Bill Shillito

I think Harvard's plan to respond to the every-decade moral panic about grade inflation by capping A's is pedagogically obscene.

Grades should be a measure of learning, not competition.

Grade caps engender competition and anxiety where we should be teaching collaboration and a love of learning.

It especially hurts because even though I've bent over backwards trying to make my classroom a place where you can make mistakes, learn from them, revise your work, and improve without it tanking your grade, a significant number of students will just bypass that and outsource their learning to AI.

I feel this. I've had to return to handwritten exams, and I kind of hate it.

AI is going to trash every single way that we had modernised university teaching and send us back to handwritten exams and vivas for every assessment. Which will bring back every problem with those assessment models (and make us seem even more out-of-touch even as we just try to do meaningful work).

Complex function domain-colouring of roots of unity, in @desmos.com
Here's the graph:
www.desmos.com/3d/hadt9rebbe
They recently enabled colouring a surface using functions, so far only in 3D grapher. Side-effect: allows complex domain colouring, using 3D tool as a 2D one! #iTeachMath #MathSky

You didn’t! But I think I’ve solidly decided on CoV so this will come in handy!

The thing that clinched it is that to derive E-L, you need to diff under the integral sign.

How do you justify that?

Fubini’s theorem!

So even if there’s no literal double integral, it explains an important step!

I love this and might do it even if it isn't the capstone, but ... I've never done 3D printing! 😂 Any recommended resources for my students (and for me)?

Then again, tbh I'm okay with giving a high-level hand-wavy look at minimal surfaces if needed, at least if I can show the *principle* behind what's going on.

I'd been thinking about minimal surfaces, but actually SOLVING those?

Oof.

I know right?!? It's like Generalization: The Course.

The question is, would there be a good way to connect it to multiple integrals? I'm also writing my own book, and I want to put this capstone at the end of the multiple integrals chapter.

By the way, by "a future class" I just mean "a future calculus class in the I-IV sequence"!

For reference, my other capstones:

* Calc I: Proving FTC (since they've seen it early, but we do limits at the end)
* Calc II: Riemann hypothesis (uses SO many things related to series!)
* Calc IV: Differential forms and exterior calculus (fundamental all the theorems!)

Calc IV contains:

* Polar / cylindrical / triple integrals, Jacobians
* Vector fields, curl / div
* Line integrals, surface integrals
* All the fundamental theorems (Gradient, Green, Divergence, Stokes)

So I'm actually trying to avoid something that's going to show up in a future class.

This wouldn't be something I'd be testing students on, by the way — just a "wow that's cool" thing to do on the last day. So it's totally okay for it to be a little bit of a stretch.

Any suggestions?

For reference, at my school, Calculus III covers:

* Thinking in higher dimensions
* Partial derivatives (+ optimization)
* Vector-valued functions
* Gradients (+ Lagrange multipliers)
* Multiple integrals (Cartesian coordinates only)

Vector calculus stuff is pushed to Calculus IV.

One thought I had was a peek into calculus of variations, since it stretches how we think of dimensions by optimizing over an infinite-dimensional space of functions.

But the problem is, I really want something that somehow involves multiple integrals, and I'm not sure how I could work that in.

Question for anyone who's taught multivariable calculus (without line/surface integrals — just up through multiple integrals).

If you were to choose a "capstone" topic that ties multiple topics together, what would you choose?

Teaching the Calculus sequence again soon and trying to plan ahead.

Do you have any vivid examples? Would love to talk about this with my proofs class!

You asked why it would help didactically. I just gave my own observations from when I’ve taught Monty Hall.

I would argue that convincing is a necessary but not sufficient component of effective explanation. The math seems to “stick” better if it aligns with experience.

I think one reason many struggle with Monty Hall is that they still feel like the probability “should be” 1/2 and so even a perfect explanation doesn’t convince them.

I think a simulation like this can create a visceral cognitive dissonance that allows that gap to be bridged. They can SEE it now.

I know this feeling well. 😂

WOOHOO! Our sandwiches paper has been published! 🥳

...in celebration, I'm going to claim that tonight's lasagna was a sandwich. 🥪

Suppose you've made the following deductions: * The murderer must be Miss Scarlett or Colonel Mustard. S ∨ M * If Miss Scarlett did it, it was in the kitchen with the lead pipe. S → (K ∧ L) If your friend shows you the kitchen card, what can you conclude? ¬K

Teaching logic tomorrow ... with Clue!

It’s ironic that the people who seem to be able to benefit the most from AI tools are the ones who don’t actually need them.

If we want to destigmatize “guess-check-revise”, we need to also revise our assessments so we’re not expecting students to factor non-monic trinomials on a time crunch with a bajillion other skills.

How can we accomplish that? Seriously, what would it take?

Unfortunately, “enjoying learning and understanding” isn’t part of the grade. 😭

The last step of subtracting from numbers with lots of zeros is done mentally.

You can easily justify it using the "Same Distance, Same Difference" strategy in @howiehua.bsky.social's video.

Although I first encountered it as a "Vedic Math" sutra — "All from nine and last from ten"!

As my contribution to sharing subtraction strategies, here's how I do subtraction nowadays, using negative numbers!

Made this up on the fly way back when I first started tutoring because I couldn't remember how to do multiple borrowing anymore and I never liked the lie of "you can't take 3 from 2".

Left: Everything bagel deviled eggs, with smoked salmon and chives. Right: Dua Belibis deviled eggs, with green onion, cilantro, and chopped peanuts.

Deviled eggs for New Year's Eve!

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