When learning maths, should I review past topics?

Question

So I've been learning maths from online video lectures and books for some time now and, as is probably quite natural, I have forgotten some of the stuff I've already read through. That'd be mostly some galois theory and "Analysis on manifolds" by Munkres.

My question is - is it more effective to read through those books once again, to make sure I remember as much as possible or should I just keep going, reading books about further, more advanced topics, assuming I have the basics to understand them?

Answer 1

If you want to engage with the actual subject, the answer is "all of the above". That is, yes, regular review of things one's already read is necessary, if only because the human mind is fallible, forgetful.

Yet one should not allow understanding one's own forgetfulness or even incomprehension to inhibit moving forward. This is true in a much stronger sense than nearly all novices perceive, namely, that the sense of many "lower-level" things can only be understood genuinely at some later moment, in terms of more sophisticated concepts (which, duh, srsly, ppl, were created for that understanding). That is, thinking that "strict logical development" is a good virtue to practice is ... misguided.

That is, mathematics is not "logic" (although "logic" is a good thing in itself...), thus, a dynamic engagement with it, if you like it at all, is the only real way. Engagement "according to rules" is artificial, probably not so good. Maybe harmless, at best. Tastes differ. "Your mileage may vary."

The last note is that there is the insidious pretense/mythology in mathematics that it is "objective" or somehow "trans-human/worldly". This conceit clouds-the-minds of many students, and is dangerous. People are people. Not transcendental, even if we might like to be. :)

Answer 2

I would suggest some type of review. I find it amazing how much more I see when I go through things again. And connections I can make with subsequent things I have encountered.

Sometimes this can go pretty fast. But I also feel if this hurts, I need it.

Lastly, you might think of it as harvesting the seeds you have planted.

Answer 3

I am giving the way i read

first time when i am reading i highlight almost any simple thing that i feel is of value

and put some sticky index's on top for any formula's i think i may forget and need to recap

and when i feel like i need a brush up i review just that material.

I suggest this idea for you and one more thing if you are forgetting something means you haven't mastered first time round well or you don't use the knowledge at all or both.

figure out what's your case in anyway if you develop a plan that works for you so that you can recap in quick time in future that will be helpful.

as you most certainly will want to review now and then.

I say for now try and recap the old books before you study further level otherwise you will not learn advanced topics 100% and you have to start all over again

Answer 4

Personally I think rereading entire books/scripts etc. that you have already studied is a waste of time and rather boring. So I would just go on and if you need to review some particular thing you can still look it up. You will also gain additional understanding of things naturally as you develop mathematical maturity through studying advanced topics.

However, if you want to refresh a whole topic I suggest you find some different, maybe slightly more advanced, material on the subject than what you used before. The reason is that changing perspective can deepen your understanding a lot and help you make new connections which you might miss otherwise.

After all, from my point of view, it is about getting a good mathematical intuition and acquiring a solid working knowledge.

Answer 5

You shouldn't read the whole book again, but review some important topics that you have forgot

Answer 6

I would begin going over the new material, and review on a need basis, as I go over new topics.