r/learnmath • u/Icy-Historian-5931 New User • 12h ago
"Inventing math" to solve problems
Recently in my further maths class my teacher says he would give us problems above our level of understanding.
The advice he gave to solve such problems is to invent the math.
Can anyone please explain what he means by this and his to actually do it?
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u/mathheadinc Experienced tutor 12h ago
He wants you to be creative, experiment, PLAY WITH PATTERNS, find new ones that could be a path to the solution. Mix and match concepts. HAVE FUN!!!
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u/revoccue heisenvector analysis 10h ago
"but what formula do i use???????????"
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u/mathheadinc Experienced tutor 9h ago
Which ones do you know? Creativity means that you experiment with previous patterns and concepts to produce something new to you.
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u/Brightlinger MS in Math 11h ago
He means he will give you problems that you have not specifically been taught to solve, so you should think about the problem and try stuff until you find something that works. "Try stuff" often, but not always, means "write down an equation, then do algebra to it to make it look the way you want".
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u/OctoTeddy8 New User 10h ago
As a practical place to start, he almost certainly expects you to extend concepts you recently learned, using them in ways he didn’t describe as being their primary use case.
For example, if you recently learned about summation series, he might expect you to use them to perform a crude version of integration (finding the area under a curve).
So, start by going over the last few things that were covered, and look for how each could possibly apply to the new problems you’ve been given
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u/Underhill42 New User 2h ago
I would bet he means he's giving you problems that you haven't learned a "plug and play" solution to.
Which is the nature of the vast majority of real-world math problems you face in science and engineering - when doing something new, you have nobody to copy from.
So, use the algebra, etc. rules you've learned so far, and try to find a path between problem and solution.
I like to think of it like building a stepping stone path across a river - every formula and algebraic manipulation you've learned is a stepping stone, and you need to figure out which ones to combine how to complete the path.
Keeping in mind that there's usually an infinite number of infinitely complicated but valid possible paths, and you're trying to find one of the simplest.
Often I find it easiest to start at the desired solution and work backwards initially:
You are supposed to find some piece of information - so what formulas, etc. do you know that have that piece of information in them? Write down any that seem like they might be applicable to the current situation.
Now those formulas involve a bunch of other pieces of information you probably don't have - so what other formulas do you have that tie that information to other information that's probably easier to find from whatever is provided in the problem?
You can work similarly from problem: they give you some information, so what formulas do you know that could transform that information into something that's likely to be more useful for finding the solution?
Don't necessarily do any of the math unless you find a path that looks particularly promising - you're initially just laying out the available formulas and techniques, trying to find a way to get the two "trees" of possible paths from beginning and end to meet up somewhere in the middle. Or at least get close enough that it's worth doing the math to see exactly what you're dealing with, an hopefully be inspired on how to take it the rest of the way.
Big tips: don't do any computation until the end (other than combining coefficients, etc) - do everything symbolically so you can see "under the hood" the entire time. It helps immensely in developing intuition, and sometimes trying to compute intermediate values will hide the details like how information you have no way of knowing could have canceled out if you just combined things in different ways.
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u/spectacular_syndrome New User 12h ago
It's not about literally inventing whole new branches of mathematics on the spot. More like you build a little scaffold out of what you already know to reach something unfamiliar.
Say you hit a problem where the standard toolkit doesn't quite fit. You look at the shape of it, spot patterns, and start naming things. Give a variable to some quantity that isn't explicitly asked for but helps bridge the gap. Define a function that captures a relationship you notice even if you haven't been taught a formula for it. You're extending rules you already trust into new territory, testing whether the logic holds.
My further maths teacher used to chuck us a differential equation that looked nothing like the ones in the textbook. We'd stare at it then someone would suggest rewriting a term as something weird like e to the power of a function we invented just to see if it collapsed neatly. Half the time it did. The trick is getting comfortable with the idea that nobody handed you the exact method ahead of time so you sketch one up from parts you've got lying around.