The learning and teaching of linear algebra: Observations and generalizations

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Abstract

This paper is about a teaching experiment (TE) with inservice secondary teachers (hereafter “participants”) in the theory of systems of linear equations. The TE was oriented within particular social and intellectual climates, and its design and implementation took into consideration a series of findings concerning the difficulties students have in linear algebra. The questions we set for this study were: (1) Did the participants in the particular TE climates construct viable knowledge in the theory of systems of linear equations? Our criteria for viable knowledge consist in evidence for the ability to (a) generate non-trivial conjectures, judged so subjectively by a mathematician, (b) prove such conjecture, and (c) move upward along the APOS conception levels. (2) What difficulties and insights did the participants experience as they constructed such knowledge?

The potential contributions of our investigation into these questions to researchers and practitioners include (a) a detailed depiction of the participants’ achievements and challenges in dealing with theoretical questions concerning linear systems in an authentic learning environment and under a tutelage oriented in a particular constructivist perspective; and (b) a field-based hypothesis about the consequences of a particular learning environment vis-à-vis construction of knowledge in linear algebra.

All of the participants had taken a linear algebra course as part of their undergraduate studies, on average 17 years prior to the TE, with an average grade of about 80%. Thus, a third question set for this study concerns retention. (3) What did the participants retain from their linear algebra courses vis-à-vis concepts, ideas, and problem solving pertaining to the theory of systems of linear equations, assuming they had constructed such knowledge during these courses?

Section snippets

Sources of student difficulties

Linear algebra is one of two courses—the other being calculus—required by virtually all science, engineering, economics, and mathematics students, the number of whom is likely to be in the hundreds of thousands. Yet, by all accounts—subjective reports by mathematicians who taught linear algebra as well as research studies—students find linear algebra difficult. An example of the subjective reports is Carlson's (1993) broadly-cited reflection on his experience of teaching elementary linear

TE's learning environment

We stipulate that outcomes of any teaching experiment are crucially impacted by—and therefore must be interpreted and understood in the context of—its learning environment. In this section, we describe the social and intellectual climates realized in the TE.

The social climate of the TE was set so that the participants—inservice secondary school teachers—re-experienced themselves as mathematics learners. They solved mathematical problems in small-working groups, discussed their solutions and

Research questions

The TE was organized around three main parameters: It focused on the theory of linear equations, was oriented within particular social and intellectual climates, and its design and implementation took into consideration a series of obstacles known to be a source of difficulties for linear algebra students. The questions we set for this study were:

  • 1.

    Did the participants in the TE, which was organized around these parameters, construct viable knowledge in the theory of systems of linear equations?

Data source and analysis

The participants in this experiment were 10 in-service secondary school teachers (hereafter, “participants”) from a large Southwestern metropolitan region. Initially, the primary reason for selecting these participants as our population of interest was educational, to promote advanced mathematical knowledge among inservice secondary mathematics teachers. We have been pursuing this professional development goal for the last 15 years with different groups of teachers and in different mathematical

Planning stage

The first unit consisted of 15 problems dealing with scenarios of heat or traffic flow, chemical reactions, economic systems of exchange of goods, center of mass, and spatial relations among planes. Most of the scenarios were linear; the rest non-linear. The aim was that through the participants’ solutions to these problems and discussions thereof, the participants will retrieve, re-construct, or newly construct the following ways of understanding.

  • (a)

    Differentiate between linear and non-linear

Planning stage

The first central idea to be developed in the second unit was the following:

  • Let S be an m × n system, with equations ε1, ε2, …, εm. For any m scalars c1, c2, …, cm, any solution of system S is a solution of the equation εΣ = c1ε1 + c2ε2 +  + cmεm.

The goal is that through a discussion of this idea the foundational concept of “linear combination” (i.e., the equation ɛΣ is a linear combination of the equations, ε1, ε2, …, εm) will be elicited, and with it, the following conclusion:

  • Given two systems S1 and S

Contributions and hypotheses

The planning, implementation, and data analysis of the TE took into considerations sources of student obstacles known in the literature. We distinguished between didactical obstacles, those which results from a narrow instruction, and epistemological obstacles, those which are unavoidable due the meaning of concepts. The didactical obstacles included inadequate attention to conceptual understanding; introduction of a hodgepodge of concepts, ideas, and symbolism early in linear algebra courses;

Acknowledgments

This study is based upon work supported by the <gs1>National Science Foundation</gs1> under Grant Nos. <gn1>0934695</gn1> and <gn1>1035503</gn1>. Any opinion expressed in this paper is of the author, not the National Science Foundation.

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