수학적 사고에 동원되는 두뇌 영역들과 이의 교육학적 의미 = Mathematical thinking, its neural systems and implication for education

다국어 초록 (Multilingual Abstract)

What is the foundation of mathematical thinking? Is it logic based symbolic language system? or does it rely more on mental imagery and visuo-spatial abilities? What kind of neural changes happen if someone`s mathematical abilities improve through practice? To answer these Questions, basic cognitive processes including long term memory, working memory, visuo-spatial perception, number processes are considered through neuropsychological outcomes. Neuronal changes following development and practices are inspected and we can show there are neural networks critical for the mathematical thinking and development: prefrontal-anterior cingulate-parietaI network. Through these inquiry, we can infer the answer to our question.

참고문헌 (Reference)

1 Changeux, J.P, "정신, 물질 그리고수학" 경문사 2002

2 김연미, "신경심리학적 이론에 근거한 수학학습장애의 유형분류 및 심층진단검사의 개발을 위한 기초연구" 한국수학교육학회 14 (14): 237-259, 2011

3 황우형, "수학교육학에서 바라본 학습심리학의 발달과정과 전망" 한국수학교육학회 42 (42): 5-136, 2003

4 Hadamard, J, "수학 분야에서 발명의 심리학" 범양사 1990

5 Baddley, A.D, "Working memory, In The psychology of learning and motivation: Advances in research and theory 8" Academic Press 47-89, 1974

6 Holmes, J, "Working memory and children's mathematical skills : Implications for mathematical development and mathematical curricula" 26 : 339-366, 2006

7 Schwanenflugel, P.J, "Why are abstract concepts hard to understand?, In The psychology of word meanings" Erlbaum 235-250, 1991

8 Mc Neil, N.M, "When theories don't add up : Disentangling the manipulatives debate" 46 (46): 309-316, 2007

9 Kosslyn, S.M, "What are you optimistic about: Today's leading thinkers on why things are good and getting better" Harper 285-287, 2007

10 Jarrold, C, "Variation in working memory due to typical and typical development, In Variation in working memory" Oxford University Press 137-161, 2007

11 Wright, R, "Training generalized skills" 15 (15): 763-771, 2008

12 Dehaene, S, "Three parietal circuits for number processing" 20 (20): 487-560, 2003

13 Heathcote, D, "The role of visuo-spatial working memory in the mental addition of multi-digit addends" 13 (13): 207-245, 1994

14 Krutetskii, V. A, "The psychology of mathematical abilities in schoolchildren" Univer-sity of Chicago Press 1976

15 Jung, R.E, "The parieto-frontal integration theory (P-FIT) of intelligence: converging neuroimaging evidence" 30 (30): 135-154, 2007

16 Dehaene, S, "The number sense" Oxford University Press 1997

17 Kong, J, "The neural substrates of arithmetic operations and procedure complexity" 22 (22): 397-405, 2005

18 Newman, S.D, "The neural bases of intelligence: a per-spective based on functional neuroimaging, In Cognition and Intelligence: Identifying the Mechanisms of the Mind" Cambridge University Press 88-103, 2005

19 Simon, T. J, "The foundations of numerical thinking in a brain without numbers" 3 (3): 363-365, 1999

20 Alloway, T. P, "The cognitive and behavioral characteristics of children with low working memory" 80 (80): 606-621, 2009

21 Qin, Y, "The change of the brain activation patterns as children learn algebra equation solving" 101 (101): 5686-5691, 2004

22 Dragansky, B, "Temporal and Spatial Dynamics of Brain Structure Changes during Extensive Learning" 26 (26): 6314-6317, 2006

23 Tall, D.O, "Symbols and the Bifurcation between Procedural and Conceptual Thinking" 1998

24 Michelli, A, "Structural plasticity in the bilingual brain" 431 : 2004

25 Lee, K, "Strategic differences in algebraic problem solving: Neuroanatomical correlates" 1155 : 163-171, 2007

26 Knauff, M, "Spatial imagery in deductive reasoning : A functional MRI study" 13 (13): 203-312, 2002

27 Barsalou, L.W, "Situating abstract concepts, In Grounding cognition: The role of perception and action in memory, language, and thinking" Cambridge University Press 129-163, 2005

28 Varma, S, "Scientific and pragmatic challenges for bridging education and neuroscience" 37 (37): 140-152, 2008

29 Thompson-Schill, S. L, "Role of left prefrontal cortex in retrieval of semantic knowledge: A re-evaluation" 94 (94): 14792-14797, 1997

30 Chan, J, "Retrieval-Induced Facilitation: Initially Nontested Material Can Benefit From Prior testing of Related Material" 135 (135): 553-571, 2006

