A 16-year veteran of high school mathematics teaching, Lamont Stewart describes a typical unit in his geometry class: I begin with a concrete example. When we were ready to start parallelism, I raised the question of the parking lot at the high school, whether it could be restriped to hold more cars. The kids measured lines, turning radii, angles. Then they proposed ideas. They were doing a heck of a lot of math.But I have to give a departmental exam, and sometimes it's a real challenge to get to the right point at the right time. Also, it's hard to give a grade to projects submitted by groups. Is it fair to give a high grade to someone who is a —al member of a productive group?
Another long-term teacher, Olivia Green, describes her geometry class: I start a unit by asking the students what they know and then posing a problem for them to investigate in the computer labs next to my classroom. They create certain figures, usually working in pairs, take measurements, and make and test conjectures. Then, we talk about their findings.
These two classroom scenarios highlight key elements of the Curriculum and Evaluation Standards for School Mathematics issued by the National Council of Teachers of Mathematics (NCTM 1989). They also reveal some of the demands placed on teachers as they struggle with new paradigms in traditional settings.
Policy initiatives from Kentucky to California are espousing curriculum based on core concepts that students understand through induction rather than memorization, teaching based on guided inquiry rather than didactic instruction, and assessment that is open-ended rather than machine-scorable. The challenge of these extensive reforms for teachers is to teach and test in new ways. For schools and districts, the challenge is to provide the necessary support for teachers as they follow these new directions.
Standards-Oriented Classrooms
Lamont Stewart and Olivia Green were among 50 innovative geometry teachers interviewed by researchers at the Harvard Graduate School of Education (Wiske et al. 1992). All were recommended by school administrators or mathematics education specialists for their success in using practices consistent with the NCTM Standards. The sample was drawn from urban, rural, and suburban schools in all parts of the country, operating in a range of education policy contexts and serving students of varied backgrounds and academic tracks.
In structured telephone interviews, the 50 teachers described how they are incorporating inquiry-based, problem-centered approaches in their classrooms. They also discussed resources and policies that help or hinder their efforts. Many reported that their classrooms have changed significantly in the past several years. For example, Paula Sandman now provides a problem box in which students submit questions about their investigations. These, rather than textbook assignments, become the basis for class discussion. Teachers use Manipulatives, computer software, and other materials to help students investigate geometry through multiple senses. John Seeley, perhaps the most daring of the interviewed teachers, discarded a hopelessly inappropriate book. With his guidance, Seeley's students develop their own text.
These examples indicate the variety of teaching techniques emerging in classrooms today. These innovative teachers use guided inquiry and inductive reasoning to help their students observe and articulate geometric principles. Students often work collaboratively to develop and defend their own conjectures rather than to memorize information presented by others. They use manipulatives and computer technologies to visualize geometric phenomena and analyze data.
The teachers we interviewed report that, over time, their classrooms develop a “feeling of family.” In these emerging learning communities students take increasing responsibility for their own education and for helping peers. When these classes hum, teachers—some of whom were on the verge of quitting after years of frustration with traditional geometry—find pleasure, purpose, and professional renewal. They speak of students who are applying their knowledge outside the classroom and pursuing higher levels of mathematics.
Obstacles to Change
Numerous barriers, however, can impede such promising developments. Teachers implementing the NCTM Standards mentioned several deterrents to change.
Incompatible texts and materials. When handed a lemon of a text, John Seeley and his students made lemonade by devising their own book. Many teachers complained of being saddled with prescriptive texts that are incompatible with the new directions in mathematics education. They must often steer a course among irrelevant or counterproductive problem sets and end-of-chapter reviews and tests. They also spend considerable time developing curriculum materials that are more responsive to the NCTM Standards. Moreover, teachers must struggle with expectations from educators, parents, and required tests that they will “cover” the traditional course of study in geometry texts.
Inaccessible technology. Unlike Olivia Green, who has ready access to computers, many teachers are constrained by either a dearth of technology or logistical hurdles that make using computers not worth the bother. Teachers reported having to reserve computer labs weeks in advance and not having appropriate software or sufficient training in its use.
Inappropriate assessments. Almost without exception, the teachers interviewed struggle with ways to measure their students' problem-solving abilities, evaluate projects of cooperative learning groups, and develop and grade open-ended assessments that allow students to demonstrate their capacity to reason about complex problems. Teachers also wish to conduct assessments as rapidly as they currently score multiple-choice tests, which are efficient but inappropriate for the new learning approaches.
