When erroneous mathematical thinking is just as "correct": The oxymoron of rational errors

The Nature of Mathematical Thinking

Volumn , Issue , 2012, Pages 55-80

  Ben Zeev, Talia ^a  

^a YALE UNIVERSITY   (United States)

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EID: 85070018831     PISSN: None     EISSN: None     Source Type: Book    
DOI: None     Document Type: Chapter

References (62)

1. Anderson, J. R. (1983). The architecture of cognition. Cambridge, MA: Harvard University Press
2. Anderson, J. R. (1989). The analogical origin of errors in problem solving. In D. Klahr and K. Kotovsky (Eds.), Complex information processing: The impact of Herbert A. Simon (pp. 343-371). Hillsdale, NJ: Lawrence Erlbaum Associates
3. Anderson, J. R. (1993). Rules of the mind. Hillsdale, NJ: Lawrence Erlbaum Associates
4. Anderson, J. R., Farrell, R., & Saurers, R. (1984). Learning to program in LISP. Cognitive Science, 8, 87-129
5. Anderson, J. R., & Thompson, R. (1989). Use of analogy in a production system architecture. In S. Vosniadou & A. Ortony (Eds.), Similarity and analogical reasoning (pp. 267-297). Cambridge, MA: Cambridge University Press
6. Bartlett, F. C. (1932). Remembering. Cambridge, MA: Cambridge University Press
7. Ben-Zeev, T. (1995a). The nature and origin of rational errors in arithmetic thinking: Induction from examples and prior knowledge. Cognitive Science, 19, 341-376
8. Ben-Zeev, T. (1995b). Rational errors and the mathematical mind. Manuscript submitted for publication
9. Brown, J. S., & Burton, R. B. (1978). Diagnostic models for procedural bugs in basic mathematical skills. Cognitive Science, 2, 155-192
10. Brown, J. S., & VanLehn, K. (1980). Repair Theory: A generative theory of bugs in procedural skills. Cognitive Science, 4, 379-426
11. Chi, M. T. H., Bassok, M., Lewis, M. W., Reimann, P., & Glaser, R. (1989). Self-explanations: How students study and use examples in learning to solve problems. Cognitive Science, 13, 145-182
12. Chi, M. T. H., Feltovich, P. J., & Glaser, R. (1981). Categorization and representation of physics problems by experts and novices. Cognitive Science, 5, 121-152
13. Chi, M. T. H., & VanLehn, K. (1991). The content of physics self-explanations. Journal of the Learning Sciences, I, 69-105
14. Davis, R. B. (1982). The postulation of certain specific, explicit, commonly-shared frames. Journal of Mathematical Behavior, 3, 167-201
15. Davis, R. B. (1984). Learning mathematics: The cognitive science approach to mathematics education. Norwood, NJ: Ablex
16. Davis, R. B., & Vinner, S. (1986). The notion of limit: Some seemingly unavoidable misconception stages. Journal of Mathematical Behavior, 5, 281-303
17. Dugdale, S. (1993). Functions and graphs-Perspectives on student thinking. InT. A. Romberg, E. Fennema, & T. Carpenter (Eds.), Integrating research on the graphical representation of functions (pp. 101-130). Hillsdale, NJ: Lawrence Erlbaum Associates
18. Duncker, K. (1945). On problem solving. Psychological Monographs, 58 (Whole No. 270)
19. Gentner, D. (1983). Structure-mapping: A theoretical framework. Cognitive Science, 7, 155-170
20. Gibson, J. J. (1979). The ecological approach to visual perception. Boston, MA: Houghton Mifflin
21. Gick, M. L., & Holyoak, K. J. (1980). Analogical problem solving. Cognitive Psychology, 12, 306-355
22. Gick, M. L., & Holyoak, K. J. (1983). Schema induction and analogical transfer. Cognitive Psychology, 15, 1-38
23. Greeno, J. G. (1980). Some examples of cognitive task analysis with instructional implications.In R. E. Snow, P. Federico, & W. E. Montague (Eds.), Aptitude, learning, and instruction (Vol. 2, pp. 1-21). Hillsdale, NJ: Lawrence Erlbaum Associates
24. Hinsley, D., Hayes, J. R., & Simon, H. A. (1977). From words to equations. In M. Just & P. Carpenter (Eds.), Cognitive processes in comprehension (pp. 89-106). Hillsdale, NJ: Lawrence Erlbaum Associates
25. Holland, J. H., Holyoak, K. J., Nisbett, R. E., & Thagard, P. R. (1986). Induction: Processes of inference, learning, and discovery. Cambridge, MA: MIT Press
26. Holyoak, K. J., & Thagard, P. R. (1989a). A computational model of analogical problem solving. In S. Vosniadou & A. Ortony (Eds.), Similarity and analogical reasoning (pp. 242-266). Cambridge, England: Cambridge University Press
27. Holyoak, K. J., & Thagard, P. R. (1989b). Analogical mapping by constraint satisfaction. Cognitive Science, 13, 295-355
28. Hudson, T. (1983). Correspondences and numerical differences between disjoint sets. Child Development, 54, 84-90
29. Kintsch, W., & Greeno, J. G. (1985). Understanding and solving word arithmetic problems. Psychological Review, 92, 109-129
30. LeFevre, J., & Dixon, P. (1986). Do written instructions need examples? Cognition and Instruction, 3, 1-30
31. Lewis, M. W., & Anderson, J. R. (1985). Discrimination of operator schemata in problem solving: Learning from examples. Cognitive Psychology, 17, 26-65
32. Matz, M. (1982). Towards a process model for high school algebra errors. In D. Sleeman & J. S. Brown (Eds.), Intelligent tutoring systems (pp. 25-49). New York: Academic Press
