Rational errors and the mathematical mind

Review of General Psychology

Volumn 2, Issue 4, 1998, Pages 366-383

  Ben Zeev, Talia ^a  

^a BROWN UNIVERSITY   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0011738645     PISSN: 10892680     EISSN: None     Source Type: Journal    
DOI: 10.1037/1089-2680.2.4.366     Document Type: Article

References (70)

1. Anderson, J. R. (1983). The architecture of cognition. Cambridge, MA: Harvard University Press.
2. Anderson, J. R. (1989). The origin of errors in problem solving. In D. Klahr & K. Kotovsky (Eds.), Complex information processing: The impact of Herbert A. Simon (pp. 343-371). Mahwah, NJ: Erlbaum.
3. Anderson, J. R. (1993). Rules of the mind. Hillsdale, NJ: Erlbaum.
4. 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, England: Cambridge University Press.
5. Ashlock, R. B. (1976). Error patterns in computation. Columbus, OH: Bell & Howell.
6. Beck, A. T. (1967). Depression: Causes and treatment. Philadelphia: University of Pennsylvania Press.
7. Beck, A. T. (1985). Theoretical perspectives on clinical anxiety. In A. H. Tuma & J. D. Maser (Eds.), Anxiety and the anxiety disorder (pp. 183-198). Mahwah, NJ: Erlbaum.
8. Beck, A. T. (1991). Cognitive therapy: A thirty-year retrospective. American Psychologist, 46, 368-375.
9. Ben-Zeev, T. (1995). The nature and origin of rational errors in arithmetic thinking: Induction from examples and prior knowledge. Cognitive Science, 19, 341-376.
10. Ben-Zeev, T. (1996). When erroneous mathematical thinking is just as "correct": The oxymoron of rational errors. In R. J. Sternberg & T. Ben-Zeev, (Eds.), The nature of mathematical thinking (pp. 55-79). Mahwah, NJ: Erlbaum.
11. Ben-Zeev, T. (1998). Spurious correlations in problem solving. Manuscript submitted for publication.
12. Blessing, S. B., & Ross, B. H. (1996). Content effects in problem categorization and problem solving. Journal of Experimental Psychology: Learning, Memory, and Cognition, 22, 792-810.
13. Bransford, J. D., Sherwood, R. D., & Sturdevant, T. (1987). Teaching thinking and problem solving. In R. J. Sternberg & J. Boykoff Baron (Eds.), Teaching thinking skills: Theory and Practice (pp. 162-181). New York: Freeman.
14. Bransford, J. D., & Stein, B. S. (1984). The IDEAL problem solver. New York: Freeman.
15. Brown, J. S., & Burton, R. B. (1978). Diagnostic models for procedural bugs in basic mathematical skills. Cognitive Science, 2, 155-192.
16. Brown, J. S., & VanLehn, K. (1980). Repair theory: A generative theory of bugs in procedural skills. Cognitive Science, 4, 379-426.
17. Brown, S. I., & Walter, M. I. (1993). Problem posing in mathematics education. In S. I. Brown & M. I. Walter (Eds.), Problem posing: Reflections and applications (pp. 16-27). Hillsdale, NJ: Erlbaum.
18. Buswell, G. T. (1926). Diagnostic studies in arithmetic. Chicago: University of Chicago Press.
19. Chi, M. T. H., & Slotta, J. D. (1993). The ontological coherence of intuitive physics. Cognition and Instruction, 10, 249-260.
20. Cox, L. S. (1975). Diagnosing and remediating systematic errors in addition and subtraction computation. The Arithmetic Teacher, 22, 151-157.
21. Davis, R. B. (1982). The diversity of errors in mathematics. Journal of Mathematical Behavior, 3, 73-77.
22. Davis, R. B., & Vinner, S. (1986). The notion of limit: Some seemingly unavoidable misconception stages. Journal of Mathematical Behavior, 5, 281-303.
23. diSessa, A. A. (1982). Unlearning Aristotelian physics: A study of knowledge-based learning. Cognitive Science, 6, 37-75.
