Teaching Mathematical Reasoning in Secondary School Classrooms (Mathematics Teacher Education)

Educational Sciences

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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. Teachers are unlikely to be able to provide an adequate explanation of concepts they do not understand, and they can hardly engage their students in productive conversations about multiple ways to solve a problem if they themselves can only solve it in a single way. Most of the investigations have been case studies, almost all involving fewer than 10 teachers, and most only one to three teachers.

In general, the researchers found that teachers. Not surprisingly, these teachers gave the students little assistance in developing an understanding of what they were doing. Some of the same studies contrasted the teaching practices of teachers with low levels of mathematical knowledge with the teaching practices of teachers who had a better command of mathematics. The teacher also needs to be sensitive to the unique ways of learning, thinking about, and doing mathematics that the student has developed. Each student can be seen as located on a path through school mathematics, equipped with strengths and weaknesses, having developed his or her own approaches to mathematical tasks, and capable of contributing to and profiting from each lesson in a distinctive way.

Teachers also need a general knowledge of how students think—the approaches that are typical for students of a given age and background, their common conceptions and misconceptions, and the likely sources of those ideas. We have described some of those progressions in chapters 6 through 8.

Using that body of evidence, researchers have also. From the many examples of misconceptions to which teachers need to be sensitive, we have chosen one: Children can develop this impression because that is how the notation is often described in the elementary school curriculum and most of their practice exercises fit that pattern. Knowing classroom practice means knowing what is to be taught and how to plan, conduct, and assess effective lessons on that mathematical content.

We have discussed these matters in chapter 9. In the sections that follow, we consider how to develop an integrated corpus of knowledge of the types discussed in this section. First, however, we need to clarify our stance on the relation between knowledge and practice. We have discussed the kinds of knowledge teachers need if they are to teach for mathematical proficiency.

Although we have used the term knowledge throughout, we do not mean it exclusively in the sense of knowing about. Teachers must also know how to use their knowledge in practice. Effective programs of teacher preparation and professional development cannot stop at simply engaging teachers in acquiring knowledge; they must challenge teachers to develop, apply, and analyze that knowledge in the context of their own classrooms so that knowledge and practice are integrated.

In chapter 4 we identified five components or strands of mathematical proficiency. Teaching is a complex activity and, like other complex activities, can be conceived in terms of similar components. Just as mathematical proficiency itself involves interwoven strands, teaching for mathematical proficiency requires similarly interrelated components. In the context of teaching, proficiency requires:. Like the strands of mathematical proficiency, these components of mathematical teaching proficiency are interrelated. In this chapter we discuss the problems entailed in developing a proficient command of teaching.

In the previous section we discussed issues relative to the knowledge base needed to develop proficiency across all components. Now we turn to specific issues that arise in the context of the components. It is not sufficient that teachers possess the kinds of core knowledge delineated in the previous section. One of the defining features of conceptual understanding is that knowledge must be connected so that it can be used intelligently.

Teachers need to make connections within and among their knowledge of mathematics, students, and pedagogy. The implications for teacher preparation and professional development are that teachers need to acquire these forms of knowledge in ways that forge connections between them. For teachers who have already achieved some mathematical proficiency, separate courses or professional development programs that focus exclusively on mathematics, on the psychology of learning, or on methods of teaching provide limited opportunities to make these connections.

Unfortunately, most university teacher preparation programs offer separate courses in mathematics, psychology, and methods of teaching that are taught in different departments. The difficulty of integrating such courses is compounded when they are located in different administrative units. It is not enough, however, for mathematical knowledge and knowledge of students to be connected; both need to be connected to classroom practice.

Teachers may know mathematics, and they may know their students and how they learn. But they also have to know how to use both kinds of knowledge effectively in the context of their work if they are to help their students develop mathematical proficiency. Similarly, many inservice workshops, presentations at professional meetings, publications for teachers, and other opportunities for teacher learning focus almost exclusively on activities or methods of teaching and seldom attempt to help teachers develop their own conceptual understanding of the underlying mathematical ideas, what students understand about those ideas, or how they learn them.

