Note: When clicking on a Digital Object Identifier (DOI) number, you will be taken to an external site maintained by the publisher.
Some full text articles may not yet be available without a charge during the embargo (administrative interval).
What is a DOI Number?
Some links on this page may take you to nonfederal websites. Their policies may differ from this site.

Karunakaran, S. S. ; & Higgins, A. (Ed.)Although a wide variety of reports on developmental math exist, to date there has not been a largescale examination of existing work from a math education point of view. Towards this goal, we analyzed 426 reports and peerreviewed journal articles relating to developmental math published between 2000 and 2020. In report, we quantify the publishers and intended audience, examine the types of outcomes reported on and, where possible, examine the type of developmental math model discussed. We find that over the last decade, less than 20% of reports on developmental math have been aimed at math education audiences. While math education publications more frequently examine math knowledge and student experiences, the overall number of reports, compared to those examining pass rates, is relatively small.more » « less

Karunakaran, S. S. ; & Higgins, A. (Ed.)This paper describes a model of student thinking around equivalence (conceptualized as any type of equivalence relation), presenting vignettes from student conceptions from various college courses ranging from developmental to linear algebra, and courses in between (e.g., calculus). In this model, we conceptualize student definitions along a continuous plane with two dimensions: the extent to which definitions are extracted vs. stipulated; and the extent to which conceptions of equivalence are operational or structural. We present examples to illustrate how this model may help us to recognize illdefined or limited thinking on the part of students even when they appear to be able to provide “standard” definitions of equivalence, as well as to highlight cases in which students are providing mathematically valid, if nonstandard, definitions of equivalence. We hope that this framework will serve as a useful tool for analyzing student work, as well as exploring instructional and curricular handling of equivalence.more » « less

Karunakaran. S. S. ; Higgins, A. (Ed.)We present the results of a classroom teaching experiment for a recently designed unit for the InquiryOriented Linear Algebra (IOLA) curriculum. The new unit addresses orthogonality and least squares using Realistic Mathematics Education design principles with the intent to implement the new unit in an IOI (InquiryOriented Instruction)style classroom. We present an analysis of students’ written responses to characterize how they thought about the notion of shortest distance, travel vectors, orthogonality, and dot product in the “Meeting Gauss” context.more » « less

Karunakaran, S. S. ; Higgins, A. (Ed.)This study presents linear algebra students’ vector conception found in the leastsquares solution context through an IOLA (InquiryOriented Linear Algebra) classroom teaching experiment. Students’ reflection writings after the classroom teaching experiment are the data source. Using the previously found student conception of vector in another study as a basic framing, the data have been analyzed to investigate how students used the word vector and what they referred to. A framework is developed as a tool to be useful in a wide range of describing student conception of a vector emphasizing their natural way of thinking of a vector and on their use of the vector.more » « less

Karunakaran, S. S. ; Higgins, A (Ed.)One important outcome of physics instruction is for students to be capable of relating physical concepts and phenomena to multiple mathematical representations. In quantum mechanics (QM), students are asked to work between multiple symbolic notations, some not previously encountered. To investigate student understanding of the relationships between expressions used in these various notations, many of which describe analogous physical concepts, a survey was distributed to students enrolled in upperdivision QM courses at multiple institutions. Network analysis techniques were shown to be useful for gaining information about how students relate these expressions. Preliminary analysis suggests that students view Dirac bras and kets as more similar to generic vectors than to their physically analogous wave function counterparts, and that Dirac bras and kets serve as a bridge between vector and wave function expressions.more » « less

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.more » « less

Karunakaran, S. S. ; Higgins, A. (Ed.)The critical role of teachers in supporting student engagement with reasoning and proving has long been recognized (Nardi & Knuth, 2017; NCTM, 2014). While some studies examined how prospective secondary teachers (PSTs) develop dispositions and teaching practices that promote student engagement with reasoning and proving (e.g., Buchbinder & McCrone, 2020; Conner, 2007), very little is known about longterm development of proofrelated practices of beginning teachers and what factors affect this development (Stylianides et al., 2017). During the supervised teaching experiences, interns often encounter tensions between balancing their commitments to the university and cooperating teacher, while also developing their own teaching styles (Bieda et al., 2015; Smagorinsky et al., 2004; Wang et al., 2008). Our study examines how sociocultural contexts of the teacher preparation program and of the internship school, supported or inhibited proofrelated teaching practices of beginning secondary mathematics teachers. In particular, this study aims to understand the observed gap between proofrelated teaching practices of one such teacher, Olive, in two settings: as a PST in a capstone course Mathematical Reasoning and Proving for Secondary Teachers (Buchbinder & McCrone, 2020) and as an intern in a highschool classroom. We utilize activity theory (Leont’ev, 1979) and Engeström’s (1987) model of an activity system to examine how the various components of the system: teacher (subject), teaching (object), the tasks (tools), the curriculum and the expected teaching style (rules), the cooperating teacher (community) and their involvement during the teaching (division of labor) interact with each other and affect the opportunities provided to students to engage with reasoning and proving (outcome). The analysis of four lessons from each setting, lesson plans, reflections and interviews, showed that as a PST, Olive engaged students with reasoning and proving through productive proofrelated teaching practices and rich tasks that involved conjecturing, justifying, proving and evaluating arguments. In a sharp contrast, as an intern, Olive had to follow her school’s rigid curriculum and expectations, and to adhere to her cooperating teacher’s teaching style. As a result, in her lessons as an intern students received limited opportunities for reasoning and proving. Olive expressed dissatisfaction with this type of teaching and her desire to enact more prooforiented practices. Our results show that the sociocultural components of the activity system (rules, community and division of labor), which were backgrounded in Olive’s teaching experience as a PST but prominent in her internship experience, influenced the outcome of engaging students with reasoning and proving. We discuss the importance of these sociocultural aspects as we examine how Olive navigated the tensions between the proofrelated teaching practices she adopted in the capstone course and her teaching style during the internship. We highlight the importance of teacher educators considering the sociocultural aspects of teaching in supporting beginning teachers developing proofrelated teaching practices.more » « less

Karunakaran, S. S. ; Higgins, A. (Ed.)Systems of equations is a core topic in linear algebra courses. Solving systems with no or infinitely many solutions tends to be less intuitive for students. In this study, we examined two students’ reasoning about the relationship between the structure of a system of linear equations and its solution set, particularly when creating systems with a certain number of equations and unknowns. Using data from a paired teaching experiment, we found that both students favored the notion of parallel planes, both geometrically and numerically, in the case of a system having no solution or infinitely many solutions. We also found that algebraic or numerical approaches were used as the main way of developing systems with a unique solution, especially in systems with more than two equations and two unknowns. In particular, one student gravitated toward geometric approaches and the other toward algebraic and numerical approaches.more » « less

Karunakaran, S. S. ; Higgins, A. (Ed.)Vector spaces are often taught with an axiomatic focus, but this has been shown to rely on knowledge many students have not yet developed. In this paper, we examine two students’ conceptual resources for reasoning about null spaces drawing on data from a paired teaching experiment. The task sequence is set in the context of a school with one directional hallways. Students’ informal reasoning about paths that leave the room populations unchanged supported more formal reasoning about null spaces. We found that one student used contextbased resources (such as ‘loops’ in hallway) to reason about null spaces, while the other student drew largely on previously formalized mathematical resources (e.g. free variables, linear dependence). The use of formal resources sometimes required recontextualization, which may function to constrain student sensemaking or afford opportunities for broadening students’ formal prior knowledge.more » « less

Karunakaran, S. S. ; Higgins, A. (Ed.)