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.

what is teaching anyway? (complicated argument 2)

Yesterday I ranted about politics, the reason being that education is crucial to our continued existence as a semi-democracy. The specter of a fascist, conspiracy-theory driven regime in Washington is so beyond horrifying to me. I noticed today that a group of teachers of the year have broken their own rules about remaining neutral in elections and written a letter denouncing Trump, appearing in the Washington Post. I’m not the only one.

So, the point is, what are we as educators going to do about the apparent inability of citizens to think critically, weigh evidence and apply the lessons of history to the present? This is a matter of urgency, and we don’t have much time to fix it.

I laid responsibility at the feet of teacher education, which is not completely fair. I was reacting to the NYT article about teacher recruitment, which frankly made me angry. What I didn’t say  was that it’s a whole-society problem, and that teacher education is simply a reflection of the culture.

I recently talked to some middle school science students about learning. They had been doing a sequence of learning activities designed to “teach” them about the electromagnetic spectrum. We first explored what happens to light when it travels through a cup of water (with a pencil in it), and what happens to light when it passes through a convex lens.

We physically (by going outside and actually doing it) modeled the tried-and-true marching soldiers demonstration (click here) to think about what happens when light waves pass through a new medium at an angle. We then did an angle of incidence/angle of reflection lab with mirrors, and the students then read about light in their textbook and answered the “section review” questions. They created posters of the electromagnetic spectrum, and finally took a test. They did pretty well.

I had very good reasons for designing the unit the way I did. There was an extended period of exploration of the phenomena in question, so that students would have a deeper understanding of what the textbook abstractions were talking about. We physically modeled the marching soldiers because movement activates the brain and enhances learning. I regretted that I did not have multimedia available in this classroom so that we could explore other parts of the electromagnetic spectrum, nor did I as a substitute have a stockroom full of equipment to tap into. Diffraction gratings would have been useful.

The interesting part was the conversation we had about learning when preparing for the test. I made the statement that the students were supposed to learn from the activities we had been doing. Someone said, “You have to teach us!” I immediately replied, “I can’t get inside your head and put knowledge in. You have to think. Learning is your responsibility.”

I reviewed the sequence of activities, and we discussed what had been the point of each. I tried to be explicit. “I’m not going to tell you what to think. You need to make sure you understand the point of each lesson. The learning happens inside your head.”

I’ve written before about students not understanding that the point of learning activities is to learn– in a blog post, “learning through discussion, what does it mean?”My conclusion is that the cultural model of learning, in which teachers tell students what to think, is the perspective through which learners understand classroom activities. They passively wait for the teacher to tell them what will be on the test.

Such a set of assumptions undermines democracy. Our students do not see themselves as anything but passive consumers of information, and not active agents whose job is to think about and determine the truthfulness of what they see and hear.

a complicated argument

It’s going to take many posts to make the argument about why teacher education is “failing.” See my post from yesterday

The first thing I want to say is that I respect one of the impulses that drives people to support Trump, that is, the feeling that corporations are completely in control of government. This is at least partially true. There are also people who will do anything to stop abortion, and are willing to support an unhinged, mentally ill, sociopathic, narcissistic fascist who probably has been responsible for any number of women getting abortions, and for all I know has paid for them. To those people I say, if you think abortion is wrong, don’t have one. Leave the rest of us alone.

I also say, “Be careful.” Democratically elected, unhinged, mentally ill, sociopathic, narcissistic fascists have, in the not so distant past, brought down on the world the fires of  tyranny, genocide and other unspeakable horrors of the 20th and 21st Centuries.

Probably the last obviously mentally ill president was Richard Nixon. His paranoia led to Watergate. Perhaps even worse, he actively prolonged the end of the Vietnam War so that he could have political advantage. Thousands of people died so he could be president. His sociopathic, narcissistic behavior led to  his ‘’enemies list,” damaging the lives of people who opposed him.

There are also the persistent and reliable reports that DJT studied Hitler’s rise to power as a model for his campaign, and has done so for many years.

The point is—It is possible to support Trump if you are ignorant of history; otherwise the red flags and warning whistles are overwhelming and truly frightening. And why are people ignorant of history?

Teacher education.

The New York Times says that the US teacher pool is filled with the not very bright. They compare us unfavorably with Finland, which allows only applicants from the top quarter of university students to apply to become teachers. Something the Times doesn’t talk about: Teachers in Finland are well enough paid, have good working conditions and  have the respect of society. But the Times doesn’t talk about that, because bastion of corporate America that it is, it doesn’t support adequate funding for education.

If Americans want good teachers, they have to pay for them.

First Donald Trump, Now This

My good friend Charles texted me that he had read about how Finland has much better teacher education, and the NY Times had an editorial stating that low quality teacher education in the US is the cause of well, I don’t know, monumentally STUPID people? (That’s not exactly what the Times said. That’s what I say.)

Image result for "teacher education"

You can read it here.

When I read Charles’ text, I laughed bitterly, anger surging through my body. So the New York Times wants to improve teacher education. Well, well, never mind all the years of editorials urging competition and charter schools as the cure for low achievement, and all the years pressing for schools to be run like businesses. Schools are getting steadily worse since the 1990’s because corporate America has discovered the profits to be made from education. See the critique of the Times education positions here.

Americans are just batshit crazy. We have a perfectly good country and are throwing it all away. Okay, so the education debacle predates the DJT debacle and probably is part of the cause. I just don’t know where to begin.