I just saw this great YouTube video giving an intuitive explanation of why e^(i*pi)=-1. The creator doesn’t use any trigonometry or calculus, or really any algebra. Just basic arithmetic and lots of great visuals. Give it a watch here:
I often find myself relying on rigorous notation and calculation when trying to explain an idea–both in the classroom and when writing papers. Of course, being correct is a priority. But being convincing is nearly as important. If a student or reader walks away thinking, “I guess I buy it. I mean, all the algebra checked out,” then I’ve given them very little basis for growth.
I want to start including more intuitive explanations in my teaching. I can still give the formal proof of a concept, but let me first try to justify the outcome naturally.
Why does sin(x)/x approach 1 as x approaches 0? Not because the squeeze theorem says it does, using two cleverly chosen other functions. But because sin(x) has values very close to x near the origin.
Why does the formula for Simpson’s Rule work? Again, not because we can do a bunch of arithmetic and change the indices on a summation. It’s because it’s a weighted average of two other approximation rules, one that over-estimates and one that under-estimates.
I can also focus on giving intuition when writing my research. Before I define some new variable that seems to come from nowhere, and before I start some algorithm on a graph that decomposes it for some reason, let me invest a couple sentences saying “Here is the goal. Here’s how we’ll reach it.” Now the reader has some idea where we’re headed and may suddenly recognize the route we’re taking to get there, instead of spending the entire proof scratching their head. It’s not enough that they agree that all the variables add up the way I claim; the reader should walk away feeling like they’ve learned a method they can reuse.
Brad Elliott's Math Blog 