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.

Place-based teacher education

Critical Place-Based Teacher Education

In the current climate of accountability and standardization in (teacher) education, the prospect of PBE taking root seems preposterous. And Dr. D. does not suggest that this particular foray into PBE is exemplary. The process of participating in the research, both by observing in classrooms in a focused and local way has been crucial, as was the literature we reviewed, and the teacher candidates’ reading of the various drafts of this paper.

We leave the reader with Jolynn’s reflections on PBE for her future students:

Given the rich history of the area, we are now left with what to do with it. We have made ourselves accountable by discovering such a rich history and now must attempt to use it to structure lessons. When beginning the process of creating a place-based lesson, we must do what has already been done in this instance, and find out the history of the place and the people. When considering a place’s history, we must also juxtapose that with the present. Each school year brings along with it changes in the type of learners, and as teachers, comfortable in the ways in which we were taught and how we learn, we must not forget this. Student surveys of their personal learning styles as well as their intelligences (Silver et al., 2000) help us to understand more about their “place.” When we begin to incorporate all of this, the term “place” begins to take on a new meaning. What is “place” to our students and how can we connect that with the environment around them and the history and culture of the territory they inhabit? For example, in a science lesson centered around the Jones County history, we would delve into the ecology of the county and how it was shaped by its history and agricultural practices that have left the area in its current state. When beginning a lesson such as this, we would need to get the students involved in the history by choosing a location that can share a story. This story will become the thread of the lesson and a historical timeline that the students will be able to reference with ease. This “story” also coincides with the culture of the area, as southern practices dictate history be passed down from person to person in a narrative format. Then we begin to involve scientific practices such as surveying the flora and fauna of the area as well as investigating areas of succession and human impact on the environment. Of course these scientific practices can be taught in other styles and formats, but by giving the students an anchor they can tie this information to, we are ensuring that the students are able to apply this information rather than simple memorization as well as instilling a sense of accountability to them for their environment and culture. This style of teaching requires that the educator go above and beyond the textbook, and certainly not be considered for those with concern for teaching to the test. But when enacted correctly, standards, evaluations, and course tests come simply and pleasantly as the teacher and student are both comfortable and confident with their knowledge of the material and their surroundings.

reflection on today’s science and math pedagogy class

For the last 6 weeks I’ve been teaching a 6-hour face to face class on Saturdays to a group of pre-service teachers in an MAT program and in-service teachers getting advanced degrees. This has been brutal for everyone concerned, to say the least. I’ve had to pare down my expectations because after about 4 hours nobody can absorb much, no matter how many times I asked them to get out of their seats and try something different.

I decided to focus on Stigler & Hiebert’s The Teaching Gap, which is old news, but not to the students, so that they would be open to questioning their cultural assumptions about what it means to teach science and mathematics. My plan was this would allow them to be open to reform ideas, such as those embodied in the Tools 4 Teaching Science out of the University of Washington.

I think this pretty much worked. In our last class today I asked the students to make concept maps. I gave each person a page of stickers with 60 nouns culled from a variety of class readings:

  • The biology people read The Beak of the Finch, which most hated because it is tedious in spots. The math people read either Jacqueline Leonard’s book on multicultural mathematics education or the Joy of X. The purpose was to increase content knowledge, which I think was modestly successful.
  • Magdalene Lampert, “When the Problem is not the Question and the Solution is not the answer.” We used the theoretical framework of this article to think about what it means to do mathematics (and science).
  • Hand et al., Negotiating Science. We used this to provide a framework for inquiry activities in science.
  • 5 Practices for Orchestrating Productive Mathematics Discussions.

That was actually quite a lot of reading for 6 weeks. We only got through Chapter 7 of The Teaching Gap, since that is the point after which the authors mostly  just repeat themselves.

I gave each student a giant Post-It and asked him or her to do a concept map using Novak and Gowin’s 1984 procedure and using their scoring scheme.

Several students struggled mightily with this format. More about struggling later.

Thinking about the maps in public

I wanted the maps to be public records of thinking, but  didn’t want to put anyone on the spot. Therefore each map was displayed on the wall and was only identified with a number. We did a gallery walk during lunch; I asked students to record on index cards what they thought was interesting or new about each map, and what was similar to what they put.

I collected those cards, and a first glance through them shows they didn’t write much. However I still think it was an important focus to help them look at other people’s maps, since they were all quite different.

We then went around to each map and talked about what the students saw in it. The easiest entry point for the students was surface features, how it was organized. Several remarked on people having chosen different starting points for the map.

About halfway through, I started pointing out things that I saw, going back to the earlier maps and comparing. Students stopped contributing much, once they realized I was going to tell them “the right answer.” Of course it wasn’t, but I was trying to get certain ways of thinking on the table. I think they also were at a loss as to what to say. The holistic approach, looking at where the maps went, what concepts had lots of links and what seemed isolated, was new to them. In making a quick instructional decision, I considered briefly that going with a cognitive apprenticeship model, in which I shared my thinking with them, was probably going to be most productive.

This was such a rich discussion, I want to get down as much as I remember and it might be an overly long and boring blog post. But I do want to capture what happened while it is still fresh in my mind.

I was aware that a person got very upset when I pointed out how a particular map showed that this set of ideas was not integrated with the rest of the map. (Someone appearing to get upset or embarrassed actually happened more than once.) I infer that the persons who seemed upset were the authors of the map in question. I was very careful to say, “This person did…” In truth, I knew who authored only 3 of the 13 maps. Because we were pretty much anonymous, so they were not singled out in public. It is interesting to me that several students seemed to feel that their maps were private performances that should not be critiqued in public. I’m an artist accustomed to having my drawings and paintings critiqued by art professors. Critiques are always scary, but they are also very powerful because they give the entire group the benefit of the instructor’s thinking.

I think of Fred Erickson repeating several times in my own grad school classes, “School is the only place where people must publicly display incompetence.” In spite of my repeating often to my own students that the right answer is how you’re thinking, not the content of what you say, the social practices surrounding school being about the right answers is still the dominant cultural model. In spite of my never saying that one of the maps was wrong, some students still interpreted my remarks as showing them up as incompetent.

I ended class saying the maps made me feel really good, that it was formative assessment for me, I can see what people got out of the class, and I’m pleased. I was. Of course the students wanted to know how I was going to grade them. I said, I’m going to write each of you a letter about your map, and I want you to write me back. (Even though class meetings are over, there is still an assignment out there to do a unit plan. Class is officially over in May.)

specific items that interested me

One of the maps had a strand that included “disciplinary language,” that was cross-linked to “writing,” which was also cross-linked to disciplinary practices. There was another independent strand that included  “discussion,” that was linked to “teacher,” and not linked to disciplinary language or practice. I didn’t say anything public about this observation; I thought it might seem too critical. I will write to the author about it. I also note that our pre-service teachers have the most difficulty with the section of EdTPA having to do with academic language. That this shows up in at least one map is confirmation to me of the power of concept mapping.

One of the students who was having a really hard time, had chosen “knowledge” for her topic and “teacher” and “student” for the next level. As I walked around, she asked me for help. She couldn’t figure out to where put any of the other  labels, and she couldn’t think of what to write as links.  I asked her to clarify, what did she define the link between teacher and knowledge to be, and what was the link between student and knowledge. She replied, The teacher has knowledge, and the student has some knowledge. Without saying anything, I thought, This is the traditional  transmission model of teaching, slightly updated to include students’ prior knowledge. I replied to her that she might try different concepts instead of student and teacher, maybe that was the problem. She was able to successfully complete the map.

to be continued…