How much tolerance do you have for puzzlement? When faced with a mathematical conundrum, do you embrace the challenge, or do you feel some trepidation at confronting the unknown?
For many of us, an unfamiliar mathematical task is sure to raise our heart rate a beat or two. As teachers, it’s easy to translate these fears onto our students, assuming that a non-routine task will leave them frustrated and sapped of energy. But children’s curiosity and perseverance always surprise me, often exceeding my own tolerance for exploring novel problems.
I was reminded of children’s tenacity when working as part of a ReimaginED teaching learning community at PS 503 in Brooklyn. We invited fifth graders to investigate a mathematical mystery. The mystery began with the interactive model below. (and here) The chart displays the letters A through J, with two draggable red counters. For each placement of the counters, one or more letters appear below the chart. Without providing any information about the meaning of the letters, we asked the children to experiment by dragging the counters and telling us what they noticed and wondered.
Before reading further, put yourself in the children’s shoes: Play with the model and take note of what you see and the questions you have. Any observation, no matter how small, is worth capturing. Give yourself at least five minutes to explore.
When Toni Cameron, Rachel Kugelman, and I interviewed the children at PS 503, here is what they noticed:
- Sometimes the output is a single letter, and sometimes it is two letters.
- You can place both counters on the same letter.
- More than one pair of letters can create the same output. For example, A and B as well as E and I both produce an output of JJ.
- When one counter rests on G, the results are especially interesting. No matter which letter you combine with G, the output is the same as what you put in. For example, G and A make A, G and D make D, G and L make L, and so on.
The third and fourth observations above suggested to students that there was an underlying logic that governed the relationship between the inputs they chose in the chart and the corresponding outputs. To build on these discoveries, we gave students the interactive model below (and here), nearly identical to the first, but now the output was labeled as “sum” and “tens” and ones” appeared above the letters.
With this additional information, students began to make some astute conjectures. Maya proposed that the “letters of the sum stand for numbers.” When asked about the numerical value of each letter, she suggested, “Because there are ten letters, A could be 1, B could be 2, C could be 3, and so on. Each letter is a number.” Toni asked Maya to put this hypothesis to the test: If A = 1 and F = 6, what letter represents the sum? Since 1 + 6 = 7, the expected sum should be G, the seventh letter of the alphabet. However, the actual sum was reported as JD. Even though Maya’s idea did not bear fruit, it was a lovely example of making and testing a hypothesis.
Our other interviewee, Tammy, focused on the unique role of G. Since G combined with any other letter produced that letter as an output, Tammy saw a connection to the role that 1 plays in multiplication. She observed, “It’s not plus, but it could be times. Because if you were to see G as 1, 1 times E is E. And 1 times B is B.” Tammy, for whatever reason, did not focus on the word “sum” that appeared next to the output, but her observation is spot-on. While the interview concluded before we could continue the conversation, I suspect that Tammy would have little trouble identifying G as 0 when reminded that the output represented a sum and not a product.
Soon, we will be presenting this mathematical mystery to entire classrooms of students and allow them to work their way through the whole challenge, determining the numerical values, 0 through 9, of all ten letters. In the meantime, can you crack this number code yourself?
You can read more about this code (and investigate a related multiplication-based code) in my earlier post.