Dr. L.N. Stout, a mathematician, on his page, discusses a number of strategies to effectively study college mathematics. He offers some valuable advises and caveats for math undergraduates, as naive as me. I only intend to outline it here:
- How is college mathematics different from school mathematics ?
College mathematics pays more emphasis on precise statements of definitions, theorems and logical a structure of proofs that are used to validate them.
It requires developing both judgement and technical competence to solve problems, unlike, school mathematics which mostly cared about technical proficiency.
- How to deal with Definitions ?
Understand what the definition says, i.e. to decipher the general class of things it talks about and then it’s logical structure.
Find out examples as distinct as possible ,both true and false, that deal with each aspect of definition.
Memorize the exact wordings of the definition.
- How to deal with Theorems, Lemmas, Corollaries ?
Understand what the theorem says. Understand the vocabulary used in the theorem. Clarify what the hypothesis are and what are the conclusions.
Determine how the theorem is used. Find example of problems that can be solved using the stated theorems.
Find out what the hypothesis are doing. Understand, why the theorem would not work if one or the other hypothesis is removed .
Memorize the statement of theorems. Theorems can be best used only when their precise statements are in mind.
- Fitting the Subject together .
Working Backwards: This involves sketching out the complete genealogy of results to determine how do they fit together or how they are connected to each other.
Make a definition theorem outline: A definition-theorem outline is an arrangement of the results so that each result is introduced before it is needed in a proof . This would contain precise statement of definition and theorems and sketch of proofs .
- How to make sense of a proof.
Proofs are ultimate test of validity in mathematics, hence, it is more than important to make a sense of them.
Understand what the theorem says: Make sure that conclusions and hypothesis are not mixed up.
Sketch an outline of proof: Omit the details and try to reach the conclusion through a chain of implications.
Fill in all the details : Concentrate on tactics, work out how to reach from one line to another, check all the hypothesis of theorems used in between the proofs and make sure you get that the final proof is correct.
- Developing technique.
Read through the theorems and examples. This would help to identify pattern in advance, while solving problems.
Work enough problems to master a technique. Practicing as many problems as possible would help to embed the technique illustrated by the problem, firmly in mind.
Work a few problems in as many as different ways as possible. Practice problems using different techniques .
Make yourself a set of randomly chosen problems. This would build your judgement technique as you would have to first decipher which technique to use in each such problem.
- Few final suggestions.
Fix class notes immediately after the class. This would help to identify the parts not understood so that can be clarified with the professor in the next class.
Mathematics requires precision, habits of clear thought and practice. So keep working.