Subspace in linear algebra: investigating students’ concept images and interactions with the formal definition

Abstract

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][152][153][154][155][156][157][158][159][160][161][162][163][164][165][166][167][168][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.

Key Mathematical Concepts in the Transition from Secondary School to University

“…Working with modelling and APOS frameworks Possani et al (2010) leveraged students' intuitive ways of thinking through a genetic composition of linear independence and systems of equations. Student use of different modes of representation in making sense of the formal notion of subspace was analysed by Wawro, Sweeney and Rabin (2011a), and their results suggest that in generating explanations for the definition, students rely on their intuitive understandings of subspace, which can be problematic but can also help develop a more comprehensive understanding of subspace.…”

Students’ Use of Computational Thinking in Linear Algebra

“…However, the question remains, why did these students use Computational thinking so much more than those in our pilot study (Wawro et al 2011)? Both groups of students took the same course, with the same textbook, and were interviewed at the end of the class.…”

“…We had included this problem as one among many interview questions in an earlier pilot project conducted near the end of a previous year's iteration of the same honors linear algebra course (Wawro et al 2011). Based on the responses at that time, we determined that the task has the potential to elicit all three modes of thinking: computational, geometric, and abstract.…”

Editorial: Focused Collection: PER in Upper-Division Physics Courses

“…Particularly promising are the opportunities for collaboration between PER and researchers from math education, especially the Research in Undergraduate Math Education (RUME) community. While there is a long history of research on student thinking in calculus [50], differential equations [51], linear algebra [52], and statistics [53], there has only recently been attention to this work among PER researchers. There have been a handful of collaborations between PER and RUME researchers [54], including some at the upper-division level [55], but we believe more work in this area would be fruitful.…”