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Trigueros, M. ; Barquero, B. ; Hochmuth, R. ; & J. Peters (Ed.)We report on a variety of innovative projects that are at different stages of development and implementation. We start by presenting a project still in development to help address Klein’s second discontinuity problem, that is, the perception of precollege teachers that the advanced mathematics courses they took at the university are of little use in the practice of their profession. Then we briefly discuss the study and research paths (SRP). This is the proposal from the Anthropological Theory of the Didactic (ATD) to foment a move from the prevailing paradigm of visiting works to that of questioning the world. This is followed by the discussion of an online course for inservice teachers, designed to help them experience, adapt, and classtest a modeling intervention, as well as reflect on institutional issues that might constrain the future application of modeling in their teaching. We end with a discussion of a project based on the idea of guided reinvention, to design and study the implementation of inquiryoriented linear algebra.more » « less

Cook, S. ; Katz, B. ; MooreRusso, D. (Ed.)Postsecondary instructors interested in inquiryoriented instruction of Linear Algebra participated in a sequence of eight onehour online work group meetings with other experienced inquiryoriented linear algebra facilitators and teachers. Recordings from three meetings were analyzed for how two participants referenced goals of instruction in preparation for teaching a new instructional unit on subspaces. We identified four goals of instruction of teaching subspaces. We discuss the intersections of several goals of instruction and possible implications for those who want to transition to inquiry oriented instructional approaches.more » « less

Cook, S. ; Katz, B. ; MooreRusso, D. (Ed.)The method of Least Square Approximation is an important topic in some linear algebra classes. Despite this, little is known about how students come to understand it, particularly in a Realistic Mathematics Education setting. Here, we report on how students used literal symbols and equations when solving a least squares problem in a travel scenario, as well as their reflections on the least squares equation in an openended written question. We found students used unknowns and parameters in a variety of ways. We highlight how their use of dot product equations can be helpful towards supporting their understanding of the least squares equation.more » « less

Cook, S. ; Katz, B. ; MooreRusso, D. (Ed.)Inquiry and active learning instructional methods have largely been regarded as equitable and beneficial for students. However, researchers have highlighted math classrooms as racialized and gendered spaces that can negatively impact marginalized students’ experiences in such spaces. In this study, I examine the development of one argument, and whose ideas are solicited and leveraged, in an inquiryoriented linear algebra course with an eye toward participatory equity. I found that gender related most to the inequity of participation in argumentation and that only men participated in generalizing activity. This study adds to the growing literature addressing equity in inquiry and active learning math settings.more » « less

Cook, S. ; Katz, B. ; MooreRusso, D. (Ed.)We discuss two Dynamic Geometry Software applets designed as part of an InquiryOriented instructional unit on determinants and share students' generalizations based on using the applet. Using the instructional design theory of Realistic Mathematics Education, our team developed a task sequence supporting students' guided reinvention of determinants. This unit leverages students' understanding of matrix transformations as distortion of space to meaningfully connect determinants to the transformation as the signed multiplicative change in area that objects in the domain undergo from the linear transformation. The applets are intended to provide students with feedback to help connect changes in the matrix to changes in the visualization of the linear transformation and, so, to changes in the determinant. Critically, the materials ask students to make generalizations while reflecting on their experiences using the applets. We discuss patterns among these generalizations and implications they have on the applets' design.more » « less

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.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.)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