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2000 51: 241Journal of Teacher EducationDeborah Loewenberg Ball

Bridging Practices: Intertwining Content and Pedagogy in Teaching and Learning to Teach  

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Journal of Teacher Education, Vol. 51, No. 3, May/June 2000Journal of Teacher Education, Vol. 51, No. 3, May/June 2000

BRIDGING PRACTICESINTERTWINING CONTENT AND PEDAGOGY INTEACHING AND LEARNING TO TEACH

Deborah Loewenberg BallUniversity of Michigan

Subject matter and pedagogy have been peculiarly and persistently divided in the conceptualizationand curriculum of teacher education and learning to teach. This fragmentation of practice leavesteachers on their own with the challenge of integrating subject matter knowledge and pedagogy inthe contexts of their work. Yet, being able to do this is fundamental to engaging in the core tasks ofteaching, and it is critical to being able to teach all students well. This article proposes three prob-lems that would have to be solved to bridge this gap and to prepare teachers who not only know con-tent but can make use of it to help all students learn. The first problem concerns identifying thecontent knowledge that matters for teaching, the second regards understanding how such knowledgeneeds to be held, and the third centers on what it takes to learn to use such knowledge in practice.

At the turn of the 20th century, John Dewey(1904/1964) articulated a fundamental tensionin the preparation of teachers: that of the“proper relationship” of theory and practice. Atthe turn of the 21st century, this tension endures.In fact, many of the same questions persist: Onone hand, to what extent does teaching andlearning to teach depend on the development oftheoretical knowledge and knowledge of sub-ject matter? On the other hand, to what extentdoes it rely on the development of pedagogicalmethod?

Clearly, the answer must be that it dependson both. Yet, across the century, this tension hascontinued to simmer, with strong views on bothsides of what is unfortunately often seen as adichotomy. Policy makers debate whetherteachers should major in education or in a disci-pline. Others argue that what matters is caringfor students and having the skills to work effec-tively with diverse learners. Dewey’s (1904/1964) conception of the relationship of subjectmatter knowledge and method was sophisti-cated and subtle, so much so that 100 years later,

his idea is still elusive. He wrote, “Scholasticknowledge is sometimes regarded as if it weresomething quite irrelevant to method. Whenthis attitude is even unconsciously assumed,method becomes an external attachment toknowledge of subject matter” (p. 160).

Dewey (1904/1964) believed that good teach-ers were those who could recognize and create“genuine intellectual activity” in students, andhe argued that methods of such activity wereintimately tied into disciplines. Subjects, hebelieved, were the embodiments of the mind,the product of human curiosity, inquiry, and thesearch for truth. A mind “which is habituated toviewing subject-matter from the standpoint ofthe function of that subject matter in connectionwith the mental signs of intellectual activitywhen exhibited in the child of four, or the youthof sixteen” would, in his view, be prepared tohear and extend students’ thinking. To do this,teachers would need to be able to study subjectmatter in ways that took it back to its “psychicalroots” (p. 162).

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Despite these prescient ideas, teacher educa-tion throughout the 20th century has consis-tently been structured across a persistent dividebetween subject matter and pedagogy. Thisdivide has many faces. Sometimes it appears ininstitutional structures as the chasm betweenthe arts and sciences and schools of education,or as the divide between universities andschools (Lagemann, 1996). Sometimes thedivide appears in the prevailing curriculum ofteacher education, separated into domains ofknowledge: educational psychology, sociologyof education, foundations, methods of teaching,and the academic disciplines corresponding toschool subjects. These knowledge chunks arecomplemented by experience: supervised prac-tice, student teaching, and practice itself. In allof these, the gap between theory and practicefragments teacher education by fragmentingteaching.

In recent years, in yet another peculiar frag-mentation, commitments to equity and con-cerns for diversity have often been seen as intension with concerns for content preparation.Yet, understanding subject matter is essential tolistening flexibly to others and hearing whatthey are saying or where they might be heading.Knowing content is also crucial to being inven-tive in creating worthwhile opportunities forlearning that take learners’ experiences, inter-ests, and needs into account. Contending effec-tively with the resources and challenges of adiverse classroom requires a kind of responsi-bility to subject matter, without which, efforts tobe responsive may distort students’ opportuni-ties to learn (Ball, 1995). Moreover, the creativityentailed in designing instruction in ways thatare attentive to difference requires substantialproficiency with the material. The overarchingproblem across these many examples is that theprevalent conceptualization and organizationof teachers’ learning tends to fragment practiceand leave to individual teachers the challenge ofintegrating subject matter knowledge andpedagogy in the contexts of their work. Weassume that the integration required to teach issimple and happens in the course of experience.In fact, however, this does not happen easily,and often does not happen at all.

