄙ ᄬᇴᇲᇳᇲᄭᄖ ƞ ƾƻƣơƫƞƶ ƫƾƾǀƣ ƺƤ Ѡъѓ: The International Journal on Mathematics Education on networking strategies for connecting theƺƽƣƿƫơƞƶ ƞƻƻƽƺƞơƩƣƾ ᄬƽƣƢƫƨƣƽ ƣƿ ƞƶᄙᄕ ᇴᇲᇲᇺᄖ ƞƢƤƺƽƢᄕ ᇴᇲᇲᇺᄭᄖ ƞƹƢ ƿƩƣ ƣƢƫƿƣƢ volume Networking of Theories as a Research Practice in Mathematics Education ᄬƫƴƹƣƽᅟƩƾƟƞƩƾ ѵ ƽƣƢƫƨƣƽᄕ ᇴᇲᇳᇶᄭ ƞƽƣ ƿƩƣ Ƹƞƫƹ ƢƺơǀƸƣƹƿƾ ƾƻƣơƫƤDŽƫƹƨ the parameters of the approach, whose aim is to provide a conceptual space

ሥር

ѿ‫ءآ؜اؖبؗآإاء‬

within which different theories are able to interact with each other. Prediger ƞƹƢƫƴƹƣƽᅟƩƾƟƞƩƾᄬᇴᇲᇳᇶᄭƫƹƤƺƽƸǀƾƿƩƞƿᅸƟDŽƹƣƿǂƺƽƴƫƹƨǂƣƸƣƞƹƽƣƾƣƞƽơƩ practices that aim at creating the dialogue and establishing relationships between parts of theoretical approaches while respecting identity of each ƞƻƻƽƺƞơƩᅺᄬƻᄙᇳᇳᇺᄭᄙƩǀƾᄕǂƣƾƣƣƿƩƞƿƺǀƽǂƺƽƴƺƹƫƹƿƣƨƽƞƿƫƹƨƿƩƣƟƫƾƺơƫƞƿƫƺƹ theory of Aha! moment creativity with the processes and theories of learning can be welcomed with the theoretical framework formulated within the last decade, which precisely addresses our need of the moment. Conceptually, networking and coordinating or combining theories is close to the process of connecting unconnected matrices of thought—the subject of bisociation theory itself. Consequently, our interest in coordinating the theory of bisociation with the theory of learning of Piaget is aimed at finding creativƫƿDŽƫƹƣǁƣƽDŽƶƣǁƣƶƺƤƾơƩƣƸƞƢƣǁƣƶƺƻƸƣƹƿƞƾƾƣƣƹƤƽƺƸƿƩƣƫƞƨƣƿᅬƞƽơƫƞƽƫƞƢᄙ Our work here focused on two integration processes of a different nature: ᇳᄙ ƹƿƣƨƽƞƿƫƺƹ ƺƤ ƿƩƣ ƿƩƣƺƽDŽ ƺƤ Ɵƫƾƺơƫƞƿƫƺƹ ǂƫƿƩ ơƶƞƾƾƽƺƺƸ ƞƹƢ ƽƣƾƣƞƽơƩ practice 2. ƹƿƣƨƽƞƿƫƺƹ ƺƤ ƿƩƣ ƿƩƣƺƽDŽ ƺƤ Ɵƫƾƺơƫƞƿƫǁƣ ơƽƣƞƿƫǁƫƿDŽ ǂƫƿƩ ƿƩƣ ƿƩƣƺƽƫƣƾ ƺƤ learning Regarding the integration of a theory or research with classroom practice, a standard Ҋ҈ practice consists in identifying elements of the relevant classroom mathematical situation with the concepts of the theory and using them in understanding and coordinating classroom events. Thus, we not only coordinate the concepts of the theory within classroom discourse but also use the newly established theoretical connections to guide our classroom actions. ƩƣǂƺƽƴƺƤƽƞƟƩǀᄬᇴᇲᇳᇸᄭᄕǂƩƺƟƽƺǀƨƩƿƺƣƾƿƶƣƽᅷƾƿƩƣƺƽDŽƿƺƿƩƣơƶƞƾƾƽƺƺƸᄕ ƨƫǁƣƾ ǀƾ ƞƹ ƣǃƞƸƻƶƣ ƺƤ ƿƩƣ ƻƽƺơƣƾƾ ƺƤ ơƺƺƽƢƫƹƞƿƫƺƹᄙ ƸƸƣƢƫƞƿƣƶDŽ ƞƤƿƣƽ ƾƩƣ understood its central component of focusing attention on the gap between two subdomains in classroom mathematics, she designed an exercise based ƺƹƺƣƾƿƶƣƽᅷƾƿƽƫƻƿDŽơƩᄕƞƾƴƫƹƨƾƿǀƢƣƹƿƾƿƺƢƫƾơǀƾƾƿƩƣƽƣƶƞƿƫƺƹƾƩƫƻƞƸƺƹƨƿƩƽƣƣ different concepts of her statistics class: Trailblazer

Outlier

Originality

‫ ؘإبؚ؜ؙ‬ѿᄙᇴᏻƽƫƻƿDŽơƩᅟƟƞƾƣƢƾƿǀƢƣƹƿƞƾƾƫƨƹƸƣƹƿ

Such triptych assignments, whose design has been suggested by bisociation ƿƩƣƺƽDŽᄕ Ƥƞơƫƶƫƿƞƿƣ ƞ ƾƿǀƢƣƹƿᅷƾ ƾƣƞƽơƩ Ƥƺƽ ơƺƹƹƣơƿƫƺƹƾ Ɵƣƿǂƣƣƹ ᄬƺƤƿƣƹ Ƥƺƽ ƾƿǀƢƣƹƿƾᄭƾƣƻƞƽƞƿƣơƺƹơƣƻƿǀƞƶƾǀƟƢƺƸƞƫƹƾƞƹƢƻƽƺƸƺƿƣƞǂƞƽƣƹƣƾƾƺƤƿƩƣơƺƹnections between relevant concepts. This results in progress of understanding.

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ѹ‫ؔ؛ؖآءإؔح‬

ƺǂƣǁƣƽᄕǂƩƞƿᅷƾƸƺƽƣƫƸƻƺƽƿƞƹƿƫƾƿƩƞƿƿƩƣƾƣƞƾƾƫƨƹƸƣƹƿƾƤƞơƫƶƫƿƞƿƣơƶƞƾƾƽƺƺƸ Ƣƫƾơǀƾƾƫƺƹᄕ ǂƩƫơƩ Ʃƣƶƻƾ Ɵƽƣƞƴ ƿƩƣ ᅸ  ơƞƹƹƺƿ Ƣƺᅺ ƾDŽƸƻƿƺƸ ƺƤ ƾƿǀƢƣƹƿƾ ƞƹƢƿƽƞƹƾƤƺƽƸƫƿƫƹƿƺƞƽƣƞƶƺƽƫƨƫƹƞƶƫƿDŽᄬƽƞƟƩǀᄕᇴᇲᇳᇸᄕƻᄙᇳᇳᇺᄭᄙƣƾƣƣƩƣƽƣƩƺǂ through the process of establishing the connections and incorporating them into the design of classroom learning, the teacher-researcher develops a richer schema of thinking about the classroom events by incorporating the concepts of the theory already grounded in classroom experience into his or her concepƿǀƞƶƤƽƞƸƣǂƺƽƴᄕƤƺƽƸƫƹƨƞƹƣǂơƺƸƻƺƹƣƹƿƺƤƿƩƫƹƴƫƹƨƿƣơƩƹƺƶƺƨDŽᄬƽƞƟƩǀᄕ ᇴᇲᇳᇸᄭᄙ We become theoretically and practically guided by the concept of bisociation, the Aha! moment. Here the central identification is in finding the bisociaƿƫǁƣƤƽƞƸƣᄬƺƽƣǁƣƹƸǀƶƿƫƾƺơƫƞƿƫǁƣƤƽƞƸƣᄭǂƫƿƩƫƹƿƩƣƸƞƿƩƣƸƞƿƫơƞƶᄧơƺƨƹƫƿƫǁƣ content of the insight as reported by a student or more generally by an individual who experienced it and analyzing the cognitive connections made through the insight, together with hidden analogies that might have brought them. ᄬƺƾƿƩƞᄛƸƺƸƣƹƿƾƫƹƩƞƻƿƣƽᇳᇹƩƞǁƣƟƣƣƹƞƹƞƶDŽDžƣƢƫƹƢƫƤƤƣƽƣƹƿơƩƞƻƿƣƽƾ ƞơơƺƽƢƫƹƨƿƺƟƫƾƺơƫƞƿƫǁƣƤƽƞƸƣƾƞƹƢƩƫƢƢƣƹƞƹƞƶƺƨƫƣƾᄙᄭƩƣơƣƹƿƽƞƶƢDŽƹƞƸƫơ effect of such an integration by a teacher-researcher is the development of a ƿƣƞơƩƣƽᅷƾƽƣƾƣƞƽơƩƞƹƢƻƽƞơƿƫơƣƫƹƿǀƫƿƫƺƹƤƺƽƤƞơƫƶƫƿƞƿƫƺƹƞƹƢƺƞƾƾƣƾƾƸƣƹƿ ƺƤƩƞᄛƸƺƸƣƹƿƾᄬƩƞƻƿƣƽᇶᄭᄙ The integration between two different theories of learning is more complex but could be seen as the generalization of the standard Ҋ҈ practice of integrating teaching practice with research presented above. We look at bisociation as a theory of learning-through-creativity of Aha! moments where learning is ƾƣƣƹƞƾƺƣƾƿƶƣƽᅷƾƻƽƺƨƽƣƾƾƺƤǀƹƢƣƽƾƿƞƹƢƫƹƨᄬƩƞƻƿƣƽᇳᄭᄙ ƹƿƣƨƽƞƿƫƹƨƿƩƣƺƽƫƣƾ of learning, each of them with a different conceptual vocabulary and differƣƹƿᄬƾƣƣƸƫƹƨƶDŽᄭơƶƞƾƾƽƺƺƸƣǁƣƹƿƾᄕƸƣƞƹƾƿƩƞƿǂƣƩƞǁƣƿƺơƺƹƢǀơƿƫƢƣƹƿƫƤƫơƞtion not only of corresponding concepts but also of components of classroom events which they refer to. Ʃƫƾ ƻƽƺơƣƾƾ ƫƾ ƾƫƸƫƶƞƽ ƿƺ ƿƩƣ ǂƺƽƴ ƺƤ ƞƶǁƣƽƾơƩƣƫƢ ᄬᇴᇲᇲᇺᄭᄕ ǂƩƺƾƣ ƨƺƞƶᄕ ƞơơƺƽƢƫƹƨ ƿƺ ƞƢƤƺƽƢ ᄬᇴᇲᇲᇺᄭᄕ ƫƾ ᅸƿƺ ƾƿǀƢDŽ ƞ ƻƞƽƿƫơǀƶƞƽ ƣƢǀơƞƿƫƺƹ ƻƽƺƟƶƣƸ ᄚ through the use of elements from two different theories.” Halverscheid was studying a modeling theory and a cognitive theory. We are studying Aha! moments through the theory of bisociative creativity and the theory of the ƫƞƨƣƿᅬƞƽơƫƞƽƫƞƢ ǂƫƿƩ ƿƩƣ ƨƺƞƶ ƺƤ ƾƩƺǂƫƹƨ ƿƩƣƫƽ ƻƞƽƞƶƶƣƶ ƾƿƞƹơƣƾ ǁƫƾᅟdžᅟǁƫƾ concept development during the creative insight. The process of integrating different theories of learning became the subưƣơƿƺƤƫƹƿƣƹƾƣƿƩƣƺƽƣƿƫơƞƶƫƹǁƣƾƿƫƨƞƿƫƺƹƾǂƫƿƩƫƹƿƩƣƶƞƾƿƢƣơƞƢƣᄬƽƣƢƫƨƣƽƣƿƞƶᄙᄕ ᇴᇲᇲᇺᄖƽƣƢƫƨƣƽѵƫƴƹƣƽᅟƩƾƟƞƩƾᄕᇴᇲᇳᇶᄭᄙƫƴƹƣƽᅟƩƾƟƞƩƾƣƿƞƶᄙᄬᇴᇲᇳᇲᄭᄕƽƣƻƺƽƿing from the Research Forum on Networking of Theories in Mathematics

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ƢǀơƞƿƫƺƹƞƿƿƩƣᇵᇶƿƩƺƹƤƣƽƣƹơƣƺƤƿƩƣ ƹƿƣƽƹƞƿƫƺƹƞƶƽƺǀƻƤƺƽƿƩƣƾDŽơƩƺƶƺƨDŽƺƤƞƿƩƣƸƞƿƫơƾ ƢǀơƞƿƫƺƹᄕƫƹƤƺƽƸǀƾƿƩƞƿᅸᄴƿᄵƩƣƹƣƿǂƺƽƴƫƹƨƺƤƿƩƣƺƽƫƣƾƫƾ regarded as a systematic way to connect theories and to reflect about networking processes and its outcomes leading to a dialogue of theories.” Radford, in ƩƫƾơƺƸƸƣƹƿƾƺƹƿƩƣᇴᇲᇲᇺƾƻƣơƫƞƶƫƾƾǀƣƺƤѠъѓ on networking strategies for ơƺƹƹƣơƿƫƹƨƿƩƣƺƽƣƿƫơƞƶƞƻƻƽƺƞơƩƣƾᄬƽƣƢƫƨƣƽƣƿƞƶᄙᄕᇴᇲᇲᇺᄭᄕƣǃƻƞƹƢƾƺƹƿƩƣƾǀƟject of the dialogue, agreeing, of course, with the assessment of its complexity ƞƹƢƾǀƨƨƣƾƿƫƹƨƿƩƞƿᅸᄴƫᄵƹƺƽƢƣƽƿƺǀƹƢƣƽƾƿƞƹƢǂƩƞƿƫƾƾƞƫƢƫƹƿƩƣƶƞƹƨǀƞƨƣL of ƞƿƩƣƺƽDŽ஥ᄕƞƿƩƣƺƽDŽ஥’ƩƞƾƿƺƿƽƞƹƾƶƞƿƣƫƿᄬƞƿƶƣƞƾƿƫƹƿƩƣƟƣƨƫƹƹƫƹƨᄭƫƹƿƺƫƿƾƺǂƹ language L’ᅺᄬƞƢƤƺƽƢᄕᇴᇲᇲᇺᄕƻᄙᇵᇳᇺᄭᄙƣƞƨƽƣƣᄙƩƫƾƫƾƻƽƣơƫƾƣƶDŽǂƩƞƿᅷƾƩƞƻƻƣƹing in several chapters of the book, which ultimately is devoted to illuminating the phenomenon of the Aha! moment from several different theories and points of view. That is possible only if we can identify the presence of bisociation within the theory or, more precisely, the presence of the bisociative frame. A bisoơƫƞƿƫǁƣƤƽƞƸƣᄬƩƞƻƿƣƽᇳᄭƫƾƞƻƞƫƽƺƤƢƫƾơƺƹƹƣơƿƣƢƻƶƞƹƣƾƺƤƽƣƤƣƽƣƹơƣǂƫƿƩƫƹ which the connecting Aha! moment insight creating a novel conception takes or may take place. Thus, the process of translation between one theory and the other does not have to be global but may refer to certain relevant concepts of ƿǂƺƿƩƣƺƽƫƣƾᄙ ƺƽƣǃƞƸƻƶƣᄕǂƫƿƩƫƹƿƩƣƢƫƾơǀƾƾƫƺƹƫƹƩƞƻƿƣƽᇳᇵᄕƿƩƣƫƾƾǀƣǂƞƾ how to identify a bisociative frame within the student/researcher discourse and how to position it within the shift of attention, the central concept of ƞƾƺƹᅷƾƿƩƣƺƽDŽƺƤƞƿƿƣƹƿƫƺƹᄙ ƹƿƩƣơƺƹƿƣǃƿƺƤƿƩƣƢƫƾơǀƾƾƫƺƹƫƹƩƞƻƿƣƽᇳᇴᄕƿƩƣ issue was how to identify, or maybe to recognize, a bisociative frame within participatory and anticipatory stages of learning. Thus, within each theoretical chapter there is a process of translating sentences and concepts between bisociation and host theory to find a fit between them, or more precisely to see how bisociation impacts the host theory and how the integration with the host theory can modify bisociation itself. One of the studies in Chapter 7 had to undertake a quite intensive process of transƶƞƿƫƹƨƿƩƣƣƶƣƸƣƹƿƾƺƤƺƣƾƿƶƣƽᅷƾƟƫƾƺơƫƞƿƫƺƹƞƹƢƺƢƣƹᅷƾᄬᇴᇲᇲᇶᄭǁƫƣǂƺƹơƽƣativity to be able to understand espoused creativity of classroom students participating in a project-based learning course. ƽƣƢƫƨƣƽƞƹƢƫƴƹƣƽᅟƩƾƟƞƩƾᄬᇴᇲᇳᇶᄭƺƤƤƣƽƞƶƞƹƢƾơƞƻƣƺƤƾƿƽƞƿƣƨƫƣƾƤƺƽơƺƹnecting theoretical approaches, which are ordered with respect to the degree of integration for the theory in question: understanding and making understandable, comparing and contrasting, combining and coordinating and integrating ƶƺơƞƶƶDŽᄕ ƞƹƢ ƾDŽƹƿƩƣƾƫDžƫƹƨᄙƩƣ ƻƽƺơƣƾƾƣƾ ƺƤ ƫƹƿƣƨƽƞƿƫƺƹ ƫƹ Ʃƞƻƿƣƽƾ ᇳᇴ ƞƹƢ ᇳᇵ ƻƽƣƾƣƹƿƣǃƞƸƻƶƣƾƺƤƶƺơƞƶƫƹƿƣƨƽƞƿƫƺƹᄕǂƩƫƶƣƩƞƻƿƣƽᇳᇳƻƽƣƾƣƹƿƾƣƶƣƸƣƹƿƾƺƤ ƿƩƣƾDŽƹƿƩƣƾƫƾƺƤƟƫƾƺơƫƞƿƫƺƹƿƩƣƺƽDŽǂƫƿƩƿƩƣƿƩƣƺƽDŽƺƤƿƩƣƫƞƨƣƿᅬƞƽơƫƞƽƫƞƢᄙ

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ƩƞƻƿƣƽᇳᇳƫƾƸƺƽƣƞƸƟƫƿƫƺǀƾƤƽƺƸƿƩƣƻƺƫƹƿƺƤǁƫƣǂƺƤơƺƹƹƣơƿƫƹƨƿǂƺƿƩƣƺƽƫƣƾᄕƺƣƾƿƶƣƽᅷƾƟƫƾƺơƫƞƿƫƺƹƿƩƣƺƽDŽƞƹƢƫƞƨƣƿᅟƟƞƾƣƢƶƣƞƽƹƫƹƨƿƩƣƺƽƫƣƾᄙƩƣƞƫƸ of the chapter is to support the premise that creativity enters into every level of ơƺƹơƣƻƿǀƞƶƢƣǁƣƶƺƻƸƣƹƿǂƫƿƩƫƹƿƩƣƫƞƨƣƿᅬƞƽơƫƞƽƫƞƢᄕƾƻƣơƫƤƫơƞƶƶDŽƫƹƿƩƣƫƹƫƿƫƞƶ level of interiorization by situating this process within the bisociative frame. By ƤƺơǀƾƫƹƨƺƹƿƩƽƣƣơƽƫƿƣƽƫƞƿƩƽƺǀƨƩƿƩƣƿƽƫƞƢᄬƾƣƞƽơƩᄕƾƿƽǀơƿǀƽƞƶơƺƹƹƣơƿƫƺƹᄕƞƹƢ ƹƺǁƣƶƻƽƺơƣƾƾᄭᄕǂƣƞƽƨǀƣƿƩƞƿƟƫƾƺơƫƞƿƫƺƹƣƹƿƣƽƾƣǁƣƽDŽƶƣǁƣƶƺƤƶƣƞƽƹƫƹƨƞƾƾƣƣƹ through the lenses of the Piaget triad theory, in other words, that the creativity of the Aha! moment is present in every act of concept development. That would mean that creativity of an Aha! moment is irreducibly connected with learning. Ʃƣ ƫƹƿƣƽᅟƿƩƣƺƽDŽ ƿƽƞƹƾƶƞƿƫƺƹ ƻƽƺơƣƾƾ ƢƣƾơƽƫƟƣƢ ƟDŽ ƞƢƤƺƽƢ ᄬᇴᇲᇲᇺᄭ ƿƞƴƣƾ ƞ more extensive nature. Of course, the translation between the terms of both theories takes place primarily for these concepts which are essential for the introduction of creativity into learning processes: ‫ؘ؟ؕؔا‬ѿᄙᇳᏻᏺƽƞƹƾƶƞƿƫƺƹƺƤƿƣƽƸƾƟƣƿǂƣƣƹƟƫƾƺơƫƞƿƫƺƹƿƩƣƺƽDŽƞƹƢƫƞƨƣƿƫƞƹƿƩƣƺƽƫƣƾ

Bisociation theory

Piagetian theories

Matrix with its codes Searching process for hidden analogies Progress in understanding Exercise of understanding Codes

Action schema Problem-solving environment Accommodation Assimilation

ƹǁƞƽƫƞƹƿƽƣƶƞƿƫƺƹƾƩƫƻƾ

ƿƾƞƫƸƫƾƹƺƿƺƹƶDŽƿƺơƺƺƽƢƫƹƞƿƣƸƣƞƹƫƹƨƫƹƢƫƤƤƣƽƣƹƿƿƩƣƺƽƫƣƾƟǀƿƞƶƾƺƿƺ compare and identify the operation of their dynamic principles, that is, of bisociation and of reflective abstraction. The coordination process presented ƫƹ Ʃƞƻƿƣƽ ᇳᇳᄕ ǂƩƫơƩ ǀƾƣƾ ƹƣƿǂƺƽƴƫƹƨ ƾƿƽƞƿƣƨƫƣƾ ƺƤ ơƺƸƻƞƽƫƹƨᄧơƺƹƿƽƞƾƿƫƹƨ ƶƣƞƢƫƹƨƿƺƶƺơƞƶƫƹƿƣƨƽƞƿƫƺƹᄬƫƴƹƣƽᅟƩƾƟƞƩƾƣƿƞƶᄙᄕᇴᇲᇳᇲᄭᄕƩƞƾƟƣƣƹƫƹƫƿƫƞƿƣƢƫƹ ƞƴƣƽᄬᇴᇲᇳᇸᄭᄕǂƩƣƽƣƿƩƣƟƫƾƺơƫƞƿƫǁƣƤƽƞƸƣƩƞƾƟƣƣƹƫƢƣƹƿƫƤƫƣƢǂƫƿƩƫƹƿƩƣƻƽƺưƣơƿƫƺƹƾƿƞƨƣᄬƫƞƨƣƿѵƞƽơƫƞᄕᇳᇻᇺᇻᄭᄙ We need to note the central role of the bisociative frame in the process of introducing the theory of bisociation into elementary levels of learning. The concept of a bisociative frame has been generalized during this process to include not only two unconnected planes of reference but also the situation when one of the planes of the bisociative frame emerges from the other, as often ƿƞƴƣƾƻƶƞơƣƢǀƽƫƹƨƿƩƣƻƽƺơƣƾƾƺƤƞƟƾƿƽƞơƿƫƺƹᄬƩƞƻƿƣƽƾᇳᇳƞƹƢᇳᇵᄭᄙƩƣƢƫƾơǀƾsion above introduces the concept of inverse bisociation, which describes that

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process of emergence, from indistinguishable connection to separation of the two components, forming by this virtue the new relationship of abstraction. Bisociative frames have been identified along the following interfaces: ᅬ Between internalization and interiorization ᅬ Between participation and anticipation ᅬ Within the projection stage, as discussed earlier More detailed descriptions of chapters are below in the theoretical part. Part 3: Bisociation and Theories of Learning Chapter 11: Bisociation, Creativity, and Interiorization An underlying premise of this work is that creativity occurs within all levels of development, that is, whenever there is conceptual reasoning that leads to insight and the construction of a novel interior structure. To support this ƻƽƣƸƫƾƣᄕǂƣƤƫƽƾƿƽƣǁƫƣǂƣƞƽƶƫƣƽǂƺƽƴƫƹǂƩƫơƩǂƣƫƹƿƣƨƽƞƿƣƢƺƣƾƿƶƣƽᅷƾƟƫƾƺơƫƞƿƫƺƹƸƣơƩƞƹƫƾƸƺƤơƽƣƞƿƫǁƫƿDŽǂƫƿƩƫƞƨƣƿᅷƾƸƣơƩƞƹƫƾƸƺƤƞơơƺƸƸƺƢƞƿƫƺƹƞƹƢ ƽƣƤƶƣơƿƫǁƣƞƟƾƿƽƞơƿƫƺƹᄬƞƴƣƽᄕᇴᇲᇳᇸᄭᄙ Ʃƫƾ ƫƹƿƣƨƽƞƿƣƢ ƤƽƞƸƣǂƺƽƴ ƫƾ ƞ Ɵƫƾƺơƫƞƿƫǁƣ ƤƽƞƸƣ ᄬƩƞƻƿƣƽ ᇳᄭ ƫƹǁƺƶǁƫƹƨ existing and emerging matrices, and we use this bisociative frame to analyze and classify student insight at all levels of development. Our classification of student insight within this bisociative frame focuses on three characteristics, ƟƞƾƣƢ ǀƻƺƹ ƺǀƽ ǀƹƢƣƽƾƿƞƹƢƫƹƨ ƺƤ ƺƣƾƿƶƣƽᅷƾ ǂƺƽƴᄘ Ƥƫƽƾƿᄕ ƿƩƣ ƾƣƞƽơƩ ƻƽƺơƣƾƾ leading to the insight; second, the connection made during the insight; and third, the novel reasoned activity that results from the insight. ƫƞƨƣƿƞƹƢƞƽơƫƞᄬᇳᇻᇺᇻᄭƢƣǁƣƶƺƻƣƢƿƩƽƣƣƾƿƞƨƣƾƺƽƶƣǁƣƶƾƺƤƾƿƽǀơƿǀƽƞƶƢƣǁƣƶopment which we use to provide an overview based upon the structural development resulting from moments of insight. However, this requires interpreting the most elementary or initial stage form of reflective abstraction interiorization within the bisociative frame. Examples of student insight in this book and from the literature on creativity and construction of meaning within mathematics are then reviewed using these three criteria at different levels of the triad. Chapter 12: Two Stage Changes in Anticipation: Cognitive Sources of Aha! Moments This chapter is an illustrative essay in showing the process of networking, more precisely, of integrating two theories: the theory of participation and anticipation with the theory of bisociation. The chapter provides an explicit search within host theory for those components or transitions points which could

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support bisociative frame. The author identifies two types of reflection where ƿƩƣƩƞᄛƸƺƸƣƹƿơƺǀƶƢƟƣƾƫƿǀƞƿƣƢƞƹƢƩƞƾƟƣƣƹƺƟƾƣƽǁƣƢᄙ ƿƫƾƿƩƣƽƣƤƶƣơƿƫƺƹ within the activity/effect dyad, which takes place when the anticipated effect is different from the actual one. ƩƣƾƣơƺƹƢƿDŽƻƣƺƤƽƣƤƶƣơƿƫƺƹƫƾƿƩƣƽƣƤƶƣơƿƫƺƹƞơƽƺƾƾƾƣǁƣƽƞƶƾǀơƩƢDŽƞƢƾᄙ ƹ Piagetian language, it is of a higher order than the previous one and leads to ǂƩƞƿƫƞƨƣƿơƞƶƶƾƿƩƣƿƽƞƹƾƾƿƞƨƣᄙ ƹƞƢƢƫƿƫƺƹᄕƿƩƣƞǀƿƩƺƽƫƢƣƹƿƫƤƫƣƾƿǂƺƿƽƞƹƾƫƿƫƺƹƾǂƫƿƩƫƹƿƩƣ ᅚƣƤƞƻƻƽƺƞơƩᄕǂƩƫơƩơƽƣƞƿƣƿƩƣƾƫƿǀƞƿƫƺƹƤƞǁƺƽƞƟƶƣƤƺƽ Aha! moments, altogether coming with six possible categories of Aha! moment insights. Besides the formal analysis of the integration process, the essay shows that the term “Aha! moment” has entered the mainstream of the participation/ anticipation matrix of thinking. Chapter 13: Aha! Moments, Bisociation, and Multifocal Attention ƩƞƻƿƣƽᇳᇵᄕƶƫƴƣƩƞƻƿƣƽᇳᇴᄕƾƣƞƽơƩƣƾƤƺƽƿƩƣƹƣƿǂƺƽƴƫƹƨƞƹƢƫƹƿƣƨƽƞƿƫƺƹƺƤƿǂƺ ƿƩƣƺƽƫƣƾᄕƿƩƣƿƩƣƺƽDŽƺƤƞƿƿƣƹƿƫƺƹᄬƞƾƺƹᄕᇳᇻᇺᇻᄕᇴᇲᇲᇵƞᄕᇴᇲᇲᇵƟᄕᇴᇲᇲᇺᄭƞƹƢƿƩƣƿƩƣƺƽDŽƺƤƟƫƾƺơƫƞƿƫƺƹᄙƫƾƺơƫƞƿƫƺƹƫƸƻƶƫƣƾƿƩƣƾƫƸǀƶƿƞƹƣƫƿDŽᄬƺƽƤƞƾƿƺƾơƫƶƶƞƿƫƺƹᄭƺƤ attention to two components of the bisociative frame while the Aha! moment insight is taking place. Consequently, the first guiding question in the chapter is whether the Mason theory of attention can accommodate simultaneous or, as can happen, a fast-oscillating multifocal attention to two separate focal frames. Such an accommodation was found within dynamics of five different types ƺƤ ƞƿƿƣƹƢƫƹƨ ᄬƞƾƺƹᄕ ᇴᇲᇲᇵƞᄕ ᇴᇲᇲᇵƟᄭᄘ ƸƺƹƞƢƫơᄕ ƢDŽƞƢƫơᄕ ƿƽƫƞƢƫơᄕ ƿƣƿƽƞƢƫơᄕ ƞƹƢ pentadic thinking, which describe stages of conceptual development within ƿƩƣƿƩƣƺƽDŽƺƤƞƿƿƣƹƿƫƺƹᄙ ƣƽƣᄕƺƣƾƿƶƣƽᅷƾƢƣƤƫƹƫƿƫƺƹƺƤƟƫƾƺơƫƞƿƫƺƹƞƾƞƾƻƺƹƿƞneous leap of insight which connects two or more unconnected frames of reference suggests that an Aha! moment comprises a sudden transition between ƢDŽƞƢƫơᄬƢƫƾơƣƽƹƸƣƹƿƞƹƢơƺƸƻƞƽƫƾƺƹᄭƞƹƢƿƽƫƞƢƫơƿƩƫƹƴƫƹƨᄬƞƻƻƽƣơƫƞƿƫƹƨƽƣƶƞƿƫƺƹƾƩƫƻƾᄭᄕƺƽƟƣƿǂƣƣƹƿƽƫƞƢƫơƞƹƢƿƣƿƽƞƢƫơƿƩƫƹƴƫƹƨᄙ ƿƻƽƺǁƫƢƣƾƢDŽƹƞƸƫơƾƺƤ change in the structure of attention. The second area of the fit between the two theories is within the shift of ƞƿƿƣƹƿƫƺƹƿƩƞƿƿƞƴƣƾƻƶƞơƣƫƹƿƩƣƞƟƾƿƽƞơƿƫƺƹƻƽƺơƣƾƾᄬƞƾƺƹᄕᇳᇻᇺᇻᄭᄙƩƣƤƫƹƢƫƹƨ is supported by the analysis of two examples of abstraction found in the work ƺƤƫƸƺƹƣƿƞƶᄙᄬᇴᇲᇳᇲᄕᇴᇲᇳᇶᄭᄕǂƩƫơƩƶƣƢƿƺƿƩƣƫƢƣƹƿƫƤƫơƞƿƫƺƹƺƤƿƩƣƫƹǁƣƽƾƣƟƫƾƺciation described earlier. Part 4: Bisociativity from Without “Without” here means outside of mathematics and mathematics teaching. More precisely, this part presents two related views, one from a neuroscientific

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orientation; the other, from the domain of artificial intelligence. Both pose wider questions, such as the role of different parts of the brain as well as different aspects of consciousness participating in an Aha! insight. ƩƞƻƿƣƽᇳᇶƢƫƾơǀƾƾƣƾƿƩƣƻƺƾƾƫƟƫƶƫƿDŽƺƤƣǃƿƣƹƢƫƹƨƿƩƣƣƾƿƞƶƿƾƿƞƨƣƞƻƻƽƺƞơƩ through preparation, incubation, illumination, and verification to creativity by adding a new stage of intimation, between incubation and illumination, whose source is fringe consciousness. The description of the laboratory tracƫƹƨƺƤƿƩƣƩƞᄛƸƺƸƣƹƿƿƩƽƺǀƨƩƣDŽƣƸƺǁƣƸƣƹƿƞƹƢƣƶƣơƿƽƺơƞƽƢƫƺƨƽƞƸᄬѻѹѽᄭ instruments reveals the possibility of a slight delay in external manifestation of the moment in relation to its actual occurrence. ƩƞƻƿƣƽᇳᇷᄕƺƹƿƩƣƺƿƩƣƽƩƞƹƢᄕƻƽƣƾƣƹƿƾƿǂƺƾƿǀƢƫƣƾƺƤƟƫƾƺơƫƞƿƫǁƣƾƿƽǀơƿǀƽƣƾᄕ which find their completion in the ѷѿƤƽƞƸƣǂƺƽƴᄙơơƺƽƢƫƹƨƿƺƺƢƣƹᄬᇴᇲᇲᇶᄭᄕ the bisociative idea of connecting the unconnected as the central process participating in progress of understanding correlates with the connectivism program of ѷѿ with its neural networks. Chapter 14: The Aha! Moment at the Nexus of Mind and Brain We present a wide spectrum of neuroscientific considerations, theories, methods, and results from the study of creativity and insight as they pertain to the Aha! moment. We frame these neuroscientific considerations within an ongoƫƹƨƢƣƟƞƿƣƫƹƿƩƣƻƩƫƶƺƾƺƻƩDŽƺƤƾơƫƣƹơƣƻƽƣơƫƻƫƿƞƿƣƢƟDŽƽƿƩǀƽƺƣƾƿƶƣƽƞƹƢ ơƺƶƶƣƞƨǀƣƾƟƣƿǂƣƣƹƻƽƣƢƺƸƫƹƞƹƿƶDŽƸƫƹƢᅟƟƞƾƣƢƻƾDŽơƩƺƾƺơƫƞƶᄬƞƹƿƫƽƣƢǀơƿƫƺƹƫƾƿᄭƞƹƢƟƽƞƫƹᅟƟƞƾƣƢƟƫƺᅟƹƣǀƽƺƶƺƨƫơƞƶᄬƽƣƢǀơƿƫƺƹƫƾƿᄭǁƫƣǂƾƺƤƩǀƸƞƹơƺƨƹƫƿƫƺƹᄙ

ƹ ƾƺ Ƣƺƫƹƨᄕ ǂƣ ǁƫƾƫƿ ƿƩƣ Ƥƺƶƶƺǂƫƹƨ Ƽǀƣƾƿƫƺƹƾ ƿƺ ƤǀƽƿƩƣƽ ƾƫƿǀƞƿƣ ƺƣƾƿƶƣƽᅷƾ theory of bisociation and explore its potential usefulness in the classroom: Are Aha! moments of insight and creativity more generally free creations of the human mind? Or can such cognitive phenomena be fully accounted for deterministically as an outcome of brain activity operating in accord with natural laws? Perhaps there is a middle path to be found between these two alternatives. Chapter 15: Bisociative Structures This chapter establishes the connection, or better, the relationship between student creativity in our classrooms and the ѷѿ ƞƻƻƽƺƞơƩ ƿƺ ơƽƣƞƿƫǁƫƿDŽᄙ ƿ ƫƾ composed of two studies, both of which find their completion in the bisociative creativity of the ѷѿƤƽƞƸƣǂƺƽƴƞƾƾƣƣƹƟDŽƺƢƣƹᄬᇴᇲᇲᇶᄭƞƹƢƿƩƣƟƫƾƺơƫƞƿƫǁƣ ƴƹƺǂƶƣƢƨƣƢƫƾơƺǁƣƽDŽᄬѸҁѺᄭƺƤƣƽƿƩƺƶƢᄬᇴᇲᇳᇴᄭᄙ The first study is interested in the investigation of the connections between ƾƿǀƢƣƹƿƾᅷ ƺǂƹ ƞƾƾƣƾƾƸƣƹƿ ƺƤ ơƽƣƞƿƫǁƫƿDŽ ƞƹƢ ƿƩƣ ƞƾƾƣƾƾƸƣƹƿ ƺƤ ơƽƣƞƿƫǁƫƿDŽ ƟDŽ

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observers of student project results. To do that, the author investigated the relaƿƫƺƹƾƩƫƻƟƣƿǂƣƣƹƺƣƾƿƶƣƽᅷƾƞƹƢƺƢƣƹᅷƾƞƻƻƽƺƞơƩƣƾƞƹƢƣƾƿƞƟƶƫƾƩƣƢƿƩƣƶƺơƞƶ translation between them. This coordination helps in establishing the explanatory framework for the differences in student understanding of espoused creativity in the context of the project-based learning course focused on coding and cryptography described in Chapter 7, where the differences between student-espoused and student-enacted creativity are discussed. The second study compares the classification of Aha! moments obtained in Chapter 4 with the types of structures obtained by ѸҁѺ and through comparing and contrasting strategies finds interesting similarities and differences between the two. The author provides examples of connecting structures found among the characteristic Aha! moments together with elements of the ƫƹƾƫƨƩƿᅷƾƢƣƾơƽƫƻƿƫƺƹǂƩƣƽƣƿƩƣƾƣơƺƹƹƣơƿƫƺƹƾƞƽƣƾƿƞƿƣƢᄙ

Notes ᇳ ƩƣƢƣƤƫƹƫƿƫƺƹǂƞƾƞƟƾƿƽƞơƿƣƢƤƽƺƸƿƩƣƤƺƶƶƺǂƫƹƨƤƽƞƨƸƣƹƿᄘ There are two ways of escaping our more or less automatized routines of thinking and behaving. The first, of course, is plunging into dreaming or dream-like states, when the codes of rational thinking are suspended. The other way is also an escape—from boredom, stagnation, intellectual predicaments, and emotional frustration—but an escape ƫƹƿƩƣƺƻƻƺƾƫƿƣƢƫƽƣơƿƫƺƹᄖƫƿƫƾƾƫƨƹƞƶƶƣƢƟDŽƿƩƣƾƻƺƹƿƞƹƣƺǀƾƤƶƞƾƩᄴƶƣƞƻᄵƺƤƫƹƾƫƨƩƿ which shows a familiar situation or event in a new light and elicits a new response to it. The bisociative act connects previously unconnected matrices of experience; it makes us “underƾƿƞƹƢǂƩƞƿƫƿƫƾƿƺƟƣƞǂƞƴƣᄕƿƺƟƣƶƫǁƫƹƨƺƹƾƣǁƣƽƞƶƻƶƞƹƣƾƞƿƺƹơƣᄙᅺᄬƺƣƾƿƶƣƽᄕᇳᇻᇸᇶᄕƻᄙᇶᇷᄭ 2 Big C creativity refers to creativity important and recognized in a society, while little C creativity is the personal creativity recognized as such. Later in the chapter we discuss the division in more detail. ᇵ ƽᄙƽǀƹƢƞƽƞƟƩǀƻƞƾƾƣƢƞǂƞDŽƻƽƣƸƞƿǀƽƣƶDŽƫƹƞƽơƩᇴᇲᇳᇵᄙ ᇶ ƩƣᇴᇲᇳᇲƣƹƾǀƾơƺƹƤƫƽƸƾƿƩƞƿƸƺƽƣƿƩƞƹƞƼǀƞƽƿƣƽƺƤƞƸƫƶƶƫƺƹƻƣƺƻƶƣƫƹƿƩƣƺǀƿƩƽƺƹǃ ƞƽƣƶƫǁƫƹƨƫƹƻƺǁƣƽƿDŽᄕƸƞƴƫƹƨƣƻᄙƺƾƣƣƽƽƞƹƺᅷƾᇳᇸƿƩơƺƹƨƽƣƾƾƫƺƹƞƶƢƫƾƿƽƫơƿƿƩƣƻƺƺƽƣƾƿƫƹƿƩƣ ƹƞƿƫƺƹᄬƣƹƾǀƾǀƽƣƞǀᄕƣƻƿƣƸƟƣƽᇵᇲᄕᇴᇲᇳᇲᄭᄙ ᇷ ƣǂƞƹƿƿƺƞơƴƹƺǂƶƣƢƨƣƿƩƣƤƺƶƶƺǂƫƹƨơƺƸƸƣƹƿƺƤƞƽƣǁƫƣǂƣƽᄘ The book reads as if the real approach to creativity is the big C approach, the study of moments where widely acclaimed original and useful ideas come up by bright individǀƞƶƾᄙᄚ ƫƽƾƿᄕƿƩƞƿơƽƣƞƿƫǁƫƿDŽƫƾǂƫƿƩƫƹƣƞơƩƫƹƢƫǁƫƢǀƞƶᄕƩƞƻƻƣƹƾƸƞƹDŽƿƫƸƣƾƢǀƽƫƹƨƿƩƣ day, and is a subjective phenomenon, meaning that what is creative for one individual may not be creative for another. These many moments of creativity may not be really ơƺƹƾơƫƺǀƾƿƺƿƩƣƻƣƽƾƺƹƞƹƢƸƞDŽƹƺƿƟƣơƺƹƾƫƢƣƽƣƢƺƤǂƫƢƣƽǀƾƣƤǀƶƹƣƾƾƞƹƢǁƞƶǀƣᄙ ƿƫƾ the heart of constructivist thinking and many have termed it little C. Besides little C there has been another term, middle C, to convey the idea of conscious creative thoughts which

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are original and useful for the social context at hand, a solution to a problem which for the student is original and useful to solve other problems but which already exists only they do not know it.

ƹƿƣƞơƩƫƹƨƞƹƢƶƣƞƽƹƫƹƨƸƞƿƩƣƸƞƿƫơƾᄕǂƣƤƫƹƢǀƾƣƤǀƶƺƹƶDŽƿǂƺƺƤƿƩƣƿƣƽƸƾᄕƶƫƿƿƶƣƞƹƢ ƟƫƨᄕǂƩƣƽƣƶƫƿƿƶƣơƫƹơƶǀƢƣƾƟƺƿƩƶƫƿƿƶƣƞƹƢƸƫƢƢƶƣơᅷƾᄙ 6 The term “Janusian” originates with Janus, the two-faced Roman god of beginnings and endings, ᅸƿƩƣƞƹƫƸƫƾƿƫơƾƻƫƽƫƿƺƤƢƺƺƽǂƞDŽƾᄬjanuaeᄭƞƹƢƞƽơƩǂƞDŽƾᄬjaniᄭᅺᄬƩƿƿƻƾᄘᄧᄧǂǂǂᄙƟƽƫƿƞƹƹƫơƞᄙơƺƸᄧ ƿƺƻƫơᄧƞƹǀƾᅟƺƸƞƹᅟƨƺƢᄭᄙ 7 ѹቀѿ҈ѽᇹᄬᇴᇲᇳᇲᄭƽƺƟƶƣƸᅟƾƺƶǁƫƹƨƫƹƣƸƣƢƫƞƶƽƫƿƩƸƣƿƫơᄘǀƸƻƾƿƞƽƿƿƺƣƤƺƽƸᄬᇣᇴᇺᄕᇲᇲᇲᄭǂƫƿƩ V. Prabhu, W. Baker, O. Dias, Bronx ѹѹ, and Hostos ѹѹ, ѹҋ҄ҏ.

References ƸƞƟƫƶƣᄕᄙᄙᄕƣƿƞƶᄙᄬᇴᇲᇲᇷᄭᄙƤƤƣơƿƞƹƢơƽƣƞƿƫǁƫƿDŽƞƿǂƺƽƴᄙAdministrative Science Quarterly, 50ᄕᇵᇸᇹᅬᇶᇲᇵᄙ ƻƻƶƣƟƞǀƸᄕᄙᄕѵƞǀƶᄕᄙᄬᇴᇲᇲᇻᄭᄙƹƣơƢƺƿƣƾƞƹƢƞƾƾƣƽƿƫƺƹƾƞƟƺǀƿơƽƣƞƿƫǁƫƿDŽƫƹƿƩƣ ǂƺƽƴƫƹƨƸƞƿƩƣƸƞƿƫơƾơƶƞƾƾƽƺƺƸᄙ ƹᄙƣƫƴƫƹᄕᄙƣƽƸƞƹᄕѵᄙƺƫơƩǀᄬ ƢƾᄙᄭᄕCreativity in mathematics and the education of gifted studentsᄬƻƻᄙᇴᇹᇳᅬᇴᇺᇶᄭᄙƣƹƾƣǀƟlishers. ƽƹƺƹᄕ ᄙᄕƺƿƿƽƫƶƶᄕᄙᄕǀƟƫƹƾƴDŽᄕ ᄙᄕƴƿƞǴᄕᄙᄕƺƞ ǀƣƹƿƣƾᄕᄙᄕƽƫƨǀƣƽƺƾᄕᄙᄕѵƣƶƶƣƽᄕᄙ ᄬᇴᇲᇳᇶᄭᄙAPOS theory: A framework for research and curriculum development in mathematics education. Springer. ƾƩƟDŽᄕ ᄙᄙᄕ ƾƣƹᄕᄙᄙᄕѵǀƽƴƣƹᄕᄙᄬᇳᇻᇻᇻᄭᄙƹƣǀƽƺƻƾDŽơƩƺƶƺƨƫơƞƶƿƩƣƺƽDŽƺƤƻƺƾƫƿƫǁƣ affect and its influence on cognition. Psychological Review, 106ᄕᇷᇴᇻᅬᇷᇷᇲᄙ ƞƴƣƽᄕᄙᄬᇴᇲᇳᇸᄭᄙƺƣƾƿƶƣƽᅷƾƿƩƣƺƽDŽƞƾƞƤƺǀƹƢƞƿƫƺƹƤƺƽƻƽƺƟƶƣƸᅟƾƺƶǁƫƹƨᄙ ƹᄙDžƞƽƹƺơƩƞᄕᄙ ƞƴƣƽᄕ ᄙ ƫƞƾᄕ ѵᄙ ƽƞƟƩǀ ᄬ Ƣƾᄙᄭᄕ The creative enterprise of mathematics teaching research: Elements of methodology and practice—From teachers to teachers ᄬƻƻᄙᇴᇸᇹᅬᇴᇺᇸᄭᄙƣƹƾƣǀƟƶƫƾƩƣƽƾᄙ ƞƽƹƣƾᄕᄙᄬᇴᇲᇲᇲᄭᄙƞƨƫơƸƺƸƣƹƿƾƫƹƸƞƿƩƣƸƞƿƫơƾᄘ ƹƾƫƨƩƿƾƫƹƿƺƿƩƣƻƽƺơƣƾƾƺƤơƺƸing to know. For the Learning of Mathematics, 20ᄬᇳᄭᄕᇵᇵᅬᇶᇵᄙ ƣƽƿƩƺƶƢᄕ ᄙ ᄙ ᄬᇴᇲᇳᇴᄭᄙ ƺǂƞƽƢƾ Ɵƫƾƺơƫƞƿƫǁƣ ƴƹƺǂƶƣƢƨƣ ƢƫƾơƺǁƣƽDŽᄙ ƹ ᄙ ᄙ ƣƽƿƩƺƶƢ ᄬ ƢᄙᄭᄕBisociative knowledge discovery: An introduction to concept, algorithms, tools, and applicationsᄬƻƻᄙᇳᅬᇳᇲᄭᄙƻƽƫƹƨƣƽᄙ ƫƴƹƣƽᅟƩƾƟƞƩƾᄕᄙᄕƽƣDŽƤǀƾᄕᄙᄕƫƢƽƺƹᄕ ᄙᄕƽDžƞƶƣƶƶƺᄕ ᄙᄕƞƢƤƺƽƢᄕᄙᄕƽƿƫƨǀƣᄕᄙᄕѵƞƟƣƹƞᄕ ᄙᄬᇴᇲᇳᇲᄭᄙƣƿǂƺƽƴƫƹƨƺƤƿƩƣƺƽƫƣƾƫƹƸƞƿƩƣƸƞƿƫơƾƣƢǀơƞƿƫƺƹᄙ ƹᄙᄙ ᄙƫƹƿƺѵ ᄙ ᄙƞǂƞƾƞƴƫᄬ ƢƾᄙᄭᄕProceedings of the 34th Conference of the International Group for the Psychology of Mathematics Education: Mathematics in different settings, Vol. 1 ᄬƻƻᄙᇳᇶᇷᅬᇳᇹᇷᄭᄙ ƹƿƣƽƹƞƿƫƺƹƞƶƽƺǀƻƤƺƽƿƩƣƾDŽơƩƺƶƺƨDŽƺƤƞƿƩƣƸƞƿƫơƾ Ƣǀơƞƿƫƺƹᄙ ƫƴƹƣƽᅟƩƾƟƞƩƾᄕᄙᄕѵƽƣƢƫƨƣƽᄕᄙᄬ ƢƾᄙᄭᄙᄬᇴᇲᇳᇶᄭᄙNetworking of theories as a research practice in mathematics education. Springer.

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ƺƢƣƹᄕᄙᄙᄬᇴᇲᇲᇶᄭᄙThe creative mind: Myths and mechanisms ᄬᇴƹƢƣƢᄙᄭᄙƺǀƿƶƣƢƨƣᄙ ƩƞƸƟƣƽƶƫƹᄕᄙᄙᄬᇴᇲᇳᇵᄭᄙ ƸƻƫƽƫơƞƶƫƹǁƣƾƿƫƨƞƿƫƺƹƾƺƤơƽƣƞƿƫǁƫƿDŽƞƹƢƨƫƤƿƣƢƹƣƾƾƫƹƸƞƿƩematics: An international perspective: A review of The elements of creativity and ƨƫƤƿƣƢƹƣƾƾƫƹƸƞƿƩƣƸƞƿƫơƾƟDŽᄙƽƫƽƞƸƞƹѵᄙ ᄙƣƣᄙJournal for Research in Mathematics Education, 44ᄬᇷᄭᄕᇺᇷᇴᅬᇺᇷᇹᄙ DžƞƽƹƺơƩƞᄕ ᄙ ᄬᇳᇻᇻᇻᄭᄙ ƶ Ƹƞƣƾƿƽƺ ơƺƹƾƿƽǀơƿƫǁƫƾƿƞ ơƺƸƺ ƫƹǁƣƾƿƫƨƞƢƺƽᄙ ʖƸƺ ƣƹƾƣʎƞƽ razones y proporciones a adolescentes. Educación Matemática, 11ᄬᇴᄭᄕᇷᇴᅬᇸᇵᄙ DžƞƽƹƺơƩƞᄕᄙᄬᇴᇲᇳᇶᄭᄙƹƿƩƣơǀƶƿǀƽƣƺƤơƽƣƞƿƫǁƫƿDŽƫƹƸƞƿƩƣƸƞƿƫơƾƣƢǀơƞƿƫƺƹᄙJournal of Teaching Innovations, 27ᄬᇵᄭᄙ DžƞƽƹƺơƩƞᄕ ᄙ ᄬᇴᇲᇳᇸᄭᄙ ƣƞơƩƫƹƨᅟƽƣƾƣƞƽơƩ ƣǂ ƺƽƴ ƫƿDŽ ƺƢƣƶ ᄬᄧ ƫƿDŽᄭᄙ ƹ ᄙ DžƞƽƹƺơƩƞᄕᄙƞƴƣƽᄕᄙƫƞƾᄕѵᄙƽƞƟƩǀᄬ ƢƾᄙᄭᄕThe creative enterprise of mathematics teaching research: Elements of methodology and practice—From teachers to teachersᄬƻƻᄙᇵᅬᇴᇳᄭᄙƣƹƾƣǀƟƶƫƾƩƣƽƾᄙ DžƞƽƹƺơƩƞᄕᄙᄕƞƴƣƽᄕᄙᄕƫƞƾᄕᄙᄕѵƽƞƟƩǀᄕᄙᄬ ƢƾᄙᄭᄙᄬᇴᇲᇳᇸᄭᄙThe creative enterprise of mathematics teaching research: Elements of methodology and practice—From teachers to teachers. Sense Publishers. DžƞƽƹƺơƩƞᄕᄙᄕѵƞưᄕᄙᄬᇴᇲᇲᇺᄭᄙƣƞơƩƫƹƨƣǃƻƣƽƫƸƣƹƿᄙ ƹᄙDžƞƽƹƺơƩƞᄬ ƢᄙᄭᄕHandbook of mathematics teaching research: Teaching experiment—A tool for teacher-researchers ᄬƻᄙᇶᇹᄭᄙƹƫǁƣƽƾƫƿDŽƺƤDžƣƾDžʖǂᄙƩƿƿƻᄘᄧᄧƸƞƿƩᄙƴǀᄙƾƴᄧƢƞƿƞᄧƻƺƽƿƞƶᄧƢƞƿƞᄧƺƺƶᄙƻƢƤ DžƞƽƹƺơƩƞᄕᄙᄕѵƽƞƟƩǀᄕᄙᄬᇴᇲᇲᇸᄭᄙƣƞơƩƫƹƨᅟƽƣƾƣƞƽơƩƫƿDŽƸƺƢƣƶᄙDydaktyka Matematyki, 29ᄕᇴᇷᇳᅬᇴᇹᇴᄙ ƣƣƶƶƫƾᄕᄙᄙᄕѵƺƶƢƫƹᄕᄙᄙᄬᇴᇲᇲᇸᄭᄙƤƤƣơƿƞƹƢƸƣƿƞᅟƞƤƤƣơƿƫƹƸƞƿƩƣƸƞƿƫơƞƶƻƽƺƟlem solving: A representational perspective. Educational Studies in Mathematics, 63, ᇳᇵᇳᅬᇳᇶᇹᄙ ƫƹƾƿƣƫƹᄕᄙᄬᇳᇻᇶᇻᄭᄙǀƿƺƟƫƺƨƽƞƻƩƫơƞƶƹƺƿƣƾᄬᄙᄙơƩƫƶƻƻᄕƽƞƹƾᄙᄭᄙ ƹᄙᄙơƩƫƶƻƻᄬ Ƣᄙᄭᄕ Albert Einstein: Philosopher-scientistᄬƻƻᄙᇳᅬᇻᇷᄭᄙƫƟƽƞƽDŽƺƤƫǁƫƹƨƩƫƶƺƾƺƻƩƣƽƾᄕᇳᇻᇶᇻᄙ ƽƞDŽᄕ ᄙᄙᄕѵƞƶƶᄕᄙᄙᄬᇳᇻᇻᇶᄭᄙǀƞƶƫƿDŽᄕƞƸƟƫƨǀƫƿDŽᄕƞƹƢƤƶƣǃƫƟƫƶƫƿDŽᄘᅸƻƽƺơƣƻƿǀƞƶᅺǁƫƣǂ of simple arithmetic. Journal for Research in Mathematics Education, 25ᄬᇴᄭᄕᇳᇳᇸᅬᇳᇶᇲᄙ ƞƢƞƸƞƽƢᄕᄙᄬᇳᇻᇶᇷᄭᄙ The psychology of invention in the mathematical field. Princeton University Press. ƞƶǁƣƽƾơƩƣƫƢᄕᄙᄬᇴᇲᇲᇺᄭᄙǀƫƶƢƫƹƨƞƶƺơƞƶơƺƹơƣƻƿǀƞƶƤƽƞƸƣǂƺƽƴƤƺƽƣƻƫƾƿƣƸƫơƞơƿƫƺƹƾ in a modelling environment with experiments. ZDM: The International Journal on Mathematics Education, 40ᄬᇴᄭᄕᇴᇴᇷᅬᇴᇵᇶᄙ ƣƽƾƩƴƺǂƫƿDžᄕᄙᄕƞƟƞơƩᄕᄙᄕѵƽƣDŽƤǀƾᄕᄙᄬᇴᇲᇳᇹᄭᄙƽƣƞƿƫǁƣƽƣƞƾƺƹƫƹƨƞƹƢƾƩƫƤƿƾƺƤƴƹƺǂƶedge in the mathematics classroom. ZDM: Mathematics Education, 49ᄬᇳᄭᄕᇴᇷᅬᇵᇸᄙ ƫƣƽƞƹᄕᄙᄕƽƞƫƹƣƽᄕᄙᄕѵƩƞǀƨƩƹƣƾƾDŽᄕᄙᄙᄬᇴᇲᇳᇵᄭᄙƫƹƴƫƹƨƽƣƾƣƞƽơƩƿƺƻƽƞơƿƫơƣᄘƣƞơƩƣƽƾ ƞƾƴƣDŽƾƿƞƴƣƩƺƶƢƣƽƾƫƹƸƞƿƩƣƸƞƿƫơƾƣƢǀơƞƿƫƺƹƽƣƾƣƞƽơƩᄙ ƹᄙᄙƶƣƸƣƹƿƾƣƿƞƶᄙᄬ Ƣƾᄙᄭᄕ Third international handbook of mathematics educationᄬƻƻᄙᇵᇸᇳᅬᇵᇻᇴᄭᄙ Springer. ƺƣƾƿƶƣƽᄕᄙᄬᇳᇻᇸᇶᄭᄙThe act of creation. Macmillan.

ѿ‫ءآ؜اؖبؗآإاء‬

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ƺǀƹƫƺƾᄕᄙᄕѵƣƣƸƞƹᄕᄙᄬᇴᇲᇳᇷᄭᄙThe eureka factor: Aha moments, creative insight, and the brain. Random House. ƫƶưƣƢƞƩƶᄕ ᄙ ᄬᇴᇲᇲᇶᄭᄙ The Aha! experience: Mathematical contexts, pedagogical experiences ᄬƩƿƩƣƾƫƾᄭᄙƫƸƺƹ ƽƞƾƣƽƹƫǁƣƽƾƫƿDŽᄙ ƫƶưƣƢƞƩƶᄕᄙᄬᇴᇲᇳᇵᄭᄙ ƶƶǀƸƫƹƞƿƫƺƹᄘƹƞƤƤƣơƿƫǁƣƣǃƻƣƽƫƣƹơƣᄞZDM: The International Journal of Mathematics, 45ᄬᇴᄭᄕᇴᇷᇵᅬᇴᇸᇷᄙ ƞƾƺƹᄕᄙᄬᇳᇻᇺᇻᄭᄙƞƿƩƣƸƞƿƫơƞƶƞƟƾƿƽƞơƿƫƺƹƞƾƞƽƣƾǀƶƿƺƤƞƢƣƶƫơƞƿƣƾƩƫƤƿƺƤƞƿƿƣƹƿƫƺƹᄙ For the Learning of Mathematics, 9ᄬᇴᄭᄕᇴᅬᇺᄙ ƞƾƺƹᄕᄙᄬᇴᇲᇲᇵƞᄭᄙƹƿƩƣƾƿƽǀơƿǀƽƣƺƤƞƿƿƣƹƿƫƺƹƫƹƿƩƣƶƣƞƽƹƫƹƨƺƤƸƞƿƩƣƸƞƿƫơƾᄙThe Australian Mathematics Teacher, 59ᄬᇶᄭᄕᇳᇹᅬᇴᇷᄙᄬƣƻƽƫƹƿƺƤƞƾƺƹᄕᇴᇲᇲᇵƟᄭ ƞƾƺƹᄕᄙᄬᇴᇲᇲᇵƟᄭᄙƿƽǀơƿǀƽƣƺƤƞƿƿƣƹƿƫƺƹƫƹƿƩƣƶƣƞƽƹƫƹƨƺƤƸƞƿƩƣƸƞƿƫơƾᄙ ƹᄙƺǁƺƿƹLJᄬ ƢᄙᄭᄕProceedings of International Symposium Elementary Maths Teachingᄬƻƻᄙ ᇻᅬᇳᇸᄭᄙƩƞƽƶƣƾƹƫǁƣƽƾƫƿDŽᄕ ƞơǀƶƿDŽƺƤ Ƣǀơƞƿƫƺƹᄙ ƞƾƺƹᄕᄙᄬᇴᇲᇲᇺᄭᄙƣƫƹƨƸƞƿƩƣƸƞƿƫơƞƶǂƫƿƩƞƹƢƫƹƤƽƺƹƿƺƤƶƣƞƽƹƣƽƾᄘƿƿƣƹƿƫƺƹᄕƞǂƞƽƣness, and attitude as sources of differences between teacher educators, teachers ƞƹƢƶƣƞƽƹƣƽƾᄙ ƹᄙƣƾǂƫơƴѵᄙƩƞƻƸƞƹᄬ ƢƾᄙᄭᄕInternational handbook of mathematics teacher education, Vol. 4: The mathematics teacher educator as a developing professionalᄬᇴƹƢƣƢᄙᄕƻƻᄙᇵᇳᅬᇷᇷᄭᄙƣƹƾƣǀƟƶƫƾƩƣƽƾᄙ ƫƾƾᄕᄙᄬᇳᇻᇻᇻᄭᄙƾƻƣơƿƾƺƤƿƩƣƹƞƿǀƽƣƞƹƢƾƿƞƿƣƺƤƽƣƾƣƞƽơƩƫƹƸƞƿƩƣƸƞƿƫơƾƣƢǀơƞƿƫƺƹᄙ Educational Studies in Mathematics, 40ᄕᇳᅬᇴᇶᄙ ƫƞƨƣƿᄕᄙᄕѵƞƽơƫƞᄕᄙᄬᇳᇻᇺᇻᄭᄙ Psychogenesis and the history of scienceᄬ ᄙ ƣƫƢƣƽᄕƽƞƹƾᄙᄭᄙ Columbia University Press. ƽƞƟƩǀᄕᄙ ᄬᇴᇲᇳᇸᄭᄙƩƣ ơƽƣƞƿƫǁƣ ƶƣƞƽƹƫƹƨ ƣƹǁƫƽƺƹƸƣƹƿᄙ ƹ ᄙ DžƞƽƹƺơƩƞᄕᄙ ƞƴƣƽᄕ ᄙ ƫƞƾᄕѵᄙƽƞƟƩǀᄬ ƢƾᄙᄭᄕThe creative enterprise of mathematics teaching research: Elements of methodology and practice—From teachers to teachers ᄬƻƻᄙ ᇳᇲᇹᅬᇳᇴᇸᄭᄙ Sense Publishers. ƽƞƟƩǀᄕᄙᄕ ѵ DžƞƽƹƺơƩƞᄕ ᄙ ᄬᇴᇲᇳᇶᄭᄙ ƣƸƺơƽƞƿƫDžƫƹƨ ƸƞƿƩƣƸƞƿƫơƞƶ ơƽƣƞƿƫǁƫƿDŽ ƿƩƽƺǀƨƩ ƺƣƾƿƶƣƽᅷƾƟƫƾƺơƫƞƿƫƺƹƿƩƣƺƽDŽᄙ ƹᄙƫơƺƶᄕᄙƣƾƿƣƽƶƣᄕᄙƫƶưƣƢƞƩƶᄕѵᄙƶƶƞƹᄬ Ƣƾᄙᄭᄕ Proceedings of the 38th Conference of the International Group for the Psychology of Mathematics Education and the 36th Conference of the North American Chapter of the Psychology of Mathematics Education, Vol. 5 ᄬƻƻᄙᇳᅬᇺᄭᄙ ᄙƩƿƿƻƾᄘᄧᄧǂǂǂᄙƻƸƣƹƞᄙƺƽƨᄧ proceedings/ ƽƣƢƫƨƣƽᄕᄙᄕƽDžƞƽƣƶƶƺᄕ ᄙᄕƺƾơƩᄕᄙᄕѵƣƹƤƞƹƿᄕᄙᄬ ƢƾᄙᄭᄙᄬᇴᇲᇲᇺᄭᄙComparing, combining, coordinating—Networking strategies for connecting theoretical approaches. Special issue of ZDM: The International Journal on Mathematics Education, 40ᄬᇴᄭᄙ ƽƣƢƫƨƣƽᄕᄙᄕѵƫƴƹƣƽᅟƩƾƟƞƩƾᄕᄙᄬᇴᇲᇳᇶᄭᄙ ƹƿƽƺƢǀơƿƫƺƹƿƺƹƣƿǂƺƽƴƫƹƨᄘƣƿǂƺƽƴƫƹƨƾƿƽƞƿƣƨƫƣƾƞƹƢƿƩƣƫƽƟƞơƴƨƽƺǀƹƢᄙ ƹᄙƫƴƹƣƽᅟƩƾƟƞƩƾѵᄙƽƣƢƫƨƣƽᄬ ƢƾᄙᄭᄕNetworking of theories as a research practice in mathematics educationᄬƻƻᄙᇳᇳᇹᅬᇳᇴᇷᄭᄙƻƽƫƹƨƣƽᄙ ƞƢƤƺƽƢᄕᄙᄬᇴᇲᇲᇺᄭᄙƺƹƹƣơƿƫƹƨƿƩƣƺƽƫƣƾƫƹƸƞƿƩƣƸƞƿƫơƾƣƢǀơƞƿƫƺƹᄘƩƞƶƶƣƹƨƣƾƞƹƢƻƺƾsibilities. ZDM: The International Journal on Mathematics Education, 40ᄬᇴᄭᄕᇵᇳᇹᅬᇵᇴᇹᄙ

ሧሤ

ѹ‫ؔ؛ؖآءإؔح‬

ƺƿƩƣƹƟƣƽƨᄕᄙᄬᇳᇻᇹᇻᄭᄙThe emerging goddess: The creative process in art, science, and other fields. University of Chicago Press. ƫƸƺƹᄕᄙᄙᄕƶƞơƞᄕᄙᄕѵǁƫƿDžǀƽᄕᄙᄬᇴᇲᇳᇶᄭᄙǂƺƾƿƞƨƣƾƺƤƸƞƿƩƣƸƞƿƫơƾơƺƹơƣƻƿƶƣƞƽƹƫƹƨᄘƢƢƫƿƫƺƹƞƶƞƻƻƶƫơƞƿƫƺƹƾƫƹƞƹƞƶDŽƾƫƾƺƤƾƿǀƢƣƹƿƶƣƞƽƹƫƹƨᄙ ƹᄙƫơƺƶᄕᄙƣƾƿƣƽƶƣᄕ ᄙƫƶưƣƢƞƩƶᄕѵᄙƶƶƞƹᄬ ƢƾᄙᄭᄕProceedings of the 38th Conference of the International Group for the Psychology of Mathematics Education and the 36th Conference of the North American Chapter of the Psychology of Mathematics Education, Vol. 5 ᄬƻƻᄙᇴᇲᇳᅬ ᇴᇲᇺᄭᄙ ᄙƩƿƿƻƾᄘᄧᄧǂǂǂᄙƻƸƣƹƞᄙƺƽƨᄧƻƽƺơƣƣƢƫƹƨƾᄧ ƫƸƺƹᄕ ᄙᄕ ƞƶƢƞƹƩƞᄕ ᄙ ơƶƫƹƿƺơƴᄕ ᄙᄕ ƴƞƽᄕ ᄙ ᄙᄕ ƞƿƞƹƞƟƣᄕ ᄙᄕ ѵ ƣƸƟƶƞƿᄕ ᄙ ᄙ ᄬᇴᇲᇳᇲᄭᄙƢƣǁƣƶƺƻƫƹƨƞƻƻƽƺƞơƩƿƺƾƿǀƢDŽƫƹƨƾƿǀƢƣƹƿƾᅷƶƣƞƽƹƫƹƨƿƩƽƺǀƨƩƿƩƣƫƽƸƞƿƩƣmatical activity. Cognition and Instruction, 28ᄬᇳᄭᄕᇹᇲᅬᇳᇳᇴᄙ ƫƸƺƹᄕ ᄙ ᄙᄕDžǀƽᄕ ᄙᄕ ƣƫƹDžᄕ ᄙᄕ ѵ ƫƹDžƣƶᄕ ᄙ ᄬᇴᇲᇲᇶᄭᄙ ǃƻƶƫơƞƿƫƹƨ ƞ ƸƣơƩƞƹƫƾƸ Ƥƺƽ conceptual learning: Elaborating the construct of reflective abstraction. Journal for Research in Mathematics Education, 35ᄬᇷᄭᄕᇵᇲᇷᅬᇵᇴᇻᄙ ƹƺǂᄕᄙ ᄙᄕƺƽƹƺᄕᄙᄕѵƞơƴƾƺƹᄕᄙᄕ

ᄙᄬᇳᇻᇻᇸᄭᄙ ƹƢƫǁƫƢǀƞƶƢƫƤƤƣƽƣƹơƣƾƫƹƞƤƤƣơƿƫǁƣƞƹƢ ơƺƹƞƿƫǁƣƤǀƹơƿƫƺƹƾᄙ ƹᄙᄙƣƽƶƫƹƣƽѵᄙᄙƞƶƤƣƣᄬ ƢƾᄙᄭᄕHandbook of educational psychologyᄬƻƻᄙᇴᇶᇵᅬᇵᇳᇲᄭᄙƞơƸƫƶƶƞƹᄙ ƽƫƽƞƸƞƹᄕᄙᄕƣƿƞƶᄙᄬᇴᇲᇳᇳᄭᄙƞƿƩƣƸƞƿƫơƞƶơƽƣƞƿƫǁƫƿDŽƞƹƢƸƞƿƩƣƸƞƿƫơƾƣƢǀơƞƿƫƺƹᄙ ƹᄙ ƽƫƽƞƸƞƹѵᄙ ᄙƣƣᄬ ƢƾᄙᄭᄕThe elements of creativity and giftedness in mathematics ᄬƻƻᄙᇳᇳᇻᅬᇳᇵᇲᄭᄙƣƹƾƣǀƟƶƫƾƩƣƽƾᄙ ƞƶƶᄕᄙᄙᄬᇳᇻᇺᇷᄭᄙƹƢƣƽƾƿƞƹƢƫƹƨƿƩƣơƞƶơǀƶǀƾᄙMathematics Teaching, 10ᄕᇶᇻᅬᇷᇵᄙ ƞƶƶᄕᄙᄙᄬᇴᇲᇲᇶᄭᄙ ƹƿƽƺƢǀơƫƹƨƿƩƽƣƣǂƺƽƶƢƾƺƤƸƞƿƩƣƸƞƿƫơƾᄙFor the Learning of Mathematics, 23ᄬᇵᄭᄙᇴᇻᅬᇵᇵᄙ ƞƶƶᄕ ᄙ ᄙ ᄬᇴᇲᇳᇵᄭᄙ How humans learn to think mathematically. Cambridge University Press. ƞƶƶᄕᄙᄙᄕƞƶƶᄕᄙᄕѵƞƶƶᄕᄙᄬᇴᇲᇳᇹᄭᄙƽƺƟƶƣƸƻƺƾƫƹƨƫƹƿƩƣƶƺƹƨᅟƿƣƽƸơƺƹơƣƻƿǀƞƶƢƣǁƣƶƺƻƸƣƹƿƺƤƞƨƫƤƿƣƢơƩƫƶƢᄙ ƹᄙƿƣƫƹᄬ ƢᄙᄭᄕA life’s time for mathematics education and problem solving: On the occasion of Andràs Ambrus’ 75th birthdayᄬƻƻᄙᇶᇶᇷᅬᇶᇷᇹᄭᄙ WTM. ƞƶƶᄕᄙᄙᄕѵƫƹƹƣƽᄕᄙᄬᇳᇻᇺᇳᄭᄙƺƹơƣƻƿƫƸƞƨƣƞƹƢơƺƹơƣƻƿƢƣƤƫƹƫƿƫƺƹƫƹƸƞƿƩƣƸƞƿƫơƾ with particular reference to limits and continuity. Educational Studies in Mathematics, 12ᄕᇳᇷᇳᅬᇳᇸᇻᄙ ƞƶƶƞƾᄕᄙᄬᇳᇻᇴᇸᄭᄙThe art of thought. Harcourt, Brace & Co.

