Mathematical pedagogy is far from common. Despite serious efforts to reform math teaching over the past 30 years, mathematics continues to be taught much as it always been with, at best, little pockets of change. Each of us probably knows some teachers who manage to engage pupils in solving genuine mathematical problems or who approach algebra conceptually or whose conceptions of what mathematics is worth teaching include topics like probability or number theory. Rarer are teachers for whom the logic of mathematical discovery and the logic of justification are coequal and focal dimensions of mathematical knowledge. Most pupils spend their time in ordinary mathematics classrooms — classrooms where mathematics is no more than a set of arbitrary rules and procedures to be memorized. In their book, The Mathematical Experience, Davis and Hersh (1981) characterize a typical mathematics class:
The program is fairly clear cut. We have problems to solve, or a method of calculation to explain, or a theorem to prove. The main work will be done in writing, usually on the blackboard. If the problems are solved, the theorems proved, or the calculations completed, then the teacher and the class know they have completed the daily task. (p. 3)
When students don't "get it," their confusions are addressed by repeating the steps in "excruciatingly fine detail," more slowly, and sometimes even louder (Davis & Hersh, p. 279). This pattern is old and dominant. Why is mathematical pedagogy, as an ideal of teaching and learning mathematics, so rarely realized? No single cause can account for the failure of past reform efforts to change the face of mathematics teaching in classrooms, and, yet, the patterns of the "ordinary" math class have dominated and continue to prevail. To set a context for my own work, below I discuss four factors that contribute to this inertia:
Factor #1: Culturally embedded views of knowledge and of teaching
Factor #2: The organization of schools and the conditions of teaching
Factor #3: Poor curriculum materials
Factor #4: Inappropriate mathematics teacher education
Factor #1: Culturally embedded views of knowledge and of teaching.
Ordinary mathematics teaching reflects a culturally grounded epistemology — what Jackson (1986) calls the mimetic tradition. Knowledge is fixed; teachers give knowledge to pupils who store and remember it. This tradition is firmly embedded in Western culture. Cohen (in press) writes that, as much as 300 years ago,
most teaching proceeded as though learning was a passive process of assimilation. Students were expected to follow their teachers' directions rigorously. To study was to imitate: to copy a passage, to repeat a teacher's words, or to memorize some sentences, dates, or numbers. Students may have posed questions in formal discourse, and perhaps even embroidered the answers. But school learning seems to have been a matter of imitative assimilation. Cohen (in press) argues that innovative approaches to teaching and learning embody assumptions about knowledge and about teaching and learning that are "a radical departure from inherited ideas and practices" in this culture. Mathematical pedagogy is such a case. Reflecting a view of knowledge as "emergent, uncertain, and subject to revision — a human creation rather than a human reception" (Cohen, in press, p. 15), mathematical pedagogy places the teacher in the role of guide and the student as inventor and practitioner of mathematics. Such ideas fly in the face of centuries-old intellectual traditions, not just in mathematics but in all disciplines. These traditions live outside the institution of schools as well, in the everyday occasions for informal teaching — parenting, for example. A view of knowledge as fixed, of teaching as transmission, and of teachers as authorities runs very deep.
Mathematical pedagogy is also at odds with the individualistic American intellectual tradition in which working independently and figuring things out for oneself is valued. Indeed, modal school practices — testing, remedial programs, individualized self-paced instruction reflect this tradition. We speak of "individual differences" and different personal "learning styles." Psychology, the science most often presumed to be foundational to teaching, reflects this same bias, thus further reinforcing the traditional focus on the individual. Ordinary mathematics classrooms reflect this in the norms of quiet individual work and of interpretations of collaboration as cheating. Jackson (1968, p. 16) describes this dominant feature of classroom life, that students must ignore those who are around them and "learn how to be alone in a crowd":
In elementary classrooms students are frequently assigned seatwork on which they are expected to focus their individual energies. During these seatwork periods talking and other forms of communication between students are discouraged, if not openly forbidden. The general admonition in such situations is to do your own work and leave others alone.
In addition, right answers are the currency of the classroom economy, earning privilege and reward. Just as the views of knowledge and of teaching discussed above are embedded in the surrounding culture, so too is this intellectual competitiveness. Individualism, however, is not the image of the academic community of discourse (e.g. Kuhn, 1962). Scientists disagree and hammer out new understandings in the pursuit of their work and in solving novel puzzles. Mathematical pedagogy, too, implies a learning community (Schwab, 1976), a group collaborating in the enjoyable pursuit of mathematical understanding. Students are expected to share ideas, to work with others, and to see the collective as the intellectual unit and arbiter. While right answers may not be the issue, intellectual progress and strength are a resource to everyone, not just to some; each person's contributions can help the community.
