Not everything is proportional: Effects of age and problem type on propensities for overgeneralization

Cognition and Instruction

Volumn 23, Issue 1, 2005, Pages 57-86

  Van Dooren, Wim ^a,b,d   De Bock, Dirk ^a,c   Hessels, An ^a   Janssens, Dirk ^a   Verschaffel, Lieven ^a  

^a UNIVERSITY OF LEUVEN   (Belgium)

^b FUND FOR SCIENTIFIC RESEARCH   (Belgium)

^c Department Medical Imaging   (Belgium)

^d Centre for Instructional Psychology and Technology   (Belgium)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 13944269902     PISSN: 07370008     EISSN: None     Source Type: Journal    
DOI: 10.1207/s1532690xci2301_3     Document Type: Article

References (56)

1. Behr, M. J., Harel, G., Post, T., & Lesh, R. (1992). Rational number, ratio, and proportion. In D. A. Grouws (Ed.), Handbook of research on mathematics thinking and learning (pp. 296-333). New York: Macmillan.
2. Ben-Zeev, T. (1998). Rational errors and the mathematical mind. General Review of Psychology, 2, 366-383.
3. Ben-Zeev, T., & Star, J. R. (2001). Spurious correlations in mathematical thinking. Cognition and Instruction, 19, 253-275.
4. Blum, W. (2002). ICMI Study 14: Applications and modelling in mathematics education - Discussion document. Educational Studies in Mathematics, 51, 148-171.
5. Brousseau, G. (1997). Theory of didactical situations in mathematics (N. Balacheff, M. Cooper, R. Sutherland, & V. Warfield, Eds. & Trans.). Dordrecht, The Netherlands: Kluwer.
6. Centre de Recherche sur l'Enseignement des Mathématiques. (2002). Des grandeurs aux espaces vectoriels: La linéarité comme fil conducteur [From quantities to vector spaces: Linearity as a leitmotiv]. Nivelles, Belgium: Author.
7. Chi, M. T. H., & Bassok, M. (1989). Learning from examples via self-explanations. In L. B. Resnick (Ed.), Knowing, learning and instruction: Essays in honor of Robert Glaser (pp. 251-282). Hillsdale, NJ: Lawrence Erlbaum Associates, Inc.
8. Cramer, K., Post, T., & Currier, S. (1993). Learning and teaching ratio and proportion: Research implications. In D. T. Owens (Ed.), Research ideas for the classroom: Middle grades mathematics (pp. 159-178). New York: Macmillan.
9. De Bock, D. (2002). The illusion of linearity: An empirical analysis of secondary school students' improper proportional reasoning in geometry problems. Unpublished doctoral dissertation, University of Leuven, Leuven, Belgium.
10. De Bock, D., Van Dooren, W., Janssens, D., & Verschaffel, L. (2002), Improper use of linear reasoning: An in-depth study of the nature and the irresistibility of secondary school students' errors. Educational Studies in Mathematics, 50, 311-334.
11. De Bock, D., Verschaffel, L., & Janssens, D. (1998). The predominance of the linear model in secondary school students' solutions of word problems involving length and area of similar plane figures. Educational Studies in Mathematics, 35, 65-83.
12. De Bock, D., Verschaffel, L., & Janssens, D. (2002). The effects of different problem presentations and formulations on the illusion of linearity in secondary school students. Mathematical Thinking and Learning, 4, 65-89.
13. De Bock, D., Verschaffel, L., Janssens, D., Van Dooren, W., & Claes K. (2003). Do realistic contexts and graphical representations always have a beneficial impact on students' performance? Negative evidence from a study on modeling non-linear geometry problems. Learning and Instruction, 13, 441-463.
14. Fischbein, E. (1999). Intuitions and schemata in mathematical reasoning. Educational Studies in Mathematics, 38, 11-50.
15. Freudenthal, H. (1983). Didactical phenomenology of mathematical structures. Dordrecht, The Netherlands: Reidel.
16. Galilei, G. (1954). Dialogues concerning two new sciences (H. Crew & A. de Salvio, Trans.) New York: Dover. (Original work published 1638)
17. Gerofsky, S. (1996). A linguistic and narrative view of word problems in mathematics education. For the Learning of Mathematics, 16(2), 36-45.
