The findings revealed that most of the students understood a variable as something whose value is unknown, whilst some other students comprehend a variable as a substitute for something whose value is unknown. Participants involved in this study were ten eighth-grade students, while the object in this study was students’ concept image in understanding the variable concept. Data were collected by means of interviews and tests, then analyzed using the content analysis method. This study is qualitative research with a phenomenological method. This study aims to analyze and describe the concept image of students in understanding the variable concept. The concept image develops continuously through experience and changes as the individual faces stimulation and maturity. After shifting from two to three dimensions, students symbolized all possible linear combinations with a parametric vector equation and reasoned about how this could represent particular points on the corresponding lines or planes.Ĭoncept image is the overall structure associated with a concept, including all mental images and their related properties and processes.

We found students' use of sliders supported their reasoning about individual linear combinations the trace function and slider animation supported their intuition for all possible linear combinations of vectors. We analyzed these with a lens of mathematization, with a particular focus on tool use. Data sources come from six individual task-based interviews with two linear algebra students. Considering tools and functions of a digital environment (specifically of GeoGebra) with design heuristics of Realistic Mathematics Education, we introduce four tasks (in R 2 and R 3) and results of pilot studies. In this paper, we focus on students' symbolizing activity and mathematization relating to linear combination and span in the context of a task sequence designed with digital tools. 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.

Subspace definition vector