The above video is an excerpt from Bill Crombie’s portion of the “Advancing Equity and Inclusion in Math Education: The Algebra Project, AI, and Mindset.” The event was hosted by Dr. Alan Lyles of the University of Baltimore, with Dr. Jing Liu of the Univ. of Maryland and Bill Crombie of Algebra Project, Inc. on April 13th, 2023. You can watch the full recording here.
Below is a transcript of the video:
Alan Lyles: I would like to turn to Mr. Bill Crombie, who will, speaking from his experience in the Algebra Project and in the classroom, address some historical aspects of attempting to do what you were working on with modern technology. And that is how can we have rigor and equity in the types of math instruction that’s available so that people will work hard and succeed rather than being held back by the things that happen outside the classroom. Bill, did I give a fair introduction for what you’re going to be talking about?
Bill Crombie: Yes. Good. And I’m going to use a whiteboard and that means I’m laying my laptop down. So I just warn you, I’m turning off my camera, or else you’ll be staring into the light on the ceiling.
Alan Lyles: I can see your whiteboard, Bill.
Bill Crombie: You see my whiteboard? Good. And now let me just make sure that I can write with it. Good.
Okay, so good afternoon everyone. As Alan said, my name is Bill Crombie. I work with the Algebra Project. The Algebra Project is a small nonprofit in Cambridge, Massachusetts. We’ve worked for some 40 odd years to raise the floor of math literacy for all children in the US public school system, but especially for those students, most underserved and underperforming.
I want to describe an event from a few years ago. That captures, excuse me, some critical issues that students and teachers face in mathematics classrooms. The event itself took all of five minutes, but will walk through it in a kind of slow motion to see what it reveals. But first, a little history. So I’m old enough to remember coming out of, you know, the new math of the sixties and the back to basic movement of the seventies. In 1980, NCTM, the National Council for Teachers of Mathematics put forward what they called an agenda for action. The agenda for action said that mathematics instruction should be centered around problem solving. It is actually considered the beginning of the standards movement leading all the way up, you know, to our present common core.
Now, I don’t know if we ever landed close to that problem solving target, but over the years mathematics, and its instruction in the classroom has been moving increasingly towards exercises in the sense that in a problem, the method of solution is not known in advance. While an exercise practices an algorithm or a rule or procedure, and unfortunately, presently, most algebra students are practicing the applications of rules, not problem solving. To engage in problem solving, you need what we call an ill-defined domain. An ill-defined domain is one where the structure and the methods of solution are not given to the learner in advance. Importantly, an ill-defined domain. Sorry for my phone. An ill-defined domain really has the space for the exercise of both imagination and creativity.
Most of the so-called problems that our students get in say, algebra one, and that’s what I’ll be focusing on, are in fact well-defined domains. They are well defined domains in the sense that everyone knows, may not be able to do, but everyone knows what the algorithms or rules are that will be employed.
They are given by the chapter title right in the algebra one book. They are given by the standards that are written on the board at the beginning of class. They typically call for the careful execution and repetition of the steps in an algorithm, they do not call for an exercise of imagination and creativity.
I have to tell you, usually when I go into a school, I take a quick informal survey of students, and this consists of asking them for your major content areas. English, language arts, history, natural science and mathematics. Where do you rank those areas as opportunities for your exercise of imagination and creativity.
And as you can probably guess, mathematics has really consistently fallen at the bottom, right, of that survey. So the story that I want to share is about how math is presently done in classrooms all over this country. More so in some classrooms and in some communities than in others. And we will, we’ll get back to that question of particularity, but more to my story a few years ago, I was working with the school in Brooklyn. I visited the school once a month. I was working with the ninth grade teachers and their students, and we were about halfway through the school year when this incident occurred. One morning I’m in an algebra one class. The students were coming in and the teacher put an equation on the board and asked the students to solve it.
You know, it’s typically called a do now or a warmup exercise, and it’s meant to get students to start thinking about the mathematics that they’ll be engaged in during that period. So the equation that the teacher put on the board was something like this. Let me see.
There you go. 10 minus x is equal to a negative fine. Now, some of you might remember your own experience of going to the board to solve an algebra equation, you know, with very little or a lot of understanding of what the past was all about. As I tell you this story, try to to to bear that memory in mind and contrast it with what happens in this story.
The teacher almost asked for a volunteer to come to the front and share what they had done. But for some reason, and I, I’m not sure if the thinking behind it, she picked a girl that didn’t even raise her hand when she asked for volunteers. The girl went up to the board, took the marker, and she knew the first step as teachers tend to say, we want to get X by itself on one side of the equation. So the girl subtracted 10 from both sides of the equation and she ended up getting something like this.
Now she, she did know that the addition of those two negative numbers gives a negative 15. So she ends up with starting with 10 minus x is equal to negative five. Minus x is equal to negative 15. She then froze like a deer staring into oncoming headlights. Now without telling her directly, the teacher tried to suggest what the next step might be.
