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Karunakaran, S. ; Higgins, A. (Ed.)We present findings from a study analyzing and comparing the strategies participants deployed in playing the game Vector Unknown and completing the Magic Carpet Ride task. Both the game and task are designed to give students an introduction to basic concepts about vectors needed for success in linear algebra. We found that participants used a diverse array of strategies, tending to favor algebraic approaches to the Magic Carpet Ride task. We also found that participants tended to try the same strategies in both tasks, but did not usually follow through with the same strategy in both contexts. These findings havemore »Free, publiclyaccessible full text available January 1, 2023

Karunakaran, S. ; Higgins, A. (Ed.)Understanding linear combinations is at the core of linear algebra and impacts their understanding of basis and linear transformations. This research will focus on how students understand linear combinations after playing a video game created to help students link the algebraic and geometric representations of linear combinations. I found that having students reflect upon the game and create their own 3D version of the game illustrated which elements of 2D understanding could be translated into 3D. Also, students' creation of easy, medium, and hard levels provided insight into how students progressively structure space.Free, publiclyaccessible full text available January 1, 2023

Karunakaran, S. ; Higgins, A. (Ed.)In this report, we characterize seven of twentyfive students’ responses to a single written homework assignment from the Spring 2021 academic semester. The homework was designed to incorporate the Vector Unknown 2D digital game to investigate how students answered questions about span and linear independence after playing various levels of the game. We present our modification of the roles and characteristics framework of Zandieh et al. (2019), our identification of students’ grammatical use of game language and math language, as well as the results of analyzing students’ homework responses using our framework.Free, publiclyaccessible full text available January 1, 2023

Karunakaran, S. S. ; Higgins, A. (Ed.)The idea of intellectual need, which proposes that learning is the result of students wrestling with a problem that is unsolvable by their current knowledge, has been used in instructional design for many years. However, prior research has not described a way to empirically determine whether, and to what extent, students’ experience intellectual need. In this paper, we present a methodology to identify students’ intellectual need and also report the results of a study that investigated students’ reactions to intellectual needprovoking tasks in firstsemester calculus classes.Free, publiclyaccessible full text available January 1, 2023

Karunakaran, S. ; Higgins, A. (Ed.)Systems of linear equations (SLE) comprise a fundamental concept in linear algebra, but there is little research regarding the teaching and learning of SLE, especially students' conceptions of solutions. In this study, we examine students’ understanding of solutions to SLE in the context of an experientially real task sequence. We interviewed two undergraduate mathematics majors, 3 who were also preservice teachers, to see how they thought about solutions to SLE, especially linear systems with multiple solutions. We found participants used their knowledge of SLE in two dimensions to think about systems in higher dimensions, sometimes ran into algebraic complications, andmore »

Karunakaran, S. ; Higgins, A. (Ed.)In this paper, we introduce an RMEbased (Freudenthal, 1991) task sequence intended to support the guided reinvention of the linear algebra topic of vector spaces. We also share the results of a paired teaching experiment (Steffe & Thompson, 2000) with two students. The results show how students can leverage their work in the problem context to develop more general notions of Null Space. This work informs further revisions to the task statements for using these materials in a wholeclass setting.

Karunakaran, S. S. ; Higgins, A. (Ed.)The abrupt switch from inperson instruction and tutoring to remote or online instruction and tutoring as a result of the COVID19 pandemic in March 2020 was difficult for even the most experienced instructor. In this paper, we explore how graduate teaching assistants (GTAs) at three different institutions responded to and experienced this change. Data was collected from surveys and focus groups conducted with graduate teaching assistants at each institution, as part of our ongoing collaborative NSFfunded project focusing on equipping mathematical sciences GTAs to become better teachers. In their responses, the graduate teaching assistants discussed topics ranging from what theymore »

Karunakaran, S. S. ; Higgins, A. (Ed.)Peer mentoring programs are one approach to improving the pedagogical development of mathematical sciences graduate students. This paper describes the peer mentoring experiences at three institutions that have implemented a multifaceted GTA professional development program. Data was collected from surveys and focus groups conducted with graduate teaching assistants at each institution regarding mentees’ ratings of their mentors, mentors’ ratings of their impact on mentees, mentors’ impressions of the benefits and challenges of peer mentoring, and mentees and mentors’ ratings of program components related to support from mentors, their TA coach, program staff, and other graduate students. Most GTAs found valuemore »

Karunakaran, S. ; Reed, Z. ; Higgins, A. (Ed.)We present results of a grounded analysis of individual interviews in which students play Vector Unknown  a video game designed to support students who are taking their first semester of linear algebra. We categorized strategies students employed while playing the game. These strategies range from lessanticipatory buttonpushing to more sophisticated strategies based on approximating solutions and choosing vectors based on their direction. We also found that students focus on numeric and geometric aspects of the game interface, which provides additional insight into their strategies. These results have informed revisions to the game and also inform our team's plan formore »

Karunakaran, S.S. ; Reed, Z. ; Higgins, A. (Ed.)The Mathematical Education of Teachers as an Application of Undergraduate Mathematics project provides lessons integrated into various mathematics major courses that incorporate mathematics teaching connections as a legitimate application area of undergraduate mathematics. One feature of the lessons involves posing tasks that require undergraduates to interpret or analyze the work of another student. This paper reports on thematic analysis of hourlong interviews for eight participants enrolled in an undergraduate abstract algebra course from two different implementation sites. We focus on student work and reactions to these interpreting or analyzing student thinking (AST) applications as they relate to their perceptions regardingmore »