Further, depending on state, district, and departmental requirements, teachers must often prepare students for standardized tests that are tangential, irrelevant, or even antithetical to the curriculum recommended in the NCTM Standards. Commonly available assessments not only often derail teachers in their move toward inquiry learning, but they also fail to provide teachers or their students with useful information about their progress. Appropriate assessment currently constitutes a gaping hole in the fabric of curricular change.
Inadequate professional development. Even teachers well versed in the Standards find that, because inquiry learning is neither sequential nor predictable, they need a more comprehensive understanding of mathematical concepts and their relationships than was necessary to convey traditional deductive geometry. In combination with intensive training, they also seek practice and coaching in meshing content knowledge with appropriate pedagogy—for example, how to form groups and keep them on task, pose questions that nudge rather than pull students, handle a student's related but unexpected proposition, and how to encourage students to take responsibility for and organize their own learning.
Incompatible educational paradigm. The shift in the assignment of responsibility in the classroom seems to be the most difficult hurdle to surmount, in part because it is the most abstract. Students often resist the notion that they must formulate questions, plan a scope of work, organize information, and share findings with classmates. Teachers, as well as students, note that the authority shift in the classroom is not generally reinforced by an equivalent shift in the larger context of the education system. Administrative directives regarding curriculum coverage, assessment expectations, or access to technology are often made without consulting teachers. This contradictory approach perplexes teachers in their quest to confer on students' authority that they themselves often lack.
Lack of time. Overcoming all of these obstacles takes time. Teachers need time to learn the content and pedagogy of the new curriculum, to participate in repeated and targeted professional development experiences, and to integrate concepts and experiment with new techniques. In addition to time, teachers appeal for patience as they begin a process that can require from 5 to 10 years to realize.
Sites That Support Change
Given these obstacles, why do teachers persist in working to align their practice with the NCTM Standards? Foremost is the gratification they receive from seeing engaged, achieving students. Teachers uniformly report that their students perform as well as others on standard tests. Even without formal assessments, teachers observe that their students understand more geometry. Moreover, they learn to formulate questions, organize information, and reason and communicate about mathematics better than most students in traditional classes.
A second reason is that several states and districts, which the researchers visited following the telephone interviews, are adopting policies that support teacher development (Wiske and Levinson 1992). Their activities and the recommendations that follow can guide educational leaders who want to support teachers struggling with a changing curriculum.
Montana. Kimberly Girard is the only mathematics teacher in a 150-student high school in rural northeast Montana. Rather than being isolated, however, she helps lead the Montana Council of Teachers of Mathematics (MCTM), conducts workshops on the Standards, responds to teachers' queries on the state's education electronic bulletin board, plans conferences with Montana's teacher-training institutions, communicates with the state Office of Public Instruction on mathematics education policy, and works on the National Science Foundation-funded Systemic Initiative for Montana Mathematics.
Girard's activities exemplify the cross-institutional professional support that characterizes mathematics reform in Montana. Based on years of common effort to develop a cadre of teacher leaders, the mathematics director of the state Department of Education, colleges and universities, schools, and the MCTM work together to foster active learning and mathematics teacher development across the state.
Kentucky. Kentucky has taken the big-picture approach to reform. As conveyed in the Kentucky Education Reform Act (KERA), systemic reform (Smith and O'Day 1991), or alignment, of assessment and curriculum characterize reforms in this state, not only in mathematics but in all curricular areas. The state assessment system will consist of multiple-choice items, performance evaluations, open-ended items, and portfolio sections, developed in conjunction with the state curriculum framework. The accountability clout of these assessments, combined with other measures, will likely trigger incentives or sanctions for schools. Thus, reform leaders presume that school-developed curriculums will also be based on these state-level measures of “what students need to know and be able to do.” The assessments are compatible with the NCTM Standards and KERA-defined curriculum goals. Most likely, therefore, both the state and local mathematics frameworks, as they are fleshed out, will parallel the Standards. The state reforms are structured so that curriculum and assessment will inextricably mesh.
Another key element of the Kentucky reforms is shared responsibility for decision making and educational improvement. As structured by Kentucky's new law, the state takes responsibility for articulating student learning goals and related “valued outcomes.” With teacher assistance, the state also coordinates development of assessments and related curriculum frameworks. Significant responsibility for implementing the goals and achieving the outcomes remains with schools and districts, which develop local curriculums, select texts, and make financial and other decisions. This shared responsibility for policy-making between state and local entities mirrors the shared responsibility for developing mathematical knowledge between teachers and students.
