Reflections on First Year Maths
I’ve been thinking about the Maths course in the Oxford Chemistry degree for about 15 years now, since I finished it in June 2008. There were important ways that it damaged my experience of learning Chemistry, and I wanted to use this blog as a reflection on my experiences. I understand that the course has changed since I did it, and that my own trajectory through my degree was not a universal experience.
To be very clear about this upfront, I feel the quality of point-of-delivery teaching on this course was high. At the same time, I feel the curriculum was poorly designed. My criticisms — developed at the end of the blog — are not of individuals but of structures.
Prior Learning
To situate my context, I did A Level Maths and AS Further Maths. The timetabling at my school meant that I had to teach myself the core Maths (two modules) for the second year of my Maths A Level. This probably made me better at independent study, but I think it also made me less confident in my abilities in the core content.
Curriculum Coverage
Looking back over my notes from first year, it seems like the Maths course covered:
Limits;
hyperbolic functions;
differentiation (including partial differentiation, implicit differentiation, stationary points);
complex numbers (including Argand diagrams, conjugates, Euler’s relation, De Moivre’s relationship, roots of complex numbers);
integration (definite integrals of odd and even functions, integration by substitution, integration by parts);
the Taylor series (including McLaurin, l’Hopital limits);
differential equations (including separable, homogenous, first- and second-order equations);
matrices (including determinants, Cramer’s rules, solving homogeneous equations, rotation matrices, eigenvalues and eigenvectors, orthogonal and inverse matrices); and
vectors (including dot products, cross products, unit vectors).
At some stage — which I suspect was just before Easter — I stopped making notes, so it’s hard for me to be sure of exactly what else we covered. I think we did partial fractions, perhaps? I remember being told that Fourier transforms used to be in the course but had been taken out (excitingly, though, when we had spectroscopy lectures it was assumed we had been taught Fourier transforms).
Assessment
The course was assessed in one unseen, calculator-free exam lasting 2.5h. Part A of the paper was eight 5-mark questions (no choice). Part B was three 20-mark questions from a choice of five.
I dug out the paper I sat. A few questions from each section are shown below, including one with an application of Maths to Physics. I feel the questions in the exam were generally easier than those I remember being set for classes, but perhaps this is a feature of my memory over-emphasising difficult parts of classes. Hard or easy, the questions were distant from chemical application.
5-mark questions from Part A of the 2008 paper
A 20-mark question from Part B of the 2008 paper.
Teaching
The didactic teaching was a series of hour-long lectures. I’m not sure how many there were, but the current Maths course has more lectures (46) than Inorganic Chemistry (40) in first year. In addition to this we had frequent classes - perhaps not weekly, but I think more than fortnightly - with a doctoral student. Maths took an *enormous* amount of time in my first year, mostly because of the work we had to do for classes.
My classes were streamed: I was with other people who had not done the full Further Maths A Level. Whatever the pedagogic merits of this approach, it left me with the impression that having Further Maths was a distinct advantage in first year. I believe this was a correct impression; if you could sail through the Maths course you freed up time to learn Chemistry.
Inferring the purpose of the Maths course
Broadly, I would characterise this Maths syllabus as highly theoretical: it was a course in x and y and z, not p and V and T. Chemical examples were very rare, though there were a few (a derivation of a Maxwell relation from my notes is shown below). It was also focused on exposure to material rather than mastering mathematical operations.
A Chemical application of first-year calculus from my notes (page 1)
A Chemical application of first-year calculus from my notes (page 2)
The theoretical flavour of the course coupled with the way that lots of the Maths never came up again later led me to conclude that the dominant rationale of the first year Maths course was really to prepare the most-able students for an optional module in advanced quantum mechanics in second year. Everyone had to do the Maths course so that a subset of the cohort could do one specific module if they wanted to. Others may infer a different purpose, but this conclusion is one I have sustained for over a decade of reflection (including five years as a lecturer in the same Department).
Criticism
To repeat: I feel that my teachers did a good job of teaching the material they set out to teach within the constraints of the curriculum. In no way am I seeking to criticise the quality of explanation in lectures or classes, for example. Nevertheless, I feel that the curriculum was poorly designed. I have three major points of criticism, which I’ve set out as questions.
Who is the course for?
Students who wanted to do enough Maths to enjoy the core Chemistry course were poorly served by a curriculum ambition to push the most-able towards advanced theoretical work. The pace of study meant that the ‘skills and drills’ of carrying out mathematical procedures were not sufficiently emphasised for people like me who needed more time to grasp ideas. Despite enduring a rigorous Maths course, I remained (and still remain) under-confident in the routine mathematics of quantum mechanics or the matrix multiplications of Group Theory.
