How to study Mathematics ?

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.

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First MOOC I completed .

When I first came to know about MOOC phenomena and tons of universities offering thousand of courses , I said to myself , ” I want to do them all . ”
I wanted to put all the plethora of knowledge all at once in my head . So, I enrolled in many from economics to computer science to mathematics. At one time there were more than 7 active courses running , excluding five subjects that I study at high school.
I ended up un-enrolling from most of these online courses and finishing none .

But then, last spring, when Coursera, Udacity and Edx were in total offering more than 350+ courses, I decided to do only one . What an achievement !
It was a course offered by Stanford University  ” Introduction to mathematical thinking” by Keith Devlin .

The previous time it was offered I wasn’t able to keep up with the pace of the course and had to quit the 7 week course, after just 2 weeks .
This time the same content was spread over 10 weeks and it was the only course that I was doing . In addition I was immensely interested in all mathematical things .
These reasons strengthened my goal to complete the course.

As the course began, I leaped to watch the lecture videos and complete the quizzes and HW assignments .  Since I already knew a few things (from it’s previous offering) I swiftly understood the content taught during first three weeks .
Among the most non trivial concepts were how formal notations in mathematics are necessary to avoid ambiguity present in English (or other languages) .  
The logical combinators and and or and their properties and their usage in mathematics. 
Third week explained equivalence and implication, whose’s true understanding wouldn’t have been possible without assignment and discussion forums .
But in the subsequent week I had a very hard time, apprehending the working of quantifiers.
Moreover assignments were filled with all type of English sentences which we had to convert in mathematical notations using Quantifiers and Logical combinators .
I was intimidated by this task because ,
1st I wasn’t well equipped with knowledge of quantifiers and second it looked like an english class. The previous assignments had been quiet the same.

Keith Devlin sir’s blog posts described it as a transition from high school to college level mathematics . And the fact that I wasn’t the only one suffering and spontaneous aid that discussion forums provided helped me move on.

I was now in the Fifth week. Here onward the interesting stuff(truly mathematical) began , “Proofs” being the first one .
Then can number theory  in the seventh week and proving theorems in this genre blew my mind .
They not only stimulated it but also gave a great deal of satisfaction to heart by justifying the effort of previous weeks as worthwhile .
The Euclid proof of existence of infinite primes was amazingly simple and completely logical. I was pleased by it’s profoundness.

At this stage I was spending more than 10 hours a week .
And probably it was in these intense hours of hard work, that I was encountering a great learning experience of my life.  Earlier I had carelessly or naturally accepted rational and real numbers . For me real number were simply a union of rational and irrational numbers . But when Keith sir declared how they came into being after thousand of years of thought , and how they were explicitly defined and distinguished from rational number , I was left astounded .
It seemed as if professor was saying ,” All you studied in school was either wrong or incomplete , so lets start it again. “

The course had almost reached its end. I was so terribly motivated . I had understood the technicalities of proof reading and grading , and participated in the peer review process . I checked three exam papers with and highest total I gave was 210/240 .

A few days later the result turned out . I had scored 223 marks ,beyond my expectations .
I had passed with a distinction (get_certificate).

“MaTh” in Quotes

” There is no permanent place in this world for ugly mathematics,” wrote Hardy; I believe that it is just as true that there is no place in this world for unenthusiastic, dour mathematicians. Do mathematics only if you are passionate about it, only if you would do it even if you had to find the time for it after a full day’s work in another job. Like poetry and music, mathematics is not an occupation but a vocation. ” – Béla Bollobás

“For neither genius without learning nor learning without genius can make a perfect artist.”

“The corpus of mathematics resembles a biological entity, which can only survive as a whole and which would perish if separated into disjoint pieces. ” – Alain Connes