CAREER: Knowledge for teaching mathematics: The impact of ...The problem of teachers’ mathematical knowledge has been a subject of research and policy for many years. While it is

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CAREER: Knowledge for teaching mathematics: The impact of mathematics courseson prospective elementary teachers’ mathematical knowledge.

Prospective teachers need mathematics courses that develop a deepunderstanding of the mathematics they will teach. The mathematicalknowledge needed by teachers at all levels is substantial, yet quite differentfrom that required by students pursuing other mathematics-relatedprofessions. Prospective teachers need to understand the fundamentalprinciples that underlie school mathematics, so that they can teach it todiverse groups of students as a coherent, reasoned activity and communicatean appreciation of the elegance and power of the subject. (Recommendation1, CBMS Conference Board of the Mathematical Sciences, 2001, p. 7)

In light of this recommendation, and others in the CBMS report, do courses forprospective teachers offer them what they need? Are these courses successful? That is, doprospective teachers learn what they need to know? These are the driving question of theresearch proposed here.

The problem of teachers’ mathematical knowledge has been a subject of research andpolicy for many years. While it is almost a truism to say that K-6 teachers’ mathematicalknowledge is inadequate, the problem is more complex than the simple assertion suggests.Fundamental questions of what mathematical knowledge is needed and how it can be taughtand learned continue to demand attention, as improvement in K-6 students’ mathematicalachievement in the US lags in national and international assessments (Braswell, Daane, &Grigg, 2003; Braswell et al., 2001; Sherman, Honegger, McGivern, & Lemke, 2003). Recentpolicy documents (Committee on Science and Mathematics Teacher Preparation, 2000;Committee on the Mathematical Education of Teachers, 1991; CBMS 2001; Leitzel, 1991;Mathematical Sciences Education Board, 1996; Mathematical Sciences Education Board &National Research Council, 2001; National Research Council & Committee on Science andMathematics Teacher Preparation, 2001; RAND Mathematics Study Panel, 2002) urgecontinuing research on teachers’ mathematical knowledge and propose agendas to addressthis issue. The research proposed here will focus on 1) opportunities to learn mathematicsprovided in undergraduate teacher preparation programs for K-6 teachers; 2) prospectiveteachers’ learning given those opportunities; and 3) how those opportunities relate to researchand policy on teachers’ mathematical knowledge.

Since Shulman’s seminal presidential address to AERA (1986), research on teachers’subject matter knowledge has escalated in every domain. Shulman and colleagues(Grossman, 1990; Shulman, 1987; Wilson, Shulman, & Richert, 1987; Wineburg & Wilson,1988) identified three kinds of knowledge of importance for teaching: content knowledge,pedagogical content knowledge (PCK), and general pedagogical knowledge. Subject matterknowledge included the kind of knowledge a professional in the field might have, at a levelappropriate for teaching. PCK included knowledge of representations that are likely to helpstudents learn the content and of typical student misconceptions and errors. Generalpedagogical knowledge included aspects of teaching not specifically related to a subject,such as organizing and managing group work. In mathematics, this division has turned out tobe less crisp than the labels might suggest, as the nature and substance of subject matterknowledge for teaching has come under scrutiny.

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Ball and colleagues (Ball, Lubienski, & Mewborn, 2001) review of research onteacher knowledge in mathematics outlines the problematic nature of its conceptualizationand study. It points out that early studies of teacher knowledge used teacher characteristics,primarily the number of mathematics courses taken, as a proxy for mathematical knowledge.These studies produced mixed results, suggesting that, while some number of mathematicscourses seems to correlate with student outcomes, there is a point at which returns arediminished or even negative. At that ill-defined point, more courses do not result in higherstudent achievement (p. 442-3). A report by Wilson, Floden, and Ferrini-Mundy (2001;2002) indicates that rigorous research about teacher preparation is sparse, both acrossdisciplines and within the field of mathematics. In the few studies that met the requirementsto be included in their review, they too found mixed results about the development ofmathematical knowledge. In one study, the number of mathematics methods courses teacherstook had larger correlations with K-12 pupil outcomes than the number of mathematicscourses while in other studies methods courses did not appear to have the same effects(Monk & King, 1994, cited in Wilson et al, 2001). In a recent study, Greenberg andcolleagues (Greenberg, Rhodes, Ye, & Stancavage, 2004) used NAEP data to investigate therelationship between teachers’ certification status and eighth grade student achievement.They found that certification status correlated with student achievement, even whencontrolling for ability, race, and socio-economic status. None of these studies, however,considers how mathematics courses or other coursework, or experiences required forcertification, change what teachers know or do to achieve better outcomes.

The mathematics community has a long history of analyzing and discussing whatmathematics teachers need to know. In recent years, the Mathematical Association ofAmerica report (Leitzel, 1991) made recommendations about topics that should be includedin teachers’ education. The recent CBMS report (2001) “augments those recommendations,by giving more attention to the mathematical conceptions of K–12 students and how theirteachers can be better prepared to address these ideas,” (p. 13). The report suggests areas ofstudy and desired outcomes, describes in some detail aspects of knowledge in each of severaltopical areas and grade level groups, and makes eleven recommendations. Whileacknowledging that quality is more important than quantity, the CBMS report recommendsthat “prospective elementary grade teachers should be required to take at least 9 semester-hours on fundamental ideas of elementary school mathematics,” (Recommendation 2, p. 8).

