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Mathematics Curriculum Implementation: Not a Beginning, Not an End

In the past, curriculum materials have been considered either incidental to bringing about profound change in classroom teaching and learning or a panacea for bypassing the shortcomings of teachers. Some viewed curriculum as an impediment to systemic change: curriculum represented the traditional, dull schooling of our own childhoods in which we memorized dates and names, formulas and facts, spewed them out for the test, and promptly forgot them. Some viewed curriculum as a library of cute ideas and nifty activities - a compendium of "what to do on Monday" - handy to have around, but not a serious component of systemic change. Others viewed curriculum as a remedy to the ills of underprepared teachers using outdated materials and teaching practices: if we could simply write the perfect curriculum and hand the teacher a script, the materials themselves would automatically improve student learning.

In developing an elementary mathematics curriculum for grades K-5 during the past eight years, we have developed a view of curriculum materials as a critical component of systemic reform - not a substitute for the teacher's responsibility and intelligence, but a tool that allows the teacher to do her best work. While the materials do not stand alone, they fill a central role in bringing about real change. At the same time, if other critical components are not attended to, the materials by themselves are not enough. In this article, we first look at the role curriculum can play in systemic reform, then examine the experiences of several school systems as they implemented new curriculum materials. Our examples are drawn from work with the K-5 elementary mathematics curriculum, Investigations in Number, Data, and Space.

Some History

In 1990, the National Science Foundation began funding projects to develop complete mathematics curricula that would embody changes in mathematics content and teaching practices described by mathematics educators, researchers, and mathematicians (MSEB, 1989; NCTM, 1989, 1991). These changes included:

• broadening of mathematics content to include substantive work at all grade levels in geometry, data, probability, patterns and functions, and other topics that had traditionally been given little attention

• a focus on the development of mathematical thinking and reasoning - a rigorous understanding of mathematical ideas that goes far beyond the memorization of formulas and definitions

• an emphasis on building on prior learning through solving a wide range of mathematical problems that require reasoning, explaining, representing, justifying, and generalizing

• an active teaching role that required the teacher to learn more mathematics content and gain a deeper understanding of how children develop their mathematical understanding

• inclusion of all students, especially those who had been traditionally underserved by the U.S. mathematics curriculum, in significant mathematics learning.

Funding of these complete curricula was a major step forward in bringing about change in mathematics teaching and learning at the classroom level. For years mathematics educators had been working towards the goals spelled out in the NCTM Standards documents. Some of these ideas had filtered down to teachers - ideas about cooperative learning, use of manipulative materials, use of more complex, open-ended problems. However, many changes made by teachers during the 1980's and the first half of this decade, while useful and sometimes leading to improved learning, were also fragmented and inconsistent. In some cases, these changes were only cosmetic: students now worked in groups, instead of individually, but still worked on exactly the same mathematics content using exactly the same approaches they had before.

One of the chief difficulties for teachers who wanted to rethink their approach to mathematics was the lack of available curriculum materials. There were, indeed, many supplementary materials available that provided good problems. Several specialty publishers had developed important niches of innovative mathematics activity books and materials. NSF itself had funded significant projects that developed supplementary materials (see, for example, Russell et al., 1989-92). Although many teachers used these materials, even using them well did not provide a coherent, year-long mathematics program.

What Curriculum Provides

So what kind of curriculum can support the kinds of changes being called for by the mathematics education community? It has become clear to us from our work developing curriculum in collaboration with many schools and teachers that curriculum must have a dual focus: it must provide significant mathematical work for students and it must provide professional development for teachers.

This point of view - that curriculum materials are a key staff development tool - is a new one in United States mathematics education. The well-documented traditional model for teaching mathematics in the U.S. is as follows: the teacher presents a new kind of problem to the class and works through one or two examples; students then practice by working many problems of this type; teachers check students' work, mark it "right" or "wrong," and assign more practice problems for homework. In this model, the teacher makes few decisions and needs only very basic knowledge of the mathematics content.

As teachers change their model of classroom teaching to one in which students are investigating mathematical relationships, constructing and testing conjectures, and justifying their approaches to solving problems, teachers are required to learn much more about the content of the mathematics and the ways in which students develop mathematical understanding. Since many currently practicing teachers did not gain this experience and knowledge in their own teacher education, they need to develop it as they teach. Sound curriculum materials must provide a significant amount of material from which the teacher can continue to learn over several years of working with the materials (Russell, 1997).

