Understanding Negative Numbers: In Praise of Tiny Games

“Is this right?” Inaya asked me, pointing to the equation she’d recorded on her paper: 5 + (-8) = -13

We are one week into our 7th grade unit on operations with positive and negative numbers. Inaya’s favorite strategy seems to be making her best guess, and then scanning my eyes for any hint of that she was right: an arched eyebrow, a slight glance to the left, or maybe a meaningful squint. Inaya reads microexpressions like the lead character in a procedural about serial killers. This girl is good.

Inaya also has some great mathematical reasoning, but it can sometimes be masked by her deep lack of confidence in her work, or some of her ‘unfinished learning’ from previous grades. When I interviewed her earlier this year, she demonstrated an impressive ability to decompose and compose numbers to solve additive situations. She stumbled when it came to multiplicative situations, and didn’t demonstrate any proportional reasoning. She seems… less than thrilled with her seventh grade experience.

I looked at her equation, (5 + -8) = -13, and blinked. That was enough for Inaya to scratch out her work. She rolled her eyes and sighed.

“Hey, hey, hey. Tell me about how you figured out your answer,” I said, encouraging Inaya to slow down.

“Why. It’s probably wrong.” More eye rolls. More sighs. I adore her and her fully 7th grade attitude! She’s so honest. I’d say the feeling isn’t always mutual. Some days, she can’t escape from our sessions fast enough.

“Just tell me what you’re thinking,” I implored. “It helps me decide what we should do next.”

“Fine,” Inaya sighed. “Whatever. So it’s 5 but then even though you’re adding it’s going down because it’s like adding 8 anchors. So then it goes down to -13.”

This is a fantastic start! Inaya was referencing the float/anchor metaphor we use in our Desmos Math Curriculum. Adding floats makes the sub go up. Adding an anchor will make the sub go down.

Desmos – Math 7, Unit 5, Lesson 2: More Floats & Anchors

Listening to What Inaya Knows

Inaya talked through a few more problems with me. Instead of correcting her, I listened with an interpretive lens. Amie Albrecht (@nomadpenguin) recently wrote a brilliant blog post about “Teaching Through Listening” that digs into when and why I might choose to listen interpretively, rather than try to correct Inaya or transform her thinking. At that moment, I wanted to make sense of what she was doing.

It was clear that she had a great understanding of the action part of the operation: adding a negative number decreases the value. Subtracting a negative number — removing an anchor — will increase the value.

However, she hated my connections to a traditional number line. More importantly, I realized that her solutions were all correct if she started with a positive number, and either added a positive number or subtracted a negative. When she had to go beyond a benchmark of zero, things got murky.

The more that I listened, the more that I realized that Inaya did not accurately sequence negative numbers. She thought -5 is larger than -3. Similarly, -10 is much bigger than -1.

Quick Teaching Point: Sequencing on a Number Line

I drew a number line with intervals of 1, that stretched from -5 to 5.

“What do you notice about the order of the numbers?”

Inaya massaged her forehead with her pointer finger — I’m so annoying! — but she played along. I swear that some of her disenchantment with her world is that she’s is such a keen observer. She notices everything, including the little injustices intrinsic to middle school.

Inaya noticed that the positive numbers go forward, and the negative numbers go backward. She noticed that 0 is exactly in the middle of 5 and -5. When I asked to compare numbers using the number line, she seemed to understand that numbers to the right are larger, and that that means that negative numbers that are closer to zero (e.g. -1) are larger than numbers that are further left (e.g. -5).

Processing, Practicing, and Internalizing

Observing a mathematical phenomenon is the first step! However, even though she came up with these ideas either on her own or with some gentle nudging, I could tell that she didn’t fully own them. If we’d left it at that, she would have forgotten it long before math class the following day. She needed some more opportunities to experiment with the ideas. She needed to process them, and eventually internalize them. We could do this through some practice.

This is where “tiny games” come into play.

Tiny Math Games

There are a few older MTBoS blog posts about tiny games.

