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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. 

This study presents linear algebra students’ vector conception found in the least-squares solution context through an IOLA (Inquiry-Oriented 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. 

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. 

Social Network Analysis is a method to analyze individuals’ social accessibility and power. We adapt it to change inequitable issues in STEM postsecondary education. Equity issues in mathematics education, such as underrepresented women and racial disparities, are prevalent. With the social capital perspective, we investigate the demographic characteristics of influential students and their social networks. Seventeen participants are undergraduate students in an inquiry-oriented linear algebra course. The number of nominations on discussion boards as “Shout-out” is data to measure influence and map the social network. By analyzing data with UCINET, we found that (1) the most influential students are non-White males and the principal components of the network are male-dominant, and (2) there is a female-dominant small cluster and female students have reciprocal networks. This study suggests further discussions of (1) how discussion boards position students with the social capital perspective and (2) intersectionality, especially for women of color. 

We present the results of a classroom teaching experiment for a recently designed unit for the Inquiry-Oriented 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 (Inquiry-Oriented 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. 

Social Network Analysis is a method to analyze individuals’ social accessibility and power. We adapt it to change inequitable issues in STEM postsecondary education. Equity issues in mathematics education, such as underrepresented women and racial disparities, are prevalent. With the social capital perspective, we investigate the demographic characteristics of influential students and their social networks. Seventeen participants are undergraduate students in an inquiry-oriented linear algebra course. The number of nominations on discussion boards as “Shout-out” is data to measure influence and map the social network. By analyzing data with UCINET, we found that (1) the most influential students are non-White males and the principal components of the network are male-dominant, and (2) there is a female-dominant small cluster and female students have reciprocal networks. This study suggests further discussions of (1) how discussion boards position students with the social capital perspective and (2) intersectionality, especially for women of color. 

Mathematical Knowledge for Teaching Proof (MKT-P) has been recognized as an important component of fostering student engagement with mathematical reasoning and proof. This study is one component of a larger study aimed at exploring the nature of MKT-P. The present study examines qualitative differences in feedback given by STEM majors, in-service and pre-service secondary mathematics teachers on hypothetical students’ arguments. The results explicate key distinctions in the feedback provided by these groups, indicating that this is a learnable skill. Feedback is cast as a component of MKT-P, making the results of this study significant empirical support for the construct of MKT-P as a type of knowledge that is unique to teachers. 

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 context-based 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 sense-making or afford opportunities for broadening students’ formal prior knowledge. 

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 have implications for instructors considering using one or both tasks in their linear algebra class.