How to train your creative thinking skills to help solve complex math problems?

If you have studied maths long enough, you would notice there is a large number of math problems that you can't just calculate away, or regurgitate proofs for. The active problem solving skills you will need to tackle some math problems will require creative thinking, and not the analytical skills normally associated with maths.
The problem being, it's difficult to develop such skills without having to go through a ridiculous number of math problems. As such, unless you get over this hurdle, math grades can hit a ceiling.

Are there any specific training/courses people go through or is there any way to improve your creative thinking to tackle math problems?
If not, how do people generally improve in say, Math A Level from a C to an A* grade?
What do people do in Further Maths, and how do they differ in they way they tackle problems or think compared to Math students?
How do people doing bachelor's, master's, and PhDs in math think differently?

Thanks

Reply 1

username4591444

21

Things like learning a language or new instrument can help bridge the neural gap between the two because the brain is used in very similar ways to science and maths, but are creative in nature (musical, literary).

I've personally found that as well learning a new language and writing it (particularly the fact that it is a new alphabet too), my overall focus and patience for learning is better, and my mental maths too. To go up grades like that you need to spend the time really understanding the mathematical rules properly and train your brain into recognising and tackling them faster. It does as you say take time and practicing a lot, but it doesn't have to be pages and pages.

That is only necessary cause the same logic can be presented to you in a maths problem/question that varies a lot.

Edit: Your creative thinking will naturally apply i would have thought, if you have truly understood the rules of that maths area

(edited 3 years ago)

Reply 2

zetamcfc

19

Original post by MindMax2000

If you have studied maths long enough, you would notice there is a large number of math problems that you can't just calculate away, or regurgitate proofs for. The active problem solving skills you will need to tackle some math problems will require creative thinking, and not the analytical skills normally associated with maths.

The problem being, it's difficult to develop such skills without having to go through a ridiculous number of math problems. As such, unless you get over this hurdle, math grades can hit a ceiling.

Are there any specific training/courses people go through or is there any way to improve your creative thinking to tackle math problems?
If not, how do people generally improve in say, Math A Level from a C to an A* grade?
What do people do in Further Maths, and how do they differ in they way they tackle problems or think compared to Math students?
How do people doing bachelor's, master's, and PhDs in math think differently?

Thanks

''The problem being, it's difficult to develop such skills without having to go through a ridiculous number of math problems. As such, unless you get over this hurdle, math grades can hit a ceiling.'' This doesn't really help with creativity, in fact I think it probably hinders it judging by questions that get asked here for 1st year undergrad maths.

''Are there any specific training/courses people go through or is there any way to improve your creative thinking to tackle math problems?'' No, there is no training course. Start being more laid back, reading in lots of areas of maths and experimenting/spitballing with other people.

''What do people do in Further Maths, and how do they differ in they way they tackle problems or think compared to Math students?'' Doubt they tackle problems differently, they might have exposure to more mathematics which will help but they still approach questions the same way.

''How do people doing bachelor's, master's, and PhDs in math think differently?'' Kinda hard to explain. I think not always having direct answers to questions helps. But fundamentally at university you deal in proofs, at A-level you deal in examples, which means you have to think abstractly.

Reply 3

MindMax2000

OP

Universities Forum Helper

21

Original post by zetamcfc

''The problem being, it's difficult to develop such skills without having to go through a ridiculous number of math problems. As such, unless you get over this hurdle, math grades can hit a ceiling.'' This doesn't really help with creativity, in fact I think it probably hinders it judging by questions that get asked here for 1st year undergrad maths.

''Are there any specific training/courses people go through or is there any way to improve your creative thinking to tackle math problems?'' No, there is no training course. Start being more laid back, reading in lots of areas of maths and experimenting/spitballing with other people.

''What do people do in Further Maths, and how do they differ in they way they tackle problems or think compared to Math students?'' Doubt they tackle problems differently, they might have exposure to more mathematics which will help but they still approach questions the same way.

''How do people doing bachelor's, master's, and PhDs in math think differently?'' Kinda hard to explain. I think not always having direct answers to questions helps. But fundamentally at university you deal in proofs, at A-level you deal in examples, which means you have to think abstractly.

Thanks for your input.

Do you have any recommendations on where to start reading for ideas (regarding the second question)? I have went through entire textbooks and dealt with all the math problems they had at A Level before, but it doesn't seem to help. I am not sure whether it's the approach or the way I think about the problems. Having said that, some of the textbooks I have used weren't particularly helpful.

I think one of my main sticking points is failing to see a simpler approach to a problem, and instead go for the more complicated way around things. Another would be failing to recognise patterns, clues, or shapes from one part of a problem that is very much applicable in another part (e.g. scaled down shapes in geometry problems), or failing to recognise you use integration in parts in certain circumstances, and product rules in others. Once I have the structure of the problem in place, the calculations tend to be straightforward for me.