Alternative forms of teacher education and professional development that attempt to teach mathematical content, psychology. The second basic component of teaching proficiency is the development of instructional routines. Just as students who have acquired procedural fluency can perform calculations with numbers efficiently, accurately, and flexibly with minimal effort, teachers who have acquired a repertoire of instructional routines can readily draw upon them as they interact with students in teaching mathematics.

Some routines concern classroom management, such as how to get the class started each day and procedures for correcting and collecting homework. Other routines are more grounded in mathematical activity. For example, teachers need to know how to respond to a student who gives an answer the teacher does not understand or who demonstrates a serious misconception.

Teachers need businesslike ways of dealing with situations like these that occur on a regular basis so that they can devote more of their attention to the more serious issues facing them. When teachers have several ways of approaching teaching problems, they can try a different approach if one does not work.

Researchers have shown that expert teachers have a large repertoire of routines at their disposal. Novice teachers, in contrast, have a limited range of routines and often cannot respond appropriately to situations. Expert teachers not only have access to a range of routines, they also can apply them flexibly, know when they are appropriate, and can adapt them to fit different situations. The third component of teaching proficiency is strategic competence. Although teachers need a range of routines, teaching is very much a problem-solving activity.

These are problems that every teacher faces every day, and most do not have readymade solutions. Conceptual understanding of the knowledge required to teach for proficiency can help equip teachers to deal intelligently with these problems. It is misleading to claim that teachers actually solve such problems in the sense of solving a mathematical problem.

There is never an ideal solution to the more difficult problems of teaching, but teachers can learn to contend with these problems in reasonable ways that take into account the mathematics that students are to learn; what their students understand and how they may best learn it; and representations, activities, and teaching practices that have proven most effective in teaching the mathematics in question or that have been effective in teaching related topics.

Teacher education and professional development programs that take into account the strategic decision making in teaching can help prepare teachers to be more effective in solving instructional problems. Teachers can learn to recognize that teaching involves solving problems and that they can address these problems in reasonable and intelligent ways.

The fourth component of teaching proficiency is adaptive reasoning. Teachers can learn from their teaching by analyzing it: Many successful programs of teacher education and professional development engage teachers in reflection, but the reflection, or perhaps more appropriately the analysis, is grounded in specific examples. In those programs, teachers engage in analyses in which they are asked to provide evidence to justify claims and assertions.

As with other complex activities, teacher learning can be enhanced by making more visible the goals, assumptions, and decisions involved in the practice of. Teachers are often asked to pose a particular mathematical problem to their classes and to discuss the mathematical thinking that they observe.

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Just as students must develop a productive disposition toward mathematics such that they believe that mathematics makes sense and that they can figure it out, so too must teachers develop a similar productive disposition. Teachers whose learning becomes generative perceive themselves as in control of their own learning. The teachers become more comfortable with mathematical ideas and ripe for a more systematic view of the subject. Teachers whose learning becomes generative see themselves as lifelong learners who can learn from studying curriculum materials 35 and from analyzing their practice and their interactions with students.

Programs of teacher education and professional development that portray to the participants that they are in control of their own learning help teachers develop a productive dispo-. Programs that provide readymade, worked-out solutions to teaching problems should not expect that teachers will see themselves as in control of their own learning.

In a teacher preparation program, teachers clearly cannot learn all they need to know about the mathematics they will teach, how students learn that mathematics, and how to teach it effectively. Consequently, some authorities have recommended that teacher education be seen as a professional continuum, a career-long process. They need to be able to adapt to new curriculum frameworks, new materials, advances in technology, and advances in research on student thinking and teaching practice.

They have to learn how to learn, whether they are learning about mathematics, students, or teaching. Teachers can continue to learn by participating in various forms of professional development. But formal professional development programs represent only one source for continued learning. We consider below examples of four such program types that represent an array of alternative approaches to developing integrated proficiency in teaching mathematics. For example, prospective elementary school teachers may take a mathematics course that focuses, in part, on rational numbers or proportionality rather than the usual college algebra or calculus.