24. diSessa, A. A. (1993). Toward an epistemology of physics. Cognition and Instruction, 10, 105-225.
25. Dreyfus, T. (1990). Advanced mathematical thinking. In P. Nesher & K. Kilpatrick (Eds.), Mathematics and cognition: A research synthesis by the International Group for the Psychology of Mathematics Education (pp. 113-134). Cambridge, MA: Cambridge University Press.
26. Duffin, J. M., & Simpson, A. P. (1993). Natural, conflicting, and alien. Journal of Mathematical Behavior, 12, 313-328.
27. Dugdale, S. (1993). Functions and graphs - Perspectives on student thinking. In T. A. Romberg, E. Fennema, & T. Carpenter (Eds.), Integrating research on the graphical representation of functions (pp. 101-130). Mahwah, NJ: Erlbaum.
28. Fuson, K. C. (1992). Research on learning and teaching addition and subtraction of whole numbers. In G. Leinhardt, R. Putnam, & R. A. Hattrup (Eds.), Analysis of Arithmetic for Mathematics Teaching (pp. 53-188). Mahwah, NJ: Erlbaum.
29. Gelman, R., & Meck, E. (1986). The notion of principle: The case of counting. In J. Hiebert (Ed.), Conceptual and procedural knowledge: The case of mathematics (pp. 29-57). Hillsdale, NJ: Erlbaum.
30. Centner, D. (1983). Structure-mapping: A theoretical framework. Cognitive Science, 7, 155-170.
31. Ginsburg, H. (1996). Toby's math. In R. J. Sternberg & T. Ben-Zeev (Eds.), The nature of mathematical thinking (pp. 175-202). Mahwah, NJ: Erlbaum.
32. Greeno, J. G. (1983). Forms of understanding in mathematical problem solving. In S. G. Paris, G. M. Olson, & H. W. Stevenson (Eds.), Learning and motivation in the classroom (pp. 83-111). Hillsdale, NJ: Erlbaum.
33. Hennessy, S. (1993). The stability of children's mathematical behavior: When is a bug really a bug? Learning and Instruction, 3, 315-338.
34. Holland, J. H., Holyoak, K. J., Nisbett, R. E., & Thagard, P. R. (1986). Induction: Processes of inference, learning, and discovery. Cambridge, MA: MIT Press.
35. Holyoak, K. J., & Thagard, P. R. (1989). A computational model of analogical problem solving. In S. Vosniadou & A. Ortony (Eds.), Similarity and analogical reasoning (pp. 242-266). Cambridge, MA: Cambridge University Press.
36. Jones, E. E., & Davis, K. E. (1965). From acts to dispositions: The attribution process in person perception. In L. Berkowitz (Ed.), Advances in experimental social psychology (Vol. 2). New York: Academic Press.
37. Jones, E. E., & Nisbett, R. (1971). The actor and the observer: Divergent perceptions of the causes of behavior. Morristown, NJ: General Learning Press.
38. Jones, E. E., & Nisbett, R. (1987). The actor and the observer: Divergent perceptions of causality. In E. E. Jones, D. E. Kanouse, H. H. Kelley, & R. E. Nisbett (Eds.), Attribution: Perceiving the causes of behavior (pp. 79-94). Morristown, NJ: General Learning Press.
39. Kahneman, D., & Tversky, A. (1973). On the psychology of prediction. Psychological Review 80, 237-251.
40. Karmiloff-Smith, A. (1986). From meta-processes to conscious access: Evidence from children's metalinguistic and repair data. Cognition, 23, 95-147.
41. Klahr, D., Langley, P., & Neches, R. (1987). Production-system models of learning and development. Cambridge, MA: MIT Press.
42. Langley, P., & Ohllson, S. (1984). Automated cognitive modeling. Proceedings of the Fourth National Conference of the American Association for Artificial Intelligence, 193-197.
43. Lankford, F. G. (1972). Some computational strategies of seventh grade pupils. Charlottesville: University of Virginia. ERIC document.
