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I am currently refreshing my Elementary Algebra using Schaum's Outlines. I find them useful as they are choc full of exercises (Which I now realize is the only way to master algebra). I am worried though that I will forget a lot of what I have learnt in a year or two as I move onto more advanced Math.

Do Mathematicians constantly practice by doing exercises to prevent them from forgetting? Or is there a point where you just know how to do Algebra similar to riding a bike or driving? I enjoy doing the exercises but I don't want to have to keep going back to basics every few years.

amWhy
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Mark
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2 Answers2

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Be sure to aim for understanding why versus emphasizing automaticity with respect to the strategies and techniques you are learning.

Why do they accomplish what they do?

What justifies their use?

What are the foundations from which the "rules" follow?

Can this be generalized to other contexts?

Asking and understanding these sorts of things, finding connections as you progress in your studies (connections between what you know and what you're newly encountered), and of course, practice!, will all help to guarantee a much more solid mastery of the processes and how to derive those techniques than will simply drilling to the point of robotic "automaticity" ...

One practical pointer or suggestion: if you have the opportunity to tutor or teach students with respect to any material/subject areas that you've mastered, even if only tutoring an hour or two a week, take it. Doing so will both aid you to more deeply understand the material, and provides an opportunity to review, use and in doing so, retain/maintain your learning.

amWhy
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    I agree that "understanding why" comes first. But some "automaticity" can be useful, I think. – Julien Mar 30 '13 at 20:12
  • @julien, Agreed: I didn't mean to frame them to be mutually exclusive ;-) – amWhy Mar 30 '13 at 20:13
  • Nice Amy. It is bed time. ;-) – Mikasa Mar 30 '13 at 20:28
  • @amWhy I agree to an certain extent, however for me I learn much better by doing Math rather than just reading. – Mark Mar 30 '13 at 20:44
  • Oh, Mark, I don't mean by understanding that it's a matter of reading: it IS a matter of DOING math! – amWhy Mar 30 '13 at 20:46
  • I see, I think your answer is spot on, thanks. Regarding the Schaum's outlines, you would suggest pairing them up with a book that has a more rigorous treatment of elementary algebra then? Is there such a book? – Mark Mar 30 '13 at 21:05
  • I have heard Gelfand's Algebra might server this purpose. – Mark Mar 30 '13 at 21:13
  • That would be a wonderful text for this purpose, indeed! – amWhy Mar 30 '13 at 21:14
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Don't worry too much. It is much like riding a bike, you don't forget.

Just make sure you try out a range of different exercises, don't get stuck with one source only. The problems for math olympiads are quite challenging, and cover a wide range of techniques and ideas. Look at their preparation materials.

It is a fine line between "I handle this well" and "automatic for me", don't go too far (you need time to look into other, more intellectually challenging, stuff too).

Look at William Chen's lecture notes and the texts by the Trillia Group, check out Pólya's classic "How to solve it".

vonbrand
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