“What Are You Hoping For?”: Facilitating the Number Boxes Game to Develop Reasoning – Learning with Teachers

Number Boxes is one of my favorite games to play. The set up looks similar to an Open Middle problem, with empty boxes that you will place digits into. Open Middle problems often involve placing set digits into the boxes, and then revising your thinking to improve your outcome — by changing where you place the numbers.

With Number Boxes, once a number has been placed, it has been placed. You can’t move it. It’s done. However, you can always revise your thinking and your strategy to improve your outcome — and maybe win the game.

To play this game, you need:The rules are roughly:
• Something to write with
(dry erase marker, pen, pencil, etc.)
• Something to write on
(whiteboard, paper, window, table, etc.)
• Something to generate random numbers
(dice, spinner, random number generator, etc.)
1) Set up a board and set the target
2) Generate a random digit
3) Place the digit (and it can’t be moved!)
4) Repeat until all boxes have been filled
5) Calculate, reason, and collaborate

You can learn more about this game on a previous blog post I wrote in May 2019, and also on a blog post by Mark Chubb.

I’ve played it a few times in the last week or so, in very different contexts, that have reminded me both of the power of this game and also how we need to focus on facilitation moves to leverage this power.

NCTM Baltimore: Multi-Digit Division

Last week, the brilliant Louisa Connaughton (@lpconnaughton) and I presented as session at NCTM Baltimore that we called “Games that Promote Equity and Mathematical Brilliance in the Upper Elementary Classroom.”

Louisa modeled a “whole class” version of the game. She asked everyone to set up a game space like this:

Number Boxes Game: __ __ ÷ __ __ with two throwaway boxes

Most participants chose to write on the dry erase posters (generously donated by Wipebook) but I saw a few people writing in notebooks or on computers. Many people seemed eager to play, and a few others looked nervous. While we had wanted to use a division game — it mirrored something from earlier in the presentation — Louisa and I wondered if we’d chosen content that was uncomfortable for a few teachers. Teachers of lower elementary grades might not have had as much experience thinking about division of larger numbers.

Originally, I thought we’d simultaneously model how easy it is to cue up a 10-digit die or spinner using google — literally type in “roll a die” — but hitting ‘escape’ to get out of our presentation caused a cascade of technological catastrophes. (Oops!) Louisa quickly asked participants to roll physical dice that were at the tables.

Table 1 rolled two dice and called out the sum to act as our digit. The first number to place on participant boards: six!

Participants dutifully placed their 6. Some spoke with people at their table about the ideal placement — was the 6 large enough to make a good dividend? (the first number in this expression) Did players think that they might get larger numbers later, and thus they wanted to throw away the 6?

Table 2 rolled the two dice at their table, and called out the sum: five!

Play continued, with one person at various tables rolling two dice.

After four rolls, one participant shared that we had a lot of “middle numbers.”

“Yeah, and we’re never going to get a 1!” another participant from across the room lamented.

That’s Not Fair

Both of these these were true, and astute mathematical observations. Something to celebrate! And something to explore. While we had intended to roll a virtual ten-sided die, which would have had equal chance of each outcome, we were instead rolling two 6-sided dice.

sample space (all possible outcomes) from rolling two six-sided dice, from PDX.edu
sample space organized as a frequency graph, from Alex Kirkpatrick

There is a \frac{1}{6} chance of rolling a sum of 7, \frac{1}{36} of rolling a sum of 2, and it is impossible to roll a sum of 1. Knowing this would impact strategy for some places. They wouldn’t want to put themselves in the position of waiting on a roll of 1 that would never happen.

I pulled out my phone, googled “roll a die,” and selected the 10-sided one. I have used this before, with dramatic flourish, to announce the digits students need to place on the board. Like a madcap Bingo Caller.

I hit “roll,” which thrust the virtual die into a spinning frenzy. A participant near the front of the room shouted out the outcome: three!

At this point, we were now one digit away from finishing this round of Number Boxes. Most participants had used both of their throwaway boxes, and were hoping to fill in one final digit in either the dividend (first number) or the divisor (second number in the expression).

What are you hoping for?

“Who is hoping that we will roll a small number?” Louisa asked. I saw several participants signing “me too” with outstretched thumbs and pinky fingers. 🤙🏽 Other people gave a thumbs up, or nodded, or raised hands.

“And who is hoping that we will roll a large number?”