In the last few years, researchers have investigated the mathematics that teachers usein practice (Ball & Bass, 2000; Ferrini-Mundy, Burrill, Floden, & Sandow, 2003), andteacher educators have designed mathematics courses specifically for prospective high schoolmathematics teachers (Usiskin, 2000; Usiskin, Peressini, Marchisotto, & Stanley, 2002).Ball and Bass (2003) present convincing evidence that mathematical content knowledge forteaching is more complex than might be immediately apparent. They suggest that teachersneed mathematical knowledge that, while clearly mathematical and not pedagogical, isnonetheless knowledge that math majors and even professional mathematicians may nothave. For example, knowing how to do multidigit multiplication with borrowing (e.g., 35 x25) is mathematical knowledge that most adults have. Knowing how to reverse engineer thatproblem to understand how the incorrect answers 245, 1055, or 105 might be derived ismathematical knowledge for teaching. Each of those answers has a logical explanationlinked to a misuse or misunderstanding of mathematics, and, Ball argues, is mathematicaleven though it would not be part of the repertoire of a typical undergraduate mathematics

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course (Ball, NPR Broadcast on May 28, 2004). Hill, Rowan, and Ball (2004) name thesecommon (mathematics that is taught and learned in the course of a good secondary orpostsecondary education) and specialized (knowledge distinct to the repertoire of a teacher)content knowledge for teaching.

Over the last two decades, there have been a number of studies of preservice teachers’knowledge of mathematics, often aimed at particular topics or processes (Ball, 1990; Borkoet al., 1992; Eisenhart et al., 1993; Even, 1990; Graeber, Tirosh, & Glover, 1989; Ma, 1998;Simon, 1993; Simon & Blume, 1994, 1996; Tirosh, Fischbein, Graeber, & Wilson, 1999; M.Wilson, 1994). In general, these studies indicate that although they may be able to do themathematics of elementary school and obtain correct answers, preservice teachers oftencannot give adequate explanations, provide representations of ideas, or generate illustrativeproblems. Their understanding of elementary mathematics appears to be quite weak, and thisis true even for students who have more extensive mathematics coursework than might betypical of elementary education majors (Borko et al., 1992). Ma (1998) contrasts theknowledge of elementary teachers in China and the US, and identifies the difference as“profound understanding of fundamental mathematics.”

Research ObjectivesIn the research proposed here, the purposes are to gain insights into mathematical

knowledge for teaching and to understand how undergraduate courses can contribute toimproving the mathematical education of elementary teachers. The research questions are:What opportunities to learn are provided to prospective elementary teachers in theirundergraduate mathematics education, and what do they learn? How do teachers’opportunities to learn mathematics relate to research and policy on teacher knowledge?The research will have three primary areas: (a) Investigating opportunities to learn currentlyprovided to prospective elementary teachers through textbooks, instructors, and courses inundergraduate mathematics; (b) Investigating teacher learning in and through these courses;(c) Investigating the coherence of courses with respect to research and policy related to theundergraduate mathematical education of prospective K-6 teachers.

Building on recent conceptions and theories of mathematical knowledge for teaching,I will investigate these constructs in three lines of work: (a) analysis of textbooks, curriculummaterials and policy documents; (b) survey research aimed at mathematics departments andinstructors in selected states; and (c) qualitative case studies at institutions in those states,including assessments of teacher learning. The objectives of the research are both basic andapplied, seeking to increase theoretical understanding of teachers’ mathematical knowledgethrough the lens of those who teach them mathematics and create materials for use in thatteaching; and to provide data useful in guiding policy, standards, and practice in theundergraduate mathematics education of K-6 teachers.

The hypotheses of the research are: (a) the undergraduate mathematics education ofprospective teachers impacts their learning of mathematics; (b) mathematics courses forprospective elementary teachers typically teach general mathematical knowledge and lessfrequently attend to specialized mathematical knowledge for teaching; (c) there is littleconsistency across courses and textbooks in what is taught to prospective elementaryteachers, and little agreement about what should be taught; and (d) teachers’ knowledge ofmathematics for teaching increases when they are provided with opportunities thatspecifically address mathematical knowledge for teaching.

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One obvious caveat to this research is that prospective teachers have opportunities tolearn mathematics outside of mathematics courses for teachers: mathematics methodscourses, mathematics courses other than those designed for teachers, and field (studentteaching) experiences are all possible sites for learning mathematics. This study will focusonly on the mathematics courses aimed at prospective teachers.

The first part of the research has already begun. I have identified 18 textbookscurrently in print or in preparation (see reference list), developed a framework for theiranalysis, and completed preliminary analysis of the books. I selected three topics for in-depth analysis: fractions, multiplication, and reasoning and proof. The rationale for thisselection is explained below, along with some of the preliminary findings.

Relationship to the State of Knowledge in the FieldThe proposed research draws on several bodies of literature including studies of

teachers’ mathematical knowledge outlined above, studies of the learning of mathematics,and textbook analysis. The latter are reviewed briefly below.

