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Challenges and Strategies in Teaching Linear Algebra pp 3–27Cite as

The Learning and Teaching of Linear Algebra Through the Lenses of Intellectual Need and Epistemological Justification and Their Constituents

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Part of the book series: ICME-13 Monographs ((ICME13Mo))

Abstract

Intellectual need and epistemological justification are two central constructs in a conceptual framework called DNR-based instruction in mathematics. This is a theoretical paper aiming at analyzing the implications of these constructs and their constituent elements to the learning and teaching of linear algebra. At the center of these analyses are classifications of intellectual need and epistemological justification in mathematical practice along with their implications to linear algebra curriculum development and instruction. Two systems of classifications for intellectual need are discussed. The first system consists of two subcategories, global need and local need; and the second system consists of five categories of needs: need for certainty, need for causality, need for computation, need for communication, and formalization, and need for structure. Epistemological justification is classified into three categories: sentential epistemological justification (SEJ), apodictic epistemological justification (ASJ), and meta epistemological justification (MEJ).

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Notes

  1. 1.

    APOS theory (Arnon et al., 2014; Dubinsky, 1991) will be used to provide conceptual bases for some of these observations. Given how widely this theory has been studied during the last three decades, there is no need to allocate more than a brief illustration to the four levels of conceptualizations, action, process, object, and schema offered by the theory and used in this paper. Briefly, consider the phrase “the coordinates of a vector of \(x\) with respect to a basis-matrix \(A\) in \(R^{n}\),” denoted by \(\left[ x \right]_{A}\). At the level of action conception, the learner might be able to deal with \(\left[ x \right]_{A}\) only in the context of a specific vector and a specific suitable basis-matrix, by following step-by-step instruction to compute the respective coordinate vector. At the level of process conception one is capable of imagining taking any vector \(x\) in \(R^{n}\), representing it as a linear combination of the columns of \(A\), and forming a column vector whose entries are the coefficient of, and are sequenced in the order they appear in, the combination. With this conceptualization, the learner is able to carry out this process in thought and with no restriction on the vector \(x\) considered. At the level of object conception, one is aware of the process of relating the two coordinate vectors as a totality, for example, in finding the relation between two coordinate vectors of \(x\), one with respect to a basis-matrix \(A_{1}\), \(\left[ x \right]_{{A_{1} }}\), and one with respect to a basis-matrix \(A_{2}\), \(\left[ x \right]_{{A_{2} }}\), whereby being able to express the relation in terms of a transition matrix \(S = A_{2}^{ - 1} A_{1}\) between the two vectors. Among the ways of thinking that are essential to cope with linear algebra, in particular, and mathematics, in general, are the abilities to construct concepts at the levels of process conception and object conception, as it is demonstrate throughout the paper. (See also Trigueros, this volume.)

  2. 2.

    A matrix whose columns form a basis for a subspace.

  3. 3.

    For the philosophical foundations of this premise, see Harel (2008c).

  4. 4.

    Questions concerning convergence are not discussed, though on rare occasions were raised by students.

  5. 5.

    Of course other contexts can be used as SEJ for the SVD Theorem. The problem of transmitting a digitized image is typically used in textbooks as an application of SVD; we, on the hand, used it as an intellectual motivation (see the distinction between “application” and “intellectual need” in Sect. 3).

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Authors and Affiliations

  1. University of California, San Diego, USA

    Guershon Harel

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  1. Guershon Harel
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Correspondence to Guershon Harel .

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Editors and Affiliations

  1. Department of Mathematics, University of Oklahoma, Norman, Oklahoma, USA

    Sepideh Stewart

  2. College of Education, Florida State University, Tallahassee, Florida, USA

    Christine Andrews-Larson

  3. Department of Mathematics, Technion—Israel Institute of Technology, Haifa, Israel

    Avi Berman

  4. Arizona State University, Mesa, Arizona, USA

    Michelle Zandieh

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Harel, G. (2018). The Learning and Teaching of Linear Algebra Through the Lenses of Intellectual Need and Epistemological Justification and Their Constituents. In: Stewart, S., Andrews-Larson, C., Berman, A., Zandieh, M. (eds) Challenges and Strategies in Teaching Linear Algebra. ICME-13 Monographs. Springer, Cham. https://doi.org/10.1007/978-3-319-66811-6_1

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