preparing teachers to learn from teachingudel.edu/~jansen/hiebertetal_jte_2007.pdf · preparing...

PREPARING TEACHERS TO LEARN FROM TEACHINGJames HiebertAnne K. MorrisDawn BerkAmanda JansenUniversity of Delaware

The authors propose a framework for teacher preparation programs that aims to help prospectiveteachers learn how to teach from studying teaching. The framework is motivated by their interestin defining a set of competencies that provide a deliberate, systematic path to becoming an effectiveteacher over time. The framework is composed of four skills, rooted in the daily activity of teach-ing, that when deployed deliberately and systematically, constitute a process of creating and test-ing hypotheses about cause–effect relationships between teaching and learning during classroomlessons. In spite of the challenges of acquiring these skills, the authors argue that the frameworkoutlines a more realistic and more promising set of beginning teacher competencies than those oftraditional programs designed to produce graduates with expert teaching strategies.

Keywords: analyzing teaching; learning to teach; teacher preparation

How should teacher preparation programs bedesigned to ensure the graduates becomeexpert teachers? A recently released volume bythe American Educational Research Association(AERA) indicates that the question is not evenclose to being answered empirically (Cochran-Smith & Zeichner, 2005). This is not a newconclusion. The absence of research-based rec-ommendations cited by authors of relevantchapters in the AERA volume (e.g., Floden &Meniketti, 2005; Zeichner & Conklin, 2005) rein-forces similar conclusions reached in earlierreviews (Evertson, Hawley, & Zlotnick, 1985;Kennedy, 1999; Raths & McAninch, 1999).

The most common approach to teacherpreparation—equipping prospective teacherswith expert teaching strategies—has been con-vincingly critiqued on theoretical and practical

grounds. Nemser (1983) argued that expectingprospective teachers to become expert class-room performers on graduation is unrealisticgiven the short time of preparation programsand the strong influences on teaching of priorexperiences. Prior experiences, acquired duringyears in classrooms as students, heavily influ-ence how prospective teachers interpret whatthey are learning and how they end up teaching(Borko & Putnam, 1996; Lortie, 1975). In addi-tion, Berliner (1994) noted the long learningcurve that characterizes expert teachers, a learn-ing curve that cannot be traversed very far dur-ing a preparation program.

In the absence of empirical and theoreticalsupport for traditional forms of teacher prepara-tion, it is appropriate to consider alternatives.One alternative approach to teacher preparation

Authors’ Note: Preparation of this article was supported, in part, by the National Science Foundation, Grant 0083429 tothe Mid-Atlantic Center for Teaching and Learning Mathematics. The opinions expressed in the article are those of theauthors and not necessarily those of the Foundation. We thank Tonya Bartell, four anonymous reviewers, and the editorsfor their comments on an earlier draft of this article. Correspondence concerning this article should be addressed to JamesHiebert, School of Education, University of Delaware, Newark, DE 19716; e-mail: [email protected].

Journal of Teacher Education, Vol. 58, No. 1, January/February 2007 47-61DOI: 10.1177/0022487106295726© 2007 by the American Association of Colleges for Teacher Education

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that has a long but less noticed history in theliterature is to design programs that prepareprospective teachers to learn from teachingwhen they enter the profession. Schaefer (1967)hinted at the possibilities of this approach whenhe outlined the advantages of structuring K-12schools as places where teachers, not juststudents, could learn. Hawkins (1973) was morespecific about the role of preparation programsin this long-term learning process:

It may be possible to learn in two or three years thekind of practice which then leads to another twentyyears of learning. Whether many of our colleges getmany of their students onto that fascinating track . . .is another matter. (p. 7)

Nemser (1983) was even more explicit aboutthe advantages of this alternative approach overconventional preparation programs:

It would be far more realistic to think about prepar-ing people to begin a new phase of learning to teach.That would orient formal preparation more towarddeveloping beginning competence and laying thefoundation for learning and teaching. (p. 157)

If a preparation program took seriously the goalof preparing teachers to learn from teaching,what would such a program look like? Whatknowledge, skills, and dispositions wouldteachers need to learn from teaching—not in aninformal, haphazard way but in an intentional,systematic way?

The purpose of this article is to outline someof the specifics of a teacher preparation programthat aims to help prospective teachers learn howto teach from studying teaching. We build onand extend the work of others (e.g., Hiebert,Morris, & Glass, 2003; Santagata, Zannoni, &Stigler, in press; van Es & Sherin, 2002) to pro-pose a set of skills that will prepare prospectiveteachers to continue learning from their practicewhen they begin teaching. We draw our exam-ples from mathematics; however, we believe theskills apply equally well to all school subjects.

