In what ways could A levels Math, Further Math, and Physics be improved?

I thought of starting this discussion since I’ve reflected on my overall A-level experience recently.

I have a few ideas myself. Feel free to give your opinion on these (and post your own ideas too!):

  1. A level Physics should require students to use more advanced mathematics than it has currently*. Calculus, for example, is simply so essential to physics at that stage that this has to be made up by having Mechanics modules in A level Math. Honestly, if someone wants to take physics without math, they are probably making the wrong choice (biology may be better option).

  2. Adding to my previous point, Mechanics should be ditched from A Level Mathematics, instead being in A level Physics, along with all the relevant experiments that really helps in understanding the concepts, which A level math and Further Math lacks.

  3. Further math is intended for those who wish to specialize in mathematics and other fields requiring high levels of mathematics. So shouldn’t it have a much higher emphasis on proof? (and have topics such as real analysis, group theory, set theory, and logic). Probability would be covered axiomatically but no things like statistical tests. (proper) linear algebra and graph theory should appear here too. Time could also be spent on formalizing the content of A level Math.

  4. Even better, instead of Further math, providing a double award A level Math intended for those who wish to specialize in mathematics and other fields requiring high levels of mathematics, with the emphasis on proof as stated above. In this way, content from A level math is covered with more rigor too. Students should be able to drop to standard A level math, if they wish.

  5. Some discrete math should replace the mechanics in A level Math, but done properly! (unlike the current decision math). I think it would benefit students to do some number theory, too.

  6. Some new (and better) options for FM, such as computational mathematics, non-Euclidian geometry, etc.

*STEP could then have question on different areas of physics (with a mathematical focus, of course).

1 Like
  1. The problem is that they want to make physics accessible to “everyone”, even if they have no intention to pursue physics in the future. (since A-level maths is an absolute prerequisite for any physics course without a foundation year) I disagree with this approach personally but I understand that it is done to ensure people aren’t scared off from physics (instead, we push back the scare until university: https://www.youtube.com/watch?v=Ah4DnxAJQ44)

  2. I’m not sure, though I don’t think it’s very important for people not intending to go into theoretical physics, people seem to view it as a necessary part of the curriculum and some university courses are probably built on the premise that students will have at least some exposure to classical mechanics. So this may be a tough one.

  3. This’d be great but I’d disagree with no statistical tests (they can be covered in a way that’s not plug and chug) - but I don’t think students would be well enough equipped to make the jump to university style maths this early. If 18/19 year olds struggle then most 16 year olds will have no chance.

  4. Ditto 3

  5. I have no problem doing discrete maths heavy on computation. What is absolutely 100% useless is doing these calculations by hand with no automation in sight. There is no circumstance whatsoever where Dijkstra’s or a quicksort will be performed by hand outside an A-level exam. What you should learn is how to practically implement these algorithms in python. (or similar) Then, you don’t only learn the maths behind these algorithms but you learn how they might be useful in practice. As it stands, Decision/Discrete maths in A-level is mostly useless and favoured by schools to inflate grades. (why it was removed from A-level normal maths) There is limited discussion of theory, and this should be made to make up more of the A-level. The module I described actually existed, kind of with MEI Decision Maths Computation: https://mei.org.uk/files/papers/DC_2018_Jue.pdf. But it looks like you were just using an Excel spreadsheet, which is a bit of a joke.

  6. Computational mathematics exists with MEI’s “Further Pure with Technology” module. It is a bit limited but exists nonetheless https://www.ocr.org.uk/Images/308778-unit-y436-further-pure-with-technology-sample-assessment-material.pdf. Some questions are expected to be answered with Python or similar.

STEP used to exist in more subjects than just maths. There used to be STEP physics and STEP chemistry, but few papers have yet been digitised and are locked in Cambridge’s archives, accessible for a fee: https://www.whatdotheyknow.com/request/step_physics_and_chemistry_paper. The 1981 STEP physics paper is available here: https://telescoper.wordpress.com/2012/02/27/cambridge-entrance-examination-physics-1981/ and the 1983 paper here: http://www.mathshelper.co.uk/Cambridge%20Admissions%20Tests%201983.pdf. (I guess Mr(s). Cleanslate can add these to the backlog if (s)he wishes!)

2 Likes

Thanks for the answer I’ll give in some thought.

Okay, here are some revisions.

  1. What about having two different A level Physics, with one having a higher mathematical demand as described in my first post and one similar to the current one?

  2. Not really sure either. I kind of want to say that all uni math applicants should take A level physics but people take only 3 or 4 A levels so I’m not sure.

  3. If statistical tests can be covered in a more interesting way then there is no reason to remove them. I personally struggle with “plug and chug” and never really understood statistical tests due to that. This is turn made me avoid statistics in further math.

I’ll have a look at the STEP physics papers.

1 Like
  1. Yeah this’d be good but then you’d have the issue of what teachers could teach it. I think unfortunately many physics teachers are shaky on their mathematical foundations as it is. I’ve not done any physics at uni so I’m not sure what could exactly could be implemented anyways.

  2. Don’t think so personally - maths and physics A-levels are almost made to be mostly unrelated and nothing in physics A-level will really help with maths at uni unless you decide to pursue physics modules. Any overlap will be seen in mechanics. (which happened to be the bit of theoretical physics I was the least interested in lol) Taking something like CS/chemistry or even something completely different like history is fine.

  3. I always found tests fairly intuitive personally but I never really followed the A-level textbook closely. The problem I see is that they often rely on a statistic having a certain distribution, a fact used without proof, (because A-level students just don’t have the background at that point, and it’d be a significant distraction introducing all the necessary concepts and lemmata) meaning the tests mostly come to “compute this statistic, compare to critical value, reject/accept” which is pretty boring. The formalism behind hypothesis tests (including the idea that hypotheses and significance levels are arbitrary and can be optimised for the results you want) is explored a bit in FS1 [with size/power/Type I and Type II errors etc.] but generally you don’t see a full formalism for tests until a second year statistics module. (I’m self-studying one at the moment) They’ve tried to combat this a little bit in the new spec and tried to make it less plug and chug, I think they’ve done well tbf.

As I’ve said before, central limit theorem isn’t even treated with a proper statement in FS1!

1 Like

You can tell that my GCSE physics (and more concerningly math) teachers didn’t know much math at all!

Before I found out formal maths and proof, I was planning to study physics at uni. But by the time I was in year 13, I already decided to study math and lost my interest in physics (also realizing I had a strong preference on classical rather than modern physics). Since I was going beyond the A level syllabus in all the subjects I took (except chemistry*), I had a better (but not perfect) idea if what the subjects are at uni than most.

I did these courses on top of A level physics, which are calculus based:

https://apstudents.collegeboard.org/courses/ap-physics-c-electricity-and-magnetism

They were really interesting courses. The mechanics course felt a bit like the mechanics in A level math and FM at times, but with the addition of experimental work.

As for math, I think this course is excellent in introducing formal mathematics to high school students.


But overall I realize I am most likely an outlier in saying that I would have highly benefited to doing maths university style from a younger age.

*dropped chemistry once I decided to do math, as I found it unnecessary. Frankly, the amount of memorization required was overwhelming for me anyway.

1 Like

Have pinned this discussion :grin:

1 Like