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The book draws on a decade's detailed study of California's ambitious and controversial program to improve mathematics teaching and learning. Researchers David Cohen and Heather Hill report that state policy influenced teaching and learning when there was consistency among the tests and other policy instruments; when there was consistency among the curricula and other instruments of classroom practice; and when teachers had substantial opportunities to learn the practices proposed by the policy.
These conditions were met for a minority of elementary school teachers in California. When the conditions were met for teachers, students had higher scores on state math tests. The book also shows that, for most teachers, the reform ended with consistency in state policy. They did not have access to consistent instruments of classroom practice, nor did they have opportunities to learn the new practices which state policymakers proposed. In these cases, neither teachers nor their students benefited from the state reform. This book offers insights into the ways policy and practice can be linked in successful educational reform and shows why such linkage has been difficult to achieve. It offers useful advice for practitioners and policymakers seeking to improve education, and to analysts seeking to understand it.
America is alive with efforts to improve its schools. Governors, state legislatures, business leaders, presidents, members of Congress, and educators have all urged improvement of schools through reform legislation and new academic standards and assessments. This remarkable reform effort began with the publication of A Nation at Risk in the early 1980s and has grown ever since. Standards-based reforms have been enacted in most states, and several states took quite aggressive action. By the early 1990s, school improvement was a leading political issue in many states; most had enacted new standards, and many had tried to put schemes into place that would hold schools accountable for students' performance. Only a few states, however, did much to support local school improvement, and by the late 1990s many policymakers and observers were beginning to worry about whether states could improve schools across the board. As we write, anxiety is increasing about whether state reforms will succeed. Reports of problems with state reforms have grown, and test scores, especially those of disadvantaged students, remain low. During a recent education summit, in Washington, D.C., the New York Times printed a front-page story reporting that many governors and some university leaders were worried that the reforms were about to plunge over a cliff (Steinberg 1999).
In this book we aim to improve understanding of how state policies that support better teaching and learning can succeed. It is based on a decade's detailed study of a high-profile California effort to improve teaching and learning in mathematics. California's was one of the first major state reforms. It was led by a coalition of state educators, business leaders, and politicians. The goal was to provide much more academically demanding work for students: not just higher standards, but more complex and intellectually demanding work, and improvement for all students, not just an academic elite. The reform offered more detailed guidance for teaching and learning-in assessments, curricular frameworks, student curricula, and professional education-than has been common for most state governments during most of our history.
Like all such reforms, California's mixed pedagogy with politics. The political elements included vigorous leadership from state education officials, pressure on local schools for better performance, education of the public about public education, and confrontation with powerful interest groups, like textbook publishers. The pedagogical elements included stronger direction to teachers and other educators about how mathematics should be taught and learned, encouragement of professionals outside government to play key roles in writing new curricula and improving professional education, revision of the state's assessments to reflect the aims of reform, and improved professional education. Pedagogical and political elements of the reform merged as state leaders tried to align curricula, assessment, professional development, and other sources of guidance to schools and teachers with each other and the new math frameworks. These features of the California reform were later adopted by other states.
When Reform Works
Our central finding is that California's effort to improve teaching and learning did meet with some success, but only when teachers had significant opportunities to learn how to improve mathematics teaching. When teachers had extended opportunities to study and learn the new mathematics curriculum that their students would use, they were more likely to report practices similar to the aims of the state policy. These learning opportunities, which often lasted for three days or more, were not typical of professional education in U.S. schools. Also, teachers whose learning was focused around study of students' work on the new state assessments were more likely to report practices similar to the aims of the state policy. And teachers who had both of those learning opportunities reported practices that were even closer to reform objectives. The shorter-term professional development of most other teachers was keyed to special topics, like diversity in mathematics classrooms and cooperative grouping. These shorter-term learning opportunities, which were typical of professional education in California and elsewhere, did not allow teachers to learn about either student curricula or their work on assessments, and these teachers did not report practices close to reformers' goals.
Our observations are based on field research on the professional education available to California teachers in the early 1990s and on evidence from a survey of nearly six hundred California elementary-school teachers, undertaken by researchers from Michigan State University and Stanford University and completed in 1994-95. We focused on professional learning, because policymakers' aims were ambitious. Reformers in California sought to introduce a very different sort of math teaching from that typically found in U.S. classrooms. A conventional lesson on comparing fractions like 3/8 and 12/20, for instance, might begin with a teacher explanation of the procedure for finding a common denominator-multiply the denominators and convert the numerator, or record sets of equivalent fractions until a common denominator is found. Students would then practice the procedure, often using worksheets or problems in texts or workbooks. They would have little chance to connect the procedure to more concrete representations of size and equivalence, to develop and use alternative methods for fractional comparisons, or to begin their study by making comparisons in the context of realistic situations.
Seeing Fractions, a student curriculum package, was produced for the reforms to replace an upper-grade textbook unit on fractions-earning it and other such units the title "Replacement Units." Seeing Fractions offers an approach that is very different from conventional math instruction, first by introducing equivalent fractions in the context of rates, an often overlooked use of fractions, and then by allowing students to develop their own ideas about this topic. Teachers would start by posing a concrete problem to their classes:
I was at the amusement park last Saturday. I bought gumballs from one machine-I got three gumballs for eight pennies. My friend JoAnne told me that she'd found a different machine where gumballs were twelve for twenty cents. How could we compare them to find out which was the better buy?
Students would work for some time, using concrete materials, called "manipulatives," to aid their thinking. After developing solution methods, students would present their methods to the class and listen as others presented their own ideas. The curriculum materials asked teachers to ensure that students could talk through their strategies with reference to the concrete model, and defend their methods to peers. Only then were teachers to show a conventional method for comparing fractions. Small groups of students then would practice the method or methods they felt comfortable with; meanwhile, the teacher monitored their work by asking about methods and solutions.
