In one of the classes on Lebesgue Integration that I was attending , there was a student / executive who was rather pained by the academic side to probability analysis. Professor was trying to prove that a vector of n dimensional normal random variables is also a normal variable if every linear combination of the n components is a normal random variable.

At some point of proving the statement, the student felt frustrated and blurted out that he wanted more practical applications than theory. At that juncture, one could easily see that professor was rather disappointed with the perspective of certain students. She felt that any learning should make a person comfortable with the theoretical underpinning as well as the practical aspect of it. One superb point that she made was, all the models, all the applications in the world would change , but it is very unlikely that the theory on which the models are based is going to undergo rapid transformation. Innovation comes from understanding theory behind the models.

People who do well are the ones who can understand the theory , extend it to new applications, tools , models etc..Makes perfect sense. right!...But not many people are ready to digest the theory at a slow pace citing time as a pretext. "I don't have time..why don't you tell me a practical application"..To think of it, that is a major issue which plagues current education system. Learning by example is good but Learning by ONLY EXAMPLES is bad. I can cite my education or at least until high school and say that there was no emphasis on proofs. I do not recollect any exam that I have written had asked me to prove a mathematically a statement. Yeah! may be a problem was massaged in to a sort of theory and was asked to prove. Like usage of mathematical induction / Matrix operations etc. In all those problems too, it was more an application of an existing technique rather than actually proving a theory or statement from axioms, lemmas, propositions etc .

Honestly, the fact that one is trying to prove a theorem from lemmas, axioms etc is not to build a certain skill or technique. It is a way to think mathematically. After all, if learning a subject does not alter the way you think , it is useless. Unfortunately, in this world, learn by example, the emphasis on theory is being relegated to the background. More and more students want examples based on theory ..This would only equip them with SKILLS /Techniques which they can sell in the job market at that time period. But what happens when models change, markets change? Will they come up with new financial innovations with out understanding the theory behind it...I doubt... The same debate I had with a friend of mine from BTech. He says that he doesn't have time for understanding what goes behind a certain language and is only interested in coding up stuff..I am certain that he is going to build a zillion applications, make a ton of money ..But will he come up with something new that alters the way applications have been built until now ?..Unlikely.

In fact a week ago I stumbled on to a similar thought aired by a person at Google who was pained to see that a lot of people want to learn a language in 21 days / 30 days/ 3 weeks/ 1 day!!!!. What the hell will a person learn in 30 days ? - was the summary of his article

I think the same thought holds good for any course one is learning at an university. A semester of a certain course would do no good if you are not

- Reading around that material. In Ian Stewart's words - Approach the material from the fringes.
- Think in paradoxes: What are the current paradoxes in the field
- How can the current theory or assumptions be extended to a new proposition / lemma ?
- How can one use the theory immediately / 1 year from now / 5 years from now / 20 years from now ?
- Is there a way to combine the theory+technique to a problem in a completely different domain? Can Lebesgue integration / stochastic calculus be used to solve a certain problem in the Internet domain ?
- Finally , this is something which I firmly firmly believe in. Can you create a graphic novel for the course you are attending ? I think this is the litmus test. If you can boil down all the concepts and principles in to a graphic novel which can be easily read by any person , that is when you have had the true learning.

Anyways , I just hope that the educational system/ students/ entire system does not get carried away by examples only. I hope it teaches students the beauty of thinking in concepts and principles , apart from the applications.