How exactly does Mathematics help me becoming more intelligent (at least, in high school)?

Question

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I just graduated from high school (not as in, right just now, but this year) and while I love Math, I know its practical applications and everything, I'm confused as to how exactly it helps me become more intelligent.

In my country, India, people boast of how tough the JEE exam is (which is the entrance test for most of the colleges/universities in my country for Science/Engineering/Math, but not Medicine/Social Sciences/Arts/Commerce). One of the major reasons for the exam being touch is not only the fact that roughly 1.2 million people each year give the test in my country, but also because the test covers advanced topics in its subjects (Physics, Chemistry and Math).

Now, the thing that has disturbed me about not only this test, but standardized tests, in general (you can even take the example of SAT Math Level 2/1), is the fact that they don't really test your ability in the subject, but they test your fluency in taking that specific test. For example, my cousin brother is preparing for that test and he figured out the answer to one of the questions in the test by substituting the value of all the options given in the solution and I felt disgusted by it. How exactly does it improve your knowledge of Math and make you better at it? Even a 10 year old can substitute, add, multiply, divide and/or subtract to find the answer. How are you better than him/her?

Throughout high school, I observed that this observation (if I'm correct) is not valid of only standardized tests, but even what they taught me in school. No problem was exactly difficult. The only thing that differentiated an easy problem from a difficult one was the fact that the person solving the problem did not know the trick it required to solve the problem. Once you get to know that, you can very well repeat it like a machine for other similar problems to come, but with different numbers/variables.

Granted that using this way, I sometimes learned a little more Mathematics, and learned another technique of solving a problem, but is this really what Math is all about? Learning new tricks to solve problems? I'd be more excited about the same than I am if I were to come up with new techniques to solve problems. But the trend, at least in my country, is to know some trick of solving the problem, repeating it and getting better at it to spit it all out during the exam.

TL;DR

In school, Math has been about applying certain techniques to solve problems. But using this, do we really learn Math? Is this really what Math is all about? Learning tricks to solve problems?

Answer 1

I'm confused as to how exactly [mathematics] helps me become more intelligent.

Well the hope is not to just gain more knowledge (including techniques or tricks), but also to gain a deeper understanding of both logical and mathematical structures. Since you have only graduated from high school, you'll not know much about these. Originally, mathematics was meant to be a tool for us to discover facts (or at least good approximations) about our world. Applied mathematics today still largely is. But how can a mental endeavour link to the real world? Via logic. If you begin with facts and at each step of reasoning you perform only sound deductions, then of course you can only deduce facts and never any falsehoods. So the question boils down to getting the deductive rules right, so that you can be confident and can convince others that your deduction is sound. First-order logic is one kind of language that almost everyone agrees is meaningful, and there are many sound deductive systems for it. That means that we can systematically check a mathematical proof in first-order logic, and be 100% sure that if the assumptions were true then the conclusion is too (when they are interpreted as statements about the world).

Sufficient experience with logic will make it very easy for you to identify (and hopefully rectify) any logical errors in any sort of work, and this is especially important when it concerns lives. The lack of rigour has claimed many lives in history (one example is https://en.wikipedia.org/wiki/Therac-25). Furthermore, it will make it far easier for you to grasp underlying structures in any problem, which will of course make it easier for you to solve it. Mere algorithms are of little use; but complete grasp of why algorithms must be the way they are to work is of much benefit, for example when you need to modify it to do something different. Ever heard of "canonical forms"? The idea of finding nice canonical forms works in many places (Disjunctive normal form, Skolem normal form, standard basis, orthonormal bases, length-lexicographic order, ...), and this idea is the actual motivation behind a lot of techniques.

The only thing that differentiated an easy problem from a difficult one was the fact that the person solving the problem did not know the trick it required to solve the problem.

Not necessarily. Some people can solve totally unfamiliar problems very quickly. They do not know the trick but can figure out the trick. However, they might have meta-tricks, meaning higher-level heuristics in helping them guess at the underlying structure. For example, the extremal principle is a meta-trick that is amazing when it works.

For example, my cousin brother is preparing for that test and he figured out the answer to one of the questions in the test by substituting the value of all the options given in the solution and I felt disgusted by it. How exactly does it improve your knowledge of Math and make you better at it?

It doesn't. I also detest multiple choice questions where you can try all the answers easily to see which works. But you cannot conflate mathematics with the mathematics that you have learnt or the mathematics that you need to do in the examination.

I slightly disagree with the other answer that if you can get a computer (Matlab) to solve a problem you should just do so. I would add that you should at least understand the precise conditions under which the computer will succeed. There is no excuse to later say "My answer was wrong because there was a bug in the computer program."! It is typical to see students all over the world take out their calculator to compute $13*99$, which any mathematician ought to see can be easily computed by making use of the fact that $99 = 100-1$. If one sees this but is still too lazy, fine, go ahead and use the calculator. But it's not good if one does not even see basic things like this. (This falls under the meta-heuristic of preferring sparse forms, which usually means lots of zeros.)

