# How can I develop “mathematical thinking” as a student? [closed]

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.

## closed as primarily opinion-based by Delta-u, Strants, José Carlos Santos, Aloizio Macedo♦Sep 26 '18 at 21:25

Many good questions generate some degree of opinion based on expert experience, but answers to this question will tend to be almost entirely based on opinions, rather than facts, references, or specific expertise. If this question can be reworded to fit the rules in the help center, please edit the question.

• Kind of trivial answer, but imo the only way to develop mathematical thinking is to do mathematics. Start with any book written as a first course for mathematics students. For example, you might be interested in the following book: link.springer.com/book/10.1007/978-1-4419-9479-0 It should be easy to follow to anyone who did mathematics in high school. – Václav Mordvinov Sep 26 '18 at 15:00
• To be honest, I don't think there is a special method that can give you that skill. It's all about practice. The more problems you will solve and understand, the better your mathematical thinking will become. Sometimes I see problems that I can't solve, then look at a solution and I don't understand how one can even think about such tricks. But I learn from these problems and next time I think in more creative ways. – Mark Sep 26 '18 at 15:02
• I can perfectly agree that the best way to learn is practicing. But how come abilities such as abstraction and induction are improved when training? – joann2555 Sep 26 '18 at 16:28
• I don't know if any such general advice is helpful here, except the usual "take courses, ask questions, do problems". Although to be successful as a mathematician, you eventually need to also learn to think independently. – Jair Taylor Sep 26 '18 at 16:54

## 2 Answers

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.

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.

• Thanks for your suggestion. Is any advanced background required for this reading? – joann2555 Sep 26 '18 at 18:02
• Most of the material of the first book seems accessible with little or no knowledge of any advanced mathematical idea. Consider also that it is not strictly required to go through all of the examples and exercises proposed. – Emanuele Bottazzi Sep 26 '18 at 20:26