Such courses are offered in many universities, but they are seldom linked to instructional practice. The lesson depicted in Box 10—1 comes from a course in which connections to practice are being made. The prospective teachers stare at the board, trying to figure out what the instructor is asking them to do. After calculating the answer to a simple problem in the division of fractions and recalling the old algorithm—invert and multiply—most of them have come up with the answer, It is familiar content, and although they have not had occasion to divide fractions recently, they feel comfortable, remembering their own experiences in school mathematics and what they learned.

But now, what are they being asked? The instructor has challenged them to consider why they are getting what seems to be an answer that is larger than either of the numbers in the original problem and. Confused, they are suddenly stuck. None of them noticed this fact before. The instructor proposes a new task: Can you come up with an example or a model that shows what is going on with dividing one and three fourths by one half? The prospective teachers set to work, some in pairs, some alone. The instructor walks around, watching them work, and occasionally asking a question.

Most have drawn pictures like those below:. I have two pizzas. My little brother eats one quarter of one of them and then I have one and three quarters pizzas left.

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My sister is very hungry, so we decide to split the remaining pizza between us. We each get pieces of pizza. I have cups of sugar. Each batch of sesame crackers takes cup of sugar. How many batches of crackers can I make? And another pair has envisioned filling -liter containers, starting with liters of water.

After about 10 minutes, the instructor invites students to share their problems with the rest of the class. One student presents the pizza situation above. Most students nod appreciatively. When a second student offers the sesame cracker problem, most nod again, not noticing the difference.

The instructor poses a question: How does each problem we heard connect with the original computation? Are these two problems similar or different, and does it matter? Through discussion the students gradually come to recognize that, in the pizza problem, the pizza has been divided in half and that the answer is in terms of fourths —that is, that the pieces are fourths of pizzas. In the case of the sesame cracker problem, the answer of batches is in terms of half cups of sugar.

In the first instance, they have represented division in half, which is actually division by two; in the second they have represented division by one half. The instructor moves into a discussion of different interpretations of division: After the students observe that the successful problems— involving the sesame crackers and the liters of water—are measurement problems, she asks them to try to develop a problem situation for that represents a sharing division.

In other words, could they make a sensible problem in which the is not the unit by which the whole is being measured, but instead is the number of units into which the whole has been divided? For homework, the instructor asks the students to try making representations for several other division situations, which she chooses strategically, and finally asks them to select two numbers to divide that they think are particularly good choices and to say why.

In this excerpt from a university mathematics course, the prospective teachers are being asked to unpack familiar arithmetic content, to make explicit the ideas underlying the procedures they remember and can perform. Repeatedly throughout the course, the instructor poses problems that have been strategically designed to expose concepts on which familiar procedures rest. A second principle is to link that work with larger mathematical ideas and structures. For example, the lesson on the division of fractions is part of a larger agenda that includes understanding division, its relationship to fractions and to multiplication, and the meaning and representation of operations.

Moreover, throughout the development of these ideas and connections, the prospective teachers work with whole and rational numbers, considering how the mathematical world looks inside these nested systems. The overriding purpose of a course like this is to provide prospective teachers with ample opportunities to learn fundamental ideas of school mathematics, how they are related, and how students come to learn them.

But the course is not about how to teach, nor about how children learn. It is explicitly and deliberately a sustained opportunity for prospective teachers to learn mathematical ideas in ways that will equip them with mathematical resources needed in teaching. Teachers do not learn abstract concepts about mathematics and children. In the programs, teachers look at problem-solving strategies of real students, artifacts of student work, cases of real classrooms, and the like.

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Furthermore, the teachers in these programs are challenged to relate what they learn to their own students and their own instructional practices. They learn about mathematics and students both in workshops and by interacting with their own students. Specific opportunity is provided for the teachers to discuss with one another how the ideas they are encountering influence their practice and how their practice influences what they are learning.

The workshop described in Box 10—2 forms part of a professional development program designed to help teachers develop a deeper understanding of some critical mathematical ideas, including the equality sign.