I'm particularly intrigued by stories about people like Richard Feyman on how he managed to solve integral problems using different methods and strategies that other physicists did not used and struggled with the problems for days and months. I've recently took an interest in YouTube videos on math problems, but their approach is different to how I would approach the problems.

Reply 4

MindMax2000

OP

Universities Forum Helper

21

Original post by leopard202

Things like learning a language or new instrument can help bridge the neural gap between the two because the brain is used in very similar ways to science and maths, but are creative in nature (musical, literary).

I've personally found that as well learning a new language and writing it (particularly the fact that it is a new alphabet too), my overall focus and patience for learning is better, and my mental maths too. To go up grades like that you need to spend the time really understanding the mathematical rules properly and train your brain into recognising and tackling them faster. It does as you say take time and practicing a lot, but it doesn't have to be pages and pages.

That is only necessary cause the same logic can be presented to you in a maths problem/question that varies a lot.

Edit: Your creative thinking will naturally apply i would have thought, if you have truly understood the rules of that maths area

Thank you for the response.

I appreciate that it takes a lot of practice. Thing is, I have went through entire textbooks and tackled all math problems they had before, and it doesn't seem to have helped. I was wondering whether it was my approach.

Reply 5

hajima

20

Original post by MindMax2000

Thanks for your input.

Do you have any recommendations on where to start reading for ideas (regarding the second question)? I have went through entire textbooks and dealt with all the math problems they had at A Level before, but it doesn't seem to help. I am not sure whether it's the approach or the way I think about the problems. Having said that, some of the textbooks I have used weren't particularly helpful.

I think one of my main sticking points is failing to see a simpler approach to a problem, and instead go for the more complicated way around things. Another would be failing to recognise patterns, clues, or shapes from one part of a problem that is very much applicable in another part (e.g. scaled down shapes in geometry problems), or failing to recognise you use integration in parts in certain circumstances, and product rules in others. Once I have the structure of the problem in place, the calculations tend to be straightforward for me.

I'm particularly intrigued by stories about people like Richard Feyman on how he managed to solve integral problems using different methods and strategies that other physicists did not used and struggled with the problems for days and months. I've recently took an interest in YouTube videos on math problems, but their approach is different to how I would approach the problems.

You can train problem-solving by doing problem-solving questions, there isn't any other way. At first, it is frustrating, you might spend a day on a problem and not get the answer, but typically you gain nothing by just looking at a marking scheme; look for inspiration and think about what you know on a fundamental level. IMO contestants typically train by doing exactly this (as well as learning niche theorems and mathematical techniques). I would advise you to start with TMUA, SMC and MAT papers, they're full of difficult problems that only require AS and GCSE maths content. From there progress to BMO1/MOG and the STEP I foundation modules, once you've learnt the A-Level Maths syllabus you can attempt full STEP I questions.

I would comment that the Maths done at undergraduate is very different from the Maths done at A-Level, you're not so concerned with calculating and evaluating as much as you are proving and deriving, so there's not much use in drawing a comparison in terms of their approach to problems because the problems are fundamentally different in style. As a year 13 studying Further Maths, further mathematicians aren't super different in their thinking but I have noticed that FM papers tend to have more problem solving in them than the standard maths papers, so you'd need to be more capable at applying concepts fluidly in order to access the near-full-marks range (it's still relatively simple to get an A* though).

Reply 6

zetamcfc

19

Original post by MindMax2000

I'm particularly intrigued by stories about people like Richard Feyman on how he managed to solve integral problems using different methods and strategies that other physicists did not used and struggled with the problems for days and months. I've recently took an interest in YouTube videos on math problems, but their approach is different to how I would approach the problems.

Well he just used existing methods that existed in mathematics that the other physicists didn't learn/remember. Quite a lot of Physics is done that way, they need to explain something mathematically and they go back to the mathematical literature to find something.

From your problems it seems you just need to practice more.

Reply 7

DFranklin

18

Original post by MindMax2000

Are there any specific training/courses people go through or is there any way to improve your creative thinking to tackle math problems?
If not, how do people generally improve in say, Math A Level from a C to an A* grade?
What do people do in Further Maths, and how do they differ in they way they tackle problems or think compared to Math students?
How do people doing bachelor's, master's, and PhDs in math think differently?

Thanks

Firstly, without wanting to sound too elitist, it has to be said that there's a huge difference in the amount (and depth) of problem solving between FM A-level, a full maths degree, and a PhD, to the point that I'm not sure you can reasonably discuss them all in the same thread. Your specific questions are about A-levels, so I'm going to focus on around that level.

For the specific question of "how do people get from a C to an A* grade?", it really is just about knowing all the material well and not making basic errors. You don't need amazing problem solving skills, but you do need to be familiar enough with the material that if you see, for example,

\sin x \cos x

you instantly see this can be rewritten as

\frac{1}{2}\sin 2x

. Because then, when faced with

\int \sin^2 x \cos^2 x \,dx

, it's almost automatic to see "I can reduce the complexity of this by rewriting as

\frac{1}{4} \int \sin^2 2x\, dx

, and then I can rewrite

\sin^2 2x

as

\frac{1}{2}(1 - \cos 4x)

", and then I have something easy to integrate. Technically there's a bit of problem solving there, but it's not a lot if you know the relevant identities like the back of your hand. If you had to search in the formula book for them, it would be a lot harder to realise what option works here.

