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  1. 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 »implications for instructors considering using one or both tasks in their linear algebra class.« less
    Free, publicly-accessible full text available January 1, 2023
  2. 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, publicly-accessible full text available January 1, 2023
  3. Karunakaran, S. ; Higgins, A. (Ed.)
    In this report, we characterize seven of twenty-five 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, publicly-accessible full text available January 1, 2023
  4. 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 need-provoking tasks in first-semester calculus classes.
    Free, publicly-accessible full text available January 1, 2023
  5. 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 »initially did not find the third dimension intuitive to think about geometrically. Our findings highlight students’ ways of reasoning with infinite solution sets, such as moving toward the notion of parametrization.« less
  6. Karunakaran, S. ; Higgins, A. (Ed.)
    In this paper, we introduce an RME-based (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 whole-class setting.
  7. Karunakaran, S. S. ; Higgins, A. (Ed.)
    The abrupt switch from in-person instruction and tutoring to remote or online instruction and tutoring as a result of the COVID-19 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 NSF-funded project focusing on equipping mathematical sciences GTAs to become better teachers. In their responses, the graduate teaching assistants discussed topics ranging from what theymore »did in their remote classrooms to the challenges they faced and supports they received from their department, university, and fellow classmates and faculty.« less
  8. 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 multi-faceted 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 »in participating in the peer mentoring program. While the mentees found their mentors to be significant to their own success and effectiveness, the mentors did not rate themselves as high as the mentees rated them with respect to their own significance in impacting the effectiveness of their mentee.« less
  9. Karunakaran, S. S. (Ed.)
    This study examined how eight students in an introduction to proof (ITP) course viewed a “cheating scandal” where their peers submitted homework containing solutions found on the web. Drawing on their weekly log entries, the analysis focuses on the students’ reasoning about the difference between acceptable and unacceptable use of internet resources in learning mathematics. One pattern was that students’ view of the relationship between beliefs about mathematics and the work of learning mathematics grounded their views of “cheating.” Specifically, some felt that an implicit didactical contract required that model solutions should be available when one learned new material. Themore »case raises the general issue of the relationship between the process of learning mathematics and the appropriate use of external resources. It suggests that instructors may need to re-examine the role of homework, especially its assessment, in their courses, so that productive struggle is valued, not avoided.« less
  10. Karunakaran, S. S. (Ed.)
    This paper reports a qualitative study of how small group problem solving was enacted differently across sections of a multi-section undergraduate introduction to proof course. Common course materials, common guidelines for instruction, and weekly instructor meetings led by a faculty course coordinator supported similar instruction across sections, including an emphasis on in-class group work. But within that shared structure, classroom observations revealed important differences in how group work was introduced, organized, and managed. Our results focus on differences in the time allotted to group work, the rationale for group work, the selection and organization of groups, and aspects of studentmore »activity and participation. We suggest that these differences shaped different opportunities to learn proof writing in small groups. These results have implications for the design and teaching of collegiate mathematics courses where group work is a regular element of classroom work.« less