Studies of mathematics learning. Over the last three decades, there has beenconsiderable research about how children learn mathematics (Kilpatrick, Swafford, &Findell, 2001). On the topics of fractions, multiplication, and reasoning and proof, focal forthis study, researchers have investigated student learning including where children havedifficulty and common misconceptions or errors (Barnett, Goldenstein, & Jackson, 1994;Behr, Harel, Post, & Lesh, 1993; Bransford, Brown, & Cocking, 1999; Carpenter, Fennema,& Romberg, 1993; Chazan, 1993; Even, 1990; Fischbein, Deri, Nello, & Marino, 1985;Fischbein & Kedem, 1982; Greer, 1993; Hanna, 1995; Harel & Confrey, 1994; Kamii &Clark, 1995; Kilpatrick et al., 2001; Knuth, 2002; Lampert, 1986; Mack, 1995; Martin &Harel, 1989; Senk, 1985; Wu, 1999). A few studies focus on preservice teachers (Graeber etal., 1989; Martin & Harel, 1989; Schram, Wilcox, Lanier, & Lappan, 1988; Simon & Blume,1996). The research provides a basis for understanding what mathematics teachers are likelyto encounter in K-6 classrooms, with insights into particular problems that could call onspecialized mathematical knowledge. For example, research on children’s learning ofmultiplication suggests that children learn to expect multiplication to “make bigger”(Fischbein et al., 1985), making the transition to operations with fractions and decimalsdifficult. Translating into mathematical knowledge for teaching means creating orrecognizing problems and representations of multiplication that do not reinforce the“multiplication makes bigger” stereotype. Thus, the research on learning is a source forhypotheses about specialized mathematics that could be useful to teachers. The CBMS(2001) report suggests that this kind of knowledge does not clearly fall into eithermathematics or pedagogy and may be neglected by both mathematics and methods courses.Research on mathematics learning also suggests what kinds of problems prospective teachersmay have in their own learning in undergraduate mathematics courses.

Studies of textbooks. Another aspect of the proposed research is a study ofundergraduate mathematics textbooks. Other studies of mathematics textbooks have focusedon textbooks and curricula for K-6 students, investigating their relationship to standards(Kulm, 1999; Kulm & Grier, 1998; Senk & Thompson, 2003; Stake & Easley, 1978); howthey influence what is taught and learned in K-12 classrooms (Elliott & Woodward, 1990;Freeman & Porter, 1989; Porter & Freeman, 1989; Sosniak & Perlman, 1990; Stodolsky,1989; Venezky, 1992); how teachers themselves learn from their pupils’ textbooks andcurriculum materials (Ball & Cohen, 1996; Ball & Feiman-Nemser, 1988; Collopy, 1999;

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Remillard, 1999); and how materials compare across international boundaries (Carter, Li, &Ferruci, 1997; Schmidt, McKnight, & Raizen, 1997; Schmidt, McKnight, Valverde, Houang,& Wiley, 1997; Stigler, Fuson, Ham, & Kim, 1986). These studies use a range of techniqueson which I will draw to analyze and compare books. At MSU, the Teacher Education Studyin Mathematics, TEDS-M, is beginning an international study of curriculum in mathematicseducation that will include undergraduate textbooks. I will coordinate with Dr. JackSchwille, the PI for the TEDS-M project, to share methods and data.

What is Missing?There is little research that explores what prospective K-6 teachers have an

opportunity to learn, in terms of mathematical topics, processes, or dispositions. Researchabout undergraduate mathematics education for these students has focused on outcomes. Ihave found no research that addresses the question of the content of their undergraduatemathematics education or whether there are patterns across institutions. Although themathematics community has provided much guidance for designing these courses, we do notknow if their recommendations are being followed.

Some scholars have suggested that the content of these courses is “obvious.” Youjust teach them what they have to teach their pupils. Recent research on teacher knowledgecited above, along with my preliminary analysis of the textbooks written for such coursesmake plain that the content is far from obvious. Although there is some consistency in whatis included in textbooks (whole numbers, fractions, decimals, operations, etc.), differencesacross texts are huge with respect to what detail is provided, how it is presented, and whatstance is taken toward mathematics and mathematics learning. Even purely mathematicalideas, such as the definition of a fraction, differ considerably. Thus, one might assumedifferences exist across courses as well.

I have also been unable to find research that analyzes undergraduate mathematicstextbooks for teachers with respect to their fit with policy recommendations, currentstandards for teaching and learning, or current research on learning mathematics. Althoughmany textbooks make the claim that they are “standards-based” or “research-based” it isunclear in some cases whether they incorporate standards or relevant research findings. Suchwork has been done with respect to K-12 curriculum materials (e.g., the Project 2061analyses, see Kulm, 1999), but not at the undergraduate level.

There is little research available on the content and pedagogy of undergraduatemathematics courses for preservice K-6 teachers. Although there have been studies ofpreservice teacher learning of specific topics (see above) and methods courses (Feiman-Nemser & Featherstone, 1992), we know little about what happens in the hundreds of coursestaught by mathematics instructors across the country. One can guess that these courses aretaught in conventional ways, with textbook problems assigned, new ideas introduced throughlecture, and some time given in class for individual practice. But is that hypothesis correct?Is teaching at the college level “by the book” (Sosniak & Perlman, 1990; Stodolsky, 1989) asit often seems to be at the secondary level? If it is by the book, which book(s), and bywhom?

Research Design, Methods, and Work PlanThe questions of the proposed research -- What opportunities to learn are provided

to prospective elementary teachers in their undergraduate mathematics education, andwhat do they learn? How do teachers’ opportunities to learn mathematics relate toresearch and policy on teacher knowledge? – will be investigated in several strands of

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work. The research consists of multi-method, multilevel nested case studies, using state,institution, course, instructor, and student as the levels of analysis. Methods will includequantitative (surveys of mathematics departments and instructors, assessments of teacherknowledge), qualitative (interviews with textbook authors, observations and interviews atselected institutions) and analytic (textbook and policy analyses) work. Data sources for theproject are both external and internal (Figure 1). The internal data will be generated by theproject, while external sources include data sets and documents to be used in analyses and inselection of sites.Study of Mathematics Textbooks

The first part of the study, already under way with support from the College ofEducation at MSU and the Center for Proficiency in Teaching Mathematics at the Universityof Michigan, is an analysis of textbooks written specifically for undergraduate mathematicscourses for prospective elementary teachers. We have analyzed the 18 books currentlyavailable with respect to their overall content and three specific topics: fractions,multiplication, and reasoning and proof (Wallace, Stylianides, & Siedel, 2004). Within eachof these topics, we consider not only the mathematical treatment of the topic itself but alsothe nature of mathematics portrayed, implicit and explicit goals for teacher learning, impliedviews of teacher knowledge, and the relationship of the topic to the larger body ofmathematics and to important ideas in and about mathematics. The rationale for each of thetopics is explained briefly below.