Factor #2: The organization of schools and the conditions of teaching.
Institutional and organizational factors in schools also present obstacles to change. Schools are charged with
multiple and competing goals, not the least of which is the goal of fostering individual excellence and advancement, while simultaneously ensuring equity and access. The rhetoric of "individualism" dominates for many teachers; yet, they know that they are expected to meet individual needs and to guarantee equity. The push to standardization and to avoid risks or experiments is great, in light of these pressures (Cohen & Neufeld, 1981; Goodlad, 1984). Conservative administrators and school boards preoccupied with test scores put pressure on teachers to emphasize "basic skills" — computation and memorization of "facts." Teachers are generally responsible for a curriculum that is both traditional and warranted by its very traditions. The structure of the school day means that teachers are isolated from one another and have little time or support for learning or trying out innovations. Time is segmented into 54-minute blocks, content must be covered, pupils must be prepared not only for tests, but also for the next level. Moreover, working with groups of 30 children makes experimenting with pedagogy risky for teachers who must also maintain order and routines. Elementary teachers must teach many other subjects in addition to mathematics; secondary teachers must teach many more students.
Teachers often do not have time to plan and organize rich experiences for pupils, nor can they afford the looseness of more exploratory curricula. They feel pressure to make sure that pupils master required content. For example, the time required for students to "get inside" a topic like measurement may seem to be at odds with ensuring that students also get to everything else. The pull toward neat, algorithmic curriculum is very strong. Teaching measurement by giving out formulae — L x W = some number of square units, and L x W x H = some number of cubic units — may seem much more efficient than hauling out containers and blocks and rulers and having students explore the different ways of answering questions of "how big" or "how much." That this results in sixth graders who think you measure water with rulers goes almost unnoticed. All these features of school organization and of the conditions of teaching are part of the context within which reforms must operate (Cuban, 1984; Sarason, 1971). Sarason (1971) argues that the massive failure of the new math was due, in large measure, to the reformers' failure to take a sufficiently broad perspective of the "regularities" of the school setting and the culture of the school:
It becomes clear that introducing a new curriculum should involve one in more than its development and delivery. It should confront one with problems that stem from the fact that the school is, in a social and professional sense, highly structured and differentiated — a fact that is related to attitudes, conceptions, and regularities of all who are in the setting . . . . Any attempt to change a curriculum independent of changing some characteristic institutional feature runs the risk of partial or complete failure. (pp. 35-36)
Without taking the wider context into account, change, if it occurs at all, is likely to be superficial — changing textbooks but not mode of instruction, for instance (Sarason, 1971). These conditions of schooling have direct consequences for efforts to effect change in the teaching of mathematics — indeed, for the concept of "implementation" itself (Farrar, Descantis, & Cohen, 1980; Wildavsky & Giadomenico, 1979). Sarason would argue that reformers who do not consider what schools are like will operate under the misguided impression that changes can be simply "put into place." Quite the contrary: Innovations are interpreted and adapted by teachers; both the intent, the enactment, and the effect of an innovation changes in the translation (Berman & McLaughlin, 1975; McLaughlin & Marsh, 1978; Sarason, 1971). Expecting that reforms can be instituted faithfully from the top down is a fantasy that ignores the loose connections between official authority and actual practice. American teachers, in fact, have considerable elbow room at the classroom level, and typically "[arbitrate] between their own priorities and the implied priorities of external policies" (Schwille, Porter, Floden, Freeman, Knappen, Kuhs, & Schmidt, 1983, p. 387).
Factor #3: Poor curriculum materials.