44. Lewis, C., & Anderson, J. R. (1985). Discrimination of operator schemata in problem solving: Learning from examples. Cognitive Psychology, 17, 26-65.
45. 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.
46. McCloskey, M., Caramazza, A., & Green, B. (1980). Curvilinear motion in the absence of external forces: Naive beliefs about the motion of objects. Science, 210, 1139-1141.
47. Norman, D. A. (1981). Categorization of action slips. Psychological Review, 88, 1-15.
48. Novick, L. R., & Holyoak, K. J. (1991). Mathematical problem solving by analogy. Journal of Experimental Psychology: Learning, Memory, and Cognition, 17, 398-415.
49. Payne, S., & Squibb, H. (1990). Algebra mal-rules and cognitive accounts of error. Cognitive Science, 14, 445-481.
50. Pinker, S. (1994). The language instinct. New York: Morrow.
51. 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.
52. Resnick, L. B., & Omanson, S. F. (1987). Learning to understand arithmetic. In R. Glaser (Ed.), Advances in instructional psychology (Vol. 3, pp. 41-95). Hillsdale, NJ: Erlbaum.
53. 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: Erlbaum.
54. Rosnick, P., & Clement, J. (1984). Learning without understanding: The effect of tutoring strategies on algebra misconceptions. Journal of Mathematical Behavior, 3, 3-27.
55. Ross, B. (1984). Remindings and their effects in learning a cognitive skill. Cognitive Psychology, 16, 371-416.
56. Ross, L. (1977). The intuitive psychologist and his shortcomings: Distortions in the attribution process. In L. Berkowitz (Ed.), Advances in experimental social psychology (Vol. 10). New York: Academic Press.
57. Schoenfeld, A. H. (1988). When good teaching leads to bad results: The disasters of "well-taught" mathematics courses. Educational Psychologist, 23, 145-166.
58. 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: Erlbaum.
59. Shaughnessy, J. M. (1985). Problem-solving derailers: The influence of misconceptions on problem-solving performance. In E. A. Silver (Ed.), Teaching and learning mathematical problem solving: Multiple research perspectives (pp. 399-415). Mahwah, NJ: Erlbaum.
60. Siegler, R. S., & Shrager, J. (1984). Strategy choices in addition and subtraction: How do children know what to do? In C. Sophian (Ed.), Origins of cognitive skills (pp. 229-293). Hillsdale, NJ: Erlbaum.
61. Simon, H. A. (1957). Administrative behavior (2nd ed.). Totowa, NJ: Littlefield, Adams.
62. Sleeman, D. (1984). An attempt to understand students' understanding of basic algebra. Cognitive Science, 8, 387-412.
63. Smith, J. P., diSessa, A. A., & Roschelle, J. (1993). Misconceptions reconceived: A constructivist analysis of knowledge in transition. The Journal of the Learning Sciences, 3, 115-163.
64. Stigler, J., Fernandez, C., & Yoshida, M. (1996). Cultures of mathematics instruction in Japanese and American elementary classrooms. In T. Rohlen & G. K. LeTendre (Eds.), Teaching and learning in Japan (pp. 213-247). Cambridge, England: Cambridge University Press.
65. 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: Erlbaum.
66. 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: Erlbaum.
67. VanLehn, K. (1987). Learning one subprocedure per lesson. Artificial Intelligence, 31, 1-40.
68. VanLehn, K. (1990). Mind bugs: The origins of procedural misconceptions. Cambridge, MA: MIT Press.
69. Young, R. M., & O'Shea, T. (1981). Errors in students' subtraction. Cognitive Science, 5, 153-177.
70. Zbrodoff, J., & Logan, G. D. (1986). On the autonomy of mental processes: A case study of arithmetic. Journal of Experimental Psychology: General, 115, 118-130.