18. Greer, B. (1993). The mathematical modelling perspective on wor(l)d problems. Journal of Mathematical Behavior, 12, 239-250.
19. Hatano, G. (1988). Social and motivational bases for mathematical understanding. New Directions for Child Development, 41, 55-70.
20. Karplus, R., Pulos, E., & Stage, E. (1983). Proportional reasoning of early adolescents. In R. Lesh & M. Landau (Eds.), Acquisition of mathematical concepts and processes (pp. 45-89). New York: Academic.
21. Kieren, T. (1988). Personal knowledge of rational numbers: Its intuitive and formal development. In J. Hiebert & M. Sehr (Eds.), Number-concepts and operations in the middle grades (pp. 53-92). Reston, VA: National Council of Teachers of Mathematics.
22. Klahr, D., & Siegler, R. (1978). The representation of children's knowledge. In H. Reese & L. P. Lipsitt (Eds.), Advances in child development and behavior (Vol. 12, pp. 62-116). New York: Academic.
23. Lave, J. (1992). Word problems: A microcosm of theories of learning. In P. Light & G. Butterworth (Eds.), Context and cognition: Ways of learning and knowing (pp. 74-92). New York: Harvester Wheatsheaf.
24. Lesh, R., & Doerr, H. (2003). Beyond constructivism: Models and modeling perspectives on mathematics problem solving, learning and teaching. Mahwah, NJ: Lawrence Erlbaum Associates, Inc.
25. Lesh, R., & Lehrer, R. (2003). Models and modeling perspectives on the development of students and teachers. Mathematical Thinking and Learning, 5, 109-129.
26. Lesh, R., Lester, F. K., & Hjalmarson, M. (2003). A models and modeling perspective on metacognitive functioning in everyday situations where problem solvers develop mathematical constructs. In R. Lesh & H. Doerr (Eds.), Beyond constructivism: Models and modeling perspectives on mathematics problem solving, learning and teaching (pp. 383-404). Mahwah, NJ: Lawrence Erlbaum Associates, Inc.
27. Lesh, R., Post, T., & Behr, M. (1988). Proportional reasoning. In J. Hiebert & M. Sehr (Eds.), Number concepts and operations in the middle grades (pp. 93-113). Reston, VA: National Council of Teachers of Mathematics.
28. Liang, K.-Y., & Zeger, S. L. (1986). Longitudinal data analysis using generalized linear models. Biometrika, 73, 13-22.
29. Linchevski, L., Olivier, A., Sasman, M. C., & Liebenberg, R. (1998). Moments of conflict and moments of conviction in generalising. In A. Olivier & K. Newstead (Eds.), Proceedings of the 22nd Conference of the International Group for the Psychology of Mathematics Education (Vol. 3, pp. 215-222). Stellenbosch, South Africa: University of Stellenbosch.
30. National Council of Teachers of Mathematics. (1989) Curriculum and evaluation standards for school mathematics. Reston, VA: Author.
31. National Council of Teachers of Mathematics. (2000). Principles and standards for school mathematics. Reston, VA: Author.
32. Nesher, P. (1980). The stereotyped nature of school word problems. For the Learning of Mathematics, 1(1), 41-48.
33. Noelting, G. (1980a). The development of proportional reasoning and the ratio concept. Part I: The differentiation of stages. Educational Studies in Mathematics, 11, 217-253.
34. Noelting, G. (1980b). The development of proportional reasoning and the ratio concept. Part II: Problem structure at successive stages: Problem-solving strategies and the mechanism of adaptive restructuring. Educational Studies in Mathematics, 11, 331-363.
35. Orton, J., & Orton, A. (1994). Students' perception and use of pattern and generalization. In J. P. da Ponte & J. F. Matos (Eds.), Proceedings of the 18th Conference of the International Group for the Psychology of Mathematics Education (Vol. 3, pp. 407-414). Lisbon, Portugal: Universidade de Lisboa.
36. Puchalska, E., & Semadeni, Z. (1987). Children's reactions to verbal arithmetical problems with missing, surplus or contradictory data. For the Learning of Mathematics, 7(3), 9-16.
