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.

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…

learning through discussion. what does it mean?

I’ve been meaning to write about this for a week or so. I had occasion recently to visit a student teacher in a 7th grade life science class. The topic for the day was sexual and asexual reproduction. The lesson started out with a Brain Pop video. The plan was for a “class discussion” and then the students would take the Brain Pop quiz.

On the previous day, the students had each been given a handout on a plant or animal that reproduces asexually. I’m not quite sure what was done with the information, although I suspect groups made presentations to the class. At various points in the lesson the teacher asked them to refer back to “their” organism from yesterday.

The participation structure for the discussion followed the recitation script (IRE). The teacher asked questions about what was in the video and asked students to connect that information to the organisms they had researched. She affirmed whether or not the responses were accurate. Actually, not one student was able provide much in the way of a response, in fact, I’m pretty sure that any student who was called on, said, something like, “Um, I, I can’t remember the name of my animal.” The student were directed to look in their “interactive notebooks,” and after several seconds of silence, would say, “Oh yeah, yeast,” or “I can’t pronounce it.”

The students were genuinely trying. It appeared to me that they were not making the connections that were the intention of the lesson plans. The “discussion” involved quite a bit of silence. At this point I decided to step in. The rationale was that students seemed not to have incorporated the research on asexually-reproducing organisms into their watching of the Brain Pop video.

I hope the student teachers I “supervise” are used to me stepping in and modeling when I think it will do some good. I interrupted and asked the students, well a week later I can’t quite remember what I said. But the purpose was for students to provide their personal experiences with observing asexual reproduction. The discussion was lively and covered much ground. As I said to Betsy (pseudonym), the student teacher, afterwards, everyone is interested in sex and reproduction. It isn’t difficult to get a discussion going. Several of the ideas matched concepts from the Brain Pop video, although they were not referred to explicitly.

After 10 minutes, Betsy resumed her lesson plan and the students took the Brain Pop quiz. An overwhelming majority of the students got most of the answers wrong, in spite of the fact that they were designed to assess the very concepts students had volunteered during the class discussion.

Now I’m still thinking about this event. Clearly the students understood the concepts in the discussion and clearly they did not connect them with the formal instruction of the Brain Pop video. I remarked to Betsy that it appeared the students did not understand that the discussion was actually learning.

This has pretty profound implications. I will be writing more about this.

What is teaching? What is online teaching?

The other day someone asked me to provide an example of my “online teaching.” I explained some of the online assignments I’ve given, but the person continued, “No, I mean teaching.” It took me a few seconds to comprehend the question, that she was asking if I put my lectures online, or annotated videos. I still probably had a deer in the headlights look. You see, I don’t lecture. I consider that I’m engaging with students in instructional conversations.

Please see my earlier posts about learning through discussion, and also the rest of the category College teaching.

This past semester I have had conversations with students about their experiences with online courses. I was surprised to hear that they thought the discussion board assignments were busywork. Since I have used discussion boards quite extensively in the past, I was intrigued by this comment. When I pressed a little more, they said they did not write what they really think, but what they thought the instructor was looking for.

This has caused me to pause before such an assignment. This semester I asked students to upload concept maps to the discussion boards, but I must say it has not been as productive as I hoped.

In the past, I assigned discussion board postings, and students would always want to know how long it had to be. I settled on 300 words. Then I noticed that students were copying lengthy quotes into their postings. I had to forbid quotes longer than 10 words, and no more than one quote per posting. However, it seems to me I have to find a better way to design discussion boards so that students understand them as learning conversations.

I think the deeper issue is that students don’t understand discussion as being a way to learn. I just had an experience of this last week when I visited a student teacher in a 7th grade classroom, which will be the subject of my next post. I again refer you to the blog of December 2010, when I started writing about learning through discussion.

Teaching about classroom management

This has always been puzzling to me. There are many “systems” and methods, but I’ve never been able to get to the heart of the matter in a way that satisfied me and helped my students.

Then this semester I was observing a student teacher in his first placement. I’ll call him Mike. Mike had been a successful math tutor, and the experience made him want to be a professional educator and high school math teacher. Mike was well connected within the community and obtained a job at a local high school while he is enrolled in a teacher education program.

The lesson was okay, not great, but then completely fell apart when Mike spent 4 minutes working one-on-one with a student at the whiteboard. By the time the 4 minutes was up, no student was doing any math. Wadded up papers were being thrown across the room, calculators slid off desks and  littered the floor. I tried to get Mike’s attention but he was focusing on helping the single student.

