by Herbert P. Ginsburg, Teachers College, Columbia University
In the kindergarten classroom of a school serving low-income New York City families, I decided to experiment with a counting lesson I had been thinking of for a while. I asked the group of 20 students if they wanted to learn a new way to count. They were excited and I said that it starts after 10. I had them first count to 10 all together and I told them that that was the boring part, which they found amusing. (I really meant that those were the memorized numbers.)
Then I asked what comes after 10 and—as expected—they said 11. I disagreed, saying that 10-one comes after 10. Then I asked what comes after 10-one? Some kids said 12, but at that point many were confused. I said, of course, 10-two comes after 10-one.
You might think that saying 11 is not what comes after 10 is not a good strategy because my denial of what the children knew to be true would just confuse them. But, in a way, that’s exactly what I wanted to do. I wanted to put them in cognitive conflict and challenge their assumptions. I hoped they would understand that while we normally use 11, a different number word based on tens and ones would make more sense mathematically.
If you don’t like my strategy, an alternative would be to say: “Let’s make believe that we are going to create a new way to count the numbers after 10. Instead of 11 we are going to say, 10-1, and instead of 12 we are going to say, 10-two, …”
Eventually we got to 10-nine and then the next number, which they said was 10-10. In retrospect I should have accepted that, but instead I said we should call that two-10. We continued counting, two-10-one … two-10-nine … three-10 and so on. (It might have been better to let them do 10-10-10, instead of three-10, and then eventually 10-10-10-10-10-10-10-10-10. Would this help them to see the need to call this nine-10?).
After we got into the thirties, I suggested we try it in Spanish. Children who spoke Spanish were amused by my ignorance and pronunciation of the Spanish numbers, and everyone seemed to enjoy the activity. “What comes after dos-diez-uno?” You could do this in any language. It’s interesting that if you did it in Chinese, the result would be essentially no different from the conventional Chinese system!
Why did the kids seem to enjoy the activity? It was different. It forced them to focus on the pattern and on counting as a thoughtful activity, rather than a memorized song. This kind of activity could be used to highlight the base 10 features of the counting system; it could be expanded to include reference to objects grouped by tens and ones, etc.
The activity seemed to be going well and the head teacher wanted me to continue, so I had the kids go to their desks and write the numbers 1-10 vertically down the left side of the paper. Some kids had not yet mastered writing the numbers, and therefore focused mainly on the mechanics of forming the numerals. But others wrote the numerals more or less OK and their challenge was to figure out how to write the numbers 10-one through two-10 in the next vertical column. They began with something like 101. I said we needed to separate the 10 and the one or they would get confused, and suggested they use a plus sign. So they wrote 10+1, 10+2, etc., in a vertical line. We did OK until we got to two-10 and then I was faced with a problem because some kids wrote 2+10. It was then that I thought I should have accepted 10-10, which could have been written as 10+10. But it was too late for that, so I asked them to write 2*10, and then 2*10+1, etc. I didn't really explain anything about this.
At this point, the assistant teacher came over and the head teacher was watching with great interest. They asked why I was doing this and I explained that you could think of 11 as 10+1, and that 18 is really 10 and eight, or 10-8. They had an aha moment and said they had never thought of counting in that way before.
Then the assistant sat down to learn the new system of counting and said she found it very interesting. I gave her a test and she passed. She was genuinely fascinated by the idea of constructing (or deconstructing?) the counting numbers in this way, and really enjoyed the Spanish version (she is a Spanish speaker). The head teacher came over and said something like, "Usually teaching math is not that interesting. We need new ways to do it." I replied that my team could develop about three weeks’ worth of activities drawing on this counting exercise; she said she would be delighted if we did that. I asked if she would like to try out even more activities like these and she accepted.
I learned that the kids enjoyed this little lesson—even though I made it up on the spot, sometimes rushing a bit too much—that they did the work with some enthusiasm and competence, and that the teachers were not only receptive but also wanted to understand the ideas.
A week later, the kids were sitting in circle time with the head teacher and a student named Roberta waved to me and said she remembered how to count. I asked her to demonstrate and she did indeed remember. She began counting from 10 (that's where my new system starts) and after 10-nine, instead of saying 10-10, she remembered two-10 and eventually counted up to four-10-six. It was hard to tell whether the other kids were still interested; Roberta was hogging all the attention. Children are proud of their learning and want to show it off.
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Herbert P. Ginsburg is the Jacob H. Schiff Emeritus Professor of Psychology and Education at Teachers College, Columbia University. He is a member of DREME’s Family Math and Early Math Resources for Teacher Educators projects.