Will Rose

"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."

see next post or tweet or whatever for the content of this image

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"?

YouTube video by William Rose Logic - Proof of the Intermediate Value Theorem

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.

🤔🤔🤔

YouTube video by William Rose MCTM Presentation: Rediscovering Formulas Through Examples

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.

A Short Course on Natural Deduction: Fitch-style proofs for Propositional and Predicate Logic Share your videos with friends, family, and the world

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.

If you go and move your shape away from the origin to a new place where it doesn't want to be, then you have to describe the relationship in terms of distances to the moved origin.

This approach has the benefit of being non-verbal and giving geometric (concrete?) meaning to the expression x-a.

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.

But students should probably just plot lots of points and learn the theorem through many repeated direct experiences with the chart data (which do not lie!). Then, after the comfort level is high, for those asking why, there's this kind of lesson. But I don't think it's a good intro to the topic.

In practice, it's just something you get comfortable with or memorize. When confronted with f(x-3), you kinda just plug in 3 for x, compute f(0), plot that and just go with the flow. The graph has already moved to the right, it's happening before my very eyes, which is exactly what I expected.

But IMO this isn't actually "helpful" in the traditional sense. I don't recommend that students use this sort of reasoning in practice. It's a rhetorical thing: math DOES make sense. It is possible to understand everything. You don't have to just memorize it as a counterintuitive trick.

With f(x-3), we subtract 3 from the clock time (move the clocks back, so to speak), so everything happens the same way, but 3 hours LATER. It's a delay. Sorry, everyone, the plane is delayed 3 hours, we can all relax for awhile. Just subtract 3 from the clock time, and follow the old plan.

With f(x+3), we add 3 to the clock time (move the clocks forward, so to speak), so everything happens the same way, but 3 hours EARLIER. Suddenly we need to leave the house 3 hours earlier than we thought! Quick, advance the clock 3 hours and then follow the old plan!

Read this and discussed with a few friends. I appreciate what you're doing, but found the old/new notation confusing.

My approach was to bring time into it.

f(x) is a plan to do certain y-values at certain TIMES.

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

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

Here's the video version:
youtu.be/h0anjfhfkjs?...

A Short Course on Natural Deduction: Fitch-style proofs for Propositional and Predicate Logic Share your videos with friends, family, and the world

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)