A closer look at a sliver of the work of teach-ing serves to remind us of its demands. Con-sider a teacher examining and preparing toteach the following deceptively simple mathproblem (Gelfand & Shen, 1993),

Write down a string of 8’s. Insert some plus signs atvarious places so that the resulting sum is 1,000.

At first glance, this task may look trivial and un-interesting. One way of solving it entails simplyadding 125 8’s together. A closer look revealsthat if several 8’s are written together (i.e., 888 or88), many more solutions are possible. Workingon the problem a little further reveals interestingand provocative patterns in the solution set. Fig-uring out how to organize the solutions is itselfan interesting component of the work; depend-ing on how they are organized, different ele-ments of the problem and its solutions arevisible.

The teacher must contemplate: Would this bea good task for my students? What would it taketo figure out the patterns and nuances? Is itworthwhile in terms of what students mightlearn? At the very least, it would be important toknow what the problem is asking, whether ithas one or many solutions, and how the solu-tions might be found. It seems obvious that thetask entails some computation, for example,verifying any one solution, but what is themathematical potential of the task? Are thereimportant ideas or processes involved in theproblem? What would it take to use this taskwell with students? It would help to know whatmight make the problem hard, and how kidsmight get stuck, and anticipate what the teachermight do if they did. Would students find thisinteresting? What might it take to hook them onit?

Perhaps, after looking at this problem, theteacher decides that it is interesting, but a littletoo difficult for her students. What would it taketo make a mathematically similar problem thatis a bit easier? At what grade levels would somemathematically equivalent but simpler versionof this problem be accessible? How might onerescale the problem, for example, for third grad-ers? First graders? Suppose, in contrast, that theteacher worries that this problem is too easy.What would it take to make a more challenging,

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but again, mathematically similar task? Whathappens to the problem if one replaces 1,000with other numbers or 8 with some other digit?How might one modify the problem so thatthere are no solutions or infinitely many solu-tions? Would this be fruitful?

This sort of analysis and preparation of a sin-gle math problem reveals how much core tasksof teaching involve significant mathematicalreasoning in the context of practice. And theserepresent only a fraction of the work that ateacher would do to make productive use of thisproblem with students. When the teacher col-lects students’ work and peruses it, the teachermay grade it, determine where her students are,or decide whether to go further. When teachershold class discussions, they make decisionsabout which (and whose) ideas to pick up andpursue and which (and whose) to let drop. Theteacher formulates probes, pushes students,offers hints, and provides explanations. Stu-dents get stuck: What does one do to help themremobilize? None of these tasks of teaching ispossible to do generically. No matter how com-mitted one is to caring for students, to takingstudents’ ideas seriously, to helping studentsdevelop robust understandings, none of thesetasks of teaching is possible without making usein context of mathematical understanding andinsight.

Herein lies a fundamental difficulty in learn-ing to teach, for despite its centrality, usable con-tent knowledge is not something teacher educa-tion, in the main, provides effectively. Althoughsome teachers have important understandingsof the content, they often do not know it in waysthat help them hear students, select good tasks,or help all their students learn. Not being able todo this undermines and makes hollow theefforts to prepare high-quality teachers who canreach all students, teach in multicultural set-tings, and work in environments that maketeaching and learning difficult. Despite fre-quently heard exhortations to teach all students,many teachers are unable to hear students flexi-bly, represent ideas in multiple ways, connectcontent to contexts effectively, and think aboutthings in ways other than their own. For exam-ple, in their study of a middle school teacher’s

attempt to teach the concept of rate, Thompsonand Thompson (1994) highlight the crucial roleplayed by language. They vividly describe thesituation of one teacher who, although heunderstood the concept of rate himself, wasrestricted in his capacity to express or discussthe ideas in everyday language. Satisfied withcomputational language for his own purposes,when these did not help students understand,he was not able to find other means of express-ing key ideas. In addition, teachers may not beable to size up their textbooks and adapt themeffectively; they may omit topics central to stu-dents’ futures or make modifications that dis-tort key ideas. They may substitute studentinterest for content integrity in making choicesabout subject matter. Knowing subject matterand being able to use it is at the heart of teachingall students.