Fractions is a key topic in elementary education because of the difficulties it presentsto K-8 students and teachers and its importance for later mathematics. Students’ proceduraland conceptual problems with fractions are well-documented (Barnett et al., 1994; Carpenteret al., 1993; Kamii & Clark, 1995; Mack, 1995). Wu (2001) has argued that correct andthorough understanding of fractions, along with computational fluency, are key to latersuccess in algebra. Tirosh and colleagues’ (1999) study of prospective elementary teacherssuggests that their knowledge of fractions is “rigid and segmented,” that they “could notproduce adequate representations of rational number concepts or operations with rationalnumbers, and lacked a representation of mathematics as an organized and structured body ofknowledge” (p. 10). Their sample included 147 prospective elementary teachers of whom 26were mathematics majors. Fractions is included in every textbook we have studied, typicallyas a focal topic in early chapters.

Multiplication is a topic that permeates the K-8curriculum. Research on learning multiplication indicatesthat it is conceptually and procedurally easy for students andteachers in some cases (whole numbers, multiplication asrepeated addition) and quite difficult in others(multiplication of negative integers) (Greer, 1993; Harel &Confrey, 1994). Multiplication as taught in US schoolsoften nurtures students’ misconceptions that interfere withlater learning. The same misconceptions are often held bypreservice teachers (Graeber et al., 1989). Conceptualcomplexities abound in understanding multiplication,including coming to reason quantitatively rather thannumerically (Thompson, 1994), and using “intensivequantities” in proportional reasoning (Kaput & West, 1994).

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Multiplication is a central topic in every textbook we have analyzed, sometimes as a stand-alone chapter (Beckmann, 2004), but more often as a strand that runs through separatechapters on number systems (e.g., Billstein et al, 2001).

Reasoning and proof are key to mathematics as a discipline and to learningmathematics with understanding (Ball & Bass, 2003; CBMS, 2001; Hanna, 1995). Althoughnot topics in the same way that fractions and multiplication are, they are fundamental todoing mathematics and thinking mathematically. The Principles and Standards of SchoolMathematics (NCTM, 2000) recommend that reasoning and proof be included as a centralelement throughout school mathematics. Research suggests, however, that reasoning andproof cause difficulty for teachers and students at all grade levels (Chazan, 1993; Fischbein& Kedem, 1982; Knuth, 2002; Martin & Harel, 1989; Senk, 1985; Simon & Blume, 1996).Research by Ma (1998) and Simon and Blume (1996) indicates that, in the US, prospectiveteachers are not inclined to seek mathematical justification, but rather are content to reasonby example and to accept faulty explanations. Preliminary analysis indicates thatundergraduate mathematics textbooks for teachers take a variety of approaches to reasoningand proof, from making reasoning the centerpiece of the text (Darken, 2003) to including itas an optional appendix (Musser, Burger, & Peterson, 2002). Most books pay some attentionto reasoning but leave much of the substance implicit in the way they do or presentmathematics.

From the textbook analysis, I am developing constructs that will be used and refinedin the author interviews and measured in surveys of mathematics departments andinstructors. The initial phase of textbook analysis is complete. We have developed analyticframes for each focal topics, analyzed each book with respect to overall topical coverage andthe focal topics, and begun comparisons and interpretations based on these data. In the nextphase of analysis, we will correlate textbooks with standards and policies (e.g., CBMS 2001and state certification requirements).

Interviews with authors of each of the textbooks will be conducted to understand theirintentions for the book’s use, especially their ideas and beliefs about the mathematicalknowledge elementary teachers need. Initial analyses suggest that several of the booksinclude much more than could reasonably be taught in a short sequence of courses (1 – 3semesters). The authors’ intentions concerning use of their books, what students might learnthrough their use, and how the books came to their current form will be explored in theinterviews. As with textbooks written for K-12 mathematics courses, some of these bookscould be seen as providing a “mile wide inch deep” approach to the mathematics ofelementary school. Others take a specific point of view such as problem solving (Masingila,Lester, & Raymond, 2002), “teaching for understanding” (Center for Research inMathematics and Science Education, 2000), or presenting mathematics as a rigorous andlogical discipline (Jensen, 2003). Based on our analysis of his/her book, I will ask eachauthor specific questions about the focal topics and probe the constructs described below.Survey of mathematics departments and instructors

The goal of this part of the research is to understand what is intended and taught inmathematics courses for K-6 preservice teachers. To accomplish this, I will first select anumber of states in which to survey instructors who teach these courses. States are animportant unit of analysis to consider when studying teacher preparation, for there issignificant variation in state policies that govern teacher certification. I will select statesusing a number of indicators related to relevant policies. First, using data from NAEP,

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SASS, and TIMSS, I will select two states in which students have historically highachievement in K-6 mathematics over the last decade and two states in which mathematicsachievement has been low. The states will be selected to meet the following conditions: (a)teachers in the state are certified primarily by in-state institutions; (b) the graduates from in-state certifying institutions tend to get jobs in-state and remain in-state (Loeb & Reininger,2004); (c) a few large institutions provide a major proportion of the teacher candidates in thestate; and (d) at these large institutions, or in the state as a whole, prospective elementaryteachers are required to take at least two mathematics courses for certification.

