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I am in the UK, studying GCSE, if that means anything to you.

What would you recommend to me, to further my understanding of mathematics, of my current level or further?

Thanks.

user511660
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  • What is your current level, if I may ask ? I'm not a guy from UK, so I don't know how's the school system there. Let me know and I'll answer you with detail. – Rebellos Dec 08 '17 at 18:07
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    It depends a lot on your amibitions and how much you’re looking to challenge yourself. I would consider Halmos’ Naive Set Theory. There are many other options. – Ittay Weiss Dec 08 '17 at 18:12
  • @Rebellos I'm not sure how to judge my level, but I can give you a few examples of things that are emphasised in the course. Solving quadratic equations (factorising, completing the square, quadratic formula), transforming curves, quadratic simultaneous equations, quadratic inequalities, circle theorems, trigonometry (sin, cos, tan in right triangles, sine rule, cosine rule), surds. – user511660 Dec 08 '17 at 18:17
  • @user511660: Right, but what you should read/do depends heavily on what you're currently comfortable with. Without any context, I would recommend (from experience!) making sure you find basic algebra easy. That is, you should practice algebraic manipulation of equations and inequalities until you find it easy, it should not be a challenge. I felt like that was ultimately the biggest difference between me and my peers at A-level (and I was the only one who went on to study mathematics at university). If, however, you've already put in this work, then this advice is obviously redundant. – Will R Dec 08 '17 at 19:29
  • @WillR Awesome, thanks for the advice. – user511660 Dec 08 '17 at 19:32
  • @user511660: (In particular, the main problem I had at your age was accidents involving signs and little things like that. Get used to doing a quick check of every line when you write it down. Getting it right first time will save lots of time and frustration in the long-run!) – Will R Dec 08 '17 at 19:36

2 Answers2

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I would recommend a two-pronged approach!

Strengthening your algebra and problem solving skills beyond what is covered in GCSE would be very valuable. Books such as 'Student Problems from the Mathematical Gazette', ISBN 0 906588 49 9, available from the Mathematical Association, would provide good challenges for you.

For general appreciation of Mathematics (which should whet your appetite for your future maths education) there are many good and popular books - for example '17 equations that changed the world' by Ian Stewart.

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I recommend that you find a book that is both enjoyable and challenging but not too challenging. If it generates curiosity in mathematics everything else should be doable. I'm not saying easy, some things will come easily to you, others will be hard work but with curiosity you will persist and overcome. One view of math is that it is the art of solving problems so my suggestions will biased that way. Here are some suggestions; they may be a little on the old side. I'm sure there are newer books that I'm not as familiar with, on the other hand you may find inexpensive copies in second bookstores such as those run by Oxfam.

How to Solve It by G. Polya
Problem books by Martin Gardner
The Lore of Large Numbers by Philip J. Davis
Countdown by Steve Olson (This book is more about competitive math problem solving than the math per se. The math is only about 30% (rough) of the book.)
The Mathematical Experience by Philip J. Davis and Rueben Hersh (The math may be only about 10% but there are many short essays that are fun to dip into.)

I will add to this list if I think of anything else. Other users may add newer books or completely disagree with me.

Stephen Meskin
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