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More advanced maths for 14 year olds: includes recommended reading
By Sarah Hunak (P2100) on Friday, February 25, 2000 - 10:15 pm:

Hiya
I'm 14 (got an 8 in my SATs) and really interested in maths but can't really get much help in expanding my knowledge from school because the only thing open to us is a pretty pathetic maths club. I've heard things about calculus and discrete maths and things but everyone on here seems to know all about them already so I was wondering if anyone would like to explain them to me in a way I'll be able to understand?
Thanks

By Richard Mycroft (P2053) on Saturday, February 26, 2000 - 02:00 pm:

Hi.
I have exactly the same problems.
I'm 14 and I got an 8 in my SAT's too (missed EP by one mark on the extension paper!!!).
We don't have any kind of maths club whatsoever.
The only time in the whole year when I find maths interesting is the IMC and following events (right now I'm waiting to see what I got in the IMC).
My parents have quite a few maths books, but they are impossibly complicated and if I ask my parents for help I can't understand a word they say - they both did maths degrees and seem to expect me to be at their level. However, how can I improve with nothing to help at my level - school too easy, other sources too hard ???
I have tried talking to my maths teacher and head of department, but they seem to resent the idea of extra work for them.
So, the outcome of this is that I am left spending hours and hours of repeated questions (the same question, just with different numbers?) when I feel I should be doing stuff much harder (and therefore more interesting)
Sorry about my lack of modesty.
Can anyone help?

By Sean Hartnoll (Sah40) on Saturday, February 26, 2000 - 04:20 pm:

I can certainly sympathise with your situation, I was there myself not so long ago. A couple of suggestions. Firstly, browse through this site and find questions that people have asked that are roughly a level you can understand, and there are plenty of them. If you don't understand something, ask. Secondly, there are some books that are accessible and interesting. For example, The Teach Yourself series has books on trigonometry, calculus and groups (more advanced) which are all interesting and which are easy to learn from.

Sean

By Sarah Hunak (P2100) on Saturday, February 26, 2000 - 04:56 pm:

Thanks for replying so quickly!
Does anyone have any more interesting books that I could read? I'll certainly be looking into the Teach Yourself series, it'll give me something to do in the holidays!

By Neil Morrison (P1462) on Saturday, February 26, 2000 - 09:35 pm:

You should try an A-level study guide.

PS: what's a SAT

By Sarah Hunak (P2100) on Saturday, February 26, 2000 - 11:06 pm:

A SAT is a national test for key stage 3 students (11-14) There are also SATs for Key stage 1 and 2 students too. You only do 3 subjects: Maths, Science and English and get a level of attainment.
At Key stage 3, the average is around 5.5 or something.

By Brad Rodgers (P1930) on Sunday, February 27, 2000 - 01:44 am:

Hi, I am also 14, and, for the most part, have the same problem. But there are a few things you can do: as said above, read every book on math you can find- this will keep you at a good and interesting pace. Also, see if you can find a magazine focused solely on math; these books will usually give you several problem you are capable of solving but that are still very, very challenging. If you are in the U.S., I would recommend Quantum. If not, you may just have to look around. Also, try to find packets of math problems from math competitions; these will generally prove to be a challenge if you can ask a teacher from a grade (or three) ahead of you for one.

As far as books, I,m afraid I don't know the name of many on pure math, but if you are interested in physical science, I reccomend trying to learn some of relativity (not the tensor analysis unless you know calculus) from either Black Holes and Time Warps, The Elegant Universe, or A Brief History of Time. All these books are excellent and very interesting if you are interested at all in physical science to go along with math.

By Amanda Turner (Agt24) on Sunday, February 27, 2000 - 08:45 pm:

An excellent book if you are interested in pure maths is "A Mathematical Mosaic : Patterns & Problem Solving" by Ravi Vakil. It has lots of interesting problems of varying difficulty and doesn't require much previous knowledge. So you should be able to find stuff relevant to your ability regardless of your level.

By Harry Smith (Harry) on Monday, February 28, 2000 - 09:52 am:

If you've just finished Key Stage 3, looking a few years ahead won't solve many of your problems. You won't see calculus until A-level, and you might not see any Discrete maths even then. Instead, and for a fairly informal treatment of some of the more interesting areas of maths, I would heartily recommend Martin Gardner's series of mathematical recreations, taken from the pages of Scientific American. Also Game, Set and Math by Ian Stewart is excellent. All of these books are cheap and great fun to read - they should give you a good idea of some of the problems mathemeticians face everyday, and they will have references in them to point you towards more rigorous or technical books on the subjects covered. Steer well clear of any undergraduate level maths books. While you may have the ability to understand them, you probably won't have covered the necessary material.

The one exception to this is discrete maths and number theory, which tend to be covered from scratch in most first year undergraduate texts. I can recommend "Discrete Maths" by Biggs, which covers number theory as well and has a modern approach. For a more traditional approach to Number Theory, "The Higher Arithmetic" by Davenport is good. When you've had a look at those, and if you're feeling brave, you could have a look at "Combinatorics" by Bela Bollobas. It is utterly fantastic, and contains some profound and beautiful results, but you might have to wait a few years before you can tackle it.

I know that all sounds daunting. In response to your first question Sarah, discrete maths is the study of seperate objects. The natural numbers (1,2,3...) are seperate, and their study is known as Number Theory, which is a branch of discrete maths. If you ask "If I have two red socks and two green socks, how many ways can I choose a pair of socks?" you are asking about seperate things, socks. This is a branch of disrete maths called Combinatorics. Studying objects like this, and asking questions like this, has led to all sorts of advances in the design of computer programs and secret codes (try looking up the "RSA Code" or Trapdoor codes" in an encyclopaedia or mathematical dictionary).

In contrast, calculus is the study of things that change in a continuous way. A parachutist falling out of a plane accelerates towards the ground - you can use simple calculus to work out how fast he is going at any given time. Calculus can also be used to find the volumes or areas of shapes, and is vital to the study of the weather. Calculus was invented by Isaac Newton to work out a theory of gravitation - this theory was subsequently used to land astronauts Neil Armstrong and Buzz Aldrin on the moon in 1969.

This is all very general - if you come across anything you don't understand while your learning about these things, post a question and we'll try to answer it. But persevere - there is a whole world of incredible mathematical things waiting to be discovered.

Good Luck

Harry