The program, modeled after Cognitively Guided Instruction CGI , which has proven to be a highly effective approach, 41 assists teachers in understanding how to help their students reason about number operations and relations in ways that enhance the learning of arithmetic and promote a smoother transition from arithmetic to algebra. Several features of this example of professional development are worth noting. Although they begin by considering how children think, the teachers. Before attending the workshop, participating teachers ask their students to find the number that they could put in the box to make the following open-number sentence a true number sentence: At the workshop, the teachers share their findings with the other participants.

These findings, which surprised most teachers, have led them to begin to listen to their students, and a number of teachers have engaged their students in a discussion of the reasons for their responses. The discussion generates insights about how children are thinking and what teachers can learn by listening to their students. The task is to decide whether the sentence is true or false. Sometimes the decision requires calculation e.

The teachers work in small groups to construct true and false number sentences they might use to elicit various views of equality. Using these sentences, their students could engage in explorations that might lead to understanding equality as a relation. The sentences could also provide opportunities for discussions about how to resolve disagreement and develop a mathematical argument.

The teachers work together to consider how their students might respond to different number sentences and which number sentences might produce the most fruitful discussion. Falkner, Levi, and Carpenter, Used by permission of the authors. The teachers also begin to ponder how notation is used and how ideas are justified in mathematics. A central feature of their discussion is that math-. The mathematical ideas and how children think about them are seen in classroom interactions.

The programs do not deal with general theories of learning. For example, to understand the different strategies that children use to solve different problems, teachers must understand the semantic differences between problems represented by the same operation, as illustrated by the sharing and measurement examples of dividing cookies described above in Box 10—1. Gains in student achievement generally have been in the areas of understanding and problem solving, but none of the programs has led to a decline in computational skills, despite their greater emphasis on higher levels of thinking.

Case examples are yet another way to build the connections between knowledge of mathematics, knowledge of students, and knowledge of practice. Although the cases focus on classroom episodes, the discussions the teachers engage in as they reflect on the cases emphasize mathematics content and student thinking.

The cases involve instruction in specific mathematical topics, and teachers analyze the cases in terms of the mathematics content being taught and the mathematical thinking reflected in the work the children produce and the interactions they engage in. Cases can be presented in writing or using multiple media such as videotapes and transcriptions of lessons.

These teachers are probing the concept of functions from several overlapping perspectives. They dig into the mathematics through close work on and analysis of the task that the teacher posed. And they revisit the mathematical ideas by looking carefully at how the teacher deals with the mathematics during the lesson.

A dozen teachers are gathered around a table. They have read a case of a teacher teaching a lesson on functions. The written case includes the task the teacher used and a detailed narrative account of what happened in the class as students worked on the problem. The teacher used the following task:.

Sara has made several purchases from a mail-order company. Sara decides that the company must be using a simple rule to determine how much to charge for shipping. Help her figure out how much it would most likely cost to ship a 1-kg package and how much each additional kilogram would cost. Before the teachers studied the case and the accompanying materials, they solved the mathematical problem themselves.

To begin the discussion, the workshop leader asks the teachers to look closely at one segment of the lesson in which two students are presenting solutions to the problem. She asks them to interpret what each student did and to compare the two solutions. The teachers launch into a discussion of the mathematics for several minutes. They note that if the given values weight, cost are graphed, the points lie on the same straight line. Reading the graph provides a solution. Also, by asking how much each additional kilogram would cost, the problem suggests there is a constant difference that can be used in solving it.

She asks them to analyze the text closely and try to categorize what the teacher is doing. This discussion yields surprises for most of the teachers. Suddenly the intricate work that the teacher is doing becomes visible. They begin to describe and name the different moves she makes. One teacher becomes intrigued with how the teacher helps students express their ideas by asking questions to support their explanations before she asks other students to comment. It is quite clear that this is no generic skill, for the mathematical sensitivity and knowledge entailed are quite visible throughout.

The teachers become fascinated with what looks like an important missed opportunity to unpack a common misconception about function. Speculating about why that happened leads them to a productive conversation about what one might do to seize and capitalize on the opportunity. The session ends with the teachers agreeing to bring back one mathematical task from their own work on functions and compare it with the task used in the case. Several are overheard to be discussing features of this problem that seem particularly fruitful and that have them thinking about how they frame problems for their students.

The group briefly discusses some ways to vary the problem to make it either simpler or more complex. The leader then closes by summarizing some of the mathematical issues embedded in the task. She points out that it is not obvious what the value of 2. It is the cost of sending a package of zero weight, an idea that does not appear anywhere in the problem itself or in real life. She also says that it is important to understand that x refers to whole numbers only.

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Finally, she notes that with a different function, the differences might not be constant. The assumption of constant differences is one suggested by the problem and common in situations like those involving shipping costs, but it is not necessarily always warranted. They gained a greater repertoire of ways to represent mathematical ideas, were able to articulate connections among mathematical ideas, and developed a deeper understanding of mathematical structures.

At least some of the teachers continued the process of learning mathematics by examining the mathematical work of their own students in their own classrooms. The case-based programs that focus on classroom instruction treat the cases as problematic situations that serve as a basis for discussion and inquiry rather than as models of instruction for the teachers to emulate.

A somewhat different approach to professional development is represented by so-called lesson study groups, which are used in Japan see Box 10—4. They design a lesson, and one member of the group teaches it while the others watch. Afterwards they discuss what happened in light of their anticipations and goals. Based on this experience, the group revises the lesson and someone else teaches it. The cycle continues of trying the lesson, discussing and analyzing how it worked, and revising it. Their knowledge becomes a basis for further learning through the study of a lesson. Lesson study groups might follow somewhat different formats and schedules than the one described above, but most meet regularly during the year and focus on improving a very few lessons with clear learning goals.

Box 10—4 The Japanese Lesson Study.

Small groups of teachers form within the school around areas of common teaching interests or responsibilities e. Each group begins by formulating a goal for the year. Sometimes the goal is adapted from national-level recommendations e.

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The more specific goal might focus on a curriculum topic that has been problematic for students in their classrooms. A few lessons then are identified that ordinarily deal with that topic, and the group begins its yearlong task to improve those lessons. Lesson study groups meet regularly, often once a week after school e. Some groups divide their work into three major phases, each taking about one third of the school year.

During the first phase, teachers do research on the topic, reading and sharing relevant research reports and collecting information from other teachers on effective approaches for teaching the topic. During the second phase, teachers design the targeted lessons often just one, two, or perhaps three lessons. During the third phase, the lessons are tested and refined.

The first test often involves one of the group members teaching a lesson to his or her class while the other group members observe and take notes. After the group refines the lesson, it might be tested with another class in front of all the teachers in the school. In this case, a follow-up session is scheduled, and the lesson study group engages their colleagues in a discussion about the lesson, receiving feedback about its effectiveness. Working directly on improving teaching is their means of becoming better teachers. Learning in ways that continue to be generative over time is best done in a community of fellow practitioners and learners, as illustrated by the Japanese lesson study groups.

Studies of school reform efforts suggest that professional development is most effective when it extends beyond the individual teacher. Professional development can create contexts for teacher collaboration, provide a focus for the collaboration, and provide a common frame for interacting with other teachers around common problems. When teachers have opportunities to continue to participate in communities of practice that support their inquiry, instructional practices that foster the development of mathematical proficiency can more easily be sustained.

The focus of teacher groups matters for what teachers learn from their interactions with others. Because of the specialized knowledge required to teach mathematics, there has been increased discussion recently of the use of mathematics specialists, particularly in the upper elementary and middle school grades. Implicit in the recommendations for mathematics specialists is the notion of the mathematics specialist in a departmental arrangement.

In such arrangements, teachers with a strong background in mathematics teach mathematics and sometimes another subject, depending on the student population, while other teachers in the building teach other subject areas. Departmentalization is most often found in the upper elementary grades 4 to 6. Other models of mathematics specialists are used, particularly in elementary schools, which rarely are departmentalized. Rather than a specialist for all mathematics instruction, a single school-level mathematics specialist is sometimes used.

