"The argument is strongly geometric, which bothers some math teachers. There are spatial relationships and visual inferences to be made. The very word 'geometry' makes some math teachers uncomfortable. Geometry."

in #MathsToday: four different propositions, each proved by induction.

Fascinatingly, the first three (divisibility statements) were all proved using different but interchangeable methods.

The full statements from the image can be found below 👇🏻

The conjugate of a sum is the sum of the conjugates and the conjugate of a product is the product of the conjugates. This is completely obvious if you make a picture like this.

Skew decagon. For all you degenerate perverts who don't have the decency to insist that the vertices of your polygon all lie in a plane.

Math Ed. Question of the Day:

What's the latest "research" and/or vibes on testing students in advanced math classes by providing a finite list of complicated proofs that they have to "learn"/memorize and then produce in timed conditions on an in person exam without notes? Do people still do this?

Take f(x) = (2+cosx)/x²

It's "decreasing" in the casual (incorrect) sense that f is roughly getting smaller and approaches 0 as x →∞. In fact, 1/x² ≤ f(x) ≤ 3/x² and the function will bounce up and down inside this "envelope".

So it's not decreasing. Is there a term for this? "Kinda decreasing"?

Settle in for an 83-min proof of the Intermediate Value Theorem.
You'll thank me later.
youtu.be/SL_7yr8kDkI?...

Chain Rule? Never heard of it.

🤔🤔🤔

What's up with high school math? Is it just a bunch of formulas? Kind of!

I gave this talk a month ago: Rediscovering Formulas Through Examples.

It's expert analysis of the core issues. You're welcome.
youtu.be/6k_SJLzcQDs

Are y=x² and x=√y functions? Are they the same function or are they inverse functions? [Note: recent explosion of controversy on this topic.]

Is there a meaningful difference b/w functions and formulas?

Are A=πr² & r=πA² the same formula or not?

Are A(r)=πr² & r(A)=πA² the same function or not?

Find the inverse of:
f(x) = (2x+1)/(3x-2)

Then... discuss all things. Explain, analyze, generalize. Don't stop until everything is obvious.

For anyone taking (or teaching!) an Intro to Logic class this semester, here's a plug for my video series on Natural Deduction that I made 2 years ago. I use a Fitch-style system. It's quite popular already! youtube.com/playlist?lis...

Final thoughts. A geometry-based approach.

Certain formulas become famous because they accurately describe the relationship between the x-coordinate and the y-coordinate on certain shapes AS MEASURED FROM THE ORIGIN.

Find the inverse of:
f(x) = (2x+1)/(3x-2)

Then... discuss all things. Explain, analyze, generalize. Don't stop until everything is obvious.

For anyone taking (or teaching!) an Intro to Logic class this semester, here's a plug for my video series on Natural Deduction that I made 2 years ago. I use a Fitch-style system. It's quite popular already! youtube.com/playlist?lis...

One great thing about teaching that no one ever talks about… free umbrellas. Pretty much just an unlimited supply of umbrellas. Become a teacher and you just don't need to worry about umbrellas ever again.

as an ultrafinitist, i think it's nonsense when people try to talk about numbers larger than the number of electrons in the universe. that's why i don't believe in 2

sin(A+B+C)