In this article we analyze how students reason about linear combinations across multiple digital environments. We present the work of three groups of undergraduate students in the Southeast United States (US) who were considered ready to take linear algebra. The students played the game Vector Unknown, reflected upon aspects of their gameplay using GeoGebra, and used that knowledge to design a level for a 3D version of the game under some constraints. We performed a grounded qualitative analysis of each student’s activity to identify key episodes of student reasoning about linear combinations using technology. Both authors reviewed the episodes and categorized them according to the type of student activity. We compared their reasoning in 2D and 3D space to understand how they made the transition and finally linked these episodes to the technology used to understand its role in building student understanding. We identify four forms of structuring space: Reasoning with Numeric Sums, Reasoning with Resultant Vectors, Reasoning with Tip-to-Tail Vectors, and Reasoning with Vectors as Points. We found that how the technology represented vectors and linear combinations influenced how students engaged in structuring space.

Game-based learning assessments rely on educational data mining approaches such as stealth assessments and quasi mixed methods that help gather data on student learning proficiency. Rarely do we see approaches where student proficiency in learning is woven into the game’s design. Educational burst games (EBGs) represent a new approach to improving learning proficiency by designing fast-paced, short, repetitive, and skill-based games. They have the potential to be effective learning interventions both during instruction in the classroom and during after-school activities such as assignments and homework. Over five years, we have developed two EBGs aimed at improving linear algebra concepts among undergraduate students. In this study, we provide the results of an in-depth evaluation of the two EBGs developed with 45 participants that represent our target population. We discuss the role of EBGs and their design constructs, such as pace and repetition, the effect of the format (2D vs. 3D), the complexity of the levels, and the influence of prior knowledge on the learning outcomes.

In this paper we advance a methodological approach for documenting the mathematical progress of learners as an integrated analysis of individual and collective activity. Our approach is grounded in and expands the emergent perspective by integrating four analytic constructs: individual meanings, individual participation, collective mathematical practices, and collective disciplinary practices. Using video data of one small group of four students in an inquiry-oriented differential equations classroom, we analyze a 10 min segment in which one small group reinvent Euler’s method, an algorithmic tool for approximating solutions to differential equations. A central intellectual contribution of this work is elaborating and coordinating the four methodological constructs with greater integration, cohesiveness, and coherence.

This study examines the mathematical activity involved in engaging with two tasks designed for introductory linear algebra: the Vector Unknown digital game and the pen-and-paper Magic Carpet Ride task. Five undergraduate students worked on both tasks, and we qualitatively analyzed their strategies using a modified version of a framework from prior literature. In the findings, we report on the seven distinct strategies seen in our data set. We found that while our participants did use some of the same strategies on both tasks, there were also certain strategies which were more characteristic of work on one task or the other. In our discussion, we consider how the design differences in the tasks may influence the strategy differences, and how our findings can be leveraged by instructors of linear algebra in selecting tasks. Finally, we conclude by discussing broader implications for mathematics education research in comparing game-based and non-game-based tasks.

Los sistemas de ecuaciones lineales (SEL) corresponden a un concepto fundamental del álgebra lineal, pero hay relativamente poca investigación, pero hay relativamente poca investigación acerca de la enseñanza y el aprendizaje de los SEL, particularmente de las concepciones de los estudiantes acerca de sus soluciones. Se ha encontrado que la resolución de sistemas con un número infinito de soluciones o sin solución tiende a ser menos intuitivo para los estudiantes, lo cual indica la necesidad de más investigación en la enseñanza y aprendizaje de este tema. Entrevistamos a dos estudiantes de matemáticas que eran también maestros en formación a través de un experimento de enseñanza por parejas para mirar cómo razonaban acerca de las soluciones de SEL en ℝ3. Presentamos los resultados enfocando en la progresión del razonamiento de los estudiantes sobre las soluciones de los SEL a través del lento de simbolización. Documentamos la progresión de su razonamiento como una acumulación de significados numéricos, algebraicos y gráficos coordinados y las simbolizaciones de sus conjuntos solución.

Systems of linear equations (SLE) comprise a fundamental concept in linear algebra, but there is relatively little research regarding the teaching and learning of SLE, especially students' conceptions of solutions. It has been shown that solving systems with no or infinitely many solutions tends to be less intuitive for students, pointing to the need for more research on the teaching and learning of the topic. We interviewed two mathematics majors who were also preservice teachers in a paired teaching experiment to see how they reasoned about solutions to SLE in R-3. We present findings focused on the progression of students' reasoning about solutions to SLE through the lens of symbolizing. We document their progression of reasoning as an accumulation of coordinated numeric, algebraic, and graphical meanings and symbolizations for solution sets.