This person, who has a deep knowledge of mathematics and how students learn it, acts as a resource for other teachers in the school. The specialist may consult with other teachers about specific issues, teach demonstration lessons, observe and offer suggestions, or provide special training sessions during the year. School-level mathematics specialists can also take the lead in establishing communities of practice, as discussed in the previous section.

Because many districts do not have enough teachers with strong backgrounds in mathematics to provide at least one specialist in every school, districts instead identify district-level mathematics coaches who are responsible for several schools. Whereas a school-level specialist usually has a regular or reduced teaching assignment, district-level specialists often have no classroom teaching assignment during their tenure as a district coach.

The constraint on all of the models for mathematics specialists is the limited number of teachers, especially at the elementary level, with strong backgrounds in mathematics. For this reason, summer leadership training programs have been used to develop mathematics specialists.

Perhaps the central goal of all the teacher preparation and professional development programs is in helping teachers understand the mathematics they teach, how their students learn that mathematics, and how to facilitate that learning. Many of the innovative programs described in this chapter make serious efforts to help teachers connect these strands of knowledge so that they can be applied in practice.

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Teachers are expected to explain and justify their ideas and conclusions. They have opportunities to develop a productive disposition toward their own learning about teaching that contrib-. Teachers are not given readymade solutions to teaching problems or prescriptions for practice. Instead, they adapt what they are learning and engage in problem solving to deal with the situations that arise when they attempt to use what they learn. Professional development beyond initial preparation is critical for developing proficiency in teaching mathematics.

However, such professional development requires the marshalling of substantial resources. One of the critical resources is time. If teachers are going engage in inquiry, they need repeated opportunities to try out ideas and approaches with their students and continuing opportunities to discuss their experiences with specialists in mathematics, staff developers, and other teachers. These opportunities should not be limited to a period of a few weeks or months; instead, they should be part of the ongoing culture of professional practice. Through inquiry into teaching, teacher learning can become generative, and teachers can continue to learn and grow as professionals.

Student achievement data were based on items developed for NAEP. In fact, it appears that sometimes content knowledge by itself may be detrimental to good teaching. In one study, more knowledgeable teachers sometimes overestimated the accessibility of symbol-based representations and procedures Nathan and Koedinger, These programs share the idea that professional development should be based upon the mathematical work of teaching.

For more examples, see National Research Council, A comprehensive guide for designing professional development programs can be found in Loucks-Horsley, Hewson, Love, Stiles, No instructional materials or specifications for practice are provided in CGI; teachers develop their own instructional materials and practices from watching and listening to their students solve problems. A major thesis of CGI is that children bring to school informal or intuitive knowledge of mathematics that can serve as the basis for developing much of the formal mathematics of the primary school mathematics curriculum.

The subject matter preparation of prospective mathematics teachers: Challenging the myths Research Report 88—3. The mathematical understandings that prospective teachers bring to teacher education. Elementary School Journal , 90 , — Educational studies in mathematics 85 2 , , Teaching and Teacher Education 27 1 , , Journal of Mathematics Teacher Education 14 6 , , Challenges of teacher development: Researching mathematics education in South Africa, , For the learning of Mathematics 27 1 , , International Group for the Psychology of Mathematics Education 2, , Articles 1—20 Show more.

Forms and substance in learner-centred teaching: The power of professional learning communities K Brodie Education as change 17 1 , , Learning about learner errors in professional learning communities K Brodie Educational studies in mathematics 85 2 , , Towards a language of description for changing pedagogy K Brodie Teaching and Teacher Education 27 1 , , Dialogue in mathematics classrooms: Beyond question-and-answer methods K Brodie Pythagoras 66 , , Re-thinking teachers' mathematical knowledge: A focus on thinking practices K Brodie Perspectives in Education 22 1 , , Learning mathematics in a second language K Brodie Educational review 41 1 , ,

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