How to Successfully Learn Mathematics:

The main ingredients determining one's success in anything they do: Curiosity, Confidence, Patience and Effort.

a. Mathematical proficiency has five strands (source):

1. Understanding: Comprehending mathematical concepts, operations, and relations—knowing what mathematical symbols, diagrams, and procedures mean.

2. Computing: Carrying out mathematical procedures, such as adding, subtracting, multiplying, and dividing numbers flexibly, accurately, efficiently, and appropriately.

3. Applying: Being able to formulate problems mathematically and to devise strategies for solving them using concepts and procedures appropriately.

4. Reasoning: Using logic to explain and justify a solution to a problem or to extend from something known to something not yet known.

5. Engaging: Seeing mathematics as sensible, useful, and doable—if you work at it—and being willing to do the work.

A simple way to carry out these five tasks in a university course: 

(I) Reading the textbook, (II) Working through examples, (III) Struggling through practice problems, (IV) Discussing difficulties with professors and other students, (V) Understanding and building on what was accomplished, and (VI) Repeating this process consistently. 

b. For each of the steps listed above, consider the method of René Descartes: 

1. Never accept anything as true unless it is clearly known to be such.

2. Divide difficult problems into as many parts as possible.

3. Proceed from the simplest problems to the more complex.

4. Make the connection complete and general so that nothing has been overlooked.

Benefits of Mathematical Reasoning:

1. Improved Problem Identification (Identifying which path to take, i.e. asking the right questions)

2. Improved Analytical/Logical Reasoning (Proceeding along a path, i.e. using problem-solving methods)

3. Improved Solution Justification (Recognizing when the end of a path is reached, i.e. providing robust and stable solutions)