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'

adical Equations Rights from Mississippi to

the Algebra Project

BERT P. MOSES

ND CHARLES E. COBB, JR.

Beacon Press BOSTON

CHAPTER 5

Pedagogy The Experience of Teachers and Students

The most pressing consistent question students had was "What has·

riding the T got to do with algebra?" Though it was a struggle to

not explain the idea of the train trips as the physiC_al event on Which

future algebra concepts would be modeled, it was worth it. My­ response to the question was usually to ask another question. "Wh~?:"

do you think the connection is?" One girl looked up at me as she

was doing her math test with an "I know you won't answer this"

smile and asked, ''Just what did our trip have to do with algebra?"

I smiled a "you're right, I'm not going to answer you" smile and _ said, "You're going to learn how to answer your own questions and::·,

in the days to come I'm sure you'll be able to answer that question.'.'·:~

Lynne God

I know what the kids can do ifI can just get theni to believe in themselves.

Johnny L. Hughes, principal, Lauier High School;:§

By I 99 I we had moved from Cynthia's home and were settled intc __ new space in Cambridge, where we are still located. By then th{ first generation of Algebra Project students had graduated, or were close to graduation, from college. Many did not want ro end the# relationship with the project as they entered the "adult" wor!,f Some in this group, like my daughter Maisha, who went on t, Harvard University after finishing high school, had stayed co nected to our work all through their college years by tutoring•~,

114

PEDAGOGY 115

'ng and working with summer algebra camps. And so then when I a senior and I had to try to figure out what I was going to do, what

'•ork I was going to do '!fier college, it took me a while because I really _ n't sure. I knew that I loved working with kids. And I knew that I •anted to do something I felt was working to address the problems offree­

,_om and equality for Black people. And I felt that the Algebra Project was ling that in a way that I didn't see a lot of other types of projects doing.

ltid so I thought about it far a while and decided I wanted to work with '•',Algebra Project. So I talked to my father.

Maisha and I walked along the Charles River discussing this ,ne afternoon. In that conversation, she also told me that she did

J§t want to work in the Boston area. I needed my space. I had grown · l;there and I spent my college years there and I just felt like I needed to ive my own space. Dad was pleased that I wanted to work with the pro}­ {, and there were choices. The Algebra Project was no longer a per-

-inal experiment conducted by me in a small corner of the King i¢liool in Cambridge. Now the project was taking its first teeter­ :_gsteps toward becoming a national effort. In 1990 a Chicago Al­

bra Project had gotten under way. We also began working in ilwaukee, Oakland, and Atlanta that year. In 1991, projects were .rted in Indianapolis, San Francisco, and Los Angeles. That year,

-.- 0, Dave Dennis, through Positive Innovations, initiated an Al­ ~ra Project "pilot" program in six Mississippi Delta schools. In ~992, a project w,as under way in New Orleans. A New York City "iroject began in 1993. That year, what we were calling the South­

Initiative of the Algebra Project, under Dave's direction, _ ached South and North Carolina;Jackson, Mississippi; and Bir­

,- ingham and Bessemer, Alabama, and Dave was getting inquiries from Arkansas and Tennessee. I«?- didn't go out looking. People began ;focal/ in and ask about the Algebra Project. As we expanded, I continued · thave meetings that included everybody. Money from the Open Society

titute helped us have those first meetings and trainings. I«? would talk ·out operating as a region with everybody working together to give sup­

.- rt to each other. And that~ the way we try and work today. This regional ipproach is based on the same philosophy used in COFO. So people 'iarted defining what they wanted and the idea was, how do you pull to-

116 RADICAL EQUATIONS

gether a structure that meets the needs? What were the needs? At first, j teachers would be saying what they needed and we would look at what sources we had. The same thing happened around parents, youth, community. All of these components began to come together. Out· of came the Southern Initiative; people found a source of strength in other, and now we ourselves are our greatest resource.

What we had learned since starting in King's open progra) was beginning to take concrete form as a curriculum intervent(9 based on experiential learning. Both were evolving through c· and error as well as analysis of traditional educational approach~ I was certain now that what existed in schools wasn't working, thJ traditional schooling was inadequate for equipping our student to function in today's society. Elite private schools and elite publi_i schools prepared their students to become America's leaders, bu. all other schools belonged to an era in which work and prepara~ tion for work were defined by factories and assembly lines. Fa~–­ tories and the cities they helped create became the models fq schools and their school systems, and the work in factories bee the model for schooling. Because the new technologies give ris, to computers' and an ever-widening use of symbol systems an,_ quantitative data, we concluded that the schools and curricula Wi had to struggle to design must put mathematical and scientific !if eracy on a par with reading and writing literacy. The validity o, this approach:, at least as far as I am concerned, has been confirmec.i in the classroom.

In my geometry class at Lanier High School in Jackson, Mis-• sissippi, the school where I now spend most of my teaching time,; we are using the TI92 graphing calculator, really a small com"· puter. My students, like many of their generation, do not read as . much as they should. Most of what they learn they learn from/; pushing buttons and seeing how images change. Their modalities/; oflearning have been attuned to this image-making process. ff) you use these graphing calculators, you arouse their interest. The i;. kids will take time to try to figure out how the calculator works; • how those buttons make images come and go on the screen. They learn something and will show you what they have learned. They ·

PEDAGOGY 117

:Id never do that with a book. Students do not open a math ,k and say, "Let me show you what I know on this page;' but , will show you what they know about a single button on a

·:phing calculator. ";'.Therefore, I think you have got to go where the kids are. And :at place is different from the place where teachers have been ght to be. Our graphing calculators completely reorganize the

:],,ject. The exercises in the chapters at the end of mathematics . :books depend on the chapters at the beginning. With graph­ } calculators you can move around in innwnerable ways; it's a )n:Jinear way of presenting information. You need teachers who i comfortable taking the subject apart and putting it back to­ }ther again on the spot in response to questions of children, And lkis a real difficulty. Recently, on June 3, 2000, in a radio address is cussing U.S. 'classrooms, President Clinton said that while 9 5 ··ercent of schools and 63 percent of classrooms are wired for In-

net access, two out of three teachers with access to a computer

_ '}'they don't feel well prepared to use it in the classroom. And the tiacher-training programs that exist do not prepare teachers for ··;4is kind of work. Worst of all, math labs in inner-city schools for the 'nost part are used to remediate students about things the technology makes 'bsolete. We need a revolution in order to get the teaching of math

. ip to where the technology and the students are. Our efforts in Cambridge to get schools committed to this ap­

Q>roach to teaching rested on a foundation of consensus among t}Parents, teachers, administrators, and community volunteers. [Parents used meetings as opportunities to talk, ask questions, and }join in planning. And as change began to take place in classrooms, , more of the pace and scope of students' mathematical studies came ', under student control. Teachers began acting more as guides and •. coaches rather than lecturers, Even these small steps had dramatic

effect as motivated students set goals. In Mary Lou's classroom, for ; example, Andrea [Harvey] spoke up at one of our first meetings and said,

"I'm going to do four lessons a week because I want to finish such-and-such by the seventh grade, so that I can finish the book by the eighth grade, so I

· ·. can be in honors geometry in the ninth grade." This was a twelve-year-old.

118 RADICAL EQUATIONS

The others looked at her-this hadn't come from a teacher-and said; 'Lire you crazy?" She said, "That'swhatJ'mgoing to do." Bob was there/-' and he started to frame far them why what Andrea had done was a very. mature and farsighted act, and how maybe they weren't ready to do that:. yet. It gave Andrea a lot of support and '!!firmationforhaving said that i;· the group. And it changed what the others were going to say next. Every.C thing.from then on was in terms of Andrea: "Well, I'm not going to dq. quite what Andrea is, but … " (Andrea majored in music and mi-';" nored in math at UMass, Amherst. She worked with YPP in¥ 1999-2000 training math literacy workers, and in summer 2ooqf was certified to teach math in the Boston schools.)

Teachers were meeting in small groups for brieflessons on spec, cific concepts, and regularly gathering for small-group review ses:c', sions. The 1986 decision that all students in the seventh and eight!(' grades should take algebra had lower-grade.teachers questioning:c whether their own math curricula adequately prepared their stu{'. dents for algebra. We began to address this issue systematically 1,y;:, beginningto implement new curricula in mathematics appropri-( ate for different age and grade levels. We knew that flexibility was key. For example, when fifth- and sixth-grade teachers tried ~?' materials-centered approach that had worked very well at the) seventh- and eighth-grade level, they found that younger chi!{; dren, accustomed to more teacher-centered .instruction, needecf~ more teacher-child and small-group interaction. Accordingly,'~ these teachers modified their classroom technique while retain,} ing the principle of encouraging greater self-reliance in finding; answers to problems.

In addition to the benefits to learning attained from having , the flexibility to modify the curriculum, the process gave partici– -/ pating teachers greater appreciation for the kind of self-reliant ed- .c' ucation efforts the project was trying to encourage in students,/' Teachers also gained a deeper sense of their own authority. Bob was{ '!!firming what we were doing while he was helping us change. He didn't'< come in and say, "J:¾?'re throwing this out, it's junk." He came in and said_;J "You guys are great. Wanna try something different?" When we asked, "How will it work?" he turned it around and asked, "1#11, how do yo

PEDAGOGY 119

/hink it should work? What do you want to have happen?" He didn't re­

dly give us a why, which admittedly was frustrating, but it also gave us ownership around it. Bob didn't have all of the answers. At first I was re­ ·• 'ly annoyed that he was making me go through this process. I kept saying, ·,13cb has an agenda. Why doesn't he tell us? We're wasting so much

ime!" But he knew that it had to come from us. He knew he couldn't im­ •ose, because he didn't know what would work. He wasn't a classroom

-eacher. He just had the vision. lf he could help us catch the vision, we •ould make it work.

With elementary level teachers now wanting their students "ready for algebra in middle school, I began developing what Would become the "transition curriculum" for sixth-grade stu­

:dents. (We are still developing and refining this curriculum. Book J, which is aimed at sixth graders, develops the concepts of inte­ ·;gers, displacements, and addition and subtraction ofintegers.) We

F)ay "transition" because the Algebra Project focuses on a cluster of – ementary but elemental concepts that are essential for bridging -.. e gap between arithmetic and algebra so that all middle school {students are prepared to succeed in the college preparatory mathe­ cimatics sequence when they reach high school (see the Appendix). }It's a "floor." Our transition curriculum is rooted ip. the conviction ;',that intellectual development is, in part, a matter of integrating ··knowledge. You want the kids to learn how to engage the in­ I,quiry process.

In other words, in the Algebra Project we are using a version of experiential learning; it starts with where the children are, ex­

"'._0:periences that they share. We get them to reflect on these drawing _f _on their common culture, then to form abstract conceptualiza­ – tions out of their reflection, and then to apply the abstraction back C on their experience. You can think of it is as a circle or clock: at

twelve noon they have an experience; quarter past they are think- _ fog about it; half past they are doing some conceptual work around their reflections; and at quarter to they are doing applica­ tions based on their conceptual work. In the Algebra Project this movement from experience to abstraction takes the form of a five­ step process that introduces students to the idea that many impor-

120 RADICAL EQUATIONS

tant concepts of elementary algebra may be accessed througb or: · dinary experiences. Each step is designed to help students bridg, the transition from real life to mathematical language and opera} tions. Because of this connection with real life, the transition cur­ riculum is not only experiential, but culturally based. The experi enc es must be meaningful in terms of the daily life and culture the students. One key pedagogical problem addressed by the cut, riculum is that of providing an environment where the studenti can explore these ideas and effectively move toward their stan, expression in school mathematics.

In academic language, this process can also be described as th Hsocial construction of mathematics." Students.learn that madri.s the creation of people-people working together and depending' on one another. Interaction, cooperation, and group commuri{ cation, therefore, are key components of this process. Studenis, .. also help generate part of the content ofinstruction as well. Theyi! participate in the physical event that will generate data which bee' comes the vehicle for introducing mathematical concepts. Coopf eration and participation in group activities, as well as personal rel' sponsibility for individual work, become important not only fo the successful functioning of the learning group, but for the gener: tion of instructional materials and various representations oft-hf data as well.

There are five crucial steps in the Algebra Project curriculuni process. ,,::I:

r. Physical Events. A trip is the central experience of the tran)[' sition curriculum. The curriculum begins with a trip; this tril'f may be a trip on a metropolitan transit system, or it may be a briic. tour or a walking tour of the students' communities. I mentioned>-, earlier that my first trip with students was on the red line of the Ti. in Cambridge. In the Mississippi Delta, students and teachers: _ boarded buses. And if there is no public transportation it is alwa~:: possible to walk. __

2. Pictorial• Representation/ Modeling. In this and the nex;t­ three steps, students move through a series oflinked and progres"' sively abstracted representations of the physical event. In ab-'

PEDAGOGY 121

~cting, we focus on certain aspects of an experience and ignore hers. Students are first asked to draw a picture or construct a

;'tide! of the event. In drawing pictures, students portray particu­ ·,aspects orfeatures of their experience and begin the process of ifraction. Students are free to choose whatever they believe is of

"-''erest and value to put in their pictures. In this way they exercise ership over these visual statements.

3. Intuitive Language/"People Talk." In this third stage stu­ 'nts are asked to discuss and write about the physical event in .eir own language. Their conversations and stories may be writ­ : in the language they use to negotiate their daily lives, but as

-i.th pictorial representation, the important point is that students iould feel they have ownership over their descriptions. These ·,ories provide a window to explore intuition and ideas. We tend -,,get two types of stories: factual reports and private, uniquely

sonal stories. Both need careful attention. The personal stories en conceal "data" of which the writer is not aware. On the

dther hand, factual reports can hide what is thematically particular – rd rich to the observer, and itis precisely that richness ofintuitive lnguage in writing and discussion that we want to capture at this tage. – 4. Structured Languagel"Feature Talk." This is not a language

}1at the kids or anyone actually speaks. It is structured or "regi­

hented" language aimed at explicitly selecting and encoding ose features of the event that are deemed important for further

'study. One ofits characteristics is it talks about some feature of the '{object (e.g., the speed of the car) rather than the object itself. (For ,a more detailed account see the Appendix.) Specifically, at this ·stage we isolate those features of experience around which we will •l,uild the mathematics. If students have not explicitly tagged )hem, the teacher introduces four fundamental and recurring · mathematical features of trips: start, finish, direction, and dis­ ftance, features that are present in all trips. In contrast to the ',mathematical reforms that followed Sputnik during the 1960s, :,when attention was focused on teaching students the vocabulary, &grammar, and structure of mathematics per se, the emphasis here is

122 RADICAL EQUATIONS

on getting students to develop mathematical models for imp or, features of particular events. A process of mathematization. (S' the Appendix.)

5. Symbolic Representation. Once students have worke; through the picture making, writing, and discussion of featur, they construct symbols to represent these ideas. At first, these bols are private, holding meaning only for their creators. Fi in teams or groups, and then in class, students present and discti . the meaning of their symbolic representations. By sharing the'•i meaning, the private symbols become public. The symbols ac ._ often redesigned ro make it easier for the rest of the class to un."' derstand.

Even now I can remember Chad, one of my seventh-grad,_ students years ago, looking up from reading a page in the firs('° chapter of a traditional algebra text with a weary sigh and a pu~~' zied expression and then saying, "It's all words." With these five; steps we are trying to ward off student frustration in "the game oft signs," and the misapprehension that mathematics is the manipu;i lation of a collection of mysterious symbols. For too many young" people mathematics, especially advanced mathematics, is a garn, of signs they are unable play. They must be helped to understan~:· the conceptual language that is encoded in the "game of signs}J When middle school students use the five-step process to con"'' struct symbolic representations of physical events-representa, _ tions that they themselves make up-they forge, through direci

– . experience, their own platform for mathematical truths. Their _ personally constructed symbolic representations, rooted in what,:' they have experienced and observed, enter into a system of ma the-··• matical truth that has content and meaning.

Teachers, ofcourse, are another part of this equation. It should} be clear by now that underlying our curriculum and pedagogy is · the expectation that teachers will make fundamental changes i11' the way they teach. They cannot simply be lecturers attempting: to pour knowledge into the heads of students who sit passivelyc'1,, like inanimate vessels. Teacher workshops and larger Algebra Prof'_ ect meetings, such as the periodic _national conferences we hold,:'

PEDAGOGY 123

re usually structured around the basics of the five-step process. Dhrough repeated experience with the model in different settings, :eachers learn to work with the five steps and to understand the ,,:ocess as a model for designing instruction. In 1991, San Fran­

}sco sixth-grade math teacher Marian Currell participated in one -,four workshops and found something that "made sense." The Al­ '•bra Project matched how I believed in teaching-having a conversation.

And working with Jim Burrus from Boston's McBer Associ­ es, we began designing what we now call our Training ofTrain­

.lrs program.Jim develop a Train the Trainer for the Efficacy Insti­ liite and I participated in it in 1988 .. Jim and I lost touch after that ·irief first contact. But as the r98os wound down it was becoming

:Jear to me that while I knew how to train teachers in our devel­ i,oping curriculum and pedagogy, I did not know how to train peo­ [~le to train teachers, a very different proposition. My movement experience in warking with adults stood me in good stead when it Jame to working with teachers but I didn't have training concepts. I-decided to look Jim up. As it happened, I bumped into Jim in a

Cl,ank one day and began explaining that now that the Algebra ''Project was spreading to other places I needed to develop effective ind systematic ways to train teachers and the trainers of teachers. •·perhaps he could help.

Jim responded on the spot by saying that I should come by his house. He did and we talked. I told Bob I would be glad to help. For me, '.the kind of work he was doing was close to what got me in this business in

, ]the first place. In 1990 I was spending most of my time training execntives ff for Merrill Lynch. Although I made money, it was not very gratifying and ''.a long way from what I intended when I started. Listening to Bob, I · ·thought this will be a nice balance to what I'm doing and I'll feel a lot better

about being tired at the end of the day. He was quickly drawn into our work in Chicago, where a new

algebra project was beginning to take shape. There, in 1990, a ,, group of educators and community people had identified six . schools they wanted the Algebra Project to come into as soon as teachers were trained. In January 1991, I invited Jim to observe a

, two-week teacher training workshop in Chicago. As a result, I be-

124 RADICAL EQUATIONS

gan holding meetings with a group ofinterested community peo, pie to help form a Chicago Algebra Project and to plan an ap~i proach to the MacArthur Foundation seeking money to design a·· training progran that would train trainers of teachers. That spring?_ a group that included Dr. Frank Davis of Lesley College, Dore-; othy Strong, director of mathematics for the Chicago Public< Schools, B. J. Walker, representing the newly formed Chicago A],,.: gebra Project, and I met with the MacArthur Foundation and ulti-'; mately got funding.

Meanwhile, at the Massachusetts College of Public and Come _ munity Service (CPCS), where l had secured a part-time teach-{ ing position, William Crombie, who was teaching math and some ·· · computer science there,. was also becoming involved with our

project. Early in 1990 Marilyn Frankenstein, who headed the .· math department at CPCS, invited me to be part of a Boston dele" gation attending a meeting of the American Mathematical Soci­ ety .in Louisville, Kentucky. I presented a paper at the session of the Humanist Mathematics Network. Bill was also part of our del­ egation. Flying back to Cambridge together, we talked and I men­ tioned a need for help from someone with training in physics. "Well, I might be able to help," Bill said, and a group of us, in­ cluding my daughter Maisha, Lynne Godfrey, and Cynthia Silva, began meeting around the curriculum, with Bill aQd Dan Smith, a physicist from Northeastern, contributing to issues of velocity and acceleration.

Demands for our time and energy were now coming from all parts of the country and Bill Crombie, still teaching at CPCS, was increasingly involved in Algebra Project weekend workshops and training sessions with teachers. One day, I asked him to consider full-time involvement with the project. He was definitely inter­ ested. He was experimenting with approaches to teaching alge­ bra, but Marilyn Frankenstein, his department head (and a sup­ porter of our project, who in fact first introduced me to Bill), had told him that while his efforts were creative, students were n_ot finishing the required course. He had fabulously creative ideas efwhat he wanted to do but our students-almost all adults-felt the pressure to

PEDAGOGY 125

/iomplete the curriculum within the term. To incorporate what Bill was ,i/oing would mean them coming in on weekends and perhaps other extra ·,imes. Bill left the program and in fall 1990 began working with us

/,full-time, visiting classrooms, working with teachers. , . It was probably inevitable that Bill was going to come to work ·.,{.,ith us full0 time. Before he joined us, one of the first workshops he i'was involved with, early in the summer of 1990, used a grant from ,/the Clark Foundation to bring teachers from San Diego, Milwau­ ;-;.kee, and Louisville to Atlanta for two weeks. I basically observed Bob li1Jt the workshop so it was all kind efinquiry-based, and I thought this was •·cool, it was fun. The mathematics questions were not formulated mathe­

matically. He was pointing to the issues at the root of every mathematics 'question, encouraging-a kifld of a general thinking about the issue there, as

;ifhe was saying, "IM, can talk about that fundamental issue first and then ' specialize the results we get to the mathematics formulation." In otherwords,

for example, the m'athematical idea ef equivalence relations-it's everyday common sense-is "niake do," because everybody makes do in one way or another. So we started with discussions around some concepts from the gen­ eral culture. One of the reasons for doing this is so everybody has access. Everybody has something to say about it. What all ef this means is that, from the first instance, everybody has a voice at the mathematical table.

Movement encourages movement. In this respect also the Al­ gebra Project was echoing the work in Mississippi -and throughout the South thirty years before. As I have tried to emphasize, the civil rights movement of the 1960s was less about challenges and protests against white power than feeling our way toward our own power and possibilities-really a series of challenges by ourselves, and our communities, to ourselves. In this new community being shaped by the Algebra Project, Bill, along with Jackie, Cynthia, Maisha, Lynne Godfrey, Jim Burrus, and others took on the task of developing methods for teachers to challenge themselves. In Chicago, Jim, who was later joined by Bill, took the lead in de­ signing our training effort. l# had to get into the guts ef the teachers or potential teachers. The Algebra Project needed to be internalized better. I could see how critical it was to link data with visceral experience. This loop was not being closed well enough. It was the Algebra Project's ability

126 RADICAL EQUATIONS

to make math fan that struck me. Part if it was games, part if it was the"; trip, and part was small group dialogue. Bill and I began writing an in, ,; structionguidefor teachers-a conceptual template. This became the basis·,, for our first Training of Trainers effort in 1992.

Because the project believes that it is essential t

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