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.

STEM Writing Template

The Common Core Standards call for students to engage in literacy practices across the content areas. I am providing the STEM Writing Template as a model so that students can write to learn science.

The following is a workshop agenda.

STEM Writing Template (Adapted from Keys et. al, (1999:1067-1069) and also Hand, Prain and Wallace (2003:20-22)

The template provides a routine or procedure, a short-cut, if you will. By following the spirit as well as the letter of the template, you will have a classroom routine which supports students’ inquiry learning of STEM subjects.

We will conduct the Jet Straw design activity as if we were a class of K-12 students. As we turn to spaghetti bridge building, we will follow the Teacher Template.

Student Template

1. Beginning ideas – What are my questions?

2. Tests and Experiments – What did I do?

3. Observations – What did I see, hear, smell, feel? (Probably don’t ask students to taste.) How did I measure what I observed?

4. Claims – What can I claim as a result of my observations?

5. Evidence – How do I know? Why am I making these claims?

6. Conferences, Science or Math Talks, Reading and Instruction – How do my ideas compare with other ideas?

7. Reflection – How have my beginning ideas changed?

8. Redesign or Extension – How can I use my new ideas to improve my design (engineering) or investigate something new?

–Hand, Prain & Wallace (2003:21).

Teacher Notes

1. Beginning ideas: Students beginning ideas do not come from thin air, but from their prior experiences. In addition, students should have the opportunity to “mess around” with the materials in order to come up with questions and ideas. It is crucial to the process that the teacher does not tell students what questions they should have. In order to accomplish your learning goals, you choose the materials and task carefully. Direct instruction can occur in Step 6, after the students have had a chance to think about the implications of the activity. When scientists are beginning their investigations, they also use what they already know, and play around to get a feel for what kinds of questions might be interesting.

2. Tests and Experiments: Students will choose how to answer their questions. Because you have provided them with a task, and materials to use the tests students perform will be scaffolded. The teacher’s job is to consult with students to make sure they are designing studies that will answer their questions. You should also encourage students to collect quantitative data, and support them in this process by asking questions. It is crucial to the process that the teacher does not tell students what tests they should carry out. If there are flaws in testing procedures, discuss them in Step 6. The decision about how to design an experiment is part of the creative process of science.

3. Observations: Students will record data in a manner that makes sense to them. It is crucial to the process that the teacher does not tell students how to record data. If students’ data collection is problematic, discuss issues in Step 6. In Step 6 students will discuss the adequacy of designs with one another. On the other hand, this is an excellent point at which to introduce science or mathematics concepts.

4. Claims: This step is likely to be difficult for many students, especially if they are accustomed to being told what they are supposed to think. This step is actually difficult for scientists! Figuring out what data means is one of the major activities of science. The question of what data means is often not obvious. It is crucial to the process that the teacher does not tell students the correct answer at this point.

5. Evidence: This is the heart of what we want students to be able to do: Use data to support claims. It is crucial to the process that the teacher pushes students to think about what their data means and turn it into evidence. In fact, most of the conversation the teacher has with individuals and small groups should have the purpose of getting students to make sense of their findings.

6. Conferences, Science or Math Talks, Reading, and Instruction:

a. Conferences: Student groups will present their findings to the class. In order for presentations to be meaningful, students’ presentations should not all be the same, and every student should have an investment in getting the information which is provided by the different groups. In terms of engineering design challenges, this can mean distributing variables to be tested, with the class listening attentively, pointing out flaws in experimental design or data analysis.

b. Science or Math Talks: The teacher provides a prompt which asks students to make sense of the activities they have completed. The students talk with each other; the teacher steps in only when the conversation bogs down.

c. Reading: Students read textbooks or other reference material about the topic they have been investigating.

d. Instruction: The teacher explains any concepts which students are still unclear about or which have been missed in the student-centered process.

7. Reflection. Students revisit their journal entries and write about how their ideas have changed. It is crucial to the process that this opportunity for reflective inquiry be included.

Partial Research Base

This STEM Writing Template comes from several strands of research, although the work of Keys, Hand, Prain, Wallace et al. is its basis.

Hand, B., Prain, V., & Wallace, C. (2002). Influences of writing tasks on students’ answers to recall and higher-level test questions. Journal of Research in Science Education 32, 19-34.

Keys, C. W., Hand, B., Prain, V., & Collins, S. (1999). Using the science writing heuristic as a tool for learning from laboratory investigations in secondary science, Journal of Research In Science Teaching, 36, 1065-1084.

Keys C. W. (1997). Revitalizing instruction in scientific genres: Connecting knowledge production with writing to learn in science. Science Education, 83, 115-130.

Kolodner, J. L.; Camp, P. J.; Crismond, D.; Fasse, B.; Gray, J. H.; & Puntambekar, S.; et al. (2003). Problem-based learning meets case-based reasoning in the middle-school science classroom: Putting Learning by DesignTM into practice. The Journal of the Learning Sciences, 12, 495-547. Retrieved May 25, 2010, from

Warren, B., & Rosebery, A. (2011). Navigating interculturality: African American male students and the science classroom. Journal of African American Males in Education, 2(1). Accessed June 8, 2012 at

Teacher Template

This template contains a series of suggested activities to involve students in meaningful learning activities. More precisely, we can defined it as socio-constructivist pedagogical scenario to promote laboratory understanding. Teacher’s are of course encouraged to adapt it to their local context.

1. Exploration of pre-instruction understanding through individual or group concept mapping.

2. Pre-laboratory activities, including informal writing, making observations, brainstorming, and posing questions.

3. Participation in laboratory activity.

4. Negotiation phase I – writing personal meanings for laboratory activity. (For example, writing journals.)

5. Negotiation phase II – sharing and comparing data interpretations in small groups. (For example, making group charts.)

6. Negotiation phase III – comparing science ideas to textbooks for other printed resources. (For example, writing group notes in response to focus questions.)

7. Negotiation phase IV – individual reflection and writing. (For example, creating a presentation such as a poster or report for a larger audience.)

8. Exploration of post-instruction understanding through concept mapping.

Hand, Prain and Wallace (2003:20)