In this case and many others, the student work of reform involved the development of new mathematical ideas and strategies, proof and justification, discussion and some practice. Students would also be working on new classroom tasks, ones which allowed for multiple interpretations or a variety of different solution methods. Teachers using this set of student materials would have to learn the new instructional methods and something about the mathematics students might invent or encounter along the way. Replacement units themselves tried to meet teachers' needs by describing such things in more detail than did typical mathematics textbooks. A teacher note on this lesson on comparing fractions, for instance, explains not only the method of finding equal denominators but also a method of finding equal numerators and a method of finding the unit cost. The lesson also identifies typical student misunderstandings of the mathematical content, and elsewhere the materials discuss the value of concrete materials, student-invented methods, and proof in mathematics. Such curricular material was meant to be instructive for both teachers and students.
To aid teachers further, these student units were accompanied by "replacement unit workshops"-two-and-a-half day sessions in which teachers would do the mathematics themselves, talk with each other about the content, and observe examples of student work on the materials. Similar workshops were developed around the California Learning Assessment System (CLAS), the test designed to assess schools' progress according to state frameworks. Teachers completed the student test items, examined student work and typical mistakes on such items, and learned about the mathematics involved.
Only a small fraction of the state's teachers had substantial opportunities to learn about the replacement units or new student assessments, but they made a big difference. Teachers whose opportunities to learn were grounded in specific curricula and assessments reported more of the sorts of practices that reformers had proposed than teachers whose opportunities to learn were not so grounded. Those practices included more sustained work on problems, fostering mathematical discussions and individual or group investigations of mathematical ideas. Teachers whose opportunities to learn were grounded in new curricula or student work reported fewer of the things reformers found undesirable-primarily worksheet drills and computational testing-than teachers whose work was not so grounded.
These workshops differed substantially from others offered in the state at the time, which were keyed to topics like diversity in mathematics classrooms and cooperative grouping methods. Most of these special-topics workshops did not allow teachers to learn about new student curricula, assessments, or students' work on either. Instead, teachers might try out one or a few "hands-on" activities, activities that in some ways embodied reform yet were not linked to one another, planned around students' response to the activities, or designed to develop a mathematical idea or process in depth. This more superficial sort of opportunity for professional growth is endemic to U.S. education.
Important as teacher learning was, reformers' main aim was to improve students' learning. We found that where teachers had opportunities to learn about student materials or assessments, students posted higher scores on the state's assessment of math achievement in elementary schools. Our conclusion was based on linking schools' average student scores on the reform-oriented 1994 state mathematics assessment with our data on teachers to learn whether state policy influenced student learning. Teachers' opportunities to learn about new student curricula and assessments were a critical link between state reform and student learning. These opportunities did not just describe the broad themes of these new policy instruments but were grounded in the actual mathematics and student work on curricula and assessments that were instruments of practice.
These findings imply a departure from conventional research on policy implementation. When researchers have tried to explain problems of implementation, they have typically pointed to complex causal links between state or federal agencies on the one hand and street-level implementers on the other. Those links often have been weakened by such things as bureaucratic difficulties, weak incentives to comply, and differences in the preferences of policymakers and implementers.
Our account adds a major category to these considerations: implementers' opportunities to learn. When California teachers had significant opportunities to learn how to improve students' learning, their practice changed appreciably and students' learning improved. Some policies depart further from existing practices than others, and the more they do, the more implementers have to learn. California's effort to improve mathematics teaching and learning was quite ambitious. Because it called for classroom practices that dramatically departed from conventional schoolwork, professionals' learning was a key element in connecting policy with practice.
One implication of this idea is that policy should be distinguished from the instruments deployed in its support. The 1985 Mathematics Framework for California Public Schools announced the policy, and if reformers had followed past example, they would have left matters there; the policy would have been little elaborated. We found, however, that teachers' knowledge of the framework was only modestly related to changes in teaching. The things that made a difference to changes in their practice were integral to instruction: curricular materials for teachers and students to use in class; assessments that enabled students to demonstrate their mathematical performance -and teachers to consider it-and instruction for teachers that was grounded in these curriculum materials and assessments. These instruments elaborated the framework ideas in much more detailed form, which was usable in instruction.
When the objectives of policies and the actions that implementers need to undertake are thus elaborated, policies are more likely to work. One reason is that everyone concerned can see more clearly what the policy calls for. The goals of the California reforms, for instance, were given instructionally salient shape in examples of students' work on the new assessment and in workshops on the replacement units. Another reason is that such instruments build a basis for implementers' learning. The replacement unit student curricula were turned into materials from which teachers could learn not only the new mathematics their students would study, but how students might make sense of the subject and how teachers might respond. The most important policy instruments in the California math reform were those which enabled teachers to work with learners in the ways reformers had envisioned.
A second implication of our analysis is that consistency among policy instruments is an important influence on the success of reforms like that in California. Inconsistencies among assessments, curricular materials, and teacher education would create opportunities for different and divergent interpretations of the policy, and that would reduce consistency in implementation and policy effects. Policymakers took steps to create such consistency at the state level. They developed curricular frameworks that set out the aims and direction for the changes in math teaching and learning.
Excerpted from Learning Policy by DAVID K. COHEN HEATHER C. HILL Copyright © 2001 by Yale University. Excerpted by permission.
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|List of Abbreviations|
|1||Policy, Teaching, and Learning||1|
|2||The New New Math in California||13|
|3||Learning About Reform||31|
|4||Teachers' Ideas and Practices||57|
|5||Teachers' Opportunities to Learn and Classroom Practice||87|
|6||Learning and Policy||127|
|8||Policy and Learning||183|