We present our proposal as a hypothesis to betested. It arises from our interpretation of the lit-erature, from our analysis of teachers’ everydaywork, and from arguments about the processof improving complex, goal-oriented skills.Although we believe the proposal is sufficiently

compelling to merit serious debate and empiri-cal testing, we acknowledge that it is withoutdirect empirical support. Later we review someindirect supportive evidence; however, weknow of no research that has shown thatteachers prepared in the way we propose per-form more effectively than those prepared inmore traditional programs. The absence of rele-vant data is due, in large part, to the fact thatpreparation programs of the kind we describehave not been operational with enough fre-quency and intensity to allow follow-up studieswith graduates. So, the case we present rests onpersuasive argument and is intended to gener-ate discussion and empirical scrutiny.

LEARN TO ANALYZE TEACHING INTERMS OF STUDENT LEARNING:THE CENTRAL CONSTRUCT

The goal of teaching is to support studentlearning. It is hard to imagine teachers becom-ing more effective over time without being ableto analyze teaching in terms of its effects onstudent learning. What did students learn, andhow and why did instruction influence suchlearning? How could lessons based on this infor-mation be revised to be more effective whenteaching them next time? We propose that assess-ing whether students achieve clear learninggoals and specifying how and why instruc-tion did or did not affect this achievement lies atthe heart of learning to teach from studyingteaching. This kind of analysis is different thanthat which focuses on particular features ofteaching or behaviors of teachers, such as askinghigher order questions or managing discourse.We propose that focusing on students’ learningand explaining such learning (or its absence) interms of instructional episodes provides a tar-geted but comprehensive and systematic path tobecoming an effective teacher over time.

We propose that two quite different kindsof knowledge, skills, and dispositions—orcompetencies—contribute to analytic expertiserequired to study and improve teaching overtime. A first kind of competence is subjectmatter knowledge for teaching. This is the kindof subject matter knowledge needed to unpackthe content learning goals for students, to

48 Journal of Teacher Education, Vol. 58, No. 1, January/February 2007

understand students’ thinking about the subject,to simplify the complex ideas of the subject inways that sustain the integrity of the subject, torepresent ideas in accessible ways for students,to pose key questions and problems, and so on.Shulman (1986) captured much of this kind ofcompetence as “pedagogical content knowl-edge,” and others have extended and refinedthese ideas, especially in mathematics (Ball,1999; Ball & Bass, 2000; Ma, 1999; Sherin, 2002).Data are beginning to move beyond the vague(and often absent) correlations between subjectmatter knowledge and teacher effects (Floden &Meniketti, 2005) to show that mathematicsknowledge for teaching influences how teachersteach and how well students learn (Borko et al.,1992; Hill & Ball, 2004; Hill, Rowan, & Ball, 2005;Sherin, 2002; Stein, Baxter, & Leinhardt, 1990). Itis plausible that improvements in particularkinds of subject matter competence contribute tobetter analyses of practice which, in turn, yieldimproved teaching.

The second kind of competence that webelieve contributes to a teacher’s ability to ana-lyze teaching is a collection of dispositions,knowledge, and reasoning skills that enabledeveloping and testing hypotheses aboutcause–effect relationships between teaching andlearning. For purposes of presentation, we cap-ture these competencies as four skills: (a) settinglearning goals for students, (b) assessing whetherthe goals are being achieved during the lesson, (c)specifying hypotheses for why the lesson did ordid not work well, and (d) using the hypothesesto revise the lesson. The framework we describeforegrounds these skills, which have receivedless attention in the literature than subject matterexpertise and which, we argue, merit furtherexploration.

Evidence from other domains indicates thatthis kind of analysis—specifying cause–effecthypotheses and testing them so that useful anddefensible claims can be drawn from theevidence—is not commonly displayed and ischallenging to develop (Moshman, 2005;Sandoval & Millwood, 2005; Smith & Osherson,1995). Even though many teachers, includingnovices, probably engage in some kind of infor-mal and intuitive reflection on the success of a les-son and on the factors that might be responsible,

we expect that fostering intentional and system-atic attention to cause–effect relationships will bea major challenge for the program we propose.

The challenges of developing analytic skills tostudy teaching mean that the teacher prepara-tion we outline is not an easy substitute for tra-ditional programs. For our proposed programto succeed, teacher educators will need to learnmore than is currently known about the level ofanalytic competence possessed by prospectiveteachers when they enter preparation programsand about how to facilitate the development ofthis competence. As we argue later in the article,we believe that, in spite of these challenges, theproposed program is feasible.

We do not claim that analyzing teaching isthe only competence that prospective teachersmust have to begin a career path that leads toincreasing effectiveness over time. They arelikely to benefit from acquiring other pedagogi-cal competencies, such as common performanceroutines—reviewing homework, introducingand summarizing the main point of a lesson, con-structing and interpreting assessments, and so on(Berliner, 1994; Brown & Borko, 1992; Leinhardt,1993). Perhaps the most effective preparationprograms will be those that balance attention todeveloping pedagogical classroom skills and theanalytic skills we describe. However, we empha-size the analytic skills, not only because theyhave received less attention but also because webelieve the core of teaching—interacting withstudents about the content—is not learned wellthrough automatizing routines or even throughacquiring expert strategies during a teacherpreparation program. Rather, it is learned throughcontinual and systematic analysis of teaching.

