[This post is a part of the fabulous Virtual Conference on Humanizing Mathematics organized by Hema Khodai and Sam Shah. More information found here.]
Look at these photos for a few seconds. What do you notice? What do you wonder?
Last weekend my family and I were driving 2+ hours north of the San Francisco Bay Area to a tiny town on the Russian River, near the coast where the river hits the ocean. It was a gorgeous drive, though lots of farmland. We passed numerous fields of grazing cows, all black and white, with no comments from the peanut gallery in the back seat. Then, we passed a field of brown cows. My 5-year-old literally started screaming, “I knew it, I knew it! Chocolate milk DOES come from cows. THERE THEY ARE! The chocolate milk cows!”
It took me a minute to realize how the farmland had changed to prompt this giddy comment. My 10-year-old seized the opportunity to start lecturing his younger brother on the impossibility of this conjecture, however I cut him off to share my own commentary.
“Isn’t it fascinating to realize what ideas someone has in their head and we have NO IDEA of those ideas without something like this (passing brown cows) to open a window onto their thinking?” My 10-year-old sighed and went back to sorting his Magic Cards.
I spent much of the rest of the car ride pondering this idea, thinking about how to ensure that my math classroom opens that window onto my students’ brutally honest thinking and reasoning as frequently as possible.
Earlier in the week, I found myself engaged in deep philosophical [mathematical] discussion with my chocolate-milk-pondering child. We were reading a book entitled “How many” which had a photo of a pizza with 6 slices of basil under the heading “half dozen.” After a lengthy discussion of all the ways to count those 6 slices, he asked what a full dozen would be. We talk about cutting things into 2 parts and each being a half. When I realized all my talk was confusing, we tried with fingers.
“Hold up 6 fingers,” I requested.
I did the same. I asked him how many fingers we had up all together and he looked at me confused about what to count first.
“Let’s put our hands together,” I suggested. And we did. Like this:
We pressed our hands together and he knew right away that 5 and 5 makes 10. Then we pressed our thumbs together and I asked him how more that was. He paused and said, “10-11-12. It’s 12!” And then he paused again, became frustrated and said that it couldn’t be 12.
I asked why.
“Because 1 and 1 is 11,” he exclaimed, now with a sigh of relief and glee at having discovered the answer. He grabbed the book from me, flipped to page 11 and showed me how 1 and 1 makes 11.
Our debate of 12 vs 11, of which this bedtime moment was only the beginning, reminded me of something Tracy Zager discussed in her workshop on ways to assess students’ mathematical understanding at OAME (Ontario Association of Mathematics Educators) this past May. She talked about doing math with Christopher Danielson, how it made her nervous at first because he has his PhD in math and she is an elementary school math coach. But she quickly realized that Christopher is endlessly fascinated by learning how other people think about the math and that when solving problems together, he is constantly delighted by watching and learning from others’ thinking.
As a parent, I felt so incredibly lucky to witness my son’s rough draft thinking (to coin a math teacher term) around how chocolate milk is made and how to quantify 11 vs 12. And because I both witnessed these moments and took the time to ask more questions, left some unanswered and delight in our conversations, my son now has a far deeper understanding of 11 vs 12. (Cuz I ain’t touching that chocolate milk notion as it’s simply too precious).
Since then, I have spent the past several days thinking about how many of these equally delightful moments I miss with my students and how can I shape my classroom community this year so that we all don’t simply value this rough draft thinking, but delight in it and know that it’s necessary for deeper understanding of the mathematics we’re learning together.
My challenge to myself, and to us all, is to develop this delight in ourselves and each and every one of our dozens (and in my case, and many of yours too, 100+) students; a delight in watching and learning from each other’s thinking.
And after pondering that challenge, if you find yourself with extra time, Google “brown cows.” Apparently my 5-year-old isn’t alone.