Everyday Examples in Linear Algebra: Individual and Collective Creativity

This paper investigates creativity in students’ constructions of everyday examples about basis in Linear Algebra. We analyze semi-structured interview data with 18 students from the U.S. and Germany with diverse academic and social backgrounds. Our analysis of creativity in students’ everyday examples is organized into two parts. First, we analyze the range of students’ creative products by investigating the mathematical variability in the more commonly mentioned examples. Second, we unpack some of the collective processes in the construction of students’ examples. We examine how creativity was distributed through the interactions among the student, the interviewers, and other artifacts and ideas. Thus, in addition to contributing to the process vs. product discussion of creativity, our work also adds to the few existing studies that focus on collective mathematical creativity. The paper closes with connections to anti-deficit perspectives in mathematics education and some recommendations for individual and collective creativity in the classroom

The evolution of student understanding of the concept of derivative

This research seeks to answer the question, "What does it mean for a student to understand the concept of derivative?" A structured way to describe an individual student's understanding of derivative is developed and applied to analyzing the evolution of that understanding for each of nine high school seniors during their year-long calculus course. The methodology is a multiple case study. Interviews, including both task-based and open-ended questions, are the primary instruments for collecting data on each student's understanding. Other data collected include tests, written questions, and classroom observations. Several theoretical frameworks contribute to the research: concept image (Tall, Vinner, and Dreyfus), process-object (Sfard; Dubinsky and colleagues), and notions of multiple representations for function, limit, and derivative. I describe the concept of derivative as three layers of process-objects: the ratio or difference quotient, the limit, and the function layers. Each layer may be observed in multiple contexts: graphical (slope), verbal description (rate of change), kinematic (e.g. velocity or acceleration), and symbolic (the symbolic difference quotient definition of derivative). A description of the connections between the various aspects of the concept of derivative comes from the work of Fischbein on paradigmatic, analogic, and diagrammatic models and the work of Lakoff on metaphor and metonymy. The major theoretical result of the dissertation is the development of a structured way of describing the concept of derivative including a diagrammatic methodology for displaying which aspects of the derivative concept a student has demonstrated. This methodology may be applied to other studies and other concepts. The major result of this study of nine students is the realization that the layers and representations of the concept of derivative do not appear to be hierarchical in that none of the nine students learn the aspects in the same order

Proofs and refutations in the undergraduate mathematics classroom

Abstract In his 1976 book, Proofs and Refutations, Lakatos presents a collection of case studies to illustrate methods of mathematical discovery in the history of mathematics. In this paper, we reframe these methods in ways that we have found make them more amenable for use as a framework for research on learning and teaching mathematics. We present an episode from an undergraduate abstract algebra classroom to illustrate the guided reinvention of mathematics through processes that strongly parallel those described by Lakatos. Our analysis suggests that the constructs described by Lakatos can provide a useful framework for making sense of the mathematical activity in classrooms where students are actively engaged in the development of mathematical ideas and provide design heuristics for instructional approaches that support the learning of mathematics through the process of guided reinvention

The lived experience of linear algebra: a counter-story about women of color in mathematics

This paper focuses on the mathematical sensemaking by women of color in the USA as part of the global effort of dismantling deficit narratives about historically marginalized groups of students. Following Adiredja's anti-deficit framework for sensemaking, this cognitive study invited a group of women of color to share their understanding of basis from linear algebra to construct a sensemaking counter-story. Extending the framework, this study examines a task that explores the boundaries and nuances of a concept to support the effort of going beyond students' deficits. Eight women extended the concept of basis (and vector spaces) to 22 distinct everyday contexts, drawing from their everyday lives as well as topics from their academic experiences. Their explanations revealed analytical codes describing roles and characteristics of a basis. These codes suggest ways that students can mobilize the concept of basis beyond its logical underpinnings. Contrasting interpretations using a deficit and an anti-deficit perspective construct a counter-story that showcases these women's creativity and flexibility in understanding the concept, and potential resources for the teaching and learning of linear algebra.12 month embargo; published online: 6 June 2020This item from the UA Faculty Publications collection is made available by the University of Arizona with support from the University of Arizona Libraries. If you have questions, please contact us at [email protected]

Delineating Aspects of Understanding Eigentheory through Assessment Development

International audienceIn this report, we share insights we have gained from developing an assessment for documenting students' understanding of eigentheory. We explain the literature and theory that influenced the assessment's development and share question examples. We frame our results in terms of three eigentheory settings (Ax = λx, (A - λI)x = 0, and eigenspaces) and four interpretations (numeric, algebraic, geometric, and verbal). Results from our analysis include students' reasoning being influenced by setting, insights into students' struggle with understanding eigenspaces, and the importance of making connections between and across various interpretations

Abstract and Linear Algebra (Chapter 8)

International audienceThe aim of the book is to provide a deep synthesis of the research field "didactics of mathematics at all levels of tertiary education", as it appears through two INDRUM conferences organised in 2016 and 2018. Chapter 8 deals with Abstract and Linear Algebra. Our goal is to account for “burning issues” within University Mathematics Education research on these mathematical domains, with its related methodological challenges, and to point out current and new avenues for research. We also aim at cross-analysing results and methodologies, thus answering the question: in which respect do these studies complement each other or contrast from each other? What are the main results, open questions, debates among researchers, elements of convergence/divergence within these studies? We thus decided to organize our synthesis according to three groups of papers (merely according to the topic), and for each group of papers to present a cross-analysis of these papers according to main themes that crystallise those burning issues (in terms of epistemological content, methodological issues, results, in fact the main objects of research that seemed to us appropriate for a vivid, illuminating and contrasted account of our data). We end this chapter by summarising what appeared to us as major advances in research on Abstract and Linear Algebra teaching and learning through the work of the INDRUM network as well as the further avenues for research that have been brought to light

Exploring everyday examples to explain basis: insights into student understanding from students in Germany

There is relatively little research specifically about student understanding of basis. Our ongoing work addresses student understanding of basis from an anti-deficit perspective, which focuses on the resources that students have to make sense of basis using everyday ideas. Using data from a group of women of color in the United States, we previously developed an analytical framework to describe student understanding about basis, including codes related to characteristics of basis vectors and roles of basis vectors in the vector space. In this paper, we utilize the methods of the previous study to further enrich our findings about student understanding of basis. By analyzing interview data from students in Germany, we found that this group of students most often used ideas that we describe by the roles generating, structuring, and traveling, and the characteristics different and essential. Some of the themes that emerged from the data illustrate common pairings of these ideas, students' flexibility in interpreting multiple roles within one everyday example, and the ways that the roles and characteristics motivate students to create additional examples. We also discuss two ways that differences between the German and English languages were pointed out by students in the interviews.Division of Undergraduate Education12 month embargo; published online: 8 February 2019This item from the UA Faculty Publications collection is made available by the University of Arizona with support from the University of Arizona Libraries. If you have questions, please contact us at [email protected]