A consequence of focusing on analytic skills isthat the center of teaching expertise shifts fromon-the-fly performance in the classroom topreparation and reflection outside the class-room. Although some researchers questionwhether preparation and real-time teachingcan be meaningfully distinguished (see Brown& Borko, 1992, for a review), we believe it is use-ful to separate the kind of planning and reflec-tion we have in mind from real-time teaching inthe classroom. Teaching expertise, from our per-spective, includes planning to learn from teach-ing (one’s own teaching and the teaching of

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others) and revising practice based on the datacollected. This is quite different from models ofteaching expertise that focus on the increasinglyfluid and intuitive nature of classroom perfor-mance (Berliner, 1994).

That analytic activity can be engaged out-side of the classroom and away from the pressof real-time classroom performance suggeststhat it might be suited to teacher preparationprograms. Numerous samples of teaching,captured in videos or transcripts of classroomlessons, can be slowed down, dissected, parsedinto meaningful chunks, and recomposed. Theskills can be scaled up either by addressing oneskill at a time or soliciting all of them initiallyon a constrained and carefully selected teach-ing episode. In general, teaching can be ana-lyzed deliberately and systematically. Teachingcan be treated as an object of study. These fea-tures encourage us to agree with Hawkins (1973)and Nemser (1983) and propose that prospectiveteachers can be prepared during their trainingprograms to learn from teaching.

A FRAMEWORK FORANALYZING TEACHING

The framework we propose consists of fourskills drawn from the daily routines of ordi-nary classroom teachers as they plan, imple-ment, and reflect on classroom lessons. Asidentified earlier, the skills are setting learninggoals for students, assessing whether the goalsare being achieved during the lesson, develop-ing hypotheses about why the lesson did or didnot work well, and revising the lesson on thebasis of these hypotheses. In this section, wedescribe these skills and argue for elevatingthem to conscious, deliberate attention.

Although these skills are rooted in the dailyactivity of teaching, they take on the nature ofresearch or inquiry skills when they are deployeddeliberately and systematically (Dewey, 1929).The compatibility of teaching and researchinghas been drawn numerous times in the litera-ture, perhaps most frequently by those promot-ing or examining the idea of teacher asresearcher (Burnaford, Fischer, & Hobson, 1996;Cochran-Smith & Lytle, 1993, 1999). Our frame-work shares some features with teacher as

researcher in its attention to collecting concreteevidence to inform decisions about teaching;however, there are important differences aswell. Our framework does not include thedevelopment of research skills needed to con-duct a range of research studies in the class-room. Rather, “research” is invoked in ourdescriptions to convey a critical and empiricaldisposition toward planning and reflecting onteaching. In this sense, our framework outlinesa research process of intentional learning fromcarefully planned experiences as part of thedaily routine of teaching. Indeed, it equatesteaching expertise, at least in part, with theresearch-like skills needed to analyze andimprove teaching.

In its inquiry stance, with an emphasis ongathering data on students’ thinking, theframework we propose also shares a centralorientation with some of the literature onreflective practice (Osterman & Kottkamp,2004). The framework is especially alignedwith interpretations like those of Cruickshankand Applegate (1980) who described reflectivepractice as a process that helps teachers thinkabout what happened during a classroom les-son, why it happened, and what could be donenext time to make it happen more successfully.As will be seen, our framework focuses reflec-tive practice squarely on the relationshipsbetween instructional practices and students’achievement of the intentional learning goals.

Skill 1: Specify the Learning Goal(s)for the Instructional Episode (WhatAre Students Supposed to Learn?)

Analyses of teaching involve assessing theeffects of an instructional episode against pre-cisely defined learning goals. An episode mightbe a task or activity that constitutes part of adaily lesson, a full daily lesson, or a sequence oflessons. By defining the learning goals for anepisode precisely and explicitly, it is possible toinvestigate whether and how instruction facili-tated or inhibited students’ achievement of thegoals.

Specifying learning goals is the first skill inthe proposed framework because until learn-ing goals are expressed clearly, further analyses


are impossible. Without explicit learning goals,it is difficult to know what counts as evidenceof students’ learning, how students’ learningcan be linked to particular instructional acti-vities, and how to revise instruction to facili-tate students’ learning more effectively in futurelessons. Formulating clear, explicit learning goalssets the stage for everything else.

Describing learning goals precisely requiresunpacking them into component goals or sub-goals. The more specifically learning goals aredescribed, the more detailed and useful the sub-sequent analysis of teaching. Consider a commonlearning goal in a sixth-grade mathematics class:“Students will understand the concept of a dec-imal fraction.” Leaving the goal at this level ofgenerality provides little guidance for subse-quent analyses. What counts as evidence thatstudents “understand the concept?” Specifyingthe goal further—”Students should recognizethat the value of a digit is determined by itsplace in the numeral, that the values of the adja-cent places are more (or less) than each other bya factor of 10, and that powers of 10 can includenegative exponents”—provides better guidancefor studying the relationships between teachingand learning during the lesson.

