achieving the best in mathematics Watch

desmondmac
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i will be starting my A levels in september and would really appreciate (as im sure others would too) if some advice could be offered with respects to suceeding in studying mathematics.

by suceeding i mean A's and Bs all around ( preferablt A's):p:
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RichE
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practising questions until material becomes so obvious you don't remember why you ever found it difficult;

trying to see connections between material, rather than learning each part as a discrete section;

thinking about what you're doing as you attempt a question, rather than unthinkingly applying formulas and techniques;

don't rely on the formula book - know them yourself.
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Loz2402
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practise practise practise (as said above), and if you are stuck on something, ask and practise practise practise!
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wave_o_mutilation
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how much work is A level maths if you have next to no natural talent. (<do you need it to do well?) .i'd like an A....
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Kolya
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(Original post by wave_o_mutilation)
how much work is A level maths if you have next to no natural talent. (<do you need it to do well?) .i'd like an A....
You can get an A with no natural talent; I am living proof of that!
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r3m0t
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(Original post by RichE)
practising questions until material becomes so obvious you don't remember why you ever found it difficult;

trying to see connections between material, rather than learning each part as a discrete section;

thinking about what you're doing as you attempt a question, rather than unthinkingly applying formulas and techniques;

don't rely on the formula book - know them yourself.
Agree, except wrt the formula book. Obviously if the formula is obvious then you should not require the formula book (e.g. in mechanics the equation s=ut+.5at^2 (not actually in the formula book!) is created by integrating a=a:
a=a
v=u+at (u is arbitrary constant, speed when t=0)
s=ut+.5at^2 (no arbitrary constant because when t=0 then s is defined as =0))
(e.g. one of the formulae for the sum of an arithmetic series is curiously similar to the one for the area of a trapezium...)
(e.g. in statistics, the E(X) and Var(X) formulae are similar for discrete and continous variables)
(e.g. in mechanics, to combine "centre of mass"s is actually a weighted average)
(e.g. the sum to infinity of a geometric series is the limit of the formula for the sum of a geometric series, as the number of terms tends to infinity)
but the formula book is there for a reason, i.e. to let you spend your time on practising the maths and not memorising the formulae. OTOH, spotting connections like the ones above can save you time and increase your confidence.

Another tip I would add is to learn proofs. The proofs of sin(a+b) and cos(a+b) are dull, but the rest (e.g. lim sin(x)/x as x->0, sums of series, differentiation identities, trig, logs) can really increase your understanding of the topic.
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yusufu
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(Original post by Lusus Naturae)
You can get an A with no natural talent; I am living proof of that!
:hmmmm: There's nothing wrong with the first clause of that sentence. The second clause is wrong .
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generalebriety
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Practise (past paper or Solomon paper questions) till your arse turns blue. If it doesn't turn blue, you haven't practised enough.

End of advice.
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Kernel
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(Original post by r3m0t)
The proofs of sin(a+b) and cos(a+b) are dull

Sorry to bore you, but what ARE the proofs of these? We were never shown them.
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AlphaNumeric
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http://mathworld.wolfram.com/Trigono...nFormulas.html

Proof by pictures
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DoMakeSayThink
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(Original post by RichE)

don't rely on the formula book - know them yourself.
To be frank, I think it's a huge waste of time. Knowing how to prove formulae is useful, but remembering them if they're already in the formula booklet is a waste of good remembering space :P Okay, so this view might be somewhat biased, but most of the decent mathematicians I know don't bother to remember that d/dx (tan^-1 x) = 1/(1+x^2). They're good mathematicians because of the though process, not the factual recall.
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AlphaNumeric
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^ While good thought processes certainly help, if you're 'a good thinker' in maths, it's not exactly a squandering of mental resources to rememer such formulas as arctanx etc. Being able to look at an equation and instantly realise "It's arcsin" because you know the expression for arcsin means you can do questions much more easier and faster instead of wasting time trying various transformations.

If someone is that good, they don't have to actively try to memorise the formula book, they already know it.
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yusufu
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(Original post by DoMakeSayThink)
To be frank, I think it's a huge waste of time. Knowing how to prove formulae is useful, but remembering them if they're already in the formula booklet is a waste of good remembering space :P Okay, so this view might be somewhat biased, but most of the decent mathematicians I know don't bother to remember that d/dx (tan^-1 x) = 1/(1+x^2). They're good mathematicians because of the though process, not the factual recall.
You obviously don't know many decent mathematicians! :p:
Your example is poor.
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Barny
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Work hard.
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generalebriety
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I assume all those decent mathematicians would be able to work it out if someone put a gun to their head and asked them to differentiate tan^-1 x though.
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AlphaNumeric
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Yes, but if someone gave you

I = \int \frac{5 dx}{8x^{2}+3}

and you weren't familiar with the expression for arctan, you'd have to do a lot of trial and error with various functions before you tried arctan to see what it differentiated into, which would waste a lot of time in an exam. Sure, you're not going to need to learn every standard result under the sun, but stuff like the inverse trig functions is just shooting yourself in the foot because they are so common.
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RichE
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(Original post by AlphaNumeric)
Yes, but if someone gave you

I = \int \frac{5 dx}{8x^{2}+3}

and you weren't familiar with the expression for arctan, you'd have to do a lot of trial and error with various functions before you tried arctan to see what it differentiated into, which would waste a lot of time in an exam. Sure, you're not going to need to learn every standard result under the sun, but stuff like the inverse trig functions is just shooting yourself in the foot because they are so common.
Exactly. In my original comment I said don't rely on the formula book - sure use it as a safety net if you're unsure whether it's plus or minus in the cos(A+B) formula - but you have to know there is a formula at all in the first place or else you'll be randomly scanning the formula booklet throughout an exam.

The example Alpha gave is good one - it won't be in the formula booklet, but if you've understood how those similar looking integrals were arrived at you'll know it's all about using trigonometric substitutions to get the denominator to a square.

At my uni, at least, you're not allowed formula sheets (nor calculators) in exams, so overreliance on the formula book might be a habit worth dropping for other reasons.
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AlphaNumeric
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(Original post by RichE)
At my uni, at least, you're not allowed formula sheets (nor calculators) so overreliance on the formula book might be a habit worth dropping for other reasons.
Definitely. The step up to degree maths can be a sharp one for many people and if you're having to be weaned off a reliance on a formula book that'll make it all the more difficult.

Having familiarity with different integral forms becomes essential as you progress through some areas of applied maths. Some integrals might need 3 or even 4 changes of variable to get to an integral you can do, each one taking a step closer to something familiar. How many people would 'recognise' (or even worse, hit with trial and error) a transformation like x = \cosh^{-1}( e^{\lambda y}) ? Or \sin x = \ln (2y+3)? Few,to none, unless they did it in small steps. Trial and error 3 times to find a function is mad, but if you're familiar with integral forms you might think "[/tex]u = e^{\lambda y}[/tex] might help" then think "Now it's just a hyperbolic integral, that's easy to do!".

Aside from remembering a minus sign here or than in things like tan(A+B) = ....., the formula book (that includes in physics!) should not be something you really want to be using if you're serious about doing a lot of maths. Near instant recognition with an integrand is much better.
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wanderer
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This is exaclt why I don't like integration. There's nothing interesting about the derivative of inverse tan, which makes it a pain in the neck to remember, despite its usefulness.
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FWoodhouse
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(Original post by wanderer)
This is exaclt why I don't like integration. There's nothing interesting about the derivative of inverse tan, which makes it a pain in the neck to remember, despite its usefulness.
You don't like integration? Excuse me while I find you a donor brain to transplant...
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