How much time should students spend studying?
Perhaps this is not a criticism of Maths in itself, but rather the broader curriculum. The University states that it expects students to work on their academic studies for 40h/week. I was figuring out how to structure my time in first year, and judge that I worked 50h (lots of this was the trial and error of figuring out how to study). I don’t think many of my students work less than 50h/week, and I believe that most work 55-60h/week. All of these numbers are too high.
Maths is particularly salient in a workload discussion because (i) it happens in first year when students are working out how to study; and (ii) the workload is perceived as being significantly lighter for students who did Further Maths at School. Whether you frame this using the Hidden Curriculum (really, we expect you to have Further Maths), survivorship bias (a professor of Chemistry saying “well my first year workload was fine”), or socioeconomic advantage (really, we expect you to have gone to a school which offers Further Maths), it is hard to see this as inclusive design. The resulting sense of panic and overwhelm which first year students experience shuts down the scope for deep learning, and is inequitably distributed. This should trouble reflective educators who have high ambitions for all of their students.
On an affective (rather than cognitive) note, I personally felt that the construction of first year destroyed my excitement about Chemistry. From second year onwards, I decided to try and do well rather than enjoy my subject. It was not possible to do both, at least not for me.
How does Maths link to (Physical) Chemistry?
The course demonstrated remarkably little urgency to support the Chemistry lectures; I believe we had two weeks of incomprehensible thermodynamics lectures before we were taught sufficient multivariate calculus to follow the mathematical reasoning (Why is that ‘d’ so curvy? Is that just his handwriting?).
This is symbolic of a broader theme in the curriculum I experienced: the jump from pure Maths to applied Maths was largely missing (I guess this will perhaps always be true of any general Maths course). I found it hard to activate the pure Maths I had learned, even when it was directly relevant to the Chemistry.
Conclusion and Recommendations
My central reflection is that I think I was taught Maths at the direct expense of learning Chemistry. The impetus for writing this blog post came from seeing a similar idea written more funnily (below).
A light-hearted tweet touching on some of the opportunity costs of studying Maths.
I sincerely believe I would have been a significantly better Chemistry student if I’d had a different experience of first year Maths. Perhaps my criticism is simply about the limitations of one student in 2008. On the other hand, maybe a curriculum where this is happening could benefit from review.
It may be that some of these suggestions have been addressed since I did the course, but I think there are a few specific actions which would have made my experience of first year Maths much better.
Who is the course for?
Asking — explicitly — who the course is intending to support would have been worth doing at some length. In particular, more ‘skills and drills’ work may be helpful for students whose experiences of learning Maths in school were built around these pedagogies.
There are some interesting resonances here with the current course: I understand that the proposed curriculum review will remove the optional quantum mechanics module, meaning that the context of the Maths course would change significantly. There will be an exciting chance for the Department to reimagine the course if it wishes, for example by thinking about students who ‘only’ have A Level Maths more deliberately.
There are additive questions here, as well as subtractive ones. We might consider training students in statistics, for example, if their main professional Maths is going to be calculating the yields of reactions (Maths is for Organic chemists, too).
How much time should students spend studying?
To ask what is necessary rather than what would be nice to have is surely the only way to rein back student workloads - it would always be nice to have more. To focus minds on the problem of student workload, one approach would have been to reduce the number of hours (lectures and classes) students spend on Maths in first year (perhaps by 30%), or fencing off substantial time for skills mastery rather than covering new material.
How does Maths link to (Physical) Chemistry?
The standard curriculum response to issues of linking is to constructively align the teaching, assessment, and intention of the course. To articulate and agree the purpose of the Maths course would be a useful start here. Is it to advance abstract reasoning? Or is it to ensure students can normalise a wavefunction?
Sequencing ‘just in time’ Maths before key PhysChem courses might be one way of aligning intention and teaching. If your Thermodynamics course needs partial differentiation, you could schedule this Maths as a sub-block of teaching within the Thermodynamics course. This approach might also help preserve objects like the advanced quantum mechanics material: students needing the advanced Maths would study it as it became relevant to them.
Change one thing?
But if I could wave a magic wand and change just one thing about the curriculum I experienced, I would personally remove the Maths exam. If we truly need Maths for Chemistry, then Maths will necessarily be assessed when we assess Chemistry.