Your thoughts! mAsterminds!

Hi guys
I was wondering what three ideas or fundamentals would you reccommend a student should know, and know very well to be strong at high school, first year university maths?

Also from your experience what has been the three most powerful and useful study ideas/principles/techniques that you have used to get the results you want... in regards to studying maths?

Thanks in advance for your answers ;)

I was told that Induction was quite important, for computer science anyways.

University maths is very different from high school maths

More about proof rather than problems. Three are very difficult to pick. Further Maths is very important

So:

1. Proof - Particularly by contradiction and by induction
2. Complex Numbers
3. Differential equations

Though on another day I might have picked something else for 2 and 3. The principles of proof are most important for university. Particularly for Analysis and Algebra

-Proofs, Proofs, a few more proofs
-Geometry (Proofs again)
-Logs/Exponentials
-Series
-BEDMAS!:)

You go into a math oriented university class with that solidified, and your prof will be ecstatically surprised... BTW, did I mention how important proofs are?

Edit: Oh, Coordinate systems, vector operations, complex numbers and MATRICES..

what type of fundamentals should one know to become good at proofs?
whats BEDMAS mean again?

Proofs take one of several forms:

1. Proof by step by step argument
2. Proof by counter example, an example that proves something is not true
3. Proof by contradiction, assume opposite is true and prove that makes no sense
4. Proof by induction, Show true for n=1, assume true for n=k implies true for n=k+1
therefore true for all natural numbers n

To become good at proofs takes much practise.

To be good at proofs you're going to need to be good at adding, subtracting, multiplying, dividing, and THINKING. Proofs ARE the fundamentals.. And practise makes perfect.. Trig Identities are a good start, as are more pure geometry problems.. Every equation you've ever been taught can be proven by breaking it down to its most basic concepts.. The reason you should focus on proofs now is because your education from grades 2 to 12 has corrupted your way of thinking, they've given you far more toolsets than you actually need and you've been accustomed to using them all.. University profs are going to expect you to be able to rebuild those toolsets with not much more than what you learned in grade 1..

I hope you forgot the smiley after your BEDMAS question..:)
Brackets
Exponents
Division\ Equivalent effect Operations
Multiply/
Addition\"
Subtract/

I did further mathematics A level at grammar school (archaic type of UK high school) and have found it absolutely indispensible. I've just finished my first year at university and can't imagine how I'd have coped if I'd never met the calculus contained within further maths. 2 modules out of 8 (a quater of my whole first year!) was basically solving ordinary differential equations and integrating tremendously awful functions.

Have you covered first and second order ordinary differential equations?
Have you covered matrices?
Have you covered all the standard forms of integrals?
Have you covered complex numbers?

Having said that, I have many friends who passed their first year having covered NONE of these things. They will probably be taught, but if you haven;t already met them it will require far more effort on your part.

It will also be assumed that you can remember simple trig rules, like sin(pi/6) = 1/2 or cos(2x) = cos²x - sin²x.

Most importantly, WELL DONE for choosing maths and I wish you good luck in your course. :)

Richard Feynman (great physicist) believed in seeing things in as many different ways as possible. He also believed in having as many different 'tools in the toolkit' as possible.

To answer your question about your first year it really depends what sort of Mathematics you are going to end up doing. Proofs are important but are not necessarily the 'be all and end all' - it really depends on what sort of Mathematics you are going to end up doing. (You are going to have do some though !).

So depending on your previous course (and how far you have already got) you will need:

Integration & Differentiation, so you can go onto:

Differential Equations

(you will need complex numbers as well for this)

In your first year try and find out about as many different 'tools / techniques' as possible for your toolkit so you can solve many different problems - there are no shortcuts to this though ! (Reading different texts is very useful here).

(Read anything by Richard Feynman to inspire you to keep going !)

Good luck with your course (we have a shortage of mathematicians in the UK)


N Pinhey

GCSE Maths Revision Cards Fast exam analysed revision for GCSE Mathematics.

Another good Source of inspiration: Nikola Tesla! If you ever wondering what kinds of crazy things you can actually realistically do with these tools, then google that guy.. :)