And as a, as an audience member watching this, it seemed like the this coaxing really went on forever. I can’t imagine how long it felt to the girl who was at the board, but here’s part of the difficulty the girl was facing. The equation was originally subtracting X from 10, but now it was subtracting X from nothing.
What in the world do we do with that? The teacher finally got her to divide both sides of the equation by minus one.
Looked something like that.
And she did know again that a negative number divided by a negative number was positive. The answer she got was X is equal to 15, and she genuinely looked relieved that it was over. It’s also important to note that the type of equation she was asked to solve is called a two-step equation because the algorithm students practice in solving these equations involves two steps, right?
In this case, there was a subtraction and a division, and of course there are one step equations that students did before they got to this. And multi-step equations that they would get to after this. This domain is highly structured in an algorithmic sense. They’re actually the, the procedure, excuse me.
The procedure is named by counting the steps. You literally counted steps as it’s being done. It’s almost like the Dancing by Numbers. If you remember those numbered footprints on the floor, it’s almost there. The Do Now is usually followed by the next algorithm and procedure to be considered in the district’s token sequence, and this scenario happens hundreds of thousands of times in classrooms across America.
This focus on algorithms and procedures has consequences. I had recently looked up data from the National Assessment of Educational Progress, their 2019 report. Which was prior to Covid, and that report says that the 12th grade NAEP mathematics achievement level has 32% of white 12th graders at or above proficiency. NAEP defines proficiency as demonstrating solid academic performance and competence in challenging subject matter. That’s less than one third of graduating white students in the US are prepared to go on to challenging subject matter In mathematics, 32% is nothing to write home about, but wait, it gets worse. Blacks performed at 8%. Hispanics at 11% and Native Americans at 9%.
If you think about that, this says that 92 out of every 100 black graduating 12th graders are not prepared for challenging mathematics or the STEM fields that depend upon that mathematics. And you know, I think we really need to picture this. If we were talking about language arts literacy, we would be referring to a time when only eight out of a hundred young people could read beyond a simple primer.
Now, I certainly don’t think that everyone should become a mathematician or choose a STEM career. I do believe that there are literally thousands of good reasons for not going into STEM. I’ve only found one bad reason for not going into a STEM field, and that one bad reason that I found is caught in the phrase I simply was not prepared.
Which all too often means I couldn’t handle the mathematics. But when we look at this, I think with percentages this low, the problem is that most students simply do not have a choice. Percentages this low go beyond the preparedness issue of individual students. This is a systemic institutional dysfunction.
You know, Bob Moses, the founder of the Algebra Project, used to characterize this situation as sharecropper education, and what he meant by this, this sharecropper education was and still is, an arrangement where communities of color, primarily communities of color, are systematically denied access to levels of literacy within the educational system, only to have those same levels of literacy used as conditions for either blocking or enabling greater participation in the system.
You know, but my story in the classroom doesn’t end there. The girl was ready to sit down. But I raised my hand and the teacher called on me and I asked her, I asked the girl, can you draw the picture that that equation describes? She actually looked relieved, that that was all I was asking her. She did and you know, this is how she did it.
Let me explain a bit, first. She and her class, were thinking about these types of equations as describing trips, believe it or not. They were thinking about equations as describing trips because the idea of a trip was something that everyone understood. It made sense from their everyday lives. They naturally understood what a trip was and what it entailed.
The idea, in this case, in this mathematics class, the idea of a trip was acting as a grounding metaphor to make the abstract symbols of algebra, concrete and meaningful. Now, it was a subtraction equation that she was working with. So instead of thinking of subtraction as takeaway, right, that was the model from elementary school, she and students in her class, were rethinking the notion of subtraction as a comparison, and we can think about this, of this idea of comparison like this. You know, you could imagine if the equation that you see to the left over there is the answer to a question on jeopardy, then what question is that an answer to?
And one way of framing this is the question, well, where is location 10?
And the immediate issue there is, well, the location of 10 depends upon where you start from it’s relative position. The equation says that we’re starting from location x. 10 compared to X is equal to negative five. The, the picture, the model the grounding metaphor that they’re using for subtraction is this idea of comparison.
And it’s the idea that the finish of a trip compared to the start of a trip, that is the location of 10 compared to the location of X, is simply the movement that takes you from the start to the finish. In this case, it’s a negative five. It’s five to the left, so the picture that she drew was this, the location of 10 compared to the location of X is one to the left and it is five units or five stops to the left.
So she, after she drew that picture, I asked her, so what is the location of X? She looked at the picture again and she immediately said, oh, X has to be 15. That is an example of math literacy. By math literacy we mean the ability to read, write, and reason with the symbol systems of mathematics. This student had a very limited ability to manipulate symbols without meaning. But she could easily read and reason about the symbols that meant something.
