6
Preparing Mathematics Teachers
The preparation of mathematics teachers in the United States has been a topic of increasingly impassioned discussion in the last 20 years. There is deep concern about the numeracy of the nation’s high school graduates, as well as concern about perceived shortages of highly qualified mathematics teachers. The organization of U.S. schooling as three stages—elementary (kindergarten through grade 4 or 5), middle (grade 5 or 6 through 8), and secondary or high (grades 9 through 12)—presents particular challenges for mathematics education. Mathematics learning at each of these levels has a distinct character, reflecting the developmental and educational needs of different age groups. For teachers, this structure has meant that different sorts of preparation are required to teach each level. Most elementary teachers are prepared to teach all subjects, while teachers at the secondary level are prepared as specialists in a particular content area. Preparation for middle grades mathematics teachers varies from place to place, and certification requirements reflect the ambiguous status of middle school. For example, many states offer grade K8 certification to teachers prepared as generalists, as well as grade 712 certification to those specifically prepared to teach mathematics.
Though the preparation of elementary, middle, and secondary level teachers may differ, expectations for all mathematics teachers have increased steadily and dramatically over the last few decades. In particular, schools now try to teach more mathematics earlier than was the case even a decade ago. The most visible evidence of this change has been the push to encourage all high school students to take both 2 years of algebra and
1 year of geometry. Many districts and even some states have made it a goal that all students take algebra I by the 8th grade.
U.S. students are not yet, as a group, meeting the higher expectations of recent years. Trends in student achievement in mathematics, as measured by the National Assessment of Educational Progress (NAEP), have shown considerable improvement since 1990, but the 2009 results showed that just 39 percent of 4th graders and 34 percent of 8th graders are performing at or above the proficient level (National Center for Education Statistics, 2009). In the mathematics portion of the Third International Mathematics and Science Study (TIMSS), U.S. 4th and 8th graders scored above the median, but the nation was not among the topperforming nations (Gonzales et al., 2008). A 1998 comparison of the performance of older students showed that U.S. students were among the lowest performing group of the 21 nations in the study (National Center for Education Statistics, 1998).
At the same time, considerable evidence indicates that many teachers, especially in grades K8, are not well prepared to teach challenging mathematics. The time allotted for mathematics content in the preparation of many elementary and middle school teachers is unlikely to be adequate, and many secondary school mathematics teachers (including those in the middle grades who are prepared as specialists) may also be receiving training that does not prepare them to teach advancedlevel mathematics (e.g., algebra, geometry, and trigonometry). Mathematics teachers may also need specific preparation for the challenge of teaching mathematics in ways that engage all students and gives them a chance to succeed. Moreover, many of those who teach mathematics in U.S. secondary schools, especially in poor and underserved communities, lack appropriate certification and adequate content preparation. These concerns have been evident for a long time, and their persistence underscores the importance of assessing the status of the preparation of mathematics teachers.^{1}
This chapter is organized as was the preceding one, beginning with a brief overview of the research base and then turning to our four key questions:

What do successful students know about mathematics?

What instructional opportunities are necessary to support successful students?

What do successful teachers know about mathematics and how to teach it?

