Is there a point in revising mathematical concepts?

The title may be somewhat misleading depending on your definition of revision: I mean looking back on notes of key concepts, rather than practicing questions.

I'm developing a note taking system whereby I use a scrap book in class on which I scribble down thoughts and fit ideas together, after which I go home and neatly and concisely distil the key points onto a notebook that I go back to for revision of concepts, written in my own "mental language". This works as a memory jogging device in lieu of browsing through a textbook whenever I need a quick reminder.

I am planning to use this method for English Language and Physics, but my question is whether it would be an appropriate method for maths and further maths.

The reason for my doubts is that to me, mathematical concepts aren't difficult to remember once fully understood. Of course, that is only for AS Level maths, which is the only experience I have of it. (I am picking up further maths for year 13, which means 9 exams of it excluding M1 which I have already sat) But then you have chain rule and product rule and theorems and formulas in C3 C4 and FPwhatever, which could point towards some vague necessity of note taking. And then all the applied modules like D1 which I have to take and probably D2 as well since I'm looking at compsci for uni, all of which sound like they would need notes.

So my question, in its bare bones, is: is keeping a neat notebook of key concepts in maths worth spending 1 or 2 hours every day on, or would that time be more wisely spent practicing questions, obligating me to refer to a textbook/revision guide every time I need clarification? Please provide reasons. Thanks!

Reply 1

BTAnonymous

22

maths is slightly different to most subjects.

this year (yr 12) I only used my class notes when I struggled with revision. everything else was practice, practice annnnnd... practice. worked for GCSEs very well, i will have to wait for the 17th and evaluate the success of this method :smile:

Reply 2

K-Man_PhysCheM

17

I did the full maths A-level this year (and will be doing full FM next year). I can't speak for Decision/discrete maths, but certainly for pure, spending a long time on notes isn't a good idea imo. Once you understand a topic, practise questions. By virtue of practise, you will come to easily remember all the things you need to know anyway (like the trig identities, various calculus rules etc...).

Nevertheless, if may be worth writing your own "formula booklet" to refer back to if you forget something, and such a set of very condensed notes would be good for revision to make sure you know everything. But practise questions are key, and there's not that much to memorise anyway given the formula booklet. Become very familiar with the real formula booklet too, maybe write brackets in your notes around the formulae in the booklet.

Reply 3

MR1999

21

I wouldn't recommend taking notes beyond learning certain rules/results. The more difficult ones will be in the formula booklet, but the easiest way to learn something is to just keep practising it.

The only thing I would recommend taking notes for is definitions - these come up frequently in D1 and a little in C3. But even then, just answering questions on definitions will probably be enough.

Reply 4

DFranklin

18

Original post by Hubrillity

The reason for my doubts is that to me, mathematical concepts aren't difficult to remember once fully understood. Of course, that is only for AS Level maths, which is the only experience I have of it. (I am picking up further maths for year 13, which means 9 exams of it excluding M1 which I have already sat) But then you have chain rule and product rule and theorems and formulas in C3 C4 and FPwhatever, which could point towards some vague necessity of note taking. And then all the applied modules like D1 which I have to take and probably D2 as well since I'm looking at compsci for uni, all of which sound like they would need notes.

I'd say you're going to do better on concepts by doing maths rather than spending time on memorization. But it's hard to pick up all the actual formulas without some degree of memorization.

At the same time, the formulas you need are in the formula booklet, so to some extent it's more important to know the "concept of the formula" (*) than the actual details.

(*) e.g. one trig formula (now in FM only, I believe) is:

\cos a \, \cos b = \frac{1}{2} (\cos(a+b)+\cos(a-b))

. But the important concept here is "if you have a product of two cos terms, you can replace it with the sum of two cos terms". Because then if you see a place where it would be useful to do that, you know to look in the formula booklet to see how to do it. But if you didn't know the concept, it would never even occur to you to look at the booklet.

So my question, in its bare bones, is: is keeping a neat notebook of key concepts in maths worth spending 1 or 2 hours every day on, or would that time be more wisely spent practicing questions, obligating me to refer to a textbook/revision guide every time I need clarification? Please provide reasons. Thanks!

Despite the above, I'd say it will help (considerably) to learn the formulas from the booklet - there may be certain things where you decide "this is too hard to remember, I'll use the booklet" but in general it will make life easier to know them. But there's no way this should take 1 to 2 hours a day. 5-10 minutes, max.

Reply 5

Notnek

21

Original post by DFranklin

(*) e.g. one trig formula (now in FM only, I believe) is:

\cos a \, \cos b = \frac{1}{2} (\cos(a+b)+\cos(a-b))

. But the important concept here is "if you have a product of two cos terms, you can replace it with the sum of two cos terms". Because then if you see a place where it would be useful to do that, you know to look in the formula booklet to see how to do it. But if you didn't know the concept, it would never even occur to you to look at the booklet.

