Dear @graciegcunning,
Since @JoelChan86 @solidangles @hobbet and @mathillustrated all tagged me, I’ll take a crack at your first 2 questions: “how did they (mathematicians) know what they were looking for in theorizing about formulas, and how did they know they were right?” 1/ https://twitter.com/graciegcunning/status/1298804338727489536

One of the great things about math is that it equips you to see the unseen: things otherwise invisible. There’s structure—patterns—all around us, governing the motion of planets, the spread of a virus, the data that Netflix uses to guess what you want to watch next. 2/

If you begin to ask ‘why’ (which you did in your videos) you are starting the journey of mathematical thinking (bravo!) that helps you see these unseen patterns. There’s great satisfaction and joy in understanding deeply *why* patterns are there: you’re unlocking mysteries! 3/

A formula is nothing more than a pattern: a relationship between quantities like the radius (of a pizza you order) & area (you eat). When you first search for a pattern you don’t know what you’ll find in advance (though with experience you might have an educated guess). 4/

But when you explore enough examples, a pattern (formula) may reveal itself. The next step is to see if you can understand *why* the conjectured pattern holds. That’s how you know the formula is true. 5/

Notice what I just said: you know a formula is true when YOU YOURSELF (not just other people) understand it. Math demands an investment of your thinking in order to behold its wonder. But it’s worth it, and better than the version of math that is just memorizing stuff. 6/

(That’s the version you were rightfully lamenting in your first video). The better version is: seeing the unseen, mysteries revealed! Who doesn’t want this superpower: to see the unseen? And not have to rely on an external authority to know what is true? 7/

I remember beginning to love math when I saw this ‘proof’ that the area of a circle is pi*r^2 (see Wikipedia image). Slice pizza into thin wedges, rearrange in rectangular way, note height is r and width is about pi*r (half circumference). So area is pi*r^2, approximately 8/

This approx gets better with thinner slices (the idea of calculus!). Of course this proof depends on knowing formulas for rectangle area & the circumference of the pizza. But math does that: makes connections between things you know, resulting in “aha!” when things make sense. 9/

So every time you see a formula, remember it’s the culmination of many repeated attempts to unlock a mystery, a triumph by each person who, like you, earnestly asks ‘why’ and endeavors to understand its rightness for themselves. End/