Geometry is used in different ways in the teaching of linear algebra. In this paper, I offer a typology of these ways, which I call varieties, and address three central questions. The first question is, What varieties of use of geometry in the teaching of linear algebra exist? This question is addressed through an analysis of six linear algebra textbooks, republished in multiple editions in the last decade or so. The analysis resulted in seven varieties, which can be used by researchers to investigate systematically the use of geometry in the teaching of linear algebra, in and outside the classroom. The second question is, What are the different impacts of these varieties on students’ ability to accomplish the following: (a) abstract geometrically-based linear-algebraic concepts into general representations? (b) Extend geometrically-based linear-algebraic concepts to their counterparts in other models, such as space of polynomials, functions, and matrices? This question is addressed through analyses of a sample of various results reported in the literature. The third question is, What might account for these impacts? A conceptual basis underlying the results discussed in this paper is theorized.
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The notion of shared scheme is taken from Davis (1984). It refers to consistencies in mathematical behaviors among different people. Davis discusses this notion in the context of arithmetic, but it can be extended to mathematics in general, including geometry. Davis’ point is that while the ideas in a person’s mind are uniquely that person’s, not fully shared by any other human being, different individuals belonging to comparable cultures share similar mathematical experiences, which account for the commonalities in these individuals’ perceptions, abstractions, and generalizations of mathematics. (See Chap. 8, A Few Commonly Shared Frames, in Davis 1984.)
The distinction between students’ geometry and instructors’ geometry is critical because it reflects the constructivist philosophy of learning in which this study is oriented. Steffe and Thompson (2000) articulate the philosophical roots of such a distinction:Students’ mathematics is something we attribute to students independently of our interactions with them. In doing so, we reason as follows: If we acknowledge that students are autonomous organisms not unlike ourselves, and if we claim that our consensual mathematical reality is the only one, then certain difficulties arise. If our, the researchers’, consensual mathematical reality is the only one, then the mathematical reality of any student would be solely in our imagination. But a student, being not unlike ourselves, can insist that his or her mathematical reality is the sole mathematical reality and that the mathematical reality of everyone else is in his or her imagination, including our, the researchers’, consensual reality. Being autonomous also, we would not want to accept that our consensual mathematical reality is merely a concoction of the student’s imagination. So, we have to accept the student’s mathematical reality as being distinct from ours. We call those mathematical realities “students’ mathematics,” whatever they might be. Students’ mathematics is indicated by what they say and do as they engage in mathematical activity, and a basic goal of the researchers in a teaching experiment is to construct models of students’ mathematics. (p. 269). This viewpoint has been a cornerstone in my research, expressed in one of the eight premises of DNR, called the subjectivity premise, which states: Any observations humans claim to have made is due to what their mental structure attributes to their environment (see Harel 2008).
This information is based on personal communication with instructors in these institutions.
This focus is opposed to an encompassing textbook analysis, such as the one offered by Okeeffee (2013).
The term variety was borrowed from biology. As in biology, my use of the term refers to a taxonomic category that ranks below a ‘species’ (in our case, ‘pedagogical use of geometry’), where members of different varieties, though belonging to the same ‘species’, differ in significant characteristics. My use of the term variety is also analogous to the use of the term in linguistics, where it refers to a specific form of a language. Likewise, my use of variety refers to a specific pedagogical use of geometry in the teaching of linear algebra.
To avoid confusion of terms when these students take a linear algebra course in college, Harel uses the term ‘space of vectors’ rather than ‘vector space’.
Banchoff, T., & Wermer, J. (1992). Linear algebra through geometry. New York: Springer.
Bogomolny, M. (2006). The role of example-generation tasks in students’ understanding of linear algebra. Ph.D. dissertation: Simon Fraser University.
Britton, S., & Henderson, J. (2009). Linear algebra revisited: An attempt to understand students’ conceptual difficulties. International Journal of Mathematical Education in Science and Technology, 40, 963–974.
Brousseau, G. (1997). Theory of didactical situations in mathematics. Dordrecht: Kluwer.
Davis, R. (1984). Learning mathematics: The cognitive science approach to mathematics education. Westport: Greenwood Publishing Group.
Dieudonné, J. (1964). Linear algebra and geometry. Boston: Houghton Mifflin Company.
Dorier, J. L. (Ed.). (2000). On the teaching of linear algebra (Vol. 23). Boston: Kluwer Academic Publishers.
Dubinsky, E. (2014). Personal communication.
Dubinsky, E., & McDonald, M. (2001). APOS: A constructivist theory of learning in undergraduate mathematics education research. In D. Holton (Ed.), The teaching and learning of mathematics at university level: An ICMI study (pp. 275–282). Dordrecht: Kluwer Academic Publishers.
Ertekin, E., Solak, S., & Yazıcı, E. (2010). The effects of formalism on teacher trainees’ algebraic and geometric interpretation of the notions of linear dependency/independency. International Journal of Mathematical Education in Science and Technology, 41, 1015–1035.
Fischbein, E. (1987). Intuition in science and mathematics: An educational approach. Dordrecht: D. Reidel.
Gol Tabaghi, S., & Sinclair, N. (2013). Using dynamic geometry software to explore eigenvectors: The emergence of dynamic-synthetic-geometric thinking. Technology, Knowledge and Learning, 18, 149–164.
