From a review for Polya's book on Amazon, the books to be read in sequence:
- Mathematical Problem Solving by Alan Schoenfeld
- Thinking Mathematically by J. Mason et al.
- The Art and Craft of Problem Solving by Paul Zeitz
- Problem Solving Strategies by Arthur Engel
- Mathematical Olympiad Challenges by Titu Andreescu
- Problem Solving Through Problems by Loren Larson
Full text of the review below:
By Abhi:
Good aspects of this book have been said by most of the other
reviewers. The main problem with such books is that for slightly
experienced problem solvers, this book probably does not provide a
whole lot of information as to what needs to be done to get better.
For instance, for a kid who is in 10th grade struggling with math,
this is a very good book. For a kid who is in his 11th grade trying
for math Olympiad or for people looking at Putnam, this book won't
provide much help.
Most people simply say that "practice makes perfect". When it comes to
contest level problems, it is not as simple as that. There are
experienced trainers like Professor Titu Andreescu who spend a lot of
time training kids to get better. There is lot more to it than simply
trying out tough problems.
The most common situation occurs when you encounter extremely tough
questions like the Olympiad ones. Most people simply sit and stare at
the problem and don't go beyond that. Even the kids who are extremely
fast with 10th grade math miserably fail. Why?
The ONE book which explains this is titled "Mathematical Problem
Solving" written by Professor Alan Schoenfeld. It is simply amazing. A
must buy. In case you have ever wondered why, in spite of being
lightning fast in solving textbook exercises in the 10th and 11th
grade, you fail in being able to solve even a single problem from the
IMO, you have to read this book. I am surprised to see Polya's book
getting mentioned so very often bu nobody ever mentions Schoenfeld's
book. It is a must read book for ANY math enthusiast and the math
majors.
After reading this book, you will possibly get a picture as to what is
involved in solving higher level math problems especially the
psychology of it. You need to know that as psychology is one of the
greatest hurdles to over when it comes to solving contest problems.
Then you move on to "Thinking Mathematically" written by J. Mason et
al. It has problems which are only few times too hard but most of the
times, have just enough "toughness" for the author to make the point
ONLY IF THE STUDENT TRIES THEM OUT.
The next level would be Paul Zeitz's The Art and Craft of Problem
Solving. This book also explains the mindset needed for solving
problems of the Olympiad kind. At this point, you will probably
realize what ExACTLY it means when others say that "problem solving is
all about practice". All the while you would be thinking "practice
what? I simply cannot make the first move successfully and how can I
practice when I can't even solve one problem even when I tried for
like a month". It is problem solving and not research in math that you
are trying to do. You will probably get a better picture after going
through the above three books.
Finally, you can move on to Arthur Engel's Problem Solving Strategies
and Titu Andreescu's Mathematical Olympiad Challenges if you managed
to get to this point. There is also problem solving through problems
by Loren Larson. These are helpful only if you could solve Paul
Zeitz's book successfully.
To conclude, if you are looking for guidance at the level of math
Olympiad, look for other books. This book won't be of much assistance.
On the other hand, if you are simply trying to get better at grade
school math, this book will be very useful.
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