When the lesson was over, we debriefed. “You have to teach the whole class,” I said. I thought a little more. “You have to teach the whole class all the time. Even when you’re teaching one student you’re still teaching the whole class.”

I’ve been considering this idea ever since. I think I’m beginning to formulate for myself the nature of the difficulties pre-service teachers have with classroom management. They usually think of teaching as standing up in front of the class explaining things. But everything a teacher does within the four walls and in preparing for the students to be studying within the four walls is teaching. In fact, the parts that are not explaining things to the whole class are much more important and much more work.

I started thinking of the chapter titles of my favorite book about teaching, Magdalene Lampert’s Teaching Problems and the Problems of Teaching: “Teaching while Preparing for a Lesson,” Teaching while Students work Independently,” “Teaching Students to be People Who Study in School.” I’ve always been drawn to these words, but in the last month the insight they contain has begun to grow in my mind.

Before Christmas I assigned the student teaching seminar to watch the entire five tedious hours of YouTube videos of Mr. Hester’s first three days of school. We had a discussion about the importance of establishing procedures. I asked my students what we can take home from the videos. There were a number of good ideas, then somebody said, “Consistency.” A little switch went off in my brain. “Why does consistency matter?” I said. The students didn’t really know. It’s what they heard. It’s what I have heard many times. I hadn’t really thought about it, other than that’s what the research says, and it’s obviously what Mr. Hester did in his videos.

An idea was growing, a connection being made. “It’s because you’re teaching the whole class. If one student acts out and you ignore it, for that one student maybe it’s not a big deal. But if you ignore it, then you’re teaching the rest of the class they can do it too. Teaching is carried out in public. You’re always teaching the whole class.”

I feel like we’re all, the teacher candidates and me, understanding the classroom management thing better now. Being consistent isn’t being mean and picky—it’s teaching. And we had a slightly magical moment when something shared fell into place.

Mr. Hester’s videos are on YouTube. We watched Day 1 first, then days 2 and 3, and then Meet Mr. Hester last. Okay, it wasn’t 5 hours. Just seemed like it. Worth it.

teaching about race and social class

This semester I’ve been teaching “the diversity course” to MAT students. My goal has been to facilitate students’ development of insights into how differences that appear natural, such as smartness in school, are actually socially constructed. It’s been tough at times, because while it’s not polite to talk about race publicly, it’s really not polite to talk about social class.

Last week a White teacher candidate, one of the more outspoken class members,  remarked, “You really don’t like White people do you?” To which I replied, “I don’t like privilege.” What I don’t like is that the ideology of whiteness denies access to social goods to people of color. Facing America’s long history of privilege for some made possible by oppression of others is painful. We still have to talk more.

But the cool thing that happened is that one of the African American students wrote me an email this weekend saying this class has really opened her eyes to what has been going on. I want to find out what she means  when we meet this week. We still have to talk more.

The sequence of readings and activities I hoped would scaffold examining of assumptions about schooling seems to be working,  although I need more evidence to say this definitively.

  • critical place-based curriculum readings
  • focused observations of classroom interactions around race, class and gender in practicum placements
  • reading the classic Ray Rist article
  • talking about it over and over
  • reading The Children in Room E-4 by Susan Eaton
  • learning about and doing Complex Instruction

We still have to talk more.

balancing equations: algebra v. chemistry

Yesterday in our content pedagogy class, we were talking about key ways of thinking in the disciplines, which we called signature pedagogies. A proposal for a signature pedagogy brought up by a chemistry teacher included balancing equations. One of the high school math teachers suggested that balancing equations is a concept in algebra also.

The goal of the discussion was to consider disciplinary ways of thinking. While it is true that there is a procedure called “balancing (or solving) equations” in these two disciplines, they involve very different ways of thinking.

In chemistry, balancing equations is grounded in the big idea that matter cannot be created or destroyed. The key  idea is that the same atoms we start with must be the ones we end up with. I used to tell students that balancing chemistry equations is just bookkeeping.

In algebra, we start with a true statement, and “balancing” equations is part of solving them. The solution, finding out what x can be, is about maintaining the integrity of the original true statement. The procedures for solving equations include balanced operations on either side of the equal sign.

Balancing mathematical equations is about logic. Balancing chemical equations is about processes which occur when different atoms react. To some extent, both require bookkeeping. However, understanding them as only bookkeeping does not go to the heart of the disciplines.

While this might seem like a philosophical quibble, I think it has important consequences for learners. When I used to tutor algebra, I would focus on the procedure for solving equations as being about maintaining the truth of the statement. When they went from a procedural understanding to a conceptual one, every young person (but one) that I tutored went from low grades to A’s. One high school student decided she didn’t need to go to class anymore, and started skipping, still getting A’s, at which point her mother ended the tutoring. What kind of commentary that is on math teaching in the US, I will not venture to say since it’s not very comprehensive data.