A recent analysis provides a glimpse of theimportance of the distinction between knowinghow to do math and knowing it in ways thatenable its use in practice. This distinction is keyto understanding how mathematics knowledgematters in good teaching. In general, astonish-ingly little empirical evidence exists to linkteachers’ content knowledge to their students’learning. One hypothesis has been that what isbeing measured as “content knowledge” (oftenteachers’ course attainment) is a poor proxy forsubject matter understanding. However, in anarticle describing their analysis of data from theNational Education Longitudinal Study of 1988,Rowan, Chiang, and Miller (1997) report strongpositive correlations between teachers’ responsesto items designed to measure the use of mathe-matical knowledge in teaching and their stu-dents’ performance (see Kennedy, Ball, & McDi-armid, 1993).1 This analysis provides someconfirmation that understanding the use ofmathematics in the work of teaching is a criticalarea ripe for further examination. It is not justwhat mathematics teachers know, but how theyknow it and what they are able to mobilizemathematically in the course of teaching.2

An important challenge for teacher educa-tion at the turn of the 21st century is to bridgethe chasm identified by Dewey (1904/1964)almost 100 years ago. Our schools are more

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diverse than ever, and we ask more of bothteachers and students. Finding ways to inte-grate knowledge and practice is essential if weare to help teachers develop the resources theyneed for their work. This is a call to preserviceteacher education, where academic studiesdominate, as well as to professional develop-ment, where opportunities to study content arefar more rare (Wilson, Theule-Lubienski, &Mattson, 1996). In neither setting is this dividebeing closed. What would it take to bring thestudy of content closer to practice and to pre-pare teachers to know and be able to use subjectmatter knowledge effectively in their work asteachers?

Three problems stand out, problems that wemust solve if we are to meet this challenge toprepare teachers who not only know contentbut can make use of it to help all students learn.The first problem concerns identifying the con-tent knowledge that matters for teaching, thesecond regards understanding how suchknowledge needs to be held, and the third cen-ters on what it takes to learn to use such knowl-edge in practice.

First, we would need to reexamine what con-tent knowledge matters for good teaching. Sub-ject matter knowledge for teaching has too oftenbeen defined by the subject matter knowledgethat students are to learn. Put simply, manyassume that what teachers need to know is whatthey teach, along with a broad perspective onwhere their students are heading. Nothing isinherently wrong with this perspective. How-ever, the lists of what teachers should know thatare produced by analyzing the school curricu-lum are long as well as arbitrary and unsubstan-tiated. Little is known about how “knowing”the topics on these lists affects teachers’ capa-bilities. The unexamined conviction that pos-sessing such knowledge is what teachers needto know has blocked the inquiry needed tobring together subject matter and practice inways that would enable teacher education to bemore effective.

Instead of beginning solely with the curricu-lum, our understanding of the content knowl-edge needed in teaching must start with prac-tice. We must understand better the work that

teachers do and analyze the role played by con-tent knowledge in that work. What are therecurrent core task domains of teachers’ work?For example, the teacher examining the 8’sproblem would have to probe the task, considerhow it might be done by particular students,decide how difficult it might be, and perhapsrescale it to make an easier or more challengingversion. In teaching it, the teacher would haveto listen carefully to what students said, inter-pret what they meant, ask them questions, givehints, and observe.

To improve our sense of what content knowl-edge matters in teaching, we would need toidentify core activities of teaching, such as figur-ing out what students know; choosing and man-aging representations of ideas; appraising,selecting, and modifying textbooks; and decid-ing among alternative courses of action, andanalyze the subject matter knowledge andinsight entailed in these activities. Thisapproach, a kind of job analysis of classroomteaching focused on the actual work that teach-ers do, could provide a view of subject matter asit is used in practice.

Turning the usual approach on its head, thisapproach would uncover what teachers need toknow and what they need to be sensitive toregarding content to teach well. This kind ofanalysis may bring some surprises. For exam-ple, in our recent work with Bass on mathemat-ics teaching (Ball & Bass, in press), we expectedto see that concepts such as place value anddecimal notation would be central, and theyhave been, as have operations and methods ofreasoning. However, beyond that, we have beenstruck by the unanticipated but recurrentprominence of certain mathematical notions.For instance, we have found that ideas aboutequivalence, similarity, and even isomorphismemerge across many instances of ordinary andextraordinary teaching and learning. We havealso uncovered salient issues involving mathe-matical language: symbolic notation, mappingamong representations, and definitions ofterms. Similarly new notions are emerging fromparallel work in the teaching and learning ofhistory and science (Rose, 1999; Wilson, inpress). Inquiries that begin with practice are

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revealing subject matter demands of teachers’work that are not seen when we begin with listsof content to be taught derived from the schoolcurriculum. These content demands emergefrom analyzing the sorts of challenges withwhich teachers must contend in the course ofpractice, as they mediate students’ ideas, makechoices about representations of content, mod-ify curriculum materials, and the like.

A second problem we would have to solveconcerns the assumption that someone whoknows content for himself or herself is able touse that knowledge in teaching. This is the prob-lem of how subject matter must be understoodto be usable in teaching. Simply increasingteachers’ opportunities to study mathematics,English, history, or physics—one easy response—will be insufficient to impact their capacities asteachers. We need to probe not only what teach-ers need to know but also what sort of contentunderstanding and insight matters in practice.

Viewed from the perspective of practice andthe actual work of teaching, at least two aspectsseem central. First is the capacity to deconstructone’s own knowledge into a less polished andfinal form, where critical components are acces-sible and visible. This feature of teaching meansthat paradoxically, expert personal knowledgeof subject matter is often ironically inadequatefor teaching. Because teachers must be able towork with content for students in its growing,unfinished state, they must be able to do some-thing perverse: work backward from matureand compressed understanding of the contentto unpack its constituent elements (Cohen, inpreparation). Knowing for teaching requires atranscendence of the tacit understanding thatcharacterizes and is sufficient for personalknowledge and performance (Polanyi, 1958).Understanding why a 6-year-old might write“1005” for “one hundred five,” and not readingit as a mistaken count, “one thousand five,”requires the capacity to appreciate the eleganceof the compressed notation system that adultsuse readily for numbers but which is not auto-matic for learners. After all, Roman numeralsfollow precisely the same structure as the youngchild’s inclination, each element with its ownnotation, CV for “one hundred five,” without

the place-value core of our system. Being able tosee and hear from someone else’s perspective,to make sense of a student’s apparent error orappreciate a student’s unconventionallyexpressed insight requires this special capacityto unpack one’s own highly compressed under-standings that are the hallmark of expert knowl-edge. Even producing a comprehensible expla-nation depends on this capacity to unpack one’sown knowledge, because an explanation worksonly if it is at a sufficient level of granularity, thatis, if it includes in it the steps necessary for thereasoning to make sense for a particular learneror a whole class, based on what they currentlyknow or do not know (Bass & Ball, in press).

Another aspect of being able to use theknowledge one has is the special sort of knowl-edge that Shulman and his colleagues (Shul-man, 1986, 1987; Wilson, Shulman, & Richert,1987) called “pedagogical content knowledge:”a special amalgam of knowledge that links con-tent and pedagogy. Included here is knowledgeof what is typically difficult for students, of rep-resentations that are most useful for teaching aspecific idea or procedure, and of ways todevelop a particular idea, for example, theordering of decimals or interpreting poetry.What are the advantages and disadvantages ofparticular metaphors or analogies? Wheremight they distort the subject matter? For exam-ple, both “take away” and “borrowing” createproblems for students’ understanding of sub-traction. These problems cannot be discernedgenerically because they require a careful map-ping of the metaphor against core aspects of theconcept being learned and against how learnersinterpret the metaphor. Knowing that subtrac-tion is a particularly difficult idea for students tomaster is not something that can be seen fromknowing the “big ideas” of the discipline. Thiskind of knowledge is not something a mathe-matician would necessarily have, but neitherwould it be familiar to a high school social stud-ies teacher. It is quite clearly mathematical, yetformulated around the need to make ideasaccessible to others. Pedagogical contentknowledge highlights the interplay of mathe-matics and pedagogy in teaching. Rooted in

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content knowledge, it comprises more thanunderstanding the content oneself.