In school, Math has been about applying certain techniques to solve problems. But using this, do we really learn Math? Is this really what Math is all about? Learning tricks to solve problems?

This is the usual state of affairs in high-school and below. It changes a bit at university level, but if you love mathematics for its own beauty, you should not stick to the curriculum. Instead attempt to explore on your own but at first guided. One suggestion would be to work through Spivak's Calculus textbook, each time trying to prove the lemma or theorem on your own before looking at the given proof. (See https://math.stackexchange.com/a/803595/21820.) Along the way, every time you have some question (especially out of curiosity), like "There are so many quantifiers in the definition of continuity; what happens when I swap them about? Does only the original capture the intended meaning? ...", try to answer your own question! (Ask on Math SE if after trying for some time you believe it's beyond your reach.)

Answer 2

This is in no way a complete answer, but I want to touch on something really important that you said.

The only thing that differentiated an easy problem from a difficult one was the fact that the person solving the problem did not know the trick it required to solve the problem.

I've done a lot of competition math, and I've also taken the IIT/JEE exam for fun (from the comfort of my own home and without any pressure of needing it to attend university).

And yes, it's true that a lot of problems become trivial if you know the "trick." In that way I liken math problems to riddles: it's painfully obvious after-the-fact, but these types of problems are much more about the journey than the destination.

There are two ways you can approach competition math or the IIT/JEE exam. One, you can memorize a whole host of tricks and hope that the tricks that show up are a subset of the tricks you learned. I don't find this fun, but this is absolutely a valid way to approach these exams. The other way is to understand the fundamentals extremely well and attempt to derive these tricks during the exam. I'm bad at memorizing, so this is what I do.

I'm not a mathematician, so take the following with a grain of salt, but the crux of mathematics research is looking at problems whose tricks have not yet been found. These tests are not meant to weed out the critical-thinkers from the memorization experts. In fact, I would argue that no reasonable test can do that job. That's the job of graduate programs, of PhD advisers, and ultimately of academia and industry.

It's up to the individual to determine the best way to take these tests, but let me make a bold statement: I don't think Gauss or Newton or Wiles were the "memorizing" types, and I don't think you'll find many famous "memorizing" types anywhere.

For example, my cousin brother is preparing for that test and he figured out the answer to one of the questions in the test by substituting the value of all the options given in the solution and I felt disgusted by it.

I disagree with this sentiment heavily. I think multiple-choice tests are bad, but this is a completely valid way of solving a problem, even if it, in your opinion, shows zero understanding of the mathematics behind it. If someone gave me a function I didn't know how to differentiate and told me to find its minimum value in two minutes, instead of being clever and trying to find some weird substitution trick, I would stick that function into MATLAB, plot its values, and tell you the minimum. You gave me a problem and I solved it time-efficiently$^1$. So what? Maybe this is the engineer versus mathematician dichotomy, but I don't think the attitude of "disgust" is how you should approach this topic.

Just to round out this discussion, remember that low-level mathematics education is not and might never be aimed at fostering professional mathematicians. Other people use math all the time, and while I think the education system should develop a sense of mathematical curiosity, this should not take precedent over teaching people useful things. This might be a jarring statement, but it's the truth in my eyes.

To answer your question, no, mathematics in college is not the same as mathematics in high school. Mathematics in high school is taught to everyone, even the would-be English major who might never use it. Mathematics in college is taught to a subset of people who self-select the course, and who intend to pursue a career related to math. This might be restricted to my university, but the abstract courses I've taken were not just rote memorization tasks - they required legitimate maturity, rigor, and critical thinking, and this is what I think you want.

$^1$Just to clarify something here: these are very specific conditions when I think just using MATLAB is okay: namely that we're under time pressure and you don't require an exact answer. If a rocket is about to blow up in two minutes unless I tell you the minimum, but it would take me more than two minutes to differentiate the function, going that route is fundamentally wrong. The caveat here is that you should have some base knowledge: I know roughly what the function should look like from its form, I know I need to find the minimum value, and I know that if MATLAB gives me $10^{1000}$ as my answer, something went wrong.

Answer 3

This is an interesting question, thoughtfully asked. I really like @SSS 's answer, and have upvoted it.

You are right that mathematics is much more than being able to pass the tests you get in school. One way to learn what mathematics is in life after school is to learn more - but not just what's in the next course

If you're curious and determined enough you might enjoy two books that turned me into a future mathematician when I discovered them in high school. One is Hugo Steinhaus's Mathematical Snapshots. It's available from Dover at a very reasonable price ( http://store.doverpublications.com/0486409147.html ), also at Amazon, where you can read parts of it. The second is Polya's Induction and Analogy in Mathematics. You can read it free at https://archive.org/stream/Induction_And_Analogy_In_Mathematics_1_#page/n0/mode/2up . If you like it you can buy a copy and read it for the rest of your life.