Algebra, the New Civil Right (1993)

by | Apr 5, 2022

Foreword by Ben Moynihan

“Se wo were fi na wosankofa a yenkyi,” or, “it is not taboo to go back and fetch what you forgot.”
― An Akan proverb from Ghana, West Africa (1)

Consider the Sankofa bird, facing forward with its beak cherishing an egg while looking back, like an arrow which must be pulled backwards before it can fly into the future, we hold the Akan Twi word, “Sankofa”—which means literally “to retrieve”—in our hearts as we look backwards so that we may move forward imbued with Bob Moses’ liberatory passion. In this spirit, we revisit an address he made almost 30 years ago.

The Sankofa Bird

Moses spoke at the Strengthening Underrepresented Minority Mathematics Achievement Intervention Programs Conference II (SUMMAC II) in November 1993. Weaving together history, common sense, and empirical research, Bob sought to spark a discussion leading towards a consensus that mathematics education be pursued as a universal literacy requirement for full participation in 21st century society.

Bob posits that all K-12 students must become fluent in the language of mathematics and be prepared for credit bearing courses in college. He calls on mathematicians and mathematics educators, teachers and school systems, and all of us working with them, to take responsibility for creating structured opportunities for young people to gain the literacies they need for the Information Age. He states, “…history has played this trick on us and put mathematicians in this … critical place around the question of democracy of this country, because this math/science tool is really assuming as important a place as reading and writing assumed in the old dispensation.”

Why are we pulling up Bob’s 1993 address? In this year of transition since his passing in July 2021, the Algebra Project is working with schools and teachers and students to carry on Bob’s vision. At the time of this talk, the Algebra Project was solely focused on facilitating middle school students’ transition from arithmetic to algebraic thinking. Since the 2000s we also have collaborated with high schools. All of this requires of us, Sankofa-like, to look backward and forward at the same time.

We seek to remember and respond to Bob’s call today: enabling students, teachers and school leaders to “raise the floor of mathematics literacy” remains a responsibility of mathematicians, mathematics educators and those of us concerned with empowering young people to fully participate in society so that democracy may survive.

With gratitude to Dr. Florence D. Fasanelli who transcribed his remarks back in 1993, the text of Bob’s speech was included in an Algebra Initiative Colloquium proceeding, and edited by Carole B. Lacampagne, William Blair and Jim Kaput, and published by the U.S. Dept. of Education in 1995 (2).
― Ben Moynihan, Interim Executive Director

Algebra, The New Civil Right
Bob Moses
Cambridge, Massachusetts

I would like to put out some ideas which face us, face the country, and also, of course, face this group. I find it ironic that mathematicians should be so centrally placed in a national issue. It seems that history has done you a disservice. It has put to you a task for which you are not prepared. Mathematicians would be the last people that I would turn to to organize the country. And yet, that’s what it seems you have to do.

One way I think about the situation we’re in is that we have this kind of moving of the plates which is associated with earthquakes. These plates move and sometimes they lead to these earthquakes and houses fall down and people get dislocated. So it seems that we are living in a time when two technological plates are rubbing up against each other, living through some of the earthquake-type phenomenon which has resulted from this-the movement from industrial technology

into a new information age based upon computer technology. There is a lot of social dislocation which is happening because of this kind of earthquake-like phenomenon. At least, that is how I see it. So, it’s again ironic that the new technology puts mathematics and science into front and center and, therefore, requires that mathematicians play a role in stemming this social dislocation for which they were not prepared. How shall we characterize this? One way that I think about it is that there is a literacy issue for citizenship. I view myself as working in a tradition in the Civil Rights Movement which is not really well known. How many people here saw “Eyes On the Prize” or some segment.

Well, “Eyes On the Prize” is a documentary of the Civil Rights Movement. It was a 6-hour documentary dealing with the Civil Rights Movement in the early sixties put on PBS about 3 or 4 years ago and is replayed every year. You really need to see it. It usually comes on in February, Black History Month time, but other stations do it at different times around the year. So you need to keep an eye out for it.

“Eves On the Prize” puts forth a certain myth about the civil rights history. I think of it as our first visual history book. It’s a product of the new technology. It puts forward the idea that we might have different kinds of people being historians; that the person with the camera will also be an historian of the future; that the new technology allows us to record enormous amounts of visual data in small spaces, and opens up the issue of “Well, what is a history book?”.

I mention it because I took issue with Henry Hampton who developed “Eves On the Prize.” I said, “Look, Henry, there is a part of the history that you are not telling. Henry’s point of view was that if someone didn’t capture it on film at the time in which it happened, then it didn’t happen That is, you couldn’t tell about it. You couldn’t put it in an interesting way on film to a mass audience. So, the “Eyes On the Prize” visual history book tells one myth about civil rights history. And it is a myth which deals with the history of great campaigns, March On Washington; Birmingham; the Voting Rights March in Selma, Alabama; Albany, Georgia; Freedom Summer-big campaigns of the Civil Rights Movement-and the person who came to symbolize such campaigns, Dr. Martin Luther King.

There is another myth about that history, and I consider myself to be a part of a legacy of that myth. That’s a myth dealing with the organizing aspect. The part that did not get on film. Remember in the 1960s the nation was cutting its teeth with its TV programs. So, you had three major networks, and they were learning how to do TV. They learned using the Civil Rights Movement and its campaigns. But they did not pick up the organizing efforts which undergird those campaigns, and that’s the tradition that I came out of.

Ella Baker, who was the person who helped found Dr. King’s organization, sort of provided the model for us. The idea that leadership could be found in and among what we call grassroots people. That it was important to keep working with grassroots people to help develop the leadership from among them. That is the tradition that I came out of in the Rights Movement. It’s an organizing tradition. It’s a tradition which tries to stake out some problem around which there is consensus and builds to see if there is a way to find a solution to the problem.

In those days the issue was the Right to Vote, the question was Political Access, and associated with both of these was a literacy question around reading and writing. In these days there is another issue which is math and science literacy. It is associated with, not political access, but economic access. At the center of it as it is constituted in our society are you folks, mathematicians, and the question about algebra. Look at the work I am doing today as a continuation of the work that we did in the sixties. That is, certain people in Mississippi were serfs, people who were living in serfdom on plantations. They basically had no control over their lives-their political lives, their economic lives, their educational lives. So, within our industrial society we had this sort of microcosm of serfdom that we permitted to thrive, and the movement used the vote and political access to try to break it up.

It seems to me that we are growing these serf-like entities or neighborhoods within our cities today. We have within our midst, I think, a process of criminalization of our neighborhoods. I find an analogy to that with what we found in the Mississippi Delta plantations. We learned some things that we could do to change Mississippi. One of the things we learned was that if somehow there was a consensus that everyone agreed that we should do item “A,” this consensus provided some base for strategy and action to try to work our way out of the problem. What everyone agreed to in Mississippi was that the vote would help.

So for a short period of time, all of the people who were acting to try to change Mississippi agreed to work together on a common program to get the vote. That enabled us to get resources from around the country to come and work with us, because they could all work on the same program.

Now, it seems to me there is a similar type of agreement today around math. That is, everyone agrees that if we can teach these children this mathematics, and let’s suppose we agree about what the mathematics is, but if we can teach these children this mathematics, then we ought to. There seems to be universal agreement about this, that if we can do it, we ought to do it.

That’s a basis. If we can get some consensus about how to work this, and granted, there is the issue that Paul Sally raised about “keeping the math honest” and the issue about “a truly objective standard,” I think those were his words. But I think this is an opening here if somehow we can get a consensus, and you people here are critical to the fashioning of such a consensus. So, I don’t want to let you off your task. That is, it is really critically important that you fashion a consensus. It isn’t fun and games. And no one else can do it. For good or for bad, this ball has been tossed in your laps. It isn’t something I would have chosen to do. As I said, the mathematicians really are not prepared for this job. Your training as mathematicians didn’t prepare you to organize.

But, the first job of the organizer is to flesh-out a consensus. You cannot move this country, unless you have a consensus around which you are going to move them. The country’s too big, too huge, too diverse, too confused. Now, that’s part of what we learned in Mississippi. We learned, and I am saying we learned it on the ground…running. We learned that if we could get resources and develop strategies to try to work our way out of this situation.

