How can I develop "mathematical thinking" as a student?

Question

In my humble opinion as a math student and considering that my main area of interest is computer science, I see that one of the most important skills required to solve problems is the mathematic thinking - a skill that involves the ability to look into a problem and extract informations.

With my background from high school, I developed a "mechanical way" to look into solutions for a problem, searching for formulas, techniques and theories that can be applied towards the solution. This can be problematic for many reasons: this "mechanical way" can lead to wrong formulas, bad abstraction of the problem, etc.

As I see, many people share this way of thinking. I'd like to receive a constructive feedback to start building confidence and think as a mathematician.

Answer 1

Peter Eccles book, An Introduction to Mathematical Reasoning, contains some very good foundational material. Solving problems and being able to check your answers is very helpful. Schaum's Outline Series for various subjects is very useful for this. (Many are available online free and used paperback versions can be found at online book stores.) Let me just add this: One of the most important mathematical tools to learn to use is the principle of induction.

Answer 2

Besides the excellent references provided by RL2 and Moo, I would suggest the two books by Polya on what he calls plausible reasoning:

George Polya. Mathematics and Plausible Reasoning Volume I: Induction and Analogy in Mathematics. Princeton University Press, 1954.

George Polya. Mathematics and Plausible Reasoning Volume II: Patterns of Plausible Inference. Princeton University Press, 1954.

Both volumes present some common pattern of mathematical thinking (for instance, in the first chapter of book I there is an interesting discussion on generalization, specialization and analogy). Each chapter is filled with examples and exercises that should help you learn some basic ideas of mathematical thinking and problem solving. There is no need to read them both from cover to cover: you should start getting acquainted with the ideas presented in the first chapters of both volumes and see how it goes from there.