Computing solutions to algebraic problems using a symbolic versus a schematic strategy

ZDM - International Journal on Mathematics Education

Volumn 42, Issue 6, 2010, Pages 591-605

  Lee, Kerry ^a   Yeong, Stephanie H M ^a,d   Ng, Swee Fong ^a   Venkatraman, Vinod ^b   Graham, Steven ^c   Chee, Michael W L ^b  

^a NANYANG TECHNOLOGICAL UNIVERSITY   (Singapore)

^b DUKE NUS MEDICAL SCHOOL   (Singapore)

^c NATIONAL UNIVERSITY OF SINGAPORE   (Singapore)

^d UNIVERSITY OF WESTERN AUSTRALIA   (Australia)

Author keywords

Algebra; Neuroimaging; Pedagogy; Word problems; Working memory

Indexed keywords

EID: 84867497927     PISSN: 18639690     EISSN: 18639704     Source Type: Journal    
DOI: 10.1007/s11858-010-0265-6     Document Type: Article

References (52)

1. Anderson, J. R. (2005). Human symbol manipulation within an integrated cognitive architecture. Cognitive Science, 29, 313-341.
2. Ansari, D. (2007). Does the parietal cortex distinguish between "10", "ten", and ten dots? Neuron, 53, 165-167.
3. Ansari, D., Lyons, I. M., van Eimeren, L., & Xu, F. (2007). Linking visual attention and number processing in the brain: The role of the temporo-parietal junction in small and large symbolic and nonsymbolic number comparison. Journal of Cognitive Neuroscience, 19, 1845-1853.
4. Behrmann, M., Geng, J. J., & Shomstein, S. (2004). Parietal cortex and attention. Current Opinion in Neurobiology, 14, 212-217.
5. Bobrow, D. G. (1968). Natural language input for a computer problem solving system. In M. Minsky (Ed.), Semantic information processing (pp. 146-226). Cambridge, MA: MIT Press.
6. Brain Innovation. (2008). Brain Voyager QX (Version 1.9) [Computer software]. Maastricht, The Netherlands: Brain Innovation.
7. Briars, D. J., & Larkin, J. H. (1984). An integrated model of skill in solving elementary word problems. Cognition and Instruction, 1, 245-296.
8. Carpenter, T. P., Moser, J. M., & Bebout, H. C. (1988). Representation of addition and subtraction word problems. Journal for Research in Mathematics Education, 19, 345-357.
9. Cohen Kadosh, R., Cohen Kadosh, K., Kaas, A., Henik, A., & Goebel, R. (2007). Notation-dependent and-independent representations of numbers in the parietal lobes. Neuron, 53, 307-314.
10. Davis, N., Cannistraci, C. J., Rogers, B. P., Gatenby, J. C., Fuchs, L. S., Anderson, A. W., et al. (2009). The neural correlates of calculation ability in children: An fMRI study. Magnetic Resonance Imaging, 27(9), 1187-1197.
11. Dehaene, S. (1997). The number sense. New York: Oxford University Press.
12. Dehaene, S., Piazza, M., Pinel, P., & Cohen, L. (2003). Three parietal circuits for number processing. Cognitive Neuropsychology, 20, 487-506.
13. Donders, F. C. (1868, 1969). On the speed of mental processes. Acta Psychologica, 30, 412-431.
14. Eger, E., Sterzer, P., Russ, M. O., Giraud, A. L., & Kleinschmidt, A. (2003). A supramodal number representation in human intraparietal cortex. Neuron, 37, 719-725.
15. Fias, W., Lammertyn, J., Reynvoet, B., Dupont, P., & Orban, G. A. (2003). Parietal representation of symbolic and nonsymbolic magnitude. Journal of Cognitive Neuroscience, 15, 47-56.
16. Genovese, C. R., Lazar, N. A., & Nichols, T. E. (2002). Thresholding of statistical maps in functional neuroimaging using the false discovery rate. Neuroimage, 15, 870-878.
17. Grabner, R. H., Ansari, D., Koschutnig, K., Reishofer, G., Ebner, F., & Neuper, C. (2009a). To retrieve or to calculate? Left angular gyrus mediates the retrieval of arithmetic facts during problem solving. Neuropsychologia, 47, 604-608.
18. Grabner, R. H., Ischebeck, A., Reishofer, G., Koschutnig, K., Delazer, M., Ebner, F., et al. (2009b). Fact learning in complex arithmetic and figural-spatial tasks: The role of the angular gyrus and its relation to mathematical competence. Human Brain Mapping, 30, 2936-2952.
19. Hernandex-García, L., Wager, T., & Jonides, J. (2004). Functional brain imaging. In H. Pashler & J. Wixted (Eds.), Stevens' handbook of experimental psychology, Vol. 4: Methodology in experimental psychology (3rd ed., pp. 175-221). New York: Wiley.
20. Huettel, S. A., Song, A. W., & McCarthy, G. (2004). Functional magnetic resonance imaging. Sunderland, MA: Sinauer Associates.
21. Ischebeck, A., Zamarian, L., Egger, K., Schocke, M., & Delazer, M. (2007). Imaging early practice effects in arithmetic. Neuroimage, 36, 993-1003.
22. Kaufmann, L., Vogel, S. E., Wood, G., Kremser, C., Schocke, M., Zimmerhackl, L. B., et al. (2008). A developmental fMRI study of nonsymbolic numerical and spatial processing. Cortex, 44, 376-385.
23. Khng, F., & Lee, K. (2009). Inhibiting interference from prior knowledge: Arithmetic intrusions in algebra word problem solving. Learning and Individual Differences, 19, 262-268.
24. Kho, T. H. (1987). Mathematical models for solving arithmetic problems. In Proceedings of fourth Southeast Asian conference on mathematical education (ICMI-SEAMS). Mathematical Education in the 1990s (Vol. 4, pp. 345-351). Singapore: Institute of Education.