Pittsburgh. Backed by an innovative district mathematics director, teachers have taken the initiative in mathematics education reform in the Pittsburgh public schools. After discovering a prototype geometry textbook based on student inquiry and consistent with the NCTM Standards, a cluster of innovative teachers developed an inquiry-oriented course. The course was so successful that the mathematics director asked them to provide workshops and authorized the purchase of the new textbook, which was adopted districtwide. The district further helped spread reform by offering summer workshops and after-school meetings for teachers.
Several factors supported change in Pittsburgh. First, the pioneer teachers took action by jettisoning an inadequate textbook. Collaborative colleagues then developed alternative materials, teaching approaches, and assessments until they found an appropriate textbook. Building and district administrators endorsed their innovations by encouraging new teaching techniques, funding the adoption of a new text, and sponsoring the extension of innovations through professional development opportunities. Finally, the teachers received support from the Urban Math Collaboratives—organizations in large cities that link schools, universities, business people, and others interested in furthering reform. The Pittsburgh initiative, therefore, tells a story of action generated at the school level and supported at the district level.
Policy Recommendations
What can educational leaders learn from these teachers and sites about policies that will support reform efforts in the classroom? Following are recommendations that derive from the research.
Provide direction with discretion. Instruction by guided inquiry defies the traditional teacher-lecture/student-swallow relationship in the classroom. Standards-oriented teachers who struggle with mandated texts or assessments that undermine this approach are engaged in a larger conflict in which the policy context contradicts the teaching agenda. While trying to increase their students' responsibility for learning, they are denied the responsibility that they need to do so. Enhancing the authority of learners in the classroom is a practice more likely to thrive in an environment in which teachers have requisite authority to make this happen.
As promoted in Kentucky, this re-allocation of functions is not tantamount to relinquishing responsibility. Leadership that is unafraid of mandating standards but willing to share authority models the spirit of the new mathematics curriculum. Teachers warned, however, that some restructuring demands can distract teachers from their primary role in instruction. Leaders should recall that, in the midst of change, student learning is the ultimate priority.
Align texts, tests, technology, and teacher development. Deciding that no book was preferable to one that contradicted his teaching model, John Seeley abandoned the assigned textbook. Similarly, Lamont Stewart and virtually every other teacher interviewed criticized irrelevant or misguided tests. Others yearned for applicable software. Many wished for help in learning how to guide inquiry.
These mismatches bespeak a lack of alignment among teachers' responsibilities, resources, tools, and skills. School, district, state, and higher education leadership can coordinate these key factors in a teacher's repertoire to enhance rather than combat one another.
Provide sustained support for professional development. The teachers who reported that they were most comfortable with and adept at integrating Standards-based curriculum and pedagogy are those who participated in extensive, cyclical staff development, including practical coaching. In particular, they valued opportunities to participate in guided inquiry of mathematics (rather than merely being told about it), observe successful teachers at work, and practice new methods with observation and coaching by experienced colleagues. Federal, state, or local resources should be used to support extended training and follow-up, released time, and joint planning for teachers, department chairs, and curriculum coordinators.
Support collegial networks. Working through a new curriculum, teachers need ongoing opportunities to consult with colleagues. Even otherwise isolated teachers gained sustenance through sharing ideas at conferences, planning sessions, inservice opportunities, and by electronic communications.
Attend to the details of school culture and structure. The particulars of school life also affect teachers' ability to incorporate new curriculum and practices. These include length of class periods (longer is better); size and grouping of classes (smaller is better); time to review and develop materials and to plan with colleagues; access to computers, calculators, and manipulatives; and leadership from supervisors in arranging substantive support for mathematics education reform. Leaders should consult with teachers on the core facets of curriculum, instruction, classroom interactions, and professional development that support teachers' growth.
The gist of the findings from these pioneering teachers and supportive settings is that progress toward the desired reforms is possible but that there are no short-cuts. Accomplishing deep changes in mathematics curriculum, teaching, and assessment requires linked leadership across levels of the education system, comprehensive attention to the key components of classroom life, and sustained support for teachers as they and their students forge new directions in mathematics teaching and learning.
