Soft question - meta-thinking and problem solving internal dialogue.

Question

For context, I am currently a first year Undergraduate student, and am currently trying to be more reflective of my mental dialogue and thought processes whenever I practice solving problems. Recently, I've been noticing that a lot of problems often have a key aspect of it that involves some form of algebraic manipulation.

My question is then, when you are looking to solve some problem or prove a result that requires some form of algebraic manipulation or computation, are the results ever "unexpected" before-the-fact? In particular, I am referring to problems where it is not immediately clear where to even begin with your working or what "frame-of-attack' to use. Thinking of this in a computational manner in terms of investment in resources (time, and frustration), and likely output (the chance of deriving some useful intermediary results, or the final solution itself), suppose I am an algorithm looking to solve problem $P$, and I have some vague ideas of approaches to try, say, three of them called $M_1$, $M_2$, and $M_3$. Now suppose that $M_2$ and $M_3$ were actual valid approaches to solving the problem, meaning that my time investment (sometimes a very long time for harder problems) was worthwhile and fruitful, whereas $M_1$, while superficially looks like a potential valid approach, is ultimately a huge waste of time and a great deal of frustration. Worse is when you don't even realise that it's not a valid approach until deciding to take a break and try $M_2$.

I've come to realise that one of the main reasons why I am often not able to "realise" that my current method $M_i$ is incorrect is because I end up "lost" in a jungle of blind algebraic manipulations, blindly shifting symbols around in hopes of deriving something useful out of it.

For example, consider this problem along with my following working:

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Show that if two positive integer numbers m and n can be written in the form $a^2 − 2b^2$ for some integer $a$ and $b$ then their product mn also can be written in the same form.

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My answer: Let $m=a^2-2b^2$, and $n-c^2-2d^2$. Then

$$\begin{align*} mn &= (a^2-2b^2)(c^2-2d^2) \\ &=a^2c^2-2a^2d^2-2b^2c^2+4b^2d^2\\ &= \text{and so on... (*) } \end{align*}$$

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Pretend for the sake of argument that I get lost at (*), and have no idea how to re-organise the string of algebraic symbols into some useful algebraic pattern. Any further attempt to rearrange the symbols in a different way in hopes of noticing some useful result in it now consistutes blindly shifting symbols around with no deeper theory involved (I'll call it "black-box algebra"). Do I continue trying to manipulate the symbols in a different way or do I ditch $M_i$ altogether?

Is "black-box algebra" ever justified, or should it instantly be seen as a sign that your chosen $M_i$ is likely fallacious? Should you always have a rough intuition of what the intermediary/final result looks like, or what potentially useful intermediary results could even arise from "playing around with the symbols".

I'm having trouble trying to capture the preciseness of what exactly it is that I'm trying to ask in words and exact terms, but hopefully everyone can somewhat understand the key intuitions to my question. I am happy to clarify if need be.

Answer 1

If a solution to a problem does not arise, you should look for an easier problem and try to solve it. Is there another problem? How do you find the problem you want to solve?? The very simple solution

First: to start with easy problems and then step by step to the next task

Second: Before solving any math problem, make sure it is correct

Third: Enjoy solving problems

Answer 2

My question is then, when you are looking to solve some problem or prove a result that requires some form of algebraic manipulation or computation, are the results ever "unexpected" before-the-fact?

This is a psychological question, not exactly a mathematical one, and also somewhat a matter of personal taste. My personal opinion is that if the result of an algebraic manipulation is unexpected, in the sense that you, personally, are surprised by the result - e.g. maybe some unexpected cancellations occur - then there's something there to understand, and you'll be more satisfied if you figure out "why" (also psychological and somewhat a matter of personal taste) the result is the way it is.

I've come to realise that one of the main reasons why I am often not able to "realise" that my current method $M_i$ is incorrect is because I end up "lost" in a jungle of blind algebraic manipulations, blindly shifting symbols around in hopes of deriving something useful out of it.

Again, as a matter of personal taste, I think this sort of thing should be avoided. Blind algebraic manipulations mean you don't understand what's going on. Of course sometimes you don't have any better ideas.

Let's take your $a^2 - 2b^2$ example. Of course there is a beautiful abstract explanation of what's going on in terms of norms (or, if you prefer, determinants), but actually we don't need to know that to solve this problem conceptually without blind algebraic manipulation. We simply ask ourselves: given that the problem tells us that there must be a way to write $(a^2 - 2b^2)(c^2 - 2d^2)$ as another sum $x^2 - 2y^2$ of the same form, what must that sum look like?

To answer this question we will make the following guess: let's guess that $x$ and $y$ are polynomials in $a, b, c, d$. (If blind algebraic manipulation were going to work this would have to be true anyway.) Given that, by comparing degrees they must be homogeneous quadratic polynomials. The positive terms on the LHS are $a^2 c^2$ and $4b^2 d^2$ which suggests that $x$ has the form $ac \pm 2bd$, and similarly the negative terms are $-2b^2 c^2$ and $-2a^2 d^2$ which suggests that $y$ has the form $bc \pm ad$. Now we just have to check which choice of signs makes the middle terms cancel, which gives

$$(a^2 - 2b^2)(c^2 - 2d^2) = \boxed{ (ac + 2bd)^2 - 2(bc + ad)^2 }$$

and we're done. This is, to my mind, a fairly conceptual way to approach the problem as written, although it has the disadvantage that it does not really explain why this identity exists in the first place; as written it is sort of a lucky coincidence that the middle terms happen to cancel.

$2$ plays no special role in this argument so we in fact deduce the more general identity

$$(a^2 + nb^2)(c^2 + nd^2) = (ac - nbd)^2 + n(bc + ad)^2$$

which is Brahmagupta's identity, and which has an abstract explanation in terms of the norm form on $\mathbb{Q}[\sqrt{n}]$.

Answer 3

TLDR; just do math, bro

The answer to this is basically, you will know when you know...

Thinking too much about how to think about math is not very useful, especially when you are not very good at math yet, you would be better off just working on problems, then the intuition for this stuff will build and then it might be useful to think about your thinking process...

Because the fact is that there are millions of problems in the world, i can make some math problem where mindless algebra is the only way to solve it (i dont think this is usually the case) and i can make problems where mindless algebra will take you nowhere...

the one thing you should know is that while working on math, you should never be mindless, because even when doing algebraic manipulations, it is detrimental to not be very attentive... i feel that doing mindless manipulations is a sign of laziness, sometimes when i do not feel like thinking too much on a geometry problem, i just try to close off my mind and complex bash it, and then after 3 hours of stupid 'blind' algebra, maybe i have solved it, maybe not, but i have surely not learned anything. Even while bashing stuff, you can use lots of clever techniques...

But honestly, all of this stuff is mostly useless, because after all, you don't know what you don't know, but also you know what you need to do to get good, that is just work hard, be passionate and enjoy math, just work on problems... because really the only way to get better and to build intuition is to work on problems yourself.

Once you start working hard and getting better all of these doubts/questions will not remain anymore...

All the best...

Answer 4

As others have said, metacognition's not very useful in practice; if you spend all your time thinking about how to think, you'll be a step behind in actually thinking, if that makes any sense. Nike's slogan applies just as much to this field as any other: Just Do It! There are a couple of genuine insights I can offer though: when it comes down to deciding between different approaches to a problem, your best guide is experience; have somebody do increasingly difficult integrals for a whole year, for example, and by the end of it they'll be able to spot a correct method as easily as breathing. Give that same person even a few months off of integrals, and they're quite likely to have forgotten their experience and may find it difficult to spot the correct method for a u-sub, for example.

When it comes to algebraic proofs, yes, you should be familiar with the form you wish to prove. Each step should bring you closer to achieving the correct form but of course that can be difficult to determine ahead of time (even for pros), so some trial-and-error is almost inevitable but still, you should be doing it with consideration and not blindly. Again, experience is a huge benefit in this regard. Of course it's not wrong to note intermediate results of this process, but the focus should always be on the main goal.

Basically, as unsatisfying an answer as this may be, the only way to make inroads on your problem is to gain as much mathematical experience as you possibly can, noting what works for different classes of problems and applying it when you see a similar one. Read as many proofs as you can and take the time to understand them; this will increase your mathematical maturity greatly. Best of luck!