Unpacking learning goals is one site in theframework where subject matter competence issure to make a difference. In fact, the skillfulspecification of learning goals could be used tohelp define the kind of subject matter knowl-edge for teaching that prospective teachers needto acquire. What does one need to know to takeapart larger goals into related, component goalsand to connect component goals to create largergoals (Ma, 1999)? Although little empirical datahave addressed the way in which subject matterknowledge influences teaching at this levelof detail, we suspect that this first skill in ana-lyzing teaching provides a critical mechanismthrough which subject matter knowledge affectsteaching.

Two caveats conclude our description of thefirst skill in the framework. Unintended learn-ing, not captured by the learning goals, is likelyto occur during an instructional episode. In thisarticle we focus on learning intended by theteacher, learning aligned with explicit learninggoals. Although unintended learning might be

important, intended learning is the kind oflearning teachers can plan for and get better atfacilitating. By definition, accidental or uninten-tional learning is unplanned and, consequently,beyond the reach of systematic study. Becausewe are proposing a preparation program forteachers, we focus on skills that can be improvedby intentional, systematic study.

A second caveat is that the skill of specifyinglearning goals takes different forms dependingon whether one is analyzing someone else’steaching or one’s own teaching. Learning goalsare easier to specify if you are the teacher.Assuming that prospective teachers also analyzethe teaching of others, say from a videotapedlesson, they probably will need supplementaryresources to identify learning goals. Goals arenot always made explicit by the teacher duringthe lesson. Written lesson plans, textbook pagesand teacher handouts, commentaries by theteacher about the lesson, and, ideally, interviewswith the teacher can help to specify the learninggoals so that the analysis process can proceed.

How can the development of prospectiveteachers’ skill in specifying learning goals bemeasured? Two criteria emerge from the previ-ous discussion. First, goal descriptions are moreuseful when they are more specific, when theyinclude subgoals and primary or general goals.Second, goal descriptions are more useful whenthey use the language of the subject. A goal of“80% correct by each student on the quiz at theend of the lesson” is not as useful as a goal of“students should construct relationshipsbetween the value of a decimal fraction, the sumof the values of each of its digits, and increasing(and decreasing) powers of 10.” Goal descrip-tions that meet these criteria are useful becausethey set the stage for what will count as evidencethat students are achieving them. This leads tothe second skill.

Skill 2: Conduct Empirical Observationsof Teaching and Learning (WhatDid Students Learn?)

When learning goals are specified, evidencecan be collected about whether, and to whatextent, each student is achieving the goals.Conducting appropriate empirical observations


to collect the evidence involves (a) appreciatingthat evidence about students’ learning is essen-tial for assessing the effects (and effectiveness)of teaching—indeed, no other information willsuffice; (b) recognizing what counts as evidencethat students are achieving the learning goals—distinguishing students’ responses that arerelevant from those that are irrelevant; and (c)knowing how to collect evidence—identifyingkey moments in a lesson where evidence ofstudents’ learning should be apparent and plan-ning ways to collect it from each student. Insummary, conducting useful empirical observa-tions requires knowing that, knowing what, andknowing how.

Knowing that evidence on students’ learningis needed to assess the effectiveness of teachingis not as obvious as it might sound. It is temptingto assess teaching effectiveness based on whatthe teacher does rather than on how the studentsrespond (Morris, in press; Santagata et al., inpress). The presence of teaching features thatalign with current reform recommendations, forexample, can be and often are interpreted as asign of effectiveness regardless of student learn-ing (Hiebert & Stigler, 2000; Spillane & Jennings,1997). And teachers often analyze their practicein terms of a smooth implementation of activitiesrather than an anticipated change in students’thinking (McCutcheon, 1980; Zahorik, 1970). Theshift from focusing on the teacher to focusing onthe students represents, by itself, a significantdevelopment of this second skill.

It is likely that prospective teachers will strug-gle to make this shift—to focus on their studentsas learners rather than on themselves as teachers.Beginning teachers are likely to be especially con-scious of their own performance. In addition,attending to students is motivated by concernabout the effectiveness of a teaching episode,and prospective teachers have not yet encoun-tered experiences that would prompt them toquestion their own effectiveness. Teachers’ self-efficacy doubts might support closer attention totheir students’ learning because the disequilib-rium teachers experience from questioning theireffectiveness can promote changes in their per-spectives (Wheatley, 2002). Knowing that evi-dence on students’ learning provides criticalinformation for improving teaching provides a

useful tool when prospective teachers encounterdoubts about their own effectiveness.