The rationale for these conditions follows from a basic hypothesis of the study, whichis that undergraduate mathematics courses for teachers make a difference. The first conditionis based on the hypothesis that high or improving achievement may be related to teachers’mathematical knowledge, which may be related to their undergraduate mathematicseducation. The second and third provide the possibility that teachers’ mathematical learningis at least in part a product of education that took place within the state. This will providesome basis for comparisons across states. The fourth condition will allow me to concentratethe research in a smaller number of institutions that likely have an impact on teachers in thestate. The final condition also provides a basis for comparison across states, and guaranteesthat institutions will offer courses that include substantial numbers of preservice teachers.States will be matched to take into account population distribution across variables such associo-economic status.

Once states are selected, I will survey all mathematics departments (via the chair oran assistant chair with responsibility for curriculum and staffing) that certify elementary andmiddle school teachers to learn what mathematics courses they offer, what they require, howthey organize and staff courses, who teaches these courses, and what the prerequisites are. Iwill also seek information about how they coordinate with mathematics methods instructionand with the certifying department. I will ask whether there are any documents availablefrom recent reviews of the institution, either for the purposes of teacher certification or forprogram review. I will include all two- and four-year institutions that participate in teacherpreparation, since many community colleges are now responsible for teaching the basicmathematics courses for elementary teachers (National Science Foundation, 1999). Fromthese surveys, I will develop a population from which to sample instructors. I will survey arandom sample of instructors within each state, over-sampling at the institutions in which Iwill conduct fieldwork. The collection of program review documents will provideinformation for looking at important variability across these sites and may provide richerinformation than a survey would allow.

It is likely that many of the courses elementary teachers are expected to take are notdesigned specifically for them. Instead, they take courses designed for general, non-technicalstudents. One result from this research will be a better understanding of what kind of courseselementary teachers take, and what those courses include. The CBMS annual survey ofmathematics departments (Lutzer, Maxwell, & Rodi, 2002) provides some information aboutthese questions. We know for example that over 40% of mathematics departments that offermathematics courses for teachers require two courses, with over 70% requiring one to threecourses. From CBMS, we also know that there were approximately 68,000 students takingsuch courses at four-year colleges and 17,000 at two-year colleges in 2000. These datasuggest that there is much to study, and many details that are as yet unexplored.

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Many aspects of these surveys will be determined with help from my advisorycommittee, including: piloting the survey; selecting states; developing the population ofinstructors; and determining the number of instructors to include in the sample. In additionto asking demographic questions about their position, experience, and background, thesurvey of instructors will measure constructs developed in the textbook analysis, includinggoals of the course, conception of the course, material use, pedagogy, conceptions ofmathematics knowledge for teaching, and assessments of student (prospective teacher)learning. I have developed a preliminary construct map (Figure 2) that shows therelationship between these constructs and the data sources and outcomes. Constructs in thediagram are explained below, after a description of the fieldwork component of the study.Fieldwork

Within the high achieving states, I will identify two institutions per state to be thesubject of detailed case studies, matching institutions across the states with respect topopulation variables. I will restrict the case studies to high achieving states since this part ofthe work will be limited in scope and, consistent with theories of case study research, usesimilar rather than divergent cases allowing for a “replication logic” (Yin, 1993, p. 34).Using data from the CBMS survey and from my own survey of departments, I will chooseinstitutions that are representative in terms of the number of courses required, the number ofgraduates, credentials of instructors, and other characteristics. At the selected institutions, Iwill conduct case studies of three sections of the mathematics courses for elementaryteachers, including interviews with the instructors, students (prospective teachers), and othermembers of the mathematics faculty. These protocols will investigate opportunities to learnand teacher learning, and will provide narrative and descriptive data about the courses.

General faculty interviews will explore the departmental participation in teachereducation, including how decisions about courses and coordination are made. Interviewswith instructors will probe in greater depth than allowed on the survey, with direct questionsabout opportunities to learn and assessment of student learning. The observation protocolwill include observing in each selected section for three hours (one week’s classes), withattention to how time is spent, how the textbook or other materials are used, how theinstructor and students interact, and what subject matter is addressed. These observationswill provide for triangulation with survey data.

An important part of the fieldwork will be assessing learning in mathematics coursesat the case study institutions. I will use items and methods from Hill and colleagues (Hill,Rowan et al., 2004; Hill, Schilling, & Ball, 2004); NCRTL (Kennedy, Ball, & McDiarmid,1993); Ma (1998); TIMSS (1999) to develop an assessment tool. I will use this tool in thecase study classes and possibly in a wider selection of classes in the case study institution.My plan is to link assessment data with instructor data. That is, I will assess students whoare in the classes of instructors selected for the survey. I will administer pre- and post-surveys to these students and will be in a position to do follow-up work with them in futureresearch.Constructs

The constructs identified in Figure 2 will be investigated across protocols andresearch sites. Preliminary definitions of these constructs and data sources for theirinvestigation are discussed here:

Goals include the intentions of the textbook, author, or instructor. For the textbookanalysis we have identified several goals that will be used in developing the surveys. These

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are: knowing the mathematics of elementary school; understanding mathematics in depth;being able to solve problems; understanding mathematics as a discipline; developing arigorous foundation for mathematics teaching and learning; and improving attitudes towardmathematics. While these goals are not mutually exclusive, books tend to emphasize one (orsome) more than others, and we expect instructors do the same. Interviews with authors willhelp solidify these categories as we probe what they intend in their development of the texts.Once categories are clear, we will write survey items to investigate the goals mathematicsinstructors have for their courses.

