Volume 4, Issue B ::: September 2000
The New York City Math Exchange Group
Helping teachers change the way they teach mathematics
by Charles Brover, Denise Deagan, and Solange Farina
Much has been written about reform in math education. The National Council of Teachers of Mathematics (NCTM) is leading an energetic and concerted campaign for reform, yet most math classes and curricula remain decidedly unreformed (Hiebert, 1999). A lack of first-rate math instruction is particularly pronounced in adult basic education (ABE), with its historic mission to teach print literacy. The New York City Mathematics Exchange Group (MEG) is the organizational expression of a limited continuing attempt to change the way math is actually taught and learned in an adult literacy community. We hope our experience may be useful to others searching for a small-scale professional development approach that brings math reform from the lofty pages of academic journals and the high intensity of conferences into the day-to-day work of ABE classrooms.
For the past eight years ABE teachers participating in MEG have been meeting monthly to shake hands, throw dice, examine fruit and vegetables, build bridges, and eat popcorn - that is, to do math. We have been learning to think about mathematics education in a new way.
MEG was the idea of Georgia Salley, a staff developer at the New York City Community Development Agency. She and a small group of teachers from community-based organizations initiated the project in 1992 as a series of workshops for teachers interested in improving mathematics instruction in adult education. The workshops evolved into a teacher collaborative that met on a regular monthly basis to discuss the sorry state of math education in our programs and to imagine ways to make it better. Early MEG meetings were also a place where teachers talked about their own experiences with school math as well as the challenges and joys of teaching math. MEG understood that many teachers carried into the classroom the burden of the alienating experiences of their own math education. Without alternative models of learning math, teachers tended to teach math the way it had been taught to them.
MEG meetings also provided cooperative, hands-on problem-solving experiences through which teachers could construct math knowledge for themselves. One teacher wrote in her journal, "Though I moan and groan about doing math problems, it's probably the best part of the meeting for me. It has allowed me to experience math in a new and different way. I'm finding that it's given me new ways of thinking about and teaching math. It's interesting because now I am beginning to integrate it into my way of teaching."
Another participant wrote, "I always thought math was math and you learned it the way I learned it through pain and agony, like everyone else. Well, I was wrong. Teaching math is a wonderful thing . . . I highlight the students' ability to overcome math instead of math overcoming them."
Thinking about Contradictions
The adult literacy community in New York in the 1990s was an interesting and perhaps unique place in which to work for educational reform. In New York, literacy education programs are staffed mainly with nonunionized, part-time moonlighters whose pay and working conditions reflect the socially marginal position of the students they serve; nevertheless, many of these teachers are professional literacy workers with considerable experience and pedagogical sophistication. Many teachers closely followed and actively participated in the "reading wars" between whole-language advocates and supporters of skills-and-drills phonics-based approaches. A core of teachers settled on a meaning-based approach to literacy learning, and a few programs even described themselves as "whole language." But this intellectually fertilizing discourse did not extend to mathematics instruction.
Many of the teachers MEG attracted struggled to make literacy learning meaningfully based upon the experience of their students. However, many of these teachers had given scant consideration to their pedagogic assumptions with regard to mathematics. Many teachers had been drawn to the field of adult basic education as a democratic political project through which they could promote educational equity and access to print literacy for those denied school-based literacy. Nevertheless, they had not fully considered the political and social implications of the congruent denial of adequate math education in an increasingly technologically driven world.
MEG found a curious compartmentalization among many first-rate literacy teachers. The teacher who decried rote memorization and shunned "decontexualized" language learning nevertheless limited math education to "math facts" taught from worksheets. The teacher who was attentive to learners' cultural and language backgrounds and sought to provide learners with meaningful opportunities to learn reading and writing offered abstract and alienating computational drills when it came time to teach the language of mathematics. Teachers who followed the ideas of the Brazilian educator, Paulo Freire, would "problematize" for literacy, but would not put problem-solving at the heart of their math instruction. Despite demands for "more math" from learners, some teachers who described themselves as "student-centered" taught no math at all.
provided a place for teachers to explore these glaring contradictions as we
began to consider whether what we knew about literacy learning also applied to
math. These discussions
often led to questions such as:
- Is math different?
- Why is math seen as abstract, not based in "real life?"
- What does it mean to think mathematically?
- What is the role of the teacher in the mathematics classroom?
One teacher wrote in her journal: "Once again I am confronted by my lack of confidence and fear of taking risks when it comes to math . . . Slowly, very slowly I am willing to play and explore with numbers. It's interesting to see how my traditional math education still intrudes into how I think and feel about math. More and more I see that good teaching that applies in BE [Basic Education]'s reading and writing is also true for math."
By openly recognizing these contradictions, teachers became engaged in the project of changing the way they taught mathematics. Our deeply held pedagogic and political assumptions about education generally helped us to embrace the approach to mathematics advocated by the NCTM's Principles and Standards for School Mathematics (2000). The NCTM Standards elevate meaning-making in mathematics. They emphasize problem-solving, cooperative learning, and math communication; they affirm the opportunity to learn and equity. What we knew about literacy learning applied also to mathematics. Teachers who previously had not heard of an organization called The National Council of Teachers of Mathematics began to develop math lessons based on the NCTM Standards for MEG meetings and their classes. They began to think of themselves as math teachers as well as literacy teachers.
Doing Math - Teaching Math
Becoming a good math teacher is not only a matter of pedagogy, it is also a matter of knowing mathematics. In many programs, adult educators, few of whom come to adult education with a math background or even much interest in math, are expected to teach all subjects to students who range from new readers to students preparing to take the tests of General Educational Development (GED). As MEG organizers quickly discovered, teachers were often as math phobic as their students. They avoided teaching math for the same reasons their students avoided learning it: they hated math. It seemed self-evident that if teachers did not know and do math, they could not teach it effectively. MEG turned its attention to teachers learning math as they were learning how to teach it. We did not adopt a strategy of progress by discrete stages: first we would master math content, and only then would we teach it. That was a luxury not supported by the conditions of our work.
MEG meetings are now organized around learning and teaching math at the level we are teaching it. We take the NCTM Standards as our guide. In MEG's collaborative, problem-solving culture, we seek to connect our experiences learning math with the possibilities for math instruction. With the help of more advanced math thinkers among us, we explore the "big ideas" of basic math and arithmetic embedded in our activities and lessons. As we become deeper mathematical thinkers, we are better able to help our students understand math concepts. Of course, the more mathematics we learn, the more we appreciate the terrible deficiency of math content in ABE programs.
MEG believes problem-solving is central to mathematics instruction. We therefore organize our own learning according to that principle. At each meeting a member or team of members presents a math question or problem appropriate to adult education classes - often drawn from the NCTM Addenda books - and guides the discussion. We have a dual goal: (1) to solve problems together, thereby raising our own level of mathematical content knowledge and problem-solving ability, and (2) to explore the implications for math instruction resulting from this process. Teachers with varying levels of experience and comfort with math work together and communicate their reasoning; all involved come away with a deeper understanding of math and more confidence in their ability to bring this understanding to their classrooms. Teachers reflect on how they solved a problem, discuss the math involved, and consider the application to particular levels and classes of learners. The meetings incorporate writing and journaling as part of the reflection process. In her journal one teacher wrote: "Attending the MEG workshops has revolutionized my math teaching. I am using manipulatives in the math lab I facilitate. I am trying to explain less and less, and ask ëWhy?' more and more." Often teachers are surprised to learn that there are different ways to solve a problem. A teacher wrote, "I still have to think more about today's problems and try to figure them out for myself. It's always amazing to me that there are several ways of solving them - of course in the bad old days there was only the teacher's way, and that was so limiting . . .I would like to co-teach a math class and use what I learn here."
One meeting engaged the classic question of why the product of two negative numbers is a positive number (see pages 16 and 17). Teachers sat around a table as the facilitator walked around placing piles of red and yellow two-color counters in front of each group. "Why do you think that multiplying a negative integer by a negative integer results in a positive product? I'd like you to take some time with your group to come up with an explanation of why this makes sense." Most groups jumped right into a discussion:
"I remember that rule but I'm not sure why it is true."
"When I learned it, I think I was just given the rule, but no explanation. I guess that is the way I've been teaching it too."
"Maybe it is like using a double negative in language. If I don't have nothing then I have something."
"But that is only true in some languages, and not in others like Spanish."
"I always have to think of this in terms of time and money."
The facilitator begins the activity: "Let's try to solve it using the two-color counters." And the MEG meeting becomes a rich, textured investigation of math content.
Expanding Our Activities
Our monthly meetings have always been the mainstay of our work. We mail detailed minutes of meetings to all members, along with math problems and activities we had worked on. As we gained confidence in our mission, MEG reached out to more practitioners in the New York literacy community. We organized workshops and made yearly presentations at the citywide Adult Basic Education Conference. In 1995, in conjunction with the New York City Professional Development Consortium, MEG offered a four-week math institute that drew participants from all the literacy-providing agencies in the city. In 1997 we began our newsletter, The Math Exchange, aimed at an audience unable to attend our meetings. The newsletter chronicles MEG's work, connects the city's literacy community to wider concerns of math education, reports on the national and regional conferences of the NCTM and Adult Numeracy Network, and highlights exemplary practice in math education. Recently MEG meetings have been scheduled at literacy program sites throughout New York City so that we can reach more people in the field. At the York College Learning Center (City University of New York [CUNY]) in Jamaica, Queens, for instance, we have provided a number of on-site workshops and continuing collaboration over the past five years. We are currently working with York College Learning Center to develop standards-based math activities and a Family Math Fair.
Their Standards and Ours
In 1993, MEG began systematic study of the 1989 National Council of Teachers of Mathematics Standards and Principles. As we became advocates for the educational values expressed in the document, a broader political discussion ensued within MEG on the topics of equity, access, and the use of standards in high-stakes educational decisions. We knew that many of our students, particularly young adult African-Americans and second-language students, had been excluded from educational opportunity by the use of inappropriate, high-stakes tests administered in the name of standards and academic excellence. We recognized that the word "standards" meant different things to different people. For most policymakers, standards seems to signify an agenda of tougher high-stakes testing. They do not seem to concern themselves with standards focused around adequate resources or equal access to opportunities to learn for all students. Instead, educational equity seems to be limited to all students taking the same test.
The NCTM had something else in mind when they used the words "standards" and "standards-based math education." The NCTM Standards outlined a process of learning and teaching that included all students; it emphasized professional development and adequate resources to support math teachers; it advocated multiple assessments for specified outcomes.
Standardized tests are now ubiquitous in US educational institutions. They are the central mechanism in a system of tracking and sorting by social class and race. The tests also drive instruction in the direction of narrowly conceived test preparation. In ABE most students understand that the standardized, norm-referenced tests of General Educational Development (GED) are the gatekeeper to the crown of graduation. In its present version the GED mathematics test emphasizes surface knowledge of school-based math and does not suggest to teachers and students that mathematics knowledge develops within a context of problem-solving and investigation. When the new series debuts in 2002, it may reflect better some of the values encouraged by the NCTM.
But do better assessment instruments by themselves create better instruction? Not without professional development to implement better curriculum adequately. MEG rejects the use of tests to deny access to educational and vocational opportunities for historically marginalized groups. We strongly believe that equity in instruction, resources, and professional development must be the horse driving the cart of improved standards. Without equity, the new Standards-based tests will only further marginalize large segments of our population.
As a group concerned with professional development in mathematics, we have been particularly sensitive to the use of bureaucratic standards to "deskill" and disempower teachers. Rigidly applied standards take decision-making out of the hands of teachers (Deagan, 1998). Mathematics teachers have a special responsibility to assess critically the effects of the current standards and accountability movement. Large-scale, standardized math tests are considered authoritative gatekeepers because they are viewed erroneously as culturally neutral and objective. Those of us who support the NCTM's approach to math education must clearly differentiate promotion of the Standards and Standards-based curriculum from the use of standards to deny democratic access and equity in education.
MEG and The Future
How do we evaluate MEG's experience thus far? What impact have we had? We can certainly point to some successes. We have created a small but active community of math learners and teachers. We know and understand a lot more math than we did when we began. We have been able to serve as a resource for particular teachers and programs, trying to provide richer, more challenging math education. While gratifying and important successes, they are slight when judged against the larger goal of making significant change in math instruction in ABE in New York City. Our resources are meager and the task is overwhelming. We are well aware that Standards-based, engaging math instruction remains the rare exception.
What do ABE math teachers in New York City need? First of all, we need support for intensive professional development in math education. We need to learn mathematics and best practices: content and pedagogy. We need to connect ABE teachers to the work of the NCTM, ensure that they are capable of implementing a Standards-based curriculum, and that we all make use of the Council's resources. We need research to understand math education in the ABE classroom better. Recent data from the National Science Foundation studies in Ohio, Pennsylvania, and Maryland show that Standards-based mathematics instruction improves student achievement as it bridges the equity gap (Kahle, 2000). We need to know which practices narrow the divide between educational haves and have-nots.
We think MEG offers a model of how a small group of motivated teachers can make progress around the edges and in certain well-lighted corners. Perhaps someday all students, including ABE students, will have the opportunity to receive first-rate math educations. That is certainly MEG's goal, and in the meantime we will keep trying to be the math teachers our students need and deserve today.
Hiebert, J. (1999) "Relationships between research and the NCTM standards." Journal for Research in Mathematics Education 30, 1.
Kahle, J. (2000), Equity Diversity Standards Conference. New York: New York University.
Council of Teachers of Mathematics (2000).
Principles and Standards for School Mathematics. Reston, VA: NCTM.
About the Authors
Brover has been a full-time literacy teacher for 12 years. He is currently the
math resource teacher at the York College Learning Center, CUNY.
Denise Deagan is the director of the Adult Basic Education Program at the Borough of Manhattan Community College, CUNY. She is also an instructor at the New School University Teaching Adult Literacy Certificate Program and at the Bedford
Hills Correctional Facility College Bound Program, NY.
Solange Farina is a staff developer at the BEGIN Managed Programs and is also the program director of the HANAC ESOL and Civics/Citizenship Programs in New York, NY. She has worked in the adult education field for more than ten years.
Feel free to contact MEG with questions, or to join their mailing list. They can be reached at email@example.com