25. Kho, T. H. (2006). Mathematical models for solving arithmetic problems. In P. Y. Lee (Ed.), Teaching secondary school mathematics: A resource book (pp. 367-377). Singapore: McGraw Hill Education.
26. Kieran, C. (1989). The early learning of algebra: A structural perspective. In S. Wagner & C. Kieran (Eds.), Research issues in the learning and teaching of algebra (pp. 33-56). Reston, VA: National Council of Teachers of Mathematics.
27. Kintsch, W., & Greeno, J. G. (1985). Understanding and solving word arithmetic problems. Psychological Review, 92, 109-129.
28. Koedinger, K. R., & Nathan, M. J. (2004). The real story behind story problems: Effects of representations on quantitative reasoning. The Journal of the Learning Sciences, 13(2), 129-164.
29. Kucian, K., von Aster, M., Loenneker, T., Dietrich, T., & Martin, E. (2008). Development of neural networks for exact and approximate calculation: A fMRI study. Developmental Neuropsychology, 33, 447-473.
30. Lee, K., Lim, Z. Y., Yeong, S. H. M., Ng, S. F., Venkatraman, V., & Chee, M. W. L. (2007). Strategic differences in algebraic problem solving: Neuroanatomical correlates. Brain Research, 1155, 163-171.
31. Lee, K., Ng, E., & Ng, S. F. (2009). The contributions of working memory and executive functioning to problem representation and solution generation in algebraic word problems. Journal of Educational Psychology, 101, 373-387.
32. Lee, K., Ng, S. F., Ng, E. L., & Lim, Z. Y. (2004). Working memory and literacy as predictors of performance on algebraic word problems. Journal of Experimental Child Psychology, 89, 140-158.
33. Lewis, A. B. (1989). Training students to represent arithmetic word problems. Journal of Educational Psychology, 81, 521-531.
34. Mayer, R. E. (1992). Thinking, problem solving, cognition (2nd ed.). New York: W. H. Freeman.
35. Mayer, R. E., & Hegarty, M. (1996). The process of understanding mathematical problems. In R. J. Sternberg & T. Ben-Zeev (Eds.), The nature of mathematical thinking (pp. 29-53). Hillsdale, NJ: Lawrence Erlbaum.
36. Ministry of Education. (2009). The Singapore model method for learning mathematics. Singapore: EPB Pan Pacific.
37. Ng, S. F. (2003). How secondary two express stream students used algebra and the model method. The Mathematics Educator, 7, 1-17.
38. Ng, S. F., & Lee, K. (2009). The model method: Singapore children's tool for representing and solving algebraic word problems. Journal for Research in Mathematics Education, 40, 282-313.
39. Ng, S. F., Lee, K., Ang, S. Y., & Khng, F. (2006). Model method: Obstacle or bridge to learning symbolic algebra. In W. Bokhorst-Heng, M. Osborne, & K. Lee (Eds.), Redesigning pedagogies (pp. 227-242). New York: Sense.
40. Owen, A. M., McMillan, K. M., Laird, A. R., & Bullmore, E. (2005). N-back working memory paradigm: A meta-analysis of normative functional neuroimaging. Human Brain Mapping, 25, 46-59.
41. Piazza, M., Izard, V., Pinel, P., Le Bihan, D., & Dehaene, S. (2004). Tuning curves for approximate numerosity in the human intraparietal sulcus. Neuron, 44, 547-555.
42. Piazza, M., Pinel, P., Le Bihan, D., & Dehaene, S. (2007). A magnitude code common to numerosities and number symbols in human intraparietal cortex. Neuron, 53, 293-305.
43. Riley, M. S., & Greeno, J. G. (1988). Developmental analysis of understanding language about quantities and of solving problems. Cognition and Instruction, 5, 49-101.
44. Rivera, S. M., Reiss, A. L., Eckert, M. A., & Menon, V. (2005). Developmental changes in mental arithmetic: Evidence for increased functional specialization in the left inferior parietal cortex. Cerebral Cortex, 15, 1779-1790.
45. Stacey, K., & MacGregor, M. (1999). Learning the algebraic method of solving problems. The Journal of Mathematical Behavior, 18, 149-167.
46. Steinberg, R. M., Sleeman, D. H., & Ktorza, D. (1991). Algebra students' knowledge of equivalence of equations. Journal for Research in Mathematics Education, 22(2), 112-121.
47. Stocco, A., & Anderson, J. R. (2008). Endogenous control and task representation: An fMRI study in algebraic problem-solving. Journal of Cognitive Neuroscience, 20, 1300-1314.
48. Venkatraman, V., Ansari, D., & Chee, M. W. (2005). Neural correlates of symbolic and non-symbolic arithmetic. Neuropsychologia, 43, 744-753.
49. Verguts, T., & Fias, W. (2004). Representation of number in animals and humans: A neural model. Journal of Cognitive Neuroscience, 16, 1493-1504.
50. Zago, L., Pesenti, M., Mellet, E., Crivello, F., Mazoyer, B., & Tzourio-Mazoyer, N. (2001). Neural correlates of simple and complex mental calculation. Neuroimage, 13, 314-327.
51. Zago, L., Petit, L., Turbelin, M. R., Andersson, F., Vigneau, M., & Tzourio-Mazoyer, N. (2008). How verbal and spatial manipulation networks contribute to calculation: An fMRI study. Neuropsychologia, 46, 2403-2414.
52. Zago, L., & Tzourio-Mazoyer, N. (2002). Distinguishing visuospatial working memory and complex mental calculation areas within the parietal lobes. Neuroscience Letters, 331, 45-49.