Learn the general case first, or the special case first?

[andytoh]

By studying the generalization of a topic, you learn all the properties that other related topics share with the topic you want to study, and thus you can more easily learn the other topics due to the foreknowledge you've gained, and also should understand the original topic more. On the other hand, by studying a special case of the topic, you learn specialized properties which does not hold in the general case and you understand the original topic more quickly.

I believe both the special case and the general case should be studied to get a full understanding of the topic. My question which is the better ORDER to optimize your understanding during the learning process?

 

[andytoh]

In case you find my question too vague, let me give you an example. Suppose you wanted to study Hilbert spaces, which is an inner product space that is complete in its norm. Should you directly jump into the study of Hilbert spaces? This would certainly make you understand Hilbert spaces quickly.

Or should you first more generally study the properties of metric spaces (and the properties of Cauchy sequences in metric spaces) which has complete normed spaces as a special case, and/or the property of normed spaces (or even more general, Banach spaces) instead of specifially the norm obtained from an inner product. By doing this, you get a better understanding of completeness and norms and so when finally going into Hilbert spaces it is all so much clearer than if you jumped into Hilbert spaces immediately.

Is it better to study Riemannian geometry first or the more general pseduo-Riemannian geometry first, if you want to apply your knowledge to general relativity (which uses the latter)?

And let us assume that time is not a factor. Which is the better order in which to study (and to teach as well)?

 

[andytoh]

I believe that learning the general case first is better, since the special case makes much more sense when you study it after the general case. Learning related topics then usually does not require backtracking and the related topics seem much more related than without having studied the general case first. However, we are usually taught the special case first though due to time constraints.

However, for some reason I find learning Riemannain geometry before pseudo-Riemannian geometry may be better, contradicting myself, and I can't pinpoint why. Also, I studied the projective space before studying the more general Grassmanian manifold and due to the complexity of the the Grassmanian, I think it was the correct order, supporting the opposite view again.

 

[Crosson]

I find that the study of the special case helps me appreciate, more so enjoy, the general case. For maximal efficiency, I would say to do the general case first, and for maximal enjoyment I would say to do the special case first.

 

[andytoh]

I find that the study of the special case helps me appreciate, more so enjoy, the general case. For maximal efficiency, I would say to do the general case first, and for maximal enjoyment I would say to do the special case first.

Wow. So general case vs. special case is efficiency vs. joy?

 

[Alkatran]

I do special cases then work out to the general case.

It's probably a byproduct of being a programmer: trying to solve the entire problem in one go is suicide.

 

[andytoh]

trying to solve the entire problem in one go is suicide.

I don't view studying the generalizations to be "solving an entire problem in one go". I think its more like postponing the special topic for later, and collecting more general information first. The special topic can then be learned much more clearly (and probably faster) after having first developed a solid general background.