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.


1. Light travels in a straight line. What is the evidence for this?

You cannot see around a corner! You only see something when the light that bounces off an object reaches your eye. You must be in the direct line of sight in order for them to reach your eye. for more information.

2. When do light rays bend? What is the evidence for this?

Light rays bend when they change medium and hit at an angle. When we did the demonstration in class, the line of students changed direction when they entered the “slower” medium. Please see if you want further clarification.

Other evidence of light rays bending was the appearance of a pencil or ruler placed in a glass beaker of water.

The part of the pencil in the water: When light reflected from the pencil reached your eye it traveled through the water and when it reached the curved surface of the glass, it changed direction when it changed medium – because the glass was curved, the light rays did not bend equally and therefore the image appeared magnified. The light rays reached your eye in a slightly different position.

The part of the pencil in the air: The light did not change medium, but traveled in air the whole time. The light rays stayed in the same relationship to each other and the image your eye perceived was not magnified.

When you looked at the pencil in the water, it appeared to be “broken” at the water line. This was because only part of the image of the pencil was magnified.

We looked through a  triangular glass prism and saw that the laser beam was bent and did not travel in a straight line. We also saw that the prism refracted sunlight into a “rainbow” of colors.

3. A lens makes light rays focus at one point. This is called a focal point. What is the evidence for this?

When you looked through the magnifying glass you were able to focus the sun’s light to a single point. Some people used a flashlight to get the same effect.

When we projected the image of the window onto a piece of paper, there was one particular distance when the image was clear and upside down. This distance was the focal distance of the lens.

We saw that the differently curved lenses had different focal points.

4. A shadow happens when light rays are blocked.

This is evidence that light travels in a straight line. provides more information.

5. You only see light when it is reflected into your eyeball.

Light can be there but we cannot see it unless it strikes something and gets reflected into the eye. A laser beam can only be seen when it strikes a surface such as a floor or the wall. However, when we placed a piece of paper in the air in front of the path of the laser light, the light reflected off the paper and traveled to our eye, so we could see it.

6. Convex lenses curve out. These are the kinds of lenses that magnify.

We saw that the magnifying glasses used convex lenses.

7. Why do objects appear black?

Black objects absorb all the light that reaches them. No light rays are reflected back to your eye.

Spring Valley High School and group-work

Okay, the events at Spring Valley High School are going to hijack the blog a little. I am so saddened by the way that discipline was hablack kidsndled in this case. That is, the case of a teacher managing his or her classroom so poorly that a policeman entered and arrested the student.

The adolescents in the photo are respectfully and interestedly taking part in a science lesson. The lesson was designed to engage students, connect to their lives, and allow them to learn from each other. This happened to have been a high-poverty, rural school, and the students in question were in a low track class.

Getting back to Spring Valley, I’m not really complaining about the violent way the student was thrown to the floor and injured, nor the way in which she defended her personal space by striking out at the person, who happened to be a policeman, who was attempting to drag her out of the seat. I’ve seen this called “punching” on television. That is ludicrous. This event is horrendous. But it begs the question. I’ve been in two schools in the South where policemen came into classrooms and removed African-American youth from the class in full view of all, and without consulting the teacher. This display of authoritarian power is unacceptable.

Don Lemon commented on the air that he doesn’t know the whole story and therefore can’t judge. Fair enough, but why is no one asking, What is going on with our schools that policemen are required to enforce order? This is crazy! And it’s totally unnecessary.

Misbehavior in school increasingly is becoming a criminal offense, resulting in a criminal record for the young person in question. I object to the practice of discipline being enforced by law officers.

As for the connection to group-work, when skillful teachers design engaging and challenging tasks for students, tasks which have some connection to their lives, and which they care about, the need for police intervention is 0. In the case of the girl being arrested for not obeying, there are many unanswered questions. The first is why did the responsible adult, the teacher, not have a relationship with the young person, which would have made calling a policeman unthinkable? It is the teacher’s job to connect with the student. That is the first and most important duty of a teacher.

why group-work is essential for learning



These children are engaged in what developmental psychologist Barbara Rogoff calls “intent participation.” Each group is learning to perform an engineering task with no teacher. The girls at upper left are constructing a book support out of newspapers and masking tape. The other two groups are constructing parachutes out of coffee filters. None of them has been taught how to complete the product. They are learning what they are already doing.

The notion that a teacher, someone with knowledge, must tell learners what they should be thinking is the single most crippling unintended result of Modern European-American style education. Most Americans leave school feeling stupid, because they are unable to figure out what teachers are telling them to think. No wonder American students continue to perform so poorly on measures of problem-solving and critical thinking. If we want children to think for themselves they must have opportunities and encouragement to do so.

I am not, of course, talking about the elite children, those who are able to guess what the teacher is thinking and reproduce it. My experience is that most such children, who get good grades and score high on standardized tests, have a very difficult time performing unstructured tasks that have more than one solution. They wait to be told what to do.

Anyway, if we want self-reliant learners, who can figure things out for themselves, children must work in groups with peers with the guidance of an adult.

More tomorrow about learning in groups…

So many children…

I’m going to be a little poetic here. By which I mean turning my attention to that which is not tangible.

I have the opportunity right now to work one-on-one with several children. It’s a chance to go back to basics: children want love, acceptance, security. They want to find out about the world, while being respected as valuable. Cliches. And so easy to lose sight of when you are engaging with big issues of school reform, achievement gaps and standardized testing.

I live across the street from an elementary school. Every school day starting about 7:30 the cars start jamming the road. Right next door to our building is a large, usually empty parking lot with signs posted everywhere: No School Parking. People park there anyway, hurry their children out of the car, shepherd them through the tangle of behemoth SUV’s driven by people who patiently or impatiently seek to thread their way through the obstacles of cars and human bodies. By the time the school bell rings at 7:50, things have quieted down, a few stragglers, and by 8:00 all is quiet again until the loudspeaker starts blaring announcements.

I remember announcements; it was an unremarkable feature of school life, just like standing in lines. Socializing children to being managed, to take their places as bodies in the machine where everything is monetized.

What would it take, I wonder, to educate (in the original meaning of the word, “to draw out”) the mass of children? They are, in fact, not the mass, but individuals looking for love. Any good teacher knows this, which is why there is so much resistance to standardization of education. The goal of education is to engage with children in the kinds of relationships that will draw out the beauty of the human soul.

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.