And, you know, from an algorithmic perspective I, I still wonder if anyone can count the steps involved in finding the solution to the equation in this way. In some ways this is much simpler than the two step algorithm, but in other ways, it’s much more sophisticated. Most importantly, when she solved the problem in this way, she actually understood what she was doing and what her answer meant.
But I, and I have to tell you, before she sat down, I asked her to look over at the calculation she had just done, and she did a kind of double take. She looked over, then looked back at the picture, she looked over again and then looked back, and then she turned around facing me and pointed to the calculation and she said, why would anyone do it that way?
And I, I have to tell you, I told her I really don’t know, but her question was important. It was implicitly raising a demand for a quality education in the mathematics classroom. It implicitly said that doing mathematics should not make students feel stupid, confused, or discouraged. It implicitly said that doing mathematics should be as easy as reading, reading what is there, and figuring out answers that make sense and it’s for that reason that Bob Moses had said that math literacy is the key to 21st century citizenship. Personally, I believe that we all need to ask that question, until we get a good answer, why would anyone do it that way?
Alan Lyles: Thank you, Bill. And we have a comment from one of my colleagues, Dr. (sp?). And she says, what a great, powerful example, Dr. Crombie. That’s I was wondering where your title came from and where you were going with it. A question for you before I, I say a few things and, and that is Bill, given the numbers that you started with what can the Algebra Project do with its pedagogy to start to turn that around?
Bill Crombie: Well, a couple of things. You know, we, we always say that learning should begin in the place where the learner is. And that is why in, if you look at the Algebra Project’s pedagogy, the most important mathematical concepts that we want students to learn. So it’s not many of them. And it may be, you know, half a dozen.
The central you know what, what, what Understanding by Design used to call the big ideas that they should carry with them. We’ve been working across those 40 years to make sure that we have an experiential foundation that is accessible to all students, and especially the lowest performing students.
Right. That’s accessible to those ideas through a concrete shared experience, you know, I think there’s now at least nominally agreement that we should not approach student learning with a deficit model. But the question is, well, so where are the strengths? And, you know, our late founder Bob Moses, he, he had a background, not only he was teaching mathematics in New York City before he went to Mississippi. When he came back from Africa in, in the eighties, he was studying the philosophy of mathematics as a graduate student at Harvard.
And both, I think both his experience in Mississippi. And his studies in the philosophy of mathematics and that experience in the classroom said that all students do come with strengths. And one of the most overlooked strengths that all students have is the logic that is embedded in the language that they speak.
And I think we aren’t really exploiting that to the, to the degree that we should. So after that experience, we don’t jump into the math if there’s a shared concrete experience. We want students to draw pictures of it, to write about it, and to talk about it in what we call people talk, because that is a strength that all students bring, and giving the student an opportunity to reflect upon their own language creates a domain of imagination and creativity, which is also an essential to doing mathematics, English, social studies or science. If, if we so strip those disciplines of that most human capability, of course students are not gonna learn well. So, those are the elements that I, I think the Algebra Project is most concerned with.
And I should mention, you know, implicit in that, in that kind of work is not a picture of the isolated student working alone. It’s the concrete shared experience. It’s students making an investment not only in their own understanding, but in the understanding of their classmates and to really build that knowledge foundation.
Alan Lyles: Bill, one more question before I say my few comments. And this one has both numbers, but also requires insightful judgment. One of my colleagues has written Baltimore City School’s most recent citywide math testing for third through eighth grades yielded a 7% proficiency, half of pre pandemic proficiency, which was 14% with a budget of approximately $20,000 per student.
Does this outcome likely reflect the teaching or the testing or both or neither?
Bill Crombie: Yes. From my opinion, I’m affirming the disjunction. Look, when, when, when you see scores that low as, as I was saying, it is, it’s a systemic, it is an institutional issue and to say it’s an institutional issue. An easy way out for the institution is to say, my teachers aren’t doing what they’re supposed to be doing.
That is not how you, how you resolve an institutional issue. I don’t, I don’t know if I’m gonna say this, but let’s keep this just between us. You know, in the US military, when there’s an institutional issue, they don’t remove the troops. They change the leadership, right? They get leadership in there, right, that can reorganize human effort. And believe me, I’m surprised to find myself using a military example, but they get leadership in there that can reorganize effective human effort to get the job done. Both teachers and students are too often the scapegoat for data like that. I mean, I, I think it basically says that all of us in some way have to collaborate, work together.
But at least from our perspective, and this is really a lesson that Bob took out of the civil rights struggle, the people that are most affected by the problem need to be active at the center of the solution, need to have agency at the center of the solution. And, you know, those are teachers who struggle with their students and those broader communities.
If it, if, if they’re not there. You don’t have the necessary condition for finding a solution.