Conceptions of the course, while related to goals, looks at somewhat different aspectsof course planning and implementation. This includes giving students opportunities to domathematics; covering a few topics in depth; covering a significant part of the topics in theelementary curriculum; giving students opportunities to solve complex problems; andhelping students learn to learn mathematics. This list will be fleshed out in the authorinterviews and survey piloting.

Mathematics content and material use will be covered on the surveys of mathematicsdepartments and instructors. We want to learn whether textbooks or other materials are used,what specific materials are used, and what topics are covered in the course. The textbookanalysis has provided us with a list of topics that we will use in the survey, relating it eitherto specific texts in the case where a textbook is used or to the general coverage of the coursesin other cases.

Pedagogy will be included in the instructor survey, aiming to find out modes ofteaching. Lecture, discussion, problem solving sessions, homework review, and othercategories will be explored, as to their use and frequency. Previous work on K-12 teachinghas made use of categorizations of classroom activity, and we will draw on these studies(e.g., Porter and colleagues work on content determinants, Porter, Floden, Freeman, Schmidt,& Schwille, 1986). Pedagogy will also be explored in field observations.

Mathematical knowledge for teaching will also be included in the instructor survey,with questions aiming to understand what the instructors think it is important for teachers toknow and whether this is different than what non-teachers might learn. This construct will be

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explored in interviews with textbook authors, textbook analyses, instructor survey andstudent assessments.

Assessment of learning will be included in the instructor survey, with questions abouthow they assess student learning, and whether they are satisfied with what students learn.Learning will be directly assessed in the instrument administered to students in theundergraduate courses in pre- and post-tests.

Opportunities to learn (OTL) will be measured in the instructor survey throughquestions that ascertain what materials they use, what topics they cover, and how much timethey spend on focal topics. OTL data will also be derived from textbooks and their authors;from surveys of mathematics departments and their instructors; and from field observations.OTL is a construct that has a two-decade history in educational research (Floden, 2002). Atthe most basic level, it was defined as whether students have had explicit chances to learn agiven topic in a given educational context. This definition has been elaborated and expandedin evermore complex instantiations in national and international assessments of students’learning, in efforts to understand differences across countries or even states or districts(McDonnell, 1995). Floden points out that OTL can be assessed at multiple levels, fromstate to district to school to classroom, with each one getting closer to what students actuallysee and do. For the proposed research, I will assess OTL at multiple levels: state andinstitutional requirements, textbooks, instructors, and classes.Data Analysis

Textbooks analysis provides data to develop constructs for the other parts of thestudy. The constructs for focal mathematical topics will include definitions, representations,sequence, and emphasis, the latter two based on the work of Schmidt (Schmidt, McKnight,Valverde et al., 1997) and Porter (Porter et al., 1986), respectively. Each textbook will havevalues on each of these dimensions, along with others to be defined. For each specific itemwithin a topic, I will have a two part code: a binary code for whether the OTL exists and avariable for its relative strength and emphasis. For example, if a textbook includes arepresentation of fractions as division, the opportunity to learn that representation has a valueof 1 for that textbook. The other value will be determined based on how much space itoccupies and how it is presented in the text. The analysis will also be used to create a scaleof correspondence with the CBMS (2001) recommendations, NCTM (2000) Standards, andstate requirements.

Author interviews will be coded with respect to the constructs and then matched withcoding for the book. For example, if the author says that he has provided a rigorousdefinition for “fraction” but I have coded it as an intuitive definition, I will reevaluate tounderstand the difference. I will ask the authors about their sense of the emphasis ofparticular items in the focal topics to confirm the prior analysis. I will use the results of thisanalysis in several ways: to describe and understand the books fully; to interpret instructorresponses to survey questions with respect to the book they use; and to aid in developing anobservation protocol for the fieldwork, and as a proxy measure of OTL in courses that usethe book. In addition, I expect author interviews to provide important data about sales, whois using their books, and new ideas for what would be interesting to learn about courses forprospective elementary teachers.

Mathematics department surveys will provide descriptive data for the four states inthe study, including how many courses are offered for elementary preservice teachers; whatcourses they take in addition to those required; who teaches the courses (by rank, experience,

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age, sex); how many sections and students they teach each year. This survey will alsoinclude items that will be analyzed using Item Response Theory with respect to theconstructs, and used in comparison to other mathematics departments, and to surveys ofinstructors within the department. I will use Hierarchical Linear Modeling techniques tomake comparisons across institutions and states.

Instructor surveys will provide descriptive data about who teaches these courses,along with data for evaluating the constructs related to general attributes of the course andtheir teaching, and the focal mathematical topics. The survey will include some very specificquestions about a small subset of the focal areas. For example, within multiplication, thesurvey will focus on multiplication of integers. I will use the same techniques for analyzingthese surveys – IRT to assess instructor’s positions with respect to the constructs, HLM tomake comparisons across classes, institutions, and states.

Observations of classes will be used to validate instructor responses (on a small scale)and as illustrative data to flesh out the meaning of the surveys. I will observe classes whenthey are working on one of the specific areas of interest within a focal topic wheneverpossible.

Teacher pre- and post- assessments will include some general questions along withspecific questions about the focal topics (e.g., multiplication of integers). These will be usedto understand what teachers have learned, using IRT techniques to analyze measures ofparticular aspects of teacher knowledge. Teacher learning will be correlated with instructorand departmental data to look for significant relationships between the constructs of interestand student learning. These will be compared across institutions and states as well.Work Plan

There are four streams of work that constitute this research: (a) Mathematics textbookanalysis and interviews with authors; (b) Mathematics department and instructor surveys;(c) Field observations and interviews; (d) Assessments of teacher learning. I have developeda five-year plan for this work as shown in Figure 3. To accomplish the proposed work, I willhave two half-time graduate assistants one of whom will help me with surveys andquantitative analysis, and the other with field work design and implementation.

I will need expertise and advice in several areas, and have created an advisorycommittee for this purpose. The board includes Drs. Robert Floden and Suzanne Wilson(education) and Dr. Joan Ferrini-Mundy (education and mathematics) from Michigan State;Drs. Deborah Ball and Ed Silver (education) and Hyman Bass (mathematics) from theUniversity of Michigan; Dr. Sybilla Beckmann (mathematics; also a textbook author) fromthe University of Georgia; Dr. Roger Howe (mathematics) from Yale University; Dr.Zalman Usiskin (mathematics education) from the University of Chicago. I will add onemore mathematician to the team. In addition, Dr. Mark Reckase from MSU’s Measurementand Quantitative Methods program has agreed to consult with me on statistical issues. I willmeet with the advisory board three times during the project, in years 1, 3 & 4, and I will callon individuals as needed to help me with specific problems. Michigan State University is thesite of many projects with which my work intersects, including the Center for the Study ofMathematics Curriculum; the Teachers for a New Era project funded by the CarnegieCorporation, which is in the process of redesigning undergraduate education for prospectiveteachers; the Michigan/Ohio Math/Science partnership PROM/SE; the international TeacherEducation Study in Mathematics, TEDS-M; and the ROLE Knowledge of Algebra forTeaching (KAT) project, on which I am a co-PI. I plan to coordinate with these projects to

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benefit from their work, and to share mine. Drs. Floden, Ferrini-Mundy, and Usiskin areeach involved in one or more of these projects, and I am a co-PI on the KAT project.

DisseminationAs indicated in the project flow chart (Figure 3), my plan is to produce research

reports at various stages of the work, beginning immediately with the textbook analyses.There has already been considerable interest in the textbook work, with proposals accepted atthe Joint meeting of the Mathematical Association of America and the AmericanMathematical Society, 2004, and NCTM (National Council of Teachers of Mathematics)2004. I will soon submit a paper on textbook analysis to the Journal for Research inMathematics Education (JRME). As the work progresses, I plan to submit articles to TheAmerican Mathematical Monthly (writing on mathematical knowledge of prospectiveelementary teachers), Research in Collegiate Mathematics Education (what is taught inundergraduate mathematics courses for elementary teachers), Journal of MathematicsTeacher Education (textbook analysis), and Educational Evaluation, and Policy Analysis(OTL and implications for policy).

Educational PlanThe objectives of my educational plan are to dual: (a) to contribute to the

improvement of the mathematical education of prospective elementary teachers at MSU and,more ambitiously, nationwide; and (b) to improve my professional skills in both teachingand research. During the five years of this research, I plan to teach both mathematics andmethods courses. Two different versions of the mathematics courses are taught at MSU, oneusing Parker & Baldridge (2003) and the other using materials developed at San Diego State(Center for Research in Mathematics and Science Education, 2000). I will gain insights intothe impact of different materials on student learning and will try different assessments notpractical on a large scale. As instructor, I can interrogate my own mathematical knowledge,a different level of mathematical knowledge for teaching and a logical next step given theresearch I propose above.

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I will also redesign a PhD seminar I teach, “Teaching and Technology.” In thiscourse, I conceptualize technology broadly as any tool for teaching, including conventionaltechnologies such as curriculum materials. I want to shift the focus from practice to teacherlearning and knowledge. One part of my previous research has been about teaching andlearning to teach with technology, an interest that has always been more about teaching thanabout technology. Technology has provided an interesting site for my research because itreveals aspects of teaching that are often obscured by routines and expertise. I see theresearch proposed here as an extension of my work on teaching and learning to teach, and, inparticular, on using resources (technologies) in teaching. The PhD course will provide avenue for new thinking and reading about teacher knowledge.

Finally, with the support of my department, I plan to enroll in an intensive summerprogram to learn more about survey development and analysis and other statisticaltechniques. Although my background is in mathematics and I taught elementary statistics as agraduate student, my knowledge of survey methods and more advanced statistical methodsneeds improvement to do the kind of work I want to do.

As a result of this educational work, I will be in a good position both to participate inand to research the mathematical education of prospective K-6 teachers and the problem ofteacher knowledge. The research and educational plan proposed here opens up a number ofvery interesting possibilities for ongoing research, discussed in the next section.

Future ResearchIt would make sense to develop the textbook analysis into an ongoing repository for

information and data about mathematics textbooks for K-6 preservice courses. This could bea database, updated regularly, that includes data about the content of the books andinformation about where books are being used. We have learned that the market changesrapidly, with books going out of print, new editions being published, and new books comingonto the market every year. A second kind of work related to the textbook studies ishistorical analysis of how these books have changed over time. A third type of work is toexpand this analysis to include methods books.

In the instructor/course strand, future research could include an intervention study,comparing results for carefully designed interventions. The survey could be extended toadditional states. Since we know so little about what is taught in these courses, data from alarger population of instructors, courses and states would be interesting and potentially usefulfor policy and practice. Another interesting possibility is studying instructors’ mathematicalknowledge.

In the case study work, an assessment tool for preservice teacher’s mathematicalknowledge is valuable as a guide to what we should teach and as feedback about the successof our teaching. Other projects at MSU (PROM/SE, TEDS-M, TNE, and KAT) are alsoworking to develop assessments of teachers’ mathematical knowledge. I will work withthem and share items and results as each of these projects progresses.

