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1. Constructing linear systems with particular kinds of solution sets.

   Smith, J. ; Lee, I. ; Zandieh, M. ; & Andrews-Larson, C. ( January 2022 , Proceedings of the Annual Conference on Research in Undergraduate Mathematics Education)

   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. 

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2. Constructing linear systems with particular kinds of solution sets

   Smith, J. ; Lee, I. ; Zandieh, M. ; Andrews-Larson, C. ( January 2022 , Proceedings of the Annual Conference on Research in Undergraduate Mathematics Education)

   Karunakaran, 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. 

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3. Two Students’ Conceptions of Solutions to a System of Linear Equations

   Smith, J. ; Lee, I. ; Zandieh, M. ; & Andrews-Larson, C. ( January 2021 , 2021 Research in Undergraduate Mathematics Education Reports)

   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, and 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. 

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4. Two Students’ Conceptions of Solutions to a System of Linear Equations

   Smith, J. ( January 2021 , Proceedings of the Annual Conference on Research in Undergraduate Mathematics Education)

   S. S. Karunakaran, & 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, who were also preservice teachers, to see how they thought about solutions to SLE in R3, especially linear systems with multiple solutions. We found participants used their knowledge of SLE in R2 to think about systems in higher dimensions, sometimes ran into algebraic complications, and 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. 

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5. Two students' conceptions of solutions to systems of linear equations

   Smith, J. ( January 2021 , Proceedings of the Annual Conference on Research in Undergraduate Mathematics Education)

   S. S. Karunakaran, & 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, who were also preservice teachers, to see how they thought about solutions to SLE in ℝ3 , especially linear systems with multiple solutions. We found participants used their knowledge of SLE in ℝ2 to think about systems in higher dimensions, sometimes ran into algebraic complications, and 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. 

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