Teaching research methodology

This is a blog post I made after class, which started out as an email to a student who had to be absent. Like most successful students, she views the purpose of class as imparting information and asked me what information she had missed. Part of my role as an instructor is to pry students’ fingers off this notion. Of course there is information that happens. We have just started reading Writing the New Ethnography by H.L. Goodall.

As for what we did in class tonight in terms of information, hmm, well, I talked about the "crisis of representation" and modern social science. I talked a little about positivism, positivistic science, the death of positivistic science with the development of quantum mechanics and Einsteinian relativity, and how this coincided with the collapse of "civilized" Europe in WWI, the ensuing worldwide depression, the Russian Revolution, WWII and then the atom bomb. The foundational assumptions of the European intellectual tradition were challenged by both science and historical catastrophe. This led to people in the social sciences questioning the enterprise of scientifically describing the experiences of others, and to the question of how it is possible to represent the experience of others. What gives anyone the right to make pronouncements about others? And besides that, how can anyone claim that any amount of data collection can give a complete picture of an experience? And furthermore, how can words in a journal article or a book accurately convey the reality of the people it describes? This is the "crisis of representation." How do we represent the world? It is not possible. We just do the best we can, acknowledging the limitations.

I would add, which I did not in class, that the research rings truest when the limitations are discussed as specifically as possible. There are always of course unknown limitations, but the researcher does the best she can with identifying them, beginning with her own biases. Another way researchers make clear what limitations might exist is by being very forthright about how the data was collected, what she was doing during the data collection, how she reacted emotionally and intellectually to the data. When observing, recording these reactions is crucial.

A researcher is her or himself the ultimate instrument of data collection. It is through his brain and the connections he makes in his mind that the data acquires meaning. The self of the researcher interacts with the selves in the research setting. Therefore it is important to acknowledge the role of the self in doing research, and to be open and self-reflective about it. This is discussed a little in Goodall Ch. 1. He’s going to talk more about it in further chapters. We talked about good writing being transformative.

In discussing students’ responses to the Goodall book, we got off onto the topic of nothing ever being good enough in academia, it is always open to critique. There is a sense that we should have done more. While this is true of society to some extent, it is a major practice in academia. Critique is the lifeblood of the academy.

We spent most of the class time reviewing each other’s research questions through a gallery walk and comments made on sticky notes. I don’t know that this is something we could tell you about. You had to be there. I deliberately design in-person class using a Vygotskian scaffolding model, as discussed in the Walqui and vanLier chapters I sent you. This means me and other students responding to each other in ways which allow for "authoring" of knowledge rather than its consumption. ("Authoring" is a term from literary criticism, originated by Mikhail Bakhtin.) My goal is that a ZPD arises spontaneously based on my intuition about what might be a productive direction for generating new knowledge. I know I miss things, and let drop threads of conversation which might be valuable, but on the whole I’m satisfied that this provides an environment for deep learning.

So we considered research questions at length, and worked together to think about how to help each other with our questions.

Science Education for All. I Mean it: Each and All

I’m reading Larry Cuban’s new book, Inside the Black Box of Classroom Practice, which has a chapter on the history of science education reform. (Note the subtitle, Change without Reform in American Education.) He quotes Jonathan Osborne, who points out that the goals of science education appear to be contradictory. Are we aiming to produce scientifically literate citizens or future scientists and engineers?

I thought about it for few minutes, and came down on the side of scientific literacy. Well,  but certainly we do need future scientists and engineers. Why can’t we have both?

If you read some of my blog posts, one of the issues I’ve been grappling with is college science teaching. Post-secondary instruction drives the whole show. Future K-12 science teachers quite naturally try to reproduce the curriculum they experience in college. Lectures, Q&A sessions, laboratory investigations, exams, quizzes, etc. A very strong body of evidence supports the notion that college science coursework is not much like what scientists actually do in their work. Science studies show real scientists engage in flights of imagination and visualization, personify inanimate entities such as electrons, and work out tough problems in dreams. College students become scientists when they join lab groups as apprentices, usually as graduate students, although undergraduate research is becoming more common.

Beth Warren and Ann Rosebery show how young children really do think like scientists, using their imagination for example, wondering out loud what it feels like for a plant to grow, comparing plant growth to the experience of outgrowing your shoes. Well meaning teachers, enculturated into school science by 16 years of “science education,” typically squelch such flights of fancy in order to prepare students for the difficult and dry science they will encounter in the future.