Implementing Curriculum: Not a Beginning

As school systems begin to make serious attempts to move the teaching and learning of mathematics in these new directions, it often becomes clear that teachers can only move so far without a coherent full-year curriculum that is compatible with the changes being undertaken. This was the case in District 2 in New York City. District 2 serves a diverse ethnic, linguistic, and socioeconomic population of 24,000 students. About six years ago, the district worked with Marilyn Burns, a well-known mathematics educator, to offer a summer institute for teachers. Over the next four years, further steps were taken to expose teachers to new ways of thinking about mathematics teaching and learning. In the fall of 1995, Lucy West was hired as mathematics coordinator for the district and charged with developing a plan for scaling up the mathematics initiative. Ms. West reports:

The teachers in the district who had attended Math Solutions [Marilyn Burns' summer staff development institutes]...tended to like the instructional strategies and the activities. However, when they returned to their classrooms in the fall, they had textbooks that were not designed to help them implement the changes they wanted to make...many of the teachers felt this was unsatisfactory and began to scrounge for materials that were more in line with the constructivist theories they were trying to implement. This proved a daunting task for most teachers. They often lamented how much easier teaching mathematics would be if there were a cohesive, comprehensive curriculum....

In the fall of 1997, the district adopted Investigations in Number, Data, and Space as its mathematics curriculum. This policy was new; previously, the district had allowed each school to choose its own textbook. District 2, like many other systems that are adopting innovative mathematics curriculum, sees the adoption of new curriculum as a critical component. However, new curriculum that embodies an approach to teaching mathematics very different from the approach of a traditional textbook cannot be implemented in a vacuum. Curriculum is not the first step. A completely unprepared staff would not be willing or able to use a curriculum such as Investigations that makes very new demands on them as mathematics teachers.

Hudson is a small town in Massachusetts with about 2,700 students. It is diverse socioeconomically with a Portuguese community that comprises about 35% of the population. Sheldon Berman, the school superintendent, reports how the implementation of mathematics curriculum was part of a long-term process:

In general, elementary teachers in Hudson have been moving in the direction of hands-on, problem-oriented methods in math and science for a number of years. Therefore, Investigations hasn't dramatically changed the nature of instruction in the district but rather furthered a direction we were already pursuing. This was clearly stated by one teacher who has been working with the curriculum for two years. She pointed out that "It means that I can teach math the same way I teach everything else - with an emphasis on exploration, dialogue, communication, problem solving, and cooperation."

If the new curriculum provides material from which teachers can learn, they are able to gain knowledge about mathematics content and about children's mathematical thinking, as Dr. Berman reports:

The most commonly mentioned benefit of the program for their [teachers'] classroom practice was that it gave them a better understanding of each student's thinking and enabled them to address misunderstandings more effectively. Teachers noted that: "Letting students explain different strategies has taught me a few things!"..."It enabled me to see a type of mathematical reasoning which had not become evident with my traditional presentation."...In addition, the curriculum helped some teachers understand math better themselves. One teacher reported that "I have a better understanding of math concepts and so can more effectively explain them to my students." Another wrote that "Math is much more interesting to teach using Investigations. I am gaining a stronger sense of mathematical thinking."

The Role of Professional Development

Just as the curriculum is not a beginning, it is not an end. Implementing a very new kind of curriculum in a school system is a critical and far-reaching step. It has the potential to provide all classrooms with a coherent core of mathematical content, a good sequence of problems to engage students in that content, and pedagogical strategies that help the teacher understand and support the mathematical progress of the entire range of students in the classroom. However, once curriculum is mandated - even if teachers have been involved in the selection process - problems inevitably arise: not all teachers want to learn how to teach from a new curriculum; not all teachers are well prepared to teach a curriculum that emphasizes mathematical reasoning; not all teachers are convinced that a change from a traditional textbook is necessary; not all teachers feel confident about learning new pedagogical approaches or explaining these changes to parents. As Lucy West comments:

Even though many teachers had been a part of the process of selecting this series and trying out some of the units, and even though teachers had asked for one main source to use as their curriculum, once...[the Investigations curriculum] was officially adopted, there was resistance and many concerns were raised.

First, only about 80 out of 800 elementary teachers had been involved in the selection process. Ninety percent of our teachers were untrained in the use of these materials. Though many teachers had taken Math Solutions courses over the summer, most had not truly incorporated the philosophy behind the instructional strategies. They still believed that teaching is telling and that math is learned through the practice of drills. Many saw the activities advocated by [Investigations] and Math By All Means units as "enrichment" to be used after students had the basics. Many teachers feared the parent reaction to no recognizable math homework and some teachers even used the opportunity to arouse parent doubts. In part, teachers also reacted to the fact that they were now being told what curriculum to use. This seemed sacrilegious in a community that was used to a lot of autonomy.

This summarizes the dilemma faced by many communities in the midst of systemic reform. New curriculum is vital in order to insure a coherent learning experience for students, to provide the best current knowledge about children's thinking, and to offer rich, significant mathematics. Yet, it is equally critical that teachers' experience and judgment be respected, that they are offered opportunities to continue their learning, and that they are given enough time to work with the new materials. For this reason, many school systems use a "replacement unit" approach that capitalizes on the unit structure of many of the new curricula and encourages teachers to choose one or two or three units to try out. In the past, when many traditional textbooks were very similar in content and structure, it was not so difficult to change from one curriculum to another. However, the change from a textbook to an innovative curriculum is quite different. Teachers are being asked to implement an unfamiliar model that they did not experience in their own education and have had little opportunity to see in action. Lucy West comments:

The format of the books was different than a text. The units required teachers to read and think about the mathematics. This was a foreign process to many teachers and one they were very uncomfortable with for two reasons. Many elementary teachers did not understand the mathematics themselves and many teachers were already feeling burdened by class size and demands in other subject areas. They perceived [Investigations] as an added burden. [R]equiring them to study and think through what your math lesson was going to be each day felt overwhelming to many.

Therefore, ongoing staff development is an essential part of the systemic approach to reform. School systems that see professional development as a long-term effort, not just a quick fix, take seriously the need for sustained support of teachers' learning. This usually takes two forms - staff development workshops for teachers and in-classroom support. Many school systems find that they need to develop a multi-faceted system of staff development offerings that provide different formats and entry points for teachers. For example, in Hudson, the first two years of staff development to support curriculum implementation included: a 5-day summer workshop led by TERC staff, a Math Solutions summer workshop, an institute sponsored by PALMS (the Massachusetts State Systemic Initiative), participation of a few teachers in a leadership development institute run by TERC staff, school-year follow-up workshops, and grade level meetings. By the end of the second year of implementation, the system recognized the need to create a position for a specialist who would develop and coordinate efforts to change mathematics and science teaching in the system. Dr. Berman writes:

This summer Hudson will be hiring an Elementary Math and Science Specialist to provide support for the implementation of Investigations as well as new science materials that the district will be piloting next fall....The teacher leaders did a good job with the time they had to offer other teachers but there was too little of this time available to meet the needs of some teachers. This will be well addressed through the training of three additional teacher leaders and the hiring of a full-time elementary math and science specialist to assist with implementation.

Similarly, District 2 has put considerable resources in place to attempt to reach their 800 elementary teachers:

We have done many things to address the issues and concerns that have come up as we plod ahead toward our goal of implementing the Investigations curriculum in every elementary classroom in our district. We have a team of 8 staff developers. Each works in 3 schools a week for a minimum of 1 day per week, in the same 4 classrooms every week for a full year. We have adopted a replacement unit strategy of implementation. We provide 3 workshops per grade level per year. Teachers receive the unit of study and a day to work with colleagues on the implementation of that unit.

The commitment to a multi-year plan for staff development and the allocation of resources to support that plan is a second crucial component of systemic reform:

It is clear to all of us who have been involved in developing curriculum that any curriculum materials, no matter how well they can be used, can also be used badly, can be misunderstood and distorted. Teachers have not necessarily been prepared, in their own mathematics education, to focus on student thinking or to see their role as partners with the curriculum in the way that we have described this partnership here. The best use of good curriculum materials is in the context of a long-term staff development program which engages teachers in ongoing reflection about students' mathematical thinking and continued work on mathematics content with their peers. [Russell, 1997, p. 252]

Parents, Too

However, teachers are not the only audience that needs support. When parents see mathematics being taught in a way that is unfamiliar to them, they naturally have questions about what is going on. When children bring homework from school that parents feel unable to help with, they feel confused and "out of the loop."

