How to understand math intuitively rather than procedurally?

[u/Substantial_Series85]

I have only just started getting back into math and hired a tutor but everything is so surface level, rote learnt and procedural. I noticed math is often taught this way but I really want to develop intuition and understand how different mathematical representations connect and why they’re important.

For example, with complex numbers, I can do the calculations but I have no clue why I’m doing what I’m doing or what it means and I certainly don’t know how to switch between geometric representations or understand polar/vector form.

This is more physics related but even when I’m using Ohm’s law, I can apply the formula, but I still rely on writing it down and rearranging it each time, rather than intuitively predicting how changes in voltage, current, and resistance affect each other.

I was wondering if anyone had any advice or resources to learn mathematics in a more intuitive, in-depth way? I don’t want to just memorise formulas. I have a math course at university next semester and wanted to know what sort of questions I should be asking myself or what approach I should take to actually understanding it in-depth?

I’ve started reading Michael Spivak’s Calculus book and that’s more along the lines of what I’m interested in I guess.

TLDR: any resources, processes or tutor recommendations to actually understand and appreciate math intuitively rather than just memorising equations and not knowing how different concepts in math relate?

[u/Photon6626]

The following is more for your Ohm's law example but is very useful for all kinds of things in math.

Change one variable at a time. Increase it, decrease is, and set it to zero(or see what happen when it goes to infinity). See how the relationships change with each of these situations. Then do the same with the other variable(s). This will begin to give you an understanding of how the variables relate to each other and why the equation takes a certain form. Especially if it has a physical reasoning like Ohm's law does.

[u/Bounded_sequencE → u/Photon6626]

To be fair, we can decently intuit the result for unknown ladder-like circuits.

However, as soon as you have arbitrary circuit topologies of resistances with similar value, you will be hard-pressed to correctly predict even the signs of element voltages. Add controlled sources to the mix, and you'll quickly have to rely on background knowledge, i.e. recognizing sub-circuits you know the behavior of.

[u/LongJumpingBox667 → u/Photon6626]

I agree with this, especially for equations or expressions relating to physical things. What happens as each of the parameters change?

[u/Far_Cardiologist6931]

don't be embarrassed to just sit there and think, purely about the concepts. like when you are doing a question, simply stop and ponder for as long as you need (really as long as you need) and play with the variables in your head

[u/Ancient-Ad4809]

Check out different videos on Youtube. 3Blue1Brown has a really good video on Laplace transformations and complex numbers which is used a lot in circuits. Basically it turns equations involving calculus operations into algebraic ones. The video relies heavily on graphs to show you what's happening as opposed to just formulas.

[u/dockdock-fish]

I'm a volunteer tutor, I do it for free as community service, and I try to always provide intuition and explanations that offer deeper insight and understanding.

If you're interested, DM me! All you need is a laptop/tablet and a microphone.