Greeno, G. (1992). Mathematical and scientific thinking in classroom and other situations. In D. Halpern (Ed.), Enhancing thinking skills in sciences and mathematics (pp. 39–61). Hillsdale: Lawrence Erlbaum Associates.
Gueudet-Chartier G (2006). Using geometry to teach and learn linear algebra. Research in collegiate mathematics education, VI, pp. 171–195.
Hannah, J., Stewart, S., & Thomas, M. (2013). Emphasizing language and visualization in teaching linear algebra. International Journal of Mathematics Education Science and Technology, 44, 475–489.
Hannah, J., Stewart, S., & Thomas, M. (2014). Teaching linear algebra in the embodied, symbolic and formal worlds of mathematical thinking: Is there a preferred order? In Oesterle, S., Liljedahl, P., Nicol, C., & Allan, D. (Eds.), Proceedings of the joint meeting of PME 38 and PME-NA 36 (Vol. 3, pp. 241–248). Vancouver: PME.
Harel, G. (1989). Applying the principle of multiple embodiments in teaching linear algebra: Aspects of familiarity and mode of representation. School Science and Mathematics, 89, 49–57.
Harel, G. (1990). Using geometric models and vector arithmetic to teach high-school students basic notions in linear algebra. International Journal for Mathematics Education in Science and Technology, 21, 387–392.
Harel, G. (1999). Students’ understanding of proofs: a historical analysis and implications for the teaching of geometry and linear algebra. Linear Algebra and Its Applications, 302–303, 601–613.
Harel, G. (2000). Three principles of learning and teaching mathematics: Particular reference to linear algebra—old and new observations. In J.-L. Dorier (Eds.), On the teaching of linear algebra (pp. 177–190). Dordrecht: Kluwer Academic Publishers.
Harel, G. (2008). A DNR perspective on mathematics curriculum and instruction. Part II: With reference to teacher’s knowledge base. ZDM, 40, 893–907.
Harel, G. (2017). The learning and teaching of linear algebra: Observations and generalizations. Journal of Mathematical Behavior, 46, 69–95.
Hillel, J. (2000). Modes of description and the problem of representation in linear algebra. In J.-L. Dorier (Ed.), On the teaching of linear algebra (pp. 191–207). Dordrecht: Kluwer Academic Publishers.
Harel, G., & Sowder, L. (1998). Students' proof schemes. In E. Dubinsky, A. Schoenfeld, & J. Kaput (Eds.), Research on Collegiate Mathematics Education (Vol. III, pp. 234–283). American Mathematical Society.
Kaplansky, I. (1974). Linear algebra and geometry. New York: Dover Publications.
Kolman, B., & Hill, D. (2007). Elementary linear algebra with applications (9th edn.). New Jersey: Prentice Hall.
Lay, D., Lay, S., & McDonald, J. (2016). Linear algebra and its applications (5rd edn.). Boston: Pearson.
Leon, S. (2015). Linear algebra with applications (9th edn.). Boston: Pearson.
Moore, G. (1995). An axiomatization of linear algebra: 1875–1940. Historia Mathematica, 22, 262–303.
Okeeffe, L. (2013). A framework for textbook analysis. International Review of Contemporary Learning Research, 1, 1–13.
Penney, R. (2015). Linear algebra: Ideas and applications. New York: Wiley.
Reid, D., & Knipping, C. (2010). Proof in mathematics education. Research, learning and teaching. Boston: Sense Publishers.
Sierpinska, A., Deyfus, T., & Hillel, J. (1999). Evaluation of a teaching design in linear algebra: The case of linear transformations. Recherches en Didactique des Mathematique, 19, 7–41.
Steffe, L. P., & Thompson, P. W. (2000). Teaching experiment methodology: Underlying principles and essential elements. In R. Lesh & A. E. Kelly (Eds.), Research design in mathematics and science education (pp. 267–307). Hillsdale: Erlbaum.
Stewart, S., & Thomas, M. (2009). A framework for mathematical thinking: The case of linear algebra. International Journal of Mathematical Education in Science and Technology, 40, 951–961.
Stewart, S., & Thomas, M. (2010). Student learning of basis, span and linear independence in linear algebra. International Journal of Mathematical Education in Science and Technology, 41, 173–188.
Strang, G. (2016). Introduction to linear algebra. (5th Edition). Wellesley-Cambridge Press Cambridge.
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.
Valverde, G., Bianchi, L. J., Wolfe, R., Schmidt, W. H., & Houang, R. T. (2002). According to the book: Using TIMSS to investigate the translation of policy into practice through the world of textbooks. London: Kluwer Academic Publishers.
Wawro, M., Sweeney, G., & Rabin, J. (2011). Subspace in linear algebra: Investigating students’ concept images and interactions with the formal definition. Educational Studies in Mathematics, 78, 1–19.
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Harel, G. Varieties in the use of geometry in the teaching of linear algebra. ZDM Mathematics Education 51, 1031–1042 (2019). https://doi.org/10.1007/s11858-018-01015-7
- Learning and teaching of linear algebra
- Use of geometry in the teaching of linear algebra
- Impact of geometry use in the teaching of linear algebra and their conceptual bases