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Tacit Models that Govern Undergraduate Reasoning about Subspaces

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Abstract

This study is concerned with the reasoning that undergraduates apply when deciding whether a prompt is an example or non-example of the subspace concept. A qualitative analysis of written responses of 438 students revealed five unconventional tacit models that govern their reasoning. The models account for whether a prompt is a subset of a vector space, whether the zero vector is included, the structure of vectors, their number in the formula for the general solution to the system of linear equations, and the corresponding coefficient matrix. Furthermore, a conception was identified in students’ responses, according to which the algebraic structure of a vector space passes from a ‘parent’ space to its subset, turning automatically it into a subspace. For many students this conception of an inheriting structure was instrumental for identifying and reasoning around subspaces. Polysemy of the prefix ‘sub’ and students’ prior experiences in identifying concept examples are used for offering explanations for the emergence of the conception.

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Notes

  1. The imprecise formulation of the question was intentional.

References

  • Biza, I., & Zachariades, T. (2010). First year mathematics undergraduates’ settled images of tangent line. The Journal of Mathematical Behavior, 29, 218–229.

    Article  Google Scholar 

  • Box, G., & Draper, N. P. (1987). Empirical model-building and response surfaces. New York: Wiley.

    Google Scholar 

  • Bruner, J. S., Goodnow, J. J., & Austin, G. A. (1956). A study of thinking. New York: John Wiley & Sons.

    Google Scholar 

  • Cornu, B. (1991). Limits. In D. Tall (ed.), Advanced mathematical thinking (pp. 153–166). Kluwer Academic Publishers.

  • Denzin, N. K., & Lincoln, Y. S. (2011). Introduction: The discipline and practice of qualitative research. In N. K. Denzin & Y. S. Lincoln (Eds.), The SAGE handbook of qualitative research (pp. 1–20). Thousand Oaks, CA: SAGE Publications.

    Google Scholar 

  • Dorier, J.-L., Robert, A., Robinet, J., & Rogalski, M. (2000). The obstacle of formalism in linear algebra. In J.-L. Dorier (Ed.), On the teaching of linear algebra (pp. 85–124). Dordrecht: Kluwer Academic Publishers.

    Google Scholar 

  • Durkin, K., & Shire, B. (1991). Lexical ambiguity in mathematical contexts. In K. Durkin & B. Shire (Eds.), Language in mathematical education (pp. 71–84). Miltone Keynes England: Open University Press.

    Google Scholar 

  • Fischbein, E. (1989). Tacit models and mathematical reasoning. For the Learning of Mathematics, 9(2), 9–14.

    Google Scholar 

  • Fischbein, E. (2001). Tacit models and infinity. Educational Studies in Mathematics, 48, 309–329.

    Article  Google Scholar 

  • Fischbein, E., Deri, M., Nello, M. S., & Marino, M. S. (1985). The role of implicit models in solving verbal problems in multiplication and division. Journal for Research in Mathematics Education, 16(1), 3–17.

    Article  Google Scholar 

  • Giraldo, V., Tall, D., & Carvalho, L. M. (2008). Using theoretical-computational conflicts to enrich the concept image of derivative. Research in Mathematics Education, 5(1), 63–78.

    Article  Google Scholar 

  • Harel, G., & Kaput, J. (1991). The role of conceptual entities and their symbols in building advanced mathematical concepts. In D. Tall (Ed.), Advanced mathematical thinking (pp. 82–94). Dordrecht: Kluwer Academic Publishers.

    Google Scholar 

  • Hershkowitz, R. (1989). Visualization in geometry—Two sides of the coin. Focus on Learning Problems in Mathematics, 11(1), 61–76.

    Google Scholar 

  • Jones, S. R., & Watson, K. L. (2017). Recommendations for a “target understanding” of the derivative concept for first semester calculus teaching and learning. International Journal of Research in Undergraduate Mathematics Education.

  • Kontorovich, I. (2016). Students’ confusions with reciprocal and inverse functions. International Journal of Mathematical Education in Science and Technology, 48(2), 278–284.

    Article  Google Scholar 

  • Kontorovich, I. (2018a). Why Johnny struggles when familiar concepts are taken to a new mathematical domain: Towards a polysemous approach. Educational Studies in Mathematics, 97(1), 5–20.

    Article  Google Scholar 

  • Kontorovich, I. (2018b). Unacceptable discrepancy: The case of the root concept. For the Learning of Mathematics, 38(1), 17–19.

    Google Scholar 

  • Kontorovich, I. (2018c). Undergraduates’ images of the root concept in R and in C. Journal of Mathematical Behavior, 49, 184–193.

    Article  Google Scholar 

  • Mamolo, A. (2010). Polysemy of symbols: Signs of ambiguity. The Montana Mathematics Enthusiast, 2-3, 247–262.

    Google Scholar 

  • Maracci, M. (2008). Combining different theoretical perspective for analyzing students' difficulties in vector space theory. ZDM The International Journal on Mathematics Education, 40, 265–276.

    Article  Google Scholar 

  • McNeil, N. M. (2007). U-shaped development in math: 7-year-olds outperform 9-year-olds on equivalence problems. Developmental Psychology, 43(3), 687–695.

    Article  Google Scholar 

  • Monaghan, J. (1991). Problems with the language of limits. For the Learning of Mathematics, 11(3), 20–24.

