torquetum | Entries tagged with vectors

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.

Miller-Young, J. (2010). How engineering students describe three-dimensional forces. In Proceedings of the Canadian Engineering Education Association.

Miller-Young, J.E. (2013). Calculations and expectations: How engineering students describe three-dimensional forces. The Canadian Journal for the Scholarship of Teaching and Learning 4(1), Article 4, 1-11.

I have grouped these two papers together since they are almost the same. The first is a 2010 conference paper and the second is a 2013 journal paper which includes all the 2010 work as well as a bit more data. The study was interested in digging into the details of how students visualise three dimensional statics problems when what they are presented with is a 2-d diagram. The data collected was students’ think-aloud processes of answering two questions, one without context and the other in a real-world context. The 2013 paper also included data on a quiz question which was part of a standard course assignment. All three problems required that the students see the page as the given vertical coordinate plane (xy in the three problems) and the third axis (z) extending out of the page in the positive direction. Points with a negative z-coordinate, in other words, are behind the plane of the page.

The students seemed to find the problems relatively difficult. The author found three main themes in student errors. (1) The students struggled to visualise points behind the plane of the page or vector which extended behind the plane of the page. The two-dimensional drawing on the flat page had to be visualised as a three-dimensional collection of vectors and the students found that particularly tricky for vectors extending backwards relative to their gaze. (2) The students did not always use the provided context to help them visualise the problems. One of the problems involved a pylon with guy ropes attaching to the ground, which was idealised as the flat xz-plane. All the ends of the guy ropes in this problem were on the xz-plane and had a y-coordinate of zero, yet some students struggled to see that. (3) The students reached too quickly for equations to try and answer questions even when there was not enough information to answer the question that way. The tendency to calculate something using a formula is ubiquitous across all maths and physics teaching and is no surprise. This final data point serves only to add to the depressing mountain of similar results.

Gray, G. L., Costanzo, F., Evans, D., Cornwell, P., Self, B., & Lane, J. L. (2005). The dynamics concept inventory assessment test: A progress report and some results. In Proceedings of the 2005 American Society for Engineering Education Annual Conference & Exposition.

The authors report on their progress in developing the Dynamics Concept Inventory, a MCQ format assessment of 30 questions on concepts used or needed by students in a mechanical engineering dynamics course. The process followed to achieve the final product was thorough, involving polling multiple lecturers of dynamics across several institutions, developing questions, piloting the instrument and going through various phases of refining the instrument. The DCI is available online by contacting the developers through the DCI website.

In this paper, the authors describe the process towards the development of the final version of the instrument and give a list of the concepts involved. They also provide much statistical evidence for the reliability and validity of the instrument. A few items on the test are pulled out for special scrutiny to illustrate clear evidence of misconceptions. The authors are clearly in favour of the test being used in pre-test/post-test format. Their website encourages this format and the DCI developers request that anyone using the test send them the raw data so that they can use the data to further verify the discriminatory power of the instrument.

It would be interesting to run the DCI on one of our cohorts of dynamics students and see if any of the results correlate with our vector assessment results.

Barniol, P., & Zavala, G. (2016). A tutorial worksheet to help students develop the ability to interpret the dot product as a projection. Eurasia Journal of Mathematics, Science & Technology Education, 12(9), 2387-2398.

Following their earlier work (2013, 2014) on determining frequent errors in vector problems, the authors developed a tutorial carefully designed to address the conceptual difficulties students experience with vector projections. The tutorial is presented in an Appendix to the paper. It consists of six sections, requiring the students to determine projections geometrically as well as using the mod(A)mod(B)cos(theta) definition, using a range of theta values. The final section of the tutorial explicitly addresses the observed confusion students experience between the scalar product and vector addition. The paper closes with an open invitation to teachers to use their tutorial. I am tempted to do just that.

De Laet, T., & De Schutter, J. (2013). Teaching inspires research and vice versa case study on matrix/vector methods for 3D kinematics. In Proceedings of the 41st SEFI Conference (pp. 1-8).

De Laet and De Schutter are robotics researchers and lecturers of 3D kinematics and statics. They observed that students struggle with the concepts and the notation of the subject and that their struggles were related to challenges roboticists experience with non-standardised coordinate representations and related software. They developed a semantics underlying the geometric relationships in 3D kinematics and a notation designed to make relationships clearer and eliminate errors experienced while working across different coordinate representations.

Neither kinematics nor robotics is a speciality of mine, so I might be phrasing my summary badly. I hope I’m correctly representing the work discussed here. The authors claim that their students have benefited from the new notation, making fewer errors than before, and that roboticists have also welcomed the new notation. I particularly liked two bits of this paper. The first bit is the explicit admission that engineers and engineering students need to be aware of the different terminology and notation which can exist across even closely related disciplines – “it is important that students are aware of the lack of standardisation and the implications this might have when reading textbooks or consulting literature” (p. 2) – which relates to my concern about vector notation. The second bit is the attention the authors pay to threshold concepts, which has long been a theory I have tried to apply to vectors, with little luck so far. Reading this paper has given me some new ideas, not least that I would probably enjoy a SEFI conference!

Barniol, P., & Zavala, G. (2014). Test of understanding of vectors: A reliable multiple-choice vector concept test. Physical Review Special Topics-Physics Education Research, 10(1), 010121-1-010121-14.

Barniol and Zavala describe a really nicely designed investigative project. In the first phase they conducted several studies over a period of four years, using open-ended problems in order to develop a taxonomy of frequent errors students make when solving vector problems. At the same time, they sought references in the literature to frequent vector errors. In the second phase, they developed a test in multiple choice format, named the "test of understanding of vectors" (TUV). They administered this test to over 400 physics students and thereafter observed the categories of errors and the frequencies of errors in different classes of problems.

I really admire the TUV, the preliminary work that went into designing it and the detailed analysis of the errors made. I feel the authors left the “so what?” question up to the reader, making a few minor suggestions about other people using the test in similar ways, but not making any broad assertions about teaching or learning or cognitive concept formation. I hope Barniol and Zavala have written further on this topic, as the work laid out in this paper is admirable and provides much food for thought.