Original post by zetamcfc

''The problem being, it's difficult to develop such skills without having to go through a ridiculous number of math problems. As such, unless you get over this hurdle, math grades can hit a ceiling.'' This doesn't really help with creativity, in fact I think it probably hinders it judging by questions that get asked here for 1st year undergrad maths.

At the level of the OP, I think you're going through that "ridiculous number of maths problems" just to solidify understanding and polish skills. Because it doesn't matter how great your problem solving skills are, if you regularly write things like

(a+b)^2 = a^2 + b^2

, you're not going to be solving many complex problems - you'll get destroyed by stupid mistakes. (Also, if the basic algebra/trig/etc. is taking 90% of your cognitive capacity, it doesn't leave much for problem solving).

At the same time, I take your point - the big issue I see with most undergrad posts is the expectation that they should get a lot of handholding. There's no "self-reliance" - no ability to search on the internet, or experiment, or even check their own results. It does seem particularly bad this year though, so I'm wondering how much that is Covid + online learning etc.

Original post by MindMax2000

Thanks for your input.

Do you have any recommendations on where to start reading for ideas (regarding the second question)? I have went through entire textbooks and dealt with all the math problems they had at A Level before, but it doesn't seem to help. I am not sure whether it's the approach or the way I think about the problems. Having said that, some of the textbooks I have used weren't particularly helpful.

If you're talking A-level, my own experience was that (on topics where I needed to improve) going through *all* the examples in Bostock + Chandler made a huge difference. It was a *lot* of work though. (This was quite a while ago, so it was "The Core Course for A-Level", I don't know what the current version is like). Again, though, this was 90% just drilling skills - not problem solving.

I think one of my main sticking points is failing to see a simpler approach to a problem, and instead go for the more complicated way around things. Another would be failing to recognise patterns, clues, or shapes from one part of a problem that is very much applicable in another part (e.g. scaled down shapes in geometry problems), or failing to recognise you use integration in parts in certain circumstances, and product rules in others. Once I have the structure of the problem in place, the calculations tend to be straightforward for me.

Would be good for you to post some actual examples.

I'm particularly intrigued by stories about people like Richard Feyman on how he managed to solve integral problems using different methods and strategies that other physicists did not used and struggled with the problems for days and months. I've recently took an interest in YouTube videos on math problems, but their approach is different to how I would approach the problems.

Firstly, you're not Richard Feynman. Looking to learn from his approach is somewhat like wanting to learn mountain climbing from one of those maniacs who free-climb 1000 ft cliffs without safety ropes - it's one thing to watch someone else do this, but something else entirely to try to do it yourself.
And just because these methods worked for Richard Feynman doesn't mean he wouldn't have done even *better* if he'd been more aware of the more standard techniques. (Various contemporaries have made this point).
It's also worth remembering these stories all tend to grow in the telling. You don't hear about the hundred integrals he couldn't solve this way.

From what I've seen of very high achieving mathematicians (senior wranglers, IMO winners, Fields medal winners), the norm is that they know an enormous amount of "standard mathematics". There are exceptions, but in general, they could spend days just listing theorems they've studied, never mind going into the actual proofs etc. that they are familiar with.

Reply 8

ran-dumb

21

Original post by MindMax2000

If you have studied maths long enough, you would notice there is a large number of math problems that you can't just calculate away, or regurgitate proofs for. The active problem solving skills you will need to tackle some math problems will require creative thinking, and not the analytical skills normally associated with maths.
The problem being, it's difficult to develop such skills without having to go through a ridiculous number of math problems. As such, unless you get over this hurdle, math grades can hit a ceiling.

Are there any specific training/courses people go through or is there any way to improve your creative thinking to tackle math problems?
If not, how do people generally improve in say, Math A Level from a C to an A* grade?
What do people do in Further Maths, and how do they differ in they way they tackle problems or think compared to Math students?
How do people doing bachelor's, master's, and PhDs in math think differently?

Thanks

For A level Maths (and Further Maths), there's very little creative element. The biggest part is understanding how things work and doing enough papers to be familiar enough for everything to be second nature.
The MAT and STEP are completely different from A levels in what skills you need to do well. From the MAT work I've done, the 'creative' part for me is sometimes just finding relevant ways to start a problem to get something I can go on (for example if I see 'find the maximum' I think differentiation, although of course sometimes it leads to a dead end and I need another idea).
Further Maths is just more theorems and methods, and doesn't really require more creativity. You'll need to be familiar and comfortable enough with the regular Maths things you've learnt, and perhaps as a result the people who do it are more mathematically inclined and have a more creative mind that helps for harder Maths, but I don't think it's to do with A level Further Maths itself

How to train your creative thinking skills to help solve complex math problems?

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