## Abstract

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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## Notes

- 1.
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.) - 2.
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). - 3.
This information is based on personal communication with instructors in these institutions.

- 4.
This focus is opposed to an encompassing textbook analysis, such as the one offered by Okeeffee (2013).

- 5.
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. - 6.
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’.

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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

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### Keywords

- 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