Yes this is so important. So many students think they'll be fine not learning e.g. the trig double angle formulae because they can be derived using formula book identities. Then often what happens is these students won't know that if they have

\cos 2x

for example then that can be written in terms of

\cos^2 x

or

\sin^2 x

.

Knowing how identities can be used is more important than knowing them off-by-heart in my opinion. And the best way to learn how identities can be used is to practice using them. So it all comes down to students putting in the effort to try as many questions as they can. I don't think I ever spent time learning the identities - they just stuck in my brain after doing loads of questions. Students who say they don't know the identities off-by-heart usually just haven't practiced using them enough.

Reply 6

Hubrillity

OP

9

Thanks a lot guys, I'm probably going to end up annotating the formula booklet with interesting ways of summarising the rule (like the sum of cos one) and then, whatever I end up struggling to remember, compile into a personal formula booklet for later reference. It's going to be a reactive note taking method as opposed to a pre-emptive one.

Thank you again, you've all been very helpful!

Reply 7

Physics Enemy

19

Back in the day there used to be essay Qs in maths exams, so your ideas on notes may have been useful.

Now not so much, besides you should have books with distilled notes. For STEP etc it may be handy, but there's material online.

I did read/think over summary notes from books nearer exam time. I'm not a fan of past paper regurgitation tho, monotonous.

(edited 6 years ago)

Reply 8

atsruser

11

Original post by Physics Enemy

Back in the day there used to be essay Qs in maths exams, so your ideas on notes may have been useful.

When was that? I can't remember such a time, but maybe it was syllabus specific?

Reply 9

Physics Enemy

19

Original post by atsruser

When was that? I can't remember such a time, but maybe it was syllabus specific?

In the 60s and possibly 70s, there were essay Qs on A-Level (and some degree) maths exams. Can't say if only certain boards did it. Shame they were ditched.

(edited 6 years ago)

Reply 10

atsruser

11

Original post by Physics Enemy

In the 60s and possibly 70s, there were essay Qs on A-Level (and some degree) maths exams. Can't say if only certain boards did it. Shame they were ditched.

I was studying A level maths at the end of the 70s, and was familiar with papers from Oxford and Cambridge, AEB, and London, and I don't think that any of them did it. It's possible that it happened in the 60s, I guess, but it sounds slightly mythological to me. The most likely possibility to my mind is that it was done under the aegis of the Nuffield stuff: they were long haired hippy types, IIRC, and would probably have thought it a good idea.

Reply 11

Physics Enemy

19

Original post by atsruser

I was studying A level maths at the end of the 70s, and was familiar with papers from Oxford and Cambridge, AEB, and London, and I don't think that any of them did it. It's possible that it happened in the 60s, I guess, but it sounds slightly mythological to me. The most likely possibility to my mind is that it was done under the aegis of the Nuffield stuff: they were long haired hippy types, IIRC, and would probably have thought it a good idea.

I saw the papers many years back on TSR, it was one question per paper, rest was hard maths. I don't see why it's hippy, the essay Qs were very stimulating.

Reply 12

atsruser

11

Original post by Physics Enemy

I saw the papers many years back on TSR, it was one question per paper, rest was hard maths. I don't see why it's hippy, the essay Qs were very stimulating.

I've seen old papers posted here and elsewhere, but I've never seen a standard A level maths paper with an essay question. Do you have a link? That would prove it once and for all.

The other possibility is that:

a) there were essay questions on the S papers of A level maths (I'm not sure)
b) you are thinking of A level physics, where there were certainly essay-style questions (or at least part questions) even in the standard papers.

As for hippy: there is too much actual maths to be learnt/done for mathematicians to be spending time writing essays. Even when they spend 100% of their time on maths, it isn't enough - there's just too much maths.

Reply 13

Physics Enemy

19

Original post by atsruser

I've seen old papers posted here and elsewhere, but I've never seen a standard A level maths paper with an essay question. Do you have a link? That would prove it once and for all.

The other possibility is that:

a) there were essay questions on the S papers of A level maths (I'm not sure)
b) you are thinking of A level physics, where there were certainly essay-style questions (or at least part questions) even in the standard papers.

As for hippy: there is too much actual maths to be learnt/done for mathematicians to be spending time writing essays. Even when they spend 100% of their time on maths, it isn't enough - there's just too much maths.

I don't have a link as it was over 10 years ago. Mr M or DFranklin might know. Definitely maths and most likely the A-Level, not Special. People still often do essays in their maths degrees nowadays tho rarely in an exam.

I think your idea of the maths essay is a bit off, it was quite specific and philosophical, not 'what do you like about Newton's work' etc.

(edited 6 years ago)

Is there a point in revising mathematical concepts?

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