Bridging curriculum design (theory) and classroom implementation (practice) is a critical issue in tertiary mathematics education. In the Inquiry-Oriented Linear Algebra project, we introduce the Design-Based Research (DBR) spiral as a mechanism to bridge theory and practice. In this chapter, we elaborate our project’s DBR spiral informed by Realistic Mathematics Education (RME) instructional design heuristics. The phases of a Design-Based Research spiral are: Design, Paired Teaching Experiment, Classroom Teaching Experiment, Online Working Group, and Web. We explicate these phases to offer insight into the process of conceptualizing and developing an RME instructional sequence focused on determinants. The Online Working Group phase involves work with instructors who were not part of the research project team. This importantly allows us to explicitly connect to research on instructional change as part of our instructional design process. Drawing on data from these instructors’ work with the determinants unit, we gain valuable insights into the ways in which differences in instructors’ instructional contexts and orientation toward mathematical goals can constrain and afford particular kinds of instructional commitments.

The 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.

We present results of a grounded analysis of individual interviews in which students play Vector Unknown — a digital game designed to introduce visualizing vectors, scaling vectors, vector addition, and vector equations, attending to the geometric and algebraic representations of vectors. The game was designed to be used at the beginning of a linear algebra course, and participants in the study were students who had not previously taken such a course. We categorize the strategies the students employed while playing the game and analyze how these strategies evolved throughout the students’ gameplaying. They range from less anticipatory button-pushing to more sophisticated strategies based on approximating solutions and choosing vectors based on their direction. We found that student focus alternated between numerical and geometric aspects of the game interface, which provided additional insight into their strategies. These results have provided revisions to the game and also informed our team’s plans for incorporating it into classroom instruction.

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 United States 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.

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.

In this paper we share a classroom implementation of a task about basis in linear algebra, which was originally developed for research about the topic. The task asks students to construct an everyday situation that captures the definition of basis, and then to critique it mathematically. Using this task, the original research study uncovered learning resources from a group of undergraduate women of color. A mathematician who was not involved in the original study was given the opportunity to work with a group of underrepresented minority students in a linear algebra course. She was inspired by the findings of the study and decided to implement the task independently in her course. She shares how she did it, how her students responded to the task, and how it helped her further develop her understanding of an anti-deficit perspective in teaching mathematics.

To contribute to the sparse educational research on student understanding of eigenspace, we investigated how students reason about linear combinations of eigenvectors. We present results from student reasoning on two written multiple-choice questions with open-ended justifications involving linear combinations of eigenvectors in which the resultant vector is or is not an eigenvector of the matrix. We detail seven themes that analysis of our data revealed regarding student responses. These themes include: determining if a linear combination of eigenvectors satisfies the equation \(A\varvec{x}=\lambda \varvec{x}\); reasoning about a linear combination of eigenvectors belonging to a set of eigenvectors; conflating scalars in a linear combination with eigenvalues; thinking eigenvectors must be linearly independent; and reasoning about the number of eigenspace dimensions for a matrix. In the discussion, we explore how themes sometimes cut across questions and how looking across questions gives insight into individuals’ conceptions of eigenspace. Implications for teaching and future research are also offered.

In this survey paper, we describe the state of the field on linear algebra research. We synthesize themes, questions, results, and perspectives emphasized in the papers that appear in this issue, as well as a selection of those published between 2008 and 2017. We highlight the extensive base of empirical research detailing how students reason about a variety of topic areas in linear algebra, as well as studies that provide evidence of promising directions for supporting students’ success through various teaching interventions and experiments in the classroom. From this survey paper, we identify areas for future research into the teaching and learning of this increasingly important area for modern applications.

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.

Solving systems of linear equations is of central importance in linear algebra and many related applications, yet there is limited literature examining the symbolizing processes students use as they work to solve systems of linear equations. In this paper, we examine this issue by analyzing final exam data from 68 students in an introductory undergraduate linear algebra course at a large public research university in the United States. Based on our analysis, we expanded our framework (Larson & Zandieh, 2013) for interpretations of matrix equations to include augmented matrices and symbolic forms commonly used in solving linear systems. We document considerable variation in students’ symbolization processes, which broadly occurred along two primary trajectories: systems trajectories and row reduction trajectories. Row reduction trajectories included at least five symbolic shifts, two of which students executed with a great deal of success and uniformity. Students’ symbolizing processes varied more in relation to the other three shifts, and these variations were often linked to trends of variable renaming, variable creation, or imagined parameter reasoning. Students were more flexible in their solution strategies when solving systems involving lines than for systems involving planes.