Knowing what counts as evidence of students’learning means distinguishing students’ respon-ses that reveal how well they are achieving thelearning goal from responses that are relevantto the learning goal but uninformative (e.g.,students nodding their heads in agreement inresponse to a relevant question from the teacher)and from responses that are informative but notrelevant to the learning goal (e.g., a student’sdescription of how to generate repeating deci-mals on the calculator when the goal is to relatethe value of a decimal number with the value ofits digits). In general, the most useful data arestudents’ responses that use specific language ofthe subject and align with the learning goals.These responses could be verbal statements bystudents during class or written work completedby students during or after class.

Knowing what counts as evidence meansdeciding not only what kind of student work tocollect but knowing what about this work revealsstudents’ achievement of the learning goal. Thisrequires a set of competencies or skills that drawdirectly on subject matter knowledge combinedwith knowledge of student thinking. First, ateacher must be ready to observe (and eventuallypredict) the kinds of strategies that students willuse to solve a problem or the kinds of responsesthey will give to a question. Without preparing tohear or read the range of possible responses, it iseasy to overlook a response that counts as evi-dence. Second, a teacher must know what a par-ticular response implies about the student’sthinking. What would the student have to knowto give this response? Without engaging in thisanalysis, it is impossible to know what responsescount as evidence of students’ learning relevantto the goal.

Obviously, the development of these skillsdoes not occur overnight. However, teacherscan become increasingly skillful at collectingand studying students’ work and using it toinform their own practice (Kazemi & Franke,2004). One practical strategy for focusing atten-tion on what counts as evidence is to study anintended lesson plan and predict how studentswill respond during the lesson, asking, before-hand, whether these responses would count as


evidence and, if not, what kinds of responseswould count as evidence.

Knowing how to collect revealing and relevantevidence builds directly on knowing what countsas evidence. Knowing what evidence to collecthelps teachers know where to look for it. Thismeans planning to collect evidence during keyparts of the lesson and then collecting it fromevery student as the lesson plays out. Planningto collect evidence involves identifying keymoments during the instructional episode thatcould, and perhaps should, reveal students’achievement of the learning goals. Parsing the les-son or episode in this way is facilitated by clearspecifications of subgoals because it is by zeroingin on where these subgoals will be addressed thatspecific potential learning moments can be iden-tified beforehand. Collecting evidence during thelesson involves looking for revealing studentevidence as planned and, at the same time, beingopen to unanticipated key moments and unex-pected student evidence. Revealing evidence islikely to include a variety of responses from indi-vidual students signaling that different studentsare at different places in their achievement ofthe learning goal. Of course, careful listening andgathering targeted written work will be rewardedonly if the instructional episode affords studentsthe opportunity to reveal what they know.

Consider again the sixth graders who arelearning about decimal fractions. The lesson planindicated that the teacher intended to focusstudents’ attention on the relative size of adjacentpositions by asking students to represent 2.36with base-10 blocks (blocks of four sizes begin-ning with a small cube, a long [10 cubes gluedtogether], a flat [10 longs], and a large cube [10flats]). When showing decimal numbers with theblocks, students had used only the large cube torepresent 1. After students showed 2.36 with theblocks, the teacher planned to ask them to repre-sent the same number with blocks but this timeto use the flat to stand for 1. The teacher expectedthat changing the unit—which block represents1—would be a key learning moment becausestudents would need to mentally detach a partic-ular block from a particular value, or from a par-ticular position in a number’s written form. This,the teacher expected, would increase the chances

that students would focus on the 10-times-biggerrelationship that is an important subgoal (identi-fied earlier). Consequently, the teacher assumedthat evidence of students’ thinking might berevealed by observing how they solved this sec-ond problem.

As the lesson unfolded, the teacher observedthat most of the students showed the correct col-lections of blocks for 2.36 using the flat as the unitbut, when asked to explain why they chose theirset, their responses varied. Some students saidthat when they were told the flat block shouldrepresent 1, they knew they needed a block 10times smaller for the tenths. Other studentsoffered a variation of this explanation by sayingthey used the long for the tenths because 10 longsmake 1 flat. However, some students simply saidthey “lined up the blocks in order.” Apparentlysome students shifted their focus, as intended,from the size of the blocks to the 10-for-1 rela-tionship between the blocks; however, somestudents just followed the pattern “pick the nextsmallest block for the next digit” and did notthink about the 10-for-1 relationship.

This example illustrates an appreciationfor students’ responses that provide evidenceof teaching effectiveness, a beginning notion ofwhat counts as evidence, and a plan for whenand how to collect evidence (around a keylearning moment). The next step (Skill 3) is todevelop a hypothesis that might account forthese findings in terms of the instructionalevents that preceded them.

