Michelle Zandieh

An Example of Inquiry in Linear Algebra: The Roles of Symbolizing and Brokering

Abstract In this paper we address practical questions such as: How do symbols appear and evolve in an inquiry-oriented classroom? How can an instructor connect students with traditional notation and vocabulary without undermining their sense of ownership of the material? We tender an example from linear algebra that highlights the roles of the instructor as a broker, and the ways in which students participate in the practice of symbolizing as they reinvent the diagonalization equation A = PDP−1.

A hypothetical learning trajectory for conceptualizing matrices as linear transformations

ABSTRACT In this paper, we present a hypothetical learning trajectory (HLT) aimed at supporting students in developing flexible ways of reasoning about matrices as linear transformations in the context of introductory linear algebra. In our HLT, we highlight the integral role of the instructor in this development. Our HLT is based on the ‘Italicizing N’ task sequence, in which students work to generate, compose, and invert matrices that correspond to geometric transformations specified within the problem context. In particular, we describe the ways in which the students develop local transformation views of matrix multiplication (focused on individual mappings of input vectors to output vectors) and extend these local views to more global views in which matrices are conceptualized in terms of how they transform a space in a coordinated way.

Modeling Perspectives in Math Education Research

Too often powerful and beautiful mathematical ideas are learned (and taught) in a procedural manner, thus depriving students of an experience in which they create and refine ideas for themselves. As a first step toward improving the current undesirable situation in undergraduate mathematics education, this chapter describes several different modeling perspectives and their implications for teaching and learning.

Examining Students’ Procedural and Conceptual Understanding of Eigenvectors and Eigenvalues in the Context of Inquiry-Oriented Instruction

This study examines students’ reasoning about eigenvalues and eigenvectors as evidenced by their written responses to two open-ended response questions. This analysis draws on data taken from 126 students whose instructors received a set of supports to implement a particular inquiry-oriented instructional approach and 129 comparable students whose instructors did not use this instructional approach. In this chapter, we offer examples of student responses that provide insight into students’ reasoning and summarize broad trends observed in our quantitative analysis. In general, students in both groups performed better on the procedurally oriented question than on the conceptually oriented question. The group of students whose instructors received support to implement the inquiry-oriented approach outperformed the other group of students on the conceptually oriented question and performed equally well on the procedurally oriented question.

Stretch Directions and Stretch Factors: A Sequence Intended to Support Guided Reinvention of Eigenvector and Eigenvalue

In this chapter, we document the reasoning students exhibited when engaged in an instructional sequence designed to support student development of notions of eigenvectors, eigenvalues, and the characteristic polynomial. Rooted in the curriculum design theory of Realistic Mathematics Education (RME; Gravemeijer, 1999), the sequence builds on student solution strategies from each problem to the next. Students’ used their knowledge of how matrix multiplication transforms space to engage in problems involving stretch factors and stretch directions. In working through these problems students reinvented general strategies for determining eigenvectors, eigenvalues, and the characteristic polynomial.

Teaching Linear Algebra

Research on students’ conceptual difficulties with linear algebra first made an appearance in the 90’s and early 2000’s (e.g. Carlson, 1997; Dorier & Sierpinska, 2001). Over the past decade, research on linear algebra has concentrated on the nature of these difficulties and students’ thought processes (e.g. Stewart &Thomas, 2009; Wawro, Zandieh, Sweeney, Larson, & Rasmussen, 2011). The aim of the discussion group is to initiate a multinational research project on how to foster conceptual understanding of Linear Algebra concepts. Key questions and issues to be discussed are listed below:

The Evolution of an Interdisciplinary Collaborative for Pre-Service Teacher Reform

The Arizona Collaborative for Excellence in Preparation of Teachers is a large National Science Foundation funded project aimed at revising science and mathematics pre-service courses at a large public university in the South-western United States. This chapter describes the collaborations of a community of university faculty in reforming a block of five pre-service mathematics and mathematics education courses. Through a series of workshops and ongoing dialogue, both the instructional delivery and curriculum for these pre-service courses has shifted to student-centred classrooms with inquiry, concept development and problem solving as central themes. The chapter provides information about the process and products of these reforms, with a major focus on providing specific insights into the role of research in guiding the curricular and instructional philosophies and decisions.