“I want a nine!” Called out someone in the back. I didn’t make it all the way over to them, but we can make assumptions about what box this person had remaining.

For example, this player might want a 9. Increasing the value of the dividend means that the divisor can “go in” more times, in thinking about how many groups of 36 “go into” __4.

This player would not want a 9. Having a large divisor and a smaller dividend would not result in a large quotient. 35 ÷ 94 is less than 1.

And it was all up to chance! I raised my phone high, and pressed “roll.” A participant at table 3 called out everyone’s fate: one!

A collective groan rippled through the conference room, punctuated by a few cheers from lucky players.

“You don’t even need to calculate it.”

“It sounds like many of you aren’t pleased with this outcome,” I smiled, perhaps smugly. “Why not?”

“Now my answer is less than one!” Called out a participant. “It’s a fraction!”

I did my best to annotate some of this thinking on the wipebook positioned on an easel near the podium. I could see that some people were heads down, trying to calculate their exact quotient. Multi-digit division can be a beast.

“And you don’t even need to calculate it to know that?” I pressed.

“No, I know that a number divided by a bigger number is going to be a fraction.”

There are lots of ways to contextualize this, with both sharing and grouping. 14 cookies shared amongst 36 people will result in everyone getting a mere fraction — basically crumbs. And if you’re trying to figure out how many groups of 36 go into 14, well… not a whole group.

I urged others to lift their heads to attend to the work of others. “You don’t even need to calculate your quotient,” I said, “to reason about whether it’s the largest. Who else has a quotient that is between 0 and 1?” So many admissions of defeat.

“And who has a quotient that is more than 1? How about between 1 and 2?”

A participant from the back offered 66 ÷ 35.

35 x 2 is 70, which is a little larger than the dividend of 66. So 35 does not go into 66 a full two times.

Louisa and I elicited a few other examples, and explored how we can compare them without necessarily calculating the precise quotient. When it came down to problems that were close, like 66 ÷ 14 and 65 ÷ 13, participants used a variety of strategies to explain why 65 ÷ 13 is larger.

66 and 65 are so close together, and 13 will ‘go in’ more times.

13 x 5 is exactly 65, so the quotient is 5. 14 x 5 is 50 + 20, or 70, and 66 is smaller. than that.

If you write them as fractions, \frac{66}{14} is smaller than \frac{65}{13} because the size of the thirteenths is larger.

Clearly, we had a lot of mathematical brilliance within the room. Everyone had something they could have been writing with, but most chose to reason it out mentally. Sometimes, the less precise answer is more fun!

Showcasing Brilliance, and Facilitating with Equity in Mind

Perhaps you’re skeptical about what this might look like in the classroom. Maybe this is the kind of brilliance we can expect only from math educators — the kind of math educators that happily choose to sit in a conference room on a Friday afternoon, chatting about multi-digit division. (We are a special breed!)

I think there were a few stock facilitation moves that made it so that Louisa and I could showcase some of the brilliance in the room.

  • What are you hoping for?”
    This encouraged participants to think about strategy and the size of numbers with an operation. They had to consider their constraints, while also dreaming of a better mathematical future. (…one where they can gloat that they bested their friend at the table, or that their table had a larger quotient than the table across the room.)
  • Can you do this without calculating?
    Sometimes, I use the Number Boxes game to practice calculation. If I were using the game for that purpose, I wouldn’t ask this question! However, if the goal is to elicit thinking around quantities, and support students tin developing the ability. to reason and employ number sense, this is a great way to nudge them in that direction.
  • Showcasing several student answers, starting with ones that we knew were way off.
    There is a lot to learn from the student work, and the element of luck can soften the blow of losing. What would you have hoped for instead of what you got? Would you have changed your strategy if we play it again? (Let’s play it again!) This has the ability to assign competence to lots of students within the room, and to affirm their mathematical identities.

This is a competitive, “winning” game, but we don’t learn only from the winners.

And there are more! But this felt like a good start, and it was electric to see the enthusiasm from the educators in the room.

Louisa and I contextualized some of these moves within the 5 practices for equity-based teaching, from The Impact of Identity in K-8 Mathematics: Rethinking Equity-Based Practices (Aguirre, Mayfield-Ingram, Martin; NCTM 2013).

Choosing a good game to play — one that is high leverage and involves simple prep — is only the beginning. Then we need to think about the student experience, and. how we can use games to develop students as mathematicians.

Tomorrow, I’ll explore what this looks like with one of my second grade students.


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