How do you know if you're thinking right?

Question

Every time I stuble upon a difficult problem in math, I can't even figure out where to look at.

finally I do some trial and error and give up, and ask in mathematics discord about this problem, every time, thinking and hoping that there's a typo in the problem.

some person who's good at math leaves a subtle hint about how to solve it. The approach hasn't ever been seen or thought by me before. I'm shaken with existential dread for 5 mins and the cycle continues.

The question is,how do I know if I'm thinking how a mathematician is supposed to think? What am I missing? Is there something more than just trying to blindly solve problems?

people say you get good at math by doing more of it. I can't figure out how to do more math when you can't even approach a problem you never saw before.

(Do keep in mind that I'm still at intermediate algebra(midschool)

Answer 1

First of all, make sure you understand what exactly the problem is asking of you. Try to condense it down into a couple of lines (if possible). That's one of the biggest problems people face I think - not actually understanding what the question wants/means. If you can't figure it out - rather than ask for a full solution - ask for the meaning of the question. Otherwise - you'll be given a solution that you also won't understand, and of what use is that?

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Other than that, if you do understand the question yet still can't make headway of it - there are some standard problem-solving procedures that you can follow. People who do these things usually do so without consciously thinking about what they are doing, so it can be very difficult for someone to understand what's going on or how the person got there in the first place. Without a concrete problem, standard problem-solving techniques are really the only things one can give you.

These things may not always work (some problems genuinely are just harder and require some radically new approaches - otherwise, Mathematics would be easy!) but some things you should definitely have in your "toolbox":

${(1)}$: Trying simpler cases. If you have a rather complicated problem - sometimes, you can find a simpler version of the problem to try. Maybe you get rid of some condition, or add a condition on top and it turns out the resultant, new problem is much simpler and easier for you to solve. Always try these. It's not necessarily a waste of time - you can sometimes find ways of connecting this simpler version and the more complicated one together

${(2)}$: Knowing your proof methods. This includes proof by contradiction, proof by induction, direct proof (that is - just argue the point directly with other known true statements), proof by contraposition, proof by construction - even proof by exhaustion. These are just a few of the many general proof methods - and the more you know, the more problems you are likely to be able to solve.

${(3)}$: Know your theorems - and most importantly (this is really very very important) - KNOW THEIR LIMITATIONS!!!. People often blindly apply a theorem without really making sure the problem even satisfies the conditions of the theorem, so be careful. Knowing your theorems gives you a whole set of true statements you can play with to get to your answer - like a puzzle!

${(4)}$: counter-examples. Maybe you have a problem that is along the lines of "Is statement ${X}$ true?" and you may think statement ${X}$ is not true. In this case - it maybe wise to try and find some examples that fit the problem statement and disprove statement ${X}$. The example could be rather complicated - but try to find the simplest of such example. Usually one exists!

${(5)}$: Try and find equivalent forms of your problem. If you cannot directly solve the problem - try and find a different problem that is exactly equivalent. You may find that problem easier.

${(6)}$: Know your standard "tricks". There are a lot of problems in Mathematics where the solution (at least the most "popular" solution) requires the use of some sort of non-trivial trick. In analysis for example, we are always "adding $0$ in a special way" by doing something like ${+c - c}$. These are just standard tricks you should have in the back of your mind.

If you try these things it'll definitely help you solve many standard problems. Definitely practice lots, and try to enjoy doing the Mathematics. If you enjoy it, and approach problems in a calm manner - you will find it so much easier. If you cannot solve it immediately - that's okay! Take it as a challenge, and keep at it. The end will be satisfying once you finally crack it. That's why I enjoy Mathematics - it's so much fun at the end when you finally have some insight and manage to solve something!

Answer 2

the most important thing is to realise there are often many ways to tackle a problem, and as you study more maths, you will see new ways of doing things, so don’t get discouraged!

This is how I would go about solving an unfamiliar problem, admittedly some of these may not always be appropriate for the type of maths you are currently doing, but may still be worth thinking about:

1. Actually understand what the question is asking you (for example, do you know the definitions of the key words or symbols being used?)

2. have an educated guess (for example, does this look like it can be written as linear equation, then see if you can do so!)

3. what do you know, and what are you aiming for. Try and work backwards and forwards from these things initially, then when you think you have a solution, write out the solution in a logical order.

4. work with different cases. For example what happens when $n=1$ or when $n$ is negative? Try and create your own examples and see if you spot a pattern.

5. think about similar problems you have solved before and try and rethink about your question in that context. For example, if it sort of looks like a quadratic equation, is there a way you can manipulate it in a way that is more familiar?

6. solve an easier case of the problem. For example is the problem easier to solve for positive integers first and then see if that helps for any real number

7. rewrite worded questions in terms of equations and symbols that you can solve.

8. break the problem down into easier chunks. For example if you only need to solve something for three different cases, then tackle those cases one at a time

9. give quantities variables so that they are easier to work with

10. systematically chose a method. This one may not be relevant, but if you are proving something you should go through the common techniques and see if any of those work best (such as induction, contradiction, contrapositive etc)

11. after you have finished check every step and make sure they follow on logically.

I hope this helps you make a start at solving some problems that you get stuck on. Have fun!

Edit: I forgot! Number 12) draw a diagram. This can be very helpful, even if the problem is not directly a geometry problem, as it just gives you a picture of what the situation actually looks like.

Answer 3

First, it might be helpful to get familiar with the various objects of mathematics. With numbers, think about odd and even numbers, primes, multiples of two, three and four. Draw pictures of square and triangular numbers. In geometry think about the properties of triangles (their angles add to $180$), squares (their angles add to $360$ and their diagonals meet at $90$ degrees) and parallel lines (opposite angles add to $180$). As you play with them you will learn their properties and develop your intuition.

Second, mathematics proceeds by logic so perhaps read something about logic. For example. I always wear my coat when it is raining, if I am not wearing my coat, it cannot be raining. This is an example of syllogism and goes back to the ancient Greeks. In mathematics, Prime numbers are only divisible by themselves and $1$, if $x$ is divisible by $3$, it cannot be prime.

In summary.

1. Think about the definitions of various mathematical objects (e.g. a prime number is only divisible by itself and $1$)

2. Develop an intuition for them by playing with them (e.g. draw a square of all the numbers from $1$ to $100$, if you cross out all the multiples of $2$, $3$, $4$ etc you will be left with the primes

3. Get interested in logic. Mathematical reasoning always follows the rules of logic.

Answer 4

Work through problems with your peers, i.e. classmates. You will be able to build on each other's strengths and weaknesses. What may seem impossible to solve to you, might seem straightforward to a buddy, and vice versa. Only by sharing knowledge are we able to learn how other people approach different problems.