Where did all the calculus go?

Taken from the September 2021 issue of Physics World. Members of the Institute of Physics can enjoy the full issue via the Physics World app.

Niki Bell argues that mathematics A-level could be reformed so that it does more to support physics students

Specialized A redesigned A-level maths could focus on the needs of physics and non-physics students. (Courtesy: iStock/SolStock)

Early in my teaching career, I had a rather uncomfortable exchange with a retired physicist. He challenged me to defend the A-level physics curriculum, which he thought had been “dumbed down” and lacked any solid, mathematical rigour. I regret not putting my thoughts across better at the time, but – with the benefit of almost 15 years teaching experience behind me – I now feel more prepared to respond.

I should add that this physicist’s views were not a one-off. Since then, I have had many similar conversations with other university physics lecturers who mourn the lack of mathematical fluency in their first-year undergraduate students. I should point out that I am a huge advocate of mathematics in physics and believe that mathematics is in fact the more important subject, given that it is at the core of new discoveries.

I was surprised to discover that the inclusion of A-level maths content in physics specifications, or syllabuses, is a more recent affair than I had anticipated. It appears to have emerged in the mid-1980s in response to changes to the maths curriculum, which previously had not been standardized. Instead, there were small but significant variations between the A-level maths syllabuses offered by different exam boards. Unfortunately, attempts to remove the differences ended up having a knock-on effect for physics.

Back in the 1970s students benefited from studying some physics topics both in their maths A-level course and in their physics A-level. This gave students more opportunity to apply their learning as well as additional contact time with teachers to perfect these skills. Universities were therefore reluctant to accept physics students who had studied maths A-level courses that featured little or no physics. Admissions tutors instead preferred to accept students who had benefited from this “double study”.

But when maths A-levels were standardized, physics content was removed to make way for other topics, such as probability and statistics, that are important in the social sciences. The reduction of physics content from the maths curriculum now meant there was less overlap between A-level physics and A-level maths. With less opportunity for those taking physics to refine their skills, physics specifications appear to have coped by increasing the onus on the student to study the material independently. Students studying the two subjects were simply no longer getting the rounded experience that they previously had received.

The uptake of physics A-level dropped over the next decade and physics went from being the most popular science to the least popular. One of the reasons suggested for the low uptake was the perception that physics was disproportionately difficult, which is supported by an analysis of student grades. Bright students performed worse in physics than they did in other subjects, which is off-putting when a student needs to strategically consider their subject choices for applying to university.

A good compromise

We need to accept that there are many other reasons to take physics A-level, besides the desire to study the subject at university. The skills it provides are useful to many other fields, which can lead to engaged, scientifically literate citizens. So it is in our interest to be as inclusive as possible. At A-level, teachers cater for a wide range of career paths, not just those progressing on to a physics degree. We deliver the subject in a way that is attractive and useful to both sets of students and accept that not all A-level physics students will be studying A-level maths. One consequence of adjusting for this balance can be a loss in mathematical rigour, particularly when it comes to notation.

The skills A-level physics provides are useful to other fields, and can lead to engaged, scientifically literate citizens

Niki Bell

It does not help either that when students get to university, lecturers often use a vastly different mathematical notation from what students were taught at school. In fact, lecturers often do not adjust their presentation of mathematics at all to accommodate students’ varying experience. This can be incredibly intimidating, and I often wish that lecturers would adapt their teaching to be more inclusive.

Maths for physics

I believe that the solution lies in recombining A-level maths and physics, by redesigning maths A-level so that students can specialize in their second year. Students do not typically get much opportunity to choose their modules in A-level, which are decided by the exam board or school leaders. But if this were changed, then those wanting to do a physics degree could select “maths for physics” in their second year of maths A-level, allowing them to focus on concepts such as mechanics and applied calculus. Those wanting to specialize in other subject areas would be able to select more appropriate mathematics, such as probability and statistics. This would mean that those not studying maths can still do physics if they wish. Redesigning the maths course this way would reintroduce the overlap between the subjects and give students more opportunity to develop skills in class, not just for physics, but for every field.

I acknowledge that there are practical problems with this model, especially for small colleges that do not have the same staff numbers as larger ones, but I believe the potential benefits are worth it. I have taught on such a course and can attest to its success. Teaching physics to students on a specialized maths course was incredibly fulfilling. Their mathematical ability was excellent, the students were confident and had ample opportunity to hone their subject-specific skills, and those progressing on to study physics at university were well prepared for the mathematical content. Achieving this was not easy. Making the most of the overlap, and creating that experience, took careful management and negotiation between departments, but outcomes for students were worth it.

I believe this is the best way forward, not only for physics students, but maths students too.

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