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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 pre-college 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 class-test 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 inquiry-oriented linear algebra. 

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 open-ended 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. 

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 open-ended 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. 

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. 

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. 

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. 

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. 

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. 

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. 

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 pre-college 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 in- service teachers, designed to help them experience, adapt, and class-test 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 inquiry-oriented linear algebra.