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Using geometry to teach and learn linear algebra. Washington, DC: Mathematical Association of America. Rasmussen (Eds.), Making the connection: Research and teaching in undergraduate mathematics (pp. The role of mathematical definitions in mathematics and in undergraduate mathematics courses. Columbus, OH: ERIC.Įdwards, B., & Ward, M. Santos (Eds.), Proceedings of PME-NA 21 (pp. Revisiting the notion of concept image/concept definition. Dorier (Ed.), On the teaching of linear algebra (pp.
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The obstacle of formalism in linear algebra. Educational Studies in Mathematics, 29, 175–197.ĭorier, J.-L., Robert, A., Robinet, J., & Rogalski, M. Meta level in the teaching of unifying and generalizing concepts in mathematics. The research act: A theoretical introduction to research methods. Teaching linear algebra: Must the fog always roll in? The College Mathematics Journal, 24(1), 29–40.ĭenzin, N. International Journal of Mathematical Education in Science and Technology, 40(7), 963–974.Ĭarlson, D. Linear algebra revisited: An attempt to understand students’ conceptual difficulties. Educational Studies in Mathematics, 68, 19–35.īritton, S., & Henderson, J. London: Sage.īingolbali, E., & Monaghan, J. Research methods in cultural anthropology. Washington, DC: The Mathematical Association of America.īernard, R. Dubinsky (Eds.), The concept of function: Aspects of epistemology and pedagogy (pp. Cognitive difficulties and teaching practices. We conclude with a discussion of this and how it may be leveraged to inform teaching in a productive, student-centered manner.Īrtigue, M. Furthermore, we found that all students interviewed expressed, to some extent, the technically inaccurate “nested subspace” conception that R k is a subspace of R n for k < n. We also present results regarding the coordination between students’ concept image and how they interpret the formal definition, situations in which students recognized a need for the formal definition, and qualities of subspace that students noted were consequences of the formal definition. Through grounded analysis, we identified recurring concept imagery that students provided for subspace, namely, geometric object, part of whole, and algebraic object. We used the analytical tools of concept image and concept definition of Tall and Vinner (Educational Studies in Mathematics, 12(2):151–169, 1981) in order to highlight this distinction in student responses. This is consistent with literature in other mathematical content domains that indicates that a learner’s primary understanding of a concept is not necessarily informed by that concept’s formal definition. In interviews conducted with eight undergraduates, we found students’ initial descriptions of subspace often varied substantially from the language of the concept’s formal definition, which is very algebraic in nature. You can think of the xy-plane (or any other plane) as a copy of R 2 embedded in R 3, but there's no canonical way to say that one particular 2D subspace is R 2.This paper reports on a study investigating students’ ways of conceptualizing key ideas in linear algebra, with the particular results presented here focusing on student interactions with the notion of subspace. R m isn't even a subset of R n, so it can't possibly be a subspace. Is Rm a subspace of Rn, where m So R n is a subspace of C n as a vector space over R. But if you're thinking of it as a vector space over C, then it isn't for example, multiplying an n-tuple with real entries by the scalar i does not give you another n-tuple with real entries. If you're thinking of C n as a vector space over R, then R n is also closed under scalar multiplication. In the case S=R n and V=C n, we do have S⊆V for the reason you say, and S is nonempty, and it is closed under vector addition. "Subspace" is a word with a specific meaning: given a vector space V and a subset S⊆V, we say that S "is a subspace" if S is nonempty, closed under vector addition, and closed under scalar multiplication. Your answer might be right depending on the details of the question, but your justification is incomplete. Is Rn a subspace of Cn? I think yes because all real numbers can be written as a complex number with the coefficient on i equal to 0.