But the difficult and dry science of high school and college is not science! In fact, professors I know complain that students who come to them don’t know how to think, imagine, or solve novel problems. By and large, school science is not preparing anybody for knowing and doing real science. Youth who later become scientists have to unlearn habits of mind which are not productive of innovation and critical thinking.

I have in front of me the latest edition of the Harvard Education Letter. In the cover article, “Changing the Face of Math,” Laura Pappano provides a good argument for reform mathematics focused on engaging youth in complex, open-ended problem solving connected with their lives. “What if our national problems with math…are more about fuzzy-sounding stuff like relationships, emotion, and identity than, well, actual math?” Students disengage from rote memorization and rote memorization of procedures does not prepare them for doing mathematics in the future.

This brings me back to the dilemma raised by Jonathan Osborne, a leading voice in science education. What if we’re looking at it wrong? Is there some other way to think about science education that does not involve a competing and mutually exclusive goals for the population of US high school students?

I believe we need a paradigm shift. What does challenging, emotionally accessible, interesting science education look like? We pretty much know, actually. It doesn’t happen because we are hesitant to abandon an admittedly flawed system which has in the past produced some pretty good results. However, continuing to exclude students of color and girls from the STEM workforce is not acceptable. Furthermore, continuing to exclude students of color and girls from the power of STEM knowledge is not acceptable.

Co-Teaching Physical Science for Teachers

Last night Rosalie facilitated a lecture-discussion on heat, developing students’ ideas about energy transfer and how to do problems. I noticed a couple of times students gave the "right" answer, but when Rosalie probed further, the students didn’t really understand what they were talking about. It would have been easy to accept the correct answer as proof of understanding and then move on. The extent of students’ not knowing was profound, and it took some time to uncover its true dimensions.

I’m reminded of diSessa’s construct of p-prims, which he describes (to the best of my recollection) as conclusions about phenomena which are not linked to other ideas, but remain as islands. When Rosalie asked students to make connections or to create a chain of causal reasoning, they were able to do so only with great difficulty and a great deal of prompting in the form of questions. She engaged individual students in extended questioning in order to scaffold putting together a cohesive whole.

I noticed that not all students were following the conversation and did not seem to understand that their colleagues’ were being questioned publicly in this way in order to get ideas onto the table for everyone to consider. Earlier in the evening students repeatedly focused on the right answer, and when someone came up with an answer that was judged to be correct, it was quickly passed around. At the time Rosalie announced that we were not really interested in the correct answer, which the students seemed to shrug off. I think that they don’t have any other perspective on science calculations, and the idea of viewing problems as a shorthand for science concepts is an entirely new idea for them.

At the end of Rosalie’s discussion of heat, I felt that we should call students’ attention to what we had been doing by investigating their ideas in depth. I had two purposes in doing this. One was to let students know that the structure of our lesson was deliberate, and that we had a particular pedagogical goal in mind. I also wanted to clue in those students who had not been paying attention that perhaps this was important. I reiterated that we were not interested in formulas, but that they should focus on understanding the problem; understanding makes the strategies for solving it obvious. Rosalie reiterated that she too is not interested in students memorizing formulas. I also explained that there had been several times during the lecture when students had appeared to give the correct answer, but Dr. Richards kept probing, and it was revealed that the students did not really understand. I tied this to the issue of deciding what to accept as evidence of learning, and asked whether they had run into this phenomenon in their field placements.

I will say that we started the evening with a wide ranging discussion of the role of energy in the body, and the way chemical energy of food is transferred through digestion and metabolism. I was expecting students would not relate the process of combustion from the lab of calories obtained by burning Cheetos with the breaking of chemical bonds within food substances. I discovered this some years ago in teaching high schoolers, when I would ask them why they need oxygen, and the students were unable to go beyond because you can’t breathe and you’ll die. What a shame it is that we don’t explore the big picture and assume that students have made connections such as the role of oxygen in both combustion and cellular respiration.

The conversation about heat contained within food revealed that students remember very little of any high school biology.

The previous night in one of my graduate classes we started exploring the idea of their perceived lack of connection between learning and completing assignments. Before Rosalie came in, I decided to see what the undergrads had to say about this topic. They basically said they had to choose: either do the assignment and get the points, or study and try to understand. I wondered whether the purpose of doing assignments is to facilitate learning.

We did not get very "far" in our discussion of heat, although we perhaps got deep. I came away from last night’s class with another piece of evidence I interpret as showing the need to explore ideas in depth, and the conviction that most science instruction merely papers over students’ confusion.