37. Reusser, K., & Stebler, R. (1997). Every word problem has a solution: The suspension of reality and sense-making in the culture of school mathematics. Learning and Instruction, 7, 309-328.
38. Rouche, N. (1989). Prouver: Amener À l'évidence ou contrôler des implications? [Proving: Bringing back to evidence or verifying the implications?]. In Commission inter-IREM Histoire et Epistémologie des Mathématiques (Ed.), La démonstration dans l'histoire (pp. 8-38). Lyon, France: Institut de Recherche sur l'Enseignement des Mathématiques.
39. Sasman, M. C., Linchevski, L., & Olivier, A. (1999). The influence of different representations on children's generalisation thinking processes. In J. Kuiper (Ed.), Proceedings of the 7th Annual Conference of the Southern African Association for Research in Mathematics and Science Education (pp. 406-415). Harare, Zimbabwe: University of Harare.
40. Schaerlaekens, A., Kohnstamm, D., & Lejaegere, M. (1999). Streeflijst woordenschat voor zesjarigen [Strived vocabulary list for six-year olds]. Lisse, The Netherlands: Swets & Zeitlinger.
41. Schoenfeld, A. H. (1988). When good teaching leads to bad results: The disasters of "well-taught" mathematics courses. Educational Psychologist, 23, 145-166.
42. Sheu, C.-F. (2000). Regression analysis of correlated binary outcomes. Behavior Research Methods, Instruments, & Computers, 32, 269-273.
43. Stacey, K. (1989). Finding and using patterns in linear generalizing problems. Educational Studies in Mathematics, 20, 147-164.
44. Stokes, M. E., Davis, C. S., & Koch, G. G. (2000). Categorical data analysis using the SAS system (2nd ed.). Cary, NC: SAS Institute.
45. Strauss, S., & Stavy, R. (1982). U-shaped behavioral growth. New York: Academic.
46. Taplin, M. (1995). Special patterning: A pilot study of pattern formation and generalization. In L. Meira & D. Carraher (Eds.), Proceedings of the 19th Conference of the International Group for the Psychology of Mathematics Education (Vol. 3, pp. 42-49). Recife, Brazil: Universidade Federal de Pernambuco.
47. Usiskin, Z. (2004). The arithmetic operations as mathematical models. In H.-W. Henn & W. Blum (Eds.), ICMI Study 14: Applications and modelling in mathematics education - Pre-conference volume (pp. 279-284). Dortmund, Germany: Universität Dortmund.
48. van den Brink, J., & Streefland, L. (1979). Young children (6-8) - Ratio and proportion, Educational Studies in Mathematics, 10, 403-420.
49. Van Dooren, W., De Bock, D., De Bolle, E., Janssens, D., & Verschaffel, L. (2003). The illusion of linearity : The role of direct versus indirect perimeter and area measures. Mediterranean Journal for Research in Mathematics Education, 2, 1-18.
50. Van Dooren, W., De Bock, D., Depaepe, F., Janssens, D., & Verschaffel, L. (2003). The illusion of linearity: Expanding the evidence towards probabilistic reasoning. Educational Studies in Mathematics, 53, 113-138.
51. Van Dooren, W., De Bock, D., Hessels, A., Janssens, D., & Verschaffel, L. (2004). Remedying secondary school students' illusion of linearity: A teaching experiment aiming at conceptual change. Learning and Instruction, 14(5), 485-501.
52. Van Lieshout, E. C. D. M., Verdwaald, A., & Van Herk, J. (1997, August). Suppression of real-world knowledge and demand characteristics in word problem solving. Paper presented at the 7th European Conference for Research on Learning and Instruction, Athens, Greece.
53. Vergnaud, G. (1983). Multiplicative structures. In R. Lesh & M. Landau (Eds.), Acquisition of mathematics concepts and processes (pp. 127-174). New York: Academic.
54. Verschaffel, L., De Corte, E., & Lasure, S. (1994). Realistic considerations in mathematical modeling of school arithmetic word problems. Learning and Instruction, 4, 273-294.
55. Verschaffel, L., Greer, B., & De Corte, E. (2000). Making sense of word problems. Lisse, The Netherlands: Swets & Zeitlinger.
56. Visser, J., Van Laerhoven, A., & Ter Beek, A. (1998). AVI toetspakket (handleiding) [AVI testing package (manual)]. Hertogenbosch, The Netherlands: KPC Groep.