caa a22 4500 445881674 CHVBK 20180317145545.0 cr unu---uuuuu 170323e20110901xx s 000 0 eng 10.1007/s10649-011-9307-4 doi (NATIONALLICENCE)springer-10.1007/s10649-011-9307-4 Subspace in linear algebra: investigating students' concept images and interactions with the formal definition [Elektronische Daten] [Megan Wawro, George Sweeney, Jeffrey Rabin] 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. 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 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. 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. Through grounded analysis, we identified recurring concept imagery that students provided for subspace, namely, geometric object, part of whole, and algebraic object. 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. Furthermore, we found that all students interviewed expressed, to some extent, the technically inaccurate "nested subspace” conception that R k is a subspace of R n for k < n. We conclude with a discussion of this and how it may be leveraged to inform teaching in a productive, student-centered manner. Springer Science+Business Media B.V., 2011 Linear algebra nationallicence Subspace nationallicence Concept image and concept definition nationallicence Undergraduate mathematics education nationallicence Wawro Megan San Diego State University, San Diego, CA, USA aut Sweeney George San Diego State University, San Diego, CA, USA aut Rabin Jeffrey University of California, San Diego, San Diego, CA, USA aut Educational Studies in Mathematics Springer Netherlands 78/1(2011-09-01), 1-19 0013-1954 78:1<1 2011 78 10649 https://doi.org/10.1007/s10649-011-9307-4 text/html Onlinezugriff via DOI 1 research-article jats NATIONALLICENCE 856 40 https://doi.org/10.1007/s10649-011-9307-4 text/html Onlinezugriff via DOI NATIONALLICENCE 700 1- Wawro Megan San Diego State University, San Diego, CA, USA aut NATIONALLICENCE 700 1- Sweeney George San Diego State University, San Diego, CA, USA aut NATIONALLICENCE 700 1- Rabin Jeffrey University of California, San Diego, San Diego, CA, USA aut NATIONALLICENCE 773 0- Educational Studies in Mathematics Springer Netherlands 78/1(2011-09-01), 1-19 0013-1954 78:1<1 2011 78 10649 Metadata rights reserved Springer special CC-BY-NC licence nationallicence BK010053 XK010053 XK010000 NATIONALLICENCE NATIONALLICENCE NL-springer