# Derivatives and Concept Images — a guest post by Kirsty Dunnett

Another guest post by Kirsty Dunnett about the difficulty of applying skills from a maths course in the context of geoscience courses, and what can be done to make it easier. Thanks for writing, Kirsty :-)

A not uncommon complaint of university teachers in at least physics and geosciences is that students cannot apply what they learnt in their maths courses when it appears in another context, so they have to re-teach the maths at the expense of being able to teach the disciplinary material. That is, stereotypically, how the story goes.

A few weeks ago, I attended a seminar where, after 25 years of curriculum development work, the conclusion seems to have been that it might be no bad idea if maths were taught with rather more attention to physical meaning(s). Their context is calculus, with particular focus on the meaning of derivatives, those pesky dx’s, and ∂x’s that sometimes look like fractions and are sometimes near a strange, elongated s (or ). The question that arises, is what the heck does ‘d’ – in whichever style, and followed by any given letter from any alphabet – mean?

And this is where the difference between disciplinary approaches becomes apparent, which Dray and colleagues have explored in several papers, two of which I read in writing this [1,2]. In the first paper [1], they report on how mathematicians understand that a derivative is obtained via an exact limit process, while physicists and engineers are perfectly happy to collect numerical data resulting from changing one of two variables while keeping the other constant and, by carefully ensuring that the changes are small, ‘measure’ the derivative (or a good approximation thereunto).

What this points to is both a problem and a solution: formally, a derivative may be the rate of change of a function at any point/time (as defined by mathematicians), but in any sort of physical context, the derivative is the instantaneous rate of change in the limit where the denominator in the ratio of changes goes to zero. In practice, for both experimental and numerical data that are always discrete, this means that one varies one variable by small [enough] amounts, while avoiding the extreme of impracticably tedious repetition.

However, the go further and consider how this manifests in different representations of – or ways of presenting – the derivative. They have developed [2] and use [1] a theoretical framework for the derivative that involves five representations — graphical, verbal, symbolic, numerical and physical — that together form a “thick derivative”. This thick derivative involves a conceptual understanding of what a derivative is in a variety of contexts. This, they argue, is what introductory calculus courses should try to enable students to learn.

The second paper [2] (which actually pre-dates the empirical discussion described above) adds just a little more to this. Specifically, in setting up the theoretical framework for the concept of the derivative the underlying idea is that this “thick derivative” is a “concept image”. A concept image is ‘the set of properties associated with a concept together with mental pictures of the concept’. It appears that this ides of a “concept image” is currently restricted to mathematics education, and is most commonly seen in connection to derivatives.

It seems to me that the idea of “concept images” could be useful or meaningful for considering student learning of almost any topic. As I understand the concept(!), concept images incorporate all ways of knowing and implicitly recognise that each student will have their own experiences contributing to the set of properties they associate with any given concept. A concept image has space for conceptual and applied knowledge, formalised and non-formalised knowledge, and makes no particular judgement about whether the elements need to be coherent, without contradiction or even, according to the accepted best-knowledge, ‘correct’. Thus learning could be thought of as being about developing, “thickening” and refining, the concept images of a variety of (often connected) topics.

Approaches such as the spiral curriculum, and discussions around continuity through a course or programme could perhaps then be seen as explicit structures through which the concept images of different topics are developed over an extended period of time. Syllabi and learning outcomes may then often be proposed additions to existing concept images, and the introduction of new ones when the topic is sufficiently distinct. As with so many things that seem to have promise for thinking about learning, how to assess or evaluate one’s students is not obvious. I guess I’ll have to think about it.

[1] Tevian Dray, Elizabeth Gire, Mary Bridget Kustusch, Corinne A. Manogue & David Roundy (2019) Interpreting Derivatives, PRIMUS, 29:8, 830-850, https://doi.org/10.1080/10511970.2018.1532934

[2] Roundy, D., Dray, T., Manogue, C. A., Wagner, J. F. & Weber, E. (2015). An extended theoretical framework for the concept of the derivative. In T. Fukawa-Connolly, N. E. Infante, K. Keene, & M. Zandieh (Eds.), Proceedings of the Eighteenth Annual Conference on Research in Undergraduate Mathematics Education (pp. 919-924). Pittsburgh, PA. https://www.exhibit.xavier.edu/mathematics_faculty/5/