    Google Scholar 

  • Movshovitz-Hadar, N., Inbar, S., & Zaslavsky, O. (1986). Students’ distortions of theorems. Focus on Learning Problems in Mathematics, 8(1), 49–57.

    Google Scholar 

  • Parraguez, M., & Oktaç, A. (2010). Construction of the vector space concept from the viewpoint of APOS theory. Linear Algebra and its Applications, 432(8), 2112–2124.

    Article  Google Scholar 

  • Pimm, D. (1991). Communicating mathematically. In K. Durkin & B. Shire (Eds.), Language in mathematical education (pp. 17–23). Miltone Keynes England: Open University Press.

    Google Scholar 

  • Pitta-Pantazi, D., Christou, C., & Zachariades, T. (2007). Secondary school students’ levels of understanding in computing exponents. Journal of Mathematical Behavior, 26, 301–311.

    Article  Google Scholar 

  • Poole, D. (2011). Linear algebra: A modern introduction (3rd ed.). Brooks/Cole, Boston: Cengage Learning.

    Google Scholar 

  • Prediger, S., Bikner-Ahsbahs, A., & Arzarello, F. (2008). Networking strategies and methods for connecting theoretical approaches: First steps towards a conceptual framework. ZDM, 40(2), 165–178.

    Article  Google Scholar 

  • Presmeg, N. (1992). Prototypes, metaphors, metonymies and imaginative rationality in high school mathematics. Educational Studies in Mathematics, 23, 595–610.

    Article  Google Scholar 

  • Roh, K. H., & Lee, Y. H. (2017). Designing tasks of introductory real analysis to bridge a gap between students’ intuition and mathematical rigor: The case of the convergence of a sequence. International Journal of Research in Undergraduate Mathematics Education, 3(1), 34–68.

    Article  Google Scholar 

  • Sarfaty, Y., & Patkin, D. (2013). The ability of second graders to identify solids in different positions and to justify their answers. Pythagoras, 34(1), 212 10 pages.

    Article  Google Scholar 

  • Schwarz, B. B., & Hershkowitz, R. (1999). Prototypes: Brakes or levers in learning the function concept? The role of computer tools. Journal for Research in Mathematics Education, 30(4), 362–389.

    Article  Google Scholar 

  • Sfard, A. (2008). Thinking as communicating: Human development, the growth of discourses and mathematizing. Cambridge: Cambridge University Press.

    Book  Google Scholar 

  • Simon, M. A. (2017). Explicating mathematical concept and mathematical conception as theoretical constructs for mathematics education research. Educational Studies in Mathematics, 94, 117–137.

    Article  Google Scholar 

  • Sinclair, N., Watson, A., Zazkis, R., & Mason, J. (2011). The structuring of personal example spaces. The Journal of Mathematical Behavior, 30, 291–303.

    Article  Google Scholar 

  • Stavy, R., & Tirosh, D. (2000). How students (mis)understand science and mathematics: Intuitive rules. New York: Teachers College Press.

    Google Scholar 

  • Tall, D., & Bakar, M. (1992). Students’ mental prototypes for functions and graphs. International Journal of Mathematics Education in Science and Technology, 23(1), 39–50.

    Article  Google Scholar 

  • Tall, D., & Vinner, S. (1981). Concept image and concept definition in mathematics with particular reference to limits and continuity. Educational Studies in Mathematics, 12, 151–169.

    Article  Google Scholar 

  • Vamvakoussi, X., & Vosniadou, S. (2004). Understanding the structure of the set of rational numbers: A conceptual change approach. Learning and Instruction, 14(5), 453–467.

    Article  Google Scholar 

  • Vinner, S. (1991). The role of definitions in teaching and learning of mathematics. In D. Tall (Ed.), Advanced mathematical thinking (pp. 65–81). Kluwer Academic Publishers.

  • Vinner, S., & Dreyfus, T. (1989). Images and definitions for the concept of function. Journal for Research in Mathematics Education, 20(4), 356–366.

    Article  Google Scholar 

  • Watson, A., & Mason, J. (2005). Mathematics as a constructive activity: Learners generating examples. Mahwah: Erlbaum.

    Google Scholar 

  • Wawro, M., Sweeney, G. F., & Rabin, J. M. (2011). Subspace in linear algebra: Investigating students’ concept images and interactions with the formal definition. Educational Studies in Mathematics, 78, 1–19.

    Article  Google Scholar 

  • Zazkis, R. (1998). Divisors and quotients: Acknowledging polysemy. For the Learning of Mathematics, 18(3), 27–30.

    Google Scholar 

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Acknowledgements

I am grateful to Chris Rasmussen and to anonymous reviewers for their thorough criticism and insightful suggestions. The help of Sze Looi Chin with proofreading the paper is greatly appreciated.

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  1. Department of Mathematics, The University of Auckland, Private Bag 92019, 38 Princes street, Auckland, 1010, New Zealand

    Igor’ Kontorovich

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  1. Igor’ Kontorovich
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Correspondence to Igor’ Kontorovich.

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Kontorovich, I. Tacit Models that Govern Undergraduate Reasoning about Subspaces. Int. J. Res. Undergrad. Math. Ed. 4, 393–414 (2018). https://doi.org/10.1007/s40753-018-0078-5

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