Moving from the least tangible to the most concrete, another factor that contributes to the continuity and conservatism of mathematics teaching is the nature of the materials from which teachers teach. Typical mathematics curriculum materials — textbooks, activity cards, kits — tend to emphasize calculation skills and a "here's how to do it" approach to mathematics. Stodolsky's (1988) analysis of elementary math textbooks suggests that concepts and procedures are often inadequately developed, with just one or two examples given, and an emphasis on "hints and reminders" to students about what to do. Even when teachers attend workshops or read about the importance of focusing on problem-solving and teaching procedures with conceptual understanding, their textbooks provide them with little guidance or support for acting on these recommendations. For example, area and perimeter, the content of the example above, are presented in terms of the formulas — L x W and 2 x L + 2 x W — with, perhaps, some pictures to illustrate. Practice is provided in calculating the area and perimeter of some rectangles, with reminders to state the answers to area problems in terms of "square units." Multiplication by ten is explained in terms of "adding a zero," algebra texts claim that vertical lines have "no slope," and rectangles are represented as figures with two long and two shorter sides. Geometry is scant, measurement more procedural than conceptual, and probability investigations relegated to the little "Time Out" boxes on a few random pages. Although recent editions of popular textbooks include new pages on "problem solving," they are often additions to rather than changes in the textbook's representation of mathematics. They tend, furthermore, to package mathematical problem solving as procedural knowledge or as a topic in its own right. Kline (1977) blasts school mathematics texts for their dogmatic presentations and failure to let students in on the struggles of mathematical discovery and activity.
Factor #4: Inappropriate mathematics teacher education.
That these three factors affect the problem of "no change" in the teaching of mathematics is unquestionable. A fourth factor, one focused on what teachers know and believe, is equally critical. The consistency of the
ordinary math class is in itself a barrier to change. Prospective teachers probably learn something from being inundated with that kind of teaching over a ten or twelve-year period — and what they learn may well include a rule-bound knowledge of mathematics. If we want schools to change, teacher education has to find out how to break into this cycle. Critics observe that pre service teacher education typically has a weak effect on teachers' knowledge and beliefs and that whatever prospective teachers learn at the university tends to be "washed out" once they get to schools. In fact, it is rather unsurprising that a handful of university courses often fails to substantially alter the knowledge and assumptions which prospective teachers have had "washed in" through years of firsthand observations of teachers. In the case of mathematics teaching, they have already clocked over 2,000 hours in a specialized "apprenticeship of observation" (Lortie, 1975, p. 61) which has instilled not only traditional images of teaching and learning but has also shaped their understandings of mathematics. As this is the mathematics they will teach, what they have learned about the subject matter in elementary and high school turns out to be a significant component of their preparation for
teaching. Thinking that they already know a lot about teaching based on their experiences in schools and on their good common sense, prospective teachers may not be disposed to inquire or to learn about teaching mathematics.
Furthermore, and equally serious, what we know about what students learn in ordinary mathematics classes suggests that prospective teachers are unlikely to know math in the ways that they will need to in order to teach. The weakness of teacher education tends to be attributed either to its truncated structure or its misfocused curriculum. Efforts to reform teacher education assume that what is taught to prospective teachers either is not optimally organized or is not enough to break with traditional modes of teaching. In response, many people have ready answers about how preservice teacher education should be changed — proposing to alter content, duration, requirements, or structure (see, for example, Holmes Group, 1986; Prakash, 1986; Carnegie Task Force, 1986). With respect to the content of teacher education, consider the preparation of mathematics teachers. Disagreements abound about what knowledge is most important to cover: Piagetian theory or a variety of perspectives on human learning? A survey of the history of mathematics or how to teach specific topics like place value? A review of basic arithmetic and algebra or exposure to nontraditional topics such as probability? Classroom management or more mathematics courses? Some advocate that elementary teachers should specialize (e.g., Elliott, 1985); others recommend the abolition of the undergraduate major in education (Holmes Group, 1986). Still alive and well are debates about the relative importance of generic skills of teaching and subject matter knowledge (e.g., Guyton & Farokhi, 1987; others).
Discussions about the ideal curriculum for teacher education, however, are premature (Lampert, in press b). They are premature because increasing the impact of teacher education on the way mathematics is taught depends on a reconsideration of our assumptions about teacher learning, not just a revision of what teacher educators deliver. While what is taught is unquestionably a critical issue for teacher education, the fact that formal teacher education is often a weak intervention is attributable in part to its failure to acknowledge that "the mind of the education student is not a blank awaiting inscription" (Lortie, 1975, p. 66) as well as its failure to confront effectively the inappropriate or insufficient understandings that prospective teachers bring. Questions of what to teach properly depend both on what learners need to know and on what the learners already know. The first three factors discussed in this chapter — cultural views of knowledge and of learning, organizational features of schools and the conditions of teaching, poor curriculum materials — all help to explain the persistence of ordinary mathematics teaching. These factors are all useful explanatory perspectives. I have chosen to focus on the last factor, contending that to ignore teachers — what they know and do as well as how they learn — dooms efforts to change the teaching of mathematics.
While widespread change in mathematics teaching is unlikely to occur by attempting only to change individuals, disciplinary-based mathematics teaching is likewise impossible without appropriately educated teachers.
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