The public parts of my notebook.
Dorothy was working with me the other day. I have been trying to get to the bottom of her misunderstanding of numbers so that I might find some solid ground to start building on. I think we may have touched the bottom, but it was a long way down.
On the table I made 2 rows of white rods, 5 in each row. As I made them, I said, "Here are 2 rows, same number of rods in each row.” She agreed. I asked how many rods I had used to make these 2 rows. She said 10. I wrote 10 on a piece of paper beside us and put a check beside it. Then I made 2 rows of 7. She agreed that the rows were equal, and told me, when I asked, that I had used 14 rods to make them. She had to count them, of course. I wrote 14 and put a check beside it.
Then I said, "Now you make some.” She pushed my rows back into the pile, and then brought out some rods, with which she made 2 rows of 6. I asked how many she had used, and she counted up to 12. I wrote this down and put a check beside it. Then I asked her to see if she could make 2 rows with the same number in each row and no rods left over, using 11 rods. She pushed her 10 rods back into the pile, then counted out 11 rods from the pile and tried to make them into 2 equal rows. After a while she said, “It won’t work.” I agreed that it wouldn’t, wrote down 11, and put a big X beside it.
Then I said, "Some numbers work, like 10 and 14, and others don’t, like 11. I’d like you to start with 6, and tell me which numbers work and which ones don’t.” After what we had been doing, these instructions were clear. She counted out 6 rods, which she made into 2 rows of 3. I wrote down 6 and checked it. Then I got my first surprise. Instead of bringing out one more rod to give herself 7, she pushed all of them back into the pile, then counted out 7 rods, and tried to make 2 equal rows out of them. After a while she said, “It won’t work.” I wrote 7, with an X beside it. Then she pushed all the rods back into the pile, counted out 8, made 2 rows of 4, and said “8 works.” Then she pushed them all back, counted out 9, could not make 2 rows, and told me so. And she followed exactly this procedure all the way up to about 14.
Then she made a big step. Having done 14, she brought out another rod to make 15, and merely added that rod to one of the rows, before telling me that 15 would not work. Again she left her rows, this time adding another rod to the short row, before telling me that 16 would work. This more efficient process she continued up into the early 20’ s-about 24, I think. Then, having found that 24 would work, she said, but without using the rods, "25 won’t work.” I wrote it, and she continued thus, with increasing speed and confidence, until we got to about 36. At this point she stopped naming the odd numbers altogether, saying only “36 works, 38 works, 40 works…” and so on up into the 50’ s, where we stopped.
We rested a bit, fooled around with the rods, did a little building with them, and then went on to the next problem. This time I made 3 equal rows, and asked her to find what numbers, beginning with 6, would work for this problem. To my surprise, she could not arrange 6 rods in 3 equal rows, arranging them instead in a 3-2-1 pattern. I helped her out, and she began to work. From the start she moved one step ahead of where she had been on the e-row problem. When I had made 6 rods into 3 rows of 2, and had written that 6 worked, she added a rod to one of the rows, told me that 7 would not work, added a rod to another row, told me that 8 would not work, added a rod to another row, and told me that 9 would work. In this way we worked our way up to about 15 or 18. Here she stopped using the rods, and said,” l9 doesn’t work, 20 doesn’t work, 21 works…“ and so on. When she got up to about 27, she just gave me the numbers that worked–30, 33, 36, and 39.
In the 4-row problem we began with 8 rods. She used the rods to tell me that 9, 10, and Il would not work, and that 12 would. Without the rods, she told me that 13, 14, and 15 would not work, and that 16 would; from there she began counting by fours–20, 24, 28, 32, etc. In the 5-row problem we began with 10 rods, and after using the rods to get to 15 she went on from there counting by fives.
Revised edition commentary:
People to whom I have described this child’s work have found it all but impossible to believe. They could not imagine that even the most wildly unsuccessful student could have so little mathematical insight, or would use such laborious and inefficient methods to solve so simple a problem. The fact remains that this is what the child did. There is no use in we teachers telling ourselves that such children ought to know more, ought to understand better, ought to be able to work more efficiently; the facts are what count. The reason this poor child has learned hardly anything in six years of school is that no one ever began where she was just as the reason she was able to make such extraordinary gains in efficiency and understanding during this session is that, beginning where she was, she was learning genuinely and on her own.
Though I have many reservations today about much of the work I did with my fifth-grade classes, I am still very pleased with this day’s work with Dorothy. I don’t think that she, any more than Ted, was internalizing, taking possession of, making her own, much of what I was showing her. But at least she was having the experience of solving problems that she understood, and knowing from the evidence of her senses that she had solved them. At least she was feeling some of the power of her own mind. The problem, my problem, probably seemed pointless and ridiculous to her, but the solution was hers.
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