Should proof form a larger part of the Maths A-levels

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_gcx
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#1
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#1
Proof is inseparable from pure maths and appears more in applied maths than one might initially expect, (applied certainly requires the same sort of creativity) yet at A-level it's presented as a separate topic, and you have plenty of students saying they "don't like proof"! Some students are shocked to find that university maths isn't what they wanted and find themselves transferring onto other degrees or taking more modules away from maths.

Should proof form a larger part of the A-level?
Is it right that proof is separated as a topic - should it come up more organically?

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tonyiptony
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#2
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#2
Not directly answering the question, but I'd like to know why people don't like proofs.
Is it "too rigorous"? Is it "too hard to start a proof"?
(Note: of course it's hard lmao, maths is hard, but that's why I like it!)

I'm going off another not-answering-the-question tangent here.
While I did not study GCE, Hong Kong maths curriculum kind of has the same problem (albeit small).
For instance, I don't like how "mathematical induction" is a topic, when it's more like a tool that can be used in many situations.
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ran-dumb
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Most people that do A level maths go on to do physics and other sciency things, a bit unnecessary to make them do any more proofs. I don't think more proofs is even the solution for uni maths, it would just put people off earlier on and they'll duck out of as many math modules as they can anyway.

I think the first time you bring out something like induction you kind of have to put it as its own topic, you can't really, in the middle of the series chapter, suddenly pull an induction proof on the sum of natural numbers out your ear and expect that to be anymore organic. And if it's not its own topic that probably means it comes up in the form of 'organic' proofs in other topics, and the only way you can have a question on it would be to get them to memorise the proof and write it out. If you want to set general induction or contradiction questions you sort of need to set up a separate topic.
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04MR17
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Perhaps the solution is introducing more proof based maths earlier in the school curriculum?

Get students used to the nature of a proof and their applications in the wider world.
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mqb2766
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Its an interesting one. Historicallly, one of the reasons why the Greek work is regarded as fundamental is the amount of work that they did to prove results and hence derive new theorems, and while its thought that a fair bit of this work was already known by other, older cultures, theres little/no evidence that they stressed the element of proof. However, learning geometry (for instance) in a Euclidean/deductive style isnt necessarily something that would encourage kids to take A level (and probably cause gcse kids to hide under their desks). Similarly induction has been mentioned a couple of times and even thats in further rather than regular maths.

Regular A level seems to cover deduction/exhaustion/counter example and contradiction, as well as cropping up a few times in the regular topics (trig identiites, arithmetic and geometric series). Which sounds a reasonable amount, but most of the contradiction questions/proofs are selected from a small set, most kids seem to approach trig identities by throwing random things together and seeing what sticks and the proofs of arithmetic and geometric series can be a couple of lines each using the gauss and recursive definition "tricks". Using the fuller notation usually used can get in the way of understanding what the terms mean.

I guess one of the problems with including more is the difficulty of examination. Assessing basic proofs generally involves selecting simple examples from a fairly small set. If youre assessing the proof of series (for instance), Id guess youre looking at things like
1/2 + 1/4 + 2/8 + 3/16 + 5/32 + 8/64 + 13/128 + ....
Its not easy to spot the value infinite sum (numerator is fibonacci, denominator is 2^n so geometric) as the "common" ratio is about 0.8 so takes a while to "converge". However, it would probably be too much "problem solving" for most kids to do it, unless a lot of signposting is used in the question.
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_gcx
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#6
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(Original post by mqb2766)
Its an interesting one. Historicallly, one of the reasons why the Greek work is regarded as fundamental is the amount of work that they did to prove results and hence derive new theorems, and while its thought that a fair bit of this work was already known by other, older cultures, theres little/no evidence that they stressed the element of proof. However, learning geometry (for instance) in a Euclidean/deductive style isnt necessarily something that would encourage kids to take A level (and probably cause gcse kids to hide under their desks). Similarly induction has been mentioned a couple of times and even thats in further rather than regular maths.

Regular A level seems to cover deduction/exhaustion/counter example and contradiction, as well as cropping up a few times in the regular topics (trig identiites, arithmetic and geometric series). Which sounds a reasonable amount, but most of the contradiction questions/proofs are selected from a small set, most kids seem to approach trig identities by throwing random things together and seeing what sticks and the proofs of arithmetic and geometric series can be a couple of lines each using the gauss and recursive definition "tricks". Using the fuller notation usually used can get in the way of understanding what the terms mean.