ѹѾѷ҆Ҋѻ҈ማ

Arthur Koestler’s Bisociation Theory Bronislaw Czarnocha

1.1

Introduction

Creativity has always been a highly valued human trait, yet its source is still ƶƞƽƨƣƶDŽơƶƺƞƴƣƢƫƹƿƩƣǀƹƴƹƺǂƹᄙ ƹƿƩƣƻƞƾƿᄕơƽƣƞƿƫǁƫƿDŽǂƞƾƞƿƿƽƫƟǀƿƣƢƿƺƿƩƣ intervention of the gods. We are told that Archimedes sacrificed a hecatomb, a hundred bulls, to the gods in gratitude for his discovery of the law of buoyancy. ƩƞƻƿƣƽᇳᇹƫƹơƶǀƢƣƾƞƢƣƾơƽƫƻƿƫƺƹƺƤƿƩƣƸƞƾƾƫǁƣƩƞᄛƸƺƸƣƹƿƺƤƫƶƶǀƸƫƹƞƿƫƺƹ by Appar, the 7th-century Tamil poet-philosopher. He attributed the sudden access to creativity he experienced to the being who “disclosed on His own accord His presence and essences” of the creativity secrets within the “Chaste Tamil of illuminating verses and compose poems and lyrics with the same” ᄬƩƞƻƿƣƽᇳᇹᄭᄙ ƻƻƞƽᅷƾƽƣƤƣƽƣƹơƣƿƺƞbeing who discloses secrets of poetry points to the unpredictable quality of creativity, over which we have no explicit control. ƩƽƺǀƨƩƿƩƣDŽƣƞƽƾᄕƫƹƿƣƽƣƾƿƫƹơƽƣƞƿƫǁƫƿDŽƩƞƾƣǃƻƞƹƢƣƢᄙ ƹƞƢƢƫƿƫƺƹƿƺƾơƫƣƹƿƫƾƿƾ and artists who experienced it directly, it attracted the interest of thinkers in general, psychologists and, very slowly, teachers and researchers. How to facilitate creativity, whose occurrence is uncertain but whose benefits are critical for pupils, has become one of the central motivations for contemporary investigations and the practice of creativity. Our Ҋ҈ƣƞƸᅷƾƞƻƻƽƺƞơƩƩƞƾƟƣƣƹƫƹƤƶǀƣƹơƣƢƟDŽƽƿƩǀƽƺƣƾƿƶƣƽᅷƾƟƫƾƺơƫƞƿƫƺƹƿƩƣƺƽDŽƺƤơƽƣƞƿƫǁƫƿDŽᄙ ƹƩƫƾƟƺƺƴThe Act of Creationᄬᇳᇻᇸᇶᄭᄕ ƺƣƾƿƶƣƽƺƻƣƹƾ his discussion with very broad strokes encompassing humor, science, and art within three columns of the triptych shown below. He wanted to underscore ƿƩƣƻƽƫƹơƫƻƶƣƿƩƞƿơƽƣƞƿƫǁƫƿDŽƫƹƣƞơƩƢƺƸƞƫƹƫƾƿƩƣƾƞƸƣƞƾƟƫƾƺơƫƞƿƫƺƹᄬƢƣƤƫƹƣƢ ƟƣƶƺǂᄭᄕǂƩƫƶƣƿƩƣƶƞƹƨǀƞƨƣƺƤƣǃƻƽƣƾƾƫƺƹƞƹƢƣƸƺƿƫƺƹƞƶơƺƹƿƣƹƿơƩƞƹƨƣƾƞƾǂƣ move from left to right side of the triptych.

ƹƺƣƾƿƶƣƽᅷƾǂƺƽƢƾᄘ

ƩƞǁƣơƺƫƹƣƢƿƩƣƿƣƽƸᅸƟƫƾƺơƫƞƿƫƺƹᅺƫƹƺƽƢƣƽƿƺƸƞƴƣƞƢƫƾƿƫƹơƿƫƺƹƟƣƿǂƣƣƹ the routine skills of thinking on a “single” plane, as it were, and the creƞƿƫǁƣƞơƿᄕǂƩƫơƩᄕƞƾ ƾƩƞƶƶƿƽDŽƿƺƾƩƺǂᄕƞƶǂƞDŽƾƺƻƣƽƞƿƣƾƺƹƸƺƽƣƿƩƞƹƺƹƣ ƻƶƞƹƣᄙᄬƻƻᄙᇵᇷᅬᇵᇸᄭ ᇙ ‫ةء؟؟؜إؘؕ؞؝؜؟؞ء؜ءآ؞‬ᄕ‫ؘ҄ؗ؜ؘ؟‬ᄕሦሤሦሥᏺᄩᏺѺ҅ѿᄘሥሤᄙሥሥሪሧᄧርራሬርሤሤረረረሪረሧረᇇሤሤሦ

ሧሦ

ѹ‫ؔ؛ؖآءإؔح‬

ƩƣƶƺƨƫơƞƶƻƞƿƿƣƽƹƺƤƿƩƣơƽƣƞƿƫǁƣƻƽƺơƣƾƾƫƾƿƩƣƾƞƸƣƫƹƞƶƶƿƩƽƣƣơƺƶǀƸƹƾᄙ ƿ consists of the discovery of hidden similarities, but the emotional climate is different in each panel. For example, in the triple comic comparison–objective hidden analogy–poetic image, the comic simile has a touch of aggressiveness. ƩƣƾơƫƣƹƿƫƾƿᅷƾƽƣƞƾƺƹƫƹƨƟDŽƞƹƞƶƺƨDŽƫƾƣƸƺƿƫƺƹƞƶƶDŽƢƣƿƞơƩƣƢᄬƫƾƹƣǀƿƽƞƶᄭᄖƿƩƣ poetic image is sympathetic or admiring, inspired by a kind of positive emoƿƫƺƹᄬƻᄙᇴᇹᄭᄙƿƿƩƣƾƞƸƣƿƫƸƣᄕƿƩƣƽƣƫƾƞơƺƸƸƺƹƿƩƽƣƞƢƺƤƸƣƞƹƫƹƨƿƩƞƿƻƞƾƾƣƾ through all three qualities. Bisociation is the spontaneous leap of insight that connects previously unconnected frames of referenceሾ by “unearthing” hidden analogies. ƺƣƾƿƶƣƽ ǀƾƣƾ ƾƣǁƣƽƞƶ ƢƫƤƤƣƽƣƹƿ ƿƣƽƸƾ ƫƹ ƞƢƢƫƿƫƺƹ ƿƺ ᅸƤƽƞƸƣƾ ƺƤ ƽƣƤƣƽƣƹơƣᅺᄘ matrices of thought, different discourses, matrices of experience, or planes of reference. All these terms convey a similar concept, perspective, or outlook. Ʃƣƹ ƺƣƾƿƶƣƽ ᄬᇳᇻᇸᇶᄭ ƾƞDŽƾ ƿƩƞƿ ƢƫƾơƺǁƣƽDŽ ƿƩƽƺǀƨƩ ƿƩƣ ǀƹƣƞƽƿƩƫƹƨ ƺƤ ƞ ᅸƩƫƢƢƣƹƞƹƞƶƺƨDŽᅺᄬƻᄙᇵᇴᇲᄭƫƾᅸƹƣǀƿƽƞƶᄕᅺƩƣƸƣƞƹƾƿƩƞƿƫƿƫƾƞᅸƹƫơƣƶDŽƟƞƶƞƹơƣƢƞƹƢ sublimated blend of motivations, where self-assertiveness is harnessed to the task; and where on the other hand heady speculations about the Mysteries of ƞƿǀƽƣƸǀƾƿƟƣƾǀƟƸƫƿƿƣƢƿƺƿƩƣƽƫƨƺǀƽƾƺƤƺƟưƣơƿƫǁƣǁƣƽƫƤƫơƞƿƫƺƹᅺᄬƻᄙᇺᇹᄭᄙ

‫ؘإبؚ؜ؙ‬ᇳᄙᇳᏻ ƺƣƾƿƶƣƽƿƽƫƻƿDŽơƩᄬƤƽƺƸ ƺƣƾƿƶƣƽᄕᇳᇻᇸᇶᄕƫƹƾƫƢƣơƺǁƣƽᄭ

As anyone who has experienced such a moment of insight knows, an Aha! moment brings a thoroughly positive, hence hardly neutral emotion with its ƣǃơƶƞƸƞƿƫƺƹƺƤᅸƩƞᄛᅺƺƣƾƿƶƣƽƢƫƾơƣƽƹƾƹƺƿƺƹƣƟǀƿƿǂƺƾƫƢƣƾƿƺƿƩƫƾƻƩƣƹƺƸᅟ enon.

ѷ‫إب؛اإ‬ҁ‫إؘ؟ائؘآ‬ᅷ‫ئ‬Ѹ‫ءآ؜اؔ؜ؖآئ؜‬Ҋ‫جإآؘ؛‬

ሧሧ

One is the triumphant explosion of tension that has suddenly become ƽƣƢǀƹƢƞƹƿƟƣơƞǀƾƣƿƩƣƻƽƺƟƶƣƸǂƞƾƾƺƶǁƣƢƺƽơƺƹơƣƻƿǀƹƢƣƽƾƿƺƺƢᄙƩƞƿᅷƾǂƩDŽ Archimedes, so the story goes, jumped out of the bathtub and ran through the town screaming “Eureka!” There is a second emotion, however: the slowly fading afterglow, the gradual catharsis of the self-transcending emotions—a quiet ơƺƹƿƣƸƻƶƞƿƫǁƣƢƣƶƫƨƩƿƫƹƿƩƣƿƽǀƿƩưǀƾƿƢƫƾơƺǁƣƽƣƢᄬƾƣƣƿƩƣƩƞƿ ƾƞƣơƿƺƽᄞ ƩƞᄛƸƺƸƣƹƿƫƹƩƞƻƿƣƽᇳᇹᄭᄙ ƩƣƤƫƽƾƿƻƩƣƹƺƸƣƹƺƹƽƣƾǀƶƿƾƤƽƺƸƿƩƣᅸ ᅺƿƩƞƿƸƞƢƣƞƢƫƾơƺǁƣƽDŽᄕƺƽƾƣƶƤᅟƞƾsertion. The second one results when a discovery leads to self-transcendence. Both phenomena generate a sense of well-being, in our case with mathematƫơƾᄙƩƞƿᅷƾǂƩDŽƿƩƣƣƤƤƫơƫƣƹƿƤƞơƫƶƫƿƞƿƫƺƹƺƤơƽƣƞƿƫǁƣƫƹƾƫƨƩƿƾƫƹƿƩƣơƶƞƾƾƽƺƺƸƫƾ so important pedagogically. These experiences are more than merely enjoyable or otherwise positive; they build a bond between the personal knowledge ơƺƹƾƿƽǀơƿƣƢƞƹƢƸƞƿƩƣƸƞƿƫơƞƶơƺƹƿƣƹƿᄬƣƣƶƶƫƾѵƺƶƢƫƹᄕᇴᇲᇲᇸᄭᄙ

1.2

Short Review of Relevant Research

ƞƶƶƞƾ ᄬᇳᇻᇴᇸᄭ ƤƺƽƸǀƶƞƿƣƢ ƺƹƣ ƺƤ Ƹƺƽƣ ƫƹƿƣƽƣƾƿƫƹƨ ƞƻƻƽƺƞơƩƣƾ ƿƺ ơƽƣƞƿƫǁƫƿDŽ with the help of Poincaré and Hadamard; these mathematicians provide striking examples of Aha! moment insights that occurred during their work. They describe a Gestalt approach to creativity composed of four well-known stages: ƻƽƣƻƞƽƞƿƫƺƹᄕ ƫƹơǀƟƞƿƫƺƹᄕ ƫƶƶǀƸƫƹƞƿƫƺƹᄕ ƞƹƢ ǁƣƽƫƤƫơƞƿƫƺƹᄙ Ʃƞƻƿƣƽ ᇳᇵ Ƣƫƾơǀƾƾƣƾ ƿƩƣƾƣƫƹƿƩƣơƺƹƿƣǃƿƺƤƿƩƣƫƸƩƞᄛƸƺƸƣƹƿᄙ The Gestalt model focused attention on the secrets of incubation and illumination as the distinct phases during which creativity takes place and insight ƫƾƤƺƽƸƣƢᄙ ƹƨƣƹƣƽƞƶᄕƣƾƿƞƶƿƽƣƾƣƞƽơƩƣƽƾơƩƞƽƞơƿƣƽƫDžƣƢƿƩƣƾƣƿǂƺƻƩƞƾƣƾƻƩƣnomenologically as the time when the mind, tired of thinking about the probƶƣƸƫƹƿƣƽƸƾƺƤƴƹƺǂƹƞƹƢƞǁƞƫƶƞƟƶƣƫƢƣƞƾᄕƢƣơƫƢƣƾƿƺƾƩǀƿƺƤƤᄙ ƿƿƣƸƻƺƽƞƽƫƶDŽ forgets about the problem at hand and focuses on other issues, preferably far removed from it. ƽơƩƫƸƣƢƣƾǂƞƾƿƞƴƫƹƨƞƟƞƿƩᄖƺƫƹơƞƽȅǂƞƾƨƣƿƿƫƹƨƺƹƿƩƣƟǀƾᄖƞƹƢƣƴǀƶƣᄕ the discoverer of the structure of the benzene molecule, was dosing by the fire ᄬƺƢƣƹᄕᇴᇲᇲᇶᄕƻᄙᇴᇷᄭǂƩƫƶƣƿƩƣƫƽƫƹƾƫƨƩƿƾǂƣƽƣƿƞƴƫƹƨƻƶƞơƣᄙƺƿƣƫƸᅷƾƢƣơƫƾƫƺƹ ᄬƩƞƻƿƣƽᇳᇲᄭƿƺƻǀƿƞƻƽƺƟƶƣƸƞƾƫƢƣƞƤƿƣƽǀƹƾǀơơƣƾƾƤǀƶƶDŽǂƺƽƴƫƹƨƞƿƫƿƞƹƢƤƺơǀƾ her attention on other problems of the exam.

ƺƽƫƸᄕƿƩƫƾƫƹơǀƟƞƿƫƺƹƾƿƞƨƣƶƞƾƿƣƢǀƹƿƫƶƾƩƣƣƽƞƾƣƢƞƶƶƻƽƣǁƫƺǀƾƹƺƿƣƾƞƹƢ started from the beginning, using a significantly different approach of creating new terms in the given series. At that moment came the illumination. Gestalt psychologists explicitly define that process as the main characterization of ƫƹƾƫƨƩƿᄘᅸᄴ ᄵƹƾƫƨƩƿƺơơǀƽƾǂƩƣƹƞƻƽƺƟƶƣƸƫƾƾƺƶǁƣƢƿƩƽƺǀƨƩƽƣƾƿƽǀơƿǀƽƫƹƨᄘƩƞƿ

ሧረ

ѹ‫ؔ؛ؖآءإؔح‬

is, if we compare the initial solution attempts with the insightful solution, ƿƩƣDŽƸǀƾƿƟƣƿƩƣƽƣƾǀƶƿƺƤƢƫƤƤƣƽƣƹƿƞƹƞƶDŽƾƣƾƺƤƿƩƣƻƽƺƟƶƣƸᅺᄬƣƫƾƟƣƽƨᄕᇳᇻᇻᇷᄕ ƻᄙᇳᇸᇵᄭᄙƩƣƻƽƺơƣƾƾƿƩƞƿƿƽƞƹƾƤƺƽƸƣƢƫƹơǀƟƞƿƫƺƹƫƹƿƺƫƶƶǀƸƫƹƞƿƫƺƹᄕƩƺǂƣǁƣƽᄕ ǂƞƾǀƹƴƹƺǂƹƿƺƣƾƿƞƶƿƿƩƣƺƽDŽᄙ ƿǂƞƾƺƣƾƿƶƣƽᅷƾǂƺƽƴƿƩƞƿƻƽƺǁƫƢƣƢƿƩƣƤƫƽƾƿ inklings in that direction. ƩƣƾƣơƺƹƢƞƻƻƽƺƞơƩƿƺơƽƣƞƿƫǁƫƿDŽơƞƸƣƢǀƽƫƹƨƿƩƣᇳᇻᇷᇲƾᄙƩƫƾƟƞơƴƶƞƾƩƸƞDŽ have been a response to behaviorist theories that started taking center stage in ƻƾDŽơƩƺƶƺƨDŽǂƩƣƹǀƫƶƤƺƽƢᄬᇳᇻᇸᇹᄭƫƹᇳᇻᇷᇲƻƽƺƻƺƾƣƢƿƩƣơƩƞƽƞơƿƣƽƫDžƞƿƫƺƹƺƤƿƩƣ thought process into “divergent” and “convergent” thinking, with the first term characterizing creative thinking. Guilford suggested the use of psychometric approaches to assess the charƞơƿƣƽƫƾƿƫơƼǀƞƶƫƿƫƣƾƺƤơƽƣƞƿƫǁƫƿDŽᄙƺƽƽƞƹơƣᄬᇳᇻᇸᇸᄭᄕƤƺƶƶƺǂƫƹƨǀƫƶƤƺƽƢᄕƢƣƾƫƨƹƣƢ ƿƩƣƣƾƿƺƤƽƣƞƿƫǁƣƩƫƹƴƫƹƨᄬҊҊѹҊᄭᄕǂƩƫơƩƽƣƼǀƫƽƣƾƻƣƽƤƺƽƸƞƹơƣƺƹǁƣƽƟƞƶ ƞƹƢ Ƥƫƨǀƽƞƶ ƿƞƾƴƾ ƿƺ Ɵƣ ƣǁƞƶǀƞƿƣƢ ƟDŽ ƤƶǀƣƹơDŽ ᄬƿƺƿƞƶ ƹǀƸƟƣƽ ƺƤ ƞƻƻƽƺƻƽƫƞƿƣ ƽƣƾƻƺƹƾƣƾᄭᄕƤƶƣǃƫƟƫƶƫƿDŽᄬƿƩƣƹǀƸƟƣƽƺƤƢƫƤƤƣƽƣƹƿơƞƿƣƨƺƽƫƣƾᄭᄕƺƽƫƨƫƹƞƶƫƿDŽᄬƽƞƽƫƿDŽ ƺƤ ƽƣƾƻƺƹƾƣᄭᄕ ƞƹƢ ƣƶƞƟƺƽƞƿƫƺƹ ᄬƞƸƺǀƹƿ ƺƤ Ƣƣƿƞƫƶ ǀƾƣƢ ƫƹ ƽƣƾƻƺƹƾƣƾᄭᄕ ƞơơƺƽƢƫƹƨ ƿƺ ƣƫƴƫƹ ƞƹƢ ƫƿƿƞᅟƞƹƿƞDžƫ ᄬᇴᇲᇳᇵᄭᄙ Ʃƣ ǂƺƽƴ ƺƤ ƫƶǁƣƽ ᄬᇳᇻᇻᇹᄭ ƞƹƢ ƣƫƴƫƹ ᄬᇴᇲᇲᇻᄭƽƣƢǀơƣƢƿƩƺƾƣơƩƞƽƞơƿƣƽƫDžƞƿƫƺƹƾƿƺƤƶǀƣƹơDŽᄕƤƶƣǃƫƟƫƶƫƿDŽᄕƞƹƢƺƽƫƨƫƹƞƶƫƿDŽᄙ ƾƣƫƴƫƹƞƹƢƫƿƿƞᅟƞƹƿƞDžƫᄬᇴᇲᇳᇵᄭƹƺƿƣᄕᅸƿƩƣƽƣƫƾƞƢƣƟƞƿƣƞƾƿƺǂƩƣƿƩƣƽƿƩƫƾ type of evaluation captures the essence of creativity.” This book is a voice in that debate. To understand the degree to which these qualities, together with psychoƸƣƿƽƫơƞƻƻƽƺƞơƩƣƾᄕơƞƻƿǀƽƣƿƩƣƣƾƾƣƹơƣƺƤơƽƣƞƿƫǁƫƿDŽᄕƫƿᅷƾƫƸƻƺƽƿƞƹƿƿƺƹƺƿƣƿƩƣ ƽƣƾǀƶƿƾ ƺƤ ƫƸ ᄬᇴᇲᇳᇴᄭᄕ ǂƩƺ ƤƺǀƹƢ ƿƩƞƿ ơƽƣƞƿƫǁƫƿDŽᄕ ƞƾ ƸƣƞƾǀƽƣƢ ƟDŽ ҊҊѹҊ, conƾƿƞƹƿƶDŽ ƢƫƸƫƹƫƾƩƣƢ Ɵƣƿǂƣƣƹ ᇳᇻᇻᇲ ƞƹƢ ᇴᇲᇲᇺ ƞƸƺƹƨ ƴƫƹƢƣƽƨƞƽƿƣƹ ƞƹƢ ƾơƩƺƺƶ ơƩƫƶƢƽƣƹƫƹƿƩƣƹƫƿƣƢƿƞƿƣƾǂƫƿƩƿƩƣƟƫƨƨƣƾƿƢƽƺƻƫƹƽƞƢƣƾᇵᄕᇶƞƹƢᇸᄙƶƣƞƽƶDŽᄕ ơƺƹƿƣƸƻƺƽƞƽDŽƣƢǀơƞƿƫƺƹƢƺƣƾǀƹƢƣƽƸƫƹƣơƩƫƶƢƽƣƹᅷƾơƽƣƞƿƫǁƫƿDŽƿƺƞǂƺƽƽƫƾƺƸƣ Ƣƣƨƽƣƣᄙ ƿᅷƾƺƹƣƺƤƺǀƽƸƞƫƹƸƺƿƫǁƞƿƫƺƹƾƤƺƽƤƺƽƸǀƶƞƿƫƹƨƩƞᄛƻƣƢƞƨƺƨDŽᄙƩƣ Ƥƺơǀƾ ƺƹ ơƽƣƞƿƫǁƫƿDŽ ơƩƺƾƣƹ ƟDŽ ƽƞƟƩǀ ƞƹƢ DžƞƽƹƺơƩƞ ᄬᇴᇲᇳᇶᄭ ǂƞƾ ƿƩƣ ƿƣƞơƩing-research classroom investigations of Aha! moments, known also as the ǀƽƣƴƞƣǃƻƣƽƫƣƹơƣᄕƟƞƾƣƢƺƹƺƣƾƿƶƣƽᅷƾᄬᇳᇻᇸᇶᄭƞƻƻƽƺƞơƩᄕǂƩƫơƩƻƺƫƹƿƾƢƫƽƣơƿƶDŽ to the stages of incubation and illumination of the Gestalt model. New directions of creativity investigations have opened within last several ƢƣơƞƢƣƾ ƿƩƽƺǀƨƩ ƹƣǀƽƺƾơƫƣƹƿƫƤƫơ ƞƹƢ ƞƽƿƫƤƫơƫƞƶ ƫƹƿƣƶƶƫƨƣƹơƣ ᄬѷѿᄭ ƞƻƻƽƺƞơƩƣƾᄙ The cognitive aspects of the neuroscientific approach are described below in ƣơƿƫƺƹƾᇳᄙᇷᄙᇵᄕᇳᄙᇷᄙᇶƞƹƢƫƹƩƞƻƿƣƽᇳᇶᄖƿƩƣƞƤƤƣơƿƫǁƣƞƾƻƣơƿƾƩƞǁƣƟƣƣƹƞƹƞƶDŽDžƣƢ ƟDŽƣƣƸƞƹƞƹƢơƺƶƶƣƞƨǀƣƾᄬƺǀƹƫƺƾѵƣƣƸƞƹᄕᇴᇲᇳᇷᄭƞƹƢƣƶƞƟƺƽƞƿƣƢƫƹƞƽƿ 2: The Aha! Moment and Affect. The impact of the ѷѿ approach is elaborated ƫƹƩƞƻƿƣƽᇳᇷᄙ

ѷ‫إب؛اإ‬ҁ‫إؘ؟ائؘآ‬ᅷ‫ئ‬Ѹ‫ءآ؜اؔ؜ؖآئ؜‬Ҋ‫جإآؘ؛‬

1.3

ሧሩ

The Aha! Experience in General

The term “Aha! moment,” which was coined by the thinkers of the Gestalt ƸƺǁƣƸƣƹƿ ƫƹ ƿƩƣ Ƥƫƽƾƿ ƩƞƶƤ ƺƤ ƿƩƣ ᇴᇲƿƩ ơƣƹƿǀƽDŽᄕ ƽƣƤƣƽƾ ƿƺ ƿƩƣ Ƣƣƣƻ ƞƤƤƣơƿƫǁƣᄧ cognitive aspect of experience often accompanied by an equally deep exclamation of “Aha!,” indicating a grasp of content and euphoria of insight. The euphoria follows the moment of truth or the flash of illumination when puzzle pieces click into place, or in our terms, when bisociated contexts fuse in a new ƾDŽƹƿƩƣƾƫƾ ᄬƾƿƞƿƣƢ ƢƫƤƤƣƽƣƹƿƶDŽᄕ ǂƩƣƹ ƢƫƾƾƫƸƫƶƞƽ ƴƹƺǂƶƣƢƨƣ ơƺƸƟƫƹƣƾ ƫƹ ƞ ƹƣǂ ǂƞDŽᄭᄙ ƺƣƾƿƶƣƽᅷƾƢƣƤƫƹƫƿƫƺƹƺƤƿƩƣơƽƣƞƿƫǁƣƸƺƸƣƹƿƫƾᄕƺƤơƺǀƽƾƣᄕơƺƹƹƣơƿƣƢƿƺƿƩƣ ƞƶƶƞƾᄧ ƞƢƞƸƞƽƢᄬᇳᇻᇴᇸᄧᇳᇻᇶᇷᄭƢƣƤƫƹƫƿƫƺƹƺƽƫƨƫƹƞƿƫƹƨƫƹƿƩƣƣƾƿƞƶƿƞƻƻƽƺƞơƩᄕ which postulates four general stages along the formulation of the creative insight: ᅬ ƽƣƻƞƽƞƿƫƺƹᄬƻƽƣƻƞƽƞƿƺƽDŽǂƺƽƴƺƹƞƻƽƺƟƶƣƸƿƩƞƿƤƺơǀƾƣƾƿƩƣƫƹƢƫǁƫƢǀƞƶᅷƾ ƸƫƹƢƺƹƿƩƣƻƽƺƟƶƣƸƞƹƢƣǃƻƶƺƽƣƾƿƩƣƻƽƺƟƶƣƸᅷƾƢƫƸƣƹƾƫƺƹƾᄭᄙ ᅬ ƹơǀƟƞƿƫƺƹᄬǂƩƣƽƣƿƩƣƻƽƺƟƶƣƸƫƾƫƹƿƣƽƹƞƶƫDžƣƢƫƹƿƺƿƩƣǀƹơƺƹƾơƫƺǀƾƸƫƹƢ ƞƹƢƹƺƿƩƫƹƨƞƻƻƣƞƽƾƣǃƿƣƽƹƞƶƶDŽƿƺƟƣƩƞƻƻƣƹƫƹƨᄭᄙ ᅬ ƶƶǀƸƫƹƞƿƫƺƹƺƽƫƹƾƫƨƩƿᄬǂƩƣƽƣƿƩƣơƽƣƞƿƫǁƣƫƢƣƞƟǀƽƾƿƾƤƺƽƿƩƤƽƺƸƫƿƾƻƽƣơƺƹƾơƫƺǀƾƻƽƺơƣƾƾƫƹƨƫƹƿƺơƺƹƾơƫƺǀƾƞǂƞƽƣƹƣƾƾᄭᄙ ᅬ ƣƽƫƤƫơƞƿƫƺƹ ᄬǂƩƣƽƣ ƿƩƣ ƫƢƣƞ ƫƾ ơƺƹƾơƫƺǀƾƶDŽ ǁƣƽƫƤƫƣƢᄕ ƣƶƞƟƺƽƞƿƣƢᄕ ƞƹƢ ƿƩƣƹ ƞƻƻƶƫƣƢᄭᄙ

ƿƫƾƫƸƻƺƽƿƞƹƿƿƺƞƢƢƞƽƣơƣƹƿƽƣƤƶƣơƿƫƺƹƺƹƞƶƶƞƾᅷƾƸƺƢƣƶƟDŽƿƣƻƩƣƹᄙƞƸƻƟƣƶƶᄙ Ʃƞƻƿƣƽ ᇳᇶ Ƣƫƾơǀƾƾƣƾ ƿƩƣ ƻƽƣƾƣƹơƣ ƺƤ ƞ ƤƫƤƿƩ ƾƿƞƨƣᄕ ƫƹƿƫƸƞƿƫƺƹᄕ Ɵƣƿǂƣƣƹ ƫƹơǀƟƞƿƫƺƹ ƞƹƢ ƫƶƶǀƸƫƹƞƿƫƺƹᄕ Ƥƺƶƶƺǂƫƹƨ ƞƢƶƣƽᅟƸƫƿƩᅷƾ ᄬᇴᇲᇳᇷᄭ ƽƣƤƶƣơƿƫƺƹᄙ Ʃƣ fifth stage might refine our understanding of the Aha! experience. ƺƣƾƿƶƣƽ ƻƺƾƫƿƾ Ʃƫƾ ƿƩƣƺƽDŽ ƺƹ ƿƩƣ ƫƹƿƣƽƻƩƞƾƣ Ɵƣƿǂƣƣƹ ƣƾƿƞƶƿᅟƟƞƾƣƢ ƞƹƢ behavior-based theories: ƩƣƣƾƿƞƶƿƾơƩƺƺƶᅷƾƺǁƣƽᅟƣƸƻƩƞƾƫƾƺƹᅸǂƩƺƶƣƹƣƾƾᄕᅺƞƹƢƿƩƣƟƣƩƞǁƫƺǀƽƫƾƿᅷƾ over-emphasis on “simple elementary processes”—the so-called S-R ᄬƾƿƫƸǀƶǀƾᅬƽƣƾƻƺƹƾƣᄭ ƾơƩƣƸƣᅭơƽƣƞƿƣƢ ƞ ơƺƹƿƽƺǁƣƽƾDŽ ƟƞƾƣƢ ƺƹ ƞ Ƥƞƶƶƞcious alternative, and prevented a true appreciation of the multi-layered ƩƫƣƽƞƽơƩƫơƺƽƢƣƽƿƺƟƣƤƺǀƹƢƫƹƞƶƶƸƞƹƫƤƣƾƿƞƿƫƺƹƾƺƤƶƫƤƣᄙᄬƻᄙᇶᇵᇴᄭ The “true appreciation of multilayered hierarchic order” can be seen in bisociaƿƫƺƹᄕǂƩƣƽƣƿƩƣƾƿƽǀơƿǀƽƣƺƤƿƩƣƻƽƺƢǀơƿƺƤƫƹƾƫƨƩƿᄬƹƣǂǀƹƢƣƽƾƿƞƹƢƫƹƨᄭƫƾƺƹƞ higher conceptual level than each matrix. Bisociation expresses the hierarchical

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ѹ‫ؔ؛ؖآءإؔح‬

ƹƞƿǀƽƣ ƺƤ ơƺƹơƣƻƿ ƢƣǁƣƶƺƻƸƣƹƿᄕ ǂƩƫơƩᄕ ƞơơƺƽƢƫƹƨ ƿƺ ƺƣƾƿƶƣƽᄕ ƫƾ ƫƿƾ ơƣƹƿƽƞƶ quality rather than the issues of arguments between Gestalt and behaviorist approaches. The critical difference between the two, as seen in the mathematics classƽƺƺƸᄕƶƫƣƾƫƹƿƩƣƫƽǁƫƣǂƺƤơƽƣƞƿƫǁƫƿDŽᄘǂƩƣƽƣƞƾǀƫƶƤƺƽƢƞƹƢƺƽƽƞƹơƣᅷƾƞƻƻƽƺƞơƩ is based on psychometric instruments that view creativity as the product of ƿƩƺǀƨƩƿᄕ ƺƣƾƿƶƣƽ ǁƫƣǂƾ ơƽƣƞƿƫǁƫƿDŽ ƞƾ ƟƺƿƩ ƿƩƣ ƻƽƺơƣƾƾ ᄬơƺƹƹƣơƿƫƹƨ ǀƹơƺƹƹƣơƿƣƢ ƤƽƞƸƣƾ ƺƤ ƽƣƤƣƽƣƹơƣᄭ ƞƹƢ ƿƩƣ ƻƽƺƢǀơƿ ᄬƾƻƺƹƿƞƹƣƺǀƾ ƶƣƞƻ ƺƤ ƫƹƾƫƨƩƿᄕ ǂƩƫơƩƟƽƫƹƨƾƫƹƿƩƣơƺƹƹƣơƿƫƺƹƿƩƽƺǀƨƩᅸǀƹƣƞƽƿƩƫƹƨᅺƞƩƫƢƢƣƹƞƹƞƶƺƨDŽᄭᄙ

ƿƫƾƿƩƣƾƣƻƞƽƞƿƫƺƹƺƤƿƩƣƿǂƺǂƫƿƩƫƹƿƩƣǀƹƫƿDŽƞƹƢǂƩƺƶƣƹƣƾƾƺƤƿƩƣƫƶƶǀƸƫƹƞƿƫƺƹƣǁƣƹƿƿƩƞƿƞƶƶƺǂƾ ƞƢƞƸƞƽƢᄬᇳᇻᇶᇷᄭƿƺƞƾƾƣƽƿƿƩƞƿ Between the work of the student who tries to solve a problem in geometry or algebra and a work of invention, one can say that there is only the difference of degree, the difference of a level, both works being of similar ƹƞƿǀƽƣᄙᄬƻᄙᇳᇲᇶᄭ Ʃǀƾᄕƫƹ ƞƢƞƸƞƽƢᅷƾǁƫƣǂᄕǂƩƫơƩƽƣƫƹƤƺƽơƣƾƺǀƽƺǂƹᄕƿƩƣơƽƣƞƿƫǁƣƻƽƺơƣƾƾƫƾ the same, while the creative product, by contrast, changes depending on the degree of the relevant knowledge and passionate commitment to creative mathematical work. Ʃƣ ƣƹƿƽDŽ ƺƤ ƺƣƾƿƶƣƽᅷƾ ƿƩƣƺƽDŽ ƺƤ Ɵƫƾƺơƫƞƿƫƺƹ ǂƫƿƩ ƫƿƾ Ƥƺơǀƾ ƺƹ ƞ ƾƻƺƹƿƞneous leap and on connecting unconnected matrices begins the third pathway toward understanding creativity, particularly the Aha! moment and the ǀƽƣƴƞƣǃƻƣƽƫƣƹơƣᄙƺƣƾƿƶƣƽᅷƾƿƩƣƺƽDŽƺƤƟƫƾƺơƫƞƿƫƺƹƫƾƤƺƶƶƺǂƣƢƟDŽƺƿƩƣƹƟƣƽƨᅷƾ ᄬᇳᇻᇹᇻƟᄭƿƩƣƺƽDŽƺƤƞƹǀƾƫƞƹƿƩƫƹƴƫƹƨᄕƞƹƢᄕƫƹƢƣƻƣƹƢƣƹƿƶDŽᄕƟDŽƿƩƣơƺƸƻǀƿƣƽơƽƣativity domain of ѷѿ with its distinct imprint of connectionist systems or neuƽƞƶƹƣƿǂƺƽƴƾᄬƣƽƿƩƺƶƢᄕᇴᇲᇳᇴᄭᄙ

1.4

The Aha! Moment: Examples of Big C, or Historical, Creativity

Ʃƣ Ʃƞᄛ ƸƺƸƣƹƿ ƫƾ ƞ ơƺƸƸƺƹ ƩǀƸƞƹ ƣǃƻƣƽƫƣƹơƣᄙ ƿᅷƾ ƾƺ ơƺƸƸƺƹ ƿƩƞƿ ƫƿ became one of the main topics of Oprah’s Lifeclass, an American primetime ƿƣƶƣǁƫƾƫƺƹƾƩƺǂᄙƹƩƞᄛ ƾƞƸƺƸƣƹƿƺƤƾǀƢƢƣƹƽƣƞƶƫDžƞƿƫƺƹǂƩƣƹƿƩƣƾƺƶǀƿƫƺƹ ƿƺƞƻƽƺƟƶƣƸᄕƾƫƿǀƞƿƫƺƹᄕƺƽƫƢƣƞǂƣᅷǁƣƟƣƣƹƨƽƞƻƻƶƫƹƨǂƫƿƩƾǀƢƢƣƹƶDŽƞƻƻƣƞƽƾƫƹ ƺǀƽƸƫƹƢᅷƾƣDŽƣƾƣƣƸƫƹƨƶDŽǂƫƿƩƺǀƿƺǀƽơƺƹƾơƫƺǀƾƻƞƽƿƫơƫƻƞƿƫƺƹᄙ ƣƶƺǂǂƣƻƽƣƾƣƹƿƾƣǁƣƽƞƶƟƫƨƩƞᄛƸƺƸƣƹƿƾᄬƺƽƩƫƾƿƺƽƫơƞƶƩƞᄛƸƺƸƣƹƿƾᄭ observed in numerous human situations: by a scribe named Appar who lived ƞƽƺǀƹƢ ᇳᄕᇶᇲᇲ DŽƣƞƽƾ ƞƨƺᄕ ƞ ơƺƹƿƣƸƻƺƽƞƽDŽ ƸDŽƾƿƫơᄖ ƟDŽ ƞ ƸƞƿƩƣƸƞƿƫơƫƞƹ ƹƞƸƣƢ

ѷ‫إب؛اإ‬ҁ‫إؘ؟ائؘآ‬ᅷ‫ئ‬Ѹ‫ءآ؜اؔ؜ؖآئ؜‬Ҋ‫جإآؘ؛‬

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Poincaré, known for his highly intuitive approach to mathematics; by Mao Tse-Tung, or Chairman Mao, leader of the Chinese communist revolution; by ƶƟƣƽƿ ƫƹƾƿƣƫƹƫƹƩƫƾᅸǀƿƺƟƫƺƨƽƞƻƩƫơƞƶƺƿƣƾᅺᄬ ƫƹƾƿƣƫƹᄕᇳᇻᇶᇻᄭᄖƞƹƢ by Oprah Winfrey, the ҊҌ personality exceedingly well familiar with Aha! moments as they take place in daily life.ሿ The collection tells us about different ways that this interesting, creative phenomenon appears in human experience and about different approaches to understanding its nature. ƩƣƤƺƶƶƺǂƫƹƨƢƣƾơƽƫƻƿƫƺƹƺƤƞƹƩƞᄛƸƺƸƣƹƿᄬƿƩƣƻƻƞƽƩƞᄛƸƺƸƣƹƿƫƹ ƩƞƻƿƣƽᇳᇹᄭƤƽƺƸƞƽƺǀƹƢᇳᄕᇶᇲᇲDŽƣƞƽƾƞƨƺǂƞƾƤƺǀƹƢƺƹƿƩƣǂƞƶƶƺƤƞƹơƫƣƹƿƽǀƫƹƾ ƫƹƞƞƸƫƶƿƣƸƻƶƣᄙ ƿƢƣƾơƽƫƟƣƾǂƩƞƿǂƣǀƹƢƣƽƾƿƞƹƢƹƺǂƞƾƞƹƩƞᄛƸƺƸƣƹƿ experienced by the poet-philosopher Appar, who suddenly understands the depth, nature, and role of his writing, which until then were in metaphysical darkness. ƫƽǀƸǀƞƫᇸᄘᇻᇳ Ʃƣ DŽƸƹƾƺƤ ǀƸƟƶƣƻƻƞƽᇳᅬᇳ ᇳᄙ panniya centamiz aRiyeen kaviyeen maaddee eNNoodu paN niRainta kalaikaLaaya tannanaiyum tan tiRattaRiyaap poRiyileenait tan tiRamum aRivittu neRiyuG kaadi annaiyaiyum attaniyaiyum poola anbaay adaiteenait todarntu ennai aaLaak koNda ten eRubiyuur malameel maaNikkattai cezunj cudaraic cenRadaiyap peRReen naanee Meaning:

ǂƞƾƾƺƫƨƹƺƽƞƹƿᄬƤǀƶƶƺƤƟƶƫƹƢƹƣƾƾƫƹƢǀơƣƢƟDŽƿƩƣƞƶƞƸᄭƿƩƞƿ ƢƫƢƹƺƿ know the Chaste Tamil of illuminating verses and compose poems and ƶDŽƽƫơƾǂƫƿƩƿƩƣƾƞƸƣᄙ ƢƫƢƹƺƿƴƹƺǂƩƺǂƿƺƞƻƻƽƣơƫƞƿƣƿƩƣƨƽƣƞƿƞƽƿƾƞƹƢ sciences brought to perfection through repeated and continuous reflecƿƫƺƹƾƺƹƿƩƣƸᄙƣơƞǀƾƣƺƤƾǀơƩƫƹơƺƸƻƣƿƣƹơƣᄕ ǂƞƾƹƺƿƞƟƶƣƿƺƞƻƻƽƣơƫate the presence of Ѹѻѿ҄ѽ and His essences. But like a mother and father full of love and care, Ѹѻѿ҄ѽ disclosed on His own accord His presence and essences and continued to be with me along with my developments always keeping me as His own subject. Now full of true understanding of

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ѹ‫ؔ؛ؖآءإؔح‬

Ѹѻѿ҄ѽᄕ ơƶƫƸƟƿƩƣƩƫƶƶƺƤ ǀƹƟƫDŽǀǀƽƞƹƢǂƫƿƹƣƾƾѸѻѿ҄ѽ as the BenevoƶƣƹƿƫƨƩƿᄙᄬƺƨƞƹƞƿƩƞƹᄕᇴᇲᇲᇶᄭ 1.4.1 Meaning Note that Appar sees the tremendous Aha! moment as a religious experience, during which Ѹѻѿ҄ѽ reveals itself to him. A contemporary mystic, who proƤƣƾƾƣƾƻƩƫƶƺƾƺƻƩDŽƺƤƹƺƹƢǀƞƶƫƿDŽƤƺƽƸǀƶƞƿƣƢƟDŽ ƹƢƫƞƹƻƩƫƶƺƾƺƻƩƣƽƾƞƸƞƹƞ Majarshi and Sri Nisargadatta, moves toward the same spiritual connection ǂƩƣƹƞƾƴƣƢƞƟƺǀƿƿƩƣƸƣƞƹƫƹƨƺƤƞƹƩƞᄛƸƺƸƣƹƿᄙ ƿƫƾᄕƩƣƾƞƫƢᄕƿƩƣƻƣƽơƣƻtion of the hole in the veil separating reality as we see it and infinite awareness. Poincaré, on the other hand, known for the collection of Aha! moments he discussed in the famous lecture he delivered to the Psychological Society of ƞƽƫƾƞƿƿƩƣ ƹƾƿƫƿǀƿȅƹȅƽƞƶƾDŽơƩƺƶƺƨƫƼǀƣƫƹᇳᇻᇲᇺᄕƞƿƿƽƫƟǀƿƣƾƿƩƣƻƩƣƹƺƸƣƹƺƹ to the work of the unconscious: “Most striking at first is this appearance of sudden illumination, a manifest sign of long, unconscious prior work. The rôle of this unconscious work in mathematical invention appears to me incontestƞƟƶƣᅺᄬƺƫƹơƞƽȅᄕᇳᇻᇺᇷƞᄕƻᄙᇴᇹᄭᄙ One sudden illumination presented at the lecture bears similarity to the concept of bisociation, which we discuss in this chapter in detail:

ƺƽƤƫƤƿƣƣƹƢƞDŽƾ ƾƿƽƺǁƣƿƺƻƽƺǁƣƿƩƞƿƿƩƣƽƣơƺǀƶƢƹƺƿƟƣƞƹDŽƤǀƹơƿƫƺƹƾƶƫƴƣ ƿƩƺƾƣ ƩƞǁƣƾƫƹơƣơƞƶƶƣƢ ǀơƩƾƫƞƹƤǀƹơƿƫƺƹƾᄙ ǂƞƾƿƩƣƹǁƣƽDŽƫƨƹƺƽƞƹƿᄖ ƣǁƣƽDŽƢƞDŽ ƾƣƞƿƣƢƸDŽƾƣƶƤƞƿƸDŽǂƺƽƴƿƞƟƶƣᄕƾƿƞDŽƣƢƞƹƩƺǀƽƺƽƿǂƺᄕƿƽƫƣƢ a great number of combinations and reached no results. One evening, ơƺƹƿƽƞƽDŽƿƺƸDŽơǀƾƿƺƸᄕ ƢƽƞƹƴƟƶƞơƴơƺƤƤƣƣƞƹƢơƺǀƶƢƹƺƿƾƶƣƣƻᄙIdeas rose in crowds; I felt them collide until pairs interlocked, so to speak, making a stable combination.ᄬƺƫƹơƞƽȅᄕᇳᇻᇺᇷƟᄕƻƻᄙᇴᅬᇵᄕƣƸƻƩƞƾƫƾƞƢƢƣƢᄭ

ƹƿƣƽƣƾƿƫƹƨƶDŽᄕƞƺᄬᇴᇲᇲᇹᄭƩƞƾƾƺƸƣƿƩƫƹƨƾǀƟƾƿƞƹƿƫƞƶƿƺƾƞDŽƞƾǂƣƶƶᄙ ƹƩƫƾƣƾƾƞDŽ “On Practice,” he explicitly includes Aha! moments in his theory of the development of human knowledge: ƾƾƺơƫƞƶƻƽƞơƿƫơƣơƺƹƿƫƹǀƣƾᄕƿƩƫƹƨƾƿƩƞƿƨƫǁƣƽƫƾƣƿƺƸƞƹᅷƾƾƣƹƾƣƻƣƽơƣƻƿƫƺƹƾ and impressions in the course of his practice are repeated many times; then a sudden change (leap) takes place in the brain in the process of cognition, and concepts are formed. Concepts are no longer phenomena, the separate ƞƾƻƣơƿƾƞƹƢƿƩƣƣǃƿƣƽƹƞƶƽƣƶƞƿƫƺƹƾƺƤƿƩƫƹƨƾᄙᄬƻᄙᇷᇷᄕƣƸƻƩƞƾƫƾƞƢƢƣƢᄭ Mao attributes the emergence of the insight to social practice as well as to the individual thinking using an old Chinese expression: “knit the brows and a ƾƿƽƞƿƞƨƣƸơƺƸƣƾƿƺƸƫƹƢᅺᄬƻᄙᇷᇷᄭᄙ

ѷ‫إب؛اإ‬ҁ‫إؘ؟ائؘآ‬ᅷ‫ئ‬Ѹ‫ءآ؜اؔ؜ؖآئ؜‬Ҋ‫جإآؘ؛‬

ሧር

ƼǀƞƶƶDŽƫƹƿƣƽƣƾƿƫƹƨƫƾ ƫƹƾƿƣƫƹᅷƾᄬᇳᇻᇶᇻᄭƿƞƴƣƺƹƿƩƣƤƺƽƸƞƿƫƺƹƺƤơƺƹơƣƻƿǀƞƶ insights in his “Autobiographical Notes”: What exactly is thinking? When at the reception of sense impressions, a memory-picture emerges, this is not yet thinking, and when such pictures form a series, each member of which calls for another, this too is not yet thinking. When, however, a certain picture turns up in many of such series then—precisely through such a return—it becomes an ordering element for such series, in that it connects series, which in themselves are unconnected, such an element becomes an instrument, a concept. ᄬƻᄙᇹᄭ The moment when a “certain picture” emerges from among many images and joins them by connecting images that are themselves unconnected is the ƸƺƸƣƹƿƺƤƫƹƾƫƨƩƿᄙƹƢƿƩƞƿƞƻƻƽƺƞơƩƫƾƞƶƽƣƞƢDŽǁƣƽDŽơƶƺƾƣƿƺƺƣƾƿƶƣƽᅷƾᄬᇳᇻᇸᇶᄭ approach: Every Aha! moment, the moment of creation, is the bisociative act that is a spontaneous leap of insight that connects previously unconnected ƤƽƞƸƣƾ ƺƤ ƽƣƤƣƽƣƹơƣ ƿƩƽƺǀƨƩ ƿƩƣ ƢƫƾơƺǁƣƽDŽ ƺƤ ƞ ᅸƩƫƢƢƣƹ ƞƹƞƶƺƨDŽᅺ ᄬƻᄙ ᇵᇴᇲᄭ between them. Einstein illustrates the role of associative thinking, which brings the whole series of unconnected but associated images. Bisociative, or in this case multiassociative thinking, is precisely the moment of insight, the discovery and creation of a hidden analogy that connects unconnected components/images of the emerging concept.

1.5

Bisociative Thinking

 Ʃƞǁƣ ơƺƫƹƣƢ ƿƩƣ ƿƣƽƸ ᅸƟƫƾƺơƫƞƿƫƺƹᅺ ƫƹ ƺƽƢƣƽ ƿƺ Ƹƞƴƣ ƞ Ƣƫƾƿƫƹơƿƫƺƹ between the routine skills of thinking on a “single” plane, as it were, and ƿƩƣơƽƣƞƿƫǁƣƞơƿᄕǂƩƫơƩᄕƞƾ ƾƩƞƶƶƿƽDŽƿƺƾƩƺǂᄕƞƶǂƞDŽƾƺƻƣƽƞƿƣƾƺƹƸƺƽƣ ƿƩƞƹƺƹƣƻƶƞƹƣᄙᄬƺƣƾƿƶƣƽᄕᇳᇻᇸᇶᄕƻƻᄙᇵᇷᅬᇵᇸᄭ

The pattern underlying bisociation is the perceiving of a situation or idea, L, in two self-consisting but habitually incompatible frames of reference Mሲ and MሳᄙᄚƩƣƣǁƣƹƿᄕƫƹǂƩƫơƩ the two intersect, is made to vibrate simultaneously on two different wavelengths, as it were. While this unusual situation lasts, L is not merely linked to one associative context, but bisociatedǂƫƿƩƿǂƺᄙᄬƺƣƾƿƶƣƽᄕᇳᇻᇸᇶᄕ ƻᄙᇵᇷᄕƣƸƻƩƞƾƫƾƫƹƿƩƣƺƽƫƨƫƹƞƶᄭ

ረሤ

ѹ‫ؔ؛ؖآءإؔح‬

‫ؘإبؚ؜ؙ‬ᇳᄙᇴᏻ An event bisociated with two ƻƶƞƹƣƾƺƤƿƩƫƹƴƫƹƨᄬƤƽƺƸ ƺƣƾƿƶƣƽᄕᇳᇻᇸᇶᄕƻᄙᇵᇷᄭ

ƣƿᅷƾƶƺƺƴơƶƺƾƣƽƞƿƟƫƾƺơƫƞƿƫƺƹᄕƿƩƣƞơƿƺƤơƽƣƞƿƫƺƹᄙƫƾƺơƫƞƿƫƺƹƫƾƿƩƣƾƻƺƹtaneous leap of insight that connects previously unconnected frames of reference, matrices of knowledge, universes of discourses or associative contexts. The content of the insight is a hidden analogy. The leap of insight is the unearthing of the hidden analogy through which the new structure of understanding on a higher conceptual and hierarchical level is introduced. ƹƫƸƻƺƽƿƞƹƿƣǃƞƸƻƶƣƺƤơƺƹƿƣƸƻƺƽƞƽDŽƟƫƾƺơƫƞƿƫǁƣƿƩƫƹƴƫƹƨƫƾƿƩƣǂƞǁƣᅬ particle duality of quantum mechanics. The example is based on two matrices ƺƤƿƩƺǀƨƩƿᄘᄬᇳᄭƿƩƣƢDŽƹƞƸƫơƾƺƤƞƻƞƽƿƫơƶƣǂƫƿƩƫƿƾƺǂƹƾƣƿƺƤƽǀƶƣƾƺƽơƺƢƣƾ ƺƤƢDŽƹƞƸƫơƾƞƾƺƣƾƿƶƣƽǂƺǀƶƢƩƞǁƣƫƿƞƹƢᄬᇴᄭƿƩƣǂƞǁƣƸƺƿƫƺƹǂƫƿƩƫƿƾǂƞǁƣ equation employing concepts of wavelength and frequency, distinct from the dynamics of the particle, which employs concepts of position and momentum. The concepts of the particle and a wave are mutually exclusive from the point of their spatial content. On one hand, we have a point particle; on the other hand, the plane wave with an infinite extension in space. When both qualities are associated with one object, they cannot be measured or defined simultaneously. The clash between these two mutually exclusive frames of reverence, or associative contexts, gave us quantum mechanics with its Hilbert space: “the emergence of new patterns of relations.” At the very basis of quantum mechanics has been the coordination of incompatible codes of the wave and particle in terms of the De Broglie relations: l = hp . An equally profound example of bisociation can be found during early child development, that moment of discovery when everything has a nameᄬƺƽƫƸƣƽᄕ ᇳᇻᇴᇻᄭᄕ ǂƩƣƹ ƿƩƣ ƿǂƺ ǀƹƽƣƶƞƿƣƢ Ƹƞƿƽƫơƣƾ ƺƤ ᅸƾƺǀƹƢƾᅺ ƞƹƢ ᅸƿƩƫƹƨƾᅺ ƾDŽƹƿƩƣƾƫDžƣ ƫƹƿƺƿƩƣơƩƫƶƢᅷƾᅸƹƞƸƫƹƨƸƞƹƫƞᅺƞƹƢƿƩƣƹǀƸƟƣƽƺƤǂƺƽƢƾǀƾƣƢƟDŽƿƩƣơƩƫƶƢ dramatically increases.

ƹƺƣƾƿƶƣƽᅷƾƻƺƣƿƫơǂƺƽƢƾᄕ Here we have the undiluted bisociative act, the sudden synthesis of the ǀƹƫǁƣƽƾƣ ƺƤ ƾƫƨƹƾ ƞƹƢ ƿƩƣ ǀƹƫǁƣƽƾƣ ƺƤ ƿƩƫƹƨƾᄙ ƹ ƫƿƾ ƾƣƼǀƣƶ ƣƞơƩ Ƹƞƿƽƫǃ

ѷ‫إب؛اإ‬ҁ‫إؘ؟ائؘآ‬ᅷ‫ئ‬Ѹ‫ءآ؜اؔ؜ؖآئ؜‬Ҋ‫جإآؘ؛‬

ረሥ

imparts a new significance, a new dimension to the other: the words begins to “live,” to “give birth to new thoughts”; and the objects begin to ᅸƼǀƫǁƣƽᅺǀƹƢƣƽƿƩƣƿƺǀơƩƺƤƿƩƣƸƞƨƫơǂƞƹƢƺƤƶƞƹƨǀƞƨƣᄙᄬƻᄙᇴᇴᇴᄭ ƿƣƹƩƺǀƾƣᄬᇳᇻᇺᇷᄭơƺƹơƣƫǁƣƢƺƤƞƹƺƿƩƣƽƶƞƽƨƣᅟƾơƞƶƣᄕƻǀƽƣƟƫƾƺơƫƞƿƫǁƣƾDŽƹƿƩƣƾƫƾ through his classroom definition of the educational act, which is also a research ƞơƿᄙǀơƩᅸƿƣƹƩƺǀƾƣƞơƿƾᅺᄬDžƞƽƹƺơƩƞƣƿƞƶᄙᄕᇴᇲᇳᇸᄖƩƞƻƿƣƽᇳᄭƞƽƣƿƩƣƞơƿƾƺƤ ƾDŽƹƿƩƣƾƫƾǂƫƿƩƫƹƿƩƣƿƣƞơƩƫƹƨᅟƽƣƾƣƞƽơƩƸƣƿƩƺƢƺƶƺƨDŽᄬƩƞƻƿƣƽᇴᄭᄕǂƩƣƽƣᄕƿƺƞ large degree, separate matrices of teaching and educational research are rich sources of bisociative insights resulting in intense cross-fertilization between the components of education profession. Teaching and research are governed by two very different codes, or sets of rules. The Stenhouse acts are those unique acts where both codes are well coordinated and adjusted to each other. Creativity then consists in combining previously unrelated mental structures in such a way that you can get more out of the emergent whole than you put in. This apparent bit of magic derives from the fact that the whole is not merely the sum of its parts and that each new synthesis leads to the emergence of new patterns of relations at the higher level of hierarchy. Consequently, this cross-fertilization appears to be the essence of creativity and justifies the term ᅸƟƫƾƺơƫƞƿƫƺƹᅺᄬƺƣƾƿƶƣƽᄕᇳᇻᇹᇻᄭᄙ ƫƶƟƣƽƿơƺƹƤƫƽƸƾƿƩƣƩƫƨƩƻƺƾƾƫƟƫƶƫƿDŽƺƤơƽƣƞƿƫǁƣ insights on such borderlines of mathematics. He states the following: We also notice that, the farther a mathematical theory is developed, the more harmoniously and uniformly does its construction proceed, and unsuspected relations are disclosed between hitherto separated branches of the scienceᄙᄬƼǀƺƿƣƢƫƹƺƽǂƣƫƹƣƿƞƶᄙᄕᇴᇲᇳᇶᄕƻᄙᇸᇶᄕƣƸƻƩƞƾƫƾƞƢƢƣƢᄭ One of the central concepts used throughout this book is that of a bisociative frame, which denotes exactly those unconnected frames of reference ǂƫƿƩƫƹǂƩƫơƩƞƩƫƢƢƣƹƞƹƞƶƺƨDŽƫƾƤƺǀƹƢƞƹƢƫƹƾƫƨƩƿƣǃƻƽƣƾƾƣƢᄬDžƞƽƹƺơƩƞƣƿ ƞƶᄙᄕᇴᇲᇳᇸᄕƻᄙᇺᄭᄙƫƾƺơƫƞƿƫǁƣƤƽƞƸƣƾƩƞǁƣƢƫƤƤƣƽƣƹƿƶƣǁƣƶƾƺƤƨƣƹƣƽƞƶƫƿDŽᄙƣƸƫƨƩƿ be discussing bisociativity of teaching-research, which bisociates between two different domains of thinking, teaching, and research, which unfortunately are ƼǀƫƿƣƢƫƾơƺƹƹƣơƿƣƢᄙƣƿƿƩƣƻƺƾƾƫƟƫƶƫƿDŽƺƤǀƹƣƞƽƿƩƫƹƨƞƩƫƢƢƣƹƞƹƞƶƺƨDŽƫƾƼǀƫƿƣ substantial because of the common attention to learning. We might also be discussing the bisociations between two mathematical domains, geometry and algebra, which have revealed deep hidden analoƨƫƣƾ ƶƣƞƢƫƹƨ ƿƺ ƞƶƨƣƟƽƞƫơ ƨƣƺƸƣƿƽDŽ ƺƽ ƞƶƨƣƟƽƞƫơ ƿƺƻƺƶƺƨDŽᄙ ƹ ƿƩƫƾ ǁƺƶǀƸƣᄕ ǂƣ pay significant attention to bisociations that are formed during the moment of insight. We discover them by the analysis of the mathematical content of

ረሦ

ѹ‫ؔ؛ؖآءإؔح‬

ƫƹƾƫƨƩƿᄙ ƺƽƣǃƞƸƻƶƣᄕƫƹƿƩƣ ƶƣƻƩƞƹƿƩƞᄛƸƺƸƣƹƿᄬƫƹƩƞƻƿƣƽᇳᇹᄕƿƩƫƾǁƺƶǀƸƣᄭᄕƿƩƣƟƫƾƺơƫƞƿƫǁƣƤƽƞƸƣƫƾơƽƣƞƿƣƢƟƣƿǂƣƣƹƿƩƣƞƶƨƣƟƽƞƫơơƺƹơƣƻƿƺƤƿƩƣ unknown and the perception of the figure of the elephant.