The most important future research, however, will be following students in the initialstudy into their induction years to look at the impact of their mathematical education on theirpractice, which, if the work proposed here is successful, I plan to do.

Merit CriteriaThe intellectual merit of this research flows from the critical need for better

mathematics teachers at all levels of education. This research will contribute tounderstanding whether prospective elementary teachers are exposed to content that they need

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for successful teaching, and whether they are learning that content. Given the urgency andexposure of the problem of teacher knowledge, it is surprising to me that research on thissubject is so rare. From the survey data, we will learn what instructors are teaching,informing policy makers and practitioners as they consider how to improve theundergraduate education of prospective teachers. We will have data about teacher learning,possibly providing insights into which models are successful in helping teachers learnrelevant mathematics.

We will have data about the fit between undergraduate mathematics courses andemerging conceptions of mathematical knowledge for teaching. The research will contributeto theories of teacher knowledge by investigating the points of view of textbook authors whohave thought long and hard about the mathematics they include in their books and how theypresent it, and instructors who deal daily with prospective teachers’ learning. This is newresearch in areas about which little is known, but which could have important implicationsfor teacher education.

I have identified four broader impacts of this work. (a) We may learn aboutparticular courses and textbooks that make a difference in teachers’ knowledge, and in thatcase, will have data that provides insights into why and how these successes occur. Thisresearch could result in a more complete understanding of the mathematics teachers need andcould influence instructors of mathematics courses for elementary teachers to teach thatmathematics directly. Based on informal discussions with two of the authors, I know thattheir purposes in writing their books were similarly ambitious (Beckmann, 2004; Parker,2003). (b) The study will produce data about textbooks, courses, and instructorcharacteristics that may be useful at a practical level to institutions, departments, andinstructors as they make critical decisions about what to teach and how to teach it. (c) Thetextbook analysis framework could also become a model with wider application, for it iscontent- (rather than market-) driven, and provides a conceptual basis for considering thequalities of textbooks. (d) Finally, the instruments -- surveys of instructors and departments,and assessments of preservice teachers -- could be more widely used for a variety ofpurposes. Hyman Bass (personal conversation) urged me to administer the instructor surveyto a random national sample, a scope which seems out of reach for the research proposedhere, but could certainly be a goal for the future.

As the Principal Investigator for this work, I am in a good position to do this researchbecause of my background as a mathematician and mathematics teacher; my ongoingparticipation in mathematics teacher education; my interest in teacher learning and teacherknowledge; and my research on use of materials in teaching. I have been continuouslyinvolved in research on mathematics teacher learning and teacher knowledge since 1995,including working with some of the leaders in the field at both the MSU and the Universityof Michigan. I am currently working on another project about teacher knowledge, the NSFROLE KAT project, which is investigating algebra knowledge for teaching. Along withFerrini-Mundy, Floden, and Senk, I am currently a co-PI on the Kat project, which started onJune 15, 2004. In prior experience with NSF, I contributed to the first ROLE Algebra project(PI’s Chazan, Ferrini-Mundy, Floden, and Senk), doing analysis of video and developing aninterview protocol for practicing algebra teachers. As a graduate student at the University ofMichigan, I worked on the NASA, NSF, DARPA Digital Libraries Initiative project andpublished an article about our early work on using the Internet as a classroom resource(Wallace, Kupperman, Krajcik, & Soloway, 2000).

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Thompson, P. W. (1994). The development of the concept of speed and its relationship toconcepts of rate. In G. Harel & J. Confrey (Eds.), The development of multiplicativereasoning in the learning of mathematics. Albany, NY: SUNY Press.

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Publications, Inc.

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List of Mathematics Textbooks (In-print and In-preparation) for Undergraduate Courses forProspective Elementary Teachers

Bassarear, T. (2005). Mathematics for elementary school teachers (3rd Edition). New York:Houghton Mifflin.

Beckmann, S. (2004). Mathematics for elementary teachers. Boston, MA: Addison Wesley.Bennett, A., & Nelson, L. T. (2003). Math for elementary teachers: A conceptual approach,

Sixth Edition. New York: McGraw-Hill.Billstein, R., Libeskind, S., & Lott, J. W. (2001). A problem solving approach to mathematics

for elementary school teachers (7th ed.). Boston, MA: Addison Wesley.Center for Research in Mathematics and Science Education. (2000). Mathematics 201. San

Diego: San Diego State University.Darken, B. (2003). Fundamental Mathematics for Elementary and Middle School Teachers.

New York: Kendall/Hunt.Devine, D. F., Olson, J., & Olson, M. Elementary mathematics for teachers. New York: Wiley

and Sons. (recently out of print)Jensen, G. R. (2003). Fundamentals of Arithmetic: (self published).Jones, P., Lopez, K. D., & Price, L. E. (1998). A mathematical foundation for elementary

teachers. Reading, MA: Addison Wesley Higher Education.Krause, E. F. (1991). Mathematics for elementary teachers: A balanced approach. Lexington,

MA: D.C. Heath and Company. (recently out of print)Long, C. T., & DeTemple, D. W. (2003). Mathematical reasoning for elementary teachers (3

ed.). Reading, MA: Addison Wesley.Masingila, J. O., Lester, F. K., & Raymond, A. M. (2002). Mathematics for elementary

teachers via problem solving. New York: Prentice Hall.Musser, G. L., Burger, W. F., & Peterson, B. E. (2002). Mathematics for elementary school

teachers: A contemporary approach (6th ed.). New York: John Wiley & Sons.O'Daffer, P. G. (1998). Mathematics for elementary school teachers. Reading, MA: Addison-

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