Christine Andrews-Larson

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.

Why Don't Teachers Understand Our Questions? Reconceptualizing Teachers' "Misinterpretation" of Survey Items

This study examined sources of inconsistency between teachers’ and researchers’ interpretations of survey items. We analyzed cognitive interview data from 12 middle school mathematics teachers to understand their interpretations of survey items focused on one aspect of their practice: the content of their advice-seeking interactions. Through this analysis we found that previously documented conceptualizations of sources of misinterpretation within teacher surveys (e.g., structural complexity, use of reform language) did not adequately account for all of the inconsistencies between the survey items and teachers’ interpretations. We found it useful to reconceptualize the broader source of many of the misinterpretations as an issue of fit between the researchers’ intended interpretation and teachers’ professional practice.

Inquiry-Oriented Instruction: A Conceptualization of the Instructional Principles

Abstract Research has highlighted that inquiry-based learning (IBL) instruction leads to many positive student outcomes in undergraduate mathematics. Although this research points to the value of IBL instruction, the practices of IBL instructors are not well-understood. Here, we offer a characterization of a particular form of IBL instruction: inquiry-oriented instruction. This characterization draws on K-16 research literature in order to explicate the instructional principles central to inquiry-oriented instruction. As a result, this conceptualization of inquiry-oriented instruction makes connections across research communities and provides a characterization that is not limited to undergraduate, secondary, or elementary mathematics education.

Roots of Linear Algebra: An Historical Exploration of Linear Systems

Abstract There is a long-standing tradition in mathematics education to look to history to inform instruction. An historical analysis of the genesis of a mathematical idea offers insight into: (i) the contexts that give rise to a need for a mathematical construct; (ii) the ways in which available tools might shape the development of that mathematical idea; and (iii) ways in which students might make sense of an idea. In this paper, I discuss historic contexts that gave rise to considerations of linear systems of equations and their solutions, as well as implications for instruction and instructional design.

Learning sorting algorithms through visualization construction

Abstract Recent increased interest in computational thinking poses an important question to researchers: What are the best ways to teach fundamental computing concepts to students? Visualization is suggested as one way of supporting student learning. This mixed-method study aimed to (i) examine the effect of instruction in which students constructed visualizations on students’ programming achievement and students’ attitudes toward computer programming, and (ii) explore how this kind of instruction supports students’ learning according to their self-reported experiences in the course. The study was conducted with 58 pre-service teachers who were enrolled in their second programming class. They expect to teach information technology and computing-related courses at the primary and secondary levels. An embedded experimental model was utilized as a research design. Students in the experimental group were given instruction that required students to construct visualizations related to sorting, whereas students in the control group viewed pre-made visualizations. After the instructional intervention, eight students from each group were selected for semi-structured interviews. The results showed that the intervention based on visualization construction resulted in significantly better acquisition of sorting concepts. However, there was no significant difference between the groups with respect to students’ attitudes toward computer programming. Qualitative data analysis indicated that students in the experimental group constructed necessary abstractions through their engagement in visualization construction activities. The authors of this study argue that the students’ active engagement in the visualization construction activities explains only one side of students’ success. The other side can be explained through the instructional approach, constructionism in this case, used to design instruction. The conclusions and implications of this study can be used by researchers and instructors dealing with computational thinking.

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.

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: