10 DEVELOPING PROFICIENCY IN TEACHING MATHEMATICS
Knowledge of Mathematics
Because knowledge of the content to be taught is the cornerstone of teaching for proficiency, we begin with it. There is a substantial body of research on teachers’ mathematical knowledge, and teachers’ knowledge of mathematic is prominent in discussions of how to improve mathematics instruction. Improving teachers’ mathematical knowledge and their capacity to use it to do the work of teaching is crucial in developing students’ mathematical proficiency.
Many recent studies have revealed that U.S. elementary and middle school teachers possess a limited knowledge of mathematics, including the mathematics they teach. The mathematical education they received, both as K-12 students and in teacher preparation, has not provided them with appropriate or sufficient opportunities to learn mathematics. As a result of that education, teachers may know the facts and procedures that they teach but often have a relatively weak understanding of the conceptual basis for that knowledge.Many have difficulty clarifying mathematical ideas or solving problems that involve more than routine calculations.3 For example, virtually all teachers can multiply multidigit numbers, but several researchers have found that many prospective and practicing elementary school teachers cannot explain the basis for multidigit multiplication using place-value concepts and the underlying properties for adding and multiplying.4 In another study,5 teachers of fourth through sixth graders scored over 90% on items testing common decimal calculations, but fewer than half could find a number between 3.1 and 3.11.
Teachers frequently regard mathematics as a fixed body of facts and procedures that are learned by memorization, and that view carries over into their instruction. Many have little appreciation of the ways in which mathematical knowledge is generated or justified. Preservice teachers, for example, have repeatedly been shown to be quite willing to accept a series of instances as proving a mathematical generalization. Nowhere in their education have they had opportunities to study and experience the nature and role of justification in mathematics, a notion central to developing mathematical knowledge. Although teachers may understand the mathematics they teach in only a superficial way, simply taking more of the standard college mathematics courses does not appear to help matters. The evidence on this score has been consistent, although the reasons have not been adequately explored. For example, a study of prospective secondary mathematics teachers at three major institutions showed that, although they had completed the upper-division college mathematics courses required for the mathematics major, they had only a cursory understanding of the concepts underlying elementary mathematics. The mathematics of the elementary and middle school curriculum is not trivial, and the underlying concepts and structures are worthy of serious, sustained study by teachers. To develop prospective teachers’ understanding of the mathematics they will teach, careful attention must be given to identifying the mathematics that teachers need in order to teach effectively, articulating the ways in which they must use it in practice and what that implies for their opportunities to learn mathematics. This sort of attention to teachers’ mathematical knowledge and its central role in practice is crucial to ensure that their study of mathematics provides teachers with mathematical knowledge useful to teaching well.
Teachers’ mathematical knowledge and student achievement. Conventional wisdom asserts that student achievement must be related to teachers’ knowledge of their subject. That wisdom is contained in adages such as “You cannot teach what you don’t know.” For the better part of a century, researchers have attempted to find a positive relation between teacher content knowledge and student achievement. For the most part, the results have been disappointing: Most studies have failed to find a strong relationship between the two. Many studies, however, have relied on crude measures of these variables. The measure of teacher knowledge, for example, has often been the number of mathematics courses taken or other easily documented data from college transcripts. Such measures do not provide an accurate index of the specific mathematics that teachers know or of how they hold that knowledge. Teachers may have completed their courses successfully without achieving mathematical proficiency. Or they may have learned the mathematics but not know how to use it in their teaching to help students learn. They may have learned mathematics that is not well connected to what they teach or may not know how to connect it. Similarly, many of the measures of student achievement used in research on teacher knowledge have been standardized tests that focus primarily on students’ procedural skills. Some evidence suggests that there is a positive relationship between teachers’ mathematical knowledge and their students’ learning of advanced mathematical concepts. There seems to be no association, however, between how many advanced mathematics courses a teacher takes and how well that teacher’s students achieve overall in mathematics. In general, empirical evidence regarding the effects of teachers’ knowledge of mathematics content on student learning is still rather sparse.
In the National Longitudinal Study of Mathematical Abilities (NLSMA),conducted during the 1960s and still today the largest study of its kind, there was essentially no association between students’ achievement and the number of credits a teacher had in mathematics at the level of calculus or beyond. Commenting on the findings from NLSMA and a number of other studies of teacher knowledge, the director of NLSMA later said,
It is widely believed that the more a teacher knows about his subject matter, the more effective he will be as a teacher. The empirical literature suggests that this belief needs drastic modification and in fact suggests that once a teacher reaches a certain level of understanding of the subject matter, then further understanding contributes nothing to student achievement.
The notion that there is a threshold of necessary content knowledge for teaching is supported by the findings of another study in 1994 that used data from the Longitudinal Study of American Youth (LSAY).12 There was a notable increase in student performance for each additional mathematics course their teachers had taken, yet after the fifth course there was little additional benefit. Data from the 1996 NAEP on teachers’ college major rather than the number of courses they had taken provide a contrast to the general trend of this line of research. The NAEP data revealed that eighth graders taught by teachers who majored in mathematics outperformed those whose teachers majored in education or some other field. Fourth graders taught by teachers who majored in mathematics education or in education tended to outperform those whose teachers majored in a field other than education.
Although studies of teachers’ mathematical knowledge have not demonstrated a strong relationship between teachers’ mathematical knowledge and their students’ achievement, teachers’ knowledge is still likely a significant factor in students’ achievement. That crude measures of teacher knowledge, such as the number of mathematics courses taken, do not correlate positively with student performance data, supports the need to study more closely the nature of the mathematical knowledge needed to teach and to measure it more sensitively.
The persistent failure of the many efforts to show strong, definitive relations between teachers’ mathematical knowledge and their effectiveness does not imply that mathematical knowledge makes no difference in teaching. The research, however, does suggest that proposals to improve mathematics instruction by simply increasing the number of mathematics courses required of teachers are not likely to be successful. As we discuss in the sections that follow, courses that reflect a serious examination of the nature of the mathematics that teachers use in the practice of teaching do have some promise of improving student performance.
Teachers need to know mathematics in ways that enable them to help students learn. The specialized knowledge of mathematics that they need is different from the mathematical content contained in most college mathematics courses, which are principally designed for those whose professional uses of mathematics will be in mathematics, science, and other technical fields. Why does this difference matter in considering the mathematical education of teachers? First, the topics taught in upper-level mathematics courses are often remote from the core content of the K-12 curriculum. Although the abstract mathematical ideas are connected, of course, basic algebraic concepts or elementary geometry are not what prospective teachers study in a course in advanced calculus or linear algebra. Second, college mathematics courses do not provide students with opportunities to learn either multiple representations of mathematical ideas or the ways in which different representations relate to one another. Advanced courses do not emphasize the conceptual underpinnings of ideas needed by teachers whose uses of mathematics are to help others learn mathematics. Instead, the study of college mathematics involves the increasing compression of elementary ideas into the more and more powerful and abstract forms needed by those whose professional uses of mathematics will be in scientific domains. Third, advanced mathematical study entails using elementary concepts and procedures without much conscious attention to their meanings or implications, thus reinforcing the making of prior learning routine in the service of more advanced work. While this approach is important for the education of mathematicians and scientists, it is at odds with the kind of mathematical study needed by teachers.
Consider the proficiency teachers need with algorithms. The power of computational algorithms is that they allow learners to calculate without having to think deeply about the steps in the calculation or why the calculations work. That frees up the learners’ thinking so that they can concentrate on the problem they are trying to use the calculation to solve rather than having to worry about the details of the calculation. Over time, people tend to forget the reasons a procedure works or what is entailed in understanding or justifying a particular algorithm. Because the algorithm has become so automatic, it is difficult to step back and consider what is needed to explain it to someone who does not understand. Consequently, appreciating children’s difficulties in learning an algorithm can be very difficult for adults who are fluent with that algorithm.
The necessary compression of ideas in the course of mathematical study also shortchanges teachers’ mathematical needs. Most advanced mathematics classes engage students in taking ideas they have already learned and using them to construct increasingly powerful and abstract concepts and methods. Once theorems have been proved, they can be used to prove other theorems. It is not necessary to go back to foundational concepts to learn more advanced ideas. Teaching, however, entails reversing the direction followed in learning advanced mathematics. In helping students learn, teachers must take abstract ideas and unpack them in ways that make the basic underlying concepts visible.For example, most adults have lost sight of the fact that there are different interpretations of division. For adults, division is an operation on numbers. Division, however, is rooted in quite different physical situations, and distinctions among those situations are important for understanding children’s thinking, developing their understanding of the meaning of division, and helping them apply that understanding to solve problems. For example, although both of the following problems can be represented as dividing 24 by 6, young children think about them in very different ways and use quite different strategies to solve them:
Jane has 24 cookies. She wants to put 6 cookies on each plate. How many plates will she need?
Jeremy has 24 cookies. He wants to put all the cookies on 6 plates. If he puts the same number of cookies on each plate, how many cookies will he put on each plate?
These two problems correspond to the measurement and sharing models of division, respectively, that were discussed in chapter 3. Young children using counters solve the first problem by putting 24 counters in piles of 6 counters each. They solve the second by partitioning the 24 counters into 6 groups. In the first case the answer is the number of groups; in the second, it is the number in each group. Until the children are much older, they are not aware that, abstractly, the two solutions are equivalent. Teachers need to see that equivalence so that they can understand and anticipate the difficulties children may have with division.
To understand the sense that children are making of arithmetic problems, teachers must understand the distinctions children are making among those problems and how the distinctions might be reflected in how the children think about the problems. The different semantic contexts for each of the operations of arithmetic is not a common topic in college mathematics courses, yet it is essential for teachers to know those contexts and be able to use their knowledge in instruction. The division example illustrates a different way of thinking about the content of courses for teachers—a way that can make those courses more relevant to the teaching of school mathematics.
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