torquetum | Entries tagged with concepts

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.

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.

van Dyke, F., Malloy, E.J. and Stallings, V. (2014). An activity to encourage writing in mathematics. Canadian Journal of Science, Mathematics and Technology Education, 14(4), 371-387.

The authors ran a very interesting study in three stages. The first stage was a short assessment of three MC questions involving the relationships between equations and their representative graphs, the first two only linear, the third linear and quadratic. The questions were quickly answered and easily graded. The second stage was giving summaries of student responses back to the class for discussion (not led by the lecturer). Directly after discussion the students were asked to write about the questions. One group was asked to write about the underlying concepts necessary to answer the questions correctly. The second group was asked to write about why students might have given incorrect answers. These written responses were also evaluated. The third stage of the study was to ask the students to answer a survey (strongly agree/disagree Likert style) testing hypotheses developed during the first two stages. The findings are interesting and point, yet again, to the student tendency to want to do calculations even if the question might not require them - “blind attraction to processes” (p. 379) - and also to the expectation that similar problems should have been encountered before. Interestingly, the students in the second writing group wrote more than those in the first, but did not make many references to actual underlying concepts. The authors stress that if you want students to write or talk about underlying concepts you need to make that explicit.

The authors present the design of this study as a way of using writing to encourage reflection without it taking a lot of time or being difficult to grade. I agree and would like to try this myself. Running effective writing assignments in a maths class can be very hard to get right. The authors make reference to cognitive conflict and how resolving a cognitive conflict can lead to cognitive growth. “It is not the intent of this article to explore the efficacy of using writing or conflict resolution in the mathematics classroom but to take that as given …” (p. 373).

Engelbrecht, J., Bergsten, C. & Kågesten, O. (2012) Conceptual and procedural approaches to mathematics in the engineering curriculum: student conceptions and performance. Journal of Engineering Education, 101(1), 138-162.

The authors develop an instrument to measure performance, confidence and familiarity with both procedural and conceptual problems in mathematics. The students were second-year engineering students in two institutions in two countries – South Africa and Sweden. The authors provide definitions of the relevant terms and take issue with some education literature using terms like “conceptual” and “knowledge” too loosely and conflating them with other terms. The paper presents detailed data and analysis, finding differences and similarities across different groups (read the paper for details), concluding that “the use of mathematics in other subjects within engineering education can be experienced differently by students from different institutions indicating that the same type of education can handle the application of mathematics in different ways at different institutions.” (p. 158/9)

Leppävirta, J. (2011). The impact of mathematics anxiety on the performance of students of electromagnetics. Journal of Engineering Education, 100(3), 424-443.

The author investigated the relationship between mathematical anxiety and performance in an electromagnetics course. There is a literature review of studies on mathematics anxiety showing, in general, that there is a correlation between high anxiety and poor performance. The causal relationship tends to be less clear, however, although there is some evidence to show that poor prior performance leads to higher anxiety which in turns impacts negatively on performance in procedural tasks. Two maths anxiety scales are discusses, the Fennema-Sherman scale and the MARS scale. Those scales and others were adapted to make the Electromagnetics Mathematics Anxiety Rating Scale (EMARS) which was used in this study. The scale had several subscales which measured perceived usefulness of the course, confidence, interpretation anxiety, fear of asking for help, and persistency. The data and results are discussed in some detail. Conclusions include that high anxiety students perform less well in procedural work than low anxiety students, but that conceptual performance is less clearly aligned with anxiety. In addition, high anxiety students felt less confident about their maths ability and also self-describe as being less persistent in solving mathematical problems. The authors close with the suggestion that assessment should be more aligned with conceptual understanding rather than procedural processes.

Moore, R.C. (1994). Making the transition to formal proof. Educational Studies in Mathematics, 27(3), 249-266.

From the abstract: “This study examined the cognitive difficulties that university students experience in learning to do formal mathematical proofs... An inductive analysis of the data revealed three major sources of the students' difficulties: (a) conceptual understanding, (b) mathematical language and notation, and (c) getting started on a proof. Also, the students' perceptions of mathematics and proof influenced their proof writing” (p. 249). The author did a grounded study of students’ difficulties with proof, by observing classroom activities and interviewing instructors and students. The author found seven major difficulties (p. 251-252):

The students did not know the definitions.

The students had little intuitive understanding of the concepts.

The students' concept images were inadequate for doing the proofs.

The students were unable, or unwilling, to generate and use their own examples.

The students did not know how to use definitions to obtain the overall structure of proofs.

The students were unable to understand and use mathematical language and notation.

The students did not know how to begin proofs.

The author explores these in greater detail, particularly concept image and concept definition, using definitions and generating examples. He finds that students need to develop concept images through diagrams, examples and graphs before being able to move to symbolic definitions and manipulations. Also, the students’ understanding of a concept seemed to be cognitively separated into different schemata, whereas the instructor had the entirely of the knowledge needed for the proof incorporated into a single schema. This makes working through a proof have a greater cognitive load for students than for the instructor. All the various difficulties seem to combine to make the starting of a proof really difficult. The author recommends a less abrupt transition to proof in the education system and perhaps a transition course on mathematical language and proof before doing courses which heavily depend on a knowledge of proof.