Meanwhile, matrices and vectors are conceptual entities linked in linear algebra. Matrices and vectors are two separate topics in the Indonesian high school mathematics curriculum (ages 16-18). Mathematical concepts emerge from a hierarchical network of interrelated structures. We conclude with a discussion of this and how it may be leveraged to inform teaching in a productive, student-centered manner. Furthermore, we found that all students interviewed expressed, to some extent, the technically inaccurate “nested subspace” conception that Rįor k < n. We also present results regarding the coordination between students’ concept image and how they interpret the formal definition, situations in which students recognized a need for the formal definition, and qualities of subspace that students noted were consequences of the formal definition. Through grounded analysis, we identified recurring concept imagery that students provided for subspace, namely, geometric object, part of whole, and algebraic object. We used the analytical tools of concept image and concept definition of Tall and Vinner (Educational Studies in Mathematics, 12(2):151–169, 1981) in order to highlight this distinction in student responses. This is consistent with literature in other mathematical content domains that indicates that a learner’s primary understanding of a concept is not necessarily informed by that concept’s formal definition. In interviews conducted with eight undergraduates, we found students’ initial descriptions of subspace often varied substantially from the language of the concept’s formal definition, which is very algebraic in nature. This paper reports on a study investigating students’ ways of conceptualizing key ideas in linear algebra, with the particular results presented here focusing on student interactions with the notion of subspace.
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