31 Rasmussen, C, "Representation and working memory in early arithmetic" 91 : 137-157, 2005

32 Dubinsky, E, "Reflective abstraction in advanced mathematical thinking, In Advanced Mathematical Thinking" Kluwer 95-126, 1991

33 Goswami, U, "Neuroscience and Education" 74 (74): 1-14, 2004

34 Prabhakaran, V, "Neural substrates of mathematical reasoning : an fMRI study of neocortical activation during performance of a necessary mathematics operations test" 15 (15): 115-127, 2001

35 Zago, L, "Neural correlates of simple and complex mental calculation" 13 (13): 314-327, 2001

36 van Nes, F, "Mathematics Education and Neuroscience: Relating spatial structures for the development of spatial sense and number sense" 4 (4): 210-229, 2007

37 Maguire, E, "London Taxi drivers and bus drivers: a structural MRI and neuropsychological analysis" 16 (16): 1091-1101, 2006

38 Geary, D. C, "Learning disabilities in arithmetic and mathematics: theoretical and empirical perspectives, In Handbook of mathematical cognition" Psychology Press 253-267, 2005

39 Delazer, M, "Learning complex arithmetic—An fMRI Study" 18 (18): 76-88, 2003

40 Alloway, T. P, "Investigating the predictive roles of working memory and IQ in academic attainment" 106 (106): 20-29, 2010

41 Schroeder, B, "Investigating a metacognitive strategy for solving indefinite integration problems in Calculus" University of Connecticut 2011

42 Krueger, F, "Integral calculus problem solving : an fMRI study" 19 (19): 1095-1099, 2008

43 Anderson, J, "Human symbol manipulation within an integrated cognitive architecture" 29 (29): 313-341, 2005

44 McGee, M, "Human spatial abi1ities: Psychometric studies and environmental, genetic, hormonal, and neurological Influences" 86 (86): 889-918, 1979

45 Barsalou, L. W, "Grounding conceptual knowledge in modalityspecific systems" 7 (7): 84-91, 1999

46 Zhu, Z, "Gender differences in mathematical problem solving patterens: A review of literature" 8 (8): 187-203, 2007

47 Goel, V, "Functional neuroanatomy of three-term relational reasoning" 39 (39): 901-909, 2001

48 Grabner, R. H, "Fact Learning in Complex Arithmetic and Figural-Spatial Tasks: The Role of the Angular Gyrus and its Relation to Mathematical Competence" 30 (30): 2936-2952, 2009

49 Thomas, M, "Evidence from cognitive neuroscience for the role of graphical and algebraic representations in understanding function" 42 (42): 607-619, 2010

50 Tulving, E, "Elements of Episodic Memory" Oxford University Press 1983

51 Paivio, A, "Dual coding theory : Retrospect and current status" 45 (45): 255-287, 1991

52 Rubinsten O, "Developmental dyscalculia: heterogeneity might not mean different mechanisms" 13 (13): 92-99, 2009

53 Rivera, S. M, "Developmental changes in mental arithmetic: Evidence for increased specialization in the left inferior parietal cortex" 15 (15): 1779-1790, 2005

54 Squire, L.R, "Declarative and non-declarative memory: Multiple brain systems supporing learning and memory, In Memory Systems" MIT Press 203-231, 1994

55 Geary, D.C, "Consequences, characterristics, characteristics, and causes of poor mathematics achevement and mathematical learning disabilities" 32 (32): 250-263, 2011

56 Rittle-Johnson, B, "Conceptual and procedural knowledge in learning mathematics: Does one lead to another?" 91 (91): 175-189, 1999

57 Dehaene, S, "Cerebral pathways for calculation: double dissociation between rote verbal and quantitative knowledge of arithmetic" 33 (33): 219-250, 1997

58 Pauli, P, "Brain potentials during mental arithmetic: Effects of extensive practice and problem difficulty" 2 : 21-29, 1994

59 Sohn, M. H, "Behavioral equivalence, but not neural equivalence–neural evidence of alternative strategies in mathematical thinking" 7 (7): 1193-1194, 2004

60 Tang, Y, "Arithemetic Processing in the brain shaped by cultures" 103 (103): 10775-10780, 2006

61 Mohler, J. L, "An instructional method for the AutoCAD modelling environment" 61 (61): 5-13, 1997

62 Terao, A, "An fMRI study of the interplay of symbolic and visuo-spatial systems in mathematical reasoning" Erlbaum 2004

63 Wynn, K, "Addition and subtraction by human infant" 358 (358): 749-750, 1992

64 Davis, G, "A memorybased model for aspects of mathematics teaching, In Proceedings of 24th Conference of the International Group for the Psychology of Mathematics Education 2" Hiroshima University 25-232, 2000