As with the first skill, criteria that can be usedto measure growth of conducting empiricalobservations of learning (Skill 2) are suggested inthe previous discussion. One criterion is crucial:The evidence should reveal clearly whether thelearning goals were achieved. This criterion ismore likely to be satisfied when the evidence ismore detailed and specific, especially when thespecificity connects the evidence with particularsubgoals. In addition, the evidence is more use-ful when it captures students’ thinking that liesbeneath surface behaviors and responses.Finally, the evidence is most useful when it rep-resents the range of thinking in the classroomrather than the selected thinking of the mostvocal students.


Skill 3: Construct Hypotheses Aboutthe Effects of Teaching on Students’Learning (How Did Teaching Help[or Not] Students Learn?)

Developing hypotheses that link teaching withlearning requires forming conjectures about howa particular instance of teaching (task, question,activity, etc.) facilitated or inhibited a particularkind of learning. Whereas Skill 2 focuses onstudents’ responses, Skill 3 shifts to the instruc-tional events that facilitated these responses. Thehypotheses are tentative claims about how theinstructional event influenced the intended learn-ing. The hypotheses specify questions that meritfurther examination in future instructionalepisodes. Hypotheses about cause–effect relation-ships can be made at different levels of localness.Hypotheses might specify connections betweenteaching and learning at particular key momentsof a lesson that focus on a specific subgoal, or theymight be framed at more general levels such asstudents’ learning at the end of a lesson connectedwith a series of teaching activities that occurredduring the lesson. Regardless of the level of local-ness, hypotheses must be stated with enoughdetail and specificity that they can be tested insubsequent instructional episodes.

How are hypotheses produced? Evidencethat students did or did not advance toward thelearning goal at a critical point in the lesson pro-vides important information about the learningend of the hypothesis. However, how does onegenerate ideas that specify the kinds of instruc-tional events that might have contributed to suchlearning or whose absence might account for thelack of learning? An expert might imagine manyalternative hypotheses. However, how are thesehypotheses generated, especially by novices, andhow are selections made among them?

A set of principles of learning and teachingcan guide the formulation and selection of usefulhypotheses that connect instruction with studentlearning. Such principles often describe theinstructional conditions under which particularkinds of learning are likely to occur. Supposeintended learning did not occur. Principles thatspecify the conditions most likely to supportlearning of that kind offer explanations for

why the learning did not occur and point toinstructional tasks, explanations, questions, andso on that might have promoted such learning.

In our work with prospective teachers, learn-ing goals for school students often involve con-ceptual understanding of mathematics. Drawingfrom theoretical and empirical work, two princi-ples that link teaching with achievement of thesegoals are (a) the explicit discussion and/orexamination of the critical mathematical rela-tionships and (b) the opportunity for students towrestle or struggle with key mathematical ideas(Brownell, 1935; Fennema & Romberg, 1999;Hiebert & Grouws, 2007; National ResearchCouncil, 2001).

If the learning goals are something other thanconceptual understanding, then other principlesmight apply. For example, rapidly paced instruc-tion, frequent feedback, and smooth transitionsfrom teacher modeling to student practice havebeen shown to facilitate quick and accurate exe-cution of procedures as measured on standard-ized tests (Brophy & Good, 1986; Hiebert &Grouws, 2007).

Principles like those associated with con-ceptual understanding or efficient executionof procedures provide lenses through which(prospective) teachers can view instructionalepisodes and generate hypotheses for whyinstruction might have, or might not have,helped students achieve particular learninggoals. In doing so, the principles provide a path-way along which empirically supported knowl-edge of learning enters the framework forstudying and improving teaching.

Apparent from this discussion is that Skill 3necessarily invokes more general knowledge oflearning and teaching, not embedded directly inthe instructional episode, to formulate usefulcause–effect hypotheses about the episode. Skill3 builds on, but moves beyond, Skill 2 by appeal-ing to principles of learning and teaching. Skill 3is an empirical process (like Skill 2) because itrequires careful, structured observations of thekey lesson events (with one eye on the teacher)and an inferential process because it requiresevaluating the potential of these events, in lightof appropriate principles, for students’ learning.This places Skill 3 as a bridge between Skills 2


and 4, a bridge between analysis of teaching andimprovement of teaching.

Return, for a moment, to the sixth-gradelesson on decimal fractions. The teacher hadobserved that asking students to show the deci-mal number 2.36 using the flat block as the unityielded at least two different ways of thinking—one focused on the 10-for-1 relationship asintended, the other focused on the decreasingsize of blocks that allowed students to evade thekey relationship. It now became clear to theteacher that changing the unit did not, by itself,require explicit attention to the 10-for-1 relation-ship. The blocks, by themselves, were notenough. Changing the unit from the large blockto the flat did not pose the dilemma, at leastfor some students, which the teacher had envi-sioned. The teacher now hypothesized that the10-for-1 relationship is not simply seen in theblocks but rather constructed by students whenthey face some kind of dilemma that cannot beresolved simply by lining up the blocks in orderof size. What kinds of questions or tasks wouldpose a dilemma for the students that could beresolved only by thinking directly about the10-for-1 relationship? As the teacher consideredpromising revisions, the teacher’s hypothesisguided the review and selection of revisionsmost likely to help all students achieve the goal.