• Tiny Games: Mathematics Edition? by Jason Dyer (April 12, 2013)
• [Confab] Tiny Math Games by Dan Meyer (April 16, 2013)
• The Difference Between Game and Drill by Jason Dyer (April 17, 2013)
• Google sheet shared by Julie Reulbach (2014)

Sadly, I was only vaguely aware of MTBoS in 2013, mostly from Lisa, the awesome 6th grade math teacher at my school. (Shoutout to @LisaSolt!) Parallel to all of this Tiny Game development, I was actually developing my own series of Tiny Games. In 2013, as part of my yearly teacher evaluation process, my evaluator insisted that I should be able to walk into any classroom K-8 and, without any advanced prep, deliver a high quality and rigorous lesson. Yes, this is a ridiculous expectation. (Never mind that I was barely even meeting baseline expectations at work that year. It was my least successful year as a teacher.)

However, as ludicrous as this expectation was, I was motivated to meet it. After a few years, I discovered that I could walk into any classroom in the building — yes, K-8! — and, with 20 seconds notice, have something reasonable (if not fully) polished ready. I have go-to resources, like 3 act tasks and open middle and number talk images, depending on what the class has been working on. And I have a series of flexible activities that I call Simple-But-High-Leverage Games. (“Low Prep, High Leverage Games” would have been a better name. Is it too late to rebrand?) As I learned years later, these are Tiny Math Games.

The Skip Count Game

I wanted Inaya to focus on sequencing with integers. (Negative rational numbers for another day!) The Skip Count Game (variations on Nim) is great for sequencing.

I told her we would be counting backwards. In the younger grades we think of this as taking away 1 each time, but, in middle school, we can think of it as skip counting by -1.

We would start at 3, and count to -15. At each turn, we could write 1, 2, or 3 numbers down. Whoever writes -15 is the winner!

Inaya nodded. She was in.

I went first. (We had moved to the hallway at this point, and didn’t have any paper handy, but a digital whiteboard would suffice.)

“I’m going to write two numbers. 3, 2…”

Inaya chose to write in purple, and kept going. She meaningfully stopped at 0. “Your turn.” Her mask obscured her face, but I imagine she was offering up a delicious 7th grade smirk with that.

I continued: -1, -2. She caught on quickly.

Yes, I decided I would let her win this round. I was pleased to hear her laugh lightly when I wrote the -12. “Are you sure you wanted to do that?” Inaya retorted.

“Why?” I asked. I have no poker face, even with a mask, so I’m sure she knew what was up.

Round 2

“I’ll start again, but this time, let’s skip count by -2s.” I wrote down 3 numbers, because I was anxious to see what she’d do beyond that 0.

In fact, Inaya forgot to write the 0. The sequence read “6, 4, 2, -2, -4…”

“Wait. I think you missed a number.”

UGH. More 7th grade eye rolls. Then: “ohhhh. Zero. Whatever.”

She used the ‘undo’ function to erase the -2 and -4, and continued.

This time, Inaya legitimately won. In fact, when she wrote down the -16, she asked me whether I wanted to give up then and there.

“Why? We haven’t reached -24,” I asked. While I was feigning innocence, I really wanted Inaya to explain her strategy.

“You’ll see.”

I dare you to find something more satisfying than when 7th grade smugness is redirected from social snark to mathematical learning.

Why Tiny Math Games (Low Prep, High Leverage)

Inaya and I played this game last Friday. On Monday, the learning about negative numbers seemed ‘stickier.’ She was still reluctant to engage in the class activity, which was an Open Middle-type puzzle using decks of cards, but she had more entry points that she’d had before. She was able to figure out that 3 + -5 was -2 without any nudging. Then she tried to charm her partner into doing a greater share of the work. It might not have been a rousing success, but this is a marathon and not a sprint, right?

Because of my familiarity both with the mathematical content and a set of quick games, I was able to match up an activity within a few seconds. It took me a few years to build up this practice.

Why did I bother developing and learning these games? For situations exactly like the moment I had with Inaya. Because these games require few materials — often just paper and pencil — and they fit multiple mathematical concepts, they can be used in a variety of settings. While I can’t promise every jaded middle schooler would want to play, there’s a chance. (And the elementary students at my school seem extremely willing to play! Here are some first graders playing the Skip Count Game when we had a spare moment. We used the game to launch a discussion around the patterns we notice, and how we can use those patterns with even larger numbers.)

In Reflection

So… listen to students. What mathematical thinking would you like to push them towards? Could that productive nudge (term c/o the fabulous Kassia Wedekind!) take the form of a game?