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I am a straight-A student (going to the ninth grade) and do nearly perfectly in math, the problem is that my school (like many other schools I suppose) makes you memorize the steps, the formulas, etc. and just apply that to similar questions, and since I’ll probably be opting for IGCSE and eventually A levels, I find it necessary to significantly improve my critical thinking skills because I never had adequate practice for that. I’m sorry for all that background, but I would like your recommendation of books/websites that would help me.

Do you think these books are appropriate:

  • How to Solve It by Polya
  • The Art of Problem Solving Vol. 1: The Basics & How to Solve It Vol. 2: And Beyond
  • Problem Solving Strategies by Engel
  • Problem-Solving through Problems
  • Mathematical Discovery: On Understanding, Learning and Teaching Problem Solving

Do you think that Solving Mathematical Problems by Tao is enough for me to significantly improve?

Chris Watson
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Tala Sadaqa
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  • Probably I will get a personal tutor once I actually begin A-levels, but for now I just wanted to acquire methods and techniques on problem-solving and do some practice. Do you have any book recommendations? – Tala Sadaqa May 27 '20 at 20:35
  • If you want to really make a name for yourself, talk to your math teacher and ace the Putnam exam while in high school: https://www.amazon.com/William-Lowell-Putnam-Mathematical-Competition/dp/1470451247/ref=sr_1_2?dchild=1&keywords=%22Putnam+exam%22&qid=1590612190&sr=8-2 – David G. Stork May 27 '20 at 20:43
  • Thank you for your comment. I am not that of a math lover, so I don't really want to make a name for myself or ace the Putnam Exams. However, I like math and now how essential mastering it is, so I am ready to set time aside and seriously work on it. I checked the book you suggested, and it seems too advanced, don't you think? – Tala Sadaqa May 27 '20 at 21:22
  • Hello Tala. I understand that you've learnt the material up to about Year 8 of a curriculum similar to that of England. Tao's and Engel's books are certainly too advanced for that. I'm not a big fan of the Art of Problem Solving series, though they might be at your level. I would suggest that there are two (non-mutually exclusive) kinds of reading that could be beneficial. Firstly, there are some good problem books for roughly your year level. The best I know of is Mathematical Circles by Fomin and Itenberg. Secondly, you could bring your knowledge of algebra up to about GCSE level (though.. – Anonymous May 27 '20 at 22:36
  • not necessarily following a specific curriculum), after which your options for collateral reading on interesting mathematical subjects will expand considerably. Suitable textbooks for this might be Basic Mathematics by Serge Lang or Algebra the Easy Way by Douglas Downing, though which one you choose isn't terribly important. If I've misunderstood and you are in fact already at GCSE level in your knowledge, then please say so and I'll make different recommendations. – Anonymous May 27 '20 at 22:40
  • Have you thought of asking your school to sign up for AMC? Also, check out https://artofproblemsolving.com/community for a great forum with people in a similar position as you. Here is a list of AMC problems: https://artofproblemsolving.com/wiki/index.php/AMC_10_Problems_and_Solutions which require much more critical thinking than rote schoolwork. To qualify for the next competition, typically one would need to solve 16-17 problems. The next competition is the AIME. Here is a list of AIME problems: https://artofproblemsolving.com/wiki/index.php/AIME_Problems_and_Solutions – mpnm May 28 '20 at 03:34
  • My opinion is a little different. Instead of recommending you problem-solving books, I would encourage you to investigate what math is. Basically, math is a language. It uses first-order logic. Before improving problem solving skills, you should probably ask yourself, are you talking about math in daily English? If yes and if you really love math, learn something at the bottom level. As you go deeper, let's say, analysis, which opens the door to modern math, you will find the issue of not knowing logical language. Knowing logic will only help you learn skills faster and understand them better. – Ziqi Fan Dec 31 '20 at 04:51

3 Answers3

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The ruling class doesn't want critical thinkers.
It wants trained workers.
Apparently you've outstripped your school's ability to educate you. So move on from the Dummy Down Dunce Dance school you're currently attending. Learn math on your own.
Choose the topics that most interest you and study them. That is where you'll excel. Skip that I've got to curriculum, it's just baggage. If you bump into something that you like but lack the background, then you'll be motivated to study the background material.
Skip that high school stuff and look for a community college that'll accept you. You'll likely be happier there. Visit a community college, meet with a math teacher, and find what opportunities be there for you. I have a friend who got so bored with high school, she attended a community college instead where she was happier. I also know a self tutored math student who was accepted into a liberal arts college based upon his success in a college math exam.
PS. Skip the most modern text books and look for books written for mathematicians where the cook book method is ignored in preference to concepts, theorems and proofs.
Good luck with your struggle living in a country where being of above average intelligence is a handicap and with your new adventure beyond high school into adult education.
PSS. If you want a tutor, I'll give you a throwaway email address by which you can contact me. I have successful experience tutoring both elbow to elbow and by phone. Being retired, I've no need of payment.
I also suggest you use this web cite for questions and when you get stuck understanding a concept or a formula.

William Elliot
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  • "If you bump into something that you like but lack the background, then you'll be motivated to study the background material." Can't agree with this more. – BLUC May 28 '20 at 05:20
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    Could you perhaps focus on the question asked and leave commentary and asides away. – quid May 28 '20 at 10:46
  • I would be very cautious when a student shows "math" talent. I would really want him/her to differentiate between math capability and the nature of rigorous math. In my opinion, Olympic math/class math is ruining most children's interest in going deeper. On the other hand, rigorous logic education enables everyone to learn math on their own, and allows them to move forward as he/she wishes. Some of them may become mathematicians, and some of them know when to stop and turn to other jobs (engineers, for example), while still keep a good sense of math. – Ziqi Fan Dec 31 '20 at 05:01
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Yes, I think the Art of Problem Solving is a good book series.

I have not heard of Solving Mathematical Problems by Tao, though, but I do know that you can practice your mathematics skills by completing AMC Tests from past years here.

https://artofproblemsolving.com/wiki/index.php/AMC_Problems_and_Solutions

Some Guy
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I wrote a book that may be helpful, as it covers the mental process of problem solving, something rarely taught in school. It has a series of engineering problems solved with the top-down approach, as a boot camp for the reader to get the hang of it. It's titled "The Top-Down Approach to Problem Solving", ISBN 979-8464073296. You can find it on amazon as paperback and ebook. I hope you like it.

Eduardo Corpeño
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