The students had built all the arrays possible to represent a given set of numbers during a recent math class. They designed one poster for each number. The poster had a title that stated the number they were working. All the possible arrays for that number were pictured using graph paper. Each array's dimensions were labeled. In the end, all the factor pairs for each number were listed.
9: 1, 3, 9
27; 1, 3, 9, 27
54: 1, 2, 3, 6, 9, 18, 27, 54
When the children had finished the two to three posters I had assigned each group, the posters were arranged on tables in preparation for a gallery walk.
As we stood around the first set of related posters I asked the students to study them and then to share what they noticed and wondered. SILENCE. MORE SILENCE.
AWKWARD, UNCOMFORTABLE, SILENCE.
No one wanted to get the conversation started because each child thought that what he or she was noticing was too obvious to share.
FINALLY after a LONG period of painful SILENCE one student shared that some numbers had more rectangular arrays than others.
Then another child shared that you could make a rectangle with a side of 1 for every rectangle and that when you did, the other factor was the number itself.
Eventually someone shared that each of the three posters shared some of the same factors.
Now we were ready to dive deep. Why is that? Is that true of every set of posters? How are the numbers for each poster related anyway? Can we make a generalization? Can we use a model to prove this generalization?
You have no idea how tempted I was to get the conversation started. It would have been all too easy for me to say, "Does anyone notice that the factors on the "9" poster are also showing up on the "27" poster and also on the "54" poster?" I was so tempted. The silence was difficult to endure. I bit my tongue.
These kids have to learn to do the work of mathematicians and their interest in math has to come from within. They have to be the ones asking the questions and making the sense.
As a teacher it is so much easier to be the one doing the noticing, the thinking, and asking all the questions. This practice just doesn't grow mathematical minds in the same way that waiting out the awkward silence and inspiring kids to notice, to wonder, to grapple, to listen, to question, to think for themselves does.
At the end of the lesson I acknowledged how difficult it is to be the first voice in the room and to feel as though you are just stating the obvious. Then I thanked the kids who were willing to do this. Sometimes it is the simplest ideas that get the conversation started. These simple observations can lead us to consider deeper mathematical concepts. Hopefully today's lesson was the start of a journey we'll continue together. I hope the kids realized that I'll be journeying along with them but that I'll be counting on them to take the lead.