This Second Grade Math Problem Won't Leave Me Alone

“Problems worthy of attack prove their worth by fighting back.” - Piet Hein

The problem seemed pretty harmless at first. It was October. I was recording a live episode at NCTM of my podcast Math Teacher Lounge with early math education expert Dr. Jennifer Bay-Williams.

She gave me a deck of playing cards. Only the numbers. One of the face cards stood in for a zero. Bay-Williams told me to deal off four cards.

She then told me to use those cards to create a two-digit subtraction problem so that it had the smallest difference those cards could produce.

In this case, I found a smallest difference of 4. She told me we were going to play the game five times but keep only four of the smallest differences. We’d add them up and compare our total score to other people in the audience. I had to decide whether or not to throw this score back or keep it. I kept it and dealt the cards again.

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First, I want to say that this particular problem has a lot of advantages over other … let’s just call them common approaches to fluency with two-digit subtraction problems.

Specifically, the Bay-Williams treatment featured more choice, optimization, collaboration, randomization, experimentation, and iteration than those common approaches. All of those features kept me engaged and working. But another difference was depth. Because I’m still thinking about that problem two months later.

For example:

• Are there general strategies for arranging cards for the smallest difference?

• What smallest differences are the most common? How often would we hope to see a 0, for example?

• What is the largest smallest difference? Put another way: what are the worst cards to deal yourself? Are there cards you may as well immediately throw back if you see them because any other set of cards will get you a smallest difference that’s the same or smaller.

I have been digging into each of these questions this week and my answers just produce more questions. I can’t quit this problem. I’m skipping meals. I am trying to free myself by passing this problem off to you.

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This tweet from the last week seems relevant here.

Agreed, except to add that trying to categorize experiences in math class as either “real world” or “not real world” is an intellectual cul-de-sac that will trap you for your entire career if you let it. Every year, you will discover anew all the ways that word problems loosely wrapped in a “real world” context fail to excite your students or help them learn.

Instead, you will open up new frontiers of engaging opportunities for your students if you realize that, for many of them, whole numbers are in their real world. They’re as real as lots of objects made of atoms and molecules and more real in lots of cases. (Give these second grade students a math problem about car insurance policies, for example.) 

I’ll stand with the Freudenthalians here and say that something is real if it’s “real in your mind,” if it’s something you can hold in your head, something you can mentally turn this way and that way, something you can place next to another thing like it and make an argument about how they’re the same or different.

In math class, “real” is a verb, not an adjective. “Real” isn’t an inherent property of a math task. “Real” is what you do with it.

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This will be my last newsletter for the year. On a gratuitously personal note, it’s been maybe ten years since I enjoyed writing like I enjoyed writing this newsletter in 2023. My love for teaching and learning has never felt stronger and words and sentences have never felt more real to me—like things I can mentally turn this way and that way—than they did this last year. I’m grateful that so many of you took the time to share your insights and sharpen my own, both in the comments and at various points in person. Rest up and let’s meet back here in 2024.

Odds & Ends

PS. Start your 2024 off right by signing up for the virtual math symposium I’ll be headlining along with some of my favorite math education thinkers here at Amplify.