One kindergarten student pulled on my shirt to ask. In this class, students were using real objects to create and represent different addition number sentences. The teacher had asked everyone to work with numbers between 1 and 10. This particular child challenged herself by putting together 10 rocks and 9 ladybugs. Now she wanted validation that she had counted the total correctly.
I could have given her instant positive feedback, but that would have ceased her mathematical thinking. Who was I to rob her of the opportunity to justify and prove this claim to herself? Her uncertainty provided a rich context for further mathematical activity. After all, it is this process that matters to me.
To support students actively doing mathematics, as opposed to simply calculating answers, I feign ignorance. It’s an effective strategy.
Me: “How did you figure that out?”
She: “Well 10 + 10 = 20 so 10 + 9 = 19 because it’s 1 less, right?”
That single word dangling at the end of her sentence clued me in to her lingering uncertainty. Clearly, this is a student who has developed some good reasoning strategies! I was impressed with her thinking and it was hard not to celebrate her good work, but I continued the conversation anyway.
Me: “It sounds like you aren’t quite sure yet, is there another way we could figure it out, just to check?”
She: “I can count it out with my fingers.”
Here she counts on from 10, using her fingers to count the extra 9 one at a time. She counted 19 in all.
She: “I got 19 again. It’s 19, right?”
There’s that word again…
Me: “You still seem unsure. Should we count the objects?
She: “Let’s count on from 10 (circling the ten rocks first with her finger and then counting the ladybugs one at a time). 10, 11, 12… I don’t know if this is going to work. I want to count all of them all over again.”
I was fascinated by her hesitation. She initially utilized a sophisticated compensation strategy and here she was questioning whether or not counting on would work! Herein lies the power of uncertainty. If she could prove to herself that there were 19 objects on the table, she would also help strengthen her understanding of the other attempted strategies. What better way to bolster understanding?
Me: “Do you want to count them all one by one?”
She: “1, 2, 3… I better put them in a line. 1, 2, 3 … 17, 18, 19! There are19! I thought there were 19!”
Me: “Sounds like you are sure now.”
She: “I want to do another one.”
There we go! She had proved it to herself, no external validation required and no more dangling question at the end of her statement. After this, she quickly turned away from me, no longer interested in further conversation. She swiftly cleaned up the 10 rocks and 9 ladybugs and chose a new set of objects to explore, independently.
I too turned away, searching for another opportunity to wander in NumberLand.
Doing Mathematics in the Classroom
I entered this profession with a love and commitment to teaching students to appreciate and enjoy mathematics. I always thought of it as a beautiful and creative subject. However, I didn’t derive my practice independently. I learned a lot by listening to others. Here are three people who have influenced my work: