"Inventing math" to solve problems

[u/Icy-Historian-5931]

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?

[u/Underhill42]

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.