What instructional opportunities are necessary to prepare successful teachers?
We continue with what is known about how mathematics teachers are currently prepared, and we end the chapter with our conclusions.
THE RESEARCH BASE
The literature on which we could rely for this chapter was an amalgam of empirical research and other kinds of work. The community of mathematics educators and mathematicians has synthesized the intellectual principles of mathematics with insights from other fields (e.g., cognitive and developmental psychology), the practicebased wisdom of classroom teachers, and the available empirical research to develop guidelines for mathematics teaching and learning. The National Council of Teachers of Mathematics (NCTM), the National Research Council (NRC), the Conference Board of the Mathematical Sciences, and, most recently, the National Mathematics Advisory Panel have published some of the most widely known documents. Each of those documents is the product of extensive efforts to collaborate, develop consensus, and distill practical guidance from theoretical models as well as research, and we have relied heavily on them. The influence of the research in learning and cognition that we discuss in Chapter 4 is evident in the reports of those groups and the field of mathematics education generally. State standards and curricula have also provided outlines of the content and skills students are expected to master. Thus, we had an array of resources on which to draw, although the empirical base is less direct than that for reading. We have attempted to describe the research base on which the points we highlight rests.
QUESTION 1:
WHAT DO SUCCESSFUL STUDENTS KNOW ABOUT MATHEMATICS?
Looking first at what students ought to learn, we found numerous sources. As part of the standards movement that began in the 1980s, states developed mathematics standards, and, along with professional societies and other interest groups, have used a variety of approaches to arrive at descriptions of the fundamental mathematical skills and knowledge that the states believe students should be taught. Although these descriptions might seem similar to most people, they reflect important differences among mathematics educators, differences that have at times been contentious. Indeed, the phrase “math wars” has been used to describe the debate over what mathematics should be taught to K12 students and how it should be taught. In particular, much debate has centered on the relative emphasis
that should be given to the mastery of basic skills and the development of conceptual understanding, though most state and other standards documents now acknowledge the importance of both.
In 1989, the NCTM became the first professional society to respond to the call for subjectmatter standards when they released Curriculum and Evaluation Standards for School Mathematics. These standards were developed through a multiyear consensus process led by committees of NCTM members. This document was followed by companion documents on teaching and assessment in mathematics, and the NCTM standards had a strong influence on the standards adopted by many states. A series of more recent publications have also been important.
Principles and Standards for School Mathematics (National Council of Teachers of Mathematics, 2000), which offered an update of the group’s earlier documents and was based on a more detailed review of the theory, research, and practice literature, has been particularly influential (also see Kilpatrick, Martin, and Schifter, 2003). Principles offers five content standards (number and operations, algebra, geometry, measurement, and data analysis and probability) and five process standards (problem solving, reasoning and proof, communication, connections, and representation). In this volume, the NCTM discusses its vision for achieving the 10 standards in four grade bands (preK through grade 2, grades 35, grades 68, and grades 912) and identifies six principles for school mathematics (equity, curriculum, teaching, learning, assessment, and technology). The high visibility of the NCTM standards is evident in the fact that nearly 85 percent of U.S. teachers surveyed as part of the Third International Mathematics and Science Study reported that they were familiar with them (though there is no hard evidence about whether the standards have changed teachers’ practice or even whether teachers have read them) (National Center for Education Statistics, 2003b).
In 2006, the NCTM released Curriculum Focal Points for PreKindergarten Through Grade 8 Mathematics: A Quest for Coherence. This document provides explicit guidance as to the most important mathematics topics that should be taught at each grade level, identifying the “ideas, concepts, skills, and procedures that form the foundation for understanding and using mathematics” (see http://www.nctm.org/standards/content.aspx?id=270 [November 2009]). The document is designed to guide states and school districts as they revise their standards, curricula, and assessment programs. As this report is being prepared, the NCTM has another task force at work on a companion document addressing high school mathematics. The NCTM documents stress that their standards are for all students, regardless of their interests or career aspirations. In general, the NCTM documents reflect an effort to achieve consensus among math
ematics teachers, mathematics educators, mathematicians, and education researchers, and they drew on the available research.
Another report, Adding it Up (National Research Council, 2001a), has also synthesized the available literature on mathematics learning. It used the topic of number as taught in grades preK through 8 as a focus in addressing the question of what constitutes mathematical proficiency. The report was based on a review of empirical research that met the committee’s standards for relevance, soundness, and generalizability, as well as other literature. The report notes that choices about the mathematics children should be taught are both reflections of “what society wants educated adults to know” and “value judgments based on previous experience and convictions [which] fall outside the domain of research” (p. 21). The report describes mathematical proficiency as having five intertwined strands: conceptual understanding, procedural fluency, strategic competence, adaptive reasoning, and productive disposition.
Efforts to address high school preparation include the American Diploma Project sponsored by Achieve, Inc., which produced benchmarks for college readiness (see http://www.achieve.org/ [February 2010]), the College Board Standards for Success, and the Common Core Project sponsored by the National Governors Association and Council of Chief State School Officers, now under way.
In Foundations for Success, the National Mathematics Advisory Panel (2008) synthesized empirical research related to students’ readiness to succeed in algebra. The panel focused on studies that used a randomized control design or statistical procedures to compensate for deviations from that model. However, because there were not enough studies of that type to address all of the panel’s questions, other research was considered as well. We note that this was a different criterion than was used by the developers of the NCTM standards or the National Research Council panel, and these differences have contributed to differences among the various reports. Some have criticized Foundations for Success for relying on a base that was excessively thin (excluding descriptive studies, for example) and thus excluding valuable findings. Others have supported the panel’s strict definition of research utility.^{2} The panel addressed many aspects of mathematics education, and among its findings and recommendations are several regarding what students should learn. The panel focused on what was needed for students to be successful in learning algebra (National Mathematics Advisory Panel, 2008, p. xix):
To prepare students for Algebra, the curriculum must simultaneously develop conceptual understanding, computational fluency, and problem
solving skills. Debates regarding the relative importance of these aspects of mathematical knowledge are misguided. These capabilities are mutually supportive, each facilitating learning of the other.
To identify the essential concepts and skills that prepare students for algebra coursework, the panel drew on a range of sources, including: the curricula for grades 1 through 8 from the countries that performed best on TIMSS, Focal Points (National Council of Teachers of Mathematics, 2000), the six highestrated state curriculum frameworks in mathematics, a 2007 survey of ACT, Inc., and a survey of algebra I teachers. The panel also reviewed state standards for algebra I and II, current school algebra and integrated mathematics textbooks, the algebra objectives in the 2005 grade 12 assessment of the NAEP, the American Diploma Project standards from Achieve, Inc., and Singapore’s algebra standards. Based on all the information gathered and professional judgment, the panel identified what they called the critical foundation of algebra and the major topics of school algebra. The panel also stressed the importance of coherence across the curriculum and the establishment of logical priorities for each year of study.
All of these documents—Principles and Standards for School Mathematics, Curriculum Focal Points, Adding It Up, and Foundations for Success—attempt to answer questions about what successful mathematics students know. But none of the documents could rely on empirical research that demonstrates that students who have mastered these domains of mathematical knowledge and skill are more productive or successful at their schoolwork or in life. As in any subject, standards for mathematics are collective decisions about which learning goals should take priority over others, not conclusions based on empirical evidence (though evidence of various sorts may influence the standards). Thus, the various descriptions of the mathematics that should be taught in K12 differ in both their perspective and the degree of detail they provide.
More important, the documents reflect important shifts in the thinking of mathematics education leaders about what students need to learn. Most notably, they show an increasing tendency to provide guidance that is both focused and concrete. They also reflect a growing consensus about the most important aspects of student learning in mathematics, which is based in part on the research on learning and thinking (see Chapter 4). For example, they reflect research that has identified the importance of learning with understanding, as opposed to memorizing isolated facts, and the importance of opportunities to engage in mathematical reasoning and problem solving. Both ideas have important implications for mathematics education, which we discuss below.
Every state has its own standards for mathematics, and there are sig
nificant differences among them (Raimi and Braden, 1998; Klein, 2005; National Research Council, 2008).^{3} Although a detailed content analysis of the similarities and differences among these standards was beyond the scope of this committee’s work, it is clear from our review that a number of important themes are consistently identified as important. Although these themes are not exclusively based on empirical research, and they have evolved through a certain amount of struggle and disagreement, a reasonable consensus has been achieved on the question of what mathematics students should be taught in grades K8.
Unfortunately, the picture is much less clear for grades 9 through 12 because the base of research on student learning related to secondary school mathematics topics and courses is relatively thin. Thus, the field tends to rely more heavily on professional judgment when deciding on curriculum for high school students. For example, views differ about the place of calculus and statistics. Yet there is consensus that for students to be successful in high school mathematics courses, they need preparation in the basic topic areas—including number, operations, and fractions—and there is little disagreement that students also need to develop conceptual understanding, procedural fluency, and confidence in their capacity to learn mathematics.
In general, successful mathematics learning entails the cumulative development of increasingly sophisticated conceptual understanding, procedural fluency, and capacity for reasoning and problem solving.^{4} Moreover, there is broad general agreement about the topics to be included in the curriculum for grades K8, though the relative emphasis they should receive and their exact placement by grade is not settled.
QUESTION 2:
WHAT INSTRUCTIONAL OPPORTUNITIES ARE NECESSARY TO SUPPORT SUCCESSFUL MATHEMATICS STUDENTS?
Turning to the question of what sorts of instructional opportunities enable students to learn mathematics effectively, we found that useful guidance comes from the research on how students learn. How Students Learn: History, Mathematics, and Science in the Classroom (National Research Council, 2005) summarizes the major findings from research on learning and cognition as they pertain to K12 teaching and learning. This report builds on previous NRC syntheses of research on learning, particularly
How People Learn (National Research Council, 2000a) (discussed in Chapter 4), and both have been influential in mathematics education.
How Students Learn begins by applying three broad principles about learning to the teaching of mathematics: that engaging students’ prior conceptions is critical to successful learning, that learning with understanding entails an integration of factual knowledge and conceptual frameworks, and that students need to learn how to monitor their learning. Thus, the authors assert that mathematics instruction should