Constructing hypotheses about causal con-nections between teaching and learning requiresan appropriate level of skepticism. Teaching isexceedingly complex, and the connectionsbetween teaching and learning, regardless ofhow well supported by evidence, are only par-tially understood. The effects of teaching occurthrough numerous interactions, only some ofwhich will be captured by any single hypothe-sis. In addition, all teaching is likely to be par-tially effective rather than completely effectiveor ineffective, and it is likely to be more effectivefor some students than others. These complicat-ing factors mean that causal hypotheses shouldbe treated as questions to be examined furtherrather than as conclusions assumed to be true.

It should be obvious that subject matterknowledge of a special kind is needed to formu-late appropriate hypotheses. Understanding thedemands that students’ responses make on dif-ferent kinds of knowledge and what instructional

cues might have triggered particular kinds ofthinking require knowing the subject deeply.However, it is a kind of knowledge that must betightly integrated with knowing how peoplethink about the subject.

Criteria that can be used to evaluate the qual-ity of the hypotheses that prospective teachersmake about the effects of teaching can be sum-marized as follows: Hypotheses are more likelyto lead to improvements in students’ learningwhen (a) they are made about students’ achieve-ment of the learning goals (rather than aboutlearning other topics), (b) they specify the teach-ing and learning with enough detail to clarifyand justify the hypothesized connectionsbetween them and with enough detail to invitetesting and refinement, (c) they appeal to well-supported principles appropriate for the partic-ular learning goals, and (d) they are expressedwith appropriate nuance and recognition of thecomplexity of teaching–learning relationships.

Skill 4: Use Analysis to ProposeImprovements in Teaching(How Could Teaching MoreEffectively Help Students Learn?)

The purpose of applying the previous threeskills is to provide the information needed tomake evidence-based decisions about how toimprove an instructional episode. To summa-rize, the skills aim to clarify the learning goals,gather information about whether students areachieving the goals, and generate hypothesesabout how instruction is (or is not) facilitatingstudents’ learning. Making revisions to improvethe instructional episode is then a matter of fol-lowing the implicit recommendations containedin the hypotheses.

A final visit to the sixth-grade lesson ondecimal fractions illustrates the process of usingwhat is gleaned from applying the first threeskills to improve teaching. After the teacher con-structed a hypothesis about why the activityof representing decimal numbers with base-10blocks prompted the intentional learning forsome students but not others, promising revi-sions to the lesson could be imagined. The chal-lenge for the teacher was to revise the tasks ormodify the sequence of tasks to ensure that


students confronted, explicitly, the 10-for-1 rela-tionship between adjacent positions of thedigits in decimal numbers. One possible revisionwould be to assign the long block the value of 1.The small cube, the smallest block, would thenhave a value of 1/10 or 0.1. To represent 2.36with blocks, students would need to invent thehundredth block—to verbally describe or drawa picture of the block needed to represent 1 hun-dredth. This would require students to describeexplicitly the relative values of the blocks ratherthan just lining them up in decreasing order ofsize. Another possible revision, that might havea similar effect, would be to ask students to usea different representation than base-10 blocks,say letting 20 circles represent 1. This wouldmean that 2 circles would represent 0.1, andstudents would need to divide 2 circles into 10equal pieces to represent 0.01.

Cause–effect hypotheses formed by applyingSkill 3 usually point the way to the revisionsdeveloped by applying Skill 4. It is difficult toimagine a carefully developed revision not pre-ceded by a hypothesis that linked a (deficient)instructional event with incomplete learning.Indeed, it is this hypothesis that provides therationale for the proposed revision. Evaluatingthe quality of proposed lesson revisions necessar-ily involves a different process than outlined forthe first three skills. In the end, the quality of arevision depends on whether the revised episodehelps all students achieve the learning goal(s)more effectively. This criterion can be difficult toapply in a teacher preparation program becauseprospective teachers cannot always test revisionsin real classrooms. We can, however, suggesttwo proximate measures. First, the proposedrevisions can be interpreted in light of principlesfor learning and teaching as described earlier.Second, it is possible to assess the revision’spotential by judging the quality of the argumentthat can be offered by the prospective teacher tosupport the recommendation (the relationshipsamong the data, the hypothesized cause–effectconnection, the principles that support thehypothesis, and the proposed revision). Like thefirst proximate measure, this does not substitutefor evidence of students’ learning during imple-mentation; however, it does reveal the soundnessof the prospective teacher’s reasoning in using

the information derived from applying the firstthree skills.

Although students’ learning is the primarycriterion for judging the effectiveness of lessonrevisions, a quite different secondary criterionalso can be applied: Does the revision increasethe teacher’s opportunities to observe whetherstudents are achieving the learning goals? If theanalysis process we have described facilitatesimprovements in teaching, then lessons thatenable more insightful and informative analysesare (at least, short-term) improvements. So, anadditional criterion to evaluate lesson revisionsis an increase in the access teachers will have tostudents’ thinking and learning at key momentsduring the lesson.