ƹƩƞƻƿƣƽᇳᇵǂƣƫƹǁƣƾƿƫƨƞƿƣƿƩƣƟƫƾƺơƫƞƿƫǁƣƤƽƞƸƣƤƺƽƸƣƢǂƫƿƩƫƹƿƩƣƾƩƫƤƿƺƤ attention between two different codes to add fractions with the same denomƫƹƞƿƺƽᄙ ƹ Ʃƞƻƿƣƽƾ ᇶ ƞƹƢ ᇳᇳ ǂƣ Ƣƫƾơǀƾƾ ƿƩƣ Ɵƫƾƺơƫƞƿƫǁƣ ƤƽƞƸƣ Ɵƣƿǂƣƣƹ ƿƩƣ processes of interiorization and internalization. Chapter 4 also discusses the Ɵƫƾƺơƫƞƿƫǁƣ ƤƽƞƸƣ Ɵƣƿǂƣƣƹ ƿǂƺ ƾƿǀƢƣƹƿƾᅷ ƿƩƫƹƴƫƹƨ Ƣǀƽƫƹƨ ƿƩƣƫƽ ơƺƶƶƞƟƺƽƞƿƫǁƣ problem-solving work. 1.5.1 Bisociation, Awareness, Habit, and Association ƺƣƾƿƶƣƽ ᄬᇳᇻᇸᇶᄭ Ʃƞƾ ƻƺƾƫƿƫƺƹƣƢ Ɵƫƾƺơƫƞƿƫǁƣ ơƽƣƞƿƫǁƫƿDŽ ǂƫƿƩƫƹ ƫƿƾ Ƣƫƞƶƣơƿƫơƞƶ relationship with automatization of skills expressed as a habit. The greatest mastery and ease we gain in the exercise of a skill, the more automatized it will tend to become because the codes of rules which control it now operate ƟƣƶƺǂƿƩƣƿƩƽƣƾƩƺƶƢƺƤƞǂƞƽƣƹƣƾƾᄬƻᄙᇳᇷᇶᄭᄙƺƹƾƣƼǀƣƹƿƶDŽᄕƩƞƟƫƿƤƺƽƸƞƿƫƺƹᄴƫƹ ƿƩƣƣǃƣƽơƫƾƣƺƤƿƩƣƾƴƫƶƶᄵƫƾƞơơƺƸƻƞƹƫƣƢƟDŽƨƽƞƢǀƞƶƢƫƸƸƫƹƨƞƹƢƢƞƽƴƣƹƫƹƨ ƺƤƿƩƣƶƫƨƩƿƺƤƞǂƞƽƣƹƣƾƾᄬƻᄙᇳᇷᇷᄭᄙƹƿƩƣƺƿƩƣƽƩƞƹƢᄕƢǀƽƫƹƨƿƩƣơƽƣƞƿƫǁƣƞơƿƺƤ ƟƫƾƺơƫƞƿƫƺƹǂƣƩƞǁƣƿƩƣǀƻƾǀƽƨƣƺƤƞǂƞƽƣƹƣƾƾᄙ ƹƾƿƣƞƢƺƤƢƫƸƸƫƹƨᄕƫƿƫƶƶǀƸƫnates the field of experience. Thus, mechanization of habits and creativity of an Aha! moment are processes moving in the opposite direction in terms of awareness, confirming that conscious and unconscious experiences do not belong to different compartƸƣƹƿƾƺƤƿƩƣƸƫƹƢᄙƞƿƩƣƽᄕƿƩƣDŽƤƺƽƸƞơƺƹƿƫƹǀƺǀƾƾơƞƶƣᄬƻᄙᇳᇷᇶᄭᄙƺƽƣƺǁƣƽᄕ whereas the creativity of the Aha! moment is bisociative, that is, springing from two sources, the mechanization of habits is one dimensional, or associative. The division between bisociative thinking as progress of understanding, and associative thinking as exercise of understanding is the central division ƻƽƺƻƺƾƣƢƟDŽƺƣƾƿƶƣƽᄕǂƩƫơƩƻƣƽƸƣƞƿƣƾƩƫƾǂƺƽƴᄙ Bisociation is the perception of the situation in relation to two incompatible matrices with their codes. Such a perception may generate an Aha! moment, induce laughter in the case of humor, or catharsis in the case of the artist. On ƿƩƣ ƺƿƩƣƽ ƩƞƹƢᄕ ƞƾƾƺơƫƞƿƫǁƣ ƿƩƫƹƴƫƹƨ ƫƾ ơƺƹƤƫƹƣƢ ƿƺ ƞ ƾƫƹƨƶƣ Ƹƞƿƽƫǃᄙ ƿ ƫƾ ƿƩƣ exercise of a more or less flexible skill, which can do only one type of task ƞƶƽƣƞƢDŽƣƹơƺǀƹƿƣƽƣƢƫƹƻƞƾƿƣǃƻƣƽƫƣƹơƣᄬƻᄙᇶᇶᄭᄙ ƽƺƸƿƩƣƻƺƫƹƿƺƤǁƫƣǂƺƤƫƞƨƣƿᅷƾƶƣƞƽƹƫƹƨƿƩƣƺƽDŽᄕƞƾƾƺơƫƞƿƫǁƣƿƩƫƹƴƫƹƨƫƾƽƣƶƞƿƣƢƿƺƻƽƺơƣƾƾƣƾƺƤƞƾƾƫƸƫƶƞƿƫƺƹ or habitual responses, whereas bisociative thinking results in a new structure. Such matrices pattern our perceptions, thoughts, or activities, which, through a condensation of learning, lead to a habit. These habits are essential for driving a car or logical thinking, for example. Without such unconscious

ѷ‫إب؛اإ‬ҁ‫إؘ؟ائؘآ‬ᅷ‫ئ‬Ѹ‫ءآ؜اؔ؜ؖآئ؜‬Ҋ‫جإآؘ؛‬

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ƩƞƟƫƿƾᄕǂƣơƺǀƶƢƣƞƾƫƶDŽƩƞǁƣƞƹƞơơƫƢƣƹƿᄬƾƺƸƣƿƩƫƹƨƿƩƞƿƺƤƿƣƹƩƞƻƻƣƹƾǂƩƫƶƣ ƶƣƞƽƹƫƹƨƿƩƞƿƾƴƫƶƶƺƽƞơƼǀƫƽƫƹƨƿƩƣƩƞƟƫƿᄭᄕƺƽƞƿƩƺǀƨƩƿơƺǀƶƢƶƺƾƣƫƿƾơƺƩƣƽence. The logic gets suspended and we get into the dreamy state governed by associative thinking, often associating along the criteria that only partially represents objects as we know them in the rational state. Associative thinking is based upon habitual responses to routine problems.

ƿơƺƸƟƫƹƣƾƣƶƣƸƣƹƿƾƤƽƺƸƿƩƣƾƞƸƣƸƞƿƽƫǃƺƤƿƩƺǀƨƩƿᄙƩƫƾƫƾƿƩƣơƺƹƿƣƹƿƺƤ associative thinking. The presence and the strength of a habit is, again, inversely related to the degree of awareness. “Awareness is the experience which diminishes and fades away with the increase of mastery of a skill exercised in monotonous condiƿƫƺƹƾᅺᄬƻᄙᇳᇷᇷᄭᄕƿƩƞƿƫƾᄕǂƩƫƶƣƿǀƽƹƫƹƨƶƣƞƽƹƫƹƨƫƹƿƺƞƩƞƟƫƿᄙƹƿƩƣƺƿƩƣƽƩƞƹƢᄕ an increase in awareness might eliminate the mastery of a skill, diminishing the habit. Students who struggle with mathematics often see it as a collection or rules to memorize and view mathematics and science instructors as authority ƣǃƻƣƽƿƾǂƩƺƢƫơƿƞƿƣƿƩƣƿƽǀƿƩƺƤƿƩƣƤƫƣƶƢᅭƿƩƞƿᅷƾƩƺǂƿƩƣƩƞƟƫƿƺƤƻƞƾƾƫǁƫƿDŽ develops. Consequently, this view of mathematics makes it difficult for learnƣƽƾƿƺƣƹƨƞƨƣƫƹƞơƿƫǁƣƻƽƺƟƶƣƸᅟƾƺƶǁƫƹƨᄬƞƤƤƣơƿƫǁƣƻƞƽƞƶDŽƾƫƾᄭᄙƺƣƾƿƶƣƽƢƣơƽƫƣƢ what he saw as an overreliance upon rules and procedures in mathematics and science lectures as well as textbooks: “the overloading with jargon, the tortuous and cramped style.” He laments this formalistic style permeates education in these fields. “The same inhuman, anti-humanistic trend pervades the climate in which science is taught as well as the classrooms and the textbooks.” His solution is guided-discovery pedagogy that exposes the learner to the joys of intuition and illumination. “To derive pleasure from the art of discovery, as from the other arts, the consumer—in this case the student—must be made to ƽƣƶƫǁƣƿƺƾƺƸƣƣǃƿƣƹƿƿƩƣơƽƣƞƿƫǁƣƻƽƺơƣƾƾᅺᄬƺƣƾƿƶƣƽᄕᇳᇻᇸᇶᄕƻᄙᇴᇸᇷᄭᄙ Clearly, habits are the essential basis of stability and ordered behavior. They also tend to become mechanized and reduce the individual to the status of a conditioned automaton. The creative act, by connecting previously unrelated dimensions of experience, enables the individual to attain to a higher ƶƣǁƣƶ ƺƤ Ƹƣƹƿƞƶ ƣǁƺƶǀƿƫƺƹᄙ ᅸ ƿ ƫƾ ƞƹ ƞơƿ ƺƤ ƶƫƟƣƽƞƿƫƺƹᅭƿƩƣ ƢƣƤƣƞƿ ƺƤ ƩƞƟƫƿ ƟDŽ ƺƽƫƨƫƹƞƶƫƿDŽᅺᄬƻᄙᇻᇸᄭᄙƹƣǃơƣƶƶƣƹƿƣǃƞƸƻƶƣƺƤƿƩƣƻƽƺơƣƾƾơƞƹƟƣƤƺǀƹƢƫƹƿƩƣ Domain Aha! moment. The imprinted, or conditioned habit of seeing the condition for the existence of the function X + ᇵ  ƿƺ Ɵƣ ǃ ኟ ᇲ ᄬƞ ƾƿƞƹƢƞƽƢ Ƹƫƾơƺƹơƣƻƿƫƺƹᄭ Ʃƞƾ Ɵƣƣƹ ƿƽƞƹƾƤƺƽƸƣƢ Ƣǀƽƫƹƨ ƿƩƣ ƸƺƸƣƹƿ ƺƤ ƫƹƾƫƨƩƿ ƫƹƿƺ ǃኟኔᇵᄙƩƞƿƫƹƾƫƨƩƿƶƫƟƣƽƞƿƣƢƾƿǀƢƣƹƿƿƩƫƹƴƫƹƨƾƺƿƩƞƿƾƩƣǂƞƾƞƟƶƣƣƞƾƫƶDŽƿƺ reach a higher level of abstraction. She could generalize the condition to the parametric function X − a .

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ѹ‫ؔ؛ؖآءإؔح‬

Ʃƣ ƽƣƶƞƿƫƺƹƾƩƫƻ Ɵƣƿǂƣƣƹ ơƽƣƞƿƫǁƫƿDŽ ᄬơƽƣƞƿƫǁƣ ƺƽƫƨƫƹƞƶƫƿDŽᄭ ƞƹƢ ƩƞƟƫƿ ƞƹƢ between bisociative and associative thinking is essential in our discussions and is addressed in several chapters of this book. The freedom from constraints and the necessity of constraints on our thinking require careful reflection from innovators and mathematicians themselves and from mathematics teachers to find the “golden mean” between the free expression of creative intuition and the constraints of the mathematical language necessary to express it without impacting the idea itself. Chapter 4 discusses this relationship in the context of old and revised Bloom taxonomies. The discussion leads to the difference “between progress in understanding—the acquisition of new insights, and the exercise of understanding at any given stage of development. Progress in understanding is achieved by the forƸǀƶƞƿƫƺƹƺƤƹƣǂơƺƢƣƾƿƩƽƺǀƨƩƿƩƣᄚƸƣƿƩƺƢƾ ᄴƺƤᄵ ᄚ ƣƸƻƫƽƫơƞƶ ƫƹƢǀơƿƫƺƹᄕ ƞƟƾƿƽƞơƿƫƺƹƞƹƢƢƫƾơƽƫƸƫƹƞƿƫƺƹᄕᄴƞƹƢᄵƟƫƾƺơƫƞƿƫƺƹᄬƻᄙᇸᇳᇻᄕƣƸƻƩƞƾƫƾƫƹƿƩƣƺƽƫƨƫƹƞƶᄭᄙƣƞƽƨǀƣƫƹƩƞƻƿƣƽᇳᇳƿƩƞƿƢƣǁƣƶƺƻƸƣƹƿƺƤƞƟƾƿƽƞơƿƫƺƹƫƹƾƺƸƣơƞƾƣƾƫƾ grounded in bisociation. The exercise of understanding, on the other hand, is the explanation of events or the use of the skill known from the past. The Fir ƽƣƣƩƞᄛƸƺƸƣƹƿᄬƩƞƻƿƣƽᇳᇹᄭƾƩƺǂƾƟƺƿƩƻƽƺơƣƾƾƣƾƫƹƢƣƿƞƫƶᄙቀ The student proceeded to solve the problem without reporting any difficulties, suggesting that she has developed an understanding of the generalization procedure that she exercised in the solution of the problem. However, after she solved the problem, she grasped the new connection between the solution as a factorized expression of the quadratic equation that completed the expression. That was her progress in understanding of the mathematical situation at hand. Similarly, the distinction between progress in understanding and the exercise of understanding can be used to distinguish between the insight through ǂƩƫơƩƿƩƣƨƣƹƣƽƞƶƫDžƞƿƫƺƹƻƽƺƟƶƣƸǂƞƾƾƺƶǁƣƢƞƾƽƣƻƺƽƿƣƢƟDŽƞƶƞƿƹƫƴƞƹƢƺƫơƩǀᄬᇴᇲᇳᇶᄭƞƹƢƿƩƣƣǃƣƽơƫƾƣƺƤƿƩƞƿǀƹƢƣƽƾƿƞƹƢƫƹƨƽƣƞơƩƣƢƞƿƿƩƣƸƺƸƣƹƿƺƤ insight to confirm the correctness of other related arithmetical relationships ƺƤƿƩƣƾƞƸƣƿDŽƻƣᄬƩƞƻƿƣƽᇳᇵᄭᄙ Such an order of events from insight to the application of that insight as an exercise of understanding reached during its course is common. All moments of insight discussed by Poincaré have that quality, when the idea appears in its full clarity and is followed by several hours of work during which the insight condenses into a habit, which at the same time exercises understanding while completing the solution. Note the feeling of a “perfect certainty.” Poincaré says: ᅸ ƢƫƢƹƺƿǁƣƽƫƤDŽƿƩƣƫƢƣƞᄖ ƾƩƺǀƶƢƹƺƿƩƞǁƣƩƞƢƿƫƸƣᄕƞƾᄕǀƻƺƹƿƞƴƫƹƨƸDŽƾƣƞƿ ƫƹƿƩƣƺƸƹƫƟǀƾᄕ ǂƣƹƿƺƹǂƫƿƩƞơƺƹǁƣƽƾƞƿƫƺƹƞƶƽƣƞƢDŽơƺƸƸƣƹơƣƢᄕƟǀƿ Ƥƣƶƿ ƞƻƣƽƤƣơƿơƣƽƿƞƫƹƿDŽᄙƹƸDŽƽƣƿǀƽƹƿƺƞƣƹᄕƤƺƽơƺƹƾơƫƣƹơƣᅷƾƞƴƣ ǁƣƽƫƤƫƣƢƿƩƣ ƽƣƾǀƶƿƞƿƸDŽƶƣƫƾǀƽƣᅺᄬƼǀƺƿƣƢƫƹƺƣƾƿƶƣƽᄕᇳᇻᇸᇶᄕƻᄙᇳᇳᇷᄭᄙƩƞƿƤƣƣƶƫƹƨƺƤơƣƽƿƞƫƹƿDŽ

ѷ‫إب؛اإ‬ҁ‫إؘ؟ائؘآ‬ᅷ‫ئ‬Ѹ‫ءآ؜اؔ؜ؖآئ؜‬Ҋ‫جإآؘ؛‬

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accompanies insight in general, while of course for some the certainty is erroneous. Returning to the discussion above, we see that the gap that separated the ƩƞƟƫƿƤƽƺƸơƽƣƞƿƫǁƫƿDŽᄬƿƩƞƿƫƾᄕƤƽƺƸƣǃƣƽơƫƾƣƫƹǀƹƢƣƽƾƿƞƹƢƫƹƨƿƺƿƩƣƻƽƺƨƽƣƾƾƺƤ ǀƹƢƣƽƾƿƞƹƢƫƹƨᄭǂƞƾƟƽƫƢƨƣƢƢǀƽƫƹƨƿƩƣƩƞᄛƸƺƸƣƹƿᄙƺƣƾƿƶƣƽᄬᇳᇻᇸᇶᄭƻƺƫƹƿƾ ƺǀƿ ƿƩƞƿ ᅸƺƽƫƨƫƹƞƶƫƿDŽ ƺƽ ᅵƫƹǁƣƹƿƫǁƣƹƣƾƾᅷ Ƹƞƴƣ ƿƩƣƫƽ ƞƻƻƣƞƽƞƹơƣ ƞƿ ƺƹƶDŽ Ƥƣǂ ƽƣƸƺǁƣƾƤƽƺƸƫƹƹƞƿƣƞƹƢƫƸƻƽƫƹƿƣƢƟƣƩƞǁƫƺǀƽᅺᄬƻᄙᇶᇻᇴᄭᄙƣƾƣƣƩƣƽƣƞơƣƽƿƞƫƹ ƞƹƞƶƺƨDŽƿƺDŽƨƺƿƾƴDŽᅷƾᄬᇳᇻᇻᇹᄭǁƫƣǂƿƩƞƿƿƩƣƤƞƾƿƣƾƿǀƹƢƣƽƾƿƞƹƢƫƹƨƺƤƞơƺƹơƣƻƿ takes place when the level of spontaneous concepts is just beneath scientific ones. 1.5.2 Bisociation and the Janusian Thinking of Rothenberg ƶƿƩƺǀƨƩƺƣƾƿƶƣƽᅷƾǂƺƽƴƺƹơƽƣƞƿƫǁƫƿDŽƞƹƢƿƩƣƞơƿƺƤơƽƣƞƿƫƺƹƞƹƢƿƩƣƩƞᄛ moment has not been well known until now, it has motivated and inspired two significant research inquiries into the nature of the creative experience. One ƺƤƿƩƣƾƣƫƾƟDŽƶƟƣƽƿƺƿƩƣƹƟƣƽƨᄕǂƩƺƫƹᇳᇻᇹᇻƻǀƟƶƫƾƩƣƢThe Emerging Goddess: The Creative Process in Art, Science, and Other Fields. The work proposes Janusian thinking consisting of “actively conceiving two or more opposite or ƞƹƿƫƿƩƣƿƫơƞƶƫƢƣƞƾᄕƫƸƞƨƣƾᄕƺƽơƺƹơƣƻƿƾƾƫƸǀƶƿƞƹƣƺǀƾƶDŽᅺᄬƺƿƩƣƹƟƣƽƨᄕᇳᇻᇹᇻƟᄕƻᄙ ᇷᇷᄭƞƾƿƩƣơƺƹƿƣƹƿƺƤơƽƣƞƿƫǁƫƿDŽᄙƩƣƺƿƩƣƽƫƾƟDŽƞƽƨƞƽƣƿᄙƺƢƣƹᄕǂƩƺƫƹᇳᇻᇻᇲ published her first edition of The Creative Mind: Myths and Mechanisms, where ƾƩƣƢƣǁƣƶƺƻƾƺƣƾƿƶƣƽᅷƾƫƢƣƞƺƤᅸơƺƹƹƣơƿƫƹƨǀƹơƺƹƹƣơƿƣƢƸƞƿƽƫơƣƾᅺƫƹƿƩƣơƺƹtext of ѷѿ, particularly in the context of neural networks. Rothenberg counterposes Janusian thinking creativity as consisting of “sevƣƽƞƶƺƻƻƺƾƫƿƣƺƽƞƹƿƫƿƩƣƿƫơƞƶƤƞơƿƺƽƾƫƹǁƺƶǁƣƢƫƹƿƩƣᄴơƽƣƞƿƫǁƣᄵơƺƨƹƫƿƫǁƣƻƽƺcess” to bisociation based on the “hidden analogy,” hence involving analogical thinking based on similarity. Moreover, he sees “the creative thinking to be priƸƞƽƫƶDŽƞơƺƹƾơƫƺǀƾƻƽƺơƣƾƾᅺᄬƻᄙᇶᇳᄭƽƞƿƩƣƽƿƩƞƹƾƿƣƣƻƣƢƫƹǀƹơƺƹƾơƫƺǀƾƸƣơƩƞƹƫƾƸƾᄕƞƾƺƫƹơƞƽȅƞƹƢƺƣƾƿƶƣƽƾǀƨƨƣƾƿƣƢᄙƺƿƩƣƹƟƣƽƨᄬᇳᇻᇹᇻƟᄭƻƽƺǁƫƢƣƾƸƞƹDŽ examples of Janusian thinking in different religions, emphasizing Buddhism and Taoism, whose symbol of yin and yang provide food for thought in their ƽƺƶƣƞƾᅸƿǂƺƾƫƸǀƶƿƞƹƣƺǀƾƞƹƢƺƻƻƺƾƣƢƤƺƽơƣƾƺƤDŽƫƹᄬƢƞƽƴƾƫƢƣᄭƞƹƢDŽƞƹƨᄬƶƫƨƩƿ ƾƫƢƣᄭᅺᄬƻᄙᇳᇶᇳᄭᄙ

ƫƨǀƽƣᇳᄙᇵƺƤƿƩƣDŽƫƹᄧDŽƞƹƨƾDŽƸƟƺƶƢƣƻƫơƿƾƿƩƣƺƻƻƺƾƫƿƫƺƹƺƤƟƶƞơƴƞƹƢǂƩƫƿƣᄕ but note how similar if not analogical are their shapes, which are responsiƟƶƣ Ƥƺƽ ƿƩƣ ơƺƸƻƶƣƿƫƺƹ ƞƹƢ ƣƹơƶƺƾǀƽƣ ƺƤ ƿƩƣ Ƥǀƶƶ ƾDŽƸƟƺƶᅷƾ ơƫƽơƶƣᄙ ƺƽƣƺǁƣƽᄕ since the opposition is along the dimension of color while the analogy is along the dimension of shape, the two criteria are neutral relative to their respective dimensions. Both opposition and analogy can “live” together within the symbol.

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ѹ‫ؔ؛ؖآءإؔح‬

‫ؘإبؚ؜ؙ‬ᇳᄙᇵᏻ ƫƹƨƞƹƢDŽƞƹƨƾDŽƸƟƺƶ

ƹƿƩƫƾƽƣƶƞƿƫƺƹƾƩƫƻƺƤƺƻƻƺƾƫƿƫƺƹƞƹƢƞƹƞƶƺƨDŽᅸƶƫǁƫƹƨᅺƾƫƢƣƟDŽƾƫƢƣǂƫƿƩƫƹ ƿƩƣơƽƣƞƿƫǁƣƫƹƾƫƨƩƿᄕǂƣƤƫƹƢƿƩƣƿǂƺƾơƫƣƹƿƫƤƫơƣǃƞƸƻƶƣƾƺƤƺƿƩƣƹƟƣƽƨᄬᇳᇻᇹᇻƞᄕ ᇳᇻᇹᇻƟᄭᄙƹƣƫƾƿƩƣᅸƩƞƻƻƫƣƾƿƿƩƺǀƨƩƿƺƤƸDŽƶƫƤƣᅺƺƤ ƫƹƾƿƣƫƹᄕƫƹơƶǀƢƣƢƫƹƩƞƻƿƣƽ ᇳᇹᄙ ƺƿƩƣƹƟƣƽƨ ƾƣƣƾ ƺƻƻƺƾƫƿƫƺƹƞƶ ƞƹǀƾƫƞƹ ƿƩƫƹƴƫƹƨ ƫƹ ƫƹƾƿƣƫƹᅷƾ Ƣƣơƫƾƫǁƣ sentence: “Thus, for an observer in free fall from the roof of a house there exists, during his fall, no gravitational field.” He sees Janusian thinking in the implicit conclusion that the observer in free fall is both in motion and at rest at the same time. Rothenberg provides the following account by Einstein as an example of Janusian thinking: ƩƣƹᄕƫƹƿƩƣDŽƣƞƽᇳᇻᇲᇹᄕ ǂƞƾǂƺƽƴƫƹƨƺƹƞƾǀƸƸƞƽDŽƣƾƾƞDŽơƺƹơƣƽƹƫƹƨ the special theory of relativity for the Yearbook for Radioactivity and Electronics ƿƽƫƣƢƿƺƸƺƢƫƤDŽƣǂƿƺƹᅷƾƿƩƣƺƽDŽƺƤƨƽƞǁƫƿƞƿƫƺƹƫƹƾǀơƩƞǂƞDŽƿƩƞƿ it would fit into the theory. Attempts in this direction showed the possibility of carrying out this enterprise, but they did not satisfy me because they had to be supported by hypotheses without physical basis. At that point there came to me the happiest thought of my life, in the following form: Just as in the case where an electric field is produced by electromagnetic induction, the gravitational field similarly has only a relative existence. Thus, for an observer in free fall from the roof of a house there exists, during his fall, no gravitational field ᄴ ƫƹƾƿƣƫƹᅷƾƫƿƞƶƫơƾᄵ—at least not in his ƫƸƸƣƢƫƞƿƣǁƫơƫƹƫƿDŽᄙ ƤƿƩƣƺƟƾƣƽǁƣƽƽƣƶƣƞƾƣƾƞƹDŽƺƟưƣơƿƾᄕƿƩƣDŽǂƫƶƶƽƣƸƞƫƹᄕ relative to him, in a state of rest, or in a state of uniform motion, indepenƢƣƹƿƺƤƿƩƣƫƽƻƞƽƿƫơǀƶƞƽơƩƣƸƫơƞƶƞƹƢƻƩDŽƾƫơƞƶƹƞƿǀƽƣᄙᄬ ƹƿƩƫƾơƺƹƾƫƢƣƽƞƿƫƺƹƺƹƣƸǀƾƿƹƞƿǀƽƞƶƶDŽƹƣƨƶƣơƿƞƫƽƽƣƾƫƾƿƞƹơƣᄙᄭƩƣƺƟƾƣƽǁƣƽƫƾ therefore ưǀƾƿƫƤƫƣƢƫƹơƞƶƶƫƹƨƫƿƾƾƿƞƿƣᅸƽƣƾƿᄙᅺᄬƺƿƩƣƹƟƣƽƨᄕᇳᇻᇹᇻƞᄕƻᄙᇵᇻᄭ

ѷ‫إب؛اإ‬ҁ‫إؘ؟ائؘآ‬ᅷ‫ئ‬Ѹ‫ءآ؜اؔ؜ؖآئ؜‬Ҋ‫جإآؘ؛‬

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ơƶƺƾƣƞƹƞƶDŽƾƫƾƺƤƿƩƣƿƣǃƿƫƹƩƞƻƿƣƽƾᇶƞƹƢᇳᇷƽƣǁƣƞƶƾƿƩƞƿƿƺƨƣƿƩƣƽǂƫƿƩƿƩƣ antithetical idea emphasized by Einstein and Janusian thinking in the context of the gravitational field, there is also the analogy between the electromagƹƣƿƫơᄬѻ҃ᄭƤƫƣƶƢƞƹƢƿƩƣƨƽƞǁƫƿƞƿƫƺƹƞƶƺƹƣǂƫƿƩƫƹǂƩƫơƩ ƫƹƾƿƣƫƹƤƺƽƸǀƶƞƿƣƾ his Janusian thinking. The analogy is bisociative; it is bisociative thinking that joined the electromagnetic field effect with gravitational one and produced the realization of motion and rest—two opposite ideas being joined. Therefore, the opposing Janusian idea of motion and rest joined within the imaginary experimental setup of Einstein has been present already in the conseƼǀƣƹơƣƾƺƤ ƞƽƞƢƞDŽᅷƾƢƫƾơƺǁƣƽDŽᄘ ƤƞƸƞƨƹƣƿƸƺǁƣƾƞƶƺƹƨƿƩƣơƺƹƢǀơƿƺƽᄕƿƩƣơǀƽrent exists, created by the electric field induced by the changing magnetic field ƺƤƿƩƣƸƞƨƹƣƿᄙ ƤƿƩƣƸƞƨƹƣƿƫƾƞƿƽƣƾƿᄬƟǀƿƿƩƣơƺƹƢǀơƿƺƽƸƺǁƣƾᄭᄕƹƺƣƶƣơƿƽƫơ field is generated. Consequently, the existence of the electric field was relative and depended on the position of the coordinate system, just as the presence of a gravitational field depended on whether the coordinate system was attached to the observer at rest or to the observer falling in the gravitational field. We see that bisociation occurs here together with Janusian thinking. From the bisociation point of view, Janusian thinking is a class of hidden analogies that are grasped at the Aha! moments of insight. Similar observations can be ƸƞƢƣơƺƹơƣƽƹƫƹƨƺƿƩƣƹƟƣƽƨᅷƾơƺƸƸƣƹƿƞƟƺǀƿƿƩƣƢƫƾơƺǁƣƽDŽƺƤƿƩƣƢƺǀƟƶƣ Ʃƣƶƫǃƾƿƽǀơƿǀƽƣᄙ ƣƾƣƣƾƞƹǀƾƫƞƹƿƩƫƹƴƫƹƨƸƞƹƫƤƣƾƿƫƹƞƿƾƺƹᅷƾƢƫƾơƺǁƣƽDŽƿƩƞƿ the base sequences of the two intertwined Ѻ҄ѷ chains were identical and complementary ᄬoppositeᄭƿƺƣƞơƩƺƿƩƣƽᄬƺƿƩƣƹƟƣƽƨᄕᇳᇻᇹᇻƟᄕƻᄙᇳᇳᇲᄭᄙ Given the base of one chain its partner was automatically determined. But ƿƩƣ ƞơƿǀƞƶ ƸƺƸƣƹƿ ƺƤ ƢƫƾơƺǁƣƽDŽᅭᅸǀƢƢƣƹƶDŽ  ƟƣơƞƸƣ ƞǂƞƽƣ ƿƩƞƿ ƞƹ ƞƢƣnine-thymine pair held together was identical in shape to a guanine-cytosine pair held together by at least two hydrogen bonds”—suggests it was a bisociaƿƫǁƣ ƫƹƾƫƨƩƿ Ƣǀƽƫƹƨ ǂƩƫơƩ ƿƩƣ ƩƫƢƢƣƹ ƞƹƞƶƺƨDŽ ᄬƾƞƸƣƹƣƾƾᄭ Ɵƣƿǂƣƣƹ ƿƩƣ Ɵƞƾƣ sequences of Ѻ҄ѷ chains was revealed. We see again the collaboration of Janusian thinking with bisociation.

ƹƾƿƣƞƢƺƤơƺǀƹƿƣƽƻƺƾƫƹƨƿƩƣƿǂƺƿƩƣƺƽƫƣƾᄕƿƩƣƿƩƣƺƽDŽƺƤƟƫƾƺơƫƞƿƫƺƹƞƹƢƿƩƣ theory of Janusian thinking, we can see them as one phenomenon in which bisociation plays the role of the process while Janusian thinking is in many but not all cases its product. One significant cause of confusion about what exactly ƫƾƺƣƾƿƶƣƽᅷƾơƺƹơƣƻƿƺƤơƽƣƞƿƫǁƫƿDŽƫƾƿƩƞƿƺƣƾƿƶƣƽᅷƾᄬᇳᇻᇸᇶᄭƢƣƤƫƹƫƿƫƺƹƺƤƟƫƾƺơƫƞƿƫƺƹƞƾᅸƾƻƺƹƿƞƹƣƺǀƾƤƶƞƾƩᄴƶƣƞƻᄵƺƤƫƹƾƫƨƩƿǂƩƫơƩᄚơƺƹƹƣơƿƾƻƽƣǁƫƺǀƾƶDŽ ǀƹơƺƹƹƣơƿƣƢƸƞƿƽƫơƣƾƺƤƣǃƻƣƽƫƣƹơƣᅺᄬƻᄙᇶᇷᄭƾƸƺƺƿƩƶDŽơƺƹƤƶƞƿƣƾƿƩƣơƽƣƞƿƫǁƣ process with the creative product, as Boden pointed out. A similar misunderƾƿƞƹƢƫƹƨ ƫƾ ƽƣƾƻƺƹƾƫƟƶƣ Ƥƺƽ ƺƿƩƣƹƟƣƽƨᅷƾ ᄬᇳᇻᇹᇻƟᄭ ƨƽƞƾƻ ƺƤ Ɵƫƾƺơƫƞƿƫƺƹ ƫƹ ƿƩƣ

ረሬ

ѹ‫ؔ؛ؖآءإؔح‬

ƫƹƿƽƺƢǀơƿƫƺƹ ƿƺ Ʃƫƾ ǂƺƽƴ ᄬƻᄙ ᇳᇴᄭ ƞƹƢ Ƥƺƽ ƫƿƾ Ƹƞƫƹ ơƽƫƿƫơƫƾƸᄘ ᅸƾ ƿƩƣ ơƺƸƻƶƣƿƣ theory of creativity, it does not explain how new aspects of creation can result from merely from the combination of previously existing and fully developed frames of reference.”

ƹƢƣƣƢᄕƾƺƸƣƩƞᄛƸƺƸƣƹƿƾƽƣƾǀƶƿƤƽƺƸƿƩƣơƺƸƟƫƹƞƿƫƺƹƺƽƞƹƣǂơƺƹƾƿƽǀơtion of existing elements, such as the Fir Tree and the Domain Aha! moments. But many are not. Connections made during Aha! moments are mathematical constructions of different levels of complexity, not merely combinations. See ƿƩƣƩƞƿ ƾƞƣơƿƺƽᄞƩƞᄛƸƺƸƣƹƿᄕǂƩƫơƩƫƹǁƺƶǁƣƾƞƟƾƿƽƞơƿƫƺƹᄕƹƺƿơƺƸƟƫƹƞƿƫƺƹᄙƫƴƣǂƫƾƣᄕƫƸᅷƾƩƞᄛƸƺƸƣƹƿƫƹǁƺƶǁƣƢƨƽƞƾƻƫƹƨƞƽƣƶƞƿƫƺƹƾƩƫƻƟƣƿǂƣƣƹ the separate patterns. We want to indicate that processes entering into connecting unconnected matrices are quite deep and point to general connecƿƫƺƹƫƾƿƾDŽƾƿƣƸƾƞƾƞƹƣǃƻƶƞƹƞƿƺƽDŽƿƺƺƶᄙƞƿǀƽƞƶƶDŽᄕƺƣƾƿƶƣƽƢƫƢƹƺƿƣǃƻƶƞƫƹƩƺǂ matrices are bisociated; however, the developments in ѷѿ suggest that his direction of thinking was correct. 1.5.3 Bisociativity, Artificial Intelligence, and Computer Creativity ǀƽƻƽƫƾƫƹƨƶDŽᄕƺƣƾƿƶƣƽᅷƾƟƫƾƺơƫƞƿƫƺƹƟƣơƞƸƣƿƩƣƫƹƾƻƫƽƞƿƫƺƹƤƺƽƿƩƣƹƣǂѷѿ domain of computational creativity. There is an irony in this unexpected computer-oriented inspiration by the theory of creativity, which expresses itself through the freedom of spontaneity in the defiance of automation or rigid following of the rules. But actions of the computer are nothing if not the expression of full automation and exactly following the rules of the computer program! Margaret Boden, one of the proponents of the new domain of computational creativity, tackles this question head on. She reminds us that the first person who came out with that objection to computer creativity was Lady Ada Lovelace, a friend of Charles Babbage, the designer of the mid-nineteenth century digital computer, the Analytical Engine. Countess Lovelace asserted: “The Analytical Engine has no pretensions to originateƞƹDŽƿƩƫƹƨᄙ ƿơƞƹƢƺᄴƺƹƶDŽᄵwhatever we know how to orderƫƿƿƺƻƣƽƤƺƽƸᅺᄬƺƢƣƹᄕᇴᇲᇲᇶᄕƻᄙᇳᇸᄕƣƸƻƩƞƾƫƾƫƹƿƩƣƺƽƫƨƫƹƞƶᄭᄙ ƺƢƣƹƽƣƾƿƞƿƣƾƺǁƣƶƞơƣᅷƾƾƿƞƿƣƸƣƹƿƞƾƤƺǀƽƢƫƤƤƣƽƣƹƿƼǀƣƾƿƫƺƹƾᄘ ᅬ Can computational ideas help us to understand how human creativity is possible? ᅬ Could computers ever do things that appear to be creative? ᅬ Could computers ever appear to recognize creativity? ᅬ Could a computer ever really be creative? ƺƢƣƹᅷƾƻƽƫƸƞƽDŽƫƹƿƣƽƣƾƿƫƾƺǁƣƶƞơƣᅷƾƤƫƽƾƿƼǀƣƾƿƫƺƹᄘƩDŽƫƾƩǀƸƞƹơƽƣƞƿƫǁƫƿDŽ possible? To structure her response, Boden introduces three types of creativƫƿDŽᄘ ơƺƸƟƫƹƞƿƺƽƫƞƶᄕ ƣǃƻƶƺƽƞƿƺƽDŽᄕ ƞƹƢ ƿƽƞƹƾƤƺƽƸƞƿƫǁƣᄙ Ʃƞƻƿƣƽ ᇳᇷ ƣǃƻƶƺƽƣƾ ƿƩƣƫƽ

ѷ‫إب؛اإ‬ҁ‫إؘ؟ائؘآ‬ᅷ‫ئ‬Ѹ‫ءآ؜اؔ؜ؖآئ؜‬Ҋ‫جإآؘ؛‬

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nature and role. Here we find a surprisingly close relationship between the creativity of students in the classroom and creativity as seen through the lens of ѷѿ. ǀƟƫƿDžƴDŽ ƣƿ ƞƶᄙ ᄬᇴᇲᇳᇴᄭ ƻƶƞơƣƾ Ɵƫƾƺơƫƞƿƫƺƹ ƫƹ ƿƩƣ Ƥƫƽƾƿ Ƽǀƣƾƿƫƺƹᄕ ǂƩƫơƩ ƨƣƹƣƽƞƿƣƾᅸǀƹƤƞƸƫƶƫƞƽơƺƸƟƫƹƞƿƫƺƹƾƺƤƤƞƸƫƶƫƞƽơƺƹơƣƻƿƾƞƹƢơƺƹƾƿƽǀơƿƾᅺᄬƻᄙᇳᇶᄭᄙ Ʃƣƽƣ ƫƾ ƞƸƟƫƨǀƫƿDŽ ƫƹ ƿƩƣ ƿƣƽƸ ᅸơƺƸƟƫƹƞƿƫƺƹƾᄙᅺ ƿ ơƞƹ Ɵƣ ǀƹƢƣƽƾƿƺƺƢ ƞƾ ƿƩƣ Fir Tree Aha! moment. That is the insight where three different concepts have taken part. The algebraic expression, quadratic equation, and factorization of trinomials were combined by strong links between them. A combination can ƞƶƾƺ Ɵƣ ƾƣƣƹ ƿƩƽƺǀƨƩ ƿƩƣ ƞƹƞƶƺƨDŽ ƺƤ ƿƩƣ ƫƹƾƿƣƫƹᅷƾ ᅸ ƞƻƻƫƣƾƿƩƺǀƨƩƿᅺ Ʃƞᄛ ƸƺƸƣƹƿƺƽƿƩƽƺǀƨƩƿƩƣƞƟƾƿƽƞơƿƫƺƹǂƫƿƩƫƹƿƩƣƩƞƿ ƾƞƣơƿƺƽᄞƩƞᄛƸƺƸƣƹƿ ᄬƫƹƩƞƻƿƣƽᇳᇹᄕƿƩƫƾǁƺƶǀƸƣᄭᄙƩƣƾƣƞƽƣƾƫƨƹƫƤƫơƞƹƿơƺƸƟƫƹƞƿƫƺƹƾƫƹƿƩƞƿƿƩƣDŽ can also have exploratory or transforming qualities. Thus, to attribute bisociƞƿƫƺƹƺƹƶDŽƿƺƿƩƣƤƫƽƾƿơƞƿƣƨƺƽDŽƢƺƣƾƹᅷƿơƺƹƾƫƢƣƽƢƫƤƤƣƽƣƹƿƿDŽƻƣƾƺƤơƺƸƟƫƹƫƹƨ leading to different modes of creativity. ǀƟƫƿDžƴDŽ ƣƿ ƞƶᄙ ᄬᇴᇲᇳᇴᄭ ƻƽƺǁƫƢƣ ƞ ƹƺƿƣǂƺƽƿƩDŽ ƣǃƞƸƻƶƣ ƺƤ ƞ ƢƫƤƤƣƽƣƹƿ ƿƣơƩnique of combining. On the one hand, they assert that “analogical reasoning is closely related to bisociative reasoning, in particular its domain-crossing conơƣƻƿǀƞƶƾƻƞơƣᄚƟƣƞƽƾƿƩƣƩƞƶƶƸƞƽƴƾƺƤƟƫƾƺơƫƞƿƫƺƹᅺᄬƻᄙᇴᇹᄭDŽƣƿƺƹƿƩƣƺƿƩƣƽᄕ they see bisociation as quite different, in general, from analogical reasoning: Perceiving a problem simultaneously from the perspective of two distinct knowledge bases, does not mean that one views the entire problem from ƺƹƣƴƹƺǂƶƣƢƨƣƟƞƾƣƞƹƢƿƩƣƹƤƽƺƸƿƩƣƺƿƩƣƽᄙ ƹƞƾƣƹƾƣᄕǂƩƣƹƟƫƾƺơƫƞtion occurs, a fraction of both knowledge bases becomes unified into a ƾƫƹƨƶƣƴƹƺǂƶƣƢƨƣƟƞƾƣƫƹƿƩƣơƺƹƿƣǃƿƺƤƿƩƣƻƽƺƟƶƣƸƞƿƩƞƹƢᄙᄬǀƟƫƿDžƴDŽ ƣƿƞƶᄙᄕᇴᇲᇳᇴᄕƻᄙᇴᇹᄕƣƸƻƩƞƾƫƾƫƹƿƩƣƺƽƫƨƫƹƞƶᄭ

ƹ ƺƿƩƣƽ ǂƺƽƢƾᄕ ƫƹ ƫƿƾ ƻǀƽƣƾƿ ƤƺƽƸ Ɵƫƾƺơƫƞƿƫƺƹ ƫƾ ƿƩƣ Ƣƫƞƶƣơƿƫơƞƶ ƾDŽƹƿƩƣƾƫƾ ƟǀƫƶƿƺǀƿƺƤƿƩƣƻƫƣơƣƾᄬƹƺƿǂƩƺƶƣƾᄭƺƤƻƽƣǁƫƺǀƾƶDŽƤƞƸƫƶƫƞƽƫƢƣƞƾᄙ ƿƾƣƣƸƾƿƩƞƿ this process of thinking nonlinearly, that is, not associatively but bisociatively, is most difficult to engage in consciously. That is why bisociativity is often misǀƹƢƣƽƾƿƺƺƢᄙ ƿ ƿƩƣ ƾƞƸƣ ƿƫƸƣᄕ ƺƣƾƿƶƣƽ ǂƞƾ ǂƣƶƶ ƞǂƞƽƣ ƺƤ ƿƩƣ ƶƫƸƫƿƾ Ƥƺƽ ƿƩƣ bisociation-based understanding of creativity: Some writers identify the creative act in its entirety with the unearthing ƺƤƩƫƢƢƣƹƞƹƞƶƺƨƫƣƾᄙᄚǀƿǂƩƣƽƣƢƺƣƾƿƩƣƩƫƢƢƣƹƶƫƴƣƹƣƾƾƩƫƢƣᄕƞƹƢƩƺǂ ƫƾ ƫƿ ƤƺǀƹƢᄞ ᄚ ᄴ ᄵƹ Ƹƺƾƿ ƿƽǀƶDŽ ƺƽƫƨƫƹƞƶ ƞơƿƾ ƺƤ ƢƫƾơƺǁƣƽDŽ ƿƩƣ ᅸƾƣƣƫƹƨᅺ ƫƾ ƫƹƤƞơƿƫƸƞƨƫƹƫƹƨᄖƫƿƫƾƢƺƹƣƫƹƿƩƣƸƫƹƢᅷƾᄕƞƹƢƸƺƾƿƶDŽƿƩƣǀƹơƺƹƾơƫƺǀƾ ƸƫƹƢᅷƾƣDŽƣᄙᄚᅸƫƸƫƶƞƽƫƿDŽᅺƫƾƹƺƿƞƿƩƫƹƨƺƤƤƣƽƣƢƺƹƞƻƶƞƿƣᄬƺƽƩƫƢƢƣƹƫƹƞ ơǀƻƟƺƞƽƢᄭᄖƫƿƫƾƞƽƣƶƞƿƫƺƹƣƾƿƞƟƶƫƾƩƣƢƫƹƿƩƣƸƫƹƢƟDŽƞƻƽƺơƣƾƾƺƤƾƣƶƣơ-

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ѹ‫ؔ؛ؖآءإؔح‬

ƿƫǁƣƣƸƻƩƞƾƫƾƺƹƿƩƺƾƣƤƣƞƿǀƽƣƾǂƩƫơƩƺǁƣƽƶƞƻƫƹƞơƣƽƿƞƫƹƽƣƾƻƣơƿᄚƞƹƢ ignoring other features. Even such a seemingly simple process as recognizing the similarity between two letters “a” written by different hands, involves processes of abstraction and generalization in the nervous sysƿƣƸǂƩƫơƩƞƽƣƶƞƽƨƣƶDŽǀƹƣǃƻƶƞƫƹƣƢᄙᄬƺƣƾƿƶƣƽᄕᇳᇻᇸᇶᄕƻᄙᇴᇲᇲᄭ ƺƢƣƹƫƹƤƺƽƸƾǀƾƿƩƞƿƺƣƾƿƶƣƽᄕƟƣƫƹƨƤǀƹƢƞƸƣƹƿƞƶƶDŽƽƫƨƩƿƺƹƿƩƣƫƾƾǀƣᄕǂƺǀƶƢ have been thrilled at the progress of knowledge here knowing that computer models of analogical thinking such as ѷѹ҃ѻ and ѷ҈ѹ҉, inspired largely by neural networks, have been designed to recognize analogies, including typographic styles and individual letters of the alphabet—a bisociative frame ᄬƺƢƣƹᄕᇴᇲᇲᇶᄕƻᄙᇳᇻᇹᄭᄙ ƽƺƸƿƩƣƻƺƫƹƿƺƤǁƫƣǂƺƤѷѿ, bisociation has directed our attention in the correct direction of understanding how analogical thinking, clearly partaking in the Aha! moments, is possible.

ƹƞƹDŽơƞƾƣᄕƺƣƾƿƶƣƽǂƣƹƿƟƣDŽƺƹƢƻǀƽƣơƺƹƹƣơƿƫǁƫƾƸƫƹThe Act of Creation ᄬᇳᇻᇸᇶᄭᄕƿƩƞƿƫƾᄕƩƣǂƣƹƿƟƣDŽƺƹƢƿƩƣƾƣƞƽơƩƺƤƞƹƞƶƺƨƫƣƾƫƹƿƺƻƽƺƟƶƣƸᅟƾƺƶǁƫƹƨ with its conscious planning based on sequential decisions. Obviously, a probƶƣƸƾƺƶǁƣƢǂƫƿƩƿƩƣƩƣƶƻƺƤƫƹƾƫƨƩƿƫƹǁƺƶǁƣƢƞƹƞƶƺƨƫơƞƶƿƩƫƹƴƫƹƨᄙ ƿƞƶƾƺƫƹǁƺƶǁƣƢ conscious reasoning and deliberate thinking, especially in the preparatory and verification stages of the Gestalt process, which cannot be understood solely with the help of connectionism. To understand such deliberate thinking, connectionism must be “combined” or maybe bisociated with another more sequential approach. Boden ᄬᇴᇲᇲᇶᄭƫƹƤƺƽƸƾǀƾƿƩƞƿᅸᄴƺᄵƹƣƺƤƿƩƣƸƺƾƿƞơƿƫǁƣƽƣƾƣƞƽơƩᅟƞƽƣƞƾƞƿƻƽƣƾƣƹƿƫƾƿƩƣ ƢƣƾƫƨƹƺƤᅵƩDŽƟƽƫƢᅷƾDŽƾƿƣƸƾᄕơƺƸƟƫƹƫƹƨƿƩƣƤƶƣǃƫƟƶƣƻƞƿƿƣƽƹᅟƸƞƿơƩƫƹƨƺƤơƺƹƹƣơƿƫƺƹƫƾƸǂƫƿƩƾƣƼǀƣƹƿƫƞƶƻƽƺơƣƾƾƫƹƨƞƹƢƩƫƣƽƞƽơƩƫơƞƶƾƿƽǀơƿǀƽƣᅺᄬƻƻᄙᇴᇺᇶᅬᇴᇺᇷᄭᄙ 1.5.4 Bisociation and Socio-Cultural Creativity ƺǂƢƺƣƾƺƣƾƿƶƣƽᅷƾƟƫƾƺơƫƞƿƫƺƹƽƣƶƞƿƣƿƺƿƩƣƢƺƸƞƫƹƺƤƾƺơƫƞƶơƽƣƞƿƫǁƫƿDŽᄕƸƞDŽƟƣ ᅸƿƩƣƶƣƞƾƿƢƣǁƣƶƺƻƣƢƞƽƣƞƫƹơƽƣƞƿƫǁƫƿDŽƽƣƾƣƞƽơƩᅺᄬƸƞƟƫƶƣᄕᇳᇻᇻᇸᄕƻᄙᇴᇸᇶᄭᄕƞƹƢ more precisely to the domain of the “cultural psychology of creativity,” which conceives creativity as a fundamentally relational, intersubjective phenomeƹƺƹᄬƶǓǁƣƞƹǀᄕᇴᇲᇳᇲᄕƻᄙᇺᇲᄭᄞ ƶǓǁƣƞƹǀᄬᇴᇲᇳᇲᄭƤƺƽƸǀƶƞƿƣƾƿƩƣƢƣƤƫƹƫƿƫƺƹƺƤơƽƣƞƿƫǁƫƿDŽǂƫƿƩƫƹơǀƶƿǀƽƞƶƻƾDŽchology perspective as a complex socio-cultural-psychological process that, through working with “culturally impregnated” materials within an intersubjective space, leads to the generation of artifacts that are evaluated as new and signifiơƞƹƿƟDŽƺƹƣƺƽƸƺƽƣƻƣƽƾƺƹƾƺƽơƺƸƸǀƹƫƿƫƣƾƞƿƞƨƫǁƣƹƿƫƸƣᄙᄬƻᄙᇺᇹᄭ

ѷ‫إب؛اإ‬ҁ‫إؘ؟ائؘآ‬ᅷ‫ئ‬Ѹ‫ءآ؜اؔ؜ؖآئ؜‬Ҋ‫جإآؘ؛‬

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The definition eliminates the cognitive individual as an independent site of creativity. We might therefore conclude that bisociation, which has been formulated ƫƹƿƩƣƣƾƾƣƹƿƫƞƶƶDŽƫƹƢƫǁƫƢǀƞƶơƺƹƿƣǃƿᄕơƞƹᅷƿƩƞǁƣƞƹDŽƿƩƫƹƨƿƺƢƺǂƫƿƩƿƩƣơƽƣƞƿƫǁƫƿDŽƾƣƣƹƣǃơƶǀƾƫǁƣƶDŽƞƾƿƩƣƾƺơƫƺơǀƶƿǀƽƞƶƻƽƺơƣƾƾᄙ ƹƤƞơƿᄕƶǓǁƣƞƹǀᄬᇴᇲᇳᇺᄭƞƾƾƣƽƿƾ that the cultural psychology of creativity takes relationships as its unit of analysis instead of isolated individuals. However, the mathematics teacher whose class includes students who function better when working independently and others who learn best through group processes and collaborative discussion cannot rely solely on the sociocultural collaborative relationships between students because of the lone thinkers. Equally important, the teacher cannot rely solely on an individual cognitive approach, because of those students who learn best through cooperation and collaboration. Thus, classroom teachers need a pedagogical approach that facilitates creativity in individuals as well as in collaborative groups. The dangers inherent in education when the two approaches are viewed as separate, ǀƹƽƣƶƞƿƣƢơƺƸƻƺƹƣƹƿƾƫƾƢƫƾơǀƾƾƣƢƣǃƿƣƹƾƫǁƣƶDŽƫƹƩƞƻƿƣƽᇷƫƹƿƩƣơƺƹƿƣǃƿƺƤ the collapse of a national core curriculum in the United States. As we will see, ƟƫƾƺơƫƞƿƫƺƹƾƞǁƣƾƿƩƣƢƞDŽᄙƶǓǁƣƞƹǀᄬᇴᇲᇳᇷᄭƻƽƺƻƺƾƣƾƿƩƞƿᅸơƽƣƞƿƫǁƣƞơƿƾƫƹǁƺƶǁƣ adopting and coordinating two or more different perspectives on the same issue or problem and, as a result, expanding our action possibilities in relaƿƫƺƹƿƺƿƩƞƿƻƞƽƿƫơǀƶƞƽƫƾƾǀƣƺƽƻƽƺƟƶƣƸᅺᄬƻᄙᇳᇹᇲᄭᄙ ƣƽƣᄕǂƣơƞƹƣƞƾƫƶDŽƽƣơƺƨƹƫDžƣ ƿƩƣ Ɵƫƾƺơƫƞƿƫǁƣ ƾƿƽǀơƿǀƽƣ Ɵƣƿǂƣƣƹ ƿǂƺ ᄬƺƽ Ƹƺƽƣᄭ ƻƣƽƾƻƣơƿƫǁƣƾ ǂƩƣƽƣ ƤƫƹƢƫƹƨ hidden analogies between the two leads to a “creative act,” a novel idea, or ƞƹƞƽƿƫƤƞơƿᄙƣƽƾƻƣơƿƫǁƣƾƞƽƣƞƴƫƹƿƺƿƩƣƤƽƞƸƣƾƺƤƽƣƤƣƽƣƹơƣᄙ ƹƤƞơƿᄕƶǓǁƣƞƹǀ ᄬᇴᇲᇳᇷᄭƞƢƺƻƿƾƣƺƽƨƣ ƣƽƟƣƽƿƣƞƢᅷƾƢƣƤƫƹƫƿƫƺƹᄘᅸƿƩƣƻƣƽƾƻƣơƿƫǁƣƫƾƿƩƣǂƺƽƶƢ in its relationship to the individual and the individual in his relationship to the ǂƺƽƶƢᅺᄬƣƞƢᄕᇳᇻᇵᇺᄕƻᄙᇳᇳᇷᄕƼǀƺƿƣƢƺƹƻᄙᇳᇸᇻᄭᄙ What we have then is an unusual situation. The bisociative theory formulated on individual creators seems able to adapt to a socially constructed understanding of creativity as a complex sociocultural-psychological process. From the classroom perspective, our assertion has no more than a heuristic quality; to be useful, we need to determine the exact connection between the individual cognitive view and the cultural psychology perspective. Ʃƞƻƿƣƽƾ ᇵ ƞƹƢ ᇷ ƾǀƨƨƣƾƿ ƿǂƺ Ƣƫƽƣơƿƫƺƹƾ Ƥƺƽ ƫƹǁƣƾƿƫƨƞƿƫƹƨ ƿƩƣ ơƺƹƹƣơƿƫƺƹ Ɵƣƿǂƣƣƹ DŽƨƺƿƾƴƫƞƹ ƫƹƿƣƽƹƞƶƫDžƞƿƫƺƹ ƞƹƢ ƫƞƨƣƿƫƞƹ ƫƹƿƣƽƫƺƽƫDžƞƿƫƺƹᄘ ᄬᇳᄭ ǂƫƿƩƫƹ ƞ ƿƣƞơƩƫƹƨ ƻƽƞơƿƫơƣᄕ ǂƩƣƽƣ ƞƽƹƣƾᅷƾ ᄬᇴᇲᇲᇲᄭ Ƣƣƾơƽƫƻƿƫƺƹ ƺƤ ƺƟƾƣƽǁƣƢ Ʃƞᄛ ƸƺƸƣƹƿƾƫƹƩƞƻƿƣƽᇵƾǀƨƨƣƾƿƾƿƩƞƿƫƹƿƣƽƹƞƶƫDžƞƿƫƺƹƞƹƢƫƹƿƣƽƫƺƽƫDžƞƿƫƺƹƫƹơƺƶƶƞƟƺƽƞƿƫǁƣƻƽƺƟƶƣƸᅟƾƺƶǁƫƹƨƞƽƣƸǀƿǀƞƶƶDŽƫƹƿƣƽƢƣƻƣƹƢƣƹƿᄖᄬᇴᄭǂƫƿƩƫƹƿƩƣƺƽƣƿical frameworks: Vygotsky considered internalization as the process through

ሩሦ

ѹ‫ؔ؛ؖآءإؔح‬

ǂƩƫơƩ ƞƹ ƫƹƢƫǁƫƢǀƞƶ ơƽƣƞƿƣƾ ƹƣǂ ơƺƹơƣƻƿƾ Ƥƺƽ ƩƫƸƾƣƶƤᄧƩƣƽƾƣƶƤ ᄬƣǃƿƣƽƹƞƶᏉƫƹƿƣƽƹƞƶᄭᄙ DŽ ơƺƹƿƽƞƾƿᄕ ơƶƞƾƾƽƺƺƸ ƿƣƞơƩƫƹƨ ƣǃƻƣƽƫƣƹơƣ ƾƩƺǂƾ ƿƩƞƿ ƫƹƿƣƽƹƞƶƫDžƞƿƫƺƹƺƤƿƩƣƣǃƿƣƽƹƞƶơƺƹơƣƻƿƢƺƣƾƹƺƿƞƶǂƞDŽƾƶƣƞƢƿƺƫƹƢƣƻƣƹƢƣƹƿƞơƿƫƺƹᄬƿƩƣ ƾƺᅟơƞƶƶƣƢƾƣơƺƹƢᅟƢƞDŽƣƤƤƣơƿᄭƽƣƶƞƿƣƢƿƺƫƹƢƫǁƫƢǀƞƶƫƹƿƣƽƫƺƽƫDžƞƿƫƺƹᄕƺƽƿƺƺǂƹƫƹƨƞ concept. That notion suggests a missing element in the process of mental concept representation. We suggest that the missing element is the individual process of interiorization, which we conjecture completes the internalization process. The investigation of the relationship between internalization and interiorization is one of the first endeavors in the follow-up creative teaching-research ᄬѹҊ҈ᄭ ơDŽơƶƣᄕ ǂƩƫơƩ ǂƣ Ʃƺƻƣ ǂƫƶƶ ƾƩƫƹƣ ƞ ƹƣǂ ƶƫƨƩƿ ƺƹ ƿƩƣ ƽƣƶƞƿƫƺƹƾƩƫƻ ƞƹƢ interaction between individual cognitive creativity and sociocultural collaborative creativity. 1.5.5 Connectionist Systems: A Couple of Words Connectionist systems are parallel-processing systems whose computational properties are very broadly modeled on the brain. Connectionist systems are not programmed in the traditional sense as much as trained, or learned from experience through self-organization. ƺƢƣƹᄬᇴᇲᇲᇶᄕƻᄙᇳᇵᇵᄭƢƣƾơƽƫƟƣƾƞƹƫƹƿƣƽƣƾƿƫƹƨƣǃƞƸƻƶƣƞƹƢƸƺƢƣƶƺƤƞơƶƞƾƾroom process. The essential feature is the capacity for pattern matching, one of the first steps in mathematical thinking. Learners can recognize a pattern they ƩƞǁƣƣǃƻƣƽƫƣƹơƣƢƟƣƤƺƽƣᄙ ƤƿƩƞƿƻƞƿƿƣƽƹƫƾƻƞƽƿƺƤƿƩƣƺƽƫƨƫƹƞƶƺƹƣᄕƿƩƣDŽơƞƹ recognize and complete it. A given input pattern can call up a range of stored different-yet-similar patterns whose activation strength varies according to their similarity. That ability might be a central resource in the Aha! moment when connections are made by comparing mental records of the past with the ƻƽƺƟƶƣƸƞƿƩƞƹƢƟDŽǀƾƫƹƨƫƸƺƹƣƿƞƶᄙᅷƾᄬᇴᇲᇳᇲᄭƞƻƻƽƺƞơƩƿƺƽƣƤƶƣơƿƫǁƣƞƟƾƿƽƞơtion. The neural networks do not construct precise definitions of every conơƣƻƿᄬƻƞƿƿƣƽƹᄭƿƩƣDŽƶƣƞƽƹᄙƩƣDŽƨƽƞƢǀƞƶƶDŽƟǀƫƶƢǀƻƽƣƻƽƣƾƣƹƿƞƿƫƺƹƾƺƤƟƽƺƞƢƶDŽ shared features of the concept concerned and can recognize individual instances of the concept despite differences in detail. An interesting process ƺƤƾǀơƩƞƟǀƫƶƢƫƹƨǀƻƫƾƞƽƸƫƶƺƤƤᅟƸƫƿƩᅷƾƽƣƫƹƾơƽƫƻƿƫƺƹƞƻƻƽƺƞơƩᄕǂƩƫơƩƩƞƾ been observed in the development of language and related drawing of children ƞƾƿƩƣDŽƨƽƺǂᄙƺƹƾơƫƺǀƾƾƣƶƤᅟƽƣƤƶƣơƿƫƺƹᄕƞƽƸƫƶƺƤƤᅟƸƫƿƩƾǀƨƨƣƾƿƾᄕƫƾƿƩƣƽƣƾǀƶƿ of the reinscription process, which allows for many levels of concept represenƿƞƿƫƺƹƿƺƨƣƿƩƣƽǂƫƿƩƣƶƣƸƣƹƿƞƽDŽƸƺƸƣƹƿƾƺƤƽƣƤƶƣơƿƫƺƹǀƻƺƹƫƿᄬƺƢƣƹᄕᇴᇲᇲᇶᄭᄙ We finish this section by introducing the Ѹѿ҉҅҄ search engine of the BisociaƿƫǁƣƣƿǂƺƽƴƾƤƺƽƽƣƞƿƫǁƣ ƹƤƺƽƸƞƿƫƺƹƫƾơƺǁƣƽDŽᄕƞƟƫƾƺơƫƞƿƫǁƣƾƣƞƽơƩƣƹƨƫƹƣ

ѷ‫إب؛اإ‬ҁ‫إؘ؟ائؘآ‬ᅷ‫ئ‬Ѹ‫ءآ؜اؔ؜ؖآئ؜‬Ҋ‫جإآؘ؛‬

ሩሧ

that uses large sets of data to investigate the creation of new concepts based on similarities of their representations in the two large different sets of data ᄬƣƽƿƩƺƶƢᄕᇴᇲᇳᇴᄖǀƟƫƿDžƴDŽƣƿƞƶᄙᄕᇴᇲᇳᇴᄭᄙ ƹƩƞƻƿƣƽᇳᇲᄕǂƣƢƫƾơǀƾƾƿƩƣƾƫƸƫƶƞƽƫƿDŽ ƺƤƿDŽƻƣƾƺƤơƺƹƹƣơƿƫƺƹƾƟƣƿǂƣƣƹƢƫƤƤƣƽƣƹƿơƺƹơƣƻƿƾƺƟƿƞƫƹƣƢƿƩƽƺǀƨƩƿƩƣƺ classification of Aha! moments and connections obtained by the Ѹѿ҉҅҄ search engine.

1.6

Conclusions

ƩƣƞƫƸƺƤƩƞƻƿƣƽᇳƩƞƾƟƣƣƹƿƺƻƺƾƫƿƫƺƹƟƫƾƺơƫƞƿƫƺƹƿƩƣƺƽDŽƫƹƿƩƣơƺƹƿƣǃƿƺƤ creativity research. We have shown the conceptual links that join bisociation, Janusian thinking, and connectionist systems of ѷѿ, which together comprise a new approach to understanding creativity in addition to Gestalt and Guilford/ Torrance approaches. An important note in the chapter concerns the possibility of a fifth stage in the Gestalt sequence, intimation, between incubation and illumination. ChapƿƣƽᇳᇶƻƽƺǁƫƢƣƾƞƹƫƹᅟƢƣƻƿƩƶƺƺƴƞƿƿƩƫƾƾƿƞƨƣᄙƣƩƞǁƣƻƽƣƾƣƹƿƣƢƣǃƞƸƻƶƣƾƺƤ Aha! moments from different intellectual/artistic domains to emphasize the wide generality of bisociation theory that show us different understandings of creativity by the creators: from the mystical Ѹѻѿ҄ѽ of the Sumerian/Tamil spirituality through the activity of infinite awareness of contemporary mysƿƫơƾƿƺƿƩƣƾƺơƫƞƶƻƽƞơƿƫơƣƺƤƞƺᅷƾƿƩƣƺƽDŽƺƤƴƹƺǂƶƣƢƨƣƤƺƽƸƞƿƫƺƹᄕƺƫƹơƞƽȅᅷƾ ơƺƹǁƫơƿƫƺƹƺƤƿƩƣƢƣơƫƾƫǁƣƽƺƶƣƺƤƿƩƣǀƹơƺƹƾơƫƺǀƾƞƹƢƿƩƣ ƫƹƾƿƣƫƹᅷƾᅸ ƞƻƻƫƣƾƿ Thought” Aha! moment. We follow with the general examples of bisociative thinking and develop the definition of the bisociative frame used extensively in Chapter 4 in the classification of types of conceptual connections reached during the insight. An important distinction between the creative progress of understanding reached through bisociation and the associative limited thinking of the exercise of understanding, which takes place within one matrix. The next part discovers from the preliminary analysis that bisociation seems ƿƺ ǀƹƢƣƽƾơƺƽƣ ƿƩƣ ƾƺơƫƺơǀƶƿǀƽƞƶ ơƺƹơƣƻƿƫƺƹ ƺƤ ơƽƣƞƿƫǁƫƿDŽ ǂƫƿƩƫƹ ƶǓǁƣƞƹǀᅷƾ ᄬᇴᇲᇳᇲᄕᇴᇲᇳᇺᄭƞƻƻƽƺƞơƩᄙƩƣƶƞƾƿƿǂƺƾƣơƿƫƺƹƾƣǃƻƶƫơƞƿƣƿƩƣƽƣƶƞƿƫƺƹƾƩƫƻƺƤƿƩƣ bisociation theory with the Janusian thinking theory of creativity by RothenƟƣƽƨᄬᇳᇻᇹᇻƟᄭᄕǂƩƣƽƣǂƣƣƸƻƩƞƾƫDžƣƿƩƣơƶƺƾƣƽƣƶƞƿƫƺƹƾƩƫƻƺƤƞƹƞƶƺƨƫơƞƶƿƩƫƹƴing encountered in bisociation with the oppositional thinking of the Janusian approach. We see here bisociation as the creative process whose product is the exact opposite of Janusian thought.

