torquetum

Stewart, S. & Thomas, M.O.J. (2009). A framework for mathematical thinking: the case of linear algebra. International Journal of Mathematical Education in Science and Technology, 40(7), 951-961.

This paper presents a framework combining two educational theories to both assist teaching and to understand student learning. One theory is Dubinsky’s APOS (action – process – object – schema) theory and the other is Tall’s three worlds of mathematics (embodied – symbolic – formal). The authors present an array where the segments of the two theories are set orthogonal to one another and, for illustration, two concepts are broken down across the array, that of adding two vectors, and multiplying a vector with a scalar.

The students whose work is analysed in the paper volunteered to participate in the study and attended supplemental tutorial classes. The topic of interest was linear algebra, specifically certain sub topics such as scalar multiplication, basis, linear independence, eigenvectors and so forth. The authors suggest that, as a teacher, one can use the array formed by the two theoretical categorisations to represent and explain a concept in a variety of ways, hopefully leading to deeper understanding. The students in the study completed six exercises designed to investigate their conceptual understanding (rather than procedural fluency). Overall student responses revealed primarily a symbolic understanding of the concepts (at both action and process levels). Embodied understanding (Tall) was not particularly evident in the data, nor understanding of the concept as object (Dubinsky). The authors point out the apparent contradiction with the assumption that the worlds in Tall’s theory are hierarchical. If so, one would expect an embodied understanding to precede symbolic, which is not evident in the data. They suggest that, distinct from Tall’s ideal model, in the real world the instructor may teach entirely symbolically and leave embodied understanding to be constructed by the student without instruction. The authors close by suggesting that instructors teach concepts from an embodied point of view as well as symbolic in order to enhance students’ understanding an enrich their representations.

In my studies of the historical development of vector analysis and its associated notation, it is interesting that vectors were seen from an embodied and object point of view long before they were well formed in symbolic notation or able to be manipulated in formal modes. The question of how to symbolically represent directed line segments in such a way that they could be added or scaled or multiplied was a sticky problem that occupied some great mathematical minds. The fact that, today, students can make the error discussed on page 955, that of scaling a vector incorrectly by misuse of the component form of the scalar product would be extremely strange to the early developers of vector analysis. It is almost as if the mathematical world has moved from embodied to symbolic to formal and the novice students of today are stranded at the end of a road they have not themselves travelled. Embodied did indeed precede symbolic, just not in the person of the individual student but in the historical development of the concept itself.

One final point of personal interest: I enjoyed the parallels between Tall’s three worlds and Piaget’s models of abstraction. The embodied world is kin to Piaget’s empirical abstraction and the other two worlds are kin to Piaget’s reflective abstraction, something which he in the original French broke down into different types of reflection (see von Glasersfeld, 1991).

Do not treat this blog entry as a replacement for reading the paper. This blog post represents the understanding and opinions of Torquetum only and could contain errors, misunderstandings or subjective views.