I guess one of the problems with including more is the difficulty of examination. Assessing basic proofs generally involves selecting simple examples from a fairly small set. If youre assessing the proof of series (for instance), Id guess youre looking at things like
1/2 + 1/4 + 2/8 + 3/16 + 5/32 + 8/64 + 13/128 + ....
Its not easy to spot the value infinite sum (numerator is fibonacci, denominator is 2^n so geometric) as the "common" ratio is about 0.8 so takes a while to "converge". However, it would probably be too much "problem solving" for most kids to do it, unless a lot of signposting is used in the question.
Personally I don't like that a lot of the "contradiction" proofs are essentially contrapositive proofs with an added unnecessary assumption, (cutting it out entirely and just rewording it a bit gives a more slick correct proof). It makes the proofs seem bizarre and confusing to students. And as you say it's mostly stuff of the same form. Induction is a bit better, the A-level boards have a "trump card" to introduce entirely unseen proofs (eg. certain inequalities or formulas for nth derivatives, the latter of which would be fairly straightforward to approach blind) rather than just the usual 3 or 4, (as was actually the case for the old P4 [?] pre-2008) but I don't think this has ever been pulled.
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_gcx
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#7
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(Original post by _gcx)
Personally I don't like that a lot of the "contradiction" proofs are essentially contrapositive proofs with an added unnecessary assumption, (cutting it out entirely and just rewording it a bit gives a more slick correct proof). It makes the proofs seem bizarre and confusing to students. And as you say it's mostly stuff of the same form. Induction is a bit better, the A-level boards have a "trump card" to introduce entirely unseen proofs (eg. certain inequalities or formulas for nth derivatives, the latter of which would be fairly straightforward to approach blind) rather than just the usual 3 or 4, (as was actually the case for the old P4 [?] pre-2008) but I don't think this has ever been pulled.
in case people don't know what I mean here, compare these proofs of "let a, b be real numbers. if a + b is irrational then at least one of a and b are irrational".

Proof by contrapositive
Spoiler:
Show
It suffices to show the contrapositive that if a and b are rational then a + b is rational. (so that if a + b is irrational, we can't have a and b rational) Writing a = p/q and b = r/s for integers p, q, r, s (q, s non-zero). Then a + b = \frac {ps + r q} {q s}. ps, rq, qs are all integers as products of integers, and ps + rq is an integer as the sum of integers, so a + b is rational.

Proof "by contradiction"
Spoiler:
Show
Let a, b be real numbers such that a + b is irrational. Suppose both a and b are rational. Then we can write a = p/q, b = r/s for integers p, q, r, s. (q, s non-zero) Then a + b = \frac {ps + r q} {q s}. ps, rq, qs are all integers as products of integers, and ps + rq is an integer as the sum of integers, so a + b is rational. But a, b are such that a + b is irrational, so we have a contradiction. So either a or b is irrational.

I suppose the difference isn't huge... I just don't like the second. Essentially you are saying "suppose the statement is false", then you prove it's true, giving a contradiction so the statement must be true. You can turn any proof into a contradiction this way.

I guess the way you prove the validity of "proof by contrapositive" is by contradiction but I hope it's clear what I mean, it doesn't make proofs flow all too well re-deriving the proof of that each time.
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themartinipolice
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#8
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no !!!! why? because I hate it lol

in all seriousness I don't think it's entirely necessary? idk about the country wide figures, but in my school less than 10% of students go on to study maths, and about 8% are further maths students (idk if there's more proof in that? I presume so). Most people take maths as a third to Biology and Chemistry, Physics and Chemistry. Second most take it in conjunction with Economics (and maybe history or business), then with computer science and something else (such as physics). And a surprisingly large amount take it as just a random subject because some secondary school teacher told them "ooh maths looks good to employers).

honestly, as much as I dislike maths I've always thought the curriculum was alright, it teaches you a broad range of topics in enough detail to be hard/A-level, but not so much detail that it feels unnecessary or beyond a-level (Which I happen to think of some other subjects). The only thing I would take out it the large data set, because there is seriously no need.
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_gcx
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#9
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my personal opinion fwiw is a bit of an impass: ideally I think maths should be served with two qualifications, one that is more methods driven and is intended to prepare someone more for physics and engineering and the other more for people who want to take maths at university, which includes basic instruction on mathematical writing and a bit of group theory/number theory of the sort you currently see in FP2, and maybe some basic "convergence of sequences" stuff. I think the IB has a similar split with Analysis and Approaches and the other one, but their first one seems only as "pure" as the current A-level iirc.