These two aspects of content knowledge helpto illuminate the territory to which Dewey(1904/1964) called attention almost a centuryago, bridging the divide between content andpedagogy. However, they do not sufficientlyillumine where and how such knowledge isused in teaching. Teaching is a practice. It is, inLampert’s (1998) terms, “a thinking practice”;that is, it integrates reasoning and knowingwith action. Our tendency to focus either on itscognitive demands (teachers’ knowledge, rea-soning, decision making, reflection) or on itsactions (teacher behavior) is yet one more recentform of fragmentation in teacher education, andin particular, in our efforts to help teachersacquire usable content knowledge.

Hence, a third problem we would have tosolve is how to create opportunities for learningsubject matter that would enable teachers notonly to know but to learn to use what they knowin the varied contexts of practice. Even withmore grounded analyses of what there is toknow and a more finely tuned conception of thenature of the understanding needed to teach,simply teaching such content may not solve theproblems of use. How do teachers use contentunderstanding in the context of practice to carryout the core activities of their work? How canwe design opportunities for learning that areaimed at helping teachers use subject matterknowledge to figure out what their studentsknow, to pose questions, to evaluate and modifytheir textbooks wisely, to design instructionaltasks, to manage class discussions, and toexplain the curriculum to parents?

Some such work along these lines is alreadyunderway. One promising possibility is todesign and explore opportunities to learn con-tent that are situated in the contexts in whichsubject matter is used, a core activity of teach-ing. For example, some teacher educators usestudent work as a site to analyze and interpretwhat students know and are learning and, in sodoing, work on the content itself.3 Anotherexample is the use of videotaping classroom les-sons or cases of classroom episodes (Lampert &Ball, 1998; Stein, Smith, Henningsen, & Silver,2000), from which, the moves made by the

teacher could be analyzed to consider theimpact on the course of the lesson, the trajectoryof the class’s work, and the opportunities forlearning for particular students and for thegroup. In both instances (using student work orusing videotapes or cases of classroom lessons),teachers or prospective teachers might engagein content-based design work, developing apossible next assignment in response to theiranalysis of students’ work, or planning a nextinstructional segment based on analysis of theclassroom episode. Each of these activities takea task of teaching that entails content knowl-edge and creates a possible site for teachers’learning of content in the contexts where theywill have to use it.

However, much more work is needed to con-tend with this endemic problem of use. Workingin specific contexts might run the risk of limitingthe generality of teachers’ learning of contentand their capacity to use it in a variety of con-texts. How can teachers be prepared to suffi-ciently know content flexibly so that they areable to make use of content knowledge with awide variety of students across a wide range ofenvironments? How could teachers develop asense of the trajectory of a topic over time orhow to develop its intellectual core in students’minds and capacities so that they eventuallyreach mature and compressed understandingsand skills?

Solving these three problems—what teachersneed to know, how they have to know it, andhelping them learn to use it, by grounding theproblem of teachers’ content preparation inpractice—could help to close the gaps that haveplagued progress in teacher education. How-ever, we should realize the challenges thatdoing this would pose. After all, Dewey(1904/1964) thought his vision at the turn of the20th century was imminently realizable. Hethought that what he was describing was “noth-ing utopian.” He suggested that “the presentmovement . . . for the improvement of range andquality of subject matter is steady and irresisti-ble” (p. 170). Almost 100 years later, as we stareat university and college catalogs that dividemethods courses from disciplinary studies frompractice, or at professional development offer-ings that are devoid of content or chock full of

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activities for kids, we should understand thatbridging these strangely divided practices willbe no small feat.

ACKNOWLEDGMENT

The ideas in this article about subject matterknowledge in teaching have been helped enor-mously through my work and discussions withHyman Bass, David Cohen, Magdalene Lam-pert, Suzanne Wilson, and Joan Ferrini-Mundy.

NOTES1. These items were developed at the National Center for Re-

search on Teacher Learning, Michigan State University.2. These ideas about the use of mathematics knowledge in

teaching draw from my work with both David K. Cohen and Hy-man Bass. See, for example, Cohen and Ball (1999) and Ball andBass (in press).

3. Several professional development curricula are built on thisidea. See, for example, Schifter (1998) and Barnett (1998).