33. Mayer, R. E. (1981). Frequency norms and structural analysis of algebraic story problems into families, categories, and templates. Instructional Science, 10, 135-175
34. Mayer, R. E. (1982). Memory for algebra story problems. Journal of Educational Psychology, 74, 199-216
35. Mayer, R. E. (1985). Implications of cognitive psychology for instruction in mathematical problem solving. In E. A. Silver (Ed.), Teaching and learning mathematical problem solving: Multiple research perspectives (pp. 123-138). Hillsdale, NJ: Lawrence Erlbaum Associates
36. Novick, L. R. (1988). Analogical transfer, problem similarity, and expertise. Journal of Experimental Psychology: Learning, Memory, and Cognition, 14, 510-520
37. Novick, L. R, & Holyoak, K. J. (1991). Mathematical problem solving by analogy. Journal of Experimental Psychology: Learning, Memory, and Cognition, 17, 398-415
38. Paige, J. M., & Simon, H. A. (1966). Cognitive processes in solving algebra word problems. In B. Kleinmuntz (Ed.), Problem solving: Research, method, and theory (pp. 51-118). New York: Wiley
39. Payne, S., & Squibb, H. (1990). Algebra mal-rules and cognitive accounts of error. Cognitive Science, 14, 445-481
40. Piaget, J. (1965). The child's conception of number. New York: Norton
41. Pirolli, P. L., & Anderson, J. R. (1985). The role of learning from examples in the acquisition of recursive programming skill. Canadian Journal of Psychology, 39, 240-272
42. Resnick, L. B. (1989). Developing mathematical knowledge. American Psychologist, 44, 162-169
43. Resnick, L. B., Nesher, P., Leonard, F., Magone, M., Omanson, S. F., & Peled, I. (1989). Conceptual bases of arithmetic errors: The case of decimal fractions. Journal for Research in Mathematics Education, 20, 8-27
44. Riley, M.S., & Greeno, J. G. (1988). Developmental analysis about understanding language about quantities and solving problems. Cognition and Instruction, 5, 49-101
45. Riley, M.S., Greeno, J. G., & Heller, J. (1983). The development of children's problem solving ability on arithmetic. In H. P. Ginsburg (Ed.), The development of mathematical thinking (pp. 153-196). New York: Academic Press
46. Rissland, E. L. (1985). Artificial intelligence and the learning of mathematics: A tutorial sampling. In E. A. Silver (Ed.), Teaching and learning mathematical problem solving: Multiple research perspectives (pp. 147-176). Hillsdale, NJ: Lawrence Erlbaum Associates
47. Rosnick, P., & Clement, J. (1980). Learning without understanding: The effect of tutoring strategies on algebra misconceptions. Journal of Mathematical Behavior, 3, 3-27
48. Ross, B. (1984). Remindings and their effects in learning a cognitive skill. Cognitive Psychology, 16, 371-416
49. Schank, R., & Abelson, R. P. (1977). Scripts, plans, goals, and understanding: An inquiry into human knowledge structures. Hillsdale, NJ: Lawrence Erlbaum Associates
50. Schoenfeld, A. H. (1988). When good teaching leads to bad results: The disasters of'well-taught' mathematics courses. Educational Psychologist, 23, 145-166
51. Schoenfeld, A. H. (1991). On mathematics as sense making: An informal attack on the unfortunate divorce of formal and informal mathematics. In J. F. Voss, D. N. Perkins, & J. W. Segal (Eds.), Informal reasoning and education (pp. 311-343). Hillsdale, NJ: Lawrence Erlbaum Associates
52. Schoenfeld, A. H., & Herrmann, D. J. (1982). Problem perception and knowledge structure in expert and novice mathematical problem solvers. Journal of Experimental Psychology: Learning, Memory, and Cognition, 8, 484-494
53. Silver, E. A. (1986). Using conceptual and procedural knowledge: A focus on relationships. In J. Hie bert (Ed.), Conceptual and procedural knowledge: The case of mathematics (pp. 181-198). Hillsdale, NJ: Lawrence Erlbaum Associates
54. Simon, H. A., & Anzai, Y. (1979). The theory of learning by doing. Psychological Review, 86, 124-140
55. Simon, H. A., & Chase, W. G. (1973). Skill in chess. American Scientist, 61, 394-403
56. Simon, H. A., & Zhu, X. (1988). Learning mathematics from examples and by doing. Cognition and ln5truction, 4, 137-166
57. Sweller, J., & Cooper, G. A. (1985). The use of worked examples as a substitute for problem solving in learning algebra. Cognition and Instruction, 2, 59-S9
58. VanLehn, K. (1983). On the representation of procedures in repair theory. In H. P. Ginsburg (Ed.), The development of mathematical thinking (pp. 201-252). Hillsdale, NJ: Lawrence Erlbaum Associates
59. VanLehn, K. (1986). Arithmetic procedures are induced from examples. In J. Hiebert (Ed.), Conceptual and procedural knowledge: The case of mathematics (pp. 133-179). Hillsdale, NJ: Lawrence Erlbaum Associates
60. VanLehn, K. (1990). Mind bugs: The origins of procedural misconceptions. Cambridge, MA: MIT Press
61. VanLehn, K., Jones, R. M., & Chi, M. T. H. (1992). A model of the self-explanation effect. Journal of the Learning Sciences, 2, 1-59
62. Young, R. M., & O'Shea, T. (1981). Errors in students' subtraction. Cognitive Science, 5, 153-177