Even from across the room, there was no mistake: Jacob and Ian were deep into a mathematical debate. Jacob had pulled his chair up to Ian’s desk, and they were hunched over opposites sides of the desk, pointing emphatically at the small paper squares on their desk.
And, because I love a good math debate, I decided to join them.
They were looking at this:
…and trying to figure out the value of each of the sections, assuming that the entire square represents 1 whole.
Everyone agreed to this constraint — this assumption that the whole square is worth 1. We later realized just how important it is to recognize and honor assumptions as we debate.
Jacob clutched tight to a ruler. “The red square is 1/64!
It’s exactly 1/2 inch long, and the whole square is exactly 4 inches long.
That means that it’s 1/8 of the length.
So it’s 1/8 times 1/8, which is 1/64.”
I hadn’t thought to be so careful with measurements. I had given this same fraction talk image (from @natbanting’s FractionTalks.com) to students two years ago. I wrote about it in one of my first blog posts: “The Intersection of Fraction Talks & Clothesline Math: Formative Assessment and the Five Practices” (Feb. 8, 2017). The students had struggled with this image, dubbed the “milk carton,” but we all assumed that the tiny square, colored in red in the image at right, was 1/9 of a quarter of the whole square, or 1/36 of the overall square.
Ian operated under the same assumption I had: that it was 1/9 of a square. “Measurements can be off,” he argued. “Maybe it’s not really 1/2.”
“It is! I’ll show you,” Jacob said, as he reached again for the ruler.
“Maybe it’s really close to 1/2, but it’s, like, not. Like it’s not 100/200, it’s 101/200 of an inch. We can’t even see the difference,” Ian stated proudly.
“But I’m really precise,” Jacob countered. “Look for yourself! It’s one half of an inch.”
Ian and I squinted, and lowered our heads for a better look. It did look like exactly 1/2″. I was surprised at the neat and tidy measurements, since the classroom teacher and I had not put any thought into the overall size of the image. We just wanted to be able to print multiple fraction squares on a page! That was it!
Meanwhile, Ian was using the ruler for another purpose: to mark off straight lines, partitioning the shape into 36 segments.
Jacob, still skeptical, pointed to the clearly crooked lines at the bottom. “How can you say these are thirty-sixths? They’re all different sizes.”
Ian was quick to draw attention to the neat squares in the top half of his work. “The ones that go around that small square are perfect. That’s because those lines helped, because there are 9 of those equal squares in each 1/4. It’s a thirty-sixth.”
They were triggering one another with fighting words: “precise,” “perfect.” Clearly, they both wanted to come out on top here, so they turned to me as the tie breaker — the mathematical authority in the room.
Meanwhile, I sat genuinely puzzled. “You know, when I did this puzzle with some students before, I assumed the same thing as Ian: that the square was 1/9. Look how neatly it fits in Ian’s work, and then it makes it easier to determine what the triangle pieces are.”
“But,” I continued. “Now that I’ve seen Jacob’s measurements, I don’t know what to think.”
“Can you look it up?” Ian implored, to which I gave a perhaps less than charitable laugh.
“I don’t think so! I can try to ask other people, but who is to say what the real answer is. Maybe there are two answers: what it’s meant to be, an what it actually is, given how difficult it is to measure with precision.”
The boys were clearly dissatisfied with my response, and Jacob looked a touch wounded that I had implied that his solution wasn’t what was “meant to be.”
Thank goodness for Twitter!
Within minutes, I had a response from the designer and curator of the Fraction Talk website, Nat Banting.
Just beautiful. I love this idea, that math tasks all operate under assumptions. As long as we make them explicit, and honor these assumptions, there is space for multiple correct “answers.”
I went back to them the next day, and shared this. They’re 10 year old boys, so they mostly shook their heads. (They wanted to know the “winner,” and any attempts I make to convince them that they’re both winners fell flat. Truthfully, beyond having two “correct” answers, both students flexed their mathematical reasoning muscles, and offered us tremendously convincing proof.
How do we make space in our classrooms for students to get used to this idea — that we can see things in different ways, and stating our assumptions increases the clarity of our ideas and also opens up space for more correct answers?
Two days later, in the same classroom, we explored the following image by Sakir Gökçebag (courtesy of Pierre Tranchemontagne/@Pierre_Tranche‘s brilliant Number Talk Images website).
How many watermelons are there?
Students mostly agreed that it was 5. We explored the many ways to visualize this, e.g. slicing up the halves to make quarters, to fill in the watermelon chunks, or putting together the halves to make wholes.
Jacob shook his head. “It’s 6,” he insisted. “There’s no way it can be 5.”
This was after four different classmates had offered up their proofs for 5.
“There’s 6, because there’s no way those chunks could be cut from 5. It’s not like the halves were cut from the 3/4 watermelon pieces. Those must have come from at least two more watermelons.”
“Oh!” Serene, a girl in the front row, looked like she was having a lightbulb moment. “I get it!”
Once again, it became important to state our assumptions. If we assume that we can put back together pieces to form a watermelon, and that we can call that a watermelon, there are 5. If we want to know how many watermelons it took to create this installation, it’s at least 6.
We name these assumptions, and we celebrate them. It’s not about winning with a correct answer, it’s about winning someone over with an idea.
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Embrace the Challenge 
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