WHY THIS FRAMEWORK?

Of the range of skills or competencies thatprospective teachers might need to becomeeffective teachers, over the long run, why didwe select these four skills? A first reason is thatthe skills are drawn from the (implicit) practiceof classroom teachers. Although we doubt thatmost U.S. classroom teachers engage in theseskills with the deliberateness we describe, theskills are not alien to them. This gives the skillsa kind of face validity and, in addition, suggeststhe skills might be sustainable, in some form, aspart of teachers’ daily and weekly practice.

A second reason for selecting these skills istheir similarity to the components of disciplinedinquiry (Dewey, 1929). This similarity indicatesthat teachers who apply these skills will beengaged in a disciplined inquiry into teaching—the precise goal of the framework. The melding ofteaching and research is a promising response tothe stubborn gap between research and practice,between the archives of researchers and the prac-tice of classroom teachers (Hiebert, Gallimore, &Stigler, 2002). It offers teachers the opportunityto accumulate knowledge for improving theirown teaching (and that of others) over time.These kinds of research-oriented teaching skills,that enable teachers to participate in the processof gathering knowledge to inform their practice,can engender a healthy and productive profes-sional identity (Franke, Carpenter, Fennema,Ansell, & Behrend, 1998; Malara & Zan, 2002).


A third reason for selecting the four skills isthat they create a framework that allows spacefor the influence of subject matter knowledge.Subject matter knowledge clearly influenceshow and how well teachers teach (Borko et al.,1992; Borko, Livingston, McCaleb, & Mauro,1988; Carlsen, 1993, 1997; Hill et al., 2005;Leinhardt & Greeno, 1986; Stein et al., 1990;Stodolsky, 1988); however, the mechanismsthrough which such knowledge enters teachers’thinking and practice are not well understood.The framework we propose identifies siteswhere subject matter knowledge could influ-ence teachers’ work, especially as they prepareto implement and then reflect on classroomlessons.

A fourth reason for selecting these skills isthat preliminary data suggest they work; that is,applying skills like those we describe leads toimprovements in teaching over time. Althoughvery little data currently exist to inform frame-works like the one we propose, the existing dataare encouraging. Goldenberg, Saunders, andGallimore (2004) and Saunders and Goldenberg(in press) reported considerable success withschoolwide efforts to improve students’ learningthat emphasize teachers’ analysis of practice. Inthe weekly teacher meetings that occurred inthese schools, teachers set learning goals forstudents, brought student work to examine,and used the findings of their analysis to reviseclassroom teaching. One teacher described theprocess this way:

First we evaluate the student work and as we eval-uate the student work we look at strengths andweaknesses. Then we decide on what kind ofinstruction we’re gonna try in the classroom. Andwe try the instruction in the classroom and then wego back and reassess to see if the instruction isworking. If it’s not working we just, we try to takea different approach until we meet those goals andthose standards and objectives. (Saunders &Goldenberg, in press, pp. 19-20)

Kazemi and Franke (2004) also described theways in which teachers can reflect on andimprove their practice by collecting and studyingstudents’ written responses to tasks designedwith particular learning goals in mind.

Cross-cultural research has uncovered analy-sis of teaching practices that form the core of

professional development activities in severalAsian countries (Lewis & Tsuchida, 1997; Ma,1999; Marton & Tsui, 2004; Paine & Ma, 1994;Stigler & Hiebert, 1999; Yoshida, 1999). Althoughmany factors account for the apparent success ofcontinued teacher learning in these countries,the relentless focus on analyzing classroom prac-tice and testing hypothesized improvements,using skills like those we described earlier,clearly support the growth of expertise amongthese teachers. That many U.S. teachers think ofteaching in terms of implementing activitiesrather than analyzing effects of instruction onstudents’ learning might be a consequence of thegeneral lack of attention in this country to theskills that compose our framework (Fernandez& Cannon, 2005).

A fifth reason for choosing this frameworklies in its potential for helping teachers movetoward more equitable instruction. With itsfocus on student thinking and on collectingconcrete evidence of students’ achievement ofthe learning goals, the framework encouragesteachers to make instructional decisions basedon each student’s learning rather than on their(sometimes biased) perceptions and expecta-tions. Collecting revealing evidence of students’thinking is facilitated by knowing one’sstudents, knowing what ideas they bring to theclassroom, and appreciating why individualstudents might differ in their thinking. Usingstudents’ thinking to revise instruction placesteachers in a better position to help each studentachieve the learning goals. We recognize, ofcourse, that equitable instruction involves addi-tional work by the teacher; however, we believethe emphasis on gathering evidence of eachstudent’s achievement of the learning goals andrevising instruction based on this evidence is afirst step toward greater equity. Ways