Transformative Teaching

Example of student work on the cookie problem.

Today I had occasion to remember one of the best adult teaching experiences of my career. I was working as an adjunct at Cal State Northridge while I was ABD. I taught a combined math and science methods class for prospective bilingual teachers. My experience teaching elementary mathematics was using the Montessori method, pretty much brilliant, but in graduate school I worked as a research assistant on a couple of elementary school math research projects, doing professional development for K-6 teachers on CGI (Cognitively Guided Instruction). I learned CGI by being part of my advisor’s research group.

When faced with the task of teaching math methods, I turned to CGI, and treated the class as though it were professional development. This was appropriate, since all but one of my class of 30 or so were already teaching on an emergency credential. Anyway, the first day the teachers introduced themselves, what grade they taught, where, and almost every one said, “My kids are really low.”

The CGI approach, at least my take on it, is not to “teach” children strategies, but to uncover and utilize the strategies they already have, and then make them public. CGI researchers developed a detailed taxonomy of the ways that children solve elementary math problems: direct modeling, grouping, invented algorithms, are some I remember. In order to disrupt the familiar, rote nature of elementary mathematics, we called adding “joining,” that is, the joining of two groups of objects. I remember “join result unknown,” “join change unknown,” “join start unknown.” Unfortunately I lent my book out, bought a new one, lent that one, etc. As a result it isn’t on my shelf any more. Children’s Mathematics by Carpenter et al., published by Heinemann. It could even be out of print by now. The key, however, is that, while the teacher may individually prompt children who get stuck, s/he doesn’t teach the class how to do it. Instead, s/he selects learners to present varied solutions to the assembled class. Children are treated as mathematical thinkers.

Anyway, so the heart of CGI professional development was showing teachers that their children do already have mathematical knowledge and will utilize their own strategies if they are publicly valued. Teachers asked their students to solve a problem that they had not been taught to solve, and then brought the student work to a group meeting, in this case our weekly class meeting.

That’s a long introduction, sorry.

The first assignment was for the K-6 teachers (we had a couple kindergarten teachers, majority first and second grade, a few third grade and one sixth) to pose the following problem to their students:  Show how you can share three cookies with four friends so that everyone gets the same amount. The class’s immediate response was, “My students won’t be able to do that. They’re too low.” I didn’t say anything except, “Well, try it and let’s see what we get.” I had a pretty good idea that the younger children would solve the problem fairly successfully, but the older ones, who had received instruction in fractions, would have more trouble.

This was exactly what happened. About 5 kindergartners out of 60 or so drew some version of ¾ of a cookie being distributed to four children. Half (and I remember this clearly, it was exactly half) of the first graders showed a mathematically correct solution, and all but one of the 100+ second graders demonstrated a correct solution, either three separate quarters of a cookie or a half plus a quarter  per child. I was interested to see that the drawings usually included the four friends, and many directly modeled three quarters going to each of four stick figures. Some of the cookies had chocolate chips in them. Also interesting to me, several of the first graders independently said, “Cut two cookies in half, and give the big one to the teacher.” I find this interesting because sharing things fairly is a major social concern of children of this age. The children did not necessarily view this as an isolated, decontextualized math problem, but as a social problem with a mathematical solution.

Of the third graders, who had been taught about fractions, about half drew normative solutions, several wrote down an incorrect algorithm (for example 1/3 X ¼) and others said they didn’t know what to do. We had no fourth and fifth graders. The sixth grade teacher brought in student work showing that only one student had been successful in applying the correct algorithm 3 x 1/4 while many had written ¼ X 1/3 = 1/12 thinking they were supposed to invert and multiply. None of the older children made drawings. Most did not attempt the problem.( for a link to the cookie problem)

We discussed that this was possible evidence that children are able to think mathematically but that school instruction divests them of this knowledge. With this hypothesis, the teachers continued to collect data on their children’s ability to solve math problems that had not been taught, and the children, especially the younger children, continued to demonstrate impressive proficiency.

The culminating project for the semester was an assignment to conduct either a science or mathematical inquiry with their students, and present the students’ work at our final class meeting. The assignment was simple: Ask your children what they want to know about a topic you are going to start. Then as a class choose one question that is investigable, and do the investigation before instruction. (We were using Wynne Harlen’s book Primary Science.) Most everyone did science, and I remember a lot of “floating and sinking” projects. Aurelio decided to do math.

Aurelio asked his kindergartners what they wanted to know about numbers 1 to 30, which was the unit they were starting. I remember one of the questions was, “Can we do it?” The question they settled on was, “Is there a faster way?” (A faster way than counting by 1’s.) So the class timed themselves counting by 1’s, 5’s and 10’s. They decided 10’s was fastest.  Aurelio’s story was very cute, but there was more: He was amazed. This was his third or fourth year teaching kindergarten, and to his astonishment, he said that every child got it. That is, every child was able to demonstrate proficiency in numbers 1 to 30, whereas in previous years, using the textbook and workbooks provided by the district, most had struggled.

This was a great moment to review learning theory, but…


I listened to the teachers proudly discussing how well their students did on their projects. At the end, I remarked that I had not heard anyone talk about how low their kids are. “How many of you still think your students are low?” Only one person raised her hand.

And that is the point of teacher education for math and science, facilitating teachers to design instruction that reinforces learners’ competence. I have  always maintained that teachers will adopt new methods if they see they will benefit their students.

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