Watertown, Massachusetts, is a suburb of Boston with a linguistically and academically diverse population. About a quarter of the students are not fluent in English, speaking 25 different languages among them, and about a quarter are classified as special needs students. Consequently, the challenge of meeting the needs of this population was one of the major considerations when the system began investigating new mathematics programs about six years ago. Claire Groden, the K-5 Mathematics Specialist, developed a carefully designed three-year implementation program with professional development as the leading component. However, the system was also convinced that the community needed to be included in these new directions. As fourth grade Watertown teacher Sharon Risso puts it:

To be successful...any new program needs to be embraced by staff, administration, and parents. We already had the support of administration and a core of teachers, but we needed to reach out to the parent community. While Investigations provides parent letters, the concerns of parents about the differences in this program from traditional math instruction and their difficulty in providing help with homework needed to be addressed in a more personal, interactive way.

To address this need, math coordinator Claire Groden, in collaboration with Watertown teachers, developed a series of workshops for parents. Each workshop focuses on one of the new curriculum units, and is designed to engage parents in activities from the unit and to help them better understand the content and philosophy of the program. Workshops are scheduled so that parents may attend two sessions in a single evening: parents may choose two workshops on units from a single grade level, or they may attend different grade level sessions in order to gain a perspective on the whole curriculum. Ms. Risso comments:

When adopting a new program, we need to respect a parent's need to share...questions and concerns, or voice their support, in an open forum. Therefore, a sharing session concludes the evening and we have found that this approach has been invaluable in addressing these issues and providing support both for and by the parents....Since these sessions are presented by the teaching staff, they give us the opportunity to not only help parents better understand the nature and philosophy of the program, but also to see what really goes on in a classroom.

No End in Sight

Systemic reform in mathematics requires a multi-year commitment that includes at least three intertwined substantive components: staff development, curriculum implementation, and community education. This article provides a few examples of school systems that have taken seriously the commitment and allocation of resources that are necessary to engage in systemic change. Significant change is fraught with complexity and problems. As Lucy West says, "This is proving to be a messy and very slow process." However, administrators and teachers report that the second year is better than the first, the third better than the second. As a tool in this process, curriculum provides a framework for student and teacher learning:

Curriculum materials, when developed through careful, extended work with diverse students and teachers, when based on sound mathematics and on what we know about how people learn mathematics, are a tool that allows the teacher to do her best work with students. [Russell, 1997, p. 248]

References

Mathematical Sciences Education Board, National Research Council. (1989). Everybody counts: A report to the Nation on the future of mathematics education. Washington, D.C.: National Academy Press.

National Council of Teachers of Mathematics. (1989). Curriculum and evaluation standards for school mathematics. Reston, VA: NCTM.

National Council of Teachers of Mathematics. (1991). Professional standards for teaching mathematics. Reston, VA: NCTM.

Russell, S. J. (1997). The role of curriculum in teacher development. In S.N. Friel & G.W. Bright, (Eds.), Reflecting on our work: NSF teacher enhancement in K-6 mathematics (pp. 247-254). Lanham, MD: University Press of America, Inc.

Russell, S.J., Corwin, R.B., Mokros, J.R., Friel, S.N., & Stone, A. (1989-92). Used numbers: Real data in the classroom.

A series of six units for grades K-6. White Plains, NY: Dale Seymour Publications.

Thanks to Sheldon Berman, Lucy West, and Sharon Risso for contributing the accounts of the work going on in their school systems. Thanks also to Myriam Steinback and Marlene Kliman for obtaining and summarizing the school system descriptions.

Susan Jo Russell is a principal scientist in the Education Research Collaborative at TERC whose work focuses on the professional development of elementary teachers and on the implementation of innovative mathematics curriculum.

Investigations in Number, Data, and Space was developed at TERC and funded in part by the National Science Foundation, ESI-9050210.