The problem is that I don't think a lot of maths teachers would be able to teach the more "pure" qualification, and that this will be mainly concentrated at top private schools and the new "maths schools" that are popping up, and so may only give advantage to the already-well-advantaged upper-middle-class students or those lucky enough to live in the catchment area for very strong state schools.
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_gcx
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#10
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#10
(Original post by themartinipolice)
no !!!! why? because I hate it lol

in all seriousness I don't think it's entirely necessary? idk about the country wide figures, but in my school less than 100% of students go on to study maths, and about 8% are further maths students (idk if there's more proof in that? I presume so). Most people take maths as a third to Biology and Chemistry, Physics and Chemistry. Second most take it in conjunction with Economics (and maybe history or business), then with computer science and something else (such as physics). And a surprisingly large amount take it as just a random subject because some secondary school teacher told them "ooh maths looks good to employers).

honestly, as much as I dislike maths I've always thought the curriculum was alright, it teaches you a broad range of topics in enough detail to be hard/A-level, but not so much detail that it feels unnecessary or beyond a-level (Which I happen to think of some other subjects). The only thing I would take out it the large data set, because there is seriously no need.
the LDS was a bit of a pointless addition - I think it was only added because stats got a bit "too easy" (70ish/75 for an A* didn't seem uncommon) and I think was criticised for not having much "practical" data analysis. But I think the solution to that is give you data to work with in the exam, rather than remember what the unit for cloud coverage is lol.
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Notnek
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(Original post by _gcx)
my personal opinion fwiw is a bit of an impass: ideally I think maths should be served with two qualifications, one that is more methods driven and is intended to prepare someone more for physics and engineering and the other more for people who want to take maths at university
There is a third category that make up the majority of A Level maths students. These are those students who won't go on to study maths, physics or engineering at university and the majority of these students will probably not use any of the maths they learn in their life/career.

Progression to further study should definitely be part of the consideration when designing the A Level curriculum but there needs to be a balance and I think the current balance in the curriculum is fine.
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Muttley79
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#12
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(Original post by 04MR17)
Perhaps the solution is introducing more proof based maths earlier in the school curriculum?

Get students used to the nature of a proof and their applications in the wider world.
Proof is in GCSE but some people just don't teach it.
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Muttley79
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(Original post by _gcx)
my personal opinion fwiw is a bit of an impass: ideally I think maths should be served with two qualifications, one that is more methods driven and is intended to prepare someone more for physics and engineering and the other more for people who want to take maths at university, which includes basic instruction on mathematical writing and a bit of group theory/number theory of the sort you currently see in FP2, and maybe some basic "convergence of sequences" stuff. I think the IB has a similar split with Analysis and Approaches and the other one, but their first one seems only as "pure" as the current A-level iirc.

The problem is that I don't think a lot of maths teachers would be able to teach the more "pure" qualification, and that this will be mainly concentrated at top private schools and the new "maths schools" that are popping up, and so may only give advantage to the already-well-advantaged upper-middle-class students or those lucky enough to live in the catchment area for very strong state schools.
Isn't that served by Maths/Further Maths? I think it's down to how the subject is taught - some schools just ignore the proog bits of GCSE and don;t encourage extension work. There's enough proof in A level if taught well.
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04MR17
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(Original post by Muttley79)
Proof is in GCSE but some people just don't teach it.
I know it's in GCSE, my suggestion is to be practicing that style of Maths earlier (ie KS3)
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Muttley79
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Some schools do but on;y NC content is mandatory ... can't see that changing again so soon.
(Original post by 04MR17)
I know it's in GCSE, my suggestion is to be practicing that style of Maths earlier (ie KS3)
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T!gger34
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#16
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Always seemed a bit of a pity that Euclidean proofs had been watered down considerably at KS3/4. Sorry about the reminiscence, but when I were t'lad, we used to spend one lesson per week copying out geometrical proofs and it never dun us no harm.
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04MR17
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(Original post by Muttley79)
Some schools do but on;y NC content is mandatory ... can't see that changing again so soon.
But the question is should it
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Muttley79
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#18
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(Original post by 04MR17)
But the question is should it
No, many schools have to rely on non-specialists teaching KS3. I wouldn't want someone without a good knowledge of maths pedagogy teaching proof.
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_gcx
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#19
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(Original post by Muttley79)
No, many schools have to rely on non-specialists teaching KS3. I wouldn't want someone without a good knowledge of maths pedagogy teaching proof.
PRSOM
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old_engineer
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#20
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I'm all in favour of proof by induction (in Further Maths), and I would always encourage an appropriate level of rigour in "show that" questions generally. However, I'm not a great fan of the specific "proof topic" questions I've seen in recent past papers. I find that they tend to be easy if you've seen (and memorised) a similar model question, but difficult to crack via general mathematical knowledge and problem solving ability otherwise. So, to me, these questions exemplify the bad old days of formulaic predictable questions solvable mostly by memory power, rather than the more modern type of question that relies more on broader problem solving ability.
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