build on and refine the mathematical understandings, intuitions, and resourcefulness that students bring to the classroom;

organize the skills and competencies required to do mathematics fluently around a set of core mathematical concepts; and

help students use metacognitive strategies when solving mathematics problems.
Addressing students’ existing ideas is important for two reasons. First, lingering misconceptions about mathematical concepts may interfere with learning. Second, students who believe, for example, that some people have the ability to “do math” and some do not, that mathematics is exclusively a matter of learning and following rules in order to obtain a correct answer, or that mathematics is exclusively a matter of reasoning (and does not also require considerable mastery of factual knowledge) are much more likely to struggle with mathematics, and perhaps give up on it. Thus, it is important that all students participate in activities that make their informal or naïve mathematical ideas and reasoning explicit so that they might examine—along with their teachers—which aspects of their thinking are valid and which are not. More generally, instruction designed to help students bridge gaps between naïve conceptions and the mathematical understanding they need to develop is important.
With regard to organizing skills around mathematical concepts, How Students Learn stresses that, in order to succeed as mathematics becomes more complex through the school years, students develop “learning paths from more informal concrete methods to abbreviated, more general, and more abstract methods” (National Research Council, 2005, p. 232). Though mastering mathematical procedures is very important, instruction that emphasizes them at the expense of developing conceptual frameworks leaves students ill equipped for algebra and higherlevel mathematics. Likewise, instruction that emphasizes conceptual understanding without corresponding attention to the development of skills may leave students unprepared for the skilloriented aspects of higherlevel mathematics. The report makes clear that there is no need to choose between the two: mathematical proficiency requires both, as well as attention to reasoning and problem solving. In
addition, instruction should help students develop the metacognitive skills and confidence to monitor and regulate their own mathematical thinking. For example, instruction that helps students use common errors as a tool for identifying misconceptions may support students’ development of problemsolving skills.
Adding It Up (National Research Council, 2001a) also addresses instruction, noting that debates about alternative approaches to teaching, such as traditional versus reform, or direct instruction versus inquiry, obscure the broader point that effective instruction is a successful interaction among three elements: teachers’ knowledge and use of mathematical content, teachers’ attention to and handling of the students, and students’ engagement in and use of mathematical tasks. Thus, while the instructional choices teachers make are important, the way they are carried out is equally so.
For example, Adding It Up cites research (e.g., Stein, Grover, and Henningsen, 1996; Henningsen and Stein, 1997) that a cognitively demanding task may become routine if the teacher specifies explicit procedures for completing it or takes over the demanding aspects as soon as students appear to struggle. The TIMSS 1999 video study identified the ability to maintain the highlevel demands of cognitively challenging tasks during instruction as the central feature that distinguished classroom teaching in countries with highperforming students from teaching in countries with lower performing students (including the United States) (National Center for Education Statistics, 2003c; Stigler and Hiebert, 2004). This research suggests that student engagement is fostered when teachers choose tasks that build on the students’ prior knowledge and guide them to the next level, rather than demonstrating exactly how to proceed, and that it is essential that students think through concepts for themselves. Other factors identified as effective include thoughtful lesson planning that tracks students’ developing understanding and allocation of sufficient time for students to achieve lesson goals.
The National Mathematics Advisory Panel made similar points, and it found no rigorous research to support claims that instruction that is either exclusively “teachercentered” or exclusively “studentcentered” is better. The panel did, however, find some evidence to support the effectiveness of cooperative learning practices and the regular use of formative assessment in elementary mathematics instruction as a tool for tailoring instruction to students’ needs. The panel called attention to the limited amount of rigorous empirical research available to answer questions about mathematics teaching and learning, and it recommended a variety of research to test hypotheses about the most effective approaches.
In sum, there is growing agreement on the specifics of what students should be taught, but there are fewer specific answers as to the best ways to teach that material. Mathematics educators have established a clear
consensus, based on research evidence, that the development of mathematical proficiency requires continual instructional opportunities for students to build their understanding of core mathematical concepts, fluency with mathematics procedures, and metacognitive strategies to guide their own mathematical learning. However, there is little empirical evidence to support detailed conclusions about precisely how this is best accomplished.
QUESTION 3:
WHAT DO SUCCESSFUL TEACHERS KNOW ABOUT MATHEMATICS AND HOW TO TEACH IT?
There is strong reason to believe that teachers’ knowledge and skills make a difference in their practice (Wenglinsky, 2002; Rockoff, 2004; Rivkin, Hanushek, and Kain, 2005; Clotfelter, Ladd, and Vigdor, 2007). Researchers have tried to disentangle the different kinds of knowledge about mathematics, about students, and about the learning process that teachers use (Ball, Lubienski, and Mewborn, 2001). Research that has searched for connections between straightforward—yet crude—measures of teacher knowledge (such as number of courses taken or degrees earned) and student learning has provided relatively little insight into questions about what skills and knowledge are most valuable for teachers (see Chapter 3). For example, research shows that high school students taught by mathematics majors outperform students taught by teachers who majored in some other field, but that research does not illuminate what it is that the teachers who majored in mathematics do in the classroom (Monk and King, 1994; Goldhaber and Brewer, 1997; Rowan, Chiang, and Miller, 1997; Wilson, Floden, and FerriniMundy, 2001; Floden and Maniketti, 2005).
Beginning in the 1980s, a growing number of scholars looked to more qualitative research—largely based on interviews and classroom observations—to provide richer pictures of the mathematical thinking teachers do when teaching. These studies supported the development of more nuanced descriptions of teachers’ knowledge and skills by illuminating the ways that the mathematical knowledge needed for teaching differs from the mathematical knowledge needed to succeed in advanced courses. The concept of pedagogical content knowledge gave a name to the knowledge of content as it applies to and can be used in teaching (see, e.g., Shulman, 1986; Ball, Lubienski, and Mewborn, 2001).
Mathematicians have always played a critical role in defining the kinds of knowledge and skills that are most useful to mathematics teachers. Several recent publications, including textbooks for aspiring elementary mathematics teachers, studies, and analytic essays, have laid out current thinking (e.g., Parker and Baldridge, 2003; Beckmann, 2004; Milgram, 2005; Wu, 2007). For example, Wu (2007) argued that, at a minimum, teachers of grades 512 must be knowledgeable about the importance of definitions, the
ubiquity of reasoning, the precision and coherence of the discipline, and the fact that the concepts and skills in the curriculum are there for a purpose (how to solve problems). Ideally, he argues, teachers in the primary grades should know all these things, too, but pedagogical knowledge carries more weight for teachers of younger students. An additional obvious difference between the requirements for elementary teachers and high school teachers is that teaching older students carries a greater demand for both technical skills and abstract reasoning.
These developments in the understanding of mathematics teaching provide a critical framework for teacher education, although they do not provide empirical support for a concrete description of precisely how to teach mathematics (Ball, Lubienski, and Mewborn, 2001). However, the sophistication of this sort of analysis of classroom teaching provides a way to understand an exceptionally complex process. The combination of this descriptive work and analyses by mathematicians provides an invaluable component of the research base.
Many of the same sources that have offered visions of what students need to learn have also made recommendations as to what mathematics teachers need to know. These summary documents draw on the range of quantitative and qualitative research available, as well as on the professional judgment of scholars and practitioners.
The NCTM has developed professional standards for mathematics teachers to accompany their content standards (National Council of Teachers of Mathematics, 1991). Similarly, states have developed standards for mathematics teachers, drawing on such resources as the standards from NCTM and the Interstate New Teacher Assessment and Support Consortium (INTASC). The focus of these resources, however, is on ensuring that teachers have studied the material covered in the standards and curricula for students. They do not, for the most part, address others kinds of knowledge and skills that might be important for teachers.
Adding It Up (National Research Council, 2001a) also describes the knowledge of mathematics, students, and instructional practices that are important for teachers. The report stresses that, to be effective, teachers not only need to understand mathematical concepts and know how to perform mathematical procedures, but also to understand the conceptual foundations of that knowledge; it is also important that they have strong confidence in their own mathematical competence. The report notes that a substantial body of work has documented the deficiencies in U.S. mathematics teachers’ base of knowledge—and that even when teachers understand the mathematics content they are responsible for teaching, they often lack deeper understanding of the way mathematical knowledge is generated and established.
The Mathematical Education of Teachers, a report prepared by the
Conference Board of the Mathematical Sciences and published by American Mathematical Society and the Mathematical Association of America (2001), offers guidelines for the preparation of mathematics teachers and is arguably the primary document that guides departments of mathematics regarding the teaching of mathematics to future teachers. The report’s recommendations are grounded in the conviction that a central goal of preservice mathematics education is to develop teachers who have excellent problemsolving and mathematicalreasoning skills themselves, and the report describes specific kinds of knowledge teachers at different grade levels need related to the topics they will teach their students. The report emphasizes how important it is that teachers understand links between the material taught in the early grades and more sophisticated concepts that will build on the earlier learning: it is not sufficient for teachers to master only the level of mathematics they will be teaching.
Moreover, the report observes that the challenges of teaching each level are distinct and require different preparation, as current certification requirements reflect. It notes in particular that teachers of middle grades mathematics often “have been prepared to teach elementary school mathematics and lack the broader background needed to teach the more advanced mathematics of the middle grades” (Conference Board of the Mathematical Sciences, 2001, see http://www.cbmsweb.org/MET_Document/chapter_4.htm [November 2009]).
The National Mathematics Advisory Panel also searched for evidence about the connections between teachers’ knowledge and instructional practice and student outcomes, and for evidence that particular instructional practices are effective. Citing evidence that variation in teacher quality may account for 12 to 14 percent of the variance in elementary students’ mathematics learning, the panel examined evidence of teachers’ knowledge that can be gleaned from certification, courses completed, and assessment results. Noting that these are imprecise measures, the panel nevertheless seconds the recommendation in Adding It Up, asserting that “teachers must know in detail the mathematical content they are responsible for teaching and its connections to other important mathematics” (National Mathematics Advisory Panel, 2008, p. xxi).
Thus, despite the lack of substantial and consistent empirical evidence, there is a growing consensus about the kinds of mathematical knowledge effective teachers have. Current research and professional consensus correspond in suggesting that all mathematics teachers, even elementary teachers, rely on a combination of mathematics knowledge and pedagogical knowledge:

mathematical knowledge for teaching, that is, knowledge not just of the content they are responsible for teaching, but also of the

broader mathematical context for that knowledge and the connections between the material they teach and other important mathematics content;

understanding of the way mathematics learning develops and of the variation in cognitive approaches to mathematical thinking; and

command of an array of instructional strategies designed to develop students’ mathematical learning that are grounded in both practice and research.
QUESTION 4:
WHAT INSTRUCTIONAL OPPORTUNITIES ARE NECESSARY TO PREPARE SUCCESSFUL MATHEMATICS TEACHERS?
How might teachers best acquire the knowledge and skills they need? The work of Deborah Ball (1990, 1991, 1993) on the knowledge of elementary school teachers, as well as Liping Ma’s (1999) influential comparison of Chinese and American teachers, which built on Ball’s work, have sparked a renewal of interest among mathematicians and mathematics teacher educators in the preparation of mathematics teachers. Other resources include guidelines for preparation programs, as well as research regarding elements of preparation that can be linked to positive outcomes for students. Nevertheless, there is relatively little empirical evidence to support guidelines for teacher preparation.
The report prepared by the Conference Board of the Mathematical Sciences (2001) offers guidelines for teachers’ mathematics preparation.^{5} The board’s guidance is based on available scholarship on mathematics education and the judgment of professional mathematicians. The board’s report offered three specific recommendations:

Prospective elementary grade teachers should be required to take at least 9 semesterhours on fundamental ideas of elementary school mathematics.

Prospective middle grades teachers of mathematics should be required to take at least 21 semesterhours of mathematics, [including] at least 12 semesterhours on fundamental ideas of school mathematics appropriate for middle grades teachers.

Prospective high school teachers of mathematics should be required to complete the equivalent of an undergraduate major in mathematics, [including] a 6hour capstone course connecting their college mathematics courses with high school mathematics.
David Monk (1994) also examined the effects of course taking on teacher effectiveness, using data from the Longitudinal Survey of American Youth. He found that students whose teachers had taken more mathematics courses performed better on achievement tests than their peers whose teachers had taken fewer such courses. He also found that courses that addressed teaching methods showed an even stronger benefit. Floden and Meniketti (2005) summarized the findings of this and other research on the effects of undergraduate coursework on teachers’ knowledge. They identified studies that examined correlations between coursework and teacher performance and correlations between coursework and student achievement and those that examined the content knowledge of prospective teachers and studies that examined what teachers learn from particular courses. Many studies focused on mathematics, and they provided support for the claim that studying collegelevel mathematics has benefits for prospective secondary level teachers. However, the research provides little clear guidance as to what the content of the coursework should be and even less guidance about content preparation for teachers of lower grades. Floden and Meniketti also note that currently available measures of teacher knowledge and of student outcomes are imprecise tools for assessing the impact of teacher education (see also Wilson, Floden, and FerriniMundy, 2001).
Ball, Hill, and Bass (2005) conducted an analysis of the role of mathematical knowledge and skills in elementary teaching in order to develop a “practicebased portrait of … mathematical knowledge for teaching” (p. 17). They then developed measures of this knowledge to use in linking it to student achievement. The researchers argue that teachers need to have “a specialized fluency with mathematical language, with what counts as a mathematical explanation, and with how to use symbols with care” (p. 21). They found that teachers need not only to be able do the mathematics they are teaching, but to “think from the learner’s perspective and to consider what it takes to understand a mathematical idea for someone seeing it for the first time” (p. 21) (see also Hill, Rowan, and Ball, 2005). Through a longitudinal study of schools engaged in reform efforts, Ball and her colleagues were able to link 1st and 3rdgrade teachers’ responses to the measure of professional knowledge with their students’ scores on the TerraNova assessment (Ball, Lubienski, and Mewborn, 2001). The results showed a significant relationship between students’ gains and their teachers’ degree of professional knowledge.
This body of work clearly supports the intuitive belief that content knowledge is important. The research on student learning described above also supports the proposition that prospective mathematics teachers should study mathematics learning and teaching methods.^{6}
Finally, a study of the preparation of middle school mathematics teachers in six nations (Schmidt et al., 2007) follows up on findings from the 1987 TIMSS study. That study had shown that, in general, middle school students in the United States were not exposed to mathematics curricula that were as focused, coherent, and rigorous as those in countries (including Korea and Taiwan) whose students scored higher on TIMSS. Schmidt and his colleagues examined teacher preparation in those countries and found that training in the highperforming countries includes extensive educational opportunities in mathematics and in the practical aspects of teaching students in the middle grades.
The relevant body of work on what instructional opportunities are most valuable for mathematics teachers is growing but thus far is largely descriptive, and it has not identified causal relationships between specific aspects of preparation programs and measures of prospective teachers’ subsequent effectiveness. Nevertheless, the field of mathematics education has established a firm consensus that to prepare effective K12 mathematics teachers, a program should provide prospective teachers with the knowledge and skills described by the Conference Board of the Mathematical Sciences:

a deep understanding of the mathematics they will teach,

courses that focus on a thorough development of basic mathematical ideas, and

courses that develop careful reasoning and mathematical “common sense” in analyzing conceptual relationships and solving problems, and courses that develop the habits of mind of a mathematical thinker.
^{6} 
Ongoing research offers the prospect of further insights. For instance, McCrory and her colleagues are investigating the link between what is taught in collegelevel mathematics classes designed for elementary teachers and what prospective teachers understand about the mathematics they are taught (see McCrory and Cannata, 2007; see also http://meet.educ.msu.edu/research.htm [February 2010]). Having surveyed 56 mathematics departments and 79 instructors, she has found that, on average, elementary teachers are expected to take two mathematics classes, although this is increasing, especially for middle school certification. Her research also indicates that instructors are committed and enthusiastic, but not necessarily knowledgeable, about mathematics education. 
HOW MATHEMATICS TEACHERS ARE CURRENTLY PREPARED
Concern about the adequacy of current teacher preparation in mathematics is unmistakable, and it is particularly sharp with regard to K8 teachers. Evidence from many sources suggests that many teachers do not have sufficient mathematical knowledge (see, e.g., Ball, 1991; Ma, 1999). Specifically, as Wu (2002) has observed, “we have not done nearly enough to help teachers understand the essential characteristics of mathematics: its precision, the ubiquity of logical reasoning, and its coherence as a discipline” (p. 2). Furthermore, there is reason to believe that teachers lack other relevant professional knowledge as well, including mathematicsspecific pedagogical knowledge.
We had two major sources of information about how teacher preparation in mathematics is currently being conducted—state requirements and coursework—although much of the information is somewhat indirect.
State Requirements
We begin with the requirements states have established for licensing mathematics teachers, which influence the goals teacher preparation programs set for themselves. According to data collected by Editorial Projects in Education, 33 of the 50 states and the District of Columbia require that high school teachers have majored in the subject they plan to teach in order to be certified, but only 3 states have that requirement for middle school teachers (data from 2006 and 2008; see http://www.edcounts.org/ [February 2010]). Fortytwo states require prospective teachers to pass a written test in the subject in which they want to be certified, and six require passage of a written test in subjectspecific pedagogy.
Limited information is available on the content of teacher certification tests. A study of certification and licensure examinations in mathematics by the Education Trust (1999) reviewed the level of mathematics knowledge necessary to succeed on the tests required of secondary mathematics teachers. The authors found that the tests rarely assessed content that exceeded knowledge that an 11th or 12th grader would be expected to have and did not reflect the deep knowledge of the subject one would expect of a collegeeducated mathematics major or someone who had done advanced study of school mathematics. Moreover, the Education Trust found that the cut scores (for passing or failing) for most state licensure examinations are so low that prospective teachers do not even need to have a working knowledge of high school mathematics in order to pass. Although this study is modest, its results align with the general perception that state tests for teacher certification do not reflect ambitious conceptions of content knowledge.
The prevalence of socalled outoffield teachers, those who are not certified in the subject they are teaching, is another indication that states sometimes find it difficult to ensure that mathematics teachers are well prepared. We discuss this issue below.
Coursework
The Conference Board of the Mathematical Sciences (CBMS) conducts a survey every 5 years of undergraduate education in mathematics in the United States, and it includes some questions about the preparation of K12 mathematics teachers (Lutzer, Maxwell, and Rodi, 2007). The most recent report describes the complexity of developing a statistical profile of undergraduate mathematics preparation programs because of their variation. However, Lutzer and colleagues report that 56 percent of programs have the same requirements for mathematics certification for all K8 teachers of mathematics, regardless of the level (e.g., kindergarten or 8th grade) those candidates intended to teach. They also found that the average number of mathematics courses required for K8 teachers was 2.1.
For teachers seeking K8 mathematics certification, 4 percent of programs do not require any mathematics courses, 63 percent require one or two courses, 33 percent require three or four courses, and none requires five or more courses. In contrast, 58 percent of programs require five or more courses for teachers of the upper elementary grades.^{7} Thus, most programs fall well short of the recommendations of The Mathematical Education of Teachers (Conference Board of the Mathematical Sciences, 2001), and some members of the mathematics community believe the subjectknowledge requirements should be even more demanding than those recommendations.
With the analyses of Florida commissioned for our study, we were able to look in more detail at the average number of mathematics credits earned by Florida teachers by certification area: see Table 61. Although these data do not provide information about the content or nature of the coursework, they do suggest significant overall exposure to mathematics, corresponding roughly to 4 threecredit courses for elementary teachers, 15 courses for teachers certified for middle school mathematics, and 19 courses for teachers certified for high school mathematics.^{8}
^{7} 
These results correspond to McCrory’s emerging results as well as the selfreports of new teachers surveyed as part of the Teacher Pathways Project being conducted by the University at Albany and Stanford University (see http://www.teacherpolicyresearch.org/TeacherPathwaysProject/tabid/81/Default.aspx [February 2010]). 
^{8} 
One possible explanation for this relatively high number of courses—in comparison with those typical in other states and programs—is that if many of Florida’s courses are remedial or elementary in nature, they would be likely to meet frequently each week, thus yielding a high number of credit hours. 
TABLE 61 Mean Mathematics Credit Hours, Florida Mathematics Teachers
In terms of the actual content of the coursework to which aspiring mathematics teachers are exposed, there are few sources.^{9} The Education Commission of the States has assembled information about whether or not states require that teacher preparation programs align their curricula with the state’s K12 curriculum standards or their standards for teachers (or both), which offers an indirect indicator (see http://www.ecs.org/). Of the 50 states and American Samoa, the District of Columbia, Guam, Puerto Rico, and the Virgin Islands, 25 require alignment with both, and 16 have no policy for either (as of 2006); another 8 require only alignment with the K12 curriculum, and 6 require only alignment with standards for teachers.
A survey of firstyear teachers in New York City that was part of our commissioned analysis included questions about their preparation and provides some additional hints about content (Grossman et al., 2008). Current elementary teachers and middle and secondary level mathematics teachers were asked about the extent to which their teacher preparation program gave them the opportunity to do and learn a variety of things, such as learning about the typical difficulties students have with aspects of mathematics. The new teachers rated their opportunities on a 5point scale, with 1 being no opportunity and 5 being extensive opportunity. The teaching activities covered in the survey were described in short phrases, so few conclusions
can be drawn about the extent to which they track with the kinds of approaches we describe above. Nevertheless, the teachers’ responses to several of the questions that seem most congruent with the kinds of teaching advocated by the mathematics education community are suggestive: see Table 62. Elementary teachers reported taking an average of 1.81 courses in mathematics; they also reported having a modest exposure to learning about the difficulties their students might have with place value or algebra. Although the survey was small in scale, it does suggest that New York City teachers who graduated recently from a teacher education program do not receive extensive exposure to these elements.
Other evidence comes from a study in which the preparation of U.S. middle school mathematics teachers was compared with that of their counterparts in other countries (Schmidt et al., 2007). The study, which compared teachers’ knowledge and skills in mathematics and mathematical pedagogy, found that U.S. teachers scored in the middle or close to the bottom in comparison with teachers in the countries whose students performed well on the TIMSS study. The results suggested possible differences in preparation across countries, and the study’s authors concluded that the
TABLE 62 New York City Teachers’ Reported Exposure to Mathematics Preparation
Opportunity to Learn Mathematics Education Approach or Strategy 
Mean Response on 15 Scale^{a} 
Elementary Teachers 

Learn typical difficulties students have with place value. 
2.71 
Practice what you learned about teaching math in your field experience. 
3.26 
How many courses did you take in the teaching of math at the college level? 
1.81 [not on 5point scale] 
Secondary Teachers 

Learn different ways that students solve particular problems. 
3.34 
Learn theoretical concepts and ideas underlying mathematical applications. 
3.36 
Learn about typical difficulties students have with algebra. 
2.5 
NOTE: Results are for teachers who attended an undergraduate teacher preparation program. Data are also available for teachers who followed other pathways. ^{a}A respondent who rated his or her exposure as a 3 would be indicating that it was roughly halfway between none at all and extensive. SOURCE: Data from Matt Ronfeldt, University of Michigan (personal communication, 2008). 
preparation available to U.S. teachers provided less “extensive educational opportunities in mathematics and in the practical aspects of teaching mathematics to students in the middle grades” (Schmidt et al., 2007, p. 2). The researchers were not able to use representative samples of teachers in each country; they relied on convenience samples. A comparative study of mathematics teachers’ preparation and mathematical knowledge using national probability samples is currently being conducted by the International Association for the Evaluation of Educational Achievement (see http://www.iea.nl/tedsm.html [November 2009]).
Many students, particularly at the secondary level, are taught mathematics by teachers who are not certified in that subject (because of a dearth of certified teachers), and these teachers are likely to have taken even fewer courses in mathematics than certified mathematics teachers. The problem is more acute in mathematics than in other subjects. The National Center for Education Statistics (2003a) reports that in the 19992000 school year, 23.0 percent of middle school students and 10.1 percent of secondary students were taught mathematics by a teacher who was not certified to teach mathematics and had not majored in it. (Note that the grades encompassed by middle school, as well as the requirements to teach at that level, vary.) Using data from the Schools and Staffing Survey, Richard Ingersoll (2008) found that 38 percent of the teachers who taught mathematics to grades 7 through 12 did not have either a major or a minor in mathematics, mathematics education, or a related field. The problem is greatest in highpoverty schools, where students are approximately twice as likely to have a mathematics teacher who is not certified in the subject. A study of the effects of teachers’ credentials on student achievement (Clotfelter, Ladd, and Vigdor, 2007) provides evidence that students whose teachers are certified in the subject they teach achieve at higher levels than students whose teachers are not, particularly in algebra and geometry.
At the state level, there is other evidence of outoffield teaching. The California Council on Science and Technology (2007) conducted an analysis of career pathways for that state’s mathematics and science teachers. They found that 10 percent of middle school and 12 percent of high school mathematics teachers were teaching out of field, and that 40 percent of novice high school mathematics teachers were not well prepared (defined as lacking a preliminary credential). The percentages are highest in lowperforming and highminority schools. They also found that California lacks the capacity to meet the growing demand for fully prepared mathematics (and science) teachers and that the state is not collecting the data necessary to monitor teacher supply and demand.
Our review of these disparate sources of information leaves us with a reasonably firm basis for concluding that many, perhaps most, K8 mathematics teachers are not adequately prepared, either because they have not
received enough mathematics and pedagogical preparation or because they have not received the right sort of preparation; the picture is somewhat less clear for middle and high school teachers. There is relatively scant specific information about precisely what it is that teacher preparation programs do, or fail to do, but there is relatively good evidence that mathematics preparation for prospective teachers provides insufficient coursework in mathematics as a discipline and mathematical pedagogy. Some might suggest that, given the prevalence of outoffield mathematics teachers, raising standards for aspiring teachers may exacerbate shortages, particularly in highpoverty areas. The committee’s view is that the more relevant question is whether there are shortages of adequately prepared teachers.
CONCLUSIONS
From our review of what mathematics teacher preparation programs ought to be doing, and the information we could find about what they are doing, three key points seem clear. First, there is a strong basis for defining clear expectations for teacher preparation programs for mathematics content and pedagogy, on the basis of some research and the considered judgments of mathematicians and mathematics educators. Second, the limited information available suggests that most programs would probably not currently meet those expectations. Third, systematic data about the content of mathematics teacher preparation are sorely lacking.
Regarding what students need to know, mathematicians and mathematics educators are in accord that successful mathematics learning is most likely when core topics in school mathematics and the five strands of mathematical proficiency identified in Adding It Up (conceptual understanding, procedural fluency, strategic competence, adaptive reasoning, and productive disposition) are interwoven at each level of schooling, and students are provided with a coherent curriculum in which clear objectives, based on a logical conception of the mathematics learning trajectory, guide each year of mathematics study. This proposition has logical implications for teacher preparation.
Conclusion 61: It is plausible that to provide students with the instructional opportunities they need to develop successfully in mathematics, teachers need preparation that covers knowledge of mathematics, of how students learn mathematics, and of mathematical pedagogy, and that is aligned with the recommendations of professional societies.
We particularly note the importance of the knowledge and skills described in Chapter 2 of The Mathematical Education of Teachers (Conference Board of the Mathematical Sciences, 2001). However, there is
currently no clear evidence that particular approaches to preparation do indeed improve teacher effectiveness, nor clear evidence about how such preparation should be carried out.
Because strong preparation in both mathematics content and mathematics pedagogy are important, it seems logical that the preparation of mathematics teachers should be the joint responsibility of faculties of education and mathematics and statistics. We recognize that this is not a simple matter because of the many competing demands that face faculty in these fields, but we believe that close collaboration among mathematics faculty and mathematics education faculty is the only realistic means of providing the necessary preparation. Such collaboration could both promote research designed to improve the education of teacher candidates and provide teacher candidates with an education that seamlessly integrates mathematics learning and pedagogical learning.
The data regarding what is currently happening in teacher preparation for mathematics is extremely limited, but the information that is available clearly indicates that such preparation is not sufficient. That is, because it appears that many preparation programs fall short of guidelines such as The Mathematical Education of Teachers recommendations, it is likely that:
Conclusion 62: Many, perhaps most, mathematics teachers lack the level of preparation in mathematics and teaching that the professional community deems adequate to teach mathematics. In addition, there are unacceptably high numbers of teachers of middle and high school mathematics courses who are teaching out of field.
Given the limited evidence base about the effectiveness of different approaches to preparing teachers of mathematics and about the nature of current preparation approaches, additional research is needed:
Conclusion 63: Both quantitative and qualitative data about the programs of study in mathematics offered and required at teacher preparation institutions are needed, as is research to improve understanding of what sorts of preparation approaches are most effective at developing effective teachers.