Now, that’s one thing, this legacy and a parallel between it and the situation we were in Mississippi, and I am just trying to show you how I look at this, these issues, and the right to vote; and political access; and political free do; and this situation we have in the country; and the idea of citizenship which now requires, not only a reading/writing tool but a math/science tool.

The subject of math literacy and economic access, that is how are we going to give hope to the young generation. I think of them as imploding. Los Angeles exploded for a brief second and everyone got concerned, concerned about the trials and all of that. But those of us who live in these neighborhoods, we are watching them implode all of the time, everyday. They are imploding. The violence and the criminalization is people eating each other up inside. There are all the issues about band-aid solutions, about how do we patch up, how do we build more jails, how do we put more police on the street—working at the problem from the back end to try and keep it manageable—keep the lid on. But working it from the front end to try and put something in place which we know has to be put in place if there is going to be some light at the end of this tunnel rests with us. The front end of this problem rests with the people who hold the key to the mathematics education of the youngsters.

That’s a new problem for you. It’s a new problem for the country. The traditional role of the mathematicians has been to find the bright young math potentials and bring them to your universities and help them become mathematicians and scientists. It hasn’t been a literacy effort. There is a difference between doing projects and doing systemic change. As a country, we don’t know how to do systemic change. We don’t have any track record. We can’t point to any school system where we put through systemic change around the math education in that school system.

How are we going to do that? That’s the issue that confronts us. I think the first step is to try and get hold of that as our issue. That’s our problem and it is related to the much larger problem which is facing the country because it’s in part going through this technological shift which is shifting the ground out from under us, My generation, we grew up with the metaphor of E pluribus unum: Out of many one. If you remember, those of you who saw “Eyes on the Prize,” the metaphor at the beginning of each series is there are black people marching and then they change into the American flag. That’s the prize. The American flag.

We were the last generation that had our eyes on that prize. The Civil Rights Movement was the last movement in this country to believe in the melting pot. That we were to create in America the identity of this American who is fashioned from all of the different peoples, somehow a new person, an “American” was to be created. The generation coming up now, they don’t have that metaphor. They don’t have any metaphor. That’s part of their problem. That’s part of our problem. They are a generation which has to create a metaphor of what it means to be an American. Does an American speak Spanish and no English? Does an American speak Chinese and no English? Is it possible to take many different peoples and find some common identity? That is, if we are going to retain our different cultural heritages, is there some unity in all this diversity? Is there enough there to hold the country together? It’s a different question that this generation has to wrestle with. And it’s a question that’s driven by the same forces that are driving math to be a central element of school education along with reading and writing.

These are very deep problems. They are not going to be solved over night. But the question is, “Is there some strategy that this group has for making sure its contribution, which really now turns out to be central, counts?”

There is discussion about process and the National Council of Teachers of Mathematics (NCTM) Standards. Irvin Vance said that one of the good things about NCTM Standards is that they raise the question about math for everyone. The technological shift is also a shift from technology that deals with physical work to technology that deals with mental labor, mental products. You have in the industrial technology, machines trying to routinize physical labor. But computers aren’t doing that. Computers are dealing with products of the mind, forcing the issue of critical thinking, because you are no longer trying to get the kids to learn how to do the “purple sheets,” as Irvin said, “the drill and kill.” Computers are forcing on education the issue that it has to produce graduates who can think in a critical way with quantitative data. This brings in process and something like the NCTM Standards.

So, we are not going to escape from this issue about process. But are we going to be able to handle it? Because if you say that, well they have to get to the content at some time. Who are we going to look to to tell what that content is, if not this group here? But do you have a consensus about it? Do you have some idea about What that content needs to be so that the children on the receiving end are viable? We are not just talking about professionals and jobs, we really are talking about democracy and citizenship. What is the content of the democracy going to be in this country?

These are heavy issues to lay on a group of mathematicians. But it seems again that history has played this trick on us and put mathematicians in this sort of critical place around the question of the democracy of this country, because this math/science tool is really assuming as important a place as reading and writing assumed in the old dispensation. And those people, we know who didn’t have that tool, they really were not citizens.

Just think about Washington, DC. On the TV last night I think they said the murder toll had gone up to over 400 and nobody blinked an eye. They have these schools and the principals are saying, “Can you really expect me to secure this school?”. It’s not conversation about education, it’s conversation about this criminalization of the schools.

I should say a word about the Algebra Project because I am able to come here and talk to you like this because we have this Algebra Project.

My family had been living in Tanzania for about 6 or 7 years, fleeing the political events of this country. Three of our children were born there and our youngest daughter, Malaika, was born in June of ’76 about a month after we came back. We wanted to put our children in the public school system, but we wanted the school system to work for the children. So, within that arrangement my job was to look after their math. So, I undertook to working with the children as they went through the grade school years with their math. We have two girls and two boys. Maisha is the oldest and Omowale, who is next, is here with us. I have to warn you he plays basketball.

We had a long discussion this morning about athletes. He is also trying to do a little math while he is here at GW (George Washington University). But there is an issue about coaches and math which surfaced this morning. Let me just say, when that issue surfaced, people were saying that the reason they use coaches to teach math is because they need math teachers. I stood up and gave a comment, an elliptical response that no, the reason they use coaches to teach math is because they think that they don’t need math teachers. So, therefore, you can use a coach to teach math.

I was serious. Part of the reason we are in the trouble we’re in is because we don’t train elementary school teachers to be math teachers. So we use coaches.

Anyway, Omo and I have slugged it out over the years about doing his math and whether he had to be in the same math course that the other guys on his high school basketball team were in. Why should he be in a different math course than the rest of his teammates? If it was okay for them to take a certain course, why wasn’t it okay for him?

So the Algebra Project grew as a family project. We slugged it out in the family. We’re still slugging it out.

Then it got into the school system when my oldest daughter was ready to do Algebra, and they weren’t offering it, so in a nutshell, I went into the classroom. I was able to do that because I had a McArthur Fellowship that came through at that time in 1982. I used it to teach Algebra. I went into the classroom for 5 years with that fellowship and taught Algebra as a parent volunteer to Maisha and Omo and then Taba and Malaika. To all my children and all of their classmates. Out of that then came the question of who takes Algebra, who gets access to Algebra, and how do we address the question of math literacy. Out of that came the Algebra Project which now has spread around the country and is trying to take root in different cities around the country.

I suppose if I think about it, I would think of it in a metaphor that the project is sort of like a young kid who is trying to stand up and is teetering and falling down a little and getting back up, falling down a little and getting back up. What I hope is that the project has the same kind of perseverance that makes young children keep getting back up. And then the same kind of perseverance that makes them eventually walk. So they keep walking until they learn how to walk. It doesn’t really matter how many times they fall down, they keep getting up and walking.

Probably part of the reason that happens is they have a lot of people around them who are also walking. Unfortunately, we don’t have that in the Algebra Project. That is, there are not a lot of projects around which are looking at this issue of literacy and how you make systemic change in schools. I am hoping that the project. will have the kind of perseverance that young people have so that it will keep standing up every time it falls and eventually learn how to walk.

In the Algebra Project we do training. As Paul Sally, who helps us in Chicago, reminds us, there are two types of training that we need to do. There is training in specific curriculum, and for us we have our own little curriculum in which we do some training. Then teachers need training in the background of mathematics. Paul has helped on this second issue; he has his own history, as you know, of training teachers. But he has also been working specifically with Algebra Project teachers in Chicago. I would like to raise this issue with you because there is a specific question: How can the mathematics community relate to a project such as the Algebra Project?

Paul has two courses which he has developed, one on number theory and one on geometry. He offers each in 10 sessions, across the spring semester to middle school teachers in the Algebra Project. He offers it to other teachers as well. These are courses which are trying to address the issue of how middle school teachers fill out their mathematics background.

Part of what is driving this for the teachers is that the Algebra Project raises for these teachers the need to fill out their mathematics background in order to successfully engage the Algebra Project in the classroom. So the question is: Can we develop across the country a network of people in the mathematics community who would work on this kind of issue? In this instance, we are trying to pull together a seminar in the summer where we get mathematicians who come to the University of Chicago to sit down in a seminar with Paul and look at the issue: How can we develop courses or procedures or methods for doing the second kind of training with middle school teachers, in this case, in the Algebra Project? That is some kind of network around the country. So this is one of the projects that the Algebra Project is trying to do. I think that I will stop here.

Questions from the Audience

Question

Is the Algebra Project about Algebra or is it preparing students to be able to succeed in Algebra and, if so, what is it in the students that you are reaching to that the rest of us haven’t been?

Answer

What is Algebra? When Omo was a sophomore and he was in Cambridge High School and he was taking Algebra II, of course he had taken geometry as a ninth grader because he had passed the citywide test for Algebra after his 8th grade, he came to me and he said: Why do I have to do Algebra II because all the other players on the basketball team are not doing Algebra II? Why do I have to do this?

We had it out head to head that this is what you have to do. You need to come out at the end of high school with your college basketball scholarship true, but you’ve got to be ready to go into college with the kind of background in these kind of subject areas.

There has got to be a product on the other end from my point of view. In other words, there is a way for the project to fail. If the project doesn’t get students who go through the college-prep math sequences and come out on the other end and enter college ready to do mathematics for which they can get college credit, then it fails.

To do that, we are making an intervention at the 6th-grade level. For historical reasons, it began at the 6th-grade level. I think of the intervention as an intervention which is saying, if a student can count and if we can get their attention, then we can get them on this college-prep track. The hardest thing is to get their attention. But if we can figure out a way to do that, then there is a way to get them on the track where they can get through their middle school years and get ready to do the college prep math sequence in high school. That’s the goal. That’s what we are setting out as a goal. We are saying that this is a goal for all of our students. That is, there’s got to be a floor. It’s not the ceiling. It’s the floor. We’re not saying anything about the 2 million that I just heard we have in this country who are gifted and talented students, and we are not doing right by them.

I am not saying anything about what should be done for the 2 million. It’s not the ceiling, it’s the floor. We are trying to say we need to put a floor under all these students and the floor, as I mentioned, has to do with these enormous issues about citizenship and democracy.

Content-wise what we have said is that at the 6th grade we are trying to help students get a more general notion of number. So if you just look at what is driving the 6th grade curriculum, it is the idea that in arithmetic, students have a question in their mind about numbers which roughly is a “How many?” question. They pick it up when they learn how to count, how many fingers, how many toes, and we’re trying to put another question in their mind about number which roughly is a “Which way?” question in a directional sense.

Having a more complex set of questions around their number question, we are trying to change their metaphor for subtraction and addition. So instead of having a “take away” metaphor for subtraction based on a “How many were left?” question, we want a metaphor for subtraction which is based on a comparison in opposite directions. How do we get direction into their subtraction concept and into their addition concept?

I really don’t care whether people say that’s Algebra or not Algebra, that’s what we are trying to do in the 6th grade. We are trying to get them to make a shift in their number concept and in their subtraction and addition concepts. Then we have some other stuff we try to follow that up with: multiplication and division in the next grade.

This floor, the college-prep math sequence, is itself a moving target. But that’s okay. The image I have here is that if you are trying to catch a bus that is moving, you just can’t stand still and as it passes by try to grab hold. You’ll loose an arm in the process. What you’ve got to do is as that bus is coming, you start running. And if you get close to the speed of the bus, then you can hop on.

The students today have got to hop onto one of those college-prep math sequences that are out there now. If they don’t, that other one you guys are trying to think of as the one that should be in place for the 21st century will whiz right on by them. They won’t have a chance in the world of getting on to that.

So in our program, what we are saying is that we need to get the kids ready for whatever is out there, they need to lock in on the idea that they should do the college-prep math sequence, and they keep locked in on that like some laser beam. However that sequence changes, however it evolves, whatever kinds of transmutations it goes under, they stick with it. And they pass the word on that this is what you’ve got to do.

We’re just saying that this is the floor. It’s got to be done. Currently 11 percent of the students in the country finish pre-calculus and 3 percent of those do calculus. We are saying whatever that is, that’s the minimum. Everybody needs to do that. The country has to take seriously that this is now a literacy effort. It’s not creaming. Everybody has got to do that.

So, here you come to a great crisis in belief. Who believes that all these kids can do this? The only thing that has given us some play is that we stepped in there with a little piece of curriculum at the 6th-grade level and said hey, these kids can do this much. And, if they can do this much then they can think of themselves and we can get them moving to do this much more. So that’s my answer to that question. I’m not sure if it’s an answer.

Question

In short, is there some elaboration on this which is available in published form, either some materials that you have developed or something that someone has written so that we can learn more about what it is that you are doing on the ground?

Answer

I will give out a phone number, but be warned we are not heavily funded. So, our office staff in Cambridge is really bare bones. The number is (617) 491-0200. That’s the phone number of Algebra Project, Inc., which is a litle small group at our central office. There is some literature which we will be glad to send you. We don’t have back logs of the student text. There is a student workbook and text. What we do is negotiate with school systems who want to do the project so they get a copy and then they print it themselves. I don’t know what would happen if we got an order for a large number of those textbooks. We are not really equipped to handle that.

There is another problem which is serious and on us. What happens to these kids in high school? It is the universities that have whatever kind of contact is going on with the high school teachers of mathematics. We are not even thinking at this point about trying to look at high school curriculum or training high school teachers. We can’t do it. We’re not equipped to do it. So, how can the university communicate with mathematicians? How can they help with this issue of working the high schools? Those are two really concrete issues that I think this group could help us with.

Question/statement

First, just a little story which perhaps might amuse you, when I told my husband that I was coming here to listen to you and have lunch he said with a smile, “You know that man is upsetting a whole neighboring state: Mississippi.” He grew up for decades with Arkansas second to the bottom in education, literacy, per capita income, whatever. The saying was “Thank God for Mississippi,” He said, “Bob Moses, he’s going to upset this apple cart!,” by the extraordinary work that you are doing in Mississippi. So that was a backhanded compliment.

Answer

Well, we might work in Arkansas.

Reply

Well, that was his next statement. Please come to Arkansas, Being a non-mathematician, one of the things that I have truly admired and respected about your work is that you have taken mathematics, through your work in using non-mathematical methods and areas and disciplines, and have made math more a part of a larger learning process. I was just wondering whether you had any plans to work with that non-mathematical world in a greater way to make it even more an integrated kind of studies.

Answer

There is an effort. We have a curriculum which we are getting ready to try to pilot which is called an African Drum and Ratio’s Curriculum. The idea in this curriculum is that children learn certain competencies on the African drum, and some of the African drum family, Then they try to abstract certain mathematical concepts out of these practices. We are targeting that for 4th and 5th grades. In doing that we are thinking that this could drive a larger integrated curriculum. There is the obvious area of selecting some African or African/Caribbean culture which becomes a part of the study at that grade level.

Then there is the issue such as information systems. That is, the drum is an ancient system of transmitting information over distance. Therefore, it opens up the idea of a larger look at what are the different ways in which information is carried today as a part of a larger interdisciplinary structure. There is also the question of rhythm, which is so prevalent in music, and also in science and in life in general. So, how would you use this as a way to drive an interdisciplinary curriculum at that grade level around those concepts?

We are trying to see if we can get some funding to actually do a training this summer. We figure we need at least 6 to 8 weeks to do a training to try to develop a basis for that as a pilot.

This raises the general question about how to teach math. There was the question this morning about process and the NCTM Standards. If you think about epistemology and how to go about teaching math, I think we inherited a way of approaching it which says that we should start with what is simple. The problem is that most of what is taken as simple is also very, very abstract. The idea of building the mathematics curriculum around what is simple and abstract is one approach. That is not the approach which we are taking.

The approach that we are taking is along the other axis. That is, if you think of simple as opposed to complex, and abstract as opposed to concrete, we are looking to develop curriculum which finds the right level of complexity in concrete events. We are not building the curriculum around concepts which are simple and abstract. We are trying to build a curriculum around events which are concrete and complex. And the question is the right level of complexity.

So, when you look at this as a jumping-off point for curriculum, then you are in very different terrain. The question then is: What are the concepts that you want to teach and what are the events which can be the jumping-off point for those concepts? It is in that sense that we are trying to engage the non-mathematical world, as you call it, in a very general sense in this curriculum process. We are just exploring this. We have just, sort of, gotten our toes wet about how to go about developing this kind of curriculum.

It would seem that one of the advantages in doing this, or approaching curriculum this way, is that you have a chance at getting the student’s attention. Then the issue is: If you can get their attention, how do you move them through to get the mathematics out? What we have done is maybe unique.

I came across the literature of the philosophy of mathematics in the writing of Quine. Some of you know Quine, he is a math logician at Harvard, Emeritus now. In his writing he talks about mathematical logic as involving a regimentation of ordinary language. So, he speaks about regimented language and regimenting ordinary language to develop the language of mathematics. There is this famous debate between Quine and Alonzo Church. Church is a math logician at Princeton, or at least he was when the debate was going on. They were arguing about the existential quantifier. Church was taking the position that the existential quantifier acquires its meaning from the axioms in which it is embedded. Quine was taking the position that the existential quantifier acquires its meaning from the logician who had in mind the ordinary language there is something such that and that’s how it gets its meaning.

I look at that discussion of Quine and Church as the 20th century version of the 19th century discussion between Hilbert and Frege. Frege, who was the chief logician in the 19th century, wrote to Hilbert that the axioms of geometry are consistent because they are true. Hilbert wrote back and said, “My God, for as long as I have been teaching, I have said exactly the opposite. The axioms of geometry are true because they are consistent.” It was Hilbert’s point of view about mathematics that won out. Church then inherited Hilbert. So the meaning of the existential quantifier is embedded in the axioms. Quine is inheriting the 20th century version of Frege. Quine doesn’t believe truth as Frege saw it. He has his own version of truth, but he is saying that meaning is attached to language and discourse and so forth.

The version of math that is taught to the kids in the school began after Sputnik in 1957. I was teaching at Horace Mann when Sputnik broke out. Right after Sputnik in 1958, I used to go down to Columbia University when Professor Fehr was holding forth, teaching students math and so forth, and I was a teacher from Horace Mann going down there to do courses. I was looking through the School Mathematics Study Group (materials) and Max Beberman’s mathematics reform effort, the Madison Project and the Syracuse project, and I was taking all of these back to my classroom at Horace Mann and doing them with the kids. Looking back on all of that work, that work was predicated on the line of thought coming from Hilbert and Church and those people on the simple and abstract.

So, Beberman was actually teaching the children about equivalence relations: that they are symmetric and transitive, and reflexive. He was building the integers by partitioning the set of natural members and getting your ordered pairs of integers from the equivalence relations. I actually taught a course at the 9th grade level. I was moon lighting those texts, so I took Beberman’s text out and had the 9th graders doing equivalence relations of all kinds, and this business about partitioning the set in constructing the integers. So, that’s one approach.

The Algebra Project is looking at a different approach. It is not the simple and the abstract and consistency, but complex and concrete and truth. Science has these things which they call observation sentences. Physics may have these big theories and so forth, but it’s got to have some kind of observation sentence against which you test these theories. My thinking is: How do we get the kids to develop a set of observation sentences which are the grounding for more abstract mathematics? Every Algebra textbook in the first few chapters has this little sentence “A minus B = A plus the opposite of B.” The book somewhere says, in algebra, subtraction is the same as addition, more or less. The student, of course, has been spending years learning that that’s not true. The student learned that subtraction is one operation and its metaphor is take away, while addition is another operation and its metaphor is piling stuff on. The book never says how these two metaphors are related. All the book says is that this syntax over here can be changed to this syntax over there. So that book is, in my mind, in the Hilbert/Church tradition, of the simple, abstract, consistency. It’s not in the tradition of Frege, Quine and looking at truth. It doesn’t give the kids anything. That little sentence I look at as a high level theoretical sentence in the Algebra I textbook. It’s not an observation sentence. There’s nothing that the kid can take from that and go test it against any experience and see whether or not it’s true.

My problem is: How do we provide for the students a whole complex set of observations sentences out of which they could abstract this high level theoretical sentence which is “A minus B = A plus the opposite of B.”? The kids need truth and experience. They need some level of complexity which they can use to test against events. That’s how I view the problem of constructing curriculum in mathematics. Viewing it that way gets us into these other areas of knowledge and human endeavor.

References:
1) https://www.masterclass.com/articles/sankofa-meaning-explained, retrieved 3/16/22
2) https://eric.ed.gov/?id=ED385437, retrieved 3/16/22

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