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ѹ‫ؔ؛ؖآءإؔح‬

Notes ᇳ ƩƣƢƣƤƫƹƫƿƫƺƹǂƞƾƞƟƾƿƽƞơƿƣƢƤƽƺƸƿƩƣƤƺƶƶƺǂƫƹƨƤƽƞƨƸƣƹƿᄘ There are two ways of escaping our more or less automatized routines of thinking and behaving. The first, of course, is plunging into dreaming or dream-like states, when the codes of rational thinking are suspended. The other way is also an escape—from boredom, stagnation, intellectual predicaments, and emotional frustration—but an escape ƫƹƿƩƣƺƻƻƺƾƫƿƣƢƫƽƣơƿƫƺƹᄖƫƿƫƾƾƫƨƹƞƶƶƣƢƟDŽƿƩƣƾƻƺƹƿƞƹƣƺǀƾƤƶƞƾƩᄴƶƣƞƻᄵƺƤƫƹƾƫƨƩƿ which shows a familiar situation or event in a new light and elicits a new response to it. The bisociative act connects previously unconnected matrices of experience; it makes us “underƾƿƞƹƢǂƩƞƿƫƿƫƾƿƺƟƣƞǂƞƴƣᄕƿƺƟƣƶƫǁƫƹƨƺƹƾƣǁƣƽƞƶƻƶƞƹƣƾƞƿƺƹơƣᄙᅺᄬƺƣƾƿƶƣƽᄕᇳᇻᇸᇶᄕƻᄙᇶᇷᄭ ᇴ ƣƣƩƿƿƻƾᄘᄧᄧǂǂǂᄙƺƻƽƞƩƸƞƨᄙơƺƸᄧƶƫƤƣᄧƞᇴᇻᇲᇻᇲᇶᇵᇸᄧƞƩƞᅟƸƺƸƣƹƿᅟƸƣƞƹƫƹƨᄧ ᇵ ƩƣơƺƶƶƣơƿƫƺƹƺƤƩƞᄛƸƺƸƣƹƿƾǀƾƣƢƫƹƿƩƣǁƺƶǀƸƣơƞƹƟƣƤƺǀƹƢƫƹƩƞƻƿƣƽᇳᇹᄙ

References ƸƞƟƫƶƣᄕᄙᄙᄬᇳᇻᇻᇸᄭᄙCreativity in context. Westview Press. ƞƽƹƣƾᄕᄙᄬᇴᇲᇲᇲᄭᄙƞƨƫơƸƺƸƣƹƿƾƫƹƸƞƿƩƣƸƞƿƫơƾᄘ ƹƾƫƨƩƿƾƫƹƿƺƿƩƣƻƽƺơƣƾƾƺƤơƺƸing to know. For the Learning of Mathematics, 20ᄬᇳᄭᄕᇵᇵᅬᇶᇵᄙ ƣƽƿƩƺƶƢᄕ ᄙ ᄙ ᄬᇴᇲᇳᇴᄭᄙ ƺǂƞƽƢƾ Ɵƫƾƺơƫƞƿƫǁƣ ƴƹƺǂƶƣƢƨƣ ƢƫƾơƺǁƣƽDŽᄙ ƹ ᄙ ᄙ ƣƽƿƩƺƶƢ ᄬ ƢᄙᄭᄕBisociative knowledge discovery: An introduction to concept, algorithms, tools, and applicationsᄬƻƻᄙᇳᅬᇳᇲᄭᄙƻƽƫƹƨƣƽᄙ ƺƢƣƹᄕᄙᄙᄬᇴᇲᇲᇶᄭᄙThe creative mind: Myths and mechanisms ᄬᇴƹƢƣƢᄙᄭᄙƺǀƿƶƣƢƨƣᄙ ƺƽǂƣƫƹᄕᄙᄕƫƶưƣƢƞƩƶᄕᄙᄕѵƩƞƫᄕ ᄙᄬ ƢƾᄙᄭᄙᄬᇴᇲᇳᇶᄭᄙMathematicians on creativity. Mathematical Association of America. DžƞƽƹƺơƩƞᄕᄙᄕƞƴƣƽᄕᄙᄕƫƞƾᄕᄙᄕѵƽƞƟƩǀᄕᄙᄬ ƢƾᄙᄭᄙᄬᇴᇲᇳᇸᄭᄙThe creative enterprise of mathematics teaching research: Elements of methodology and practice—From teachers to teachers. Sense Publishers. ƣƣƶƶƫƾᄕᄙᄙᄕѵƺƶƢƫƹᄕᄙᄙᄬᇴᇲᇲᇸᄭᄙƤƤƣơƿƞƹƢƸƣƿƞᅟƞƤƤƣơƿƫƹƸƞƿƩƣƸƞƿƫơƞƶƻƽƺƟlem solving: A representational perspective. Educational Studies in Mathematics, 63, ᇳᇵᇳᅬᇳᇶᇹᄙ ǀƟƫƿDžƴDŽᄕᄙᄕʢƿƿƣƽᄕᄙᄕơƩƸƫƢƿᄕᄙᄕѵƣƽƿƩƺƶƢᄕᄙᄙᄬᇴᇲᇳᇴᄭᄙƺǂƞƽƢƾơƽƣƞƿƫǁƣƫƹƤƺƽƸƞƿƫƺƹƣǃƻƶƺƽƞƿƫƺƹƟƞƾƣƢƺƹƺƣƾƿƶƣƽᅷƾơƺƹơƣƻƿƺƤƟƫƾƺơƫƞƿƫƺƹᄙ ƹᄙᄙƣƽƿƩƺƶƢ ᄬ ƢᄙᄭᄕBisociative knowledge discovery: An introduction to concept, algorithms, tools, and applicationsᄬƻƻᄙᇳᇳᅬᇵᇳᄭᄙƻƽƫƹƨƣƽᄙ ƫƹƾƿƣƫƹᄕᄙᄬᇳᇻᇶᇻᄭᄙǀƿƺƟƫƺƨƽƞƻƩƫơƞƶƹƺƿƣƾᄬᄙᄙơƩƫƶƻƻᄕƽƞƹƾᄙᄭᄙ ƹᄙᄙơƩƫƶƻƻᄬ Ƣᄙᄭᄕ Albert Einstein: Philosopher-scientistᄬƻƻᄙᇳᅬᇻᇷᄭᄙƫƟƽƞƽDŽƺƤƫǁƫƹƨƩƫƶƺƾƺƻƩƣƽƾᄙ ƶǓǁƣƞƹǀᄕᄙᄙᄬᇴᇲᇳᇲᄭᄙƞƽƞƢƫƨƸƾƫƹƿƩƣƾƿǀƢDŽƺƤơƽƣƞƿƫǁƫƿDŽᄘ ƹƿƽƺƢǀơƫƹƨƿƩƣƻƣƽƾƻƣơƿƫǁƣ of cultural psychology. New Ideas in Psychology, 28ᄬᇳᄭᄕᇹᇻᅬᇻᇵᄙ ƶǓǁƣƞƹǀᄕᄙ ᄙ ᄬᇴᇲᇳᇷᄭᄙ ƽƣƞƿƫǁƫƿDŽ ƞƾ ƞ ƾƺơƫƺơǀƶƿǀƽƞƶ ƞơƿᄙ Journal of Creative Behavior, 49ᄬᇵᄭᄕᇳᇸᇷᅬᇳᇺᇲᄙ

ѷ‫إب؛اإ‬ҁ‫إؘ؟ائؘآ‬ᅷ‫ئ‬Ѹ‫ءآ؜اؔ؜ؖآئ؜‬Ҋ‫جإآؘ؛‬

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ƶǓǁƣƞƹǀᄕᄙᄙᄬᇴᇲᇳᇺᄭᄙƽƣƞƿƫǁƫƿDŽƫƹƻƣƽƾƻƣơƿƫǁƣᄘƾƺơƫƺơǀƶƿǀƽƞƶƞƹƢơƽƫƿƫơƞƶƞơơƺǀƹƿᄙ Journal of Constructivist Psychology, 31ᄬᇴᄭᄕᇳᇳᇺᅬᇳᇴᇻᄙ ǀƫƶƤƺƽƢᄕᄙᄙᄬᇳᇻᇸᇹᄭᄙThe nature of human intelligence. McGraw-Hill. ƞƢƞƸƞƽƢᄕᄙᄬᇳᇻᇶᇷᄭᄙ The psychology of invention in the mathematical field. Princeton University Press. ƫƸᄕᄙ ᄙᄬᇴᇲᇳᇴᄭᄙƩƣơƽƣƞƿƫǁƫƿDŽơƽƫƾƫƾᄘƩƣƢƣơƽƣƞƾƣƫƹơƽƣƞƿƫǁƣƿƩƫƹƴƫƹƨƾơƺƽƣƾƺƹƿƩƣ Torrance Test of Creative Thinking. Creativity Research Journal, 23ᄬᇶᄭᄕᇴᇺᇷᅬᇴᇻᇷᄙ ƺƣƾƿƶƣƽᄕᄙᄬᇳᇻᇸᇶᄭᄙThe act of creation. Macmillan. ƺƣƾƿƶƣƽᄕᄙᄬᇳᇻᇹᇻᄭᄙJanus: A summing up. Pan Books. ƺǀƹƫƺƾᄕᄙᄕѵƣƣƸƞƹᄕᄙᄬᇴᇲᇳᇷᄭᄙThe eureka factor: Aha moments, creative insight, and the brain. Random House. ƣƫƴƫƹᄕᄙᄬᇴᇲᇲᇻᄭᄙ ǃƻƶƺƽƫƹƨƸƞƿƩƣƸƞƿƫơƞƶơƽƣƞƿƫǁƫƿDŽǀƾƫƹƨƸǀƶƿƫƻƶƣƾƺƶǀƿƫƺƹƿƞƾƴƾᄙ ƹ ᄙƣƫƴƫƹᄕᄙƣƽƸƞƹᄕѵᄙƺƫơƩǀᄬ ƢƾᄙᄭᄕCreativity in mathematics and the education of gifted studentsᄬƻƻᄙᇳᇴᇻᅬᇳᇶᇷᄭᄙƣƹƾƣǀƟƶƫƾƩƣƽƾᄙ ƣƫƴƫƹᄕᄙᄕѵƫƿƿƞᅟƞƹƿƞDžƫᄕᄙᄬᇴᇲᇳᇵᄭᄙƽƣƞƿƫǁƫƿDŽƞƹƢƸƞƿƩƣƸƞƿƫơƾƣƢǀơƞƿƫƺƹᄘƩƣƾƿƞƿƣ of the art. ZDM: The International Journal of Mathematics, 45ᄬᇴᄭᄕᇳᇷᇻᅬᇳᇸᇸᄙ ƺƨƞƹƞƿƩƞƹᄕᄙᄬᇴᇲᇲᇶᄭᄙƩƣƩDŽƸƹƾƺƤ ǀƸƟƶƣƻƻƞƽᄙBakti TamilᄴƻƺƣƿƽDŽᄵᄙ https://sites.google.com/site/baktitamil/appar-tevaram/humble-appar ƺƽƫƸƣƽᄕ ᄙᄬᇳᇻᇴᇻᄭᄙThe growth of reason: A study of the role of verbal activity in the growth of the structure of the human mindᄙᄙƞǀƶᄙ ƞƺᄕᄙᅟᄙᄬᇴᇲᇲᇹᄭᄙƹƻƽƞơƿƫơƣᄙ ƹᄙĺƫ̜ƣƴᄬ Ƣᄙᄭᄕ On practice and contradiction: Mao TseTung ᄬƻƻᄙᇷᇴᅬᇸᇸᄭᄙƣƽƾƺƺƺƴƾᄙ ƣƞƢᄕᄙ ᄙᄬᇳᇻᇵᇺᄭᄙThe philosophy of the actᄬᄙᄙƺƽƽƫƾᄕ ƢᄙᄭᄙƹƫǁƣƽƾƫƿDŽƺƤƩƫơƞƨƺ Press. ƞƶƞƿƹƫƴᄕᄙᄕѵƺƫơƩǀᄕᄙᄬᇴᇲᇳᇶᄭᄙƣơƺƹƾƿƽǀơƿƫƺƹƺƤƺƹƣƸƞƿƩƣƸƞƿƫơƞƶƫƹǁƣƹƿƫƺƹᄘ ƺơǀƾ ƺƹƾƿƽǀơƿǀƽƣƾƺƤƞƿƿƣƹƿƫƺƹᄙ ƹᄙƫƶưƣƢƞƩƶᄕᄙƣƾƿƣƽƶƣᄕᄙƫơƺƶᄕѵᄙƶƶƞƹᄬ Ƣƾᄙᄭᄕ Proceedings of the 38th Conference of the International Group for the Psychology of Mathematics Education and the 36th Conference of the North American Chapter of the Psychology of Mathematics Education, Vol. 4ᄬƻƻᄙᇵᇹᇹᅬᇵᇺᇶᄭᄙ ᄙ https://www.pmena.org/proceedings/ ƞƶƞƿƹƫƴᄕᄙᄕѵƺƫơƩǀᄕᄙᄬᇴᇲᇳᇷᄭᄙ ǃƻƶƺƽƫƹƨƫƹƾƫƨƩƿᄘ ƺơǀƾƺƹƾƩƫƤƿƾƺƤƞƿƿƣƹƿƫƺƹᄙFor the Learning of Mathematics, 35ᄬᇴᄭᄕᇻᅬᇳᇶᄙ ƺƫƹơƞƽȅᄕ ᄙᄬᇳᇻᇺᇷƞᄭᄙƞƿƩƣƸƞƿƫơƞƶơƽƣƞƿƫƺƹᄙ ƹᄙƩƫƾƣƶƫƹᄬ ƢᄙᄭᄕThe creative process: Reflections on the inventions in arts and sciences ᄬƻƻᄙᇴᇴᅬᇵᇳᄭᄙ University of California Press. ƺƫƹơƞƽȅᄕ ᄙᄬᇳᇻᇺᇷƟᄭᄙPapers on Fuchsian functionsᄬᄙƿƫƶƶǂƣƶƶᄕƽƞƹƾᄙᄭᄙƻƽƫƹƨƣƽᄙ ƽƞƟƩǀᄕᄙᄕ ѵ DžƞƽƹƺơƩƞᄕ ᄙ ᄬᇴᇲᇳᇶᄭᄙ ƣƸƺơƽƞƿƫDžƫƹƨ ƸƞƿƩƣƸƞƿƫơƞƶ ơƽƣƞƿƫǁƫƿDŽ ƿƩƽƺǀƨƩ ƺƣƾƿƶƣƽᅷƾƟƫƾƺơƫƞƿƫƺƹƿƩƣƺƽDŽᄙ ƹᄙƫơƺƶᄕᄙƣƾƿƣƽƶƣᄕᄙƫƶưƣƢƞƩƶᄕѵᄙƶƶƞƹᄬ Ƣƾᄙᄭᄕ Proceedings of the 38th Conference of the International Group for the Psychology of Mathematics Education and the 36th Conference of the North American Chapter of the

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ѹ‫ؔ؛ؖآءإؔح‬

Psychology of Mathematics Education, Vol. 5ᄬƻƻᄙᇳᅬᇺᄭᄙ ᄙƩƿƿƻƾᄘᄧᄧǂǂǂᄙƻƸƣƹƞᄙƺƽƨᄧ proceedings/ ƺƿƩƣƹƟƣƽƨᄕᄙᄬᇳᇻᇹᇻƞᄭᄙ ƫƹƾƿƣƫƹᅷƾơƽƣƞƿƫǁƣƿƩƫƹƴƫƹƨƞƹƢƿƩƣƨƣƹƣƽƞƶƿƩƣƺƽDŽƺƤƽƣƶƞƿƫǁƫƿDŽᄘ A documented report. American Journal of Psychiatry, 136ᄬᇳᄭᄕᇵᇺᅬᇶᇵᄙ ƺƿƩƣƹƟƣƽƨᄕᄙᄬᇳᇻᇹᇻƟᄭᄙThe emerging goddess: The creative process in art, science, and other fields. University of Chicago Press. ƞƢƶƣƽᅟƸƫƿƩᄕ ᄙ ᄬᇴᇲᇳᇷᄭᄙƞƶƶƞƾᅷ Ƥƺǀƽᅟƾƿƞƨƣ ƸƺƢƣƶ ƺƤ ƿƩƣ ơƽƣƞƿƫǁƣ ƻƽƺơƣƾƾᄘ ƺƽƣ ƿƩƞƹ meets the eye? Creativity Research Journal, 27ᄬᇶᄭᄕᇵᇶᇴᅬᇵᇷᇴᄙ ƫƶǁƣƽᄕ ᄙᄙᄬᇳᇻᇻᇹᄭᄙ ƺƾƿƣƽƫƹƨơƽƣƞƿƫǁƫƿDŽƿƩƽƺǀƨƩƫƹƾƿƽǀơƿƫƺƹƽƫơƩƫƹƸƞƿƩƣƸƞƿƫơƞƶƻƽƺƟlem solving and problem posing. ZDM, 29ᄬᇵᄭᄕᇹᇷᅬᇺᇲᄙ ƫƸƺƹᄕ ᄙᄕ ƞƶƢƞƹƩƞᄕ ᄙᄕ ơƶƫƹƿƺơƴᄕ ᄙᄕ ƴƞƽᄕ ᄙ ᄙᄕ ƞƿƞƹƞƟƣᄕ ᄙᄕ ѵ ƣƸƟƞƿᄕ ᄙ ᄙ ᄬᇴᇲᇳᇲᄭᄙƢƣǁƣƶƺƻƫƹƨƞƻƻƽƺƞơƩƿƺƾƿǀƢDŽƫƹƨƾƿǀƢƣƹƿƾᅷƶƣƞƽƹƫƹƨƿƩƽƺǀƨƩƿƩƣƫƽƸƞƿƩƣmatical activity. Cognition and Instruction, 28ᄬᇳᄭᄕᇹᇲᅬᇳᇳᇴᄙ ƿƣƹƩƺǀƾƣᄕᄙᄬᇳᇻᇺᇷᄭᄙResearch as the basis for teaching: Readings from the work of Lawrence Stenhouse ᄬᄙǀƢƢǀơƴѵᄙ ƺƻƴƫƹƾᄕ Ƣƾᄙᄭᄙ ƣƫƹƣƸƞƹƹ Ƣǀơƞƿƫƺƹƞƶƺƺƴƾᄙ ƿƣƽƹƟƣƽƨᄕᄙᄙᄕѵǀƟƞƽƿᄕᄙ ᄙᄬᇳᇻᇻᇻᄭᄙƩƣơƺƹơƣƻƿƾƺƤơƽƣƞƿƫǁƫƿDŽᄘƽƺƾƻƣơƿƾƞƹƢƻƞƽƞƢƫƨƸƾᄙ ƹᄙᄙƿƣƽƹƟƣƽƨᄬ ƢᄙᄭᄕHandbook of creativityᄬƻƻᄙᇴᇷᇳᅬᇴᇹᇴᄭᄙƞƸƟƽƫƢƨƣƹƫversity Press. ƺƽƽƞƹơƣᄕ ᄙᄙᄬᇳᇻᇸᇸᄭᄙThe Torrance tests of creative thinking: Norms-technical manual. Research edition. Verbal tests, forms A and B. Figural tests, forms A and B. Personnel Press. DŽƨƺƿƾƴDŽᄕᄙᄙᄬᇳᇻᇻᇹᄭᄙThought and languageᄬƶƣǃƺDžǀƶƫƹᄕ ƢᄙᄕᇳᇲƿƩƣƢᄙᄭᄙ ƽƣƾƾᄙ ƞƶƶƞƾᄕᄙᄬᇳᇻᇴᇸᄭᄙThe art of thought. Harcourt, Brace & Co. ƣƫƾƟƣƽƨᄕᄙᄙᄬᇳᇻᇻᇷᄭᄙƽƺƶƣƨƺƸƣƹƞƿƺƿƩƣƺƽƫƣƾƺƤƫƹƾƫƨƩƿƫƹƻƽƺƟƶƣƸƾƺƶǁƫƹƨᄘƿƞǃƺƹƺƸDŽƺƤƻƽƺƟƶƣƸƾᄙ ƹᄙᄙƿƣƽƹƟƣƽƨѵᄙ ᄙƞǁƫƢƾƺƹᄬ ƢƾᄙᄭᄕThe nature of insight ᄬƻƻᄙᇳᇷᇹᅬᇳᇻᇸᄭᄙ ƽƣƾƾᄙ

҆ѷ҈Ҋማ Bisociation in the Classroom

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ѹѾѷ҆Ҋѻ҈ሜ

Teaching-Research Analysis: The Constructivist Teaching Experiment as a Methodology of Teaching Bronislaw Czarnocha

2.1

Introduction

2.1.1 Cycles in the History of Mathematics Education The recent history of mathematics education in the United States displays an interesting cyclical structure, from the beauty of the New MathƺƤƿƩƣᇳᇻᇸᇲƾᄕ which emphasized the formal aspects of mathematics, to its Hegelian antithesis of the Back-to-Basics MovementƺƤƿƩƣᇳᇻᇹᇲƾƞƹƢᇳᇻᇺᇲƾᄙƩƣƹƫƿƟƣơƞƸƣ clear that the rote learning of Back to BasicsƢƫƢƹᅷƿƢƣǁƣƶƺƻƾƿǀƢƣƹƿƾᅷƹǀƸƟƣƽ ƾƣƹƾƣƞƹƢơƺƹƾƣƼǀƣƹƿƶDŽƢƫƢƹᅷƿƩƣƶƻƾƿǀƢƣƹƿƾƢƣǁƣƶƺƻƻƽƺƟƶƣƸᅟƾƺƶǁƫƹƨƾƿƽƞƿƣgies, there was a paradigm shift toward the process of “understanding,” leading to the constructivist paradigm. ƩƣơƺƹƾƿƽǀơƿƫǁƫƾƿƞƻƻƽƺƞơƩᄕǂƩƫơƩƽƣƞơƩƣƢƫƿƾƞƻƺƨƣƣƫƹƿƩƣᇳᇻᇻᇲƾᄕƸƫƨƩƿ ƩƞǁƣƟƣƣƹơƺƹƾƫƢƣƽƣƢƿƩƣƾDŽƹƿƩƣƾƫƾƺƤƿƩƣƻƽƣǁƫƺǀƾƿƩƣƾƫƾƞƹƢƞƹƿƫƿƩƣƾƫƾᄙ Ƥƾƺᄕ soon after coming to the surface of professional attention, it split along its own ơƺƹƿƽƞƢƫơƿƫƺƹ ƺƤ ƫƹƢƫǁƫƢǀƞƶ ơƺƨƹƫƿƫǁƣ ƢƣǁƣƶƺƻƸƣƹƿᄕ ƟƞƾƣƢ ƺƹ ƫƞƨƣƿᅷƾ ǂƺƽƴᄕ ƞƹƢƾƺơƫƺơǀƶƿǀƽƞƶƢƣǁƣƶƺƻƸƣƹƿᄕƟƞƾƣƢƺƹDŽƨƺƿƾƴDŽᅷƾᄕǂƩƣƽƣƿƩƣǀƹƫƿƺƤƫƹǁƣƾƿƫgation became the whole class. Over the years, mathematics education has struggled mightily to find a philosophy that would cut across its divisions while providing enough flexibility ƿƺƞơơƺƸƸƺƢƞƿƣƹƣǂƢƣǁƣƶƺƻƸƣƹƿƾƞƹƢƞƻƻƽƺƞơƩƣƾƿƺƶƣƞƽƹƫƹƨᄙƩƞƿᅷƾƶƞơƴing in the approach is the bisociative view, which derives its strength from each component of the ideological divide and thus leads to a new creative synthesis. At present, mathematics education is in the midst of such a cycle, one characterized by the constructivist approach. The main tenets of constructivism ƞƽƣᄬᇳᄭƿƩƞƿƴƹƺǂƶƣƢƨƣƫƾƹƺƿƻƞƾƾƫǁƣƶDŽƽƣơƣƫǁƣƢᄖƿƩƣƶƣƞƽƹƣƽƸǀƾƿƟƣƻƣƽƾƺƹƞƶƶDŽ and actively engaged in the process of learning and in building up necessary ơƺƨƹƫƿƫǁƣƾƿƽǀơƿǀƽƣƾᄬƿƣƤƤƣѵƫƣƨƣƶᄕᇳᇻᇻᇴᄭƞƹƢᄬᇴᄭƿƩƞƿƣƢǀơƞƿƫƺƹƸǀƾƿƟƣƨƫƹ ƞƿƿƩƣƶƣǁƣƶƺƤƞƾƿǀƢƣƹƿᅷƾƸƞƿƩƣƸƞƿƫơƞƶƽƣƞƶƫƿDŽᄙƞƴƣƹƞƿƤƞơƣǁƞƶǀƣᄕƿƩƣƾƣƫƢƣƞƾ make sense to instructors, researchers, and the public. The problem lies in their realization. Constructivist tenets have become the basis of the first US national curriculum for mathematics in history, the Common Core State Standards for ᇙ ‫ةء؟؟؜إؘؕ؞؝؜؟؞ء؜ءآ؞‬ᄕ‫ؘ҄ؗ؜ؘ؟‬ᄕሦሤሦሥᏺᄩᏺѺ҅ѿᄘሥሤᄙሥሥሪሧᄧርራሬርሤሤረረረሪረሧረᇇሤሤሧ

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ѹ‫ؔ؛ؖآءإؔح‬

ƞƿƩƣƸƞƿƫơƾᄬѹѹ҉҉҃ᄭᄕǂƩƫơƩƫƾƾƿƞƶƶƣƢƞƿƻƽƣƾƣƹƿƫƹƫƿƾƣƤƤƣơƿƫǁƣƹƣƾƾƿƺƽƞƫƾƣ student passing rates on standardized exams. That may not necessarily be a ƟƞƢƿƩƫƹƨᄕƞơơƺƽƢƫƹƨƿƺơƶǀƾƴƣDŽᄬᇴᇲᇳᇺᄭᄙ A more important reason for resistance to the ѹѹ҉҉҃ standards is their extreme reliance on standardized tests primarily to assess the quality of the ƿƣƞơƩƣƽƾᅷ ǂƺƽƴᄕ ǂƩƫơƩ ƶƞƿƣƽ ƾƣƽǁƣƾ ƞƾ ƞ Ɵƞƾƫƾ Ƥƺƽ ƣƸƻƶƺDŽƸƣƹƿ Ƣƣơƫƾƫƺƹƾᄙ ƹ ᇴᇲᇳᇺᄕƞǂƽƫƿƣƽƻǀƟƶƫƾƩƣƢƫƹForbes probed the issue more deeply, noting that “the Common Core State Standards initiative did provide more structure to how mathematics is being taught, and it did not, in and of itself, change the ƞƻƻƽƺƞơƩƿƺƿƣƞơƩƫƹƨƸƞƿƩƣƸƞƿƫơƾᅺᄬƽǀƨƣƽᄕᇴᇲᇳᇺᄭ ƩƺƾƣƿǂƺƤƫƹƢƫƹƨƾƞƽƣƾƿƽƫƴƫƹƨƶDŽƾƫƸƫƶƞƽƿƺƫƸƺƹᅷƾᄬᇳᇻᇻᇷᄭƺƟƾƣƽǁƞƿƫƺƹᇴᇵ years earlier, with which we take serious issue in this chapter: Although constructivism provides a useful framework for thinking about mathematics learning in classrooms and therefore can contribute in important ways to the effort to reform classroom mathematics teaching, it does not tell us how to teach mathematics; that is, it does not stipulate ƞƻƞƽƿƫơǀƶƞƽƸƺƢƣƶᄙᄬƻᄙᇳᇳᇶᄭ Our discussion here is important for two reasons. On the one hand, teachers are often held responsible for the apparent failure of ѹѹ҉҉҃. On the other hand, a constructivist approach does not change teaching or tell us how to teach mathematics. This is documented by a researcher steeped in the underlying constructivist approach to learning; the Forbes writer agrees with this somewhat surprising statement. No wonder the contemporary landscape of mathematics education in the United States is confusing. Although we have a well-established researchbased constructive framework of learning trajectories that underlies ѹѹ҉҉҃ learning, we also have classroom mathematics teachers following scripted lesƾƺƹƻƶƞƹƾƿƺƣƤƤƣơƿǀƞƿƣƻƺƾƿǀƶƞƿƣƢƿƽƞưƣơƿƺƽƫƣƾᄙƻƻƣƹƢƫǃᇳƨƫǁƣƾƞƹƣǃƞƸƻƶƣƺƤ ƿƩƣƾơƽƫƻƿƤƽƺƸƿƩƣƿƣƞơƩƣƽƾᅷƸƞƿƣƽƫƞƶƾƫƹƣǂƺƽƴƺƸƸƺƹƺƽƣǀƽƽƫơǀƶǀƸᄙ While the constructivist approach does not explicitly endorse or promote the use of scripted lessons, such lessons are nevertheless introduced in the disbelief that teachers can follow the approach on their own. The scripts tell the teacher what to ask students, what their answers might be, and even how to respond to them!ሾ ƣƶƫƞƹơƣƺƹƾơƽƫƻƿƾƸƣƞƹƾƿƩƞƿƿƣƞơƩƣƽƾƞƽƣƣǃƻƣơƿƣƢƿƺƻƽƺƸƺƿƣƾƿǀƢƣƹƿƾᅷ independent construction of relevant math concepts without having conƾƿƽǀơƿƣƢƿƩƣƸƾǀƤƤƫơƫƣƹƿƶDŽǂƣƶƶƟDŽƿƩƣƸƾƣƶǁƣƾƿƺƨǀƫƢƣƿƩƣƾƿǀƢƣƹƿᄛ ƹƽƣƞƶƫƿDŽᄕ teachers can do this only by relying on their intuition and experience integrated

Ҋ‫ؚء؜؛ؘؖؔ‬ᅟ҈‫؛ؖإؘؔئؘ‬ѷ‫ئ؜ئج؟ؔء‬

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with the relevant research knowledge, not by following scripted lesson plans. This is because these oft-repeated scripts will introduce superficial automaƿƫƺƹƫƹƿƣƞơƩƣƽƾᅷƸƫƹƢƾƹƺƿǀƹƶƫƴƣƿƩƣƞǀƿƺƸƞƿƫƺƹƺƤƩƞƽƶƫƣƩƞƻƶƫƹƫƹƿƩƣ movie Modern Times. By introducing the use of scripts, we are underestimating ƿƩƣƤƺƽơƣƺƤƩƞƟƫƿƤƺƽƸƣƢƟDŽƾǀơƩƻƽƞơƿƫơƣᄙơơƺƽƢƫƹƨƿƺƺƣƾƿƶƣƽᄕƩƞƟƫƿƫƾƞƹƿƫthetical to creativity, in particular to constructive creativity. These introductory superficial comments and quotes suggest that teachers ƞƽƣǀƾƫƹƨƾơƽƫƻƿƣƢƶƣƾƾƺƹƾƟƣơƞǀƾƣƿƩƣơƺƹƾƿƽǀơƿƫǁƫƾƿƞƻƻƽƺƞơƩƢƺƣƾƹᅷƿƾƻƣơƫƤDŽ the methodology of teaching, a view we take issue with. The five sections in this chapter explore this idea further: ᅬ ƣơƿƫƺƹᇴᄙᇳơƩƞƶƶƣƹƨƣƾƿƩƫƾƞƾƾƣƽƿƫƺƹƞƹƢƤƫƹƢƾƫƿƫƹơƺƽƽƣơƿᄙƺƹƾƿƽǀơƿƫǁƫƾƸ ƢƺƣƾƾƻƣơƫƤDŽƞƿƣƞơƩƫƹƨƸƣƿƩƺƢƺƶƺƨDŽᄘƿƣƞơƩƫƹƨᅟƽƣƾƣƞƽơƩᄬҊ҈ᄭᄙ ᅬ Section 2.2 pursues this investigation further to find out why, despite the existence of constructivist teaching methodology, Ҋ҈ is not adopted in ƾơƩƺƺƶƾƞƹƢƻƽƣƻƞƽƞƿƺƽDŽơƺƶƶƣƨƣƾᄙ ƹƾƿƣƞƢᄕƿƣƞơƩƣƽƾƞƽƣƩƞƟƫƿǀƞƿƣƢƿƺǀƾƫƹƨ scripted lessons. The answer may be linked to the hierarchical nature of our academic profession, where we observe a substantial social and professional divide between academic researchers and classroom teachers. ᅬ ƣơƿƫƺƹᇴᄙᇵƽƣƫƹƿƽƺƢǀơƣƾƿƩƣҊ҈ᄧƫƿDŽƸƺƢƣƶᄕǂƩƫơƩơƞƹơƽƺƾƾƿƩƣƢƫǁƫƢƣ and facilitate the bisociative classroom creativity of teachers and mathematical creativity of their students. ᅬ Section 2.4 explores the principle of improvement of learning of the Ҋ҈/ ƫƿDŽƸƺƢƣƶ ᅬ ƣơƿƫƺƹᇴᄙᇷƣǃƻƶƺƽƣƾƟƫƾƺơƫƞƿƫǁƣƤƣƞƿǀƽƣƾƺƤƿƩƣƿƣƞơƩƫƹƨƣǃƻƣƽƫƸƣƹƿǂƫƿƩƫƹ the Ҋ҈ᄧƫƿDŽƸƣƿƩƺƢƺƶƺƨDŽᄕǂƩƫơƩƫƾǂƣƶƶƾǀƫƿƣƢƿƺǂƫƢƣƫƸƻƶƣƸƣƹƿƞƿƫƺƹ in view of continuing communication problems between teachers and researchers of mathematics education.

2.2

The Irreducible Methodological Unity of Constructivist Research and Constructivist Teaching

2.2.1 Completing the Paradigm Shift ƾƫƞƶƞƣƿƞƶᄙᄬᇳᇻᇻᇹᄭƣǃƻƶƺƽƣƿƩƣƞƢǁƣƹƿƺƤƼǀƞƶƫƿƞƿƫǁƣƽƣƾƣƞƽơƩƫƹƸƞƿƩƣƸƞƿƫơƾ ƣƢǀơƞƿƫƺƹƞƹƢƢƣƾơƽƫƟƣƫƿᄕƤƺƶƶƺǂƫƹƨǀƩƹᄬᇳᇻᇹᇲᄭᄕƞƾƞƻƞƽƞƢƫƨƸƾƩƫƤƿƞǂƞDŽƤƽƺƸ “statistical comparison of control and experimental populations according to Ƣƣƾƫƨƹƾ ƻƽƺƻƺƾƣƢ ᄚ ƺǁƣƽ ᇸᇲ DŽƣƞƽƾ ƞƨƺ Ƥƺƽ ƿƩƣ ƻǀƽƻƺƾƣ ƺƤ Ƹƞƴƫƹƨ Ƣƣơƫƾƫƺƹƾ ƞƟƺǀƿƞƨƽƫơǀƶƿǀƽƞƶƞơƿƫǁƫƿƫƣƾᅺᄬƻᄙᇵᇺᄭᄙƩƣƻƞƽƞƢƫƨƸƾƩƫƤƿƾƿƞƽƿƣƢƿƞƴƫƹƨƻƶƞơƣ ǂƩƣƹ ƫƿ ƟƣơƞƸƣ ƺƟǁƫƺǀƾ ƿƩƞƿ ƽƺƿƣ ƶƣƞƽƹƫƹƨ ƺƤ ƞƽƫƿƩƸƣƿƫơ ƞƹƢ ƞƶƨƣƟƽƞ ƢƫƢƹᅷƿ support the development of number sense or problem-solving, which were

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ѹ‫ؔ؛ؖآءإؔح‬

increasingly required by social and political education stakeholders. Thus, a crisis developed, one that demanded an alternative paradigm that would be accepted by the educational community.

ƺƽǀƩƹᄬᇳᇻᇹᇲᄭᄕ ƻƞƽƞƢƫƨƸƫƾƿƩƣơƺƶƶƣơƿƫƺƹƺƤǀƹƢƣƽƾƿƞƹƢƫƹƨƾᄬƣǃƻƶƫơƫƿƺƽƫƸƻƶƫơƫƿᄭƺƹ the part of the individual or a group of individuals about the kinds of things one does when conducting research in a particular field, the types of questions that are to be asked, the kinds of answers that are to be expected and the kinds of methods to be used. ơơƺƽƢƫƹƨƿƺǀƩƹƿƩƣƻƞƽƞƢƫƨƸƾƩƫƤƿƾƢƺƹƺƿơƺƸƣƼǀƫơƴƶDŽƺƽƣƞƾƫƶDŽƟǀƿƿƩƣDŽ ƿƣƹƢƿƺƟƣƾƩƞƽƻᄙƩƣDŽƞƽƣơƞǀƾƣƢƟDŽǂƩƞƿǀƩƹơƞƶƶƾƞơƽƫƾƫƾƾƿƞƿƣᄕǂƩƫơƩƩƞƾ two different manifestations: ᅬ Discovery is so far-reaching that it cannot be assimilated into the present state of knowledge. ᅬ There is a developing dissatisfaction with the current paradigm that reaches a level where answers to certain question often basic to the field cannot be easily answered. ƩƣƻƞƽƞƢƫƨƸƾƩƫƤƿƿƩƞƿƿƺƺƴƻƶƞơƣƫƹƿƩƣᇳᇻᇺᇲƾƞƹƢᇳᇻᇻᇲƾǂƞƾƺƤƿƩƣƾƣơƺƹƢ type. Starting from the work of Piaget and Vygotsky that forms the constructivist approach to learning, the essential aspect of the new paradigm in mathematics education has brought a profound new understanding of how learning takes place in the mind of the student. This paradigm, which outlines two different but closely related pathways ƿƩƽƺǀƨƩƫƞƨƣƿᅷƾƢƣǁƣƶƺƻƸƣƹƿƞƶơƺƨƹƫƿƫǁƣƞƻƻƽƺƞơƩƞƹƢDŽƨƺƿƾƴDŽᅷƾƾƺơƫƺᅟơƺƹstructivist approach, has become the most influential learning approach to date. From the teaching-research point of view, it provided an entry into a powerful methodology whose aim has been a pedagogy where each educaƿƫƺƹƞƶƞơƿƫƾƞƶƾƺƞƽƣƾƣƞƽơƩƞơƿᅭƿƩƣƿƣƹƩƺǀƾƣƞơƿᄬƿƣƹƩƺǀƾƣᄕᇳᇻᇺᇷᄭᄙƣƿƿƩƣ paradigm is incomplete because it has not developed the constructivist teaching implications of the constructivist teaching experiment. 2.2.2 How Did It Happen? ƫƸƺƹᄬᇳᇻᇻᇷᄭƞƾƾƣƽƿƾƿƩƞƿ ᄴƞᄵƶƿƩƺǀƨƩ ơƺƹƾƿƽǀơƿƫǁƫƾƸ ƻƽƺǁƫƢƣƾ ƞ ǀƾƣƤǀƶ ƤƽƞƸƣǂƺƽƴ Ƥƺƽ ƿƩƫƹƴƫƹƨ about mathematics learning in classrooms and therefore can contribute in important ways to the effort to reform classroom mathematics teach-

Ҋ‫ؚء؜؛ؘؖؔ‬ᅟ҈‫؛ؖإؘؔئؘ‬ѷ‫ئ؜ئج؟ؔء‬

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ing, it does not tell us how to teach mathematics; that is, it does not stipǀƶƞƿƣƞƻƞƽƿƫơǀƶƞƽƸƺƢƣƶᄙᄬƻᄙᇳᇳᇶᄭ ƺ ƞƾƾƣƾƾ ƿƩƣ ơƺƽƽƣơƿƹƣƾƾ ƺƤ ƫƸƺƹᅷƾ ƾƿƞƿƣƸƣƹƿᄕ ǂƣ Ƹǀƾƿ ƣǃƞƸƫƹƣ Ʃƺǂ ơƺƹstructivist researchers investigate the process of classroom learning. The origin of constructive teaching experiments is found in the work of Dewey, Piaget, and Vygotsky. The traces of the methodology of the constructive teaching ƣǃƻƣƽƫƸƣƹƿ ơƞƹ Ɵƣ ƤƺǀƹƢ ƫƹ ƣǂƣDŽᅷƾ ᄬᇴᇲᇳᇳᄭ ƣǃƻƣƽƫƣƹƿƫƞƶ ƶƣƞƽƹƫƹƨᄕ ǂƩƫơƩ ƶƣƢ to later approaches, such as problem-based learning and the inquiry method. ơơƺƽƢƫƹƨƿƺ ǀƹƿƫƹƨᄬᇳᇻᇺᇵᄭᄕƿƩƣơƺƹơƣƻƿƺƤƞƿƣƞơƩƫƹƨƣǃƻƣƽƫƸƣƹƿǂƞƾƤƺƽmulated by the Vygotskian school in Russia, where “it grew out of the need to study changes occurring in mental structure under the influence of instrucƿƫƺƹᄙᅺ ơơƺƽƢƫƹƨ ƿƺ ƿƣƤƤƣ ᄬᇳᇻᇻᇳᄭ ƞƹƢ ƺƟƟ ƞƹƢ ƿƣƤƤƣ ᄬᇳᇻᇺᇵᄭᄕ ƿƩƣ ơƣƹƿƽƞƶ ƿƺƺƶ of the constructive researcher is the constructive teaching experiment, whose distinguishing feature is “that the researcher acts as a teacher.” Cobb and Steffe ᄬᇳᇻᇺᇵᄭƫƹƤƺƽƸǀƾƿƩƞƿƿƩƣƫƹƿƣƽƣƾƿƺƤƞƽƣƾƣƞƽơƩƣƽᅟƿƣƞơƩƣƽƞƾƞƽƣƾƣƞƽơƩƣƽƶƫƣƾ ƫƹᅸƫƹǁƣƾƿƫƨƞƿƫƹƨǂƩƞƿƸƫƨƩƿƨƺƺƹƫƹơƩƫƶƢƽƣƹᅷƾƩƣƞƢƾᅺƞƹƢƫƹᅸƩDŽƻƺƿƩƣƾƫDžƫƹƨ what the child might learn.” Using a teaching-experiment technique, the experimenter must “find ways and means of fostering this learning” and “based on current interpretation of ƿƩƣơƩƫƶƢᅷƾƶƞƹƨǀƞƨƣᄕƩƣƺƽƾƩƣƩƞƾƿƺƸƞƴƣƺƹᅟƿƩƣᅟƾƻƺƿƢƣơƫƾƫƺƹƾơƺƹơƣƽƹƫƹƨ situations to create and critical questions to ask, he or she acts in this capacity ƞƾƞƿƣƞơƩƣƽᅺᄬƺƟƟѵƿƣƤƤƣᄕᇳᇻᇺᇵᄭᄙƺǀƾᄕƿƩƫƾƸƣƞƹƾƿƩƞƿƞơƺƹƾƿƽǀơƿƫǁƫƾƿƿƣƞơƩing experiment constitutes a whole, or a phenomenon, in which constructivist research and constructivist teaching are inseparably connected. Hence teaching, in fact constructivist teaching, is the irreducible component of construcƿƫǁƫƾƿƽƣƾƣƞƽơƩᄖƫƿơƞƹᅷƿƟƣƞǁƺƫƢƣƢᄙƩƣơƺƹƾƿƽǀơƿƫǁƣƽƣƾƣƞƽơƩƣƽƸǀƾƿƞơƿƞƾƞ teacher to create a learning environment conducive to hypothesized learning.

ƹƺƿƩƣƽǂƺƽƢƾᄕơƺƹƾƿƽǀơƿƫǁƣơƶƞƾƾƽƺƺƸƿƣƞơƩƫƹƨƫƾƤƺƽƸǀƶƞƿƣƢƿƺƨƣƿƩƣƽǂƫƿƩ constructive research. What is missing is the medium of introducing it into mathematics classrooms at large. DžƞƽƹƺơƩƞ ᄬᇳᇻᇻᇻᄭ ƽƣƞƶƫDžƣƢ ƿƩƞƿ ƿƩƣ ƽƣƾƣƞƽơƩƣƽᅟƿƣƞơƩƣƽ ƽƣƶƞƿƫƺƹƾƩƫƻ ƻƽƞơticed during a teaching experiment can be inverted and transformed to the teacher-researcher relationship by using the very same teaching experiment as teaching-research. Consequently, the constructive teaching experiment ơƞƹƟƣǀƾƣƢƞƾƿƣƞơƩƫƹƨƸƣƿƩƺƢƺƶƺƨDŽᄙƞƿǀƽƞƶƶDŽᄕƿƩƣƿƣƞơƩƣƽᅷƾƨƺƞƶƾƢǀƽƫƹƨƿƩƣ ƿƣƞơƩƫƹƨƣǃƻƣƽƫƸƣƹƿƢƫƤƤƣƽƤƽƺƸƿƩƣƽƣƾƣƞƽơƩƣƽᅷƾƨƺƞƶƾᄙƾƞƿƣƞơƩƣƽᄕƿƩƣƿƣƞơƩer-researcher is interested in “finding means and ways to foster what a student needs to learn to reach a particular moment of discovery or to understand a ƻƞƽƿƫơǀƶƞƽƣƶƣƸƣƹƿƺƤƿƩƣƻƽƣƾơƽƫƟƣƢơǀƽƽƫơǀƶǀƸᅺᄬDžƞƽƹƺơƩƞᄕᇳᇻᇻᇻᄭᄙ

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ѹ‫ؔ؛ؖآءإؔح‬

ƾƞƽƣƾƣƞƽơƩƣƽᄕƿƩƣƿƣƞơƩƣƽᅟƽƣƾƣƞƽơƩƣƽᅷƾƫƹƿƣƽƣƾƿƫƾƨƣƹƣƽƞƿƣƢƟDŽƴƹƺǂƶƣƢƨƣ ƿƩƞƿƾƫƹơƣƾǀơƩƸƺƸƣƹƿƾƺơơǀƽƺƹƶDŽǂƫƿƩƫƹƾƿǀƢƣƹƿƾᅷƞǀƿƺƹƺƸƺǀƾơƺƨƹƫƿƫǁƣ mathematical structures, the teacher needs to investigate these structures ƢǀƽƫƹƨƞƹƫƹƾƿƽǀơƿƫƺƹƞƶƾƣƼǀƣƹơƣᄙ ƹƿƩƫƾơƞƻƞơƫƿDŽᄕƩƣƺƽƾƩƣƞơƿƾƞƾƞƽƣƾƣƞƽơƩƣƽ ᄬDžƞƽƹƺơƩƞᄕᇳᇻᇻᇻᄭᄙ ƩƣƢƫƤƤƣƽƣƹơƣƫƾƾǀƟƿƶƣDŽƣƿƻƽƺƤƺǀƹƢᄙƩƣƿƣƞơƩƣƽᅷƾƨƺƞƶƢǀƽƫƹƨƿƩƣƿƣƞơƩing experiment is not to find out what the child might learn in an experimental situation, but what the child needs to learn to understand a given curriculum concept. This distinction specifies the difference between interiorization and ƫƹƿƣƽƹƞƶƫDžƞƿƫƺƹᄖƫƿƫƾƿƩƣƽƣƤƶƣơƿƫƺƹƺƤǂƩƞƿƞƴƣƽơƞƶƶƾƫƹƩƞƻƿƣƽᇷƿƩƣƢƫƤƤƣƽence between cognitive reality of the curriculum and constructivist methodology, which, in the extreme case of radical constructivism, assumes that the only reality present is the reality of the child. The difference in goals determines the difference in the design-and-delivery conduct of the teaching experiment. The teacher must design mathematical situations to facilitate the construction of student discovery, or solving a problem on her own, leading to understanding a relevant curricular concept. At the same time, the researcher must design the teaching experiment to investigate the construction of the concept as suggested by the research interest. Both want to facilitate and investigate pertinent mental constructions but for different purposes. As teachers, both need to make on-the-spot decisions while teaching about the construction of ƶƣƞƽƹƫƹƨ ƾƫƿǀƞƿƫƺƹƾ ƞƹƢ ơƽƫƿƫơƞƶ Ƽǀƣƾƿƫƺƹƾᄙ ƺƿƩ ƿƩƣ ƽƣƾƣƞƽơƩƣƽᅟƿƣƞơƩƣƽ ᄬ҈Ҋᄭ ƞƹƢƿƩƣƿƣƞơƩƣƽᅟƽƣƾƣƞƽơƩƣƽᄬҊ҈ᄭƸǀƾƿƫƹǁƣƾƿƫƨƞƿƣƾƿǀƢƣƹƿƶƣƞƽƹƫƹƨƞƹƢƿƩǀƾᄕ based on their current understanding of student language and actions, act as researchers and teachers but with different goals. ƩƣƽƣƤƺƽƣᄕ ǂƣ ƤƫƹƢ ƫƸƺƹᅷƾ ᄬᇳᇻᇻᇷᄭ ƞƾƾƣƽƿƫƺƹ ƿƩƞƿ ơƺƹƾƿƽǀơƿƫǁƫƾƸ ᅸƢƺƣƾ ƹƺƿ tell us how to teach mathematics” not entirely true. ƺƹƾƿƽǀơƿƫǁƫƾƸ Ƣƺƣƾ ƫƹƤƺƽƸ ƿƣƞơƩƣƽƾ Ʃƺǂ ƿƺ ƿƣƞơƩ ƸƞƿƩƣƸƞƿƫơƾᄙ ƿ ƞƶƾƺ proposes that the teaching-research model can be used within the context of the constructive teaching experiment aimed at improving classroom learning.

ƸƻƽƺǁƣƸƣƹƿƺƤơƶƞƾƾƽƺƺƸƶƣƞƽƹƫƹƨᄕƫƹƿǀƽƹᄕƹƣơƣƾƾƫƿƞƿƣƾơƶƞƾƾƽƺƺƸƫƹǁƣƾƿƫƨƞtions of student thinking and their mental constructions. The results of classroom investigations influence the pedagogy of teaching in current and future classrooms. The methods of a constructive teaching experiment conducted by ƿƣƞơƩƣƽƾƞƽƣƢƣǁƣƶƺƻƣƢƫƹDžƞƽƹƺơƩƞƣƿƞƶᄙᄬᇴᇲᇳᇸᄖƞƽƿƾᇵƞƹƢᇶᄭᄙ To clarify possible misunderstandings, we want to assert at the outset that we are not suggesting that constructivism aims to guide teaching by claiming that “if you do certain things in the classroom consistent with constructivist philosophy, then such and such will happen.” On the contrary, constructivism

Ҋ‫ؚء؜؛ؘؖؔ‬ᅟ҈‫؛ؖإؘؔئؘ‬ѷ‫ئ؜ئج؟ؔء‬

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states the principle of teaching: “teach in such a way that children have the opportunity to construct/discover by themselves significant components of the mathematical curriculum we want them to learn.” This principle is sufficiently general to embrace not only teaching-research but also the variety of teaching methods, such as discovery-and-inquiry methods, problem-solving and problem-posing approaches, and collaborative ƻƽƺƟƶƣƸ ƻƺƾƫƹƨᄙ ƺƹƾƣƼǀƣƹƿƶDŽᄕ ƺǀƽ ƻƽƺƤƣƾƾƫƺƹ Ƣƺƣƾƹᅷƿ Ʃƞǁƣ ƿƺ ƫƹƢƺơƿƽƫƹƞƿƣ ƿƣƞơƩƣƽƾǂƫƿƩƞƾơƽƫƻƿƣƢƶƣƾƾƺƹᄙ ƹƾƿƣƞƢᄕƺǀƽƻƽƺƤƣƾƾƫƺƹơƞƹƞƹƢƾƩƺǀƶƢƩƣƶƻ ǀƾ ƿƺ Ƣƣǁƣƶƺƻ ƫƹƿƺ ƿƣƞơƩƣƽᅟƽƣƾƣƞƽơƩƣƽƾᄙ ǀƿ ƫƿ Ƣƺƣƾƹᅷƿᄙ Ʃƣ ƽƣƞƾƺƹƾ ǂƩDŽ ƿƩƫƾ ƢƺƣƾƹᅷƿƩƞƻƻƣƹƽǀƹƢƣƣƻᄙ ƫƶƶƫƞƸ ƞƽƽƫƹƨƿƺƹᄕƿƩƣƿƣƞơƩƣƽᅟƽƣƾƣƞƽơƩƣƽƫƹƞƹƺƣƾƿᅷƾᄬᇴᇲᇲᇸᄭơƺƶƶƣơƿƫƺƹᄕ states the following: Teachers do informal research in their classrooms all the time. We try a new lesson activity, evaluation form, seating arrangement, grouping of students, or teaching style. We assess, reflect, modify, and try again, as we consider the perceived consequences of the change we made. The general Ҋ҈ƿƺƺƶƾƞƽƣƿƩƣƽƣƞƸƺƹƨƿƣƞơƩƣƽƾᄙƩƞƿᅷƾƸƫƾƾƫƹƨƫƾƿƩƣơƺƹƾƿƽǀơtivist viewpoint. This can be easily acquired through participation in the constructive teaching experiment based on the collaboration between classroom ƿƣƞơƩƣƽƾƞƹƢƣƢǀơƞƿƫƺƹƞƶƽƣƾƣƞƽơƩƣƽƾᄙ ƹƤƞơƿᄕƾƺơƫƺᅟơƺƹƾƿƽǀơƿƫǁƫƾƿƾƺƟƟƞƹƢ ƞơƴƾƺƹᄬᇴᇲᇳᇳᄭƾǀƨƨƣƾƿƿƩƞƿƾǀơƩƿƣƞơƩƣƽᅟơƺƶƶƞƟƺƽƞƿƫǁƣƹƣƿǂƺƽƴƾƞơƼǀƫƽƣƿƩƣƫƽ depth if they are led by veteran teachers and focus on “underlying principles of the approach, the nature of mathematics, and how students learn,” which is exactly the aim of a teacher-researcher in the context of the Ҋ҈ᄧƫƿDŽƸƺƢƣƶᄙ DžƞƽƹƺơƩƞƣƿƞƶᄙᄬᇴᇲᇳᇸᄭƨƫǁƣƞƹƣǃƞƸƻƶƣƺƤƾǀơƩơƺƶƶƞƟƺƽƞƿƫƺƹƫƹơƩƞƻƿƣƽᇷᄙᇴᄙ ƺƽƣƺǁƣƽᄕƫƣƽƞƹƣƿƞƶᄙᄬᇴᇲᇳᇵᄭƢƫƾơǀƾƾƿƩƣƞǀƿƩƺƽƾᅷƞǂƞƽƣƹƣƾƾƺƤƿƩƣƫƾƾǀƣƞƹƢ ultimately turn to teaching-research without naming it as such: ƣƞơƩƣƽƾƞƽƣƽƣƨƞƽƢƣƢᄴƫƹƿƩƣơƩƞƻƿƣƽᄵƞƾƩƞǁƫƹƨƞƸƞưƺƽƽƺƶƣƫƹƿƩƣƢƣǁƣƶopment of mathematics teaching and students learning. Nevertheless, in much mathematics education research, teachers are viewed as recipients, and sometimes even as means to generate or disseminate knowledge, thus conserving a distinctive gap between research and practice. The theme of this chapter is to regard teachers as key stakeholders in ƽƣƾƣƞƽơƩᄬƫᄙƣᄙᄕƞƾᄬơƺᄭƻƽƺƢǀơƣƽƾƺƤƻƽƺƤƣƾƾƫƺƹƞƶƞƹƢᄧƺƽƾơƫƣƹƿƫƤƫơƴƹƺǂƶƣƢƨƣᄭƫƹƺƽƢƣƽƿƺƸƞƴƣƿƩƣƶƫƹƴƟƣƿǂƣƣƹƽƣƾƣƞƽơƩƞƹƢƻƽƞơƿƫơƣƸƺƽƣƤƽǀƫƿful for both sides.

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ѹ‫ؔ؛ؖآءإؔح‬

The exploration of several examples leads the authors to the presence of three important dimensions of research where teachers are key stakeholders: reflective, inquiry-based activity with respect to teaching action; a significant action-research component accompanied by the creation ƺƤ ƽƣƾƣƞƽơƩ ƞƽƿƫƤƞơƿƾ ƟDŽ ƿƩƣ ƿƣƞơƩƣƽƾ ᄬƾƺƸƣƿƫƸƣƾ ƞƾƾƫƾƿƣƢ ƟDŽ ǀƹƫǁƣƽƾƫƿDŽ ƽƣƾƣƞƽơƩƣƽƾᄭᄖƞƹƢƿƩƣƢDŽƹƞƸƫơƢǀƞƶƫƿDŽƺƤƽƣƾƣƞƽơƩƞƹƢƻƽƺƤƣƾƾƫƺƹƞƶƢƣǁƣƶopment.

ƿƾƣƣƸƾƿƩƞƿƿƩƣƽƣƫƾƞƢƫƽƣơƿƻƞƿƩǂƞDŽưƺƫƹƫƹƨƞơƺƸƸƫƿƿƣƢơƶƞƾƾƽƺƺƸƿƣƞơƩƣƽᅷƾǂƺƽƴǂƫƿƩƿƩƣơƺƹƾƿƽǀơƿƫǁƣƤƽƞƸƣǂƺƽƴᄕƣƾƻƣơƫƞƶƶDŽƫƹƿƩƣơƺƹƿƣǃƿƺƤƿƩƣơƺƹstructive teaching experiment. How and why, then, has the work of the teacher within the constructivist curriculum been reduced to contemporary script teaching—the antithesis of constructivism, leading ultimately to the collapse of the promising framework as the foundation of mathematics education?

2.3

The Obstacle Is Social, Not Pedagogical

Through its different manifestations in different countries, the history of teaching-research points to a serious social obstacle along the straightforward route from classroom teacher professional to the development of a coherent teaching methodology.

ƹᇳᇻᇹᇻᄕǂƩƫƶƣƤƺƽƸǀƶƞƿƫƹƨƿƩƣƻƽƫƹơƫƻƶƣƾƞƹƢƻƽƞơƿƫơƣƺƤƿƣƞơƩƫƹƨᅟƽƣƾƣƞƽơƩ in his inaugural lecture at the University of East Anglia, Stenhouse stated the reason for the failure of research “to contribute effectively to the growth of professional understanding and to the improvement of professional practice was the reluctance of educational researchers to engage teachers as partners in, ƞƹƢơƽƫƿƫơƾƺƤᄕƿƩƣƽƣƾƣƞƽơƩƽƣƾǀƶƿƾᅺᄬƿƣƹƩƺǀƾƣᄕᇳᇻᇺᇷᄭᄙƩƣƫƸƻƺƽƿƞƹơƣƺƤơƽƣƞƿing a respectful, constructive attitude between researcher and teacher, which recognizes the equally fundamental role of practice and research, is a necessary condition for the bisociative formation of Stenhouse acts. The reason for the “reluctance of educational researchers to engage teachers as partners” ǂƞƾƤƺƽƸǀƶƞƿƣƢƟDŽƫƿƿƸƞƹƹᄬᇳᇻᇻᇻᄭᄙƫƨƹƫƤƫơƞƹƿƶDŽᄕƿƩƺƾƣƽƣƞƾƺƹƾƞƻƻƣƞƽƣƢƫƹ ƫƣƽƻƫƹƾƴƞ ƞƹƢ ƫƶƻƞƿƽƫơƴᅷƾ ǁƺƶǀƸƣ ᄬᇳᇻᇻᇺᄭ ƿƫƿƶƣƢ Mathematics Education as a Research Domain: A Search for Identity, thus incorporating the teacher-excluƾƫƺƹƻƽƫƹơƫƻƶƣᄬƟƣƶƺǂᄭƫƹƿƺƿƩƣƫƢƣƹƿƫƿDŽƺƤƸƞƿƩƣƸƞƿƫơƾƣƢǀơƞƿƫƺƹƞƾƞƽƣƾƣƞƽơƩ ƢƺƸƞƫƹƺƤƿƩƣᇴᇳƾƿơƣƹƿǀƽDŽᄙ Wittmann focuses on the design of teaching units, the predecessors of learning trajectories, the best of which, he admits, were designed by teachers ƞƹƢƻǀƟƶƫƾƩƣƢƫƹƿƣƞơƩƫƹƨưƺǀƽƹƞƶƾᄚƞƾƾǀơƩƿƩƣDŽǂƣƽƣƩƞƽƢƶDŽƹƺƿƫơƣƢƟDŽƿƩƣ

Ҋ‫ؚء؜؛ؘؖؔ‬ᅟ҈‫؛ؖإؘؔئؘ‬ѷ‫ئ؜ئج؟ؔء‬

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research community. The reason offered for the absence of recognizing teachƣƽƾᅷǂƺƽƴƞƹƢƾǀƟưƣơƿƸƞƾƿƣƽDŽƫƾƾƫƨƹƫƤƫơƞƹƿᄘ in contrast to “research” the design of teaching units has been considered a mediocre task normally done by teachers and textbook authors. Why should anyone anxious for academic respectability stoop to designing and put himself on one level with teachers? The answer has been clear: ƩƣƺƽƾƩƣǀƾǀƞƶƶDŽǂƺǀƶƢƹᅷƿᄙ Of course, we can ask, what self-respecting teacher would be interested in ǂƺƽƴƫƹƨǂƫƿƩƞƽƣƾƣƞƽơƩƣƽǂƩƺƤƣƣƶƾƿƩƣƹƣƣƢƿƺᅸƾƿƺƺƻᅺƿƺƿƩƣƿƣƞơƩƣƽᅷƾƶƣǁƣƶƺƤ ƣǃƻƣƽƿƫƾƣᄞƩƣƞƹƾǂƣƽƩƞƾƟƣƣƹơƶƣƞƽᄘƩƣƺƽƾƩƣǀƾǀƞƶƶDŽǂƺǀƶƢƹᅷƿᄙ

ƹƿƩƣƾƻƣơƫƞƶƫƾƾǀƣƺƤMathematical Thinking and Learning devoted to teachƫƹƨ ǀƹƫƿƾ ơƞƶƶƣƢ ƩDŽƻƺƿƩƣƿƫơƞƶ ƶƣƞƽƹƫƹƨ ƿƽƞưƣơƿƺƽƫƣƾᄕ ƿƣƤƤƣ ᄬᇴᇲᇲᇶᄭ ƞƾƴƾ ǂƩƺƾƣ ƽƣƾƻƺƹƾƫƟƫƶƫƿDŽƫƿƫƾƿƺơƺƹƾƿƽǀơƿƞƶƣƞƽƹƫƹƨƿƽƞưƣơƿƺƽDŽᄙ ƾƫƿƞƿƣƞơƩƣƽᅷƾƺƽƽƣƾƣƞƽơƩƣƽᅷƾƽƣƾƻƺƹƾƫƟƫƶƫƿDŽᄞƺᄙ ƿƫƾƿƩƣƽƣƾƻƺƹƾƫƟƫƶƫƿDŽƺƤƿƣƞơƩƣƽᅟƽƣƾƣƞƽơƩƣƽƾƟƣơƞǀƾƣ they alone can integrate classroom teaching with the results of research.ሿ ƿƣƤƤƣᅷƾƿƞƴƣƺƹƿƩƣƾƣƫƾƾǀƣƾƫƾƢƫƤƤƣƽƣƹƿᄙ ƣƾƞDŽƾƿƩƞƿƟƣơƞǀƾƣƺƤƿƩƣƻƽƫƹơƫƻƶƣ of self-reflexivity of radical constructivism, his knowledge of learning trajectories is constantly changing. He is constructing knowledge as he constantly interacts with students. Therefore, rather than consider himself satisfied with his knowledge and thereby consider the construction of trajectories as the responsibility of the teachers, he takes the responsibility back to himself to ơƺᅟơƺƹƾƿƽǀơƿƿƩƣƿƽƞưƣơƿƺƽƫƣƾǂƫƿƩơƩƫƶƢƽƣƹᄬƾƿǀƢƣƹƿƾᄭᄙ To co-construct learning trajectories with students, teachers need to be teacher-researchers. Thus, instead of extending a hand to teachers for the formation of an adequate teaching-research profile, Steffe takes the research aspect of work back to himself, leaving teachers free to follow the script. He can easily do this because of the difference in status and academic respectability. Consequently, the gap between practice and research widened cognitively ƞƹƢ ƾƺơƫƞƶƶDŽ ƿƺ ƿƩƣ ƻƺƫƹƿ ǂƩƣƽƣ ƹƨƶƫƾƩ ᄬᇴᇲᇳᇲᄭᄕ ƶƫƴƣ ƿƣƹƩƺǀƾƣ ᇴᇷ DŽƣƞƽƾ ƣƞƽlier, expresses skepticism that theory-driven research can be relevant to and improve the teaching and learning of mathematics in the classroom: The elevation of theory and philosophy in mathematics education scholarship could be considered somewhat contradictory to the growing concerns for enhancing the relevance and usefulness of research in mathematics education. These concerns reflect an apparent skepticism that theory-driven research can be relevant to and improve the teaching and learning of mathematics in the classroom. Such skepticism is not ƾǀƽƻƽƫƾƫƹƨᄚơƶƞƫƸƾƿƩƞƿƿƩƣƺƽƣƿƫơƞƶơƺƹƾƫƢƣƽƞƿƫƺƹƾƩƞǁƣƶƫƸƫƿƣƢƞƻƻƶƫơƞ-

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ѹ‫ؔ؛ؖآءإؔح‬

tion in the reality of the classroom or other learning contexts have been ƹǀƸƣƽƺǀƾᄙᄬƻᄙᇸᇸᄭ Let us explore the roots of the “apparent” skepticism that theory-driven ƽƣƾƣƞƽơƩ ơƞƹ ƫƸƻƽƺǁƣ ƿƣƞơƩƫƹƨ ƞƹƢ ƶƣƞƽƹƫƹƨᄙƫƿƿƸƞƹƹ ᄬᇳᇻᇻᇻᄭ ƣǃƻƶƞƫƹƾ ǂƩDŽ teachers get disillusioned with their profession. He asserts that, the design of substantial teaching units, and particularly of substantial curricula, is a most difficult task that needs to be carried out by the experts in the field. By no means can it be left to teachers, though teachers can certainly make important contributions within a framework carried out by the experts, particularly when they are members or in close ơƺƹƹƣơƿƫƺƹǂƫƿƩƞƽƣƾƣƞƽơƩƿƣƞƸᄙᄚᄴᄵƿƣƞơƩƣƽơƞƹƟƣơƺƸƻƞƽƣƢƸƺƽƣ to a conductor than to a composer.

ƹƺƿƩƣƽǂƺƽƢƾᄕƺǀƽơƺƶƶƣƞƨǀƣǂƞƹƿƾƿƺƾƿƽƫƻƿƣƞơƩƣƽƾƺƤƿƩƣƫƽơƽƣƞƿƫǁƣƽƺƶƣƫƹ designing classroom teaching in favor of the executive role of the conductor.ቀ ǀƽơƺƶƶƣƞƨǀƣǂƞƹƿƾƿƺƾƣƻƞƽƞƿƣƿƩƣƿƣƞơƩƣƽᅷƾƫƹƿǀƫƿƫƺƹƞƹƢƴƹƺǂƶƣƢƨƣƤƽƺƸ the design of units they are supposed to teach. How could someone not have a serious dose of skepticism about the effectiveness of research into learning ƫƹƸƞƿƩƣƸƞƿƫơƾơƶƞƾƾƽƺƺƸƾƫƤƿƩƣƽƣƾƣƞƽơƩƢƺƣƾƹᅷƿƿƞƴƣƿƩƣơƶƞƾƾƽƺƺƸƞƾƿƩƣ variable that primarily only teachers really know? The comparison of a teacher to the conductor of an orchestra conjures the image of teaching based on a ƾơƽƫƻƿᄬƺƤƽƣƞƢDŽƸǀƾƫơᄭᄖƫƿƩƫƨƩƶƫƨƩƿƾƿƩƣƽƣƞƶƫƿDŽƿƩƞƿƿƣƞơƩƣƽƾᅷơƽƣƞƿƫǁƫƿDŽᄕǂƩƫơƩ in the past was their professional domain, has been taken away from them and transformed into the scripted tool of information. Wittmann is correct in identifying the main obstacle along the straightforward route from classroom teacher professional to the development of a ơƺƹƾƿƽǀơƿƫǁƫƾƿ ƿƣƞơƩƫƹƨ ƸƣƿƩƺƢƺƶƺƨDŽᄙ ƿ ƫƾ ƾƺơƫƞƶ ƾƿƞƿǀƾ ƺƽ ƞơƞƢƣƸƫơ ƽƣƾƻƣơƿƞƟƫƶƫƿDŽᄙƩƣƹƞơƞƢƣƸƫơƿƩƣƺƽƫƣƾƞƽƣƽƞƹƴƣƢƩƫƨƩƣƽƿƩƞƹƿƣƞơƩƣƽƾᅷƣǃƻƣƽƫƣƹơƣƾᄕ this hierarchy makes instruction difficult because of an inherent opposition, whether this is directly, as in the case of Wittmann, or indirectly, as in the case of Steffe. With the difference in social status comes the drive to strip teachers of the creative/investigative aspect of the design of teaching units, reducing teachers to followers of lesson scripts.