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mathematical communities. A response to TimothyLensmire and John Pryor. Harvard Educational Review,65, 670-677.

Ball, D. L., & Bass, H. (in press). Making believe: The collec-tive construction of public mathematical knowledge inthe elementary classroom. In D. Phillips (Ed.), Construc-tivism in Education. Chicago: University of ChicagoPress.

Barnett, C. (1998). Mathematics Case Methods Project.Journal of Mathematics Teacher Education, 1(3), 349-356.

Cohen, D. K. (in preparation). Teaching practice and its pre-dicaments. Unpublished manuscript, University ofMichigan, Ann Arbor.

Cohen, D. K., & Ball, D. L. (1999). Instruction, capacity, andimprovement. (CPRE Research Report No. RR-043).Philadelphia: University of Pennsylvania, Consortiumfor Policy Research in Education

Dewey, J. (1964). John Dewey on education (R. Archambault,Ed.). Chicago: University of Chicago Press. (Originalwork published 1904)

Gelfand, I. M., & Shen, A. (1993). Algebra. Boston:Birkhäuser.

Kennedy, M. M., Ball, D. L., & McDiarmid, G. W. (1993). Astudy package for examining and tracking changes in teach-ers’ knowledge (Technical Series 93-1). East Lansing, MI:National Center for Research on Teacher Education.

Lagemann, E. (1996). Contested terrain: A history of educationresearch in the United States, 1890-1990. Commissionedpaper, Spencer Foundation, Chicago, IL.

Lampert, M. (1998). Studying teaching as a thinking prac-tice. In J. Greeno & S. G. Goldman (Eds.), Thinking prac-tices (pp. 53-78). Hillsdale, NJ: Lawrence Erlbaum.

Lampert, M., & Ball, D. L. (1998). Mathematics, teaching, andmultimedia: Investigations of real practice. New York:Teachers College Press.

Polanyi, M. (1958). Personal knowledge: Towards a post-critical philosophy. Chicago: University of Chicago Press.

Rose, S. L. (1999). Understanding children’s historical sense-making: A view from the classroom. Unpublished doctoraldissertation, Michigan State University, East Lansing.

Rowan, B., Chiang, F., & Miller, R. (1997). Using researchon employees’ performance to study the effects ofteachers on students’ achievement. Sociology of Educa-tion, 70(4), 256-284.

Schifter, D. (1998). Learning mathematics for teaching:From the teacher’s seminar to the classroom. Journal forMathematics Teacher Education, 1(1), 55-87.

Shulman, L. S. (1986). Those who understand: Knowledgegrowth in teaching. Educational Researcher, 15, 4-14.

Shulman, L. S. (1987). Knowledge and teaching: Founda-tions of the new reform. Harvard Educational Review, 57,1-22.

Stein, M. K., Smith, M. S., Henningsen, M., & Silver, E.Exploring cognitively challenging mathematical tasks: Acasebook for teacher development. New York: TecahersCollege Press.

Thompson, P., & Thompson, A. (1994). Talking about ratesconceptually, Part I: A teacher’s struggle. Journal forResearch in Mathematics Education, 25, 279-303.

Wilson, S. M. (in press). Research on history teaching. In V.Richardson (Ed.), Handbook of Research on Teaching. NewYork: Macmillan.

Wilson, S. M., Theule-Lubienski, S., & Mattson, S. (1996,April). Where’s the mathematics? The competing commit-ments of professional development. Paper presented at theannual meeting of the American Educational ResearchAssociation, New York.

Wilson, S. M., Shulman, L. S., & Richert, A. (1987). 150 dif-ferent ways of knowing: Representations of knowledgein teaching. In J. Calderhead (Ed.), Exploring teacherthinking (pp. 104-124). Sussex, UK: Holt, Rinehart &Winston.

Deborah Loewenberg Ball is a professor of mathe-matics education and teacher education at the Universityof Michigan. Her work as a researcher and teacher educa-tor is rooted in practice, drawing directly and indirectly onher experience as a classroom teacher. Ball’s work focuseson studies of instruction and of the processes of learning toteach. She also investigates efforts to improve teachingthrough policy, reform initiatives, and teacher education.She has published articles on teacher learning and teachereducation, the role of subject matter knowledge in teachingand learning to teach, endemic challenges of teaching, andthe relations of policy and practice in instructionalimprovement.

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