Are mathematical processes enough?

Question

I am currently studying IB higher level maths, and by looking at how my classmates learn, I think I can see an important distinction between the ways different people learn maths.

Now, most people seem to learn the processes (or algorithms) required, without necessarily looking at any proofs or understanding how we arrive at these processes in the first place.

But, others try and gain more of an understanding of the way the maths we study is shown to be true by studying proofs and thinking about the derivation of the algorithms. These people also seem to think more about how we can use these ideas also. Necessarily, this approach takes significantly more time.

A good example which shows this distinction would be the learning of differentiation. A lot of my classmates simply learnt the straight rules we use to find derivatives of different functions. Others though, looked more at differentiation from first principles and how this is used to prove these rules which we can use to differentiate a given function. Of course they also learnt the rules, but they also thought in more detail about how we show these rules to be true.

I fall into the latter of these groups. When studying maths, i find it hard to simply accept the rules given without trying to understand it better and look at proofs. The problem with this though is that it takes ALOT more time. I find myself struggling to keep up in terms of doing questions and getting through the course with the majority who learn the algorithms and are then done with it. But, at the moment the most effective method if one wishes to succeed in tests seems to be simply learning the rules in order to do question, not necessarily having a firm grip on the fundamental concepts.

My question to anyone experienced in mathematics is this:

What method of learning is better: simply learning methods, or also studying proofs and trying to get a firmer understanding of the mathematical concepts?

Is it worth at this point spending the extra time beyond learning the algorithms when (due to the nature of IB), i dont have a great deal of time to spare?

(P.S sorry for asking such a long-winded question :))

Answer 1

I have the same exact situation as you do. I also think that the best way to learn math is to actually understand it, rather than just to memorize it. However, since both of us are preparing for a STANDALIZED TEST, we should care more about learning the methods first. The best way is to first learn the methods (by memorizing, just like your classmates). This will ensure you a good grade on your class exams and the IB test. Then you can study the proofs after you learned the methods (If you have time for it). In this way, you will do well on the IB test and also have a deep understanding of math .

(There is nothing wrong with learning the methods first then the proof, you can still stimulate your mathematical mind in this way).



If you are considering to study math at a higher level (math research), studying proofs will definitely help you. If not, simply knowing the methods would help you deal with standalized tests and real life well enough.

Answer 2

Nice question.

It all depends on how you want to focus on mathematics.

Seriously, if you know how to count, a simple pocket calculator can give you all the answers that you need in life.



But if you want to do math and understand how it works, you have to start from basic principles. Learning theorems without proof is either for not going into details, because sometimes even the simplest sounding theorems require a 15 page proof, or just because you want to solve things without going into it too much.



The latter, however, makes it so that you can not advance into manipulating mathematic entities as you like and it's useless to start working on advanced theory.



It's like programming:

You can copy well known codes and use them to write programs that work perfectly. For practical use, this is going to be fine. But you can also spend time breaking down these codes, analysing the steps, calculating efficiency etcetera. This allows you to master coding to the max, but is a waste of time if you just want to design webpages.



But like a professor once told me: you can't expect to prove everything in a proof you do. It would require 100s of pages to explain even the simplest of theorems.



So the best way is to choose which statements you understand and can agree with them being truthful and use them to proof the things you doubt of. Or try to proof what is relevant and assume what you know.



(like continuity: a proof which uses the definition of continuity need not have 20 pages of explanation how continuity is defined that way to achieve the result)





As for the direct advise:

Study the proofs you've been given and be able to explain them to yourself. Please refrain from studying them by heart without understanding what is happening, use pen and paper to write things out.

Ask questions about where certain conditions are needed in a proof and see if you can construct counter examples when you leave them out. If that gets you stuck, ask the teacher for help.

(trust me, I'm a doctoral math student, this is what got me so far)

Answer 3

Easy to look up a method and apply it to a few questions job done move on. Probably find that next week you would not know where to start in answering the same questions.

Answer 4

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