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I'm prepared for the competitive exam like this (a sample question).

In order to solve the problems, first to familiarize with the prerequisite for the each concept. It's ok!

My problem is: If I'm working certain problems on group theory, then it will take two days or a week. Ok! No problem. After the week, I can move on to the next concept like integration, so it will take some time. But in the third week, I lack some concepts bit about group theory (even sometimes, I'm forgetting what I'm doing in the first week). So I feel I waste lot of time. I really want to learn algebra, analysis, topology, and all that simultaneously (sorry if this word does not make sense) or to first learn some bit about algebra and then analysis? I don't no how to move on?

Suppose the question booklet start with group theory. That is, the first question involves group theory, the second one involves like analyticity and the 17th question (for example) involves again group theory etc.

My question is: Is the way to doing problems one by one (in the order)? or By selecting the first one and do all the problems related to group theory and then move on the next concept (like analyticity)? or anything else?

What is the best strategy to prepare this type of exam?

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Forgetting something is normal. What you have to do then is to remind yourself of what you once remembered actively. It is much easier the second time. It is easy to forget things after focusing on them for a week. A more permanent understanding of the topics needs a longer time frame than a couple of weeks.

This kind of thing is learned by repetition, and you have to keep repeating. Even if you are satisfied with your algebra skills after one intensive week of algebra, do not give up doing exercises in algebra. Otherwise you will forget easily. Every day, do a little bit of algebra on the side while learning integration.

Ideally, you should study new things that build on old things. That way you are constantly reminded of earlier material and have to keep rehearsing it. For example, when you study proofs with $\epsilon$ and $\delta$ in analysis, you will be constantly reminded of how to deal with absolute values and basic algebra.

Designing the curriculum so that you build on previous knowledge is not trivial, especially if you don't already know what prerequisites each topic has. Therefore I can only suggest to try to apply old knowledge when possible.

The most important thing is to keep all knowledge active. The best way is to do exercises with mixed topics every once in a while, so that you have to keep using all the tools you have picked up over the last weeks. You will sometimes forget something. That's fine; then just go back and relearn it.

Do not try to learn the different topics in isolation. (They are not isolated in the example exam, either.) Many of the boundaries are artificial, which will come more and more evident as you progress in your studies.

In addition to learning how to mechanically execute tasks, learn why the methods work. Insight and understanding is very valuable.

You might also be interested in the question "How can you be perfect at maths (highschool)?" and its answers.

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  • $\begingroup$ Thank you sir! Thanks for the nice advice! But I have still one doubt: Solving exercises is necessary(not sufficient) for learning mathematics. So If I'm try solve linear algebra problems , Is necessary to do all the exercises in the text book? like in Axler's linear algebra done right? or relevant questions only? or anything else? $\endgroup$
    – user444830
    Mar 25 '18 at 14:13
  • $\begingroup$ I really click the bounty button +100, but it says " you may award this bounty in 13 hours". Sorry for the inconvenience..I think it will work later... $\endgroup$
    – user444830
    Mar 25 '18 at 14:26
  • $\begingroup$ @LearningMathematics It is not necessary to do all the exercises. It is enough to know that you can do all of them. Do many of them, some from each section. If you can read an exercise and immediately tell how it would be solved, there might not be a need to solve it. I recommend reading all or most exercises this way. I think it would be silly to insist to solve all problems. After all, the collection of problems is an arbitrary choice by the book's author. (The bounty will work later as the message says. There is a mechanism to avoid it being given too soon.) $\endgroup$
    – Joonas Ilmavirta
    Mar 25 '18 at 14:40
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I think the one good advice that you can get is the fact that the best learning strategy is totally dependent on the learner himself. Joonas has already mentioned some good points, but at the end of the day, it is only you who can evaluate which is the best path to take. In short:

  • Trust yourself and your judgement about which learning method is more effective.
  • Have self esteem, and don't lose confidence in yourself if you forgot something. Forgetting stuff is totally natural and happens all the time for all of us.
  • Usually a lot of courage and perseverance is needed for these kinda exams. So you need a good motivation and strong desire to nail it.
  • One common misconception that many people have is that mathematics is all about being smart and fast in problem solving. On the contrary the most important thing that learning math needs is hard work and perseverance. So keep it up and try again.
  • You also need a good hobby for your spare times. Preferably a physical one.
  • Gamification of the learning process is a great help and boosts your progress.

I might add some other stuff later if I remember them!

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  • $\begingroup$ Thanks! This one helps too...! $\endgroup$
    – user444830
    Mar 25 '18 at 14:22

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