# The Academy Award for 2022’s Best Math Lesson Goes to ...

### Deconstructing Dan Finkel's Rosenthal Prize-winning math lesson.

The Rosenthal Prize, dubbed by many (me) the Academy Award of math lessons, awards $25,000 to the author of a math lesson every year. Dan Finkel won this year’s prize and just released his winning entry, “Billiard Ball Problem.”

While I have no insight into how the Rosenthal committee makes their decisions, I do think it’s a very good problem that gives us some very useful insights into how to design very good problems for math students. Here are three.

### Draw on a student’s concrete knowledge.

The lesson starts with a billiard ball sailing across a table at a 45° degree angle and then spends three days asking the question “What’s going to happen?” in different forms.

“What’s going to happen?” is a question that every middle school student asks themselves hundreds of times a day. Every middle school student has experienced objects in motion and ricocheting off surfaces, and every middle school student understands on an intuitive level the physical forces at work here.

In her introduction to *Rehumanizing Mathematics for Black, Indigenous, and Latinx Students*, Rochelle Gutierrez urges teachers to draw on a student’s “voice, vision, touch, and intuition.” Every student—whatever their experience with mathematical abstractions like angles and proportions—has an intuition for where that billiard ball might go next. That’s a great launch.

### The context *becomes* real.

Is this a “real world” math problem? Please do not attempt to answer that question. It has ensnared and defanged countless promising curriculum designers who proceed to drape math on top of contexts that clearly have no use for it, designers who miss the fact that math *itself—*a world of numbers and shapes and patterns—is a world that is deeply *real* to students.

More to the point, contexts aren’t real or fake in any kind of Platonic sense. They *become* real as we inhabit and explore them, an observation made by Hans Freudenthal and many others.

In Day 2 of Finkel’s lesson, he senses that our work on the “What happens?” question has made the billiard context more real to us. We understand the trajectory of the bounce better. We understand the range of possibilities better. Finkel sees all those cognitive resources, and invites us to play with them like toys. “Can you predict which corner the billiard ball will end in?” he now asks, a question that wouldn’t have felt as interesting or accessible on Day 1.

### Create a need for new math, and a connection to the old math.

On Day 3, Finkel encourages students to *prove* some of their conjectures. But proof logic requires an ability to manipulate abstractions that eludes many adults, to say nothing of kids.

Finkel is operating with a full palette, though, and understands there are colors between a) deductive proof and b) verification through drawing. And that’s to ask students a question that makes verification through drawing unappealing, one that requires a leap with logic.

If full arguments prove too difficult for students, ask students to write down a solution for how to use the patterns and conjectures to

make a prediction for what corner a billiard ball would end up in on a 36 by 98 table. This is too big to do by hand, so they’ll have to apply the conjectures from Day 2.

Congrats to Dan on a very nice lesson. Check the lesson out yourself and let us know any features you like about it.

**2023 Feb 23**

Check out Dan Finkel’s comment here.

### What Else?

**Dylan Kane**describes “What I Mean When I Say ‘Blogging is Dead’” which is a lovely time capsule and ode to a time of teacher development that was very meaningful for a bunch of us.**The Desmos curriculum team**(my colleagues) just completed their Algebra 1 curriculum and are clearly operating at the height of their powers. I recommend you check out Lizard Lines, Finding Desmos, Solutions and Sheep, Shelley the Snail, and then get yourself a free trial of the whole 6-8 & A1 curriculum.

Thanks, Dan, for your thoughtful analysis of this lesson!

I wanted to mention that there are three Rosenthal prize-winning lessons chosen every year. Of these three, mine was ranked second; so in the analogy of Academy Awards, perhaps mine won Best Director. The grand prize winner (Best Film) was some kind of divisibility rule game based on Uno. I'm very keen to see that lesson. The third prize winner (Best Screenplay?) was a game based on Sumo wrestling, where players use plane transformations to push their opponent out of a given space. I haven't read that one yet, but I did talk to the author.

I'll also mention that I have the mistake of launching this lesson by rushing to the second day of it too early, assuming that the intuition about a bouncing ball in a rectangle would lead middle schoolers to draw it's path on graph paper easily after 5 or 10 minutes of thinking about it. However, the distance from an intuition about a bouncing ball and the ability to reliably draw it on graph paper takes enough students a full class to practice with that the scaffolding of the first day is critical to the success of the lesson. That was a hard-won lesson for me.

It's a great activity! I got a lot of mileage out of it in my staff development days since it's a wonderful discovery lesson which turns on many aha moments. I was inspired to create a Scratch version. The link is in my blog entry: https://dmcpress.org/2020/04/01/an-online-math-activity-pool-paths/