2.4

Ҋ҈/NYCity Model and the Improvement of Classroom Learning

This section is a report from the work of the mathematics Teaching-Research ƣƞƸᄬҊ҈ƣƞƸᄭƺƤƿƩƣƽƺƹǃᄕǂƩƫơƩƽƣơƣƹƿƶDŽƻǀƟƶƫƾƩƣƢƞƾǀƸƸƞƽDŽƺƤƫƿƾǂƺƽƴ

Ҋ‫ؚء؜؛ؘؖؔ‬ᅟ҈‫؛ؖإؘؔئؘ‬ѷ‫ئ؜ئج؟ؔء‬

ሪር

ƺǁƣƽƿƩƣƶƞƾƿᇳᇷናDŽƣƞƽƾƺƤƫƿƾƣǃƫƾƿƣƹơƣᄬDžƞƽƹƺơƩƞƣƿƞƶᄙᄕᇴᇲᇳᇸᄭᄙƩƣƫƽƸƣƿƩƺƢƺƶƺƨDŽᄕƿƩƣƣƞơƩƫƹƨᅟƣƾƣƞƽơƩƣǂƺƽƴƫƿDŽᄬҊ҈ᄧƫƿDŽᄭƸƺƢƣƶᄕƩƞƾƟƣƣƹƢƣǁƣƶoped through the careful composition of ideas centered on action research ᄬƣǂƫƹᄕᇳᇻᇶᇸᄭᄕǂƫƿƩƿƩƣƫƢƣƞƾơƣƹƿƣƽƣƢƺƹƿƩƣơƺƹơƣƻƿƺƤƿƩƣƿƣƞơƩƫƹƨƣǃƻƣƽiment of the Vygotskian school in Russia, where it “grew out of the need to study changes occurring in mental structures under the influence of instrucƿƫƺƹᅺᄬ ǀƹƿƫƹƨᄕᇳᇻᇺᇵᄭᄙ We take its focus on the improvement of classroom practice by the classroom teacher and its cyclical instruction/analysis methodology from action research, and we take the idea of the large-scale experimental design based on ƞƿƩƣƺƽDŽƺƤƶƣƞƽƹƫƹƨƞƹƢƫƹǁƺƶǁƫƹƨƸƞƹDŽƾƫƿƣƾᅭƢƫƤƤƣƽƣƹƿơƶƞƾƾƽƺƺƸƾᄬDžƞƽƹƺơƩƞᄕᇳᇻᇻᇻᄖDžƞƽƹƺơƩƞѵƽƞƟƩǀᄕᇴᇲᇲᇸᄭƤƽƺƸDŽƨƺƿƾƴDŽᅷƾƿƣƞơƩƫƹƨƣǃƻƣƽƫƸƣƹƿᄙ DŽƨƺƿƾƴDŽᅷƾƿƣƞơƩƫƹƨƣǃƻƣƽƫƸƣƹƿƸƣƿƩƺƢƺƶƺƨDŽƫƹƿƽƺƢǀơƣƢƿƩƣƻƺƾƾƫƟƫƶƫƿDŽƺƤ viewing the classroom teacher as a member of a collaborative research team investigating the usefulness of research-based classroom integration. The action-research perspective allows us to see the same teacher as an individual teacher-researcher who, simultaneously with teaching, investigates student ƶƣƞƽƹƫƹƨᄬᅸơƩƞƹƨƣƾƺơơǀƽƽƫƹƨƫƹƸƣƹƿƞƶƾƿƽǀơƿǀƽƣƾᅺᄭǀƹƢƣƽƿƩƣƫƸƻƞơƿƺƤƩƫƾƺƽ her teachings. The result of this methodology is insight into the relationships between ƿƣƞơƩƫƹƨƞƹƢƶƣƞƽƹƫƹƨᄙ ƺǂƣǁƣƽᄕƣǂƫƹᅷƾᄬᇳᇻᇶᇸᄭƞơƿƫƺƹƽƣƾƣƞƽơƩƾǀƻƻƶƫƣƾƺƹƣ more essential quality: the quest for improvement of learning and teaching. The Ҋ҈ᄧƫƿDŽƸƺƢƣƶƫƾƿƩƣƻƽƺơƣƾƾƺƤƫƹǁƣƾƿƫƨƞƿƫƹƨƾƿǀƢƣƹƿƶƣƞƽƹƫƹƨƾƫƸǀƶtaneously with teaching by the classroom teacher with the aim of improving learning in the very same classroom, and beyond. One of its central ideas in that definition is the improvement of learning in the classroom because it is that quality that governs the dynamics and inner workings of the methodology. The “improvement” demands that each teaching experiment has at least two cycles. The interphase of two cycles, meaning the time after the analysis of data and results and before planning the next edition of the intervention, is the space where teachers need to think and reflect on how to improve either the methodology or the results. ƩƣƿƣƞơƩƣƽƾᅷƣƤƤƺƽƿƾƫƹƿƩƫƾƫƹƿƣƽƻƩƞƾƣƹƺƢƣƞƽƣƟƞƾƣƢƺƹƿƩƣƽƣơƺƨƹƫƿƫƺƹƺƤ and the constructivist belief that students need to construct or discover certain aspects of mathematics themselves. This is when teachers can become creative through the bisociative frame created between the framework of the first cycle and the framework of the desired outcome in the next. A series of such cycles of improvement in the design of the creative learning environment ơƞƹƟƣƾƣƣƹƤƽƺƸƿƩƣǂƺƽƴƺƤƽƞƟƩǀᄬᇴᇲᇳᇸᄖƞƽƿᇴᄭᄙ ƞƿǀƽƞƶƶDŽᄕƫƿᅷƾƣƼǀƞƶƶDŽƣƞƾDŽƿƺơƺƹƾƫƢƣƽƸƺƽƣƿƩƞƹƿǂƺƾǀơƩơDŽơƶƣƾƤƺơǀƾƣƢ ƺƹ ƿƩƣ ƫƸƻƽƺǁƣƸƣƹƿ ƺƤ ƞ ƿƣƞơƩƣƽᅷƾ ƞƽƿƫƤƞơƿᄖ Ƥƺƽ ƣǃƞƸƻƶƣᄕ ƿƩƞƿ ƺƤ ƞ ƶƣƞƽƹƫƹƨ

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ѹ‫ؔ؛ؖآءإؔح‬

trajectory. Thus, the “improvement” imposes the structure of iteration on the teaching experiment. The structure of iteration is the point of contact between the teaching experiment as the methodology of teaching and the teaching experiment as the instrument of research, especially when research investiƨƞƿƣƾ ƶƣƞƽƹƫƹƨ ƿƽƞưƣơƿƺƽƫƣƾ ᄬDžƞƽƹƺơƩƞᄕ ᇴᇲᇳᇵᄭᄙ ƩƽƺǀƨƩ ƿƩƣ ƫƿƣƽƞƿƫƺƹ ƺƤ ƿƩƣ ƞƽƿƫƤƞơƿ ᄬƞ ƶƣƞƽƹƫƹƨ ƿƽƞưƣơƿƺƽDŽᄕ ƞ ƿƣƞơƩƫƹƨ ǀƹƫƿᄕ ƞ ƸƣƿƩƺƢƺƶƺƨDŽᄭ ƫƹ ƢƫƤƤƣƽƣƹƿ classrooms with consecutive improvements, it becomes generalized through ƻƽƞơƿƫơƣᄬƽƞƟƩǀᄕᇴᇲᇳᇸᄭᄙ The “improvement” of learning impacts the conduct of a single cycle as well. The teaching experiment, as woven into the curriculum of the class, is composed of teaching-research episodes focused on understanding and improving relevant mathematical concepts in different parts of the curriculum. There are several methods of assessing the success of the classroom intervention; what ǂƣᅷƽƣƫƹƿƣƽƣƾƿƣƢƫƹƫƾƿƩƣƞƾƾƣƾƾƸƣƹƿƺƤơƺƹơƣƻƿǀƞƶǀƹƢƣƽƾƿƞƹƢƫƹƨᄙ ƩƣƽƣƞƽƣơƺƹơƣƻƿǀƞƶƿƣƾƿƾƿƩƞƿơƩƣơƴƣƶƣƸƣƹƿƾƺƤǀƹƢƣƽƾƿƞƹƢƫƹƨᄬƶƺǀƹƫᄕ ᇴᇲᇳᇺᄭƞƹƢƻƽƺƻƣƽƶDŽƢƣƾƫƨƹƣƢƾƣƿƾƺƤƻƽƺƟƶƣƸƾᄬƤƺƽƣǃƞƸƻƶƣᄕ ƣƽƾƩƴƺǂƫƿDžƣƿƞƶᄙᄕ ᇴᇲᇳᇹᄭᄙ ƺǂƣǁƣƽᄕƫƤǂƣƞƽƣƫƹƿƣƽƣƾƿƣƢƫƹƿƩƣƻƽƺơƣƾƾƿƩƽƺǀƨƩǂƩƫơƩƿƩƣơƺƹơƣƻƿƫƾ misunderstood or not perceived at all, then we must investigate student thinking-in-action during the class and address the misunderstood issue. We can do this by designing a problem or question concerning the issue that will reveal student thinking. That is how we obtain information about the state of student understanding. Having gained that information, we need to design questions, hints, or prompts that will clarify the issue, followed by a set of problems to ᅸƸƞƴƣƫƿƾƿƫơƴᄙᅺ ƿƫƾƻƽƣơƫƾƣƶDŽƞƿƿƩƫƾưǀƹơƿǀƽƣǂƩƣƽƣƿƩƣơƩƺƫơƣƺƤƞƻƣƢƞƨƺƨƫơƞƶ strategy takes place within the given student population. We can tell students what the correct approach should be, or we can leave it to students to grasp it themselves with the help of our facilitation techniques. As teacher-researchers in the South Bronx community, we have found that ᅸƿƣƶƶƫƹƨᅺƢƺƣƾƹᅷƿǂƺƽƴᄙƩƞƿᅷƾǂƩDŽǂƣƤƞǁƺƽƿƩƣƾƣơƺƹƢƞƻƻƽƺƞơƩᄕƹƺƿƹƣơƣƾƾƞƽƫƶDŽƟƣơƞǀƾƣǂƣƶƫƴƣƿƩƣơƺƹƾƿƽǀơƿƫǁƣƻƩƫƶƺƾƺƻƩDŽƟǀƿƟƣơƞǀƾƣǂƣᅷƽƣơƺƸƸƫƿƿƣƢ ƿƺƿƩƣƫƸƻƽƺǁƣƸƣƹƿƺƤƺǀƽƾƿǀƢƣƹƿƾᅷƶƣƞƽƹƫƹƨᄙ ƿƫƾƿƩƣƼǀƞƶƫƿDŽƺƤƫƸƻƽƺǁƣƸƣƹƿ in the definition of teaching-research that naturally leads Ҋ҈ᄧƫƿDŽƿƺǂƞƽƢƞ constructivist approach. Finally, we have one more step: to find a way in which individually constructed knowledge becomes shared classroom knowledge, which in turn brings us into ƿƩƣƢƺƸƞƫƹƺƤƿƩƣƾƺơƫƺᅟơƺƹƾƿƽǀơƿƫǁƫƾƿƞƹƢƾƺơƫƺơǀƶƿǀƽƞƶƞƻƻƽƺƞơƩƣƾᄬƩƞƻƿƣƽƾᇳƞƹƢᇷᄭᄙƣƾƣƣƿƩƞƿƫƸƻƽƺǁƣƸƣƹƿǂƫƿƩƫƹƿƩƣƿƣƞơƩƫƹƨᅟƽƣƾƣƞƽơƩƸƣƿƩƺƢology inherited from one of its roots, action research, can integrate-in-action the radical constructivist approach with the socio-constructivist one.

Ҋ‫ؚء؜؛ؘؖؔ‬ᅟ҈‫؛ؖإؘؔئؘ‬ѷ‫ئ؜ئج؟ؔء‬

2.5

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Teaching Experiment, Bisociation, and the Stenhouse Acts

2.5.1 Bisociative Quality of the њј/NYCity Model A significant broadening and deepening of the Ҋ҈ᄧƫƿDŽƸƣƿƩƺƢƺƶƺƨDŽơƞƸƣ recently with the realization that it also has the quality of bisociative creativƫƿDŽᄘƫƿƫƾƞƟƫƾƺơƫƞƿƫǁƣƤƽƞƸƣᄬƩƞƻƿƣƽᇳᄭᄙƫƾƺơƫƞƿƫǁƫƿDŽᄕƿƩƣƿƩƣƺƽDŽƺƤƿƩƣƩƞᄛ ƸƺƸƣƹƿƤƺƽƸǀƶƞƿƣƢƟDŽƺƣƾƿƶƣƽƫƹThe Act of CreationᄬᇳᇻᇸᇶᄭᄕǂƞƾƫƹƿƽƺƢǀơƣƢ ƫƹƿƺƸƞƿƩƣƸƞƿƫơƾơƶƞƾƾƽƺƺƸƾƟDŽƽƞƟƩǀᄬᇴᇲᇳᇸᄭᄙƫƹơƣƿƩƣƹƫƿƩƞƾƟƣƣƹǂƫƢƣƶDŽ investigated by the Ҋ҈ƣƞƸƺƤƿƩƣƽƺƹǃᄬƞƾƢƺơǀƸƣƹƿƣƢƫƹƿƩƫƾǁƺƶǀƸƣᄭᄙƩƣ ơƺƹƿƣƹƿƺƤƿƩƣƫƹƾƫƨƩƿƫƾƿƩƣᅸƩƫƢƢƣƹƞƹƞƶƺƨDŽᅺᄬƺƣƾƿƶƣƽᄕᇳᇻᇸᇶᄕƻᄙᇵᇴᇲᄭƿƩƞƿƫƾ ƢƫƾơƺǁƣƽƣƢᄬƺƽơƺƹƾƿƽǀơƿƣƢᄕƺƽƫƸƞƨƫƹƣƢᄭǂƩƫƶƣƿƩƣƫƹƾƫƨƩƿƿƞƴƣƾƻƶƞơƣᄙ The bisociative frame has been defined as two disconnected frames of reference with a high possibility that “hidden analogies” exist between them. The bisociative frame is a structure within which the creation of an Aha! moment manifests itself. Considering the observation that teaching and research have usually been separate worlds, discourses, or methodologies, both focused on the same process of learning, there must be within them quite a few “hidden ᄴƞƾDŽƣƿᄵƞƹƞƶƺƨƫƣƾᄕᅺǂƩƫơƩƞƽƣƿƩƣƾǀƟưƣơƿƺƤƿƣƞơƩƫƹƨᅟƽƣƾƣƞƽơƩƫƹƨƣƹƣƽƞƶƞƹƢ in particular of the Ҋ҈ᄧƫƿDŽ ƸƺƢƣƶ ǂƫƿƩ ƫƿƾ ƩƫƨƩƶDŽ ơƽƣƞƿƫǁƣ ƸƣƿƩƺƢƺƶƺƨDŽ ᄬDžƞƽƹƺơƩƞᄕᇴᇲᇳᇸᄭƟƣơƞǀƾƣƺƤƫƿƾƟƞƶƞƹơƣƢƫƹƿƣƨƽƞƿƫƺƹƺƤƽƣƾƣƞƽơƩƞƹƢƻƽƞơƿƫơƣᄙ Ʃƣ Ƣƣƾƫƽƣ Ƥƺƽ ƿƩƣ ƫƸƻƽƺǁƣƸƣƹƿ ƺƤ ƶƣƞƽƹƫƹƨ ƶƣƢ ƽƞƟƩǀ ᄬᇴᇲᇳᇸᄭ ƿƺ ƿƩƣ Ƥƫƽƾƿ ƫƹơƺƽƻƺƽƞƿƫƺƹƺƤƺƣƾƿƶƣƽᅷƾƟƫƾƺơƫƞƿƫǁƫƿDŽƫƹƿƺƸƞƿƩƣƸƞƿƫơƾơƶƞƾƾƽƺƺƸƾƢƣƾơƽƫƟƣƢ in detail in the previous volume, The Creative Enterprise of Mathematics Teaching ResearchᄬDžƞƽƹƺơƩƞƣƿƞƶᄙᄕᇴᇲᇳᇸᄭᄙ ƢǀơƞƿƫƺƹƞƶƽƣƾƣƞƽơƩƞƹƢơƶƞƾƾƽƺƺƸƿƣƞơƩƫƹƨ are two domains of educational activity separated from each other. Because of the common educational aspects of that activity, many have hidden analogies.

ƿƿƞƴƣƾƞơƽƣƞƿƫǁƣƣƤƤƺƽƿƿƺƸƣƽƨƣƿƩƣƿǂƺᄙ Teaching-research is therefore a bisociative framework from which we can expect many creative Aha! moment insights while discovering hidden analogies between educational research and classroom practice. These insights reveal hidden moments of unity between teaching practice and educational research, as ƣƹǁƫƾƫƺƹƣƢƟDŽƿƣƹƩƺǀƾƣᄬᇳᇻᇺᇷᄭᄙƩƣƿƣƹƩƺǀƾƣƞơƿƫƾƞƹᅸƞơƿᄴǂƩƫơƩƫƾᄵƞƿƺƹơƣƞƹ ƣƢǀơƞƿƫƺƹƞƶƞơƿƞƹƢƞƽƣƾƣƞƽơƩƞơƿᅺᄬƻᄙᇷᇹᄭᄙƩƣƾƣƸƺƸƣƹƿƾƺƤǀƹƫƿDŽƞƽƣƟƣơƺƸƫƹƨ the foundation of a successful Aha! pedagogy—one of the aims of this book. Classroom teaching experiments have many variants. For example, teachers can design a new approach to learning, introduce it into the classroom during the academic year or semester, and then assess the results at the end. The assessment in such a case might be quantitative, qualitative, or both. ConƤƽƣDŽ ᄬᇴᇲᇲᇲᄭ ƫƹƿƽƺƢǀơƣƢ ƿƩƣ ơƺƹưƣơƿǀƽƣᅟƢƽƫǁƣƹ ƿƣƞơƩƫƹƨ ƣǃƻƣƽƫƸƣƹƿ ᄬѹѺҊѻᄭᄕ

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ѹ‫ؔ؛ؖآءإؔح‬

ǂƩƫơƩƞƶƶƺǂƣƢƩƣƽƿƺƫƹƿƽƺƢǀơƣơƩƞƹƨƣƾƢǀƽƫƹƨƿƩƣƣǃƻƣƽƫƸƣƹƿᄕǂƩƫƶƣƣƶƶDŽ ƞƹƢƣƾƩᄬᇴᇲᇲᇲᄭƫƹƿƽƺƢǀơƣƢƿƩƣƿƫƣƽƣƢᅟƿƣƞơƩƫƹƨƣǃƻƣƽƫƸƣƹƿơƺƸƻƺƾƣƢƺƤƿǂƺ or three teaching experiments within one experimental setup. One teaching experiment addressed student learning; the second one addressed teachers conducting the teaching experiment. We can also conduct a shorter teaching ƣǃƻƣƽƫƸƣƹƿƢǀƽƫƹƨƺƹƣƺƽƾƣǁƣƽƞƶƻƣƽƫƺƢƾƺƤƿƩƣơƺǀƽƾƣᄙ ƹƨƣƹƣƽƞƶᄕƿƣƞơƩƣƽƾ conduct a teaching experiment whenever they introduce a change in teaching methods, organization, or methods of motivating students. However, when we are interested in a teaching experiment that will allow us to form Stenhouse acts involving explicitly bisociative teaching, we need to choose methods that can link both the teaching process and the research into student learning while it takes place. Among different methods of teaching, we have identified four approaches or perspectives that may create the conditions for the formation of the Stenhouse acts: the guided-discovery method of teaching, the problem-solving approach, careful process of scaffolding student ǀƹƢƣƽƾƿƞƹƢƫƹƨᄕƞƹƢƿƣƞơƩƫƹƨᅟƽƣƾƣƞƽơƩƫƹƿƣƽǁƫƣǂƾᄬ ǀơƩƾѵDžƞƽƹƺơƩƞᄕᇴᇲᇳᇸᄭᄙ Each of these approaches allows for both processes: teaching through facilitation and at the same time researching student thinking in those circumstances. Each approach can be placed in the context of a constructive teaching experiment because each of them allows for authentic student construction. The guiding idea of the instructor practicing any of the methods is to make sure that students always have enough space: the wait time within the discourse to be able to express their authentic thinking. That concern is often expressed in the design of the pedagogical situation ƟDŽƢƣƾơƽƫƻƿƫƺƹƾƺƤƿƩƣƿƣƞơƩƣƽᅷƾƽƺƶƣƞƾƞƨǀƫƢƣƽƞƿƩƣƽƿƩƞƹƫƹƤƺƽƸƣƽƺƤƿƩƣƾƺƶǀtion to the problem through the well-posed question or the hint. The descripƿƫƺƹƾƺƤƻƽƺƟƶƣƸᅟƾƺƶǁƫƹƨƿƣƞơƩƫƹƨƣǃƻƣƽƫƸƣƹƿƾƟDŽƞƽƹƣƾᄬᇴᇲᇲᇲᄭᄕƺƺƹᄬᇴᇲᇳᇴᄭᄕ ƺƽ ƞƶƞƿƹƫƴ ƞƹƢ ƺƫơƩǀ ᄬᇴᇲᇳᇶᄭ ǂƩƫơƩ ƶƣƢ ƿƺ Ʃƞᄛ ƸƺƸƣƹƿ ƫƹƾƫƨƩƿƾ ƫƹơƶǀƢƣ assurance that the problems were solved by students themselves with the instructor providing no more than observations, hints, or guiding comments. Similarly, the scaffolding approaches that produce an Aha! moment insight reveal the same approach of hints and leading questions where the final sucơƣƾƾƺƤǀƹƢƣƽƾƿƞƹƢƫƹƨơƺƸƣƾƢƫƽƣơƿƶDŽƤƽƺƸƾƿǀƢƣƹƿƾᄙƩƞƻƿƣƽƾᇵƞƹƢᇷơƺƹƿƫƹǀƣ the discussion of these topics. 2.5.2

Simultaneous Acts of Teaching and Research Conducted by the Teacher-Researcher These simultaneous acts of teaching and research are somewhat different in ƣƞơƩơƞƾƣᄙ ƹƣƞơƩᄕƩƺǂƣǁƣƽᄕƞƽƣƹƣǂƤƣƞƿǀƽƣƾƞƹƢơƩƞƽƞơƿƣƽƫƾƿƫơƾƺƤƞƹƩƞᄛ moment to be investigated. Note that each problem that produced Aha!

Ҋ‫ؚء؜؛ؘؖؔ‬ᅟ҈‫؛ؖإؘؔئؘ‬ѷ‫ئ؜ئج؟ؔء‬

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moment insights designed for collaborative student work was situated within a possible gap in knowledge between the reference frames. ƞƽƹƣƾᅷƾƻƽƺƟƶƣƸƾǂƣƽƣƢƣƾƫƨƹƣƢƿƺƣƹƨƞƨƣƟƺƿƩƿƩƣƽƣƤƣƽƣƹơƣƤƽƞƸƣƺƤƢƫƤferentiation and the frame of integration, two essentially different discourses theoretically joined by the fundamental theorem of calculus, whose concepƿǀƞƶǀƹƢƣƽƾƿƞƹƢƫƹƨƫƾƸƣƞƨƣƽƞƸƺƹƨƤƫƽƾƿᅟDŽƣƞƽơƞƶơǀƶǀƾƾƿǀƢƣƹƿƾᄙƺƺƹᅷƾƻƽƺƟlems were designed to engage the framework of a graph of a function and a graph of the derivative of the function, two quite different concepts joined by ƿƩƣƢƣƽƫǁƞƿƫǁƣƼǀƺƿƫƣƹƿᄙƩƣƻƽƺƟƶƣƸᅷƾƨƺƞƶǂƞƾƿƺơƽƣƞƿƣƞƟƽƫƢƨƣƟƣƿǂƣƣƹƿƩƣ two by formulating the analytic condition for existence of the maxima and minima of the function, of which the students had only an intuitive sense. We see that the first research question the teacher-researcher encounters is, What types of problems can engage student understanding across these two frames of reference, generally separated in student minds? A somewhat similar inquiry is undertaken by the instructors/mentors who facilitate Aha! moments during a ƾơƞƤƤƺƶƢƫƹƨƢƫƞƶƺƨǀƣǂƫƿƩƾƿǀƢƣƹƿƾᄕƞƾƫƹƿƩƣƣǃƞƸƻƶƣƾƾƩƺǂƹƫƹƩƞƻƿƣƽᇳᇹᄙ 2.5.3 The Dialogue Domain The instructor facilitating understanding of the domain of the square root function addressed a similar question. His goal was to facilitate the change ƺƤǀƹƢƣƽƾƿƞƹƢƫƹƨƺƤƿƩƣƾƿǀƢƣƹƿƤƽƺƸǃኟᇲƞƾƿƩƣơƩƞƽƞơƿƣƽƫƾƿƫơƾƺƤƿƩƣƾƻƣơƫƤƫơ Ƥǀƹơƿƫƺƹ ƿƺ ƿƩƞƿ ƺƤ ǃ ኟ ኔᇵᄙ  ƾƻƣơƫƞƶ Ƥƣƞƿǀƽƣ ƺƤ ƿƩƫƾ Ƣƫƞƶƺƨǀƣ ƫƾ ƿƩƣ ǀƾƣ of suggestions and hints leading to the cognitive conflict between original student anticipation and the result of well-chosen questions/hints. As Simon ƣƿƞƶᄙᄬᇴᇲᇲᇶᄭƾǀƨƨƣƾƿᄕƩƺǂƣǁƣƽᄕƿƩƣơƺƨƹƫƿƫǁƣơƺƹƤƶƫơƿƢƺƣƾƹƺƿƩƞǁƣƿƺƶƣƞƢƿƺ ǀƹƢƣƽƾƿƞƹƢƫƹƨᄙƺƽƣƺǁƣƽᄕƿƣƤƤƣƞƹƢᅷƸƟƽƺƾƫƺᄬᇳᇻᇻᇷᄭƾǀƨƨƣƾƿƿƩƞƿƣǁƣƹƫƤƿƩƣ student recognizes a contradiction within the cognitive conflict, there is no guarantee that the accommodation will be in the desired direction. ƣƿᅷƾƽƣơƞƶƶƿƩƣƻƞƽƿƺƤƿƩƣƢƫƞƶƺƨǀƣƺƤƿƩƣƢƺƸƞƫƹƺƤƿƩƣƾƼǀƞƽƣƽƺƺƿƤǀƹơtion f ( x ) = x + ᇵ Aha! moment for illustration: ᄘ ᄘ ᄘ ᄘ ᄘ

ᄬᇸᄭᅸ ƺǂƞƟƺǀƿxኙኔᇵᄞᅺ ᄬᇹᄭƿǀƢƣƹƿᄕƞƤƿƣƽƞƸƫƹǀƿƣƺƤƿƩƺǀƨƩƿᄘᄬ7ᄭᅸ ƿǂƺƽƴƾƩƣƽƣᄙᅺ ᄬᇺᄭᅸ ƺǂƞƟƺǀƿxኙኔᇴᄞᅺ ᄬᇻᄭᅸ ƿǂƺƽƴƾƩƣƽƣƿƺƺᄙᅺ ᄬ ᇳᇲᄭƸƺƸƣƹƿƶƞƿƣƽƿƩƣƾƿǀƢƣƹƿƞƢƢƾᄘᅸƩƺƾƣxᅷƾǂƩƫơƩƞƽƣƾƸƞƶƶƣƽ ƿƩƞƹኔᇵơƞƹᅷƿƟƣǀƾƣƢƩƣƽƣᄙᅺ

ƩƞƿǂƞƾƿƩƣƾƫƿǀƞƿƫƺƹƫƹƿƩƣƢƫƞƶƺƨǀƣƞƤƿƣƽƶƫƹƣᄬᇹᄭᄖƿƩƣƾƿǀƢƣƹƿƹƺƿƫơƣƢƿƩƣ ơƺƹƿƽƞƢƫơƿƫƺƹƟǀƿƢƫƢƹᅷƿƽƣƾƺƶǁƣƫƿᄙ ƿƫƾƫƸƻƺƽƿƞƹƿƿƺƽƣƸƣƸƟƣƽƿƩƞƿƿƩƣƨƺƞƶƺƤ

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ѹ‫ؔ؛ؖآءإؔح‬

the teacher-researcher was to transform understanding from “x must be posiƿƫǁƣᄕǃኟᇲᅺƿƺᅸǃƸǀƾƿƟƣƶƞƽƨƣƽƿƩƞƹƺƽƣƼǀƞƶƿƺኔᇵᄕǃኟኔᇵᄙᅺ ƹƢƣƣƢᄕƿƩƣơƺƨƹƫƿƫǁƣ conflict did not on its own create the needed accommodation. Note, however, that the first insight, the beginning of the needed accomƸƺƢƞƿƫƺƹᄕ ơƺƸƣƾ ƞƤƿƣƽ Ƽǀƣƾƿƫƺƹ ᄬᇺᄭ ƤƺƶƶƺǂƣƢ ƟDŽ ƿƩƣ ƞƹƾǂƣƽ ƿƺ ƫƿ ƫƹ ƶƫƹƣ ᄬᇻᄭᄙ That means that only one more piece of information, the minimum possible on ƿƩƣƻƽƺƻƣƽƶDŽơƩƺƾƣƹƹǀƸƣƽƫơƞƶơƺƹƿƣǃƿᄕǂƞƾƹƣơƣƾƾƞƽDŽƿƺƟƽƫƹƨƿƺƿƩƣƶƣƞƽƹƣƽᅷƾ attention so that she constructs needed accommodation through the moment of insight. We might say, quite correctly, that the structure of the dialogue initiated by the teacher-researcher was of such a nature that it required just one more piece of well-chosen information.

ƹƺǀƽƺƻƫƹƫƺƹᄕƿƩƞƿǂƞƾƟƣơƞǀƾƣƿƩƣƫƹƤƺƽƸƞƿƫƺƹƞƾƴƣƢƤƺƽƫƹƼǀƣƾƿƫƺƹᄬᇺᄭᄕ ƞƤƿƣƽƿƩƣƾƿǀƢƣƹƿƟƣơƞƸƣƞǂƞƽƣƺƤƿƩƣơƺƨƹƫƿƫǁƣơƺƹƤƶƫơƿƫƹƶƫƹƣᄬᇹᄭᄕƶƣƢƿƺᅸƻƺƾƫƿƫǁƣᄕᅺƿƩƣƾƞƸƣƞƹƾǂƣƽƿƺƼǀƣƾƿƫƺƹᄬᇸᄭᄙ ƣƹơƣᄕƿƩƣƾƫƸƫƶƞƽƫƿDŽƺƤƿƩƣƾƫƿǀƞƿƫƺƹ ǂƞƾ ƣƾƿƞƟƶƫƾƩƣƢᄙ ƿ ƾƻƞƹƹƣƢ ƿǂƺ ǀƹơƺƹƹƣơƿƣƢ ƤƽƞƸƣƾ ƺƤ ƽƣƤƣƽƣƹơƣᄕ ƿƩƣ ƺƹƣ before and the one after the student was aware of the cognitive conflict. The ƩƞᄛƸƺƸƣƹƿƨƣƹƣƽƞƶƫƹƾƫƨƩƿᄘᅸƩƺƾƣǃᅷƾǂƩƫơƩƞƽƣƾƸƞƶƶƣƽƿƩƞƹኔᇵơƞƹᅷƿƟƣ used here” was built, in our opinion, on that precisely hidden analogy. This analysis provides a precise example of integrating the teaching craft with the research knowledge within a small teaching experiment. The notion of cognitive conflict guiding the initial steps of the teacher-researcher comes ƺƤơƺǀƽƾƣƤƽƺƸƫƞƨƣƿᅷƾƿƩƣƺƽDŽƞƹƢƽƣƾƣƞƽơƩᄙƩƣƺƽƨƞƹƫDžƞƿƫƺƹƺƤƿƩƣƢƫƞƶƺƨǀƣ ƿƺƫƿƾƾǀơơƣƾƾƤǀƶƞơơƺƸƸƺƢƞƿƫƺƹǂƞƾƟƞƾƣƢƾƺƶƣƶDŽƺƹƿƩƣƿƣƞơƩƣƽᅷƾơƽƞƤƿᄙ ƩƣƤǀƶƶƢƫƞƶƺƨǀƣᄕƞƸƞƹƫƤƣƾƿƞƿƫƺƹƺƤƿƣơƩƹƺƶƺƨDŽƺƤƿƩƫƹƴƫƹƨᄬƽƞƟƩǀᄕᇴᇲᇳᇸᄭᄕ where codes of the teaching matrix integrate with codes of the research discourse is “the educational act which is at once the research act” as well, that is, the Stenhouse act. The success of dialogue as a constructive teaching methodology was possible due to the precise synthesis of research and teaching knowledge across two domains of knowledge which, unfortunately, rarely have much in common.

2.6

Conclusions

Through a short discussion of the history of cycles in US mathematics education, we have introduced a fundamental difficulty in instituting the first national curriculum in mathematics. The curriculum is based on investigations into learning trajectories of ơƺƹơƣƻƿ ƢƣǁƣƶƺƻƸƣƹƿ ƻƽƺƻƺƾƣƢ ƟDŽ ƫƸƺƹ ᄬᇴᇲᇲᇶᄭ ƞƹƢ ƫƹƾƻƫƽƣƢ ƟDŽ ǁƞƽƫƺǀƾ constructivist researchers. Although a constructive approach through learning trajectories is sound, in our judgment as teacher-researchers, classroom

Ҋ‫ؚء؜؛ؘؖؔ‬ᅟ҈‫؛ؖإؘؔئؘ‬ѷ‫ئ؜ئج؟ؔء‬

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implementation is incomplete and requires support by scripted lesson plans. The contradiction between the soundness of approach aimed at facilitating ƾƿǀƢƣƹƿƾᅷƞǀƿƩƣƹƿƫơƿƩƺǀƨƩƿƞƹƢƞơƿƫƺƹƾƫƹƢƣǁƣƶƺƻƫƹƨƿƩƣƫƽǀƹƢƣƽƾƿƞƹƢƫƹƨƺƤ ƸƞƿƩƣƸƞƿƫơƾƞƹƢƻƽƺƟƶƣƸᅟƾƺƶǁƫƹƨƞƹƢƿƩƣƞǀƿƺƸƞƿƫDžƞƿƫƺƹƺƤƿƣƞơƩƣƽƾᅷƿƩƫƹƴing through prescribed scripted lesson plans has been justified by the stateƸƣƹƿƾƿƩƞƿᅸơƺƹƾƿƽǀơƿƫǁƫƾƸᄚƢƺƣƾƹᅷƿƿƣƶƶǀƾƩƺǂƿƺƿƣƞơƩƸƞƿƩƣƸƞƿƫơƾᄙᅺ The difficulties of implementation arise from an absence of recognition of the teaching methodology inherent in the constructivist approach, teaching-research. We took issue with that justification and demonstrated that constructivism does in fact tell us how to teach in the same terms with which it ƻƽƺǁƫƢƣƾᅸƿƩƣᄴƽƣƾƣƞƽơƩᄵƤƽƞƸƣǂƺƽƴƤƺƽƿƩƫƹƴƫƹƨƞƟƺǀƿƸƞƿƩƣƸƞƿƫơƾƶƣƞƽƹƫƹƨᄙᅺ

ƹƞơƺƹƾƿƽǀơƿƫǁƫƾƿƿƣƞơƩƫƹƨƣǃƻƣƽƫƸƣƹƿᄕƫƤƿƩƣƽƣƾƣƞƽơƩƣƽƞơƿƾƞƾƞƿƣƞơƩƣƽᄕƿƩƣ activity constitutes constructivist teaching. Our investigations suggest that the issue is neither a lack of teacher knowledge nor the absence of information on how to teach constructively. Rather, the real issue is the hierarchic divide between academic researchers and classroom teachers. We therefore propose a teaching model consistent with the constructive ƞƻƻƽƺƞơƩᄘ ƿƩƣ ƿƣƞơƩƫƹƨᅟƽƣƾƣƞƽơƩ ᄬҊ҈ᄭ ƸƣƿƩƺƢƺƶƺƨDŽᄙ ƫƴƣ ƞơƿƫƺƹ ƽƣƾƣƞƽơƩᄕ Ҋ҈ also disregards the epistemological question of whether students discover or construct their own mathematical reality. To complete the argument, we have ƻƽƺǁƫƢƣƢƣǃƞƸƻƶƣƾƺƤƟƫƾƺơƫƞƿƫǁƣƾDŽƹƿƩƣƾƫƾƺƤƿƩƣƿƣƞơƩƣƽᅷƾƽƣƾƣƞƽơƩƞƹƢƻƽƺfessional craft knowledge in the context of a constructive teaching experiment.

Notes ᇳ ƣƞƽƣƞǂƞƽƣƿƩƞƿƿƩƣǀƾƣƺƤƾơƽƫƻƿƣƢƶƣƾƾƺƹƾǀƾƣƢƟDŽƿƣƞơƩƣƽƾƫƹƿƩƣƫƽơƶƞƾƾƽƺƺƸƩƞƾƩƞƢ mixed reactions among colleagues. Some of us see this practice as a positive element, especially in terms of professional development. Others, like ourselves, see the practice as leading to habit development, which closes off the possibility of originality in the practice of ƿƩƣƸƣƿƩƺƢƺƶƺƨDŽᄙƩƫƶƣƿƩƣƩƞƟƫƿƫƾᅸƿƩƣƩƣƫƽƿƺƺƽƫƨƫƹƞƶƫƿDŽᄕᅺƞƾƺƣƾƿƶƣƽƺƟƾƣƽǁƣƾᄕƫƿᅷƾƞƶƾƺ ƫƿƾƢƣƞƢƶDŽƣƹƣƸDŽᄙƣƿᅷƾƽƣơƞƶƶƿƩƞƿᅸᄴƿᄵƩƣơƽƣƞƿƫǁƣƞơƿᄚƫƾƞƹƞơƿƺƤƶƫƟƣƽƞƿƫƺƹᅭƿƩƣƢƣƤƣƞƿƺƤ ƩƞƟƫƿƟDŽƺƽƫƨƫƹƞƶƫƿDŽᅺᄬƺƣƾƿƶƣƽᄕᇳᇻᇸᇶᄕƻᄙᇻᇸᄭᄙ ᇴ ǀƶƶƼǀƺƿƣƤƽƺƸƿƣƫƣƽᄬᇳᇻᇻᇷᄭᄘ But, who is “the teacher,” and whose responsibility is it to construct learning trajectoƽƫƣƾᄞ ƹƽƞƢƫơƞƶơƺƹƾƿƽǀơƿƫǁƫƾƸᄕƿƩƣƻƽƫƹơƫƻƶƣƺƤƾƣƶƤᅟƽƣƤƶƣǃƫǁƫƿDŽơƺƸƻƣƶƾƸƣƿƺơƺƹƾƫƢƣƽ ƸDŽƺǂƹƴƹƺǂƶƣƢƨƣƞƾơƺƹƾƿƞƹƿƶDŽƟƣƫƹƨơƺƹƾƿƽǀơƿƣƢƞƾ ƫƹƿƣƽƞơƿǂƫƿƩƾƿǀƢƣƹƿƾƞƾƿƩƣDŽ construct knowledge. So, rather than consider my own knowledge of how children learn mathematics as “good enough,” and thereby consider the construction of learning trajecƿƺƽƫƣƾƞƾƿƩƣƽƣƾƻƺƹƾƫƟƫƶƫƿDŽƺƤƻƽƞơƿƫơƫƹƨƿƣƞơƩƣƽƾᄕ ơƺƹƾƫƢƣƽƿƩƣơƺƹƾƿƽǀơƿƫƺƹƺƤƶƣƞƽƹƫƹƨ trajectories in the context of “the idea of worlds being constructed, or even autonomously ƫƹǁƣƹƿƣƢᄕ ƟDŽ ƫƹƼǀƫƽƣƽƾ ǂƩƺ ƞƽƣ ƾƫƸǀƶƿƞƹƣƺǀƾƶDŽ ƻƞƽƿƫơƫƻƞƹƿƾ ƫƹ ƿƩƺƾƣ ƾƞƸƣ ǂƺƽƶƢƾᄙᅺ ᄬƞƾ ơƫƿƣƢƫƹƿƣƤƤƣᄕᇴᇲᇲᇶᄕƻᄙᇳᇵᇲᄭ

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ѹ‫ؔ؛ؖآءإؔح‬

ᇵ ƹƹƺǂƞDŽƢƺǂƣƢƫƸƫƹƫƾƩƿƩƣǁƞƶǀƣƞƹƢơƽƣƞƿƫǁƫƿDŽƺƤƿƩƣƺƽơƩƣƾƿƽƞơƺƹƢǀơƿƺƽᄙƣƞƽƣƸƣƽƣƶDŽ ƻƺƫƹƿƫƹƨ ƿƺ ƫƿƿƸƞƹƹᅷƾ Ƣƫƾƿƫƹơƿƫƺƹ Ɵƣƿǂƣƣƹ ƿƩƣ ơƽƣƞƿƫǁƫƿDŽ ƺƤ ƿƩƣ ơƺƸƻƺƾƣƽᄕ ǂƩƺ ơƽƣƞƿƣƾ something new, and the work of the conductor, who interprets that new piece. Using the musical analogy, the teaching-researcher then is most aptly compared with the jazz musician who simultaneously creates music during the improvisation while playing according to the rules of music.

References ƶƺǀƹƫᄕᄙᄬᇴᇲᇳᇺᄭᄙƽƞơƿƫơƣƢƺƣƾƹƺƿƞƶǂƞDŽƾƸƞƴƣƾDŽƺǀƻƣƽƤƣơƿᄙMathematics Teaching Research Journal Online, 10ᄬᇳᄭᄕᇶᅬᇳᇷᄙ ƾƫƞƶƞᄕᄙᄕƽƺǂƹᄕᄙᄕƣƽƫƣƾᄕᄙᄙᄕǀƟƫƹƾƴDŽᄕ ᄙᄕƞƿƩƣǂƾᄕᄙᄕѵƩƺƸƞƾᄕᄙᄬᇳᇻᇻᇹᄭᄙ framework for research and curriculum development in undergraduate mathematics education. MAA Notes, 44ᄕᇵᇹᅬᇷᇶᄙ ƞƽƹƣƾᄕᄙᄬᇴᇲᇲᇲᄭᄙƞƨƫơƸƺƸƣƹƿƾƫƹƸƞƿƩƣƸƞƿƫơƾᄘ ƹƾƫƨƩƿƾƫƹƿƺƿƩƣƻƽƺơƣƾƾƺƤơƺƸing to know. For the Learning of Mathematics, 20ᄬᇳᄭᄕᇵᇵᅬᇶᇵᄙ ƺƟƟᄕᄙᄕѵƞơƴƾƺƹᄕᄙᄬᇴᇲᇳᇳᄭᄙƺǂƞƽƢƾƞƹƣƸƻƫƽƫơƞƶƶDŽƨƽƺǀƹƢƣƢƿƩƣƺƽDŽƺƤƞơƿƫƺƹƤƺƽ improving the quality of mathematics teaching at scale. Mathematics Teacher Education and Development, 13ᄬᇳᄭᄕᇸᅬᇵᇵᄙ ƺƟƟᄕ ᄙᄕ ѵ ƿƣƤƤƣᄕ ᄙ ᄙ ᄬᇳᇻᇺᇵᄭᄙ Ʃƣ ơƺƹƾƿƽǀơƿƫǁƫƾƿ ƽƣƾƣƞƽơƩƣƽ ƞƾ ƿƣƞơƩƣƽ ƞƹƢ ƸƺƢƣƶ builder. Journal for Research in Mathematics Education, 14ᄬᇴᄭᄕᇺᇵᅬᇻᇶᄙ ƺƹƤƽƣDŽᄕᄙᄕѵƞơƩƞƹơƣᄕᄙᄬᇴᇲᇲᇲᄭᄙƽƞƹƾƤƺƽƸƞƿƫǁƣƿƣƞơƩƫƹƨƣǃƻƣƽƫƸƣƹƿƾƿƩƽƺǀƨƩơƺƹưƣᅟ ơƿǀƽƣᅟƢƽƫǁƣƹƽƣƾƣƞƽơƩƢƣƾƫƨƹᄙ ƹᄙ ᄙƣƶƶDŽѵᄙᄙƣƾƩᄬ Ƣƾᄙᄭᄕ Handbook of research design in mathematics and science educationᄬƻƻᄙᇴᇵᇳᅬᇴᇸᇸᄭᄙƞǂƽƣƹơƣ ƽƶƟƞǀƸƾƾƺciates. DžƞƽƹƺơƩƞᄕ ᄙ ᄬᇳᇻᇻᇻᄭᄙ ƶ Ƹƞƣƾƿƽƺ ơƺƹƾƿƽǀơƿƫǁƫƾƿƞ ơƺƸƺ ƫƹǁƣƾƿƫƨƞƢƺƽᄙ ʖƸƺ ƣƹƾƣʎƞƽ razones y proporciones a adolescentes. Educación Matemática, 11ᄬᇴᄭᄕᇷᇴᅬᇸᇵᄙ DžƞƽƹƺơƩƞᄕᄙᄬᇴᇲᇳᇵᄭᄙ ƿƣƽƞƿƫƺƹƞƾƿƩƣơƺƸƸƺƹƸƣƿƩƺƢƺƶƺƨƫơƞƶƿƺƺƶƤƺƽƞơƞƢƣƸƫơƞƹƢ ơƶƞƾƾƽƺƺƸƽƣƾƣƞƽơƩᄙ ƹᄙᄙƫƹƢƸƣƫƣƽѵᄙ ƣƫƹDžƣᄬ ƢƾᄙᄭᄕMathematics learning across the life span: Proceedings of the 37th Conference of the International Group for the Psychology of Mathematics Education (PME 37), Vol. 5 ᄬƻᄙᇴᇳᇻᄭᄙƣƫƟƹƫDž ƹƾƿƫƿǀƿƣ for Science and Mathematics Education. DžƞƽƹƺơƩƞᄕ ᄙ ᄬᇴᇲᇳᇸᄭᄙ ƣƞơƩƫƹƨᅟƽƣƾƣƞƽơƩ ƣǂ ƺƽƴ ƫƿDŽ ƺƢƣƶ ᄬᄧ ƫƿDŽᄭᄙ ƹ ᄙ DžƞƽƹƺơƩƞᄕᄙƞƴƣƽᄕᄙƫƞƾᄕѵᄙƽƞƟƩǀᄬ ƢƾᄙᄭᄕThe creative enterprise of mathematics teaching research: Elements of methodology and practice—From teachers to teachersᄬƻƻᄙᇵᅬᇴᇳᄭᄙƣƹƾƣǀƟƶƫƾƩƣƽƾᄙ DžƞƽƹƺơƩƞᄕᄙᄕƞƴƣƽᄕᄙᄕƫƞƾᄕᄙᄕѵƽƞƟƩǀᄕᄙᄬ ƢƾᄙᄭᄙᄬᇴᇲᇳᇸᄭᄙThe creative enterprise of mathematics teaching research: Elements of methodology and practice—From teachers to teachers. Sense Publishers. DžƞƽƹƺơƩƞᄕ ᄙᄕ ѵ ƽƞƟƩǀᄕ ᄙ ᄬᇴᇲᇲᇸᄭᄙ ƣƞơƩƫƹƨᅟƽƣƾƣƞƽơƩ ƫƿDŽ ƸƺƢƣƶᄙ Dydaktyka Matematyki, 29ᄕᇴᇷᇳᅬᇴᇹᇴᄙ

Ҋ‫ؚء؜؛ؘؖؔ‬ᅟ҈‫؛ؖإؘؔئؘ‬ѷ‫ئ؜ئج؟ؔء‬

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ƣǂƣDŽᄕᄙᄬᇴᇲᇳᇳᄭᄙDemocracy and education. Simon and Brown. ƹƨƶƫƾƩᄕᄙᄬᇴᇲᇳᇲᄭᄙƩƣƺƽƣƿƫơƞƶᄕơƺƹơƣƻƿǀƞƶᄕƞƹƢƻƩƫƶƺƾƺƻƩƫơƞƶƤƺǀƹƢƞƿƫƺƹƾƤƺƽƸƞƿƩƣƸƞƿƫơƾƣƢǀơƞƿƫƺƹƽƣƾƣƞƽơƩᄘƫƸƣƶƣƾƾƹƣơƣƾƾƫƿƫƣƾᄙ ƹᄙƽƫƽƞƸƞƹѵᄙ ƹƨƶƫƾƩᄬ Ƣƾᄙᄭᄕ Theories of mathematics education: Seeking new frontiersᄬƻƻᄙᇸᇷᅬᇸᇸᄭᄙƻƽƫƹƨƣƽᄙ

ǀơƩƾᄕ ᄙᄕѵDžƞƽƹƺơƩƞᄕᄙᄬᇴᇲᇳᇸᄭᄙƣƞơƩƫƹƨƽƣƾƣƞƽơƩƫƹƿƣƽǁƫƣǂƾᄙ ƹᄙDžƞƽƹƺơƩƞᄕᄙ ƞƴƣƽᄕᄙƫƞƾᄕѵᄙƽƞƟƩǀᄬ ƢƾᄙᄭᄕThe creative enterprise of mathematics teaching research: Elements of methodology and practice—From teachers to teachers ᄬƻƻᄙ ᇳᇹᇻᅬᇳᇻᇹᄭᄙƣƹƾƣǀƟƶƫƾƩƣƽƾᄙ ƣƽƾƩƴƺǂƫƿDžᄕ ᄙᄕ ƞƟƞơƩᄕ ᄙᄕ ѵ ƽƣDŽƤǀƾᄕ ᄙ ᄬᇴᇲᇳᇹᄭᄙ ƽƣƞƿƫǁƣ ƽƣƞƾƺƹƫƹƨ ƞƹƢ ƾƩƫƤƿƾ ƺƤ knowledge in the mathematics classroom. ZDM: Mathematics Education, 49ᄬᇳᄭᄕ ᇴᇷᅬᇵᇸᄙ ǀƹƿƫƹƨᄕᄙᄙᄬᇳᇻᇺᇵᄭᄙ ƸƣƽƨƫƹƨƸƣƿƩƺƢƺƶƺƨƫƣƾƤƺƽǀƹƢƣƽƾƿƞƹƢƫƹƨƫƹƿƣƽƹƞƶƻƽƺơƣƾƾƣƾ ƨƺǁƣƽƹƫƹƨơƩƫƶƢƽƣƹᅷƾƸƞƿƩƣƸƞƿƫơƞƶƟƣƩƞǁƫƺǀƽᄙAustralian Journal of Education, 27ᄬᇳᄭᄕ ᇶᇷᅬᇸᇳᄙ ƣƶƶDŽᄕᄙ ᄙᄕѵƣƾƩᄕᄙᄙᄬ Ƣƾᄙᄭᄙᄬᇴᇲᇲᇲᄭᄙ Handbook of research design in mathematics and science education. Lawrence Erlbaum Associates. ƫƣƽƞƹᄕᄙᄕƽƞƫƹƣƽᄕᄙᄕѵƩƞǀƨƩƹƣƾƾDŽᄕᄙᄙᄬᇴᇲᇳᇵᄭᄙƫƹƴƫƹƨƽƣƾƣƞƽơƩƿƺƻƽƞơƿƫơƣᄘƣƞơƩƣƽƾƞƾƴƣDŽƾƿƞƴƣƩƺƶƢƣƽƾƫƹƸƞƿƩƣƸƞƿƫơƾƣƢǀơƞƿƫƺƹƽƣƾƣƞƽơƩᄙ ƹᄙᄙƶƣƸƣƹƿƾƣƿ ƞƶᄙ ᄬ Ƣƾᄙᄭᄕ Third international handbook of mathematics education ᄬƻƻᄙ ᇵᇸᇳᅬᇵᇻᇴᄭᄙ Springer. ƺƣƾƿƶƣƽᄕᄙᄬᇳᇻᇸᇶᄭᄙThe act of creation. Macmillan. ƽǀƨƣƽᄕᄙᄬᇴᇲᇳᇺᄕƣƻƿƣƸƟƣƽᇷᄭᄙƩDŽƢƫƢƿƩƣƞƻƻƽƺƞơƩƿƺƿƣƞơƩƫƹƨƸƞƿƩơƩƞƹƨƣǂƫƿƩ Common Core? ForbesᄙƩƿƿƻƾᄘᄧᄧǂǂǂᄙƤƺƽƟƣƾᄙơƺƸᄧƾƫƿƣƾᄧƼǀƺƽƞᄧᇴᇲᇳᇺᄧᇲᇻᄧᇲᇷᄧǂƩDŽᅟƢƫƢᅟ ƿƩƣᅟƞƻƻƽƺƞơƩᅟƿƺᅟƿƣƞơƩƫƹƨᅟƸƞƿƩᅟơƩƞƹƨƣᅟǂƫƿƩᅟơƺƸƸƺƹᅟơƺƽƣᄧᄞƾƩኙᇷᇷᇸƣᇷᇴᇲᇺᇻƤƤᇴ ǀƩƹᄕᄙᄬᇳᇻᇹᇲᄭᄙThe structure of scientific revolutions. University of Chicago Press. ƣǂƫƹᄕᄙᄬᇳᇻᇶᇸᄭᄙơƿƫƺƹƽƣƾƣƞƽơƩƞƹƢƸƫƹƺƽƫƿDŽƻƽƺƟƶƣƸƾᄙJournal of Social Issues, 2ᄬᇶᄭᄕ ᇵᇶᅬᇶᇸᄙ ơƶǀƾƴƣDŽᄕᄙᄬᇴᇲᇳᇺᄕơƿƺƟƣƽᇳᇺᄭᄙƺƸƸƺƹƺƽƣƢƺƣƾƹᅷƿƾƣƣƸƿƺƟƣǂƺƽƴƫƹƨᄘƩƞƿƸƞDŽ be just fine. ForbesᄙƩƿƿƻƾᄘᄧᄧǂǂǂᄙƤƺƽƟƣƾᄙơƺƸᄧƾƫƿƣƾᄧƹƣƞƶƸơơƶǀƾƴƣDŽᄧᇴᇲᇳᇺᄧᇳᇲᄧᇳᇺᄧơƺƸƸƺƹᅟơƺƽƣᅟƢƺƣƾƹƿᅟƾƣƣƸᅟƿƺᅟƟƣᅟǂƺƽƴƫƹƨᅟƿƩƞƿᅟƸƞDŽᅟƟƣᅟưǀƾƿᅟƤƫƹƣᄧᇱᇷƟơᇶᇻơᇲᇺᇸᇻᇻᇵ ƞƶƞƿƹƫƴᄕᄙᄕѵƺƫơƩǀᄕᄙᄬᇴᇲᇳᇶᄭᄙƣơƺƹƾƿƽǀơƿƫƺƹƺƤƺƹƣƸƞƿƩƣƸƞƿƫơƞƶƫƹǁƣƹƿƫƺƹᄘ ƺơǀƾ ƺƹƾƿƽǀơƿǀƽƣƾƺƤƞƿƿƣƹƿƫƺƹᄙ ƹᄙƫƶưƣƢƞƩƶᄕᄙƣƾƿƣƽƶƣᄕᄙƫơƺƶᄕѵᄙƶƶƞƹᄬ Ƣƾᄙᄭᄕ Proceedings of the 38th Conference of the International Group for the Psychology of Mathematics Education and the 36th Conference of the North American Chapter of the Psychology of Mathematics Education, Vol. 4 ᄬƻƻᄙᇵᇹᇹᅬᇵᇺᇶᄭᄙ ᄙ https://www.pmena.org/proceedings/ ƽƞƟƩǀᄕᄙ ᄬᇴᇲᇳᇸᄭᄙƩƣ ơƽƣƞƿƫǁƣ ƶƣƞƽƹƫƹƨ ƣƹǁƫƽƺƹƸƣƹƿᄙ ƹ ᄙ DžƞƽƹƺơƩƞᄕᄙ ƞƴƣƽᄕ ᄙ ƫƞƾᄕѵᄙƽƞƟƩǀᄬ ƢƾᄙᄭᄕThe creative enterprise of mathematics teaching research: Elements of methodology and practice—From teachers to teachers ᄬƻƻᄙ ᇳᇲᇹᅬᇳᇴᇸᄭᄙ Sense Publishers.

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ѹ‫ؔ؛ؖآءإؔح‬

ƫƣƽƻƫƹƾƴƞᄕ ᄙᄕ ѵ ƫƶƻƞƿƽƫơƴᄕ ᄙ ᄬ Ƣƾᄙᄭᄙ ᄬᇳᇻᇻᇺᄭᄙ Mathematics education as a research domain: A search for identity. Springer. ƫƸƺƹᄕᄙᄬᇳᇻᇻᇷᄭᄙƣơƺƹƾƿƽǀơƿƫƹƨƸƞƿƩƣƸƞƿƫơƞƶƻƣƢƞƨƺƨDŽƤƽƺƸƿƩƣơƺƹƾƿƽǀơƿƫǁƣƻƣƽspective. Journal for Research in Mathematics Education, 26ᄬᇴᄭᄕᇳᇳᇶᅬᇳᇶᇷᄙ ƫƸƺƹᄕᄙᄬᇴᇲᇲᇶᄭᄙ ǃƻƶƫơƞƿƫƹƨƞƸƣơƩƞƹƫƾƸƤƺƽơƺƹơƣƻƿǀƞƶƶƣƞƽƹƫƹƨᄘ ƶƞƟƺƽƞƿƫƹƨƿƩƣ construct of reflective abstraction. JRME, 35ᄬᇷᄭᄕᇵᇳᇷᅬᇵᇴᇻᄙ ƽƫƽƞƸƞƹᄕᄙᄕѵ ƹƨƶƫƾƩᄙᄬᇴᇲᇳᇲᄭᄙǀƽǁƣDŽƫƹƨƿƩƣƺƽƫƣƾƞƹƢƻƩƫƶƺƾƺƻƩƫƣƾƺƤƸƞƿƩƣƸƞƿƫơƾ ƣƢǀơƞƿƫƺƹᄙ ƹᄙƽƫƽƞƸƞƹѵᄙ ƹƨƶƫƾƩᄬ ƢƾᄙᄭᄕTheories of mathematics education: Seeking new frontiersᄬƻƻᄙᇹᅬᇵᇴᄭᄙƻƽƫƹƨƣƽᄙ ƿƣƤƤƣᄕᄙᄙᄬᇳᇻᇻᇳᄭᄙƩƣơƺƹƾƿƽǀơƿƫǁƫƾƿƿƣƞơƩƫƹƨƣǃƻƣƽƫƸƣƹƿᄘ ƶƶǀƾƿƽƞƿƫƺƹƾƞƹƢƫƸƻƶƫơƞƿƫƺƹƾᄙ ƹ ᄙǁƺƹƶƞƾƣƽƾƤƣƶƢᄬ ƢᄙᄭᄕRadical constructivism in mathematics education ᄬƻƻᄙᇳᇹᇹᅬᇳᇻᇶᄭᄙƶǀǂƣƽᄙ ƿƣƤƤƣᄕᄙᄙᄬᇴᇲᇲᇶᄭᄙƹƿƩƣơƺƹƾƿƽǀơƿƫƺƹƺƤƶƣƞƽƹƫƹƨƿƽƞưƣơƿƺƽƫƣƾƺƤơƩƫƶƢƽƣƹᄘƩƣơƞƾƣƺƤ commensurate fractions. Mathematical Thinking and Learning, 6ᄬᇴᄭᄕᇳᇴᇻᅬᇳᇸᇴᄙ ƿƣƤƤƣᄕ ᄙ ᄙᄕ ѵ ᅷƸƟƽƺƾƫƺᄕ ᄙ ᄙ ᄬᇳᇻᇻᇷᄭᄙ ƺǂƞƽƢ ƞ ǂƺƽƴƫƹƨ ƸƺƢƣƶ ƺƤ ơƺƹƾƿƽǀơƿƫǁƫƾƿ teaching: A reaction to Simon. Journal for Research in Mathematics Education, 26ᄬᇴᄭᄕ ᇳᇶᇸᅬᇳᇷᇻᄙ ƿƣƤƤƣᄕᄙᄙᄕѵƫƣƨƣƶᄕ ᄙᄙᄬᇳᇻᇻᇴᄭᄙƹƽƣƤƺƽƸƫƹƨƻƽƞơƿƫơƣƫƹƸƞƿƩƣƸƞƿƫơƾƣƢǀơƞƿƫƺƹᄙ Educational Studies in Mathematics, 23ᄬᇷᄭᄕᇶᇶᇷᅬᇶᇸᇷᄙ ƿƣƫƣƽᄕ ᄙᄬᇳᇻᇻᇷᄭᄙ ƽƺƸǀƹƫǁƣƽƾƫƹƨƿƺơƺƹǁƣƽƾƫƹƨᄘƹƣơƺƶƺƨƫơƞƶơƺƹƾƿƽǀơƿƫƺƹƫƾƿƞƻƻƽƺƞơƩ ƿƺƶƣƞƽƹƫƹƨƞƹƢƸǀƶƿƫƻƶƣƢƣƾơƽƫƻƿƫƺƹᄙ ƹᄙᄙƿƣƤƤƣѵᄙƞƶƣᄬ ƢƾᄙᄭᄕConstructivism in educationᄬƻƻᄙᇸᇹᅬᇺᇶᄭᄙƞǂƽƣƹơƣ ƽƶƟƞǀƸƾƾƺơƫƞƿƣƾᄙ ƿƣƹƩƺǀƾƣᄕᄙᄬᇳᇻᇺᇷᄭᄙResearch as the basis for teaching: Readings from the work of Lawrence Stenhouse ᄬᄙǀƢƢǀơƴѵᄙ ƺƻƴƫƹƾᄕ Ƣƾᄙᄭᄙ ƣƫƹƣƸƞƹƹ Ƣǀơƞƿƫƺƹƞƶƺƺƴƾᄙ ƞƹƺƣƾƿᄕᄙᄙᄬ ƢᄙᄭᄙᄬᇴᇲᇲᇸᄭᄙTeachers engaged in research: Inquiry into mathematics classrooms, Grades 9–12ᄙ ᅟ ƹƤƺƽƸƞƿƫƺƹƨƣᄙ ƫƿƿƸƞƹƹᄕ ᄙᄙᄬᇳᇻᇻᇺᄭᄙƞƿƩƣƸƞƿƫơƾƣƢǀơƞƿƫƺƹƞƾƞᅸƢƣƾƫƨƹƾơƫƣƹơƣᄙᅺ ƹᄙƫƣƽƻƫƹƾƴƞ ѵᄙƫƶƻƞƿƽƫơƴᄬ ƢƾᄙᄭᄕMathematics education as a research domain: A search for identityᄬƻƻᄙᇺᇹᅬᇳᇲᇵᄭᄙƻƽƫƹƨƣƽᄙ ƺƺƹᄕᄙᄬᇴᇲᇳᇴᄭᄙƞƻƻƫƹƨƸƞƿƩƣƸƞƿƫơƞƶƶƣƞƻƾƺƤƫƹƾƫƨƩƿᄙ ƹᄙƩƺᄬ ƢᄙᄭᄕSelected regular lectures from the 12th International Congress on Mathematical Education ᄬƻƻᄙ ᇻᇳᇷᅬᇻᇵᇴᄭᄙƻƽƫƹƨƣƽᄙƢƺƫᄘᇳᇲᄙᇳᇲᇲᇹᄧᇻᇹᇺᅟᇵᅟᇵᇳᇻᅟᇳᇹᇳᇺᇹᅟᇸᇇᇷᇳ

Appendix 1 Scripted Lessons ƣƾƾƺƹᇳ Ask students to discern a pattern from their calculations, form a conjecture, and work to justify their conjecture. They should find that a negative value raised to an even exponent results in a positive value since the product of two negative values

Ҋ‫ؚء؜؛ؘؖؔ‬ᅟ҈‫؛ؖإؘؔئؘ‬ѷ‫ئ؜ئج؟ؔء‬

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yields a positive product. They should also find that having an even number of negative factors means each factor pairs with another, resulting in a set of positive products. Likewise, they should conclude that a negative number raised to an odd exponent always results in a negative value. Example 1 of scripting the lessonᄴƻᄙᇳᇸᄵ ǃƣƽơƫƾƣᇳᇳ Will these products be positive or negative? How do you know? ᄬኔ1ᄭኗᄬኔ1ᄭኗᄚኗᄬኔ1ᄭኙᄬኔᇳᄭሾሿᄬኔᇳᄭƸǀƶƿƫƻƶƫƣƢƟDŽƫƿƾƣƶƤᇳᇴƿƫƸƣƾ This product will be positive. Students may state that they computed the product and it was positive. If they say that, let them show their work. Students may say that the answer is positive because the exponent is positive; however, this would not be acceptable in view of the next example. ᄬኔ1ᄭኗᄬኔ1ᄭኗᄚኗᄬኔ1ᄭኙᄬኔᇳᄭሾቀᄬኔᇳᄭƸǀƶƿƫƻƶƫƣƢƟDŽƫƿƾƣƶƤᇳᇵƿƫƸƣƾ This product will be negative. Students may state that they computed the product and it was negative. If so, ask them to show their work. Based on the discussion of the last problem, you may need to point out that a positive exponent does not always result in a positive product. Example 2 of scripting the lesson ᄴƻᄙᇴᇲᄵ ƽƫƿƣƞƹƣǃƻƽƣƾƾƫƺƹǂƫƿƩᄬኔ1ᄭƞƾƫƿƾƟƞƾƣƿƩƞƿǂƫƶƶƻƽƺƢǀơƣƞƻƺƾƫƿƫǁƣƻƽƺƢǀơƿᄕ and explain why your answer is valid. Accept any answer with ᄬኔ1ᄭ to an exponent that is even. ᇵᄙƽƫƿƣƞƹƣǃƻƽƣƾƾƫƺƹǂƫƿƩᄬኔ1ᄭƞƾƫƿƾƟƞƾƣƿƩƞƿǂƫƶƶƻƽƺƢǀơƣƞƹƣƨƞƿƫǁƣƻƽƺƢuct, and explain why your answer is valid. Accept any answer withᄬኔ1ᄭ to an exponent that is odd. ᄬƺǀƽơƣᄘ ǀƽƣƴƞƞƿƩѵ ƹƨƞƨƣ߂ߍᄕᇴᇲᇳᇷᄕƻƻᄙᇳᇸᄕᇴᇲᄭᄙ Comments: Scripting reveals that it is addressed to teachers who have insufficient understanding of relevant mathematics or of elementary pedagogical Ƹƺǁƣƾᄙ ƿƫƹƾƿƽǀơƿƾǂƩƞƿƾƿǀƢƣƹƿƞƹƾǂƣƽƾƞƽƣơƺƽƽƣơƿƺƽƹƺƿᄕǂƩƞƿƿƺƢƺǂƩƣƹ the incorrect answer is given, and what type of answer to accept.

ѹѾѷ҆Ҋѻ҈ም

Classroom Facilitation of Aha! Moment Insights Bronislaw Czarnocha and William Baker

3.1

Introduction: Review of the Theoretical Literature

ƩƫƾƢƫƾơǀƾƾƫƺƹƟǀƫƶƢƾƺƹƿƩƣƻƽƣǁƫƺǀƾǂƺƽƴƺƤƽƞƟƩǀᄬᇴᇲᇳᇸᄭᄕǂƩƺƤƺǀƹƢƿƩƞƿ persistent use of insight generated by Aha! moments in her classes significantly ƫƸƻƽƺǁƣƢƩƣƽƾƿǀƢƣƹƿƾᅷƸƺƿƫǁƞƿƫƺƹƞƹƢƞƿƿƫƿǀƢƣƿƺǂƞƽƢƶƣƞƽƹƫƹƨƸƞƿƩƣƸƞƿƫơƾ while producing measurable improvements in learning outcomes. ƽƞƟƩǀƞƶƫƨƹƣƢƩƣƽƿƣƞơƩƫƹƨƸƣƿƩƺƢƺƶƺƨDŽǂƫƿƩƺƣƾƿƶƣƽᅷƾƿƩƣƺƽDŽƺƤƟƫƾƺơƫƞƿƫƺƹᄬᇳᇻᇸᇶᄭƞƹƢƢƣǁƣƶƺƻƣƢƿƩƣƟƣƨƫƹƹƫƹƨƾƺƤǂƩƞƿƾƩƣƿƣƽƸƣƢƩƞᄛƻƣƢƞƨƺƨDŽᄕ that is, pedagogy designed to facilitate Aha! moments in mathematics classes. ƩƫƾơƺƺƽƢƫƹƞƿƫƺƹƺƤƿƣƞơƩƫƹƨƽƣƾƣƞƽơƩǂƫƿƩƺƣƾƿƶƣƽᅷƾƿƩƣƺƽDŽƿƞƴƣƾƻƶƞơƣƾƺƹ ƿǂƺƶƣǁƣƶƾᄘᄬᇳᄭƞƾƞƤƽƞƸƣǂƺƽƴƿƺƞƹƞƶDŽDžƣơƺƨƹƫƿƫǁƣƞƾƻƣơƿƾƺƤƟƫƾƺơƫƞƿƫƺƹƫƹƢǀơƫƹƨƿƩƣƢƣƶƫơƞƿƣƾƩƫƤƿƺƤƞƿƿƣƹƿƫƺƹᄬƞƾƺƹᄕᇳᇻᇺᇻᄭᄕƽƣƼǀƫƽƣƢƤƺƽƞƟƾƿƽƞơƿƫƹƨƞƹƣǂ ơƺƹơƣƻƿƫƺƹᄕƞƹƢᄬᇴᄭƞƾƿƩƣƟƫƾƺơƫƞƿƫǁƣƤƽƞƸƣǂƺƽƴƺƤƿƣƞơƩƫƹƨᅟƽƣƾƣƞƽơƩƸƣƿƩodology as it applies to the facilitation of Aha! moments for both individuals and groups in a classroom setting. ƺƣƾƿƶƣƽƣƶƞƟƺƽƞƿƣƾƺƹƩƫƾƢƣƤƫƹƫƿƫƺƹƺƤƟƫƾƺơƫƞƿƫƺƹᄬƩƞƻƿƣƽᇳᄭᄘ

 Ʃƞǁƣ ơƺƫƹƣƢ ƿƩƣ ƿƣƽƸ ᅸƟƫƾƺơƫƞƿƫƺƹᅺ ƫƹ ƺƽƢƣƽ ƿƺ Ƹƞƴƣ ƞ Ƣƫƾƿƫƹơƿƫƺƹ between the routine skills of thinking on a “single” plane, as it were, and ƿƩƣơƽƣƞƿƫǁƣƞơƿᄕǂƩƫơƩᄕƞƾ ƾƩƞƶƶƿƽDŽƿƺƾƩƺǂᄕƞƶǂƞDŽƾƺƻƣƽƞƿƣƾƺƹƸƺƽƣ ƿƩƞƹƺƹƣƻƶƞƹƣᄙᄬƻƻᄙᇵᇷᅬᇵᇸᄭ These leaps of insight are further described as the unearthing of “hidden analogies” between two or more previously unconnected frames of reference joined ƟDŽƿƩƣƫƹƾƫƨƩƿƺƤƿƩƣƩƞᄛƸƺƸƣƹƿᄬDžƞƽƹƺơƩƞƣƿƞƶᄙᄕᇴᇲᇳᇸᄭᄙƺƣƾƿƶƣƽᅷƾƢƣƤƫƹƫtion of bisociation suggests that its cognitive content of connecting unconnected frames of reference is the process of building a cognitive schema—a deeper process of understanding. Equally important is the affective transformation that often takes place ƾƫƸǀƶƿƞƹƣƺǀƾƶDŽǂƫƿƩƩƞᄛƸƺƸƣƹƿƾᄬƩƞƻƿƣƽƾᇻƞƹƢᇳᇲᄭᄘᅸƩƣơƽƣƞƿƫǁƣƞơƿᄚ ƫƾƞƹƞơƿƺƤƶƫƟƣƽƞƿƫƺƹᅭƿƩƣƢƣƤƣƞƿƺƤƩƞƟƫƿƟDŽƺƽƫƨƫƹƞƶƫƿDŽᅺᄬƺƣƾƿƶƣƽᄕᇳᇻᇸᇶᄕƻᄙ ᇻᇸᄭᄕ ǂƩƫơƩ ƩƣƶƻƣƢ ƿƽƞƹƾƤƺƽƸ ƾƿǀƢƣƹƿƾᅷ ƹƣƨƞƿƫǁƣ ƩƞƟƫƿƾ ƞƹƢ ƞƿƿƫƿǀƢƣƾ ƿƺǂƞƽƢ ƸƞƿƩƣƸƞƿƫơƾƫƹƽƞƟƩǀᅷƾơƶƞƾƾᄙƣƾƫƨƹƫƹƨƩƞᄛƻƣƢƞƨƺƨDŽƽƣƼǀƫƽƣƾƞƢƣƿƞƫƶƣƢ ᇙ ‫ةء؟؟؜إؘؕ؞؝؜؟؞ء؜ءآ؞‬ᄕ‫ءؘؗ؜ؘ؟‬ᄕሦሤሦሥᏺᄩᏺ‫؜آؗ‬ᄘሥሤᄙሥሥሪሧᄧርራሬርሤሤረረረሪረሧረᇇሤሤረ

ѹ‫ؠآآإئئؔ؟‬Ѽؔ‫ؙآءآ؜اؔا؜؟؜ؖ‬ѷ‫ؔ؛‬ᄛ҃‫اءؘؠآ‬ѿ‫ئا؛ؚ؜ئء‬

ሬሥ

understanding of how to facilitate Aha! moments as well as assess the depth of ƴƹƺǂƶƣƢƨƣᄬƺᄭƽƣƞơƩƣƢƢǀƽƫƹƨƿƩƣƫƹƾƫƨƩƿᄙ Teaching research based on the creative synthesis of the roles of teacher and researcher within mathematics education to inspire creativity with an ǀƹƢƣƽƾƣƽǁƣƢƻƺƻǀƶƞƿƫƺƹǂƞƾƿƩƣƻƽƫƸƞƽDŽƤƺơǀƾƺƤDžƞƽƹƺơƩƞᄬᇴᇲᇳᇸᄭƞƹƢƽƞƟƩǀᄬᇴᇲᇳᇸᄭᄙƩƣƣƤƤƺƽƿƿƺƢƣƸƺơƽƞƿƫDžƣơƽƣƞƿƫǁƫƿDŽƽƣƼǀƫƽƣƾƽƣƿƩƫƹƴƫƹƨơƽƣƞƿƫǁƫƿDŽ research, which is often seen through the lens of gifted individuals, to apply it toward insights of students in any mathematics classroom. Questions that arise ƩƣƽƣƞƹƢƫƹƩƞƻƿƣƽᇳᇳƫƹơƶǀƢƣƿƩƣƤƺƶƶƺǂƫƹƨᄘƩƞƿƫƾƿƩƣƹƞƿǀƽƣƺƤƸƺƸƣƹƿƾƺƤ insights—the Aha! experience—for individual problem-solvers and student teams in a classroom? To what extent can insights, especially those during shared learning, be characterized as bisociative Eureka moments? How can student insights be fostered and strengthened? From a slightly different angle these questions become: What is the nature of new concepts and connections made through individual insights as opposed to those experienced during shared learning? A related question is, How can we promote insights through interiorization and help students internalize the connection concepts learned so they can independently call forth new actions without assistance? We can observe the interplay between these two processes during collaboƽƞƿƫǁƣƻƽƺƟƶƣƸᅟƾƺƶǁƫƹƨᄕƞƾƢƣƾơƽƫƟƣƢƟDŽƞƽƹƣƾᄬᇴᇲᇲᇲᄭᄙƹƞƿƩƣƺƽƣƿƫơƞƶƶƣǁƣƶ the question is, How can we create a learning environment that promotes creƞƿƫǁƣƸƞƿƩƣƸƞƿƫơƞƶƿƩƫƹƴƫƹƨᄞ ƾƾǀơƩƞƹƣƹǁƫƽƺƹƸƣƹƿƾǀƟƾǀƸƣƢǂƫƿƩƫƹƻƽƺƟlem-solving or more general methods of inquiry?

3.2

The Creative Learning Environment (ѹ҂ѻ)

ƹƿƩƫƾƾƣơƿƫƺƹᄕǂƣƶƺƺƴƞƿơƽƣƞƿƫǁƫƿDŽƿƩƽƺǀƨƩƿƩƣƾDŽƹƿƩƣƾƫƾƺƤơƽƣƞƿƫǁƫƿDŽƽƣƾƣƞƽơƩ and constructivist research on learning. Creativity research has primarily been restricted to the realm of gifted students; however, our interest has been to democratize creativity, that is, to bring it into the everyday learning of students in the mathematics classroom. One important issue is how to facilitate the insights of rank-and-file students as they create meaning in mathematics. This focus includes a cognitive ƞƾƻƣơƿƞƹƞƶDŽDžƣƢƿƩƽƺǀƨƩƿƩƣƟƫƾƺơƫƞƿƫǁƣƤƽƞƸƣǂƺƽƴƺƤƺƣƾƿƶƣƽƞƹƢƞƹƞƤƤƣơƿƫǁƣ ƤƽƞƸƣǂƺƽƴ ƺƤ ƫƶưƣƢƞƩƶ ᄬᇴᇲᇳᇵᄭ ƞƹƢ ƣƣƶƶƫƾ ƞƹƢ ƺƶƢƫƹ ᄬᇴᇲᇲᇸᄭ ƿƺ ơƽƣƞƿƣƞƻƺƾƫƿƫǁƣƞƤƤƣơƿƿƺǂƞƽƢƸƞƿƩƣƸƞƿƫơƾᄙᄬ ƺƽƞƹƣǃƩƞǀƾƿƫǁƣƞƹƞƶDŽƾƫƾƺƤƿƩƫƾ ƿƺƻƫơᄕƾƣƣƩƞƻƿƣƽƾᇻƞƹƢᇳᇲᄙᄭƩƣƶƣƞƽƹƫƹƨƻƽƺơƣƾƾƫƹǂƩƫơƩƾƿǀƢƣƹƿƾơƽƣƞƿƣ meaning of mathematics has been the central focus of constructivist teaching

ሬሦ

ѹ‫ؗءؔؔ؛ؖآءإؔح‬Ѹؔ‫إؘ؞‬

experiments in which the researcher acts as a teacher with the goal of developing and clarifying a model for schema development. The use of constructivist research to help analyze creativity is important. ƾƫƶưƣƢƞƩƶᄬᇴᇲᇳᇵᄭƹƺƿƣƾᄕƿƩƣƞƶƶᅟƿƺƺᅟơƺƸƸƺƹƤƺơǀƾƺƹƻƽƺƢǀơƿƾƺƤơƽƣƞƿƫǁƫƿDŽ tends to divert attention away from the creative process. “Note that such a use of assessment of end product pays very little attention to the actual process ƿƩƞƿƟƽƫƹƨƾƿƩƫƾƻƽƺƢǀơƿƤƺƽƿƩᅺᄬƻᄙᇴᇷᇷᄭᄙ Even more to the point, what creativity research considers focusing on the process often has little to do with how learners restructure their cognitive schemes, which is the primary focus of constructivist research. For example, ƣƫƴƫƹƞƹƢƫƿƿƞᅟƞƹƿƞDžƫᄬᇴᇲᇳᇵƞᄭƫƹƿƽƺƢǀơƣƢƿƩƣᇴᇲᇳᇵƾƻƣơƫƞƶƫƾƾǀƣƺƤѠъѓ on ơƽƣƞƿƫǁƫƿDŽƞƹƢƸƞƿƩƣƸƞƿƫơƾƣƢǀơƞƿƫƺƹᄬƣƫƴƫƹѵƫƿƿƞᅟƞƹƿƞDžƫᄕᇴᇲᇳᇵƟᄭƟDŽƹƺƿing that it featured four contributions on creative products, four on personality ƿƽƞƫƿƾƺƤơƽƣƞƿƫǁƣƫƹƢƫǁƫƢǀƞƶƾᄕƞƹƢƺƹƶDŽƿǂƺƿƩƞƿƿƺǀơƩƺƹƿƩƣƻƽƺơƣƾƾᄙ ƹƿƩƣ ƤƫƽƾƿƺƤƿƩƣƾƣƻƽƺơƣƾƾᅟƺƽƫƣƹƿƣƢƞƽƿƫơƶƣƾᄕƫƶưƣƢƞƩƶᄬᇴᇲᇳᇵᄭƢƣƞƶƾƻƽƫƸƞƽƫƶDŽǂƫƿƩƿƩƣ ƞƤƤƣơƿƫǁƣơƺƸƻƺƹƣƹƿƞƾƾƺơƫƞƿƣƢǂƫƿƩƿƩƣƩƞᄛƸƺƸƣƹƿᄙ ƹƿƩƣƾƣơƺƹƢᄕƺƫơƞ ƞƹƢƫƹƨƣƽᄬᇴᇲᇳᇵᄭƤƺơǀƾƺƹƢƣƿƣơƿƫƹƨơƺƨƹƫƿƫǁƣƤƶƣǃƫƟƫƶƫƿDŽǂƫƿƩƨƫƤƿƣƢƾƿǀƢƣƹƿƾ during problem-solving; however, the actual restructuring process that leads to this flexibility is not discussed in either article. 3.2.1 Educational Focus on Creative Product One issue that arises is that many mathematics educators do not consider classroom learning of regular students a creative process because there is no ơƽƣƞƿƫǁƣƣƹƢƻƽƺƢǀơƿᄙƩƫƾƽƣƞƾƺƹƫƹƨƫƾƻƽƺƸƺƿƣƢƟDŽƿƩƣƽƣǁƫƾƣƢƶƺƺƸᅷƾƿƞǃƺƹƺƸDŽƺƤƹƢƣƽƾƺƹƞƹƢƽƞƿƩǂƺƩƶᄬᇴᇲᇲᇳᄭƿƩƞƿƻƶƞơƣƾơƽƣƞƿƫǁƫƿDŽᄕƤƺƽƸƣƽƶDŽƾDŽƹƿƩƣsis, as the last level of cognitive thought, as opposed to its previous place before ƿƩƣƣǁƞƶǀƞƿƫƺƹƺƤƿƩƣƻƽƺƢǀơƿƞƨƞƫƹƾƿƾƿƞƹƢƞƽƢƾᄬƩƞƻƿƣƽᇶᄭᄙ

ƹƿƩƫƾƾơƣƹƞƽƫƺᄕƞơƽƣƞƿƫǁƣƺƟưƣơƿƸǀƾƿƣǃƫƾƿƞƹƢƟƣƣǁƞƶǀƞƿƣƢƟƣƤƺƽƣơƽƣƞƿƫǁƫƿDŽƫƾƞơơƣƻƿƣƢƞƾƿƞƴƫƹƨƻƶƞơƣᄙ ƤǂƣƣǃƿƣƹƢƿƩƣƹƺƿƫƺƹƺƤƣƹƢƻƽƺƢǀơƿƿƺ include the creative idea that propels our understanding to the higher level, the assessment of creativity becomes more balanced. This applies to a “material end product” as the ready solution and to an “intellectual end product” as new understanding. The view that creativity takes place only at the end of a critical-thinking process involving analysis, synthesis, and evaluation of ideas ƫƹƺǀƽƺƻƫƹƫƺƹƤǀƽƿƩƣƽƾƿƩƣƸDŽƿƩᄬƩƣƤƤƫƣƶƢᄕᇴᇲᇳᇹᄭƿƩƞƿƺƹƶDŽƤƶǀƣƹƿᅟƨƫƤƿƣƢƾƿǀdents can be creative in mathematics and science. 3.2.2 Importance of Creative Insight The creativity-as-product view takes the focus away from bisociative insight as the mechanism by which a solver crosses from incubation to illumination.

ѹ‫ؠآآإئئؔ؟‬Ѽؔ‫ؙآءآ؜اؔا؜؟؜ؖ‬ѷ‫ؔ؛‬ᄛ҃‫اءؘؠآ‬ѿ‫ئا؛ؚ؜ئء‬

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ƹƾƿƣƞƢᄕƫƿƤƺơǀƾƣƾƺƹƿƩƣǁƣƽƫƤƫơƞƿƫƺƹƾƿƞƨƣƞƾƢƣƿƣƽƸƫƹƫƹƨǂƩƣƿƩƣƽƞƹƫƢƣƞƫƾ creative. This shift of attention away from illumination through direct experience has negative consequences on student affect. That is, the student intuits that creative insights are not relevant unless they are acknowledged by an authority or external group, and this suppresses parƿƫơƫƻƞƿƫƺƹƟDŽƾƿǀƢƣƹƿƾǂƩƺƶƞơƴƾƣƶƤᅟơƺƹƤƫƢƣƹơƣᄙ ƹƾƿƣƞƢᄕƤƺơǀƾƫƹƨƺƹƾƿǀƢƣƹƿ insights leads to wider class participation. For example, Hershkowitz et al. ᄬᇴᇲᇳᇹᄭƻƽƣƾƣƹƿƺƹƣƾƿǀƢƣƹƿǂƩƺƻƽƺǁƫƢƣƢƿƩƣơƺƽƽƣơƿƾƺƶǀƿƫƺƹƟǀƿǂƞƾǀƹƞƟƶƣ to articulate a clear explanation; shortly thereafter another student gave a ơƺƩƣƽƣƹƿƽƣƞƾƺƹƤƺƽƿƩƣƤƫƽƾƿƾƿǀƢƣƹƿᅷƾƞơƿƫƺƹƾƟƞƾƣƢƺƹƿƩƣƤƫƽƾƿƾƿǀƢƣƹƿᅷƾƫƹơƺherent statements. The authors-researchers considered the second student creative but not the ƤƫƽƾƿƺƹƣᄕƿƩǀƾƞƢƩƣƽƫƹƨƿƺƿƩƣƻƽƺƢǀơƿǁƫƣǂƺƤơƽƣƞƿƫǁƫƿDŽᄙ ƹơƺƹƿƽƞƾƿᄕǂƣơƺƹƾƫƢƣƽƿƩƣƤƫƽƾƿƾƿǀƢƣƹƿᅷƾƫƹƾƫƨƩƿƞƹƫƸƻƺƽƿƞƹƿƞơƿƫǁƞƿƫƺƹƿƩƞƿƤƞơƫƶƫƿƞƿƣƢƾƩƞƽƣƢ creativity. ƺƣƾƿƶƣƽᄬᇳᇻᇸᇶᄭƢƣƻƶƺƽƣƢƿƩƣƾƿƞƿƣƺƤƣƢǀơƞƿƫƺƹǂƫƿƩƫƿƾƤƺơǀƾƺƹƻƽƺơƣƢǀƽƣƾ and rules that he felt dehumanized learning by stripping it of any chance that the learners may experience their own creativity. On the metaphysical level, he ƞƨƽƣƣƢǂƫƿƩƺƫƹơƞƽȅᅷƾǁƫƣǂƿƩƞƿơƽƣƞƿƫǁƫƿDŽƫƹƸƞƿƩƣƸƞƿƫơƾƫƹǁƺƶǁƣƢƨƣƿƿƫƹƨƫƹ touch with a higher creative-intuitive force and that this experience should be the goal of education. “To derive pleasure from the act of discovery, as from the other arts, the consumer—in this case the student—must be made to re-live, ƿƺƾƺƸƣƣǃƿƣƹƿᄕƿƩƣơƽƣƞƿƫǁƣƻƽƺơƣƾƾᅺᄬƻᄙᇴᇸᇷᄭᄙ ƺƣƾƿƶƣƽᅷƾǁƫƣǂƺƤƾƺơƫƣƿDŽᅷƾƶƞơƴƺƤƫƹƿƣƽƣƾƿƫƹƿƩƣƢƫƾơƺǁƣƽDŽƻƽƺơƣƾƾƫƹƸƞƿƩ ƞƹƢƾơƫƣƹơƣᄕƞƶƿƩƺǀƨƩƾƿƞƿƣƢƫƹᇳᇻᇸᇶᄕƫƾƻƽƺƻƩƣƿƫơƫƹƢƣƾơƽƫƟƫƹƨƿƺƢƞDŽᅷƾƽƣƞƶƫƿDŽᄙ ƺƢƣƽƹƸƞƹƶƫǁƣƾƫƾƺƶƞƿƣƢƫƹƩƫƾƞƽƿƫƤƫơƫƞƶƣƹǁƫƽƺƹƸƣƹƿᄚᄴƢǀƣƿƺᄵƩƫƾ lack of comprehension of the forces which make it work—of the princiƻƶƣƾǂƩƫơƩƽƣƶƞƿƣƩƫƾƨƞƢƨƣƿƾƿƺƿƩƣƤƺƽơƣƾƺƤƹƞƿǀƽƣᄙᄚDŽƟƣƫƹƨƣƹƿƫƽƣƶDŽ dependent on science, yet closing his mind to it, he lives the life of an ǀƽƟƞƹƟƞƽƟƞƽƫƞƹᄙᄬƻᄙᇴᇸᇶᄭ ƺƣƾƿƶƣƽơƣƽƿƞƫƹƶDŽơƺƹƾƫƢƣƽƣƢƢƫƾơƺǁƣƽDŽƺƤƣǃƫƾƿƫƹƨƸƞƿƩƣƸƞƿƫơƞƶƴƹƺǂƶƣƢƨƣ for oneself the goal of education, and he focused his bisociative theory almost ƣǃơƶǀƾƫǁƣƶDŽƺƹƞƹƫƹƢƫǁƫƢǀƞƶᅷƾơƺƹơƣƻƿƢƣǁƣƶƺƻƸƣƹƿǂƫƿƩƸƫƹƫƸƞƶƞƾƾƫƾƿƞƹơƣᄘ “Minor, subjective bisociative processes do occur on all levels, and are the main ǁƣƩƫơƶƣƺƤǀƹƿǀƿƺƽƣƢƶƣƞƽƹƫƹƨᅺᄬƻᄙᇸᇷᇺᄭᄙƺƣƾƿƶƣƽᅷƾƫƹƾƫƨƩƿƾǀƹƢƣƽƶƫƣƸǀơƩƺƤƿƩƣ constructivist work of today, especially for those interested in active learning: the process through which students create their own meaning of mathematics. Mathematics taught without student input or without a focus on student

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ѹ‫ؗءؔؔ؛ؖآءإؔح‬Ѹؔ‫إؘ؞‬

insight promotes the meaningless void of memorized formulas and rules. For ƺƣƾƿƶƣƽᄕƿƩƣƩƞᄛƣǃƻƣƽƫƣƹơƣƫƾƿƩƣƞƹƿƫƢƺƿƣᄙ ƹƿƩƫƾƾƣơƿƫƺƹᄕǂƣƾƣƣƴƿƺǀƹƢƣƽstand how to facilitate this process. 3.2.3 Creative Learning Environment: Classroom ƣƽƾƩƴƺǂƫƿDžƣƿƞƶᄙᄬᇴᇲᇳᇹᄭƩƞǁƣƾƿǀƢƫƣƢơƽƣƞƿƫǁƣƽƣƞƾƺƹƫƹƨƶƣƞƢƫƹƨƿƺƞƟƾƿƽƞơƿƫƺƹƫƹƿƩƣơƶƞƾƾƽƺƺƸᄙƩƣƫƽǂƺƽƴǂƞƾƨǀƫƢƣƢƟDŽƟƾƿƽƞơƿƫƺƹƫƹƺƹƿƣǃƿᄬѷѿѹᄭᄕ essentially a constructivist theory that involves schema development by “reorƨƞƹƫDžƫƹƨƻƽƣǁƫƺǀƾƸƞƿƩƣƸƞƿƫơƞƶơƺƹƾƿƽǀơƿƾᄚᄴƞƹƢᄵƫƹƿƣƽǂƣƞǁƫƹƨƿƩƣƸƫƹƿƺ a single process of mathematical thinking so as to lead to the construction of ƴƹƺǂƶƣƢƨƣƿƩƞƿƫƾƹƣǂƿƺƿƩƣƶƣƞƽƹƣƽᅺᄬƻᄙᇴᇸᄭᄙ The new construction of knowledge or abstraction results from the epistemic processes of first recognition—previous relevant schemes and then building with them—often guided by elaboration questions on the part of ƿƩƣƸƣƹƿƺƽᄕƿƣƞơƩƣƽᄕƺƽƽƣƾƣƞƽơƩƣƽᄬ ƣƽƾƩƴƺǂƫƿDžƣƿƞƶᄙᄕᇴᇲᇲᇳᄭᄙ ƣƽƾƩƴƺǂƫƿDžƣƿ ƞƶᄙ ᄬᇴᇲᇳᇹᄭ ơƺƹƾƫƢƣƽ ơƽƣƞƿƫǁƣ ƽƣƞƾƺƹƫƹƨ ƹƺǁƣƶ ǂƩƣƹ ƫƿ ƫƾ ƹƺƿ ƟƞƾƣƢ ƺƹ ƞơƿƫǁƫƿDŽ presented previously in the classroom. More specifically, creative reasoning requires not only novel activity but also activity supported by reasoned mathematical arguments. This chapter, however, does not analyze the structural change that occurs through ѷѿѹ but rather the shifts of knowledge among students. That is how ƾƿǀƢƣƹƿƾƽƣƶƞƿƣƿƺƞƹƢƟǀƫƶƢƺƹƣƞơƩƺƿƩƣƽᅷƾƫƹƾƫƨƩƿƾᄙ ƹƿƩƫƾƾƫƿǀƞƿƫƺƹᄕƸǀơƩƺƤ ƿƩƣƿƣƞơƩƣƽᅷƾƽƺƶƣƫƾƿƺƩƫƨƩƶƫƨƩƿƾƿǀƢƣƹƿƽƣƞƾƺƹƫƹƨᄕƿƩƞƿƫƾᄕƿƺƻƺƾƣƼǀƣƾƿƫƺƹƾ that bring about elaboration rather than clarification of the concepts being learned. That said, the shifts of knowledge between students observed by HersƩƴƺǂƫƿDžƣƿƞƶᄙᄬᇴᇲᇳᇹᄭƺƽơƩƣƾƿƽƞƿƣƢƾƴƫƶƶƤǀƶƶDŽƟDŽƿƩƣƿƣƞơƩƣƽơƶƣƞƽƶDŽƽƣƾǀƶƿƣƢƫƹƿƩƣ building up of conceptual knowledge that fit the ѷѿѹ framework. ƞƴƣƽ ᄬᇴᇲᇳᇸƟᄭ ƻƽƣƾƣƹƿƾ ƞ ƾƫƸƫƶƞƽ ơƽƣƞƿƫǁƣ ƾƩƞƽƣƢᅟƶƣƞƽƹƫƹƨ ƣǃƻƣƽƫƣƹơƣ ƫƹ which the teacher navigates student efforts to grasp the reasons behind peer work and peer questions. This teacher navigated dialogue in a class bisociation as students understand the relationship between different methods to solve a percent increase problem. 3.2.4 Shared Moments of Insight, Interiorization, and Internalization A question that arises for instructors is the strength of classroom or socially induced insights by following the reasoning of others. Are these assimilations ǂƩƞƿƺƣƾƿƶƣƽƽƣƤƣƽƾƿƺƞƾƞƹƣǃƣƽơƫƾƣƫƹƣǃƫƾƿƫƹƨǀƹƢƣƽƾƿƞƹƢƫƹƨᄕƺƽƞƽƣƿƩƣDŽ ƞơơƺƸƸƺƢƞƿƫƺƹƾᄕǂƩƞƿƺƣƾƿƶƣƽơƞƶƶƾƻƽƺƨƽƣƾƾƫƹǀƹƢƣƽƾƿƞƹƢƫƹƨᄞƿƣƤƤƣᄬᇴᇲᇲᇳᄭ talks about a student who could explain the actions of another student yet could not engage in such activity independently.

ѹ‫ؠآآإئئؔ؟‬Ѽؔ‫ؙآءآ؜اؔا؜؟؜ؖ‬ѷ‫ؔ؛‬ᄛ҃‫اءؘؠآ‬ѿ‫ئا؛ؚ؜ئء‬

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The conclusion Steffe reached is that only when we understand the reason for our own actions done independently can we state with confidence that ƫƹƿƣƽƫƺƽƫDžƞƿƫƺƹƩƞƾƺơơǀƽƽƣƢᄙ ƹơƺƹƿƽƞƾƿᄕƸƺƸƣƹƿƾƺƤƾƩƞƽƣƢơƽƣƞƿƫǁƣƫƹƾƫƨƩƿᄕ although demonstrating an understanding of connections, does not guarantee ơƞƻƞƟƫƶƫƿDŽƤƺƽƫƹƢƣƻƣƹƢƣƹƿƞơƿƫƺƹᄬƫƹƿƣƽƫƺƽƫDžƞƿƫƺƹᄭᄙƩƣƻƺƫƹƿƫƾƿƩƞƿƾƿǀƢƣƹƿƾ who borrow or internalized connections, concepts, or schema during a discussion have not interiorized the actions until they can independently perform them and use them in solving related problems. ƩƞƻƿƣƽᇷƞƹƢƿƩƣƹƣǃƿƾƣơƿƫƺƹƾƢƫƾơǀƾƾƿƩƣƫƹƿƣƽƻƶƞDŽƟƣƿǂƣƣƹƫƹƿƣƽƹƞƶƫDžƞtion and interiorization to facilitate insights leading to independent understanding. The analysis of interiorization and internalization processes in the context of student collaboration allows us to refine our understanding of creative processes during student interactions. This should be a focus of educational research on improving the quality of learning mathematics in classrooms. 3.2.5 Critical Thinking and Creativity ƺƣƾƿƶƣƽᅷƾ ǀƹƢƣƽƾƿƞƹƢƫƹƨᄕ ƫƹ ƿƩƣ ơƺƹƿƣǃƿ ƺƤ ƻƽƺƟƶƣƸᅟƾƺƶǁƫƹƨᄕ ƫƾ ƿƩƞƿ ơƽƣƞƿƫǁity—bisociation—occurs during the synthesis of the codes: the guiding principles or rules that govern the use of previously unrelated frames of reference. ƫƸƺƹᅷƾ ᄬᇴᇲᇳᇹᄭ ǀƹƢƣƽƾƿƞƹƢƫƹƨ ƫƾ ƿƩƞƿ ƞơơƺƸƸƺƢƞƿƫƺƹᅟơƺƹơƣƻƿ ƢƣǁƣƶƺƻƸƣƹƿ occurs when we understand the logical necessity of our actions. The focus on the underlying reason behind our actions during these processes begs for a distinction between creativity and critical thinking, neither of which is well defined. ƣơƩƾƶƣƽ ƣƿ ƞƶᄙ ᄬᇴᇲᇳᇺᄭ ƢƣƾơƽƫƟƣ ƿƩƣ ƽƣƶƞƿƫƺƹƾƩƫƻ Ɵƣƿǂƣƣƹ ơƽƣƞƿƫǁƣ ƫƹƾƫƨƩƿƾ and critical thinking. They suggest that during problem-solving, innovative solutions and strategies are followed by periods that require evaluation and Ƣƣơƫƾƫƺƹ Ƹƞƴƫƹƨ ᄬơƽƫƿƫơƞƶ ƿƩƫƹƴƫƹƨᄭᄙ ƹ ƫƹƿƣƨƽƞƿƣƢ ƶƞDŽƣƽƫƹƨ ƞƻƻƽƺƞơƩ ƿƺ ơƽƣƞƿƫǁƫƿDŽ ƞƹƢ ơƽƫƿƫơƞƶ ƿƩƫƹƴƫƹƨ ƫƾ ƣǃƻƽƣƾƾƣƢ ƟDŽ ƞƿƹƞƹƫƹƨƾƫƩ ƞƹƢ DŽƞƹƣ ᄬᇴᇲᇳᇹᄭ in which student creativity is expressed through conjecturing by analogical reasoning.

ƹƺƿƩƣƽǂƺƽƢƾᄕƽƣƞƾƺƹƫƹƨƟDŽƞƹƞƶƺƨDŽƫƾƟƣƿǂƣƣƹƿƩƣƻƽƺƟƶƣƸƾƿƽǀơƿǀƽƣƻƣƽceived by the student and their previous knowledge. Distinctions and similarities between previous knowledge and a problem situation in the Fir Tree Aha! moment lead the student to discover a hidden analogy by generalizing previous knowledge, when the student understood how and why a previously unrelated schema was applicable to the new problem situation.

ƹƿƩƣƢƫƾơƺǁƣƽDŽƺƤƞƩƫƢƢƣƹƞƹƞƶƺƨDŽᄕƿƩƣƾƺƶǁƣƽƫƾƟƶƺơƴƣƢᄘǀƹƞƟƶƣƿƺǀƾƣ any existing scheme to assimilate the new situation until, through Aha!, she

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ѹ‫ؗءؔؔ؛ؖآءإؔح‬Ѹؔ‫إؘ؞‬

recognizes that a past construct of her conjecture applies. At this point student synthesizes the codes of past and present into a new action during the Eureka moment. Critical thinking is involved during the search for a similar or relevant structure to the given problem and in the evaluation of the result. Creativity is essentially found in the moment of insight in the recognition of a previously hidden analogy. The insight may be based on a hunch—intuition, guessing, or conjecture based on analogical reasoning. Furthermore, the truth of the hidƢƣƹƞƹƞƶƺƨDŽƹƣơƣƾƾƫƿƞƿƣƾơƽƫƿƫơƞƶƿƩƫƹƴƫƹƨƣǁƞƶǀƞƿƫƺƹᄬƞƿƹƞƹƫƹƨƾƫƩѵDŽƞƹƣᄕ ᇴᇲᇳᇹᄭᄙ 3.2.6 Stages of Creativity ƩƞƻƿƣƽᇳƢƫƾơǀƾƾƣƢƿƩƣƣƾƿƞƶƿƸƺƢƣƶƺƤơƽƣƞƿƫǁƫƿDŽᄙ ƹƿƩƫƾƸƺƢƣƶƺƤơƽƣƞƿƫǁity, preparation involves critical thinking as the solver reviews his past records ƺƤƻƽƺƟƶƣƸƾƫƿǀƞƿƫƺƹƾǂƫƿƩƞƾƿƽǀơƿǀƽƣƾƫƸƫƶƞƽƿƺƿƩƣƻƽƣƾƣƹƿƾƫƿǀƞƿƫƺƹᄙ ƹƿƩƣ classroom, moments of insight are often not observed because student conjectures are not always built on lengthy incubation as when an individual ponders a problem at length.

ƹ ơƺƹƿƽƞƾƿ ƿƺ Ɵƫƾƺơƫƞƿƫǁƣ ƫƹƾƫƨƩƿƾ ƨƞƫƹƣƢ ƿƩƽƺǀƨƩ ƫƹƢƫǁƫƢǀƞƶ ƻƽƣƻƞƽƞƿƫƺƹ and incubation, insights during shared learning experiences may not have the ƾƞƸƣơƣƽƿƞƫƹƿDŽƹƺƽƿƩƣƾƞƸƣƶƞƾƿƫƹƨƻƺǂƣƽᄙ ƹƢƣƣƢᄕƿƩƣDŽƸƞDŽƟƣƽƞƿƩƣƽƤƶƣƣƿƫƹƨ in nature. The final stage—verification—is typically considered critical thinking often led by the teacher to determine the validity of the insight. We shall ƾƞDŽƸƺƽƣƞƟƺǀƿƿƩƣƿƣƞơƩƣƽᅷƾƽƺƶƣƶƞƿƣƽƫƹƿƩƫƾơƩƞƻƿƣƽᄙƿƻƽƣƾƣƹƿᄕǂƣƿǀƽƹƺǀƽ attention to how mathematics educational research, especially constructivist research, characterize the problem-solving or inquiry environment in which student insight is observed. 3.2.7 Theoretical Environment for Creativity and Concept Development ƹƢƣƽƾƺƹᄬᇳᇻᇻᇷᄭǁƫƣǂƾƞƶƶơƺƨƹƫƿƫƺƹƞƾƿƞƴƫƹƨƻƶƞơƣƫƹƞƨƺƞƶᅟƢƫƽƣơƿƣƢᄕƻƽƺƟƶƣƸᅟƾƺƶǁƫƹƨ ƣƹǁƫƽƺƹƸƣƹƿᄖ ơƺƹƾƿƽǀơƿƫǁƫƾƿƾᄕ ƫƹơƶǀƢƫƹƨ ƺƣƾƿƶƣƽᄕ ǂƺǀƶƢ Ƥƺƽ ƿƩƣ most part agree with this statement. Many constructivists follow the Piagetian ƹƺƿƫƺƹƿƩƞƿƞơơƺƸƸƺƢƞƿƫƺƹƺơơǀƽƾᄬᇳᄭƞƤƿƣƽƿƩƣƾƺƶǁƣƽƾƣƶƣơƿƾƞƹƞơƿƫƺƹƞƹƢ ƿƩƣƽƣƾǀƶƿƫƾƹƺƿƾƣƣƹƞƾƩƣƶƻƤǀƶƿƺǂƞƽƢƺƟƿƞƫƹƫƹƨƿƩƣƨƺƞƶᄕƾƺƿƩƞƿᄬᇴᄭƞƻƣƽƿǀƽbation or disequilibrium then takes place that can lead to creativity. At this point distinctions between traditional creativity and concept development models arise. As noted, constructivists often use the term “disequilibrium” to denote a state in which existing schema is considered no longer appropriate for a problem situation and perturbation leading to a search for a

ѹ‫ؠآآإئئؔ؟‬Ѽؔ‫ؙآءآ؜اؔا؜؟؜ؖ‬ѷ‫ؔ؛‬ᄛ҃‫اءؘؠآ‬ѿ‫ئا؛ؚ؜ئء‬

ሬራ

new approach begins. Thus, constructivists do not necessarily require drastic restructuring as do the Gestalt theorists. Ʃƣ ƫƽƽƣƣƩƞᄛƸƺƸƣƹƿƞƹƢƿƩƣƩƞƿ ƾƞƣơƿƺƽᄞƩƞᄛƸƺƸƣƹƿƻƽƺǁƫƢƣ examples in which the bisociative insight takes place without such a perturƟƞƿƫƺƹᄙƫƸƺƹƣƿƞƶᄙᄬᇴᇲᇳᇲᄭƞƹƢƫƸƺƹᄬᇴᇲᇳᇶᄭƽƣƤƶƣơƿǀƻƺƹơƩƫƶƢƽƣƹǂƩƺƫƹƿƩƣ process of following a learning trajectory or carefully selected sequence of problems designed to induce concept development under the tutelage of a researcher-teacher observed that they do so without noticeable perturbation. “The key ǂƞƾ DŽƶƣᅷƾ ƞƟƾƿƽƞơƿƫƺƹ ƽƣƼǀƫƽƣƢ ƹƺ ƶƣƞƻ ƺƤ ƫƹƾƫƨƩƿ ƺƽ ƻƽƺƟƶƣƸᅟƾƺƶǁƫƹƨ ƟƽƣƞƴƿƩƽƺǀƨƩᄙ ǀƽƿƩƣƽᄕƫƿƽƣƼǀƫƽƣƢƹƺƫƹƻǀƿƤƽƺƸƿƩƣƿƣƞơƩƣƽƺƽƺƿƩƣƽƾƿǀƢƣƹƿƾᅺᄬƻᄙᇳᇻᇹᄭᄙ We note that although these children display no perturbation, one could ƞƽƨǀƣƿƩƣƽƣǂƞƾƞƢƫƾƿƫƹơƿƾƩƫƤƿƫƹƿƩƣơƩƫƶƢƽƣƹᅷƾƻƣƽơƣƫǁƣƢƾƿƽǀơƿǀƽƣᄙƣƽƩƞƻƾ ƞƾƺƹᅷƾ ᄬᇳᇻᇺᇻᄭ ƿƣƽƸƫƹƺƶƺƨDŽ ƺƤ ƞ ᅸƢƣƶƫơƞƿƣ ƾƩƫƤƿ ƫƹ ƞƿƿƣƹƿƫƺƹᅺ ƫƾ Ƹƺƽƣ ƞƻƻƽƺpriate in describing this process. The Fir Tree Aha! moment is the insight in which bisociation, or more correctly trisociation, occurs not as the result of the block during the solving of the problem but from the need to integrate ƹƣǂƴƹƺǂƶƣƢƨƣǂƫƿƩƣǃƫƾƿƫƹƨƾơƩƣƸƞᄙƩƣƢƫƾơǀƾƾƫƺƹƫƹƩƞƻƿƣƽᇳᇵƾƩƺǂƾƩƺǂ abstraction may require the formation of the bisociative frame within the shift of attention. 3.2.8 Empirical Methodology: Qualitative Assessment of Creative Process The question of the mutual relationships between mathematical creativity and mathematical giftedness and the possibility of developing creativity in ƞƶƶƾƿǀƢƣƹƿƾƽƣƸƞƫƹƾƺƻƣƹᄙƩƫƾƢǀƞƶƫƿDŽƫƾơƺƹƹƣơƿƣƢƿƺDŽƨƺƿƾƴDŽᅷƾǁƫƣǂƺƤơƽƣativity as one of the important mechanisms of new knowledge construction, ǂƫƿƩƞƢƫƾƿƫƹơƿƫƺƹƟƣƿǂƣƣƹơƽƣƞƿƫǁƫƿDŽƿƩƞƿƶƣƞƢƾƿƺƩƫƾƿƺƽƫơƞƶƢƫƾơƺǁƣƽƫƣƾᄬƩƫƾƿƺƽƫơƞƶơƽƣƞƿƫǁƫƿDŽᄭƞƹƢơƽƣƞƿƫǁƫƿDŽƿƩƞƿơƺƹƿƽƫƟǀƿƣƾƿƺƿƩƣƞƢǁƞƹơƣƸƣƹƿƺƤƣƞơƩ ƾƿǀƢƣƹƿᅷƾƶƣƞƽƹƫƹƨᄬƻƣƽƾƺƹƞƶơƽƣƞƿƫǁƫƿDŽᄭᄙ One issue that arises is that outside of the work done by constructivist during empirical teaching experiments little is known about how to observe, analyze, and assess the structural process of the insight and learning. Thus, like a constructivist, instead of using measures such as flexibility, originality, and fluency to assess the creative process, we use qualitative assessment of individual students and social classroom interactions with not-necessarily gifted students. The nature of understanding grasped during the Aha! experience itself needs to be understood and clarified during attempts to facilitate this experience, in ƿƩƣơƶƞƾƾƽƺƺƸᄙƩƫƾƸƣƿƩƺƢǂƣƽƣƤƣƽƿƺƞƾƣƸƻƫƽƫơƞƶᄬƼǀƞƶƫƿƞƿƫǁƣᄭƽƣƾƣƞƽơƩƿƺ ƾƿǀƢDŽơƽƣƞƿƫǁƫƿDŽƺƤƞƶƶƾƿǀƢƣƹƿƾᄕƺƽƞƾƽƞƟƩǀᄬᇴᇲᇳᇸᄭǂƺǀƶƢơƩƞƽƞơƿƣƽƫDžƣƫƿƞƾƿƩƣ creativity potential within each student.

ሬሬ 3.3

ѹ‫ؗءؔؔ؛ؖآءإؔح‬Ѹؔ‫إؘ؞‬

Empirical Approach

The empirical approach involves the analysis of several collected descriptions ƺƤᄬᇳᄭƩƞᄛƸƺƸƣƹƿƾƤƺǀƹƢƫƹƸƞƿƩƣƢǀơƞƿƫƺƹƶƫƿƣƽƞƿǀƽƣᄕᄬᇴᄭƢƣƾơƽƫƻƿƫƺƹƾƺƤ ƩƞᄛƸƺƸƣƹƿƾƤƞơƫƶƫƿƞƿƣƢƫƹƺǀƽƸƞƿƩƣƸƞƿƫơƾơƶƞƾƾƣƾᄕƞƹƢᄬᇵᄭƸƞƿƩƣƸƞƿƫơƞƶ teaching/learning moments described in the literature where bisociation has ƟƣƣƹƸƫƾƾƣƢᄬƩƞƻƿƣƽᇳᇵᄭᄙƣƹƿƽƞƶƿƺƺǀƽƞƻƻƽƺƞơƩƩƣƽƣƫƾƿƩƣƨƞƻƫƹǀƹƢƣƽstanding suggested by the definition of bisociation; if the frames of experience are initially unconnected, then there must be a gap, an absence of connection between them. The gaps we observe in the collection of Aha! moments can be separated into two basic categories: the absence of a solution to a complex problem and the absence of understanding of a concept. Consequently, the collection of reported Aha! moments divides into two types: Type A, mathematical situations involving complex problems assigned to students for collaborative or individual independent work, and Type B, mathematical situations involving scaffolding of student understanding by a mentor. ơƞƤƤƺƶƢƫƹƨ ƺƤ ƾƿǀƢƣƹƿ ǀƹƢƣƽƾƿƞƹƢƫƹƨ ơƞƹ ƿƞƴƣ ƻƶƞơƣ ƿƩƽƺǀƨƩ ᄬᇳᄭ ƿƩƣ Ƣƫƞlogue of the teacher or mentor with the latter providing hints and asking questions but leaving enough intellectual freedom for the student to be able to ƞƽƽƫǁƣƞƿƩƣƽƺǂƹǀƹƢƣƽƾƿƞƹƢƫƹƨᄬƿƩƣ ƶƣƻƩƞƹƿƩƞᄛƸƺƸƣƹƿᄭƞƹƢᄬᇴᄭƿƩƽƺǀƨƩ the careful design of the problem sequence, which leads the student toward illumination. As the discussion proceeds we explore different aspects of facilitating Aha! moments, such as the role of the teacher, the uncertainty connected with occurrence, the difficulties in the recognition of the Aha! moment, as well as student interaction within which bisociation often takes place. Despite the interest of the profession in the creativity of the Eureka experience in the classroom, there have not been many precise descriptions of the circumstances of their occurrence in the math education literature. We have found six ƺƤƿƩƣƸᄬƫƶưƣƢƞƩƶᄕᇴᇲᇲᇶᄖƞƢƣưᄕᇳᇻᇻᇻᄖƞƽƹƣƾᄕᇴᇲᇲᇲᄖƺƺƹᄕᇴᇲᇳᇴᄖƞƶƞƿƹƫƴѵƺƫơƩǀᄕᇴᇲᇳᇶᄖƞƴƣƽᄕᇴᇲᇳᇸƞᄖƺƶƢƫƹᄕᇴᇲᇳᇺᄭᄕƞƹƢƿƩƣDŽƞƽƣƾǀƻƻƶƣƸƣƹƿƣƢƟDŽƿƩƣƩƞᄛ moments facilitated in the classrooms of the authors, and the Aha! moment descriptions found in the sources outside of math education literature. The ơƺƶƶƣơƿƫƺƹƫƹƩƞƻƿƣƽᇳᇹᄕƿƩƺǀƨƩƶƫƸƫƿƣƢᄕƫƾƢƫǁƣƽƾƣƣƹƺǀƨƩƿƺƞƶƶƺǂƤƺƽƢƽƞǂƫƹƨ some instructional and research conclusions. The collection naturally divides into two subcollections in accordance to the type of hole in understanding within the mathematical situation: the ƞƟƾƣƹơƣƺƤƞƾƺƶǀƿƫƺƹƿƺƞƻƽƺƟƶƣƸᄬDŽƻƣᄭƞƹƢƿƩƣƞƟƾƣƹơƣƺƤǀƹƢƣƽƾƿƞƹƢƫƹƨ ƺƤƞơƺƹơƣƻƿᄬDŽƻƣᄭᄙ

ѹ‫ؠآآإئئؔ؟‬Ѽؔ‫ؙآءآ؜اؔا؜؟؜ؖ‬ѷ‫ؔ؛‬ᄛ҃‫اءؘؠآ‬ѿ‫ئا؛ؚ؜ئء‬

ሬር

3.3.1 Type A The Type A reports describe Aha! moments that occurred while solving complex open problems designed in advance. ᇵᄙᇵᄙᇳᄙᇳ ƺƸƻƶƣǃƽƺƟƶƣƸᄕƺƶƶƞƟƺƽƞƿƫǁƣƣƿƿƫƹƨ

ƹ ᅸƞƨƫơƞƶ ƺƸƣƹƿƾ ƫƹ ƞƿƩƣƸƞƿƫơƾᅺ ᄬᇴᇲᇲᇲᄭᄕ ƞƽƹƣƾ Ƣƫƾơǀƾƾƣƾ ƞƹ ƫƹƿƣƨƽƞƿƣƢ ƸƞƿƩƣƸƞƿƫơƾơƺǀƽƾƣƾƩƣƿƞǀƨƩƿƿƺƾƿǀƢƣƹƿƾƫƹƿƩƣᇳᇳƿƩƨƽƞƢƣƫƹƣƶƟƺǀƽƹƣᄕǀƾtralia. The course, which was taken as an elective, contained some elements of calculus. Barnes examined the collaborative work between teams of students, when interaction was one of the facilitation methods employed by the instructor.

‫ؘإبؚ؜ؙ‬ᇵᄙᇳᏻƣƿƤƺƽơƺƹƤƣơƿƫƺƹƞƽDŽƟƺǃᄬƤƽƺƸƞƽƹƣƾᄕᇴᇲᇲᇲᄕƻᄙᇵᇷᄭ

This is how the problem is described: ᄴ ᄵƞơƩƨƽƺǀƻǂƞƾƨƫǁƣƹƞƾƩƣƣƿƺƤƻƞƻƣƽᇴᇲơƸƟDŽᇵᇲơƸᄕǂƫƿƩƫƹƾƿƽǀơtions on how to fold it into an open box suitable for holding candy. Then they were asked to work out how to vary the positions of the folds to ƺƟƿƞƫƹƞƶƞƽƨƣƽƟƺǃᄙᄬƻᄙᇵᇷᄭ The choice of the problem has a chance to create the bisociative frame for the students not necessarily because of its mathematical complexity but because of the gap between their preparation and the mathematical concepts needed ƿƺƾƺƶǁƣƿƩƣƻƽƺƟƶƣƸᄙƞƽƹƣƾᅷƾᅸƢƫƾơƶƞƫƸƣƽᅺƺƹƿƩƞƿƫƾƾǀƣƫƾǁƣƽDŽƾƫƨƹƫƤƫơƞƹƿƞƹƢ plays a conceptual role in the design of the teaching experiment:

ƿƫƾƫƸƻƺƽƿƞƹƿƿƺƹƺƿƣƿƩƞƿƿƩƣƾƣƾƿǀƢƣƹƿƾƩƞƢƹƣǁƣƽƩƞƢƞƤƺƽƸƞƶƶƣƾƾƺƹƺƹƸƞǃƫƸǀƸᅬƸƫƹƫƸǀƸƻƽƺƟƶƣƸƾᄙƩƫƾƿƞƾƴǂƞƾƿƩƣƿƩƫƽƢƫƹƞƾƣƽƫƣƾ

ርሤ

ѹ‫ؗءؔؔ؛ؖآءإؔح‬Ѹؔ‫إؘ؞‬

designed to introduce them to such problems and to have them develop techniques for solving them and reflect on the concepts used. Earlier in the year the class worked on a project to find gradients of graphs. As a result of this, they knew the definition of a derivative and how to calculate derivatives of polynomials, but their only knowledge about maxima and minima was what they had worked out in the course of the previous ƿǂƺƿƞƾƴƾᄙƩƣDŽƩƞƢƟƣƣƹƨƫǁƣƹƹƺƽǀƶƣƾƺƽƻƽƺơƣƢǀƽƣƾᄙᄬƻᄙᇵᇷᄭ What the author wanted to ensure was that the class organization at that moment was closest to the condition of “untutored learning” as is characterƫƾƿƫơƤƺƽƿƩƣƺơơǀƽƽƣƹơƣƺƤƟƫƾƺơƫƞƿƫƺƹƾᄬƺƣƾƿƶƣƽᄕᇳᇻᇸᇶᄕƻᄙᇸᇷᇺᄭᄙƞƽƹƣƾƽƣƻƺƽƿƾ several possible Aha! moments during the collaborative process of solving by the group of three students, Maria, Niedra, and Simon, all involving more or less clear bisociative frames. The first insight was during an initial incubation period in which the students were unsure how to proceed. From the diagram, they had formulated ƿǂƺƣƼǀƞƿƫƺƹƾᄘᇵǃናᇴDŽኙᇵᇲƞƹƢƩኙᇴኔDžƟǀƿƿƩƣƽƣǂƣƽƣƿƺƺƸƞƹDŽǁƞƽƫƞƟƶƣƾƤƺƽ them to make sense of these equations. At this point, Maria noticed that z was always half x; the other two students took a long time to understand her insight even though Maria gradually repeated it in a slightly different context. ƞƽƹƣƾᄬᇴᇲᇲᇲᄭơƺƹơƶǀƢƣƢƿƩƞƿƿƩƣƾƿǀƢƣƹƿƾᄕƫƹơƶǀƢƫƹƨƞƽƫƞᄕǂƣƽƣƹƺƿƫƹƫtially able to process her realization but once they did, it set in motion the path toward a solution. We differ with Barnes in the analysis of the first interaction; we see this as an example of an individual bisociation by Maria through her connection of two previously unconnected domains. The one domain was ƩƣƽƽƣƞƾƺƹƫƹƨƞƟƺǀƿ ƫƨǀƽƣᇵᄙᇳᅭᅸƿƩƞƿDžǂƞƾƞƶǂƞDŽƾƩƞƶƤǃᄙᅺƩƣƺƿƩƣƽƢƺƸƞƫƹᄕ ƣǃƻƽƣƾƾƣƢƶƞƿƣƽᄕǂƞƾƞƽƫƞᅷƾƾƻƞƿƫƞƶƫƸƞƨƫƹƞƿƫƺƹᄕᅸ ưǀƾƿƿƩƺǀƨƩƿƫƿƾᄴƿƩƣǁƞƶǀƣƺƤ DžᄵᄴƩƞƾᄵƨƺƿƿƺƟƣƩƞƶƤƺƤǃƾƺƫƿǂƫƶƶƤƺƶƢǀƻᄙᅺ Ʃƫƾ ơƺƺƽƢƫƹƞƿƫƺƹᄕ ƺƽ ƽƣƞƾƺƹƫƹƨᄕ ƾǀƨƨƣƾƿƾ ƿƩƞƿ ƞƽƫƞᅷƾ ƫƹƿƣƽƫƺƽƫDžƞƿƫƺƹ Ʃƞƾ ƺơơǀƽƽƣƢᄙƣơƞǀƾƣƺƤƿƩƞƿơƺƹƹƣơƿƫƺƹᄕƫƿƫƹƿƣƽƣƾƿƣƢƿƩƣƺƿƩƣƽƿǂƺƾƿǀƢƣƹƿƾᄙ ƿ became part of a shared creative moment only after Maria interiorized her ƫƹƾƫƨƩƿ ƞƹƢ ǂƩƣƹ ƿƩƣ ƺƿƩƣƽ ƿǂƺ ƾƿǀƢƣƹƿƾ ǂƣƽƣ ƽƣƞƢDŽ ƿƺ ƫƹƿƣƽƹƞƶƫDžƣ ƞƽƫƞᅷƾ ƫƹƤƺƽƸƞƿƫƺƹᄙ ƹƢƣƣƢᄕ ƫƿ ƿƺƺƴ ƫƸƺƹ Ƥƺǀƽ ƾƣơƺƹƢƾ ƿƺ ƨƽƞƾƻ ƫƿᄙ Ʃƫƾ ƣǃƞƸƻƶƣ underscores the need for the conversing learners to have interlocking zones of ƻƽƺǃƫƸƞƶƢƣǁƣƶƺƻƸƣƹƿᄬҐ҆Ѻƾᄭᄙ The second insight occurred during an intense dialogue between two boys ƞƤƿƣƽƿƩƣDŽƩƞƢƨƽƞƻƩƣƢƿƩƣƶƫƹƣᇵǃናᇴDŽኙᇵᇲᄙƩƣƿǂƺƟƺDŽƾᄕƫƸƺƹƞƹƢƫƣƢƽƞᄕ began to express possible values of x and y as “domain” and “range” and had ƢƣƿƣƽƸƫƹƣƢƿƩƞƿƿƩƣƢƺƸƞƫƹǁƞƶǀƣƺƤǃǂƞƾƟƣƿǂƣƣƹᇲƞƹƢᇳᇲᄙƩƣƹƿƩƣƿƣƞơƩƣƽ ƞƾƴƣƢƿƩƣƸƿƺƣǃƻƶƞƫƹƿƩƣƫƽƽƣƞƾƺƹƫƹƨᄕƫƸƺƹƾƿƞƿƣƢᄘᅸƣơƞǀƾƣƿƩƣƽƣᅷƾƿƩƽƣƣƺƤ

ѹ‫ؠآآإئئؔ؟‬Ѽؔ‫ؙآءآ؜اؔا؜؟؜ؖ‬ѷ‫ؔ؛‬ᄛ҃‫اءؘؠآ‬ѿ‫ئا؛ؚ؜ئء‬

ርሥ

ƿƩƣƸᄕ ƞƹƢ ƫƤ ƫƿ ǂƞƾ ƿƣƹᄕ ƿƩƣƽƣᅷƢ Ɵƣ ᇵᇲᄕ ƞƹƢ ƿƩƣƽƣᅷƾ ƹƺ ƽƺƺƸ Ƥƺƽ DŽᅷƾᄕ ƞƹƢ ƞƹDŽthing above.” As Simon explained, Niedra understood more fully the connection between the possible domain and the range values with the possible dimensions of the box. He commented on his realization in the following words: Well, Simon was the one who came up with it and he just realized that ƫƤDŽƺǀƾƺƶǁƣƤƺƽDŽᄕƿƩƞƿᅷƶƶƨƫǁƣDŽƺǀƞƽǀƶƣᄕƸƞƴƣƫƿƻƺƾƾƫƟƶƣƿƺƨƽƞƻƩƫƿᄕƞƹƢ ƿƩƣƽƣƤƺƽƣƟƣƞƟƶƣƿƺƾƣƣǂƩƣƽƣƿƩƣƽƣƫƾƞƽƣƾƿƽƫơƿƣƢƢƺƸƞƫƹᄙᄚƣƾᄕǂƣƶƶᄕ once Simon showed it to us, and we could all see what he was doing with ƫƿᄕƫƿưǀƾƿƟƣơƞƸƣơƶƣƞƽᄙᄬƻᄙᇵᇹᄭ This remark suggests a similar situation as before: one student interiorizes his understanding, while his partner internalizes the information as his own realization. The “hidden analogy” here were symbols x and y, which were present in ƞƶƶƿƩƽƣƣơƺƸƻƺƹƣƹƿƾᄙ ƹƿƩƫƾơƞƾƣᄕƿƩƣƟƫƾƺơƫƞƿƫƺƹǂƞƾƟƣƿǂƣƣƹƿƩƣƞƶƨƣƟƽƞƫơ equation with two variables x and y with its graph and between the possible domain.

ƿᅷƾƫƹƿƣƽƣƾƿƫƹƨƿƺƹƺƿƣƿƩƞƿƿƩƣƿƣƞơƩƣƽᅷƾƽƺƶƣƫƹƻƽƺƸƺƿƫƹƨƿƩƫƾƩƞᄛƸƺƸƣƹƿ was to ask the students to verbalize their actions during a build-up period, during which they had condensed the problem into a graph on the xy-coordiƹƞƿƣƻƶƞƹƣᄙƹƣƾƿǀƢƣƹƿƽƣƤƶƣơƿƫƹƨƺƹƿƩƣƺƿƩƣƽᅷƾƣǃƻƶƞƹƞƿƫƺƹƶƣƢƿƺƞƹƫƹƾƫƨƩƿ on how their graphical representation could be used to draw inferences about ƿƩƣƢƺƸƞƫƹƺƤǃƿƩƞƿǂƞƾƿƩƣǂƫƢƿƩƺƤƿƩƣƟƺǃᄙ ƹƿƩƫƾơƞƾƣᄕƿƩƣƫƹƾƫƨƩƿǂƞƾ ƟƞƾƣƢƺƹƫƸƺƹᅷƾƣǃƻƶƞƹƞƿƫƺƹᄖƩƺǂƣǁƣƽᄕƞƤƿƣƽƿƩƣƾƿǀƢƣƹƿƩƞƢƾƩƞƽƣƢƿƩƣǂƺƽƴ leading up to the explanation and insight, both had shared a preparation stage of the Gestalt sequence together. ƩƞƿƫƾǂƩDŽƿƩƣƫƹƾƫƨƩƿǂƞƾƾƿƽƺƹƨᄬƞƹƞơơƺƸƸƺƢƞƿƫƺƹᄭᄬƾƣƣƩƞƻƿƣƽᇳᇳᄭᄙ We can distinguish the process of interiorization of the connection between ƿƩƣƨƽƞƻƩƞƹƢƿƩƣơƺƹơƣƻƿƺƤƫƸƺƹᅷƾƢƺƸƞƫƹᄕƤƺƶƶƺǂƣƢƟDŽƫƿƾƫƹƿƣƽƹƞƶƫDžƞƿƫƺƹ by Naidra and Maria. This demonstrates that a shared learning experience can lead to creative moments of insight when the members are active particiƻƞƹƿƾƫƹƿƩƣƻƽƣƻƞƽƞƿƫƺƹᄧƟǀƫƶƢƫƹƨǀƻƻƽƺơƣƾƾᄙ ƹƿƩƫƾơƞƾƣᄕƞƹƫƹƢƫǁƫƢǀƞƶƫƹƾƫƨƩƿ ᄬƿƩƽƺǀƨƩƫƹƿƣƽƫƺƽƫDžƞƿƫƺƹᄭƺƤƫƸƺƹǂƩƣƹǁƣƽƟƞƶƫDžƣƢƿƺƿƩƣƫƹƾƿƽǀơƿƺƽƶƣƢƞƫƢƽƞ ƿƺƞƢƣƣƻƣƽǀƹƢƣƽƾƿƞƹƢƫƹƨƺƤƫƸƺƹᅷƾƫƹƾƫƨƩƿᄬƿƩƽƺǀƨƩƫƹƿƣƽƹƞƶƫDžƞƿƫƺƹᄭᄙ The third insight occurred during a building-up phase when the group realized it still had too many variables and began to construct a table of possible values for x, y, h, and the resulting volume. The students used the table to find values that maximized the volume and had already expressed y and h as variables dependent on x. They gradually turned their attention to their knowledge of calculus, pondering how taking a derivative of some unknown

ርሦ

ѹ‫ؗءؔؔ؛ؖآءإؔح‬Ѹؔ‫إؘ؞‬

function could be related to what they were doing. Again, having too many variables to construct such a function was a shared dilemma. At this point, in an intense moment of insight, Simon bisociated the equation V = xyh with ƿƩƣƫƽƞƶƽƣƞƢDŽơƺƹƾƿƽǀơƿƣƢƣƼǀƞƿƫƺƹƾƺƤDŽƞƹƢƩᄬƫƹƿƣƽƸƺƤǃᄭƿƺơƺƹƢƣƹƾƣƿƩƣ ƨƽƺǀƻᅷƾǂƺƽƴƫƹƿƺƞƾƫƹƨƶƣƣƼǀƞƿƫƺƹƤƺƽǁƺƶǀƸƣƫƹƿƣƽƸƾƺƤǃᄙ Simon expressed his thinking leading up to this realization as follows: “Well basically volume, equals, x times y times h. So, volume, if we want just one, we ơƞƹƽƣƶƞƿƣƫƿƞƶƶƿƺƺƹƣǁƞƽƫƞƟƶƣᄕƿƩƞƿᅷƾx ƞƹƢƿƩƣƹǂƣᅷƶƶƩƞǁƣƿǂƺǁƞƽƫƞƟƶƣƾƿƺ graph.” Barnes noted that the students were excited and described the table ƞƾƨƫǁƫƹƨƿƩƣƸƟƣƞǀƿƫƤǀƶƫƹƾƫƨƩƿƫƹƿƺƿƩƣƻƽƺƟƶƣƸᄙƫƣƢƽƞƽƣƾƻƺƹƢƾᄘᅸƩƿƩƞƿᅷƾ ƟƽƫƶƶƫƞƹƿᄙᅺƫƣƢƽƞǂƞƾưǀƾƿƞƾƣǃơƫƿƣƢƞƾƫƸƺƹƞƹƢơƶƣƞƽƶDŽǀƹƢƣƽƾƿƺƺƢƫƸƺƹᅷƾ insight. When asked by the instructor, Niedra responded: Ƹᄕ ƿƩƫƹƴƿƩƞƿǂƞƾƫƸƺƹƞƨƞƫƹᄕǀƸᄕǂƣᄕ ơƺǀƶƢƾƣƣǂƩƣƽƣƿƩƣǁƺƶǀƸƣ was going to be x times y ƿƫƸƣƾᇴᇲƸƫƹǀƾz. [He gave a detailed explanaƿƫƺƹᄙᄵƫƸƺƹƾƞǂƿƩƞƿᄕơƞƸƣǀƻǂƫƿƩƫƿᄕƞƹƢƿƩƣƹ ƾƞǂƫƿᄖƩƣƣǃƻƶƞƫƹƣƢƫƿ ƿƺƸƣᄙᄬƻᄙᇵᇺᄭ ƩƫƾƫƾƞƹƣǃƞƸƻƶƣƺƤƞƾƩƞƽƣƢᅟƶƣƞƽƹƫƹƨƣǃƻƣƽƫƣƹơƣǂƩƣƽƣƫƸƺƹᅷƾƫƹƾƫƨƩƿǂƞƾ fully understood as soon as he discussed it with the others. Stated differently, ƫƣƢƽƞᅷƾǀƹƢƣƽƾƿƞƹƢƫƹƨƺƤƫƸƺƹᅷƾƫƹƾƫƨƩƿǂƞƾƫƸƸƣƢƫƞƿƣᄕƞƾƺƻƻƺƾƣƢƿƺƣƞƽƶƫƣƽ ǂƩƣƹ Ʃƣ ǀƹƢƣƽƾƿƺƺƢ ƿƩƣ ƾƫƨƹƫƤƫơƞƹơƣ ƺƹƶDŽ ƞƤƿƣƽ ƫƸƺƹᅷƾ ƣǃƻƶƞƹƞƿƫƺƹƾ ƿƺ the teacher. This sequence demonstrates some important principles for creative learning environments. First, although creative insights frequently occur during shared-learning experiences, and often they are of individual in nature but the transfer of learning between the individuals takes place by internalization through the collaborative “thinking together.” Second, for shared learning to lead to creative insights, the members must by active participants. Third, creative insights can occur and be shared only when the participants are at the ƽƫƨƩƿƾƿƞƨƣᄕƶƣǁƣƶᄕƺƽƻƶƞơƣƿƺǀƹƢƣƽƾƿƞƹƢƣƞơƩƺƿƩƣƽᅷƾƫƹƾƫƨƩƿƾᄙ ƫƹƞƶƶDŽᄕƞƾƹƺƿƣƢ ƫƹƩƞƻƿƣƽᇷᄕƞƹƫƸƻƺƽƿƞƹƿƽƺƶƣƺƤƿƩƣƿƣƞơƩƣƽƫƾƿƺƽƣƼǀƫƽƣƾƿǀƢƣƹƿƾƿƺƣǃƻƶƞƫƹ and justify their insights not only to help them clarify their understanding but also to help their partners understand their actions. ᇵᄙᇵᄙᇳᄙᇴ ƺƸƻƶƣǃƽƺƟƶƣƸᄘƩƣ ƫƽƽƣƣƫƹƞƹ ƹƢƫǁƫƢǀƞƶƣƿƿƫƹƨ Below we have the discussion of the student from a remedial intermediate algebra class designed for students whose mathematical proficiency is below the college level of mathematics.

ѹ‫ؠآآإئئؔ؟‬Ѽؔ‫ؙآءآ؜اؔا؜؟؜ؖ‬ѷ‫ؔ؛‬ᄛ҃‫اءؘؠآ‬ѿ‫ئا؛ؚ؜ئء‬

ርሧ

The complex problems at the appropriate level were given each week to students to work on at home for extra credit; the Fir Tree of generalization was one of them. The general instruction for these assignments was to pay attention to possible Aha! moments. To get a full credit for the Aha! moment, the student had to describe with mathematical precision when exactly that insight happened and what exactly was understood through it. The exercise had several steps leading to the generalization of the formula representing the sequence of a picture with systematically increasing number of tree branches. Ʃƣ ƫƽƽƣƣƩƞᄛƸƺƸƣƹƿᄬƩƞƻƿƣƽᇳᇹᄭᄕǂƩƫƶƣƿƩƣơƺƹƾƣƼǀƣƹơƣƺƤƿƩƣơƺƸplex problem solution, is a transitional situation introducing us to Type B mathematical situations where insight is characterized by the conceptual learning ᄬƾƣƣƫƸƺƹᄕᇴᇲᇳᇶᄖƫƸƺƹƣƿƞƶᄙᄕᇴᇲᇳᇲᄭƽƞƿƩƣƽƿƩƞƹƿƩƽƺǀƨƩƻƽƺƟƶƣƸᅟƾƺƶǁƫƹƨᄙ ƹ general, Type B presents the typical classroom situations where the insight occurs during the interaction between teacher/mentor and student. Fir Tree Aha! moment is especially important because it shows that an Aha! moment does not have to occur in the context of problem-solving requiring ƻƣƽƿǀƽƟƞƿƫƺƹᄙ ƿƾƫƸƻƺƽƿƞƹơƣƫƾǀƹƢƣƽƾơƺƽƣƢƟDŽƾƣǁƣƽƞƶơƩƞƻƿƣƽƾǂƩƫơƩƞƹƞƶDŽDžƣ it from different points of view. The Aha! moment took place while solving the homework problem and was recognized by the student as such. The instructor had conducted a classroom investigation of different methods to facilitate an Aha! moment. Students were informed about and discussed the occurrence of such moments in their lives. They were asked to pay attention when they occurred and to provide a precise description of the mathematics situation when it occurred.

ƹƿƩƫƾƾƿƞƹƢƞƽƢƨƣƹƣƽƞƶƫDžƞƿƫƺƹƻƽƺƟƶƣƸᄕƞƾƣƼǀƣƹơƣƺƤƽƣơƿƞƹƨƶƣƾƾDŽƸƟƺƶƫDžƫƹƨƿƩƣ ƫƽƽƣƣƫƾǁƫƾǀƞƶƶDŽƻƽƣƾƣƹƿƣƢǂƫƿƩƿƩƣƩƣƶƻƺƤᇴᄕᇸᄕᇳᇴᄕᇴᇲǀƹƫƿƾƼǀƞƽƣƾƤƺƽ ƿƩƣƹǀƸƟƣƽƺƤƿƩƣƤƫƨǀƽƣƹኙᇳᄕᇴᄕᇵᄕᇶƞƾƿƩƣƢƫƞƨƽƞƸƟƣƨƫƹƾƿƺƨƽƺǂᄬƩƞƻƿƣƽ ᇳᇹᄭᄙƩƣƾƺƶǁƣƽƫƾƞƾƴƣƢƤƫƽƾƿƩƺǂƸƞƹDŽƽƣơƿƞƹƨƶƣƾƤƺƽƹኙᇳᇲᄕƿƩƣƹǂƩƞƿƫƾƿƩƣ general form of the equation as the function of “n.” One student determined an equation for this pattern as, the nth diagram Ʃƞƾƹᄬƹናᇳᄭƽƣơƿƞƹƨƶƣƾᄙ ƹƾƿǀƢƣƹƿᅷƾǂƺƽƢƾᄘ ƹƾǂƣƽᇱᇷ

ƿƫƾƽƣƞƶƶDŽǂƽƺƹƨƟƣơƞǀƾƣ ƿƞƨƣᇳᇸኙᇳᇸᄬᇳᇸናᇳᄭኙᇴᇹᇴ ƿƞƨƣᇳᇹኙᇳᇹᄬᇳᇹናᇳᄭኙᇵᇲᇸ

ርረ

ѹ‫ؗءؔؔ؛ؖآءإؔح‬Ѹؔ‫إؘ؞‬

The fir tree would never have 274 unit squares.

ƩƞƢƞƿƽƣƸƣƹƢƺǀƾƩƞᄛƸƺƸƣƹƿᄙ ưǀƾƿƽƣƞƶƫDžƣƢƿƩƞƿƿƩƣƤƺƽƸǀƶƞ ƨƺƿ ƤƽƺƸƿƩƣƻƞƿƿƣƽƹƾǂƞƾƞƤƞơƿƺƽƫDžƣƢƣǃƻƽƣƾƾƫƺƹƞƹƢƫƤ ƸǀƶƿƫƻƶDŽƫƿᄬƹᄬƹና ᇳᄭኙǀƹƫƿƾƾƼǀƞƽƣᄭ ǂƺǀƶƢƩƞǁƣƾƺƸƣƿƩƫƹƨƶƫƴƣƞƹƞƶƨƣƟƽƞƫơƣǃƻƽƣƾƾƫƺƹ exactly a trinomial expression that can be factorized as well, and it equals real numbers, for example: ƹᄬƹናᇳᄭኙᇳᇴ

ƹᇄᇴናƹኙᇳᇴ

ƹᇄᇴናƹኔᇳᇴኙᇲ ᏺƹᇄᇴናƹኔᇴᇲኙᇲ

ƹᇄᇴናƹኔᇷᇸኙᇲ

ƹᇄᇴናƹኔᇻᇲ

ᄬƹናᇶᄭᄬƹኔᇵᄭኙᇲᄬƹናᇷᄭᄬƹኔᇶᄭኙᇲ

ᄬƹናᇺᄭᄬƹኔᇹᄭኙᇲ

ᄬƹናᇳᇲᄭᄬƹኔᇻᄭ

ƹƢǂƩƣƹƫƿơƺƸƣƾƿƺƹᇄᇴናƹኔᇴᇹᇶᄕƫƿơƞƹƹƺƿƟƣƤƞơƿƺƽƫDžƣƢᄙ Here, the Aha! moment took place as a further refinement of the solution, which was obtained before the insight took place through the standard process of abstraction and generalization of a pattern. Once the solver abstracted ƿƩƣƽƣƶƞƿƫƺƹƾƩƫƻƹᄬƹናᇳᄭƞƾƽƣƻƽƣƾƣƹƿƫƹƨƿƩƣǁƞƶǀƣƺƤƾƼǀƞƽƣƾƞƿƾƿƞƨƣᅸƹᅺƾƩƣ coordinated this with the problem situation to gain a further insight. Thus, her motivation could not have been to solve the problem but rather to find a more ƨƣƹƣƽƞƶƸƣƿƩƺƢƺƤƾƺƶǁƫƹƨᄬƤƺƽƸƺƽƣƺƹƿƩƞƿƾǀƟưƣơƿᄕƾƣƣƩƞƻƿƣƽᇺᄭᄙƩƣƽƣƞƶƫDžƣƢƿƩƞƿƿƩƣ ƫƽƽƣƣǂƫƶƶơƺƹƿƞƫƹƾƼǀƞƽƣƾƫƤƿƩƣƽƣƫƾƞᅸƹᅺƾǀơƩƿƩƞƿƹᄬƹናᇳᄭ = N and at this point reflection upon solution activity to find such an “n” leads her to synthesize understanding of how to solve this with her understanding of factoring trinomials. Again, this abstraction creates a new code or scheme in which she reduces the problem to whether we can factor the trinomial nሿ + ƹኔኙᇲᄙᄬ ƺƽƞƸƺƽƣƢƣƿƞƫƶƣƢƞƹƞƶDŽƾƫƾᄕƾƣƣƩƞƻƿƣƽᇶᄙᄭ Within the Fir Tree Aha! moment there might have been not one jump but two, possibly simultaneous, leaps of insight or a single leap encompassing several related concepts. One of them connected the factorized algebraic expresƾƫƺƹƹᄬƹናᇳᄭǂƫƿƩƿƩƣƼǀƞƢƽƞƿƫơƣƼǀƞƿƫƺƹƹᄬƹናᇳᄭኙᄕƞƹƢƿƩƣƺƿƩƣƽƺƹƣǂƞƾ between the quadratic equation and factorization as the method of solving it. 3.3.2 Type B Type B classroom mathematical situations arise spontaneously during a regular class and when preparing for or during student discussions about a mathematics They also arise in the face of some fundamental absence of knowledge or discovery of a misconception. Problems and questions leading to the Aha!

ѹ‫ؠآآإئئؔ؟‬Ѽؔ‫ؙآءآ؜اؔا؜؟؜ؖ‬ѷ‫ؔ؛‬ᄛ҃‫اءؘؠآ‬ѿ‫ئا؛ؚ؜ئء‬

ርሩ

moments discussed below have been designed spontaneously in action of scaffolding to facilitate understanding of the relevant concept by the student. ƩƞƻƿƣƽᇳᇳơƺƹƾƫƢƣƽƾƾǀơƩƾƫƿǀƞƿƫƺƹƾƿƣƞơƩƫƹƨƸƺƸƣƹƿƾᄖƩƣƽƣᄕǂƣơƞƶƶƿƩƣƸ “teaching/learning moments” to indicate the closure of the certain learning process within it. Note that these and similar dialogues make out certain indivisible wholes because they contain the birth of the conceptual insight. Both took place during a dialogue with either the teacher or a more knowledgeable peer. The content of the insight during the illumination was a sudden grasp of the relevant concepts. The Elephant Aha! moment reports the dialogue between two pupils in ƽƞƢƣᇵᄕǂƩƫƶƣƿƩƣƢƺƸƞƫƹƩƞᄛƸƺƸƣƹƿƽƣƻƺƽƿƾƿƩƣƢƫƞƶƺƨǀƣƟƣƿǂƣƣƹƞƾƿǀƢƣƹƿ and instructor focused on the elimination of a mathematical misconception. Note the fundamentally different attitude and participation of the teacher/ mentor in both cases in comparison with the problem-solving environment. Not only is the instructor present continuously to student thinking but also directs, through scaffolding, understanding of the concept. leaving enough cognitive space for the student to grasp the it on his or her own.

ƹƾǀơƩƸƺƸƣƹƿƾᄕƿƩƣƿƣƞơƩƣƽᄧƸƣƹƿƺƽƫƾᅸƞƿƺƹƣᅺǂƫƿƩƿƩƣƾƿǀƢƣƹƿƺƽᅸƺƹƿƩƣ same wave length.” Both metaphors convey the high level of intellectual unity of the teacher/mentor with the student at that moment. Although similar to collaborative insights observed, their contents are different. Being “on the same wavelength” between two collaborators working on the ƻƽƺƟƶƣƸƫƾơƞƻƿǀƽƣƢƟDŽƿƩƣƨƽƺǀƻᅷƾƢƣƾơƽƫƻƿƫƺƹƺƤƞƹƩƞᄛƸƺƸƣƹƿƟDŽƫƶưƣƢƞƩƶ ᄬᇴᇲᇲᇶᄭᄕǂƩƺƽƣƻƺƽƿƾƺƹƿƩƣƾƫƸǀƶƿƞƹƣƺǀƾƣǃƻƣƽƫƣƹơƣƺƤƿƩƣƾƞƸƣƩƞᄛƸƺƸƣƹƿ by both collaborators. The description suggests that we have a spectrum of different circumstances when the insight takes place from purely individual ƫƹƾƫƨƩƿƾᄬƤƺƽƣǃƞƸƻƶƣᄕƿƩƣƫƸƩƞᄛƸƺƸƣƹƿƞƹƢƿƩƣƩƞƿ ƾƞƣơƿƺƽᄞƩƞᄛ ƸƺƸƣƹƿᄭ ƿƩƽƺǀƨƩ ƢƫƤƤƣƽƣƹƿ ƸƣƿƩƺƢƾ ƺƤ Ƹƣƹƿƺƽƫƹƨ ƞ ƾƿǀƢƣƹƿ ƟDŽ ƞ ƿƣƞơƩƣƽᄕ ƞ mentor or a classmate to the same insight experienced simultaneously by two ơƺƶƶƞƟƺƽƞƿƺƽƾƢƣƾơƽƫƟƣƢƟDŽƫƶưƣƢƞƩƶᄬᇴᇲᇲᇶᄕơƩƞƻƿƣƽᇹᄕƻƻᄙᇵᇴᅬᇵᇶᄭᄙ ᇵᄙᇵᄙᇴᄙᇳ Ʃƣ ƶƣƻƩƞƹƿƩƞᄛƺƸƣƹƿ

ƩƞƢƞƹƺƻƻƺƽƿǀƹƫƿDŽƿƺƶƫƾƿƣƹƿƺƞƢƫƾơǀƾƾƫƺƹƟƣƿǂƣƣƹƿǂƺƣƹƿƩǀƾƫƞƾƿƫơƾƿǀdents solving a standard word problem: The sum of two numbers is 76. One of ƿƩƣƹǀƸƟƣƽƾƫƾᇳᇴƸƺƽƣƿƩƞƹƿƩƣƺƿƩƣƽᄙ ƫƹƢƟƺƿƩƹǀƸƟƣƽƾᄙ ƿǂƞƾƞƻƽƺƟƶƣƸ ƤƽƺƸƣƸƞƢƣƹƫᅷƾƾƣƿƺƤƻƽƺƟƶƣƸƾƤƺƽƿƩƣᇵƽƢƨƽƞƢƣƞƹƢƺƹƣƩƞƢƿƺƾƺƶǁƣƫƿǀƾƫƹƨ equations. This is where the difficulty appeared. ƽDžƣƸƣƴᄬƽƣƞƢƾƩƣƸƫƴᄭǂƽƺƿƣƿƩƣƣƼǀƞƿƫƺƹƞƾǃናᄬǃናᇳᇴᄭኙᇹᇸᄙƺƾƺƶǁƣƫƿ was a bit of a problem for him, but he dealt with it. He drew an interval and then the following dialogue took place between him and his friend, Bart:

ርሪ

ѹ‫ؗءؔؔ؛ؖآءإؔح‬Ѹؔ‫إؘ؞‬

ᇳᄙ 2. ᇵᄙ  4. ᇷᄙ 6. 7. ᇺᄙ ᇻᄙ  ᇳᇲᄙ ᇳᇳᄙ ᇳᇴᄙ ᇳᇵᄙ ᇳᇶᄙ  ᇳᇷᄙ ᇳᇸᄙ ᇳᇹᄙ

ᇳᇺᄙ ᇳᇻᄙ ᇴᇲᄙ ᇴᇳᄙ

22. ᇴᇵᄙ 24.

P: That is that number. He extended this interval by almost the same ƶƣƹƨƿƩᄕƞƹƢƞƹƺƿƩƣƽƺƹƣƶƫƴƣƿƩƞƿᄙƹƢƿƩƫƾƫƾƿƩƞƿƹǀƸƟƣƽƻƶǀƾᇳᇴᄙ ᄘƹƢƿƩƫƾƞƶƶƿƺƨƣƿƩƣƽƫƾƣƼǀƞƶƿƺᇹᇸᄚ ᄘƺᄕƿƩƫƾƫƾƞƹƣƼǀƞƿƫƺƹᄙᅷDŽƺǀǀƹƢƣƽƾƿƞƹƢᄚ ᄴơƺǀƶƢƹƺƿƞơơƣƻƿƫƿᄙᄵ ᄘƩDŽ ƢƫƢ DŽƺǀ Ƣƽƞǂ ƿƩƫƾ ƫƹƿƣƽǁƞƶᄞƺǀ Ƣƺƹᅷƿ ƴƹƺǂ ǂƩƞƿ ƫƿᅷƾ ƾǀƻposed to be yet. ᄘƩƞƿᅷƾƹƺƿƫƸƻƺƽƿƞƹƿᄙ B: Why 76? ᄘᅷƞǀƾƣƿƩƞƿᅷƾǂƩƞƿƫƾƫƹƿƩƣƻƽƺƟƶƣƸᄙ ᄘƩƞƿǃᄕƿƩƞƿǃƞƢƢƾᇳᇴƞƹƢƿƩƞƿᅷƾƾǀƻƻƺƾƣƢƿƺƟƣᇹᇸᄞ P: Look, instead of x there is a little square in the book. ᄴƾƩƺǂƾƿƩƣƶƫƿƿƶƣƾƼǀƞƽƣƫƹƿƩƣƟƺƺƴᄙᄵ B: Aha, but here, here something else is written. ᄘǀƿƫƿơƺǀƶƢƟƣƞƾƩƣƽƣᄙƹƢƹƺǂ ƞƸƫƹƻǀƿƿƫƹƨƞƹǀƸƟƣƽƫƹƿƺ this square. B: A number?! Why into the square? ᄘ ƺᄕ ƫƿᅷƾ ƫƹƿƺ ƿƩƣ ǂƫƹƢƺǂᄙ ƹƿƺ ƿƩƫƾ ǂƫƹƢƺǂ  ƫƹƻǀƿ ƿƩƣ ƹǀƸƟƣƽ which comes out here. B: But here is a square. ᄴƫƹƾƫƾƿƣƢᄙᄵ ᄘ ƿᅷƾƹƺƿƞƾƼǀƞƽƣᄕƟǀƿƞǂƫƹƢƺǂᄕƞƹƢƺƹƣƫƹƻǀƿƾƿƩƣƹǀƸƟƣƽƾƫƹƿƺ that window. B: How so? ᄘǂƺ ǂƫƹƢƺǂƾ ƞƽƣ ƣƼǀƞƶ ƿƺ ᇸᇶᄕ ƞƹƢ ƺƹƣ ǂƫƹƢƺǂ ƫƾ ƣƼǀƞƶ ƿƺ ᇵᇴᄙ ƣƶƶᄕƹƺǂᄕDŽƺǀƾǀƟƿƽƞơƿᇳᇴƤƽƺƸƟƺƿƩƾƫƢƣƾᄕƞƹƢDŽƺǀƾƣƣƿƩƞƿƿƩƣƿǂƺ windows are equal to 64. B: But are there numbers in the windows? ᄘǂƺǂƫƹƢƺǂƾƞƽƣᇸᇶᄕƾƺƺƹƣǂƫƹƢƺǂƫƾᇵᇴᄙ B: Window!? ᄘƩƞƿᅷƾƽƫƨƩƿᄕƞǂƫƹƢƺǂᄙƺƺƴƩƣƽƣᄘƞƹƣƶƣƻƩƞƹƿƞƹƢƞƹƣƶƣƻƩƞƹƿ is equal to 64. Therefore, what is one elephant equal to? Two elephants are equal to 64. So, one elephant is equal to what? ᄘƹƣƶƣƻƩƞƹƿᄞ ƸƸᄕ ƾƣƣᄙƹƣƣƶƣƻƩƞƹƿƣƼǀƞƶƾᇵᇴᄙ ǀƹƢƣƽƾƿƞƹƢ ƹƺǂᄚƾƺƹƺǂƿƩƣƣƼǀƞƿƫƺƹᄚ ᄘ ƤƿǂƺƣƶƣƻƩƞƹƿƾƞƽƣƣƼǀƞƶƿƺᇸᇲᄕƿƩƣƹƺƹƣƣƶƣƻƩƞƹƿƫƾƣƼǀƞƶƿƺ what? B: An elephant? ҅ҁᄕƺƹƣƣƶƣƻƩƞƹƿƣƼǀƞƶƾᇵᇲᄙ ƾƣƣƫƿƹƺǂᄙᄚƺǂ ƣƼǀƞƿƫƺƹᄚƞƞƞƞƞƞƞᄛ

ѹ‫ؠآآإئئؔ؟‬Ѽؔ‫ؙآءآ؜اؔا؜؟؜ؖ‬ѷ‫ؔ؛‬ᄛ҃‫اءؘؠآ‬ѿ‫ئا؛ؚ؜ئء‬

ርራ

The gap in understanding within the elephant dialogue is between the algebraic equation and the meaning of the unknown. During the moment of ƫƹƾƫƨƩƿᄕ ǂƣ Ƣƺƹᅷƿ Ʃƞǁƣ ƿƩƣ ƻƽƺơƣƾƾ ƺƤ ƽƣƾƿƽǀơƿǀƽƫƹƨ ƞƾ ƸǀơƩ ƞƾ ơƺƹƾƿƽǀơƿƫƹƨ ᄬƺƽƾƿƽǀơƿǀƽƫƹƨᄭƞƹƣǂƾƿƽǀơƿǀƽƣƟDŽơƺƹƹƣơƿƫƹƨƿƩƣƞƶƨƣƟƽƞƫơƣƼǀƞƿƫƺƹǂƫƿƩƫƿƾ ᅸƣƶƣƻƩƞƹƿᅺƸƣƞƹƫƹƨᄙᄬƶƿƩƺǀƨƩǂƣƢƺƺƟƾƣƽǁƣƿƩƣƻƽƺơƣƾƾƺƤƽƣƾƿƽǀơƿǀƽƫƹƨƟDŽ ƿƩƣƸƣƹƿƺƽᄙᄭ ƺƿƣƿƩƣǂƺƽƴƺƤƿƩƣᇳᇲᅟDŽƣƞƽᅟƺƶƢƸƣƹƿƺƽᄕǂƩƺƶƣƞƢƾᄕǀƹƾǀơơƣƾƾƤǀƶƶDŽᄕƩƫƾƻƣƣƽ through two standard examples aimed at facilitating understanding of the ơƺƹơƣƻƿƺƤƿƩƣǀƹƴƹƺǂƹᄙ ƿƸǀƾƿƩƞǁƣƟƣƣƹƫƹƿǀƫƿƫǁƣƶDŽơƶƣƞƽƿƺƩƫƸƿƩƞƿƞƤƿƣƽ such two pedagogical failures he must come up with something that would get through to his friend. Here we have the process of restructuring by P. Note also ƿƩƞƿƿƩƣƸƣƹƿƺƽơƩƣơƴƣƢƩƫƾƤƽƫƣƹƢᅷƾƤƫƹƞƶǀƹƢƣƽƾƿƞƹƢƫƹƨƺƤƿƩƣơƺƹơƣƻƿᄙ

ƫƽƾƿƩƣƾƿƞƿƣƾƿƩƣƻƽƺƟƶƣƸƫƹƶƫƹƣᄬᇴᇳᄭᄘᅸƞƹƣƶƣƻƩƞƹƿƞƹƢƞƹƣƶƣƻƩƞƹƿƫƾƣƼǀƞƶ ƿƺ ᇸᇶᄙ ƺᄕ ƺƹƣ ƣƶƣƻƩƞƹƿ ƫƾ ƣƼǀƞƶ ƿƺ ǂƩƞƿᄞᅺ Ƥƿƣƽ  ƞƹƾǂƣƽƣƢ ƫƹ ƿƩƣ ƶƫƹƣ ᄬᇴᇴᄭ ƿƩƞƿƺƹƣƣƶƣƻƩƞƹƿƣƼǀƞƶƾᇵᇴᄕƩƣᄕƿƩƣƸƣƹƿƺƽᄕƾƿƫƶƶƫƹƿƩƣƶƫƹƣᄬᇴᇵᄭƞƾƴƾᄕᅸ Ƥƿǂƺ ƣƶƣƻƩƞƹƿƾƞƽƣᇸᇲᄕƿƩƣƹƺƹƣƣƶƣƻƩƞƹƿƫƾƣƼǀƞƶƿƺǂƩƞƿᄞᅺƹƶDŽƿƩƣƹᄕƩƣƞƶƶƺǂƾ his friend to approach the solution to the full problem, which the latter victoriously accomplished. The mentor was checking, like a seasoned teacher, understanding of the concept by the mentee. This double checking after which comes the correct ƾƺƶǀƿƫƺƹƫƾƞƻƞƿƿƣƽƹƿƩƞƿƺƤƿƣƹƽƣƻƣƞƿƾƫƿƾƣƶƤƫƹơƶƞƾƾƽƺƺƸƾᄙ ƹƤƞơƿᄕƿƩƣƤƫƽƾƿƿǂƺ unsuccessful pedagogical visual attempts at line intervals and with boxes/windows of the elephant dialogue also ended with success at the third attempt. Consequently, we have here coordinated two simultaneous insights, one by the student in reaching the solution, another by the mentor by coming up with the correct hint/metaphor to facilitate it. ᇵᄙᇵᄙᇴᄙᇴ ƺƨƹƫƿƫǁƣƹƞƶDŽƾƫƾ ǀƻƫƶᄴƞƽƿᄵƢƺƣƾƹƺƿǀƹƢƣƽƾƿƞƹƢƿƩƣơƺƹơƣƻƿƺƤƿƩƣǀƹƴƹƺǂƹƫƹƿƩƣơƺƹƿƣǃƿ ƺƤ ƞ ƾƫƸƻƶƣ ƶƫƹƣƞƽ ƣƼǀƞƿƫƺƹᄙ ǀƻƫƶ ᄴƾƩƣƸƫƴᄵᄕ ƞƤƿƣƽ ƾƺƶǁƫƹƨ ƿƩƣ ƣƼǀƞƿƫƺƹ ƟDŽ himself, explains the process to B. He uses two standard methods, one after another. First, he did this with the help of line intervals representing unknown and ƿƩƣƹǀƸƟƣƽᇳᇴᄙƣơƺƹƢᄕƩƣƢƫƢƿƩƫƾǂƫƿƩƿƩƣƩƣƶƻƺƤƿƩƣƾƼǀƞƽƣƾᄬƺƽǂƫƹƢƺǂƾᄭ representing the unknown. However, the metaphorical relationship between ƣƫƿƩƣƽƺƤƿƩƣǀƾƣƢƞƽƿƫƤƞơƿƾƞƹƢƿƩƣǀƹƴƹƺǂƹƫƾƶƺƾƿƺƹᄙ ƿƫƾƫƹƿƣƽƣƾƿƫƹƨƿƩƞƿ P used two visual models unsuccessfully; only the third visual metaphor of the elephant facilitated a powerful Aha! moment within the bisociative frame created by the logico/algebraic and perceptual matrices of thinking. The hidden

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ѹ‫ؗءؔؔ؛ؖآءإؔح‬Ѹؔ‫إؘ؞‬

analogy was between the x as the emerging concept of the unknown, and, speculatively, the not very well known elephant.

ƹƿƣƽƸƾƺƤƶƺƺƸᅷƾƿƞǃƺƹƺƸDŽᄕƣƹƿƣƽƾƿƩƣƻƞƿƩǂƞDŽƟƣƶƺǂƿƩƣƶƣǁƣƶƺƤơƺƸprehension or understanding, and the Aha! moment carries him into the level of synthesis, where he synthesizes the x with elephant to grasp the concept of the unknown in the context of linear equation we have here a mild Aha! ƸƺƸƣƹƿᄕǂƫƿƩƿƩƣƺƻƣƹƫƹƨƿƺǂƞƽƢƿƩƣƹƺƽƸƞƶƶƣǁƣƶᄬƩƞƻƿƣƽᇶᄭᄙ The extent of the grasp can be judged in that P did not stop after first instant ƺƤᅷƾǀƹƢƣƽƾƿƞƹƢƫƹƨᄘƞƹƣƶƣƻƩƞƹƿƞƹƢƞƹƣƶƣƻƩƞƹƿƫƾƣƼǀƞƶƿƺᇸᇶᄕƺᄕƺƹƣƣƶƣƻƩƞƹƿƫƾƣƼǀƞƶƿƺǂƩƞƿᄞƞƹƾǂƣƽƾᄘƹƣƣƶƣƻƩƞƹƿƣƼǀƞƶƾᇵᇴᄕƞƹƢǂƞƹƿƾƿƺƞƻƻƶDŽ this new understanding to the full equation: “so now the equation.” ƨƫǁƣƾƩƫƸDŽƣƿƞƹƺƿƩƣƽƻƽƺƟƶƣƸƫƹƿƩƣơƺƹƿƣǃƿƺƤᅸƣƶƣƻƩƞƹƿᅺᄘᅸ ƤƿǂƺƣƶƣƻƩƞƹƿƾƞƽƣƣƼǀƞƶƿƺᇸᇲᄕƿƩƣƹƺƹƣƣƶƣƻƩƞƹƿƫƾƣƼǀƞƶƿƺǂƩƞƿᄞᅺᅷƾơƺƽƽƣơƿƞƹƾǂƣƽ ᄬƺƹƣƣƶƣƻƩƞƹƿƣƼǀƞƶƾᇵᇲᄭƾǀƨƨƣƾƿƾƿƩƞƿǀƹƢƣƽƾƿƺƺƢƿƩƣǀƹƴƹƺǂƹƫƹƿƩƣƸǀƶtiplicative case 2x = a, and successfully applied his understanding to the full ƣƼǀƞƿƫƺƹᄙơơƺƽƢƫƹƨƿƺƿƩƣƫƞƨƣƿᅬƞƽơƫƞƽƫƞƢᄕǂƩƞƿǂƣƩƞǁƣƩƣƽƣƫƾƸƺǁƣƸƣƹƿƤƽƺƸƿƩƣƫƹƿƽƞƾƿƞƨƣƺƤƿƩƣǀƹơƺƹƹƣơƿƣƢƫƹᅷƾƸƫƹƢᄬƾƫƿǀƞƿƫƺƹƾƺƤƫƹƿƣƽǁƞƶƾᄕƟƺǃƣƾᄧǂƫƹƢƺǂƾᄕƞƹƢƿƩƣƣƶƣƻƩƞƹƿᄭƿƺƿƩƣƫƹƿƣƽƾƿƞƨƣǂƩƣƽƣᄕƿƩƽƺǀƨƩƿƩƣ connections, interiorization of the concept takes place. ƩƫƾƣƻƫƾƺƢƣƽƣƻƽƣƾƣƹƿƾǂƩƞƿƽƞƟƩǀᄬᇴᇲᇳᇸᄭƽƣƤƣƽƾƿƺƞƾƞƟƫƾƺơƫƞƿƫǁƣƹƞƿǀƽƣ of a creative learning environment when P realizes that the usual symbols or artifacts that typically introduce the concept of a variable such as visual symƟƺƶƾƺƤƫƹƿƣƽǁƞƶƾƺƽƟƺǃƣƾᄧǂƫƹƢƺǂƾǂƣƽƣƹƺƿƾǀƤƤƫơƫƣƹƿᄙ ƹƾƿƣƞƢᄕƩƣǀƾƣƢƞơƺƹcrete visual symbol of the elephant that happened to be in the room, and this ƽƣƞƶƫDžƞƿƫƺƹƶƣƢƿƺƿƩƣƺƿƩƣƽƾƿǀƢƣƹƿᅷƾƾǀơơƣƾƾƤǀƶƨƽƞƾƻƺƤƿƩƣǀƹƴƹƺǂƹơƺƹơƣƻƿᄙ This problem is noteworthy because it demonstrates a teaching-research bisociation as two realizations were occurring, one by the learner and another by his peer-tutor. The one by the student-learner who finally makes the abstraction required to consider an unknown as an object with the analogy of the elephant as being some unknown quantity; the other realization is by his peer tutor who realizes several times that the analogy he has presented is not being understood by the other student and thus he changes his analogy from line to box to window and finally to an elephant, thereby demonstrating natural ƿƣƞơƩƫƹƨƾƴƫƶƶƾᄙ ƿƫƾƞƹƣǃƞƸƻƶƣƺƤƿƩƣƩƞᄛƸƺƸƣƹƿƽƣƞƶƫDžƞƿƫƺƹƟDŽƿƩƣƾƿǀƢƣƹƿ and by the tutor in one mathematical situation. ᇵᄙᇵᄙᇴᄙᇵ ƩƣƺƸƞƫƹƩƞᄛƺƸƣƹƿ Ʃƣ ƻƽƺƟƶƣƸ ƾƿƞƽƿƾ ǂƫƿƩ ƿƩƣ Ƥǀƹơƿƫƺƹ Ƥᄬǃᄭ ኙ እᄬǃ ና ᇵᄭᄙƩƣ ƿƣƞơƩƣƽ ƞƾƴƣƢ ƿƩƣ students during the review: “Can all real values of x be used for the domain of ƿƩƣƤǀƹơƿƫƺƹእᄬǃናᇵᄭᄞᅺ

ѹ‫ؠآآإئئؔ؟‬Ѽؔ‫ؙآءآ؜اؔا؜؟؜ؖ‬ѷ‫ؔ؛‬ᄛ҃‫اءؘؠآ‬ѿ‫ئا؛ؚ؜ئء‬

ᇳᄙ 2. ᇵᄙ 4. ᇷᄙ 6. 7. ᇺᄙ ᇻᄙ ᇳᇲᄙ ᇳᇳᄙ ᇳᇴᄙ ᇳᇵᄙ ᇳᇶᄙ

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ᄬƾƿǀƢƣƹƿᄭᄘƺᄕƹƣƨƞƿƫǁƣǃᅷƾơƞƹƹƺƿƟƣǀƾƣƢᄙ ᄬƿƣƞơƩƣƽᄭᄘ ƺǂƞƟƺǀƿǃኙኔᇷᄞ S: No good. ᄘ ƺǂƞƟƺǀƿǃኙኔᇶᄞ S: No good either. ᄘ ƺǂƞƟƺǀƿǃኙኔᇵᄞ ᄴƿǀƢƣƹƿƿƩƫƹƴƾƤƺƽƞƸƫƹǀƿƣᄙᄵ ᄘ ƿǂƺƽƴƾƩƣƽƣᄙ ᄘ ƺǂƞƟƺǀƿǃኙኔᇴᄞ ᄘ ƿǂƺƽƴƾƩƣƽƣƿƺƺᄙ ᄴƿǀƢƣƹƿƿƩƫƹƴƾƤƺƽƞƸƫƹǀƿƣᄙᄵ ᄘƩƺƾƣǃᅷƾǂƩƫơƩƞƽƣƾƸƞƶƶƣƽƿƩƞƹኔᇵơƞƹᅷƿƟƣǀƾƣƢƩƣƽƣᄙ ᄘ ƺǂƞƟƺǀƿƨᄬǃᄭኙእᄬǃኔᇳᄭᄞ ᄴƿǀƢƣƹƿƿƩƫƹƴƾƤƺƽƞƸƫƹǀƿƣᄙᄵ ᄘƸƞƶƶƣƽƿƩƞƹᇳơƞƹᅷƿƟƣǀƾƣƢᄙᅺ ᄘ ƹƿƩƞƿơƞƾƣᄕƩƺǂƞƟƺǀƿƩᄬǃᄭኙእᄬǃኔƞᄭᄞ S: Smaller than a cannot be used.

ƹƿƩƣƾƣơƺƹƢƢƫƞƶƺƨǀƣᄕƿƩƣƢƺƸƞƫƹᄕƿƩƣƨƞƻƺƤǀƹƢƣƽƾƿƞƹƢƫƹƨᄕƫƾƫƹƿƩƣƸƫƾconception of the student and the correct condition arrived by the student in ƶƫƹƣᇳᇲᄙƽƫƨƫƹƞƶƶDŽƾƣƣƹƞƾƿƩƣƻǀƽƣƩƞᄛƸƺƸƣƹƿᄕƫƿơƺǀƶƢƩƞǁƣƟƣƣƹƞƩDŽƟƽƫƢᄕ one in which the student could have reasoned out the answer from the available data. ᇵᄙᇵᄙᇴᄙᇵ ƺƨƹƫƿƫǁƣƹƞƶDŽƾƫƾ ƩƣƟƫƾƺơƫƞƿƫƺƹƿƞƴƣƾƻƶƞơƣƫƹƶƫƹƣᇳᇲᄕǂƩƣƹƿƩƣƾƿǀƢƣƹƿƤƫƹƢƾƩƫƢƢƣƹƞƹƞƶƺgies in the concrete examples discussed earlier during the dialogue, which are ƞƟƾƿƽƞơƿƣƢƞƹƢƨƣƹƣƽƞƶƫDžƣƢᄙƾƫƸƫƶƞƽƻƽƺơƣƾƾƿƞƴƣƾƻƶƞơƣƫƹƶƫƹƣƾᇳᇳᅬᇳᇶᄙ The student displays a familiar misconception within the matrix of finding ƿƩƣƢƺƸƞƫƹƺƤƞƤǀƹơƿƫƺƹƾǀơƩƞƾƤᄬǃᄭኙእǃƫƹƞƻƻƶƫơƞƿƫƺƹƿƺƿƩƫƾƻƽƺƟƶƣƸƾƫƿǀƞƿƫƺƹᄕǂƩƫơƩƫƹǁƺƶǁƣƾƞƿƽƞƹƾƤƺƽƸƞƿƫƺƹƺƤƿƩƣƻƽƣǁƫƺǀƾƾƫƿǀƞƿƫƺƹᄬƿƩƣƞƽƨǀƸƣƹƿ ǂƫƿƩƫƹƿƩƣƾƼǀƞƽƣƽƺƺƿᄕǃናᇵᄭᄙ

ƹƶƫƹƣƾᇸƞƹƢᇺƿƩƣƫƹƾƿƽǀơƿƺƽǀƾƣƾơƺƹơƽƣƿƣơƺǀƹƿƣƽƣǃƞƸƻƶƣƾƿƺƻƽƺǁƫƢƣƞ ƻƣƽƿǀƽƟƞƿƫƺƹᄕƺƽƞơƞƿƞƶDŽƾƿᄕƤƺƽơƺƨƹƫƿƫǁƣơƺƹƤƶƫơƿƞƹƢơƩƞƹƨƣᄙᅸᄴᄵƣƽƿǀƽƟƞƿƫƺƹƫƾ ƺƹƣƺƤƿƩƣơƺƹƢƫƿƫƺƹƾƿƩƞƿƾƣƿƿƩƣƾƿƞƨƣƤƺƽơƺƨƹƫƿƫǁƣơƩƞƹƨƣᅺᄬƺƹƶƞƾƣƽƾƤƣƶƢᄕ ᇳᇻᇺᇻᄕƻᄙᇳᇴᇹᄭᄙƩƣƻƣƽƿǀƽƟƞƿƫƺƹᄕǂƩƫơƩơƺƹƿƽƞƢƫơƿƾƾƿǀƢƣƹƿơƺƹǁƫơƿƫƺƹᄕƫƾƸƺƽƣ ƻƽƺƹƺǀƹơƣƢ ƺƹ ƿƩƣ ƟƞơƴƨƽƺǀƹƢ ƺƤ ǀƣƾƿƫƺƹƾ ᇴ ƞƹƢ ᇵᄕ ǂƩƺƾƣ ƞƹƾǂƣƽƾ ƞƨƽƣƣ ǂƫƿƩƿƩƣƾƿǀƢƣƹƿᅷƾƺƽƫƨƫƹƞƶƾƿƞƿƣƸƣƹƿᄙ

ƹƶƫƹƣƾᇸƿƩƽƺǀƨƩᇻᄕƿƩƣƾƿǀƢƣƹƿƽƣƤƶƣơƿƾƺƹƿƩƣƽƣƾǀƶƿƾƺƤƿƩƣƾƺƶǀƿƫƺƹƞơƿƫǁƫƿDŽᄙDŽơƺƸƻƞƽƫƹƨƿƩƣƽƣƾǀƶƿƾᄬƽƣơƺƽƢƾᄭᄕƿƩƣƾƿǀƢƣƹƿƞƟƾƿƽƞơƿƾƞƻƞƿƿƣƽƹᅮƿƩƣ

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ѹ‫ؗءؔؔ؛ؖآءإؔح‬Ѹؔ‫إؘ؞‬

ƶƣƞƽƹƣƽᅷƾơƺƸƻƞƽƫƾƺƹƾƺƤƿƩƣƽƣơƺƽƢƾƞƶƶƺǂƾƤƺƽƽƣơƺƨƹƫƿƫƺƹƺƤƻƞƿƿƣƽƹƾᄬƫƸƺƹ ƣƿƞƶᄙᄕᇴᇲᇲᇶᄭᄙ Evidence of the abstraction of the principle that the transformation of the ƞƽƨǀƸƣƹƿǃናᇵƟƣƹƣƞƿƩƿƩƣƾƼǀƞƽƣƽƺƺƿƾƩƫƤƿƾƿƩƣƢƺƸƞƫƹᇵƿƺƿƩƣƶƣƤƿƫƾƨƫǁƣƹ ƫƹƶƫƹƣᇳᇲᄙƺƽƣơƺƹơƽƣƿƣƶDŽᄕơƺƹƾƫƢƣƽƞƿƫƺƹƾƫƹƩƞƻƿƣƽᇴƾǀƨƨƣƾƿƿƩƞƿƫƿǂƞƾƿƩƣ ƸƣƹƿƞƶơƺƸƻƞƽƫƾƺƹƺƤƿƩƣƞƹƾǂƣƽƾƿƺƿƩƣƼǀƣƾƿƫƺƹƾƫƹᇸƞƹƢᇺƿƩƞƿƽƣƾǀƶƿƣƢ ƫƹƿƩƣƟƫƾƺơƫƞƿƫƺƹƫƹᇳᇲᄕǂƫƿƩƿƩƣƩƣƶƻƺƤƿƩƣƩƫƢƢƣƹƞƹƞƶƺƨDŽƺƽƽƞƿƩƣƽƩƫƢƢƣƹ ᅸƾƞƸƣƹƣƾƾᅺƺƤƿƩƣƞƹƾǂƣƽƾƫƹᇹƞƹƢᇻᄙƩƣƻƽƣƾƣƹơƣƺƤƾǀơƩƿƽƫƻƶƣƾƿƺƨƣƿƩƣƽ with introductory analysis of their meaning and scope has been noted by DžƞƽƹƺơƩƞᄬᇴᇲᇳᇵᄭᄙ This bisociation, by connecting previous unconnected cases, has propelled ƿƩƣƾƿǀƢƣƹƿƫƹƿƺƿƩƣƾƣơƺƹƢƾƿƞƨƣᄬƫƹƿƣƽᄭƺƤƿƩƣƫƞƨƣƿᅬƞƽơƫƞƽƫƞƢᄙ ƺƿƣƿƩƣƸƺƸƣƹƿƾƺƤƾƫƶƣƹơƣƟƣƿǂƣƣƹƶƫƹƣƾᇸƞƹƢᇹᄕᇻƞƹƢᇳᇲᄕƞƹƢᇳᇳƞƹƢᇳᇴᄙ ƺƣƾƿƶƣƽƺƟƾƣƽǁƣƾƿƩƞƿƽƞƿƾᄕơƞƿƾᄕƞƹƢƢƺƨƾƾƫƸƫƶƞƽƶDŽƾƩƺǂƿƩƫƾƟƽƫƣƤƾǀƾƻƣƹƾƫƺƹ of activity, “this attitude of concentrated attention before they act out hypothƣƾƫƾᄬᄞᄭƿƩƣDŽƩƞǁƣƤƺƽƸƣƢᅭǂƩƫơƩƸƫƨƩƿƟƣơƺƽƽƣơƿƺƽƹƺƿᅺᄬƻᄙᇷᇸᇹᄭᄙ ƣƹơƣᄕƿƩƣ ƾƿǀƢƣƹƿᅷƾƽƣƞƶƫDžƞƿƫƺƹƫƹƶƫƹƣᇳᇲᄕƟƣƤƺƽƣƿƩƣƩƞᄛƸƺƸƣƹƿơƞƹƟƣƾƣƣƹᄕƫƾƽƣƤƶƣơƿƫǁƣƞƟƾƿƽƞơƿƫƺƹƫƹƿƩƣƾƣƹƾƣƺƤƫƸƺƹƣƿƞƶᄙᄬᇴᇲᇲᇶᄭᄙ ƩƣƾƿǀƢƣƹƿᅷƾƤǀƶƶƞơƿƫǁƫƿDŽǂƫƿƩƫƹƶƫƹƣƾᇵƞƹƢᇳᇲᄕǂƩƣƹƿƩƣƾƺƶǀƿƫƺƹƞơƿƫǁƫƿDŽ ƺƤ ƾǀƟƾƿƫƿǀƿƫƺƹ ǂƞƾ ƻƽƺưƣơƿƣƢ ƫƹƿƺ ƞƹƢ ơƺƺƽƢƫƹƞƿƣƢ ǂƫƿƩ ƿƩƣ ƾƺƶǁƣƽᅷƾ ƴƹƺǂƶedge of the domain of the square root function, is also an example of reflective ƞƟƾƿƽƞơƿƫƺƹᄕƞơơƺƽƢƫƹƨƿƺƫƞƨƣƿƞƹƢƞƽơƫƞᄬᇳᇻᇺᇻᄭᄙ

ƹƶƫƹƣƾᇳᇳƞƹƢᇳᇴᄕƿƩƣƻƣƽƿǀƽƟƞƿƫƺƹƟƽƺǀƨƩƿƞƟƺǀƿƟDŽƿƩƣƿƣƞơƩƣƽᅷƾƼǀƣƾƿƫƺƹƾ ƶƣƞƢƾƿƩƣƾƿǀƢƣƹƿƿƺƣƹƿƣƽƿƩƣƾƣơƺƹƢƾƿƞƨƣƺƤƿƩƣƫƞƨƣƿᅬƞƽơƫƞƽƫƞƢᄙƩǀƾᄕ ƿƩƣƾƿǀƢƣƹƿᅷƾǀƹƢƣƽƾƿƞƹƢƫƹƨƺƤƤƫƹƢƫƹƨƿƩƣƢƺƸƞƫƹƺƤƞƾƼǀƞƽƣƽƺƺƿƩƞƾǀƹƢƣƽgone constructive generalization to accommodate transformation, providing ƣǁƫƢƣƹơƣƺƤƿƩƣƾƿƽǀơƿǀƽƞƶǀƹƢƣƽƾƿƞƹƢƫƹƨƹƺƿƣƢƟDŽƤƞƽƢᄬᇳᇻᇻᇳᄭƞƹƢƿƩƣƿƩƫƽƢ ƾƿƞƨƣƺƤƿƩƣƽƫƞƢᄬƫƞƨƣƿѵƞƽơƫƞᄕᇳᇻᇺᇻᄭᄙ ƹƞƶDŽƾƫƾƺƤƺƿƩƽƺǀƨƩƶƺƺƸᅷƾƿƞǃƺƹƺƸDŽƤƺƽƿƩƫƾơƞƾƣƽƣǁƣƞƶƾƿƩƣƢƫƤƤƫculty in stable assessment of student progress. The dialogue goes through the application stage facilitated by the instructor: the student jumps over the analysis stage to land in synthesis as an abstraction and generalization, followed by application of the new understanding and a second synthesis. So, the Aha! moment is again taking place, twice through iteration in between levels of ƶƺƺƸᅷƾƿƞǃƺƹƺƸDŽᄙ As we have seen, both Type B situations share certain characteristics. The posed questions and hints during facilitation are formulated on the spot instead of prepared in advance; their role is to scaffold student understanding so that authentic moment of insight can occur.

ѹ‫ؠآآإئئؔ؟‬Ѽؔ‫ؙآءآ؜اؔا؜؟؜ؖ‬ѷ‫ؔ؛‬ᄛ҃‫اءؘؠآ‬ѿ‫ئا؛ؚ؜ئء‬

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Authenticity is assured at every step of the scaffolding by always leaving ƣƹƺǀƨƩơƺƨƹƫƿƫǁƣƢƫƾƿƞƹơƣᄕƣƹƺǀƨƩƺƤƞƨƞƻƟƣƿǂƣƣƹƿƩƣƾƿǀƢƣƹƿᅷƾƴƹƺǂƶƣƢƨƣ ƞƿƣƞơƩƾƿƣƻƞƹƢơƺƽƽƣơƿǀƹƢƣƽƾƿƞƹƢƫƹƨƺƤƿƩƣơƺƹơƣƻƿƿƩƽƺǀƨƩƿƩƣƾƿǀƢƣƹƿᅷƾ ơƺƨƹƫƿƫǁƣ ƣƤƤƺƽƿᄙ Ʃƣ ƿƣƞơƩƣƽᅷƾ Ƹƞƫƹ ƾƴƫƶƶ ƫƾ ƿƺ Ƣƫƾơƣƽƹ ƿƩƣ ƶƣǁƣƶ ƺƤ ƾƿǀƢƣƹƿ understanding and to find and provide a hint or question to advance the stuƢƣƹƿᅷƾƨƽƞƾƻƺƤƿƩƣƾǀƟưƣơƿᄙƾƞƽƹƣƾᄬᇴᇲᇲᇲᄭƾǀƨƨƣƾƿƾᄕƿƩƣƩƫƹƿƾƩƺǀƶƢƟƣƞƿ ƿƩƣƶƣǁƣƶƺƤơƩƞƶƶƣƹƨƣƫƹƿƩƣƿƞƾƴᄕƹƣƫƿƩƣƽƿƺƺƨƽƣƞƿƹƺƽƿƺƺƾƸƞƶƶᄙƩƣƿƣƞơƩƣƽᅷƾ ƹƣǃƿƾƿƣƻƢƣƻƣƹƢƾƺƹƿƩƣƾƿǀƢƣƹƿᅷƾƞƹƾǂƣƽᄙ ƤƿƩƣƾƿǀƢƣƹƿƢƺƣƾƹᅷƿǀƹƢƣƽƾƿƞƹƢᄕ the teacher needs to ask the next question that will diminish the cognitive gap without closing it. ƩƣƿƣƞơƩƣƽᅷƾƽƺƶƣƫƾƿƺƣƹƨƞƨƣƫƹƞơƶƺƾƣƞƹƢƻƽƣơƫƾƣƢƫƞƶƺƨǀƣǂƫƿƩƿƩƣƾƿǀdent while asking well-aimed questions that relate to concept discovery. This ƫƾƿƩƣƣǃƞơƿƺƻƻƺƾƫƿƣƺƤƿƩƣƿƣƞơƩƣƽᅷƾƣƤƤƺƽƿƫƹƿƩƣơƞƾƣƺƤơƺƸƻƶƣǃơƺƶƶƞƟƺƽƞƿƫǁƣ problems given to the whole class. There the teacher prepares the problems in advance, hoping that the structure will lead to student understanding. The teacher, no more than a benevolent observer, tries to be the least involved in student thinking apart from asking some clarifying questions to assess the degree of student independence in the solving process. As we indicate below, that posture also requires some natural limits.

3.4

Recognition of the Aha! Moment

ƺƸƣƹƿƾƺƤƫƹƾƫƨƩƿƞƽƣƺƤƿƣƹǀƹƣǃƻƣơƿƣƢᄕƞƹƢᄕƺƤơƺǀƽƾƣᄕƿƩƞƿᅷƾƻƞƽƿƺƤƿƩƣƫƽ charm. The literature offers numerous descriptions of flashes of insight that happen in the shower, while driving a car, baking a cake, or sipping coffee— in unexpected situations far removed from the problem. Consequently, while ƺƹƿƩƣơƽƣƞƿƫǁƫƿDŽƤƞơƫƶƫƿƞƿƫƺƹƻƞƿƩǂƞDŽƫƹƿƩƣơƶƞƾƾƽƺƺƸǂƣƢƺƹᅷƿƴƹƺǂƫƤƩƞᄛ ƸƺƸƣƹƿǂƫƶƶƞƻƻƣƞƽᄙƣƞƽƣǀƹơƣƽƿƞƫƹᄙ ƿƫƾƿƩƣƿƣƞơƩƣƽᅷƾƩƺƻƣƞƹƢƞƾƻƫƽƞƿƫƺƹ for it to occur that guides his or her steps in posing questions or offering during scaffolding. Sometimes an experienced teacher can notice the rising ripples of creativity in the classroom that do not gather enough momentum to become a complete ƫƹƾƫƨƩƿƫƹƿƩƣơƶƞƾƾƽƺƺƸᄬ ƣƽƾƩƴƺǂƫƿDžƣƿƞƶᄙᄕᇴᇲᇳᇹᄭᄙǀƿƾƺƸƣƿƫƸƣƾƿƩƣDŽƢƺƹƺƿᄙ How do we recognize them? A critical remark here motivated by this chapter is to emphasize that bisociative moments of insight often occur in the context of problem-solving, but not merely. They also occur as the result of the desire to understand a concept.

ƹƿƩƣơƶƞƾƾƽƺƺƸᄕǂƣƿƩƣƿƣƞơƩƣƽƾƸǀƾƿƻƞDŽƞƿƿƣƹƿƫƺƹƿƺƻƽƺƟƶƣƸᅟƾƺƶǁƫƹƨƟǀƿ

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ѹ‫ؗءؔؔ؛ؖآءإؔح‬Ѹؔ‫إؘ؞‬

not be fixated on it. We must pay equal attention to absence of understanding ƺƽƸƫƾơƺƹơƣƻƿƫƺƹƺƤƞơƺƹơƣƻƿᄙƽƣƞƿƫǁƫƿDŽƺƤƿƣƹƩƫƢƣƾƫƹƿƩƞƿƞƟƾƣƹơƣᄙ ƹƢƣƣƢᄕ the recognition of Aha! moment insight is an important issue for the process of facilitation. The literature discusses two signs of their occurrence, an affective one and a ơƺƨƹƫƿƫǁƣƺƹƣᄙƫƶưƣƢƞƩƶᄬᇴᇲᇳᇵᄭƫƹǁƣƾƿƫƨƞƿƣƢƿƩƣƽƺƶƣƞƹƢƹƞƿǀƽƣƺƤƿƩƣƻƺƾƫƿƫǁƣ affect, disregarding, based on his data, the importance of the cognitive aspect. ƞƽƹƣƾᄬᇴᇲᇲᇲᄭƫƾƞƶƾƺƨǀƫƢƣƢƟDŽƞƤƤƣơƿƿƺƫƢƣƹƿƫƤDŽƩƞᄛƸƺƸƣƹƿƾƫƹƩƣƽƣǃƻƣƽƫƸƣƹƿƞƶơƶƞƾƾᄕƽƣƸƫƹƢƫƹƨǀƾƞƿƿƫƸƣƾƺƤƿƩƣƾƿǀƢƣƹƿᅷƾƽƺƶƣƫƹǀƹƢƣƽƾƿƞƹƢƫƹƨƿƩƣ process. Here we focus on an excellent report of an Aha! moment in calculus, which, however, raises several doubts. 3.4.1 Yoon Caroline: “Mapping Mathematical Leaps of Insight” (2012) This excellent and thorough report nonetheless leaves certain doubts about whether the author is describing a true Aha! moment. This doubt provides an opportunity to sharpen our understanding of the recognition difficulty. The report describes leaps of insight of undergraduate students collaboratively participating in an experimental calculus project whose aim was to ƫƹǁƣƾƿƫƨƞƿƣƾƿǀƢƣƹƿƾᅷơƺƹƾƿƽǀơƿƫƺƹƺƤơƞƶơǀƶǀƾơƺƹơƣƻƿƾᄙ All students had knowledge of a basic calculus course at the high school level. The discussion emphasizes the placement of the problem between two separated frames of thinking and the need for the detailed account of the interaction between collaborative teams. ƺƺƹ ᄬᇴᇲᇳᇴᄭ ƢƣƾƫƨƹƣƢ ƞ ơƺƸƻƶƣǃ ƻƽƺƟƶƣƸ Ƥƺƽ ƿƩƣ ƻƽƺưƣơƿᄕ ǂƩƫơƩ ƫƹǁƺƶǁƣƢ the relationship between the graph of the gradient derivative and the graph of the function. A real-world context was given by representing it as a “tramping track gradient” and as a “distance-height graph of the track” that involved two pairs of bisociative frames: between reality and mathematics, and between a derivative of the function and the function itself. The described problem had two phases: ᅬ ƩƞƾƣᇳᄘƫǁƣƹƞơǀƽǁƣƺƤƿƩƣƢƫƾƿƞƹơƣǁƣƽƾǀƾƩƣƫƨƩƿƺƤƿƩƣƿƽƞƸƻƫƹƨƿƽƞơƴᄕ construct the gradient graph. ᅬ Phase 2: Given a new gradient graph, find the corresponding graph of distance versus height for that case. Since all participating students had a basic knowledge of calculus, the problem ǂƞƾᄬơƺǀƶƢƟƣᄕƾƩƺǀƶƢƟƣᄭǂƫƿƩƫƹƿƩƣƫƽDžƺƹƣƺƤƻƽƺǃƫƸƞƶƢƣǁƣƶƺƻƸƣƹƿᄬҐ҆Ѻᄭᄙ Therefore, the gap between unconnected frames of reference was less between the level of students and the level of the problem as it was in the previous case.

ƹƿƩƫƾƻƽƺƟƶƣƸᄕƿƩƣƨƞƻƫƾƟƣƿǂƣƣƹơƺƢƣƾᄬƽǀƶƣƾᄭƢƣƾơƽƫƟƫƹƨƿƩƣƻƺƾƫƿƫƺƹƺƤ

ѹ‫ؠآآإئئؔ؟‬Ѽؔ‫ؙآءآ؜اؔا؜؟؜ؖ‬ѷ‫ؔ؛‬ᄛ҃‫اءؘؠآ‬ѿ‫ئا؛ؚ؜ئء‬

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minima and maxima, increasing and decreasing values of the function, and ƟƣƿǂƣƣƹƿƩƣơƺƢƣƾᄬƽǀƶƣƾᄭƺƤƢƣƽƫǁƞƿƫǁƣƾƺƤƿƩƣƤǀƹơƿƫƺƹơƺƽƽƣƾƻƺƹƢƫƹƨƿƺƿƩƣƾƣ points and intervals. The important feature of the problem was in presenting the problem of ơƺƹƾƿƽǀơƿƫƹƨ ƿƩƣ ƨƽƞƻƩ ƺƤ ƿƩƣ ƨƽƞƢƫƣƹƿ ƺƤ ƿƩƣ ƨƫǁƣƹ Ƥǀƹơƿƫƺƹ ƫƹ ƿƩƣ Ʃƞƾƣ ᇳ ᄬDŽƻƣᄭƻƽƣƾƣƹƿƫƹƨƿƩƣƻƽƺƟƶƣƸǂƩƫơƩƞƾƴƣƢƤƺƽƿƩƣƫƹǁƣƽƾƣƞƻƻƽƺƞơƩᄕƹƞƸƣƶDŽ constructing the graph of the function from the given graph of the gradient in ƩƞƾƣᇴᄙơơƺƽƢƫƹƨƿƺǀƟƫƹƾƴDŽᄬᇳᇻᇻᇳᄭᄕƞơƺƹƾƿƽǀơƿƫƺƹƺƤƿƩƣƫƹǁƣƽƾƣƻƽƺơƣƾƾ from the given process is an act of reflective abstraction. We could therefore expect manifestations of student authentic thinking, reflecting on the issue while solving the problem. ƺƺƹᄬᇴᇲᇳᇴᄭǀƾƣƾƿƩƣơƽƫƿƣƽƫƺƹƺƤƽƣƾƿƽǀơƿǀƽƫƹƨƞƾƿƩƣƶƫƿƸǀƾƻƞƻƣƽƤƺƽƫƹƾƫƨƩƿ formulated explicitly by Gestalt psychologists as the basis for recognition and ơƶƞƾƾƫƤƫơƞƿƫƺƹᄘᅸ ƹƾƫƨƩƿƺơơǀƽƾǂƩƣƹƞƻƽƺƟƶƣƸƫƾƾƺƶǁƣƢƟDŽƽƣƾƿƽǀơƿǀƽƫƹƨᄘƩƞƿ ƫƾᄕƫƤǂƣơƺƸƻƞƽƣƿƩƣƫƹƫƿƫƞƶƾƺƶǀƿƫƺƹᅷƾƞƿƿƣƸƻƿᄬƾᄭǂƫƿƩƿƩƣƫƹƾƫƨƩƿƤǀƶƾƺƶǀƿƫƺƹᄕ ƿƩƣDŽƸǀƾƿƟƣƿƩƣƽƣƾǀƶƿƺƤƞƢƫƤƤƣƽƣƹƿƞƹƞƶDŽƾƣƾƺƤƿƩƣƻƽƺƟƶƣƸᅺᄬƣƫƾƟƣƽƨᄕᇳᇻᇻᇷᄕ ƻᄙᇳᇸᇵᄭᄙ This approach comes with important qualifications. The question is whether the solution can come only through insight or be solved by systematic reasoning. When both are possible, such Aha! moments of insight are called “hybrid moments” of insight. The decision about which one occurred must be made by the teacher/research/mentor based on observations of student and interviews with the student. ƺƺƹ ƾǀƨƨƣƾƿƾ ƿƩƞƿ ƿƩƣ ƸƺƸƣƹƿ ƺƤ ƫƹƾƫƨƩƿ ƞƤƿƣƽ ƿƩƣ ƾƿƽǀơƿǀƽƣ ƺƤ ƾƿǀƢƣƹƿ thinking about the meaning of x-intercept of the gradient became “stronger, ƢƣƣƻƣƽƞƹƢƸƺƽƣƽƺƟǀƾƿᅺᄬƻᄙᇷᇷᇳᄭƫƹƿƩƣƾƣƹƾƣƺƤƟƣƿƿƣƽơƺƹƹƣơƿƫǁƫƿDŽƟƣƿǂƣƣƹ ơƺƹơƣƻƿƾᄕƸǀƶƿƫƶƞDŽƣƽƾƿƽǀơƿǀƽƣᄕƞƹƢƟƣƿƿƣƽƣǃƻƶƞƹƞƿƺƽDŽƻƺǂƣƽƿƩƞƹƟƣƤƺƽƣᄙƺƺƹ ᄬᇴᇲᇳᇴᄭƟƞƾƣƢƿƩƣơƺƹơƶǀƾƫƺƹƿƩƞƿƞƶƣƞƻƺƤƫƹƾƫƨƩƿƺơơǀƽƽƣƢƺƹƿƩƣơƺƸƻƞƽƫƾƺƹ of two graphs, the latter of which demonstrated much clearer understanding ƺƤƿƩƣƽƣƶƞƿƫƺƹƾƩƫƻƟƣƿǂƣƣƹƿƩƣƾƫƨƹᄬƾᄭƞƹƢDžƣƽƺǁƞƶǀƣƾƺƤƿƩƣƾƣơƺƹƢƢƣƽƫǁƞtive graph to predict the shape of the original function. The participants demonstrated a significantly improved ability to reason logically about the rule or code that related the two graphs as well as how this relationship reflects on the real life proposed situation. As this bisociative realƫDžƞƿƫƺƹƺơơǀƽƽƣƢƫƹƞƾƩƺƽƿƾƻƞƹƺƤƿƫƸƣᄕƺƺƹᄬᇴᇲᇳᇴᄭơƺƹơƶǀƢƣƢƫƿǂƞƾƞᅸƶƣƞƻƺƤ ƫƹƾƫƨƩƿᄙᅺƣƿᄕǂƣƢƺƹᅷƿƩƞǁƣƞƹDŽƫƹƢƫơƞƿƫƺƹƾƺƤƿƩƣƞƤƤƣơƿᄕǂƩƫơƩƺƤƿƣƹƫƹƢƫơƞƿƣƾ the occurrence of the insight. ƺƺƹ ᄬᇴᇲᇳᇴᄭ ƽƣƻƺƽƿƾ ƿƩƣ ƻƽƺơƣƾƾ ƺƤ ƽƣƾƿƽǀơƿǀƽƫƹƨ ƫƹ ƿƩƣ ǂƺƽƴ ƺƤ ƞ ƻƞƫƽ ƺƤ students from intuitive and spontaneous levels to the general rule involving ƨƽƞƢƫƣƹƿƾƞƹƢƢƣƿƣƽƸƫƹƞƿƫƺƹƺƤƸƞǃƫƸƞƞƹƢƸƫƹƫƸƞƺƤƿƩƣƤǀƹơƿƫƺƹᄙ ƹƢƣƣƢᄕ

ሥሤረ

ѹ‫ؗءؔؔ؛ؖآءإؔح‬Ѹؔ‫إؘ؞‬

there might have a been a moment of insight since the generalization involved only one example of the generalized statement and therefore it could not be derived by comparison with other cases. Moreover, according to Harel and Tall ᄬᇳᇻᇻᇳᄭᄕƺƹƣƺƤƿǂƺƿDŽƻƣƾƺƤƨƣƹƣƽƞƶƫDžƞƿƫƺƹƾƫƾƞᅸƽƣơƺƹƾƿƽǀơƿƫǁƣƨƣƹƣƽƞƶƫDžƞƿƫƺƹ ᄴǂƩƫơƩᄵ ƺơơǀƽƾ ǂƩƣƹ ƿƩƣ ƾǀƟưƣơƿ ƽƣơƺƹƾƿƽǀơƿƾ ƣǃƫƾƿƫƹƨ ƾơƩƣƸƞ ƫƹ ƺƽƢƣƽ ƿƺ widen its applicability range.” Nonetheless, we still have doubts about this moment because of the absence of the important quality of discontinuity in thinking characteristic for the Aha! ƸƺƸƣƹƿƫƹƾƫƨƩƿᄙ ƹƨƣƹƣƽƞƶᄕƟǀƿƞƶƿƩƺǀƨƩƹƺƿƞƶǂƞDŽƾᄕƿƩƣơƩƞƽƞơƿƣƽƫƾƿƫơƺƤơƺƹƢƫƿƫƺƹƾƤƺƽƩƞᄛƸƺƸƣƹƿƿƺƺơơǀƽᄬᅸƿƩƣDŽƸǀƾƿƟƣƿƩƣƽƣƾǀƶƿƺƤƞƢƫƤƤƣƽƣƹƿƞƹƞƶDŽƾƣƾƺƤƿƩƣƻƽƺƟƶƣƸᅺᄭᄙƩƣƻƽƣƾƣƹƿƣƢƽƣƻƺƽƿƢƺƣƾƹƺƿƢƣƾơƽƫƟƣƞƹDŽƢƫƾơƺƹƿƫƹǀƫƿDŽ ƺƤƿƩƫƹƴƫƹƨƺƽƢƫƤƤƣƽƣƹơƣƫƹƞƹƞƶDŽƾƫƾƟDŽƾǀƟưƣơƿƾᄙƣƢƺƹᅷƿƴƹƺǂǂƩƣƿƩƣƽƿƩƣ increase in connectivity and robustness of the graphs came out of different analyses that broke from the previous approach. To decide this question, we have to know more about how the problem was solved by these two students. The absence of the information suggests that there might have been too large of a distance between the instructor and workƫƹƨƾƿǀƢƣƹƿƾᄙ ƿƻƺƫƹƿƾƺǀƿƿƩƣƿƣƹƾƫƺƹƫƹƿƩƣƟƫƾƺơƫƞƿƫǁƣƢǀƞƶƫƿDŽƺƤƿƩƣƿƣƞơƩer-researcher role. On one hand, as a teacher we want to make sure students work on their own; therefore, the teacher needs to maintain distance. On the other hand, as a researcher, we need to create methods of observation that ƨƫǁƣƞơƺƸƻƽƣƩƣƹƾƫǁƣƽƣơƺƽƢƺƤƾƿǀƢƣƹƿƾᅷƞơƿƫƺƹƾƫƹƺƽƢƣƽƿƺƟƣƞƟƶƣƿƺƢƣƽƫǁƣ definite conclusions.

3.5

Conclusions

ƹ ƺǀƽ ƽƣǁƫƣǂ ƺƤ ƣƾƿƞƟƶƫƾƩƫƹƨ ƞ ơƽƣƞƿƫǁƣ ƶƣƞƽƹƫƹƨ ƣƹǁƫƽƺƹƸƣƹƿ ᄬѹ҂ѻᄭᄕ ǂƣ described two different modes of organizing the ѹ҂ѻ, which led to a successful Aha! experience: collaborative or individual work on a complex mathematical problem and a classroom teaching-research dialogue. The two modes of organization of a ѹ҂ѻ are, of course, extreme along the spectrum of the variable of the teacher involvement. The space in between the two extrema of the teachƣƽᅷƾƫƹǁƺƶǁƣƸƣƹƿƫƾƢƫƾơǀƾƾƣƢƫƹƣơƿƫƺƹᇵᄙᇵᄙ We discussed the cognitive content of each type while pointing to the essential role of the teacher, who, during the interaction with the student leading to the moment of insight, must transform herself/himself into the teacher-researcher in order to be able to facilitate creativity. Answering the research question, we see the presence of two distinct ways of facilitation within Aha! pedagogy, through open complex collaborative probƶƣƸƾ ƢƣƾƫƨƹƣƢ Ƥƺƽ ƿƩƣ ƾơƺƻƣ ƺƤ ƾƿǀƢƣƹƿƾᅷ Ґ҆Ѻs, and through scaffolding the

ѹ‫ؠآآإئئؔ؟‬Ѽؔ‫ؙآءآ؜اؔا؜؟؜ؖ‬ѷ‫ؔ؛‬ᄛ҃‫اءؘؠآ‬ѿ‫ئا؛ؚ؜ئء‬

ሥሤሩ

elementary mathematical concepts. We see that the role of the teacher in both is significantly different, in the open problem environment the teacher keeps herself/himself in the background serving as “consultant.”

ƹƿƩƣƾƸƞƶƶƨƽƺǀƻƾƣƿƿƫƹƨƺƤƞƽƹƣƾᄬᇴᇲᇲᇲᄭᄕƿƩƣƤƫƽƾƿƫƹƾƫƨƩƿƢƣƸƺƹƾƿƽƞƿƣƾ an incubation process that involves insight followed by individual interiorization. Then, as the student becomes able to verbalize their insight, other members of the group are able to internalize this insight. The role of a teacher, in supporting small group or whole class discussion, is to make sure the reasoning that underlies an insight is elaborated upon. Excellent examples of stimulating ơƺƶƶƣơƿƫǁƣƽƣƞƾƺƹƫƹƨƻƽƺơƣƾƾƞƽƣƢƫƾơǀƾƾƣƢƫƹƩƞƻƿƣƽᇷƫƹƿƩƣơƺƹƿƣǃƿƺƤƫƹǁƣƾƿƫƨƞƿƫƺƹƾƺƤơƽƣƞƿƫǁƣƽƣƞƾƺƹƫƹƨᄬƾƣƣƞƶƾƺ ƣƽƾƩƴƺǂƫƿDžƣƿƞƶᄙᄕᇴᇲᇳᇹᄭᄙ This elaboration of reasoning both promotes interiorization by the individual and internalization of the insights by other students. The final insight docǀƸƣƹƿƣƢƟDŽƞƽƹƣƾᄬᇴᇲᇲᇲᄭƫƾƟƺƿƩƞƹƫƹƢƫǁƫƢǀƞƶƽƣƞƶƫDžƞƿƫƺƹƿƩƞƿƿƩƣǁƞƽƫƺǀƾ formulas for height and width could be condensed into one formula for volume, and a collaborative insight, the result of collaborative thinking together. This moment of insights demonstrates the value of wait time during incubation as the students pondered the question without input from the teacher, who acted more as a consultant. ƫƴƣ ƞƽƹƣƾᅷƾ ǂƺƽƴᄕ ƿƩƣ ƫƽ ƽƣƣ ƻƽƺƟƶƣƸ ƾƩƺǂƾ ƿƩƣ ǁƞƶǀƣ ƺƤ ƞ ǂƣƶƶᅟơƺƹstructed, open-ended problem to foster student insight. Once an equation was determined, the student engaged in inverse reasonƫƹƨƿƩƞƿƶƣƢƿƺƤǀƽƿƩƣƽƫƹƾƫƨƩƿƺƹƞƾƿƽǀơƿǀƽƞƶƶƣǁƣƶᄙ ƹƿƩƫƾơƞƾƣᄕƿƩƣƫƹơǀƟƞƿƫƺƹ process was not aided by teacher input.

ƹƿƩƣƢƺƸƞƫƹƻƽƺƟƶƣƸᄕƿƩƣƿƣƞơƩƣƽᅷƾƽƺƶƣƫƾƸƺƽƣƻƽƺƹƺǀƹơƣƢᄕDŽƣƿƿƩƣƿƣƞᅟ ơƩƣƽƾƣƣƴƾƿƺƴƣƣƻƢƫƽƣơƿƿƽƞƹƾƤƣƽƺƤƿƩƺǀƨƩƿƿƺƞƸƫƹƫƸǀƸᄙ ƹƾƿƣƞƢᄕƿƩƣƨƺƞƶ is to provide subtasks that, upon completion will allow the student to grasp their incorrect generalization. This role is a natural extension of constructivist teaching experiment methodology to the classroom, as it seeks to promote individual interiorization. As noted, to be effective, the teacher must know what the student misconception is, and spontaneously decide what subtask is appropriƞƿƣǂƫƿƩƫƹƿƩƣƾƿǀƢƣƹƿᅷƾҐ҆Ѻ. The elephant problem provides a nice example of perseverance. The methodology employed is simple; the peer simply continues to try different analogies until finally it clicks. What exactly made it work? Was the elephant analogy better than those used previously, or was it that a period of incubation was coming to completion, or both? ƺƺƹᅷƾᄬᇴᇲᇳᇴᄭƿƣƞơƩƫƹƨƣǃƻƣƽƫƸƣƹƿƾǀƨƨƣƾƿƾƿƩƣƹƣƣƢƿƺƹƣƨƺƿƫƞƿƣƿƩƣƞƸƺǀƹƿ of distance required to fulfill both the role of the teacher and the role of the ƽƣƾƣƞƽơƩƣƽᄙ ƹƿƩƣƾơƞƤƤƺƶƢƫƹƨƣƹǁƫƽƺƹƸƣƹƿᄕƿƩƣƿƣƞơƩƣƽƫƾƤǀƶƶDŽƫƹǁƺƶǁƣƢƫƹƿƩƣ present student thinking and his/her role is to ask such questions which may lead to understanding through the insight. We observe what seems to be an

ሥሤሪ

ѹ‫ؗءؔؔ؛ؖآءإؔح‬Ѹؔ‫إؘ؞‬

inherent uncertainty in the processes of facilitation, which appears differently in each approach. However, the mathematical situation of Type B occurs spontaneously and is difficult to catch. Close cognitive analyses of exchanges between members of collaborative teams allows the analysis of their dialogues with the help of the interiorization and internalization which had come to the surface during the writing process.

References ƹƢƣƽƾƺƹᄕᄙᄙᄬᇳᇻᇻᇷᄭᄙCognitive psychology and its implicationsᄬᇶƿƩƣƢᄙᄭᄙᄙ ᄙ ƽƣƣƸƞƹᄙ ƹƢƣƽƾƺƹᄕᄙᄙᄕƣƢƣƽᄕᄙᄙᄕƫƸƺƹᄕ ᄙᄙᄕ ƽƫơƾƾƺƹᄕᄙᄙᄕѵƶƞƾƣƽᄕᄙᄬᇳᇻᇻᇺᄭᄙƞƢƫơƞƶ constructivism and cognitive psychology. Education, 1ᄬᇳᄭᄕᇴᇴᇹᅬᇴᇹᇺᄙ ƹƢƣƽƾƺƹᄕᄙᄙᄕѵƽƞƿƩǂƺƩƶᄕᄙᄙᄬ ƢƾᄙᄭᄙᄬᇴᇲᇲᇳᄭᄙA taxonomy for learning, teaching and assessing: A revision of Bloom’s taxonomy of educational objectives. Longman. ƞƴƣƽᄕᄙᄬᇴᇲᇳᇸƞᄭᄙƺƣƾƿƶƣƽᅷƾƿƩƣƺƽDŽƞƾƞƤƺǀƹƢƞƿƫƺƹƤƺƽƻƽƺƟƶƣƸᅟƾƺƶǁƫƹƨᄙ ƹᄙDžƞƽƹƺơƩƞᄕᄙ ƞƴƣƽᄕ ᄙ ƫƞƾᄕ ѵᄙ ƽƞƟƩǀ ᄬ Ƣƾᄙᄭᄕ The creative enterprise of mathematics teaching research: Elements of methodology and practice—From teachers to teachers ᄬƻƻᄙᇴᇸᇹᅬᇴᇺᇸᄭᄙƣƹƾƣǀƟƶƫƾƩƣƽƾᄙ ƞƴƣƽᄕᄙᄬᇴᇲᇳᇸƟᄭᄙƽƺƻƺƽƿƫƺƹƞƶƽƣƞƾƺƹƫƹƨƞƹƢƻƣƽơƣƹƿᄙ ƹᄙDžƞƽƹƺơƩƞᄕᄙƞƴƣƽᄕᄙ ƫƞƾᄕѵᄙƽƞƟƩǀᄬ ƢƾᄙᄭᄕThe creative enterprise of mathematics teaching research: Elements of methodology and practice—From teachers to teachers ᄬƻƻᄙ ᇵᇳᇹᅬᇵᇴᇺᄭᄙ Sense Publishers. ƞƴƣƽᄕᄙᄕ ƫƞƾᄕ ᄙᄕ DžƞƽƹƺơƩƞᄕ ᄙᄕ ѵ ƺƩƺᄕ ᄙ ᄬᇴᇲᇳᇸᄭᄙ ƽƣƞƿƫƹƨ ƞơƿƫƺƹ ƾơƩƣƸƞ ƟƞƾƣƢ upon conceptual knowledge. Didactica Mathematicae, 38ᄕᇷᅬᇵᇳᄙ ƞƽƹƣƾᄕᄙᄬᇴᇲᇲᇲᄭᄙƞƨƫơƸƺƸƣƹƿƾƫƹƸƞƿƩƣƸƞƿƫơƾᄘ ƹƾƫƨƩƿƾƫƹƿƺƿƩƣƻƽƺơƣƾƾƺƤơƺƸing to know. For the Learning of Mathematics, 20ᄬᇳᄭᄕᇵᇵᅬᇶᇵᄙ DžƞƽƹƺơƩƞᄕᄙᄬᇴᇲᇳᇵᄭᄙƽƫƞƢƺƤƫƞƨƣƿƞƹƢƞƽơƫƞᄕƤƞƫƽDŽƿƞƶƣƾƞƹƢƶƣƞƽƹƫƹƨƿƽƞưƣơƿƺƽƫƣƾᄙ

ƹᄙƟǀDžᄕIᄙ ƞƾƣƽᄕѵᄙᄙƞƽƫƺƿƿƫᄬ ƢƾᄙᄭᄕProceedings of the Eighth Congress of the European Society for Research in Mathematics Education (CERME 8)ᄬƻƻᄙᇴᇺᇻᇴᅬ ᇴᇺᇻᇵᄭᄙƫƢƢƶƣ ƞƾƿƣơƩƹƫơƞƶƹƫǁƣƽƾƫƿDŽƞƹƢ  ᄙ DžƞƽƹƺơƩƞᄕ ᄙ ᄬᇴᇲᇳᇸᄭᄙ ƣƞơƩƫƹƨᅟƽƣƾƣƞƽơƩ ƣǂ ƺƽƴ ƫƿDŽ ƺƢƣƶ ᄬᄧ ƫƿDŽᄭᄙ ƹ ᄙ DžƞƽƹƺơƩƞᄕᄙƞƴƣƽᄕᄙƫƞƾᄕѵᄙƽƞƟƩǀᄬ ƢƾᄙᄭᄕThe creative enterprise of mathematics teaching research: Elements of methodology and practice—From teachers to teachersᄬƻƻᄙᇵᅬᇴᇳᄭᄙƣƹƾƣǀƟƶƫƾƩƣƽƾᄙ DžƞƽƹƺơƩƞᄕᄙᄕƞƴƣƽᄕᄙᄕƫƞƾᄕᄙᄕѵƽƞƟƩǀᄕᄙᄬ ƢƾᄙᄭᄙᄬᇴᇲᇳᇸᄭᄙThe creative enterprise of mathematics teaching research: Elements of methodology and practice—From teachers to teachers. Sense Publishers. ƣƣƶƶƫƾᄕᄙᄙᄕѵƺƶƢƫƹᄕᄙᄙᄬᇴᇲᇲᇸᄭᄙƤƤƣơƿƞƹƢƸƣƿƞᅟƞƤƤƣơƿƫƹƸƞƿƩƣƸƞƿƫơƞƶƻƽƺƟlem solving: A representational perspective. Educational Studies in Mathematics, 63, ᇳᇵᇳᅬᇳᇶᇹᄙ

ѹ‫ؠآآإئئؔ؟‬Ѽؔ‫ؙآءآ؜اؔا؜؟؜ؖ‬ѷ‫ؔ؛‬ᄛ҃‫اءؘؠآ‬ѿ‫ئا؛ؚ؜ئء‬

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ƫƞƾᄕᄙᄬᇴᇲᇳᇸᄭᄙƞƿƣƞƹƢƻƽƺƻƺƽƿƫƺƹƿƣƞơƩƫƹƨƾƣƼǀƣƹơƣᄙ ƹᄙDžƞƽƹƺơƩƞᄕᄙƞƴƣƽᄕᄙ ƫƞƾᄕѵᄙƽƞƟƩǀᄬ ƢƾᄙᄭᄕThe creative enterprise of mathematics teaching research: Elements of methodology and practice—From teachers to teachers ᄬƻƻᄙ ᇴᇻᇻᅬᇵᇳᇷᄭᄙ Sense Publishers. ǀƟƫƹƾƴDŽᄕ ᄙᄬᇳᇻᇻᇳᄭᄙƺƹƾƿƽǀơƿƫǁƣƞƾƻƣơƿƾƺƤƽƣƤƶƣơƿƫǁƣƞƟƾƿƽƞơƿƫƺƹƫƹƞƢǁƞƹơƣƢƸƞƿƩƣƸƞƿƫơƾᄙ ƹᄙᄙƿƣƤƤƣᄬ ƢᄙᄭᄕEpistemological foundations of mathematical experience ᄬƻƻᄙᇳᇸᇲᅬᇴᇲᇴᄭᄙƻƽƫƹƨƣƽᄙ Eureka Math & Engage߂ߍᄙᄬᇴᇲᇳᇷᄭᄙNew York State Common Core Mathematics Curriculum. Grade 8 Mathematics module 1: Teacher materials. https://www.engageny.org/ ƽƣƾƺǀƽơƣᄧƨƽƞƢƣᅟᇺᅟƸƞƿƩƣƸƞƿƫơƾᅟƸƺƢǀƶƣᅟᇳ ƺƶƢƫƹᄕᄙᄙᄬᇴᇲᇳᇺᄭᄙƺƸƣƹƿƾƺƤƸƞƿƩƣƸƞƿƫơƞƶƫƹƾƫƨƩƿᄙ ƹᄙƞƽơƫƹƫƞƴѵᄙDžƞƽƹƺơƩƞᄬ Ƣƾᄙᄭᄕ Proceedings of the CUNY Research Summit: Creativity in STEMᄬƻƻᄙᇳᇹᅬᇴᇳᄭᄙ ơƞƢƣƸƫơƺƽƴƾᄙƩƿƿƻƾᄘᄧᄧƞơƞƢƣƸƫơǂƺƽƴƾᄙơǀƹDŽᄙƣƢǀᄧƩƺᇇƻǀƟƾᄧᇹᇳ ƞƽƣƶᄕᄙᄕѵƞƶƶᄕᄙᄬᇳᇻᇻᇳᄭᄙƩƣƨƣƹƣƽƞƶᄕƿƩƣƞƟƾƿƽƞơƿᄕƞƹƢƿƩƣƨƣƹƣƽƫơƫƹƞƢǁƞƹơƣƢƸƞƿƩematics. For the Learning of Mathematics, 11ᄬᇳᄭᄕᇵᇺᅬᇶᇴᄙ ƣƽƾƩƴƺǂƫƿDžᄕᄙᄕơƩǂƞƽDžᄕᄙᄙᄕѵƽƣDŽƤǀƾᄕᄙᄬᇴᇲᇲᇳᄭᄙƟƾƿƽƞơƿƫƺƹƫƹơƺƹƿƣǃƿᄘ ƻƫƾƿƣƸƫơ actions. Journal for Research in Mathematics Education, 32ᄕᇳᇻᇷᅬᇴᇴᇴᄙ ƣƽƾƩƴƺǂƫƿDžᄕᄙᄕƞƟƞơƩᄕᄙᄕѵƽƣDŽƤǀƾᄕᄙᄬᇴᇲᇳᇹᄭᄙƽƣƞƿƫǁƣƽƣƞƾƺƹƫƹƨƞƹƢƾƩƫƤƿƾƺƤƴƹƺǂƶedge in the mathematics classroom. ZDM: Mathematics Education, 49ᄬᇳᄭᄕᇴᇷᅬᇵᇸᄙ ƞƢƣưᄕᄙᄬᇳᇻᇻᇻᄭᄙƹƣƶƣƻƩƞƹƿᅭƽǂƩƞƿǀƾƣơƞƹƟƣƸƞƢƣƺƤƸƣƿƺƹDŽƸDŽᄞMatematyka, 2. ƺƣƾƿƶƣƽᄕᄙᄬᇳᇻᇸᇶᄭᄙThe act of creation. Macmillan. ƣƫƴƫƹᄕᄙᄕѵƫƿƿƞᅟƞƹƿƞDžƫᄕᄙᄬᇴᇲᇳᇵƞᄭᄙƽƣƞƿƫǁƫƿDŽƞƹƢƸƞƿƩƣƸƞƿƫơƾƣƢǀơƞƿƫƺƹᄘƩƣƾƿƞƿƣ of the art. ZDM: The International Journal of Mathematics, 45ᄬᇴᄭᄕᇳᇷᇻᅬᇳᇸᇸᄙ ƣƫƴƫƹᄕᄙᄕѵƫƿƿƞᅟƞƹƿƞDžƫᄕᄙᄬ ƢƾᄙᄭᄙᄬᇴᇲᇳᇵƟᄭᄙƽƣƞƿƫǁƫƿDŽƞƹƢƸƞƿƩƣƸƞƿƫơƾƣƢǀơƞƿƫƺƹᄙ Special Issue of ZDM: The International Journal on Mathematics Education, 45ᄬᇴᄭᄙ ƫƶưƣƢƞƩƶᄕ ᄙ ᄬᇴᇲᇲᇶᄭᄙ The Aha! experience: Mathematical contexts, pedagogical experiencesᄬƩƿƩƣƾƫƾᄭᄙƫƸƺƹ ƽƞƾƣƽƹƫǁƣƽƾƫƿDŽᄙ ƫƶưƣƢƞƩƶᄕᄙᄬᇴᇲᇳᇵᄭᄙ ƶƶǀƸƫƹƞƿƫƺƹᄘƹƞƤƤƣơƿƫǁƣƣǃƻƣƽƫƣƹơƣᄞZDM: The International Journal of Mathematics, 45ᄬᇴᄭᄕᇴᇷᇵᅬᇴᇸᇷᄙ ƫƻƸƞƹᄕᄙᄬᇴᇲᇲᇵᄭᄙThinking in education. Cambridge University Press. ƞƾƺƹᄕᄙᄬᇳᇻᇺᇻᄭᄙƞƿƩƣƸƞƿƫơƞƶƞƟƾƿƽƞơƿƫƺƹƞƾƞƽƣƾǀƶƿƺƤƞƢƣƶƫơƞƿƣƾƩƫƤƿƺƤƞƿƿƣƹƿƫƺƹᄙ For the Learning of Mathematics, 9ᄬᇴᄭᄕᇴᅬᇺᄙ ƞƶƞƿƹƫƴᄕᄙᄕѵƺƫơƩǀᄕᄙᄬᇴᇲᇳᇶᄭᄙƣơƺƹƾƿƽǀơƿƫƺƹƺƤƺƹƣƸƞƿƩƣƸƞƿƫơƞƶƫƹǁƣƹƿƫƺƹᄘ ƺơǀƾ ƺƹƾƿƽǀơƿǀƽƣƾƺƤƞƿƿƣƹƿƫƺƹᄙ ƹᄙƫƶưƣƢƞƩƶᄕᄙƣƾƿƣƽƶƣᄕᄙƫơƺƶᄕѵᄙƶƶƞƹᄬ Ƣƾᄙᄭᄕ Proceedings of the 38th Conference of the International Group for the Psychology of Mathematics Education and the 36th Conference of the North American Chapter of the Psychology of Mathematics Education, Vol. 4 ᄬƻƻᄙᇵᇹᇹᅬᇵᇺᇶᄭᄙ ᄙ https://www.pmena.org/proceedings/ ƽƞƟƩǀᄕᄙ ᄬᇴᇲᇳᇸᄭᄙƩƣ ơƽƣƞƿƫǁƣ ƶƣƞƽƹƫƹƨ ƣƹǁƫƽƺƹƸƣƹƿᄙ ƹ ᄙ DžƞƽƹƺơƩƞᄕᄙ ƞƴƣƽᄕ ᄙ ƫƞƾᄕѵᄙƽƞƟƩǀᄬ ƢƾᄙᄭᄕThe creative enterprise of mathematics teaching research: Elements of methodology and practice—From teachers to teachers ᄬƻƻᄙ ᇳᇲᇹᅬᇳᇴᇸᄭᄙ Sense Publishers.

ሥሤሬ

ѹ‫ؗءؔؔ؛ؖآءإؔح‬Ѹؔ‫إؘ؞‬

ƞƿƹƞƹƫƹƨƾƫƩᄕᄙᄕѵDŽƞƹƣᄕᄙᄬᇴᇲᇳᇹᄭᄙƺƹưƣơƿǀƽƫƹƨǁƫƞƞƹƞƶƺƨƫơƞƶƽƣƞƾƺƹƫƹƨơƺƹƾƿƽǀơƿƾ ordinary students into like gifted student. Journal of Physics: Conference Series, 943ᄬᇳᄭᄕ ᇲᇳᇴᇲᇴᇷᄙ ƤƞƽƢᄕᄙᄬᇳᇻᇻᇳᄭᄙƹƿƩƣƢǀƞƶƹƞƿǀƽƣƺƤƸƞƿƩƣƸƞƿƫơƞƶơƺƹơƣƻƿƫƺƹƾᄘƣƤƶƣơƿƫƺƹƾƺƹƻƽƺcesses and objects as different sides of the same coin. Educational Studies in Mathematics, 22ᄬᇳᄭᄕᇳᅬᇵᇸᄙ ƩƣƤƤƫƣƶƢᄕ ᄙ ᄙ ᄬᇴᇲᇳᇹᄭᄙ ƞƹƨƣƽƺǀƾ ƸDŽƿƩƾ ƞƟƺǀƿ ᅸƨƫƤƿƣƢᅺ ƸƞƿƩƣƸƞƿƫơƾ ƾƿǀƢƣƹƿƾᄙ ZDM: Mathematics Education, 49ᄬᇳᄭᄕᇳᇵᅬᇴᇵᄙ ƫƸƺƹᄕᄙᄙᄬᇴᇲᇳᇶᄭᄙƹƣƸƣƽƨƫƹƨƿƩƣƺƽDŽƤƺƽƢƣƾƫƨƹƺƤƸƞƿƩƣƸƞƿƫơƞƶƿƞƾƴƾƣƼǀƣƹơƣƾᄘ ƽƺƸƺƿƫƹƨƽƣƤƶƣơƿƫǁƣƞƟƾƿƽƞơƿƫƺƹƺƤƸƞƿƩƣƸƞƿƫơƞƶơƺƹơƣƻƿƾᄙ ƹᄙƫơƺƶᄕᄙƣƾƿƣƽƶƣᄕ ᄙƫƶưƣƢƞƩƶᄕѵᄙƶƶƞƹᄬ ƢƾᄙᄭᄕProceedings of the 38th Conference of the International Group for the Psychology of Mathematics Education and the 36th Conference of the North American Chapter of the Psychology of Mathematics Education, Vol. 5 ᄬƻƻᄙᇳᇻᇵᅬ ᇴᇲᇲᄭᄙ ᄙƩƿƿƻƾᄘᄧᄧǂǂǂᄙƻƸƣƹƞᄙƺƽƨᄧƻƽƺơƣƣƢƫƹƨƾᄧ ƫƸƺƹᄕᄙᄙᄬᇴᇲᇳᇹᄭᄙ ǃƻƶƫơƞƿƫƹƨƸƞƿƩƣƸƞƿƫơƞƶơƺƹơƣƻƿƞƹƢƸƞƿƩƣƸƞƿƫơƞƶơƺƹơƣƻƿƫƺƹ as theoretical constructs for mathematics education research. Educational Studies in Mathematics, 94ᄬᇴᄭᄕᇳᇳᇹᅬᇳᇵᇹᄙ ƫƸƺƹᄕ ᄙᄕ ƞƶƢƞƹƩƞᄕ ᄙᄕ ơƶƫƹƿƺơƴᄕ ᄙᄕ ƴƞƽᄕ ᄙ ᄙᄕ ƞƿƞƹƞƟƣᄕ ᄙᄕ ѵ ƣƸƟƞƿᄕ ᄙ ᄙ ᄬᇴᇲᇳᇲᄭᄙƢƣǁƣƶƺƻƫƹƨƞƻƻƽƺƞơƩƿƺƾƿǀƢDŽƫƹƨƾƿǀƢƣƹƿƾᅷƶƣƞƽƹƫƹƨƿƩƽƺǀƨƩƿƩƣƫƽƸƞƿƩƣmatical activity. Cognition and Instruction, 28ᄬᇳᄭᄕᇹᇲᅬᇳᇳᇴᄙ ƫƸƺƹᄕ ᄙ ᄙᄕDžǀƽᄕ ᄙᄕ ƣƫƹDžᄕ ᄙᄕ ѵ ƫƹDžƣƶᄕ ᄙ ᄬᇴᇲᇲᇶᄭᄙ ǃƻƶƫơƞƿƫƹƨ ƞ ƸƣơƩƞƹƫƾƸ Ƥƺƽ conceptual learning: Elaborating the construct of reflective abstraction. Journal for Research in Mathematics Education, 35ᄬᇷᄭᄕᇵᇲᇷᅬᇵᇴᇻᄙ ƿƣƤƤƣᄕᄙᄙᄬᇴᇲᇲᇳᄭᄙƹƣǂƩDŽƻƺƿƩƣƾƫƾơƺƹơƣƽƹƫƹƨơƩƫƶƢƽƣƹᅷƾƤƽƞơƿƫƺƹƞƶƴƹƺǂƶƣƢƨƣᄙJournal of Mathematical Behavior, 20ᄬᇵᄭᄕᇴᇸᇹᅬᇵᇲᇹᄙ ƺƫơƞᄕᄙᄕѵƫƹƨƣƽᄕ ᄙᄙᄬᇴᇲᇳᇵᄭᄙƽƺƟƶƣƸƸƺƢƫƤƫơƞƿƫƺƹƞƾƞƿƺƺƶƤƺƽƢƣƿƣơƿƫƹƨơƺƨƹƫƿƫǁƣ flexibility in school children. ZDM: The International Journal of Mathematics, 45ᄬᇴᄭᄕ ᇴᇸᇹᅬᇴᇹᇻᄙ ƺƹƶƞƾƣƽƾƤƣƶƢᄕ ᄙᄬᇳᇻᇻᇺƞᄭᄙƺƨƹƫƿƫƺƹᄕơƺƹƾƿƽǀơƿƫƺƹƺƤƴƹƺǂƶƣƢƨƣᄕƞƹƢƿƣƞơƩƫƹƨᄙ ƹConstructivism in science educationᄬƻƻᄙᇳᇳᅬᇵᇲᄭᄙƻƽƫƹƨƣƽᄙ ƺƹƶƞƾƣƽƾƤƣƶƢᄕ ᄙᄬᇳᇻᇻᇺƟᄕƣƻƿƣƸƟƣƽᄭᄙScheme theory as a key to the learning paradox. ƞƻƣƽƻƽƣƾƣƹƿƣƢƞƿƿƩƣᇳᇷƿƩƢǁƞƹơƣƢƺǀƽƾƣᄕƽơƩƫǁƣƾƣƞƹƫƞƨƣƿᄕƣƹƣǁƞᄕǂƫƿzerland. ƞƶƶƞƾᄕᄙᄬᇳᇻᇴᇸᄭᄙThe art of thought. Harcourt, Brace & Co. Wechsler, S. M., Saiz, C., Rivas, S. F., Vendramini, C. M. M., Almeida, L. S., Mundim, M. ᄙᄕѵ ƽƞƹơƺᄕᄙᄬᇴᇲᇳᇺᄭᄙƽƣƞƿƫǁƣƞƹƢơƽƫƿƫơƞƶƿƩƫƹƴƫƹƨᄘ ƹƢƣƻƣƹƢƣƹƿƺƽƺǁƣƽƶƞƻƻƫƹƨ components? Thinking Skills and Creativity, 27ᄕᇳᇳᇶᅬᇳᇴᇴᄙ ƣƫƾƟƣƽƨᄕᄙᄙᄬᇳᇻᇻᇷᄭᄙƽƺƶƣƨƺƸƣƹƞƿƺƿƩƣƺƽƫƣƾƺƤƫƹƾƫƨƩƿƫƹƻƽƺƟƶƣƸƾƺƶǁƫƹƨᄘƿƞǃƺƹƺƸDŽƺƤƻƽƺƟƶƣƸƾᄙ ƹᄙᄙƿƣƽƹƟƣƽƨѵᄙ ᄙƞǁƫƢƾƺƹᄬ ƢƾᄙᄭᄕThe nature of insight ᄬƻƻᄙᇳᇷᇹᅬᇳᇻᇸᄭᄙ ƽƣƾƾᄙ

ѹ‫ؠآآإئئؔ؟‬Ѽؔ‫ؙآءآ؜اؔا؜؟؜ؖ‬ѷ‫ؔ؛‬ᄛ҃‫اءؘؠآ‬ѿ‫ئا؛ؚ؜ئء‬

ሥሤር

ǀᄕ ᄙᄙᄕǀᄕ ᄙᄙᄕƩƣƹᄕ ᄙᄙᄕѵƩƣƹᄕ ᄙᄙᄬᇴᇲᇳᇶᄭᄙ ǃƻƶƺƽƫƹƨƿƩƣơƽƫƿƫơƞƶƫƹƤƶǀƣƹƿƫƞƶ factors of creativity for college students: A multiple criteria decision-making approach. Thinking Skills and Creativity, 11ᄕᇳᅬᇴᇳᄙ ƺƺƹᄕᄙᄬᇴᇲᇳᇴᄭᄙƞƻƻƫƹƨƸƞƿƩƣƸƞƿƫơƞƶƶƣƞƻƾƺƤƫƹƾƫƨƩƿᄙ ƹᄙƩƺᄬ ƢᄙᄭᄕSelected regular lectures from the 12th International Congress on Mathematical Education ᄬƻƻᄙ ᇻᇳᇷᅬᇻᇵᇴᄭᄙƻƽƫƹƨƣƽᄙƢƺƫᄘᇳᇲᄙᇳᇲᇲᇹᄧᇻᇹᇺᅟᇵᅟᇵᇳᇻᅟᇳᇹᇳᇺᇹᅟᇸᇷᇳ

ѹѾѷ҆Ҋѻ҈ሞ

Assessment of the Depth of Knowledge Acquired during an Aha! Moment Insight Bronislaw Czarnocha

4.1

Introductionሾ

The search for creativity in Aha! moments grew out of our efforts to improve ƾƿǀƢƣƹƿƶƣƞƽƹƫƹƨƺƤƸƞƿƩƣƸƞƿƫơƾƫƹƿƩƣƺǀƿƩƽƺƹǃᄕƞƹƞƽƣƞƺƤƿƩƣƣǂƺƽƴ ƫƿDŽƟƺƽƺǀƨƩƺƤƿƩƣƽƺƹǃᄙƫƿƩƿƩƣƞƽƣƞᅷƾƢƣơƫƾƫǁƣƸƞưƺƽƫƿDŽƺƤƞƿƫƹƺƞƹƢƤƽƫơƞƹƫƸƸƫƨƽƞƹƿƺƽƫƨƫƹƾᄕᇸᇲነƿƺᇺᇲነƺƤǂƩƺƸƞƽƣƤƣƸƞƶƣᄕƸƞƹDŽƺƤƺǀƽƾƿǀƢƣƹƿƾ are the first in their immediate family to attend college. As shown below, this is precisely the population of the underrepresented and underserved. Our focus on Aha! moments—flashes of insight known to the general population from their life experiences under many different guises—has been of serǁƫơƣƫƹƿƩƣƢƣƸƺơƽƞƿƫDžƞƿƫƺƹƺƤơƽƣƞƿƫǁƫƿDŽᄬƽƞƟƩǀѵDžƞƽƹƺơƩƞᄕᇴᇲᇳᇶᄭᄙƣơƞǀƾƣ of wide awareness of such moments, and one might say through their selfƣǃƻƶƞƹƞƿƺƽDŽƢƣƤƫƹƫƿƫƺƹᄬƞƶƫƨƩƿƟǀƶƟᄛᄕƾǀƢƢƣƹơƶƞƽƫƿDŽᄕƺƽǂƩƞƿƻƽƞƩơƞƶƶƾƞᅸƟƫƹƨᄕ ƟƫƹƨᄕƟƫƹƨƸƺƸƣƹƿᅺᄭᄕƿƩƣƾƿǀƢDŽƞƹƢƻƣƢƞƨƺƨƫơƞƶƻƽƞơƿƫơƣƺƤƩƞᄛƸƺƸƣƹƿƾƺƤƤƣƽ entry into the creativity of all students, not just those identified as gifted. Moreover, the profound connection between these moments of insight with positive affect creates a new bond between the student and mathematics. The democratization of creativity means finding the tools and language with which to describe, analyze, and facilitate the creativity of “regular students,” ǂƩƺƩƞǁƣƟƣƣƹƶƞƽƨƣƶDŽƺǁƣƽƶƺƺƴƣƢƫƹơƽƣƞƿƫǁƫƿDŽƶƫƿƣƽƞƿǀƽƣᄬƽƫƽƞƸƞƹƣƿƞƶᄙᄕᇴᇲᇳᇳᄭᄙ The discussion below touches on two reasons why our field of mathematƫơƾƣƢǀơƞƿƫƺƹƹƣƣƢƾƹƣǂᄕƞƶƿƣƽƹƞƿƫǁƣƫƹƾƿƽǀƸƣƹƿƾƤƺƽƞƾƾƣƾƾƫƹƨơƽƣƞƿƫǁƫƿDŽᄘᄬᇳᄭ because contemporary methods exclude the underrepresented and underƾƣƽǁƣƢƤƽƺƸƿƩƣƻƺƺƶƺƤƿƩƣƨƫƤƿƣƢƞƹƢƼǀƫƿƣƻƺƾƾƫƟƶDŽơƽƣƞƿƫǁƣᄕƞƹƢᄬᇴᄭƟƣơƞǀƾƣ there is an absence of measuring instruments that may detect dynamic changes of understanding during a creative insight.

ƹƿƩƣƶƞƾƿƾƣơƿƫƺƹƺƤƿƩƣơƩƞƻƿƣƽᄕǂƣƻƽƺƻƺƾƣƞƹƣǂƼǀƞƶƫƿƞƿƫǁƣƞƾƾƣƾƾƸƣƹƿ ƺƤƿƩƣƢƣƻƿƩƺƤƴƹƺǂƶƣƢƨƣᄬƺᄭƽƣƞơƩƣƢƢǀƽƫƹƨƩƞᄛƸƺƸƣƹƿƫƹƾƫƨƩƿƾᄙǀƽ work is guided by Michael Atiyah, who observed that “The depth of the expeƽƫƣƹơƣƢƣƻƣƹƢƾƺƹƩƺǂƻƽƺƤƺǀƹƢƿƩƣǀƶƿƫƸƞƿƣƽƣƾǀƶƿƫƾᅺᄬƺƽǂƣƫƹƣƿƞƶᄙᄕᇴᇲᇳᇶᄕ ƻᄙᇴᄭᄙ ᇙ ‫ةء؟؟؜إؘؕ؞؝؜؟؞ء؜ءآ؞‬ᄕ‫ءؘؗ؜ؘ؟‬ᄕሦሤሦሥᏺᄩᏺ‫؜آؗ‬ᄘሥሤᄙሥሥሪሧᄧርራሬርሤሤረረረሪረሧረᇇሤሤሩ

ҁ‫ؘؘؚؗ؟تآء‬ѷ‫ءؚؔء؜إبؘؗؗإ؜بؤؖ‬ѷ‫ؔ؛‬ᄛ҃‫اءؘؠآ‬ѿ‫ا؛ؚ؜ئء‬

4.2

ሥሥሥ

Underrepresented and Underserved

This section is the outgrowth of reflections by the Teaching-Research Team ᄬҊ҈ƣƞƸᄭƺƤƿƩƣƽƺƹǃƫƹƺǀƽƞƿƿƣƸƻƿƾƿƺƫƹƿƽƺƢǀơƣ҉Ҋѻ҃ creativity into the ҄҉Ѽ ѿ҄ѹ҂ҋѺѻ҉ projects, a network of approximately a hundred chosen ҉Ҋѻ҃ projects aimed at increasing participation by underrepresented student populations in the United States. The Ҋ҈ Team of the Bronx has been investigating creativity for more than eight years. The ҄҉Ѽ ѿ҄ѹ҂ҋѺѻ҉ሿ program attracted our attention because not a single project took creativity as the principal medium through which to increase student participation in ҉Ҋѻ҃ơƞƽƣƣƽƾᄕƞƹƢƺƹƶDŽᇺነƺƤƿƩƣƞƟƾƿƽƞơƿƾ presented listed creativity as a by-product of their projects. This absence of attention to creativity was especially striking against the background of increased demand for creative thinking and increased innovation in the workplace. A growing number of voices from industry lists creativity as one of the primary expectations for new hires, yet mathematics education is trailing far behind, especially in the context of the underserved. The situation calls for the increased role of creativity in contemporary ҉Ҋѻ҃ domains.

ƿƾ ƫƸƻƺƽƿƞƹơƣ ơƞƹ Ɵƣ ƣƞƾƫƶDŽ ƞƾơƣƽƿƞƫƹƣƢ ƤƽƺƸ ƹǀƸƣƽƺǀƾ ƾƿƞƿƣƸƣƹƿƾ ƞƹƢ research observations, similar to the following: ᅬ According to a major new ѿѸ҃ƾǀƽǁƣDŽƺƤƸƺƽƣƿƩƞƹᇳᄕᇷᇲᇲơƩƫƣƤƣǃƣơǀƿƫǁƣ ƺƤƤƫơƣƽƾ ƤƽƺƸ ᇸᇲ ơƺǀƹƿƽƫƣƾ ƞƹƢ ᇵᇵ ƫƹƢǀƾƿƽƫƣƾ ǂƺƽƶƢǂƫƢƣᄕ ơƩƫƣƤ ƣǃƣơǀƿƫǁƣƾ believe that—more than rigor, management discipline, integrity, or even vision—successfully navigating an increasing complex world will require ơƽƣƞƿƫǁƫƿDŽᄬѿѸ҃ᄕᇴᇲᇳᇲᄭᄙ ᅬ ƹ ǀƶDŽ ᇵᇳᄕ ᇴᇲᇳᇷᄕ ƽƣƞƿƫǁƫƿDŽ ƫƹ ƹƨƫƹƣƣƽƫƹƨ ᄩ ƻƽƫƹƨƣƽƫƹƴ ƹƺƿƣƢ ƿƩƞƿ ơƽƣativity is concerned with the generation of effective and novel solutions to problems. Engineering is concerned more specifically with generating technological solutions to problems. Engineering, in short, is fundamentally a ƻƽƺơƣƾƾƺƤơƽƣƞƿƫǁƣƻƽƺƟƶƣƸᅟƾƺƶǁƫƹƨᄙƽƺƻƶƣDŽᄬᇴᇲᇳᇷᄭƞƾƾƣƽƿƾƿƩƞƿᄕᅸᄴƟᄵƣơƞǀƾƣ creativity is concerned with the generation of effective, novel solutions, creƞƿƫǁƫƿDŽƞƹƢƣƹƨƫƹƣƣƽƫƹƨƞƽƣᄕƫƹƣƾƾƣƹơƣᄕƿǂƺƾƫƢƣƾƺƤƿƩƣƾƞƸƣơƺƫƹᅺᄬƻᄙᇴᄭᄙ ᅬ ƺƺƻƣƽƞƹƢ ƣƞǁƣƽƶƺᄬᇴᇲᇳᇵᄭƞƾƾƣƽƿƿƩƞƿƞơơƺƽƢƫƹƨƿƺƿƩƣƞƿƫƺƹƞƶơƞƢƣƸDŽ of Engineering, students need to begin associating the possibilities in ҉Ҋѻ҃ fields with the need for creativity and real-world problem-solving skills. ᅬ Recent research has focused on the necessity of emphasizing the use of creativity and design in attracting girls to ҉Ҋѻ҃ academic and career fields. One research study that examined the important factors in the different levels of science achievement between men and women documented that creativity

ሥሥሦ

ѹ‫ؔ؛ؖآءإؔح‬

ǂƞƾƞơƽƫƿƫơƞƶƣƶƣƸƣƹƿƤƺƽǂƺƸƣƹƿƺƟƣơƺƸƣƾǀơơƣƾƾƤǀƶƾơƫƣƹƿƫƾƿƾᄬǀƟƺƿƹƫƴ ƣƿƞƶᄙᄕᇳᇻᇻᇵᄭᄙƹƺƿƩƣƽƾƿǀƢDŽƾǀƻƻƺƽƿƣƢƿƩƫƾƞƽƨǀƸƣƹƿƟDŽƞƾƾƣƽƿƫƹƨƿƩƞƿƿƩƣ obstacle that prevented women from pursuing careers in science was not a ƶƞơƴƺƤƞƟƫƶƫƿDŽƟǀƿƿƩƣƾǀƻƻƽƣƾƾƫƺƹƺƤơƽƣƞƿƫǁƫƿDŽᄬ ƹƹƞƸƺƽƞƿƺᄕᇳᇻᇻᇺᄭᄙ ƩƣƽƣƫƾƿƩƣƻƽƺƟƶƣƸᄞƩDŽƫƾƹᅷƿƿƩƣƹƣƣƢƤƺƽơƽƣƞƿƫǁƫƿDŽƽƣơƺƨƹƫDžƣƢƟDŽƿƩƣƻƽƺgrams that address underrepresented and underserved populations despite the obvious need in society? Who are underrepresented and underserved? What is the relationship between those students who are underrepresented and those who are underserved? According to Penn State, the definition of underrepresented minority ᄬҋ҈҃ᄭ ƫƾ Ƥƞƽ ƤƽƺƸ ơƽDŽƾƿƞƶ ơƶƣƞƽᄙ ƺǂƣǁƣƽᄕ ƿƩƣ ҋ҈҃ designation is relatively consistent among schools. An underrepresented minority may be defined as a group whose percentage of the population in a group is lower than their percentage of the population in the country.ቀƫƴƫƻƣƢƫƞᅷƾ ƢƣƤƫƹƫƿƫƺƹ ƫƾ ƞ Ɵƫƿ more specific and tells us that underrepresented groups in science, technology, engineering, and mathematics in the United States include women and some ᄴƣƿƩƹƫơᄵƸƫƹƺƽƫƿƫƣƾᄙቁ ƞƿƞƤƽƺƸƿƩƣƹƫƿƣƢƫƹƨƢƺƸƞƢƢƾƿƺƿƩƫƾƢƣƤƫƹƫƿƫƺƹƿƩƣǀƹƢƣƽƽƣƻƽƣƾƣƹƿƞtion of students from lower socioeconomic social classes and neighborhoods; data from Poland emphasizes underrepresentation of women in ҉Ҋѻ҃ profesƾƫƺƹᄬᇵᇹነƤƣƸƞƶƣƿƺᇸᇵነƸƞƶƣᄭᄙ ƹƾǀƸƸƞƽDŽᄕǀƹƢƣƽƽƣƻƽƣƾƣƹƿƣƢƻƺƻǀƶƞƿƫƺƹƾ are students of certain ethnic backgrounds, women, and students whose parents are from low-income households. Who are underserved students and in what way are they underserved? For ƽƣƣƹ ᄬᇴᇲᇲᇸᄭᄕ ƩƫƾƿƺƽƫơƞƶƶDŽ ǀƹƢƣƽƾƣƽǁƣƢ ƾƿǀƢƣƹƿƾ ƞƽƣ ƶƺǂᅟƫƹơƺƸƣ ƾƿǀƢƣƹƿƾᄕ those who are first in their families to attend college, and students of color. He finds two approaches in education that reinforce the quality of being underƾƣƽǁƣƢᄘ ᄬᇳᄭ ƨƽƣƞƿ ơƩƞƶƶƣƹƨƣƾ ƫƹ ƿƩƣ ƻƫƻƣƶƫƹƣ ƶƣƞƢƫƹƨ ƤƽƺƸ ƴƫƹƢƣƽƨƞƽƿƣƹƾ ƞƹƢ schools toward college and university education because of policies, programs, and curricula that place underserved students at a greater disadvantage, and ᄬᇴᄭƿƩƣƺǁƣƽƽƣƶƫƞƹơƣƺƹƞƢƣƤƫơƫƿƸƺƢƣƶᄕƫƹǂƩƫơƩƸƫƹƺƽƫƿDŽᄕƶƺǂᅟƫƹơƺƸƣᄕƞƹƢ first-generation college students are characterized as lacking the skills and abilities necessary to succeed in higher education. Ʃƫƾ Ƥƺơǀƾ ƺƹ ƢƣƤƫơƫƿƾ ƣƸƻƩƞƾƫDžƣƾ ƾƿǀƢƣƹƿƾᅷ ƫƹƞƟƫƶƫƿƫƣƾ ƽƞƿƩƣƽ ƿƩƞƹ ƿƩƣƫƽ abilities and encourages policies and programs that view underserved students as “less than” their peers who have traditionally populated colleges and universities. We see that underrepresentation of minority, lower socioeconomic, female, and first-generation students is conditioned by their being underserved through

ҁ‫ؘؘؚؗ؟تآء‬ѷ‫ءؚؔء؜إبؘؗؗإ؜بؤؖ‬ѷ‫ؔ؛‬ᄛ҃‫اءؘؠآ‬ѿ‫ا؛ؚ؜ئء‬

ሥሥሧ

the particular challenges while transiting along the educational pipeline as well as by the prevalence of deficit pedagogy. Both routes filter out the future underserved by imposing standards extraneous to creativity. As a result, creativity is kidnapped from minority, lower socioeconomic background, female, and first-generation students.

4.3

Creativity Kidnapped

How do both conditions play out in the context of research on and pedagogy of creativity in mathematics classrooms? How do they kidnap creativity? That is, how is creativity “abducted and held captive?” 4.3.1 Exclusion of Underserved Students ƫƶưƣƢƞƩƶᄬᇴᇲᇳᇵᄭƿƽƞơƣƾƸǀơƩƺƤƿƩƣƽƣƾƣƞƽơƩƺƹơƽƣƞƿƫǁƫƿDŽƫƹƸƞƿƩƣƸƞƿƫơƾƿƺ the work of Henri Poincaré, noting that his thoughts, “stand to this day as the most insightful and reflective instances of illumination as well as one of the most thorough treatments of the topic of mathematical discussion, creativity ƞƹƢ ƫƹǁƣƹƿƫƺƹᅺ ᄬƻᄙ ᇴᇷᇶᄭᄙ ƺƣƾƿƶƣƽ ᄬᇳᇻᇸᇶᄭ Ƽǀƺƿƣƾ ƶƺƹƨ ƻƞƾƾƞƨƣƾ ƺƤ ƺƫƹơƞƽȅ ƿƺ portray a metaphysical quality of intuition that is intimately involved in the creative process and the need for all students to experience their own intuƫƿƫƺƹƫƹƿƩƫƾơƽƣƞƿƫǁƣƻƽƺơƣƾƾᄙƺƣƾƿƶƣƽᄬᇳᇻᇸᇶᄭƻƺƾƿǀƶƞƿƣƾƞƾƻƣơƫƤƫơƸƣơƩƞƹƫƾƸ called bisociation that underlies the illumination of an Aha! moment within the Gestalt model, which combines often unconscious intuition and the conscious problem-solving process.

ƺƽƺƣƾƿƶƣƽƞƾǂƣƶƶƞƾƫƶưƣƢƞƩƶᄬᇴᇲᇳᇵᄭᄕƿƩƫƾƻƽƺơƣƾƾƩƞƾƞƹƞƤƤƣơƿƫǁƣơƺƸƻƺnent. Research on creativity, however, soon departed from its roots in intuition and the process leading to illumination and instead became focused only on the novel or original products of creativity and the methods for assessing the potential for such thought. Mathematical talent and giftedness are of course very precious; however, the contemporary process of identifying giftedness reveals an important example of underserving, which leads to underrepresentation as discussed above. ƣơƣƹƿƻǀƟƶƫơƞƿƫƺƹƾƺƹơƽƣƞƿƫǁƫƿDŽᄬƣƫƴƫƹѵƽƫƽƞƸƞƹᄕᇴᇲᇳᇸᄖƣƫƴƫƹƣƿƞƶᄙᄕᇴᇲᇳᇹᄭ emphasize the creativity of gifted students defined through fluency-achieveƸƣƹƿƺƹƾƿƞƹƢƞƽƢƫDžƣƢƿƣƾƿƾƾǀơƩƞƾƿƩƣƿƞƹƤƺƽƢᅬƫƹƣƿѿ҇ƿƣƾƿᄬƣƫƴƫƹƣƿƞƶᄙᄕ ᇴᇲᇳᇹᄖƞƽƻᄕᇴᇲᇳᇹᄭᄙơơƺƽƢƫƹƨƿƺƞƨƹƣƽƞƹƢƫƸƸƣƽƸƞƹᄬᇳᇻᇺᇸᄭᄕƿǂƺƺƿƩƣƽƿƣƾƿƾ commonly used to identify giftedness by high achievement are the ScholasƿƫơƻƿƫƿǀƢƣƣƾƿᄬ҉ѷҊᄭƞƹƢƿƩƣ ƞƸƟǀƽƨƣƽƣƾƿƤ˦ƽƞƿƩƣƸƞƿƫƾơƩƣƣƨƞƟǀƹƨ ᄬѾҊ҃ѸᄭᄕƞƾƣƿƺƤƾƣǁƣƹƻƽƺƟƶƣƸƾƢƣƾƫƨƹƣƢƣƾƻƣơƫƞƶƶDŽƤƺƽƿƞƶƣƹƿƾƣƞƽơƩᄙ

ሥሥረ

ѹ‫ؔ؛ؖآءإؔح‬

Research on gifted students often focuses on their original or novel products ƞƾ ƸƣƞƾǀƽƣƢ ƟDŽ ƿƣƾƿƾ ƟƞƾƣƢ ƺƹ ƿƩƣ ƫƢƣƞƾ ƺƤ ƿǂƺ Ƹƣƽƫơƞƹ ƻƾDŽơƩƺƶƺƨƫƾƿƾᄘ ᄬᇳᄭ ƺDŽᄙǀƫƶƤƺƽƢᅷƾƢƫƾƿƫƹơƿƫƺƹƟƣƿǂƣƣƹơƺƹǁƣƽƨƣƹƿƞƹƢƢƫǁƣƽƨƣƹƿƿƩƫƹƴƫƹƨᄕƞƹƢ ᄬᇴᄭ ƶƶƫƾᄙƺƽƽƞƹơƣᅷƾǂƺƽƴƺƹơƩƞƽƞơƿƣƽƫƾƿƫơƾƾǀơƩƞƾƤƶǀƣƹơDŽᄕƤƶƣǃƫƟƫƶƫƿDŽᄕƞƹƢ ƹƺǁƣƶƿDŽᄙƣƹƺƿƣƿƩƞƿƺƣƾƿƶƣƽᄬᇳᇻᇸᇶᄭƞƨƽƣƣƾǂƫƿƩƿƩƣƹƺǁƣƶơƽƫƿƣƽƫƞǂƩƫƶƣƩƣ disapproves of fluency as something we can learn without understanding and is skeptical about flexibility if it is learned without understanding. This means that giftedness is traditionally found among students who are very good in school measurements of achievement. This approach filters out those students who might be creatively gifted but who are not fluent in mathematics language and procedures. What does it mean to be mathematically ơƽƣƞƿƫǁƣƟǀƿƹƺƿƤƶǀƣƹƿƫƹƸƞƿƩƣƸƞƿƫơƞƶƶƞƹƨǀƞƨƣᄞ ƿƺƤƿƣƹƸƣƞƹƾƤƫƹƢƫƹƨƺǀƽselves in a remedial classroom filled with many other talented students who are unaware of their creative potential, which may get destroyed under the ƾƿƽƣƾƾƞƾƾƺơƫƞƿƣƢǂƫƿƩƣƸƺƿƫƺƹƞƶƺƽƸƞƿƣƽƫƞƶƻƺǁƣƽƿDŽᄙƩƞƿᅷƾǂƩDŽƽƞƟƩǀᄬᇴᇲᇳᇸᄭ notes based on her teaching-research experience that “the creativity in teaching remedial mathematics is teaching gifted students how to access their own ƨƫƤƿƣƢƹƣƾƾᅺᄬƻᄙᇳᇴᇲᄭᄙ How can we understand the creativity of an Aha! moment without fluency and flexibility as its defining aspects? Eliminating both qualities from the assessment of creativity eliminates the impact of habits through which fluency and flexibility are developed and which tend to restrict the originality ƺƤƿƩƣƶƣƞƽƹƣƽᄙ ƹƾƿƣƞƢᄕƫƤǂƣƤƺơǀƾƺƹƿƩƣƼǀƞƶƫƿƞƿƫǁƣƞƾƾƣƾƾƸƣƹƿƺƤƞƶƣƞƽƹƣƽᅷƾơƺƹƾƿƽǀơƿƫƺƹƢǀƽƫƹƨƿƩƣƸƺƸƣƹƿƺƤƫƹƾƫƨƩƿᄕǂƣƩƞǁƣƞơƩƞƹơƣƿƺƶƣƞƽƹƿƩƣ nature of that construction without in any way influencing the originality of ƿƩƞƿƻƣƽƾƺƹᅷƾơƽƣƞƿƫǁƫƿDŽᄙ ƩƞƽƺƹƫơƩƞƣƶᅟƩƞƢǂƣƶƶᄬᇴᇲᇳᇲᄭƻƽƺǁƫƢƣƾƞƸƺƽƣƨƣƹƣƽƞƶƾDŽƾƿƣƸƫơƞƹƞƶDŽƾƫƾ of the filtering process: The overrepresentation of White students, as opposed to historically undeserved students, in gifted programs exists because of traditional characteristics associated with gifted children versus gifted behaviors ƞƿƿƽƫƟǀƿƣƢƿƺơǀƶƿǀƽƞƶƢƫƤƤƣƽƣƹơƣƾƺƽƣǃƻƣƽƫƣƹơƣƾᄙᄚᄬƞƶƢǂƫƹᄕᇴᇲᇲᇷᄖƞƹƹƫƹƨᄕᇴᇲᇲᇸᄭᄙƽƞƢƫƿƫƺƹƞƶƟƣƩƞǁƫƺƽƾƞƿƿƽƫƟǀƿƣƢƿƺƞơƞƢƣƸƫơƨƫƤƿƣƢƹƣƾƾƞƽƣ “high grades, high scores on standardized achievement and aptitude ƿƣƾƿƾᄕƞƹƢƾƿƽƺƹƨơƶƞƾƾƽƺƺƸƻƣƽƤƺƽƸƞƹơƣᅺᄬƽƫƨƨƾƣƿƞƶᄙᄕᇴᇲᇲᇺᄕƻᄙᇳᇵᇴᄭᄖDŽƣƿ ƽƫƨƨƾƣƿƞƶᄙᄬᇴᇲᇲᇺᄭƹƺƿƣƢƿƩƞƿƽƞơƫƞƶƞƹƢơǀƶƿǀƽƞƶơǀƾƿƺƸƾƫƹƤƶǀƣƹơƣƿƩƣ manifestations of advanced behaviors not comparable to the norm, often ơƞǀƾƫƹƨƿƩƣƸƫƾƫƢƣƹƿƫƤƫơƞƿƫƺƹƺƤơǀƶƿǀƽƞƶƶDŽƢƫǁƣƽƾƣƾƿǀƢƣƹƿƾᄙᄬƻᄙᇳᇲᇳᄭ

ҁ‫ؘؘؚؗ؟تآء‬ѷ‫ءؚؔء؜إبؘؗؗإ؜بؤؖ‬ѷ‫ؔ؛‬ᄛ҃‫اءؘؠآ‬ѿ‫ا؛ؚ؜ئء‬

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She does assert that some researchers have argued that testing creates a cultural bias through ƞƹ ƺǁƣƽƿ ƽƣƶƫƞƹơƣ ƺƹ ƾƿǀƢƣƹƿƾᅷ ƸƞƿƩƣƸƞƿƫơƞƶ ƞƹƢ ƶƫƹƨǀƫƾƿƫơ ƞƟƫƶƫƿƫƣƾ ƞƾ well as ѿ҇ƿƣƾƿƾơƺƽƣƾᄬƽƺƾƾѵƽƺƾƾᄕᇴᇲᇲᇷᄖƞƽƢƹƣƽᄕᇴᇲᇲᇸᄭᄙƩƣƞƿƫƺƹƞƶ ƾƾƺơƫƞƿƫƺƹƤƺƽƫƤƿƣƢƩƫƶƢƽƣƹᄬᇴᇲᇲᇺᄭƞƢǁƺơƞƿƣƢƤƺƽƿƩƣƾǀƾƿƞƫƹƣƢƣǃƻƶƺration, adaptation, and re-evaluation of alternative assessment instruments and practices that grant equal opportunities to all potentially ƨƫƤƿƣƢơƩƫƶƢƽƣƹᄙᄬƻᄙᇳᇲᇴᄭ Our task in this chapter is to formulate an alternative approach for assessing ƿƩƣƢƣƻƿƩƺƤƴƹƺǂƶƣƢƨƣᄬƺᄭƽƣƞơƩƣƢƢǀƽƫƹƨƩƞᄛƸƺƸƣƹƿƾᄕǂƩƫơƩƞƻƻƶƫƣƾ to all students including, of course, “all potentially gifted students.” Before we ƻƽƺƻƺƾƣƿƩƣƸƣƿƩƺƢᄕǂƣƶƺƺƴƞƿǂƩDŽƿƩƣƾƿƞƹƢƞƽƢƫƹƾƿƽǀƸƣƹƿƾƺƤƺᄕƾǀơƩ ƞƾƶƺƺƸᅷƾᄬᇳᇻᇷᇸᄭƿƞǃƺƹƺƸDŽƞƹƢƿƩƣƽƣǁƫƾƣƢƶƺƺƸᅷƾƿƞǃƺƹƺƸDŽᄬƹƢƣƽƾƺƹѵ ƽƞƿƩǂƺƩƶᄕᇴᇲᇲᇳᄭᄕƞƽƣƫƹƞƢƣƼǀƞƿƣƤƺƽƺǀƽƻǀƽƻƺƾƣᄙƩƣơƩƞƹƨƣƤƽƺƸƶƺƺƸᅷƾ ƿƞǃƺƹƺƸDŽ ƿƺ ƿƩƣ ƽƣǁƫƾƣƢ ƶƺƺƸᅷƾ ƿƞǃƺƹƺƸDŽ ƽƣǁƣƞƶƾ ƣƶƣƸƣƹƿƾ ƺƤ ƿƩƣ ƢƣƤƫơƫƿ approach mentioned above. 4.3.2 Example of the Deficit Approach—Holding Captive ƹƺƿƫơƣƞƟƶƣƞƾƾƞǀƶƿƺƹơƽƣƞƿƫǁƫƿDŽƾƿƞƽƿƣƢƫƹᇴᇲᇲᇳǂƩƣƹƹƢƣƽƾƺƹƞƹƢƽƞƿƩǂƺƩƶ ǂƫƿƩ ƿƩƣƫƽ ƿƣƞƸ ƻǀƟƶƫƾƩƣƢ ƞ ƽƣǁƫƾƣƢ ƶƺƺƸᅷƾ ƿƞǃƺƹƺƸDŽ ƟƞƾƣƢ ƺƹ ƹƣǂ ƽƣƾƣƞƽơƩ ơƺƹƢǀơƿƣƢ Ɵƣƿǂƣƣƹ ᇳᇻᇻᇷ ƞƹƢ ᇴᇲᇲᇲ ᄬƹƢƣƽƾƺƹ ѵ ƽƞƿƩǂƺƩƶᄕ ᇴᇲᇲᇳᄭᄙ ƩƣƻƽƺơƣƾƾƽƣƾǀƶƿƣƢƫƹƾƫƨƹƫƤƫơƞƹƿơƩƞƹƨƣƾƿƺƿƩƣƺƽƫƨƫƹƞƶƶƺƺƸᅷƾƿƞǃƺƹƺƸDŽᄙ

ƫƨǀƽƣᇶᄙᇳơƺƸƻƞƽƣƾƿƩƣƿǂƺᄙƩƣƽƣƞƽƣƿƩƽƣƣƣƾƾƣƹƿƫƞƶơƩƞƹƨƣƾƫƹƿƩƣƽƣǁƫƾƣƢ ƿƞǃƺƹƺƸDŽᄘᄬᇳᄭƿƩƣǀƾƣƺƤǁƣƽƟƾƫƹƾƿƣƞƢƺƤƹƺǀƹƾᄕᄬᇴᄭƿƩƣơƩƞƹƨƫƹƨƺƤƾDŽƹƿƩƣƾƫƾ ƿƺơƽƣƞƿƫǁƫƿDŽᄕƞƹƢᄬᇵᄭƿƩƣơƩƞƹƨƣƺƤƺƽƢƣƽƤƽƺƸƾDŽƹƿƩƣƾƫƾᏉƣǁƞƶǀƞƿƫƺƹƫƹƿƩƣƺƽƫƨƫƹƞƶƶƺƺƸᅷƾƿƞǃƺƹƺƸDŽƿƺƣǁƞƶǀƞƿƫƺƹᏉơƽƣƞƿƫǁƫƿDŽƫƹƿƩƣƽƣǁƫƾƣƢƺƹƣᄙƩƣƸƞƫƹ ƫƾƾǀƣƞƾƫƿƽƣƶƞƿƣƾƿƺƴƫƢƹƞƻƻƫƹƨơƽƣƞƿƫǁƫƿDŽƫƾƹƺƿƫƹƻƺƫƹƿᇳƺƽᇴᄕƞƾơƽƣƞƿƫǁƫƿDŽ often involves, of course, synthesis. Ʃƣ ƴƫƢƹƞƻƻƫƹƨ ƻƽƺơƣƾƾ ƾƿƞƽƿƾ ƫƹ ƻƺƫƹƿ ᇵᅭƿƩƣ ơƩƞƹƨƣ ƺƤ ƺƽƢƣƽ Ɵƣƿǂƣƣƹ evaluation and synthesis/creativity in the revised taxonomy. Note that evaluation in the original taxonomy, which comes after synthesis, is characterized by ƿƩƣƾƿƞƿƣƸƣƹƿᅸǀƢƨƣǁƞƶǀƣƺƤᄴƺƟƿƞƫƹƣƢᄵƸƞƿƣƽƫƞƶƤƺƽƞƨƫǁƣƹƻǀƽƻƺƾƣᅺǂƩƫƶƣ the evaluation in the revised one introduces a different element, significantly, before reaching creativity on the pyramid: make judgments based on criteria and standards.

ƹ ƺƿƩƣƽ ǂƺƽƢƾᄕ ǂƩƣƽƣƞƾ ƿƩƣ ƾDŽƹƿƩƣƾƫƾᄧơƽƣƞƿƫǁƫƿDŽ ƻƽƺơƣƾƾ ǂƞƾ ƾƫƿǀƞƿƣƢ ƾƺƶƣƶDŽƟƣƿǂƣƣƹƿƩƣơƽƣƞƿƺƽƞƹƢƿƩƣƻǀƽƻƺƾƣƺƤƶƺƺƸᅷƾƿƞǃƺƹƺƸDŽᄕƿƩƣƽƣǁƫƾƣƢ

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ѹ‫ؔ؛ؖآءإؔح‬

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