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Vectors - Zavala and Barniol, 2013

Zavala, G., & Barniol, P. (2013). Students’ understanding of dot product as a projection in no-context, work and electric flux problems. In AIP Conference Proceedings (Vol. 1513, No. 1). American Institute of Physics. American Institute of Physics, Melville NY United States.

Zavala and Barniol ran a project investigating students’ understanding of the projection role of the dot product. First they gave three isomorphic problems to a class of physics students, with approximately 140 students seeing each of the three problems. The following semester they interviewed 14 of these students, who solved the three problems (and two more) while thinking aloud. The three problems all involved the same arrangement of vectors requiring the same projection, however one was no-context and the others were in context, namely work and electric flux. The investigation found that the students had a weakly constructed concept of the projection role of the dot product. The students were more likely to answer the question correctly in the contextualised problems than in the no-context problem, however even at best only 39% of the students answered correctly.

The authors observe that a majority of the students chose one of the two responses which described scalar quantities, rather than the four other MCQ options which described vector quantities. However in the no-context problem that majority is a disappointingly low 57%. Problematically, I would use the term “projection” differently to Zavala and Barniol, where projection of a vector onto another vector is still a vector, not a scalar quantity. The projection of A onto B in my lexicon is the vector component of A in the direction of B. Zavala and Barniol mean by projection what we elsewhere (Craig and Cleote, 2015) have referred to as “the amount” of A in the direction of B (p. 22) . So, given my definition, there is only one available MCQ option which describes a scalar quantity (option a, a popular incorrect option). I have to assume that the students participating in the study were familiar with the authors’ definition, however, and would have seen that MCQ option as describing a scalar quantity.

The authors cite other work reporting students’ difficulties in connecting concepts and formal representations. They see this dot product projection difficulty as part of that more general situation. “In this article we demonstrate that this failure to make connections is very serious with regard to dot product projection’s formal representation” (p. 4).

Not much has been written on students’ difficulties with the dot product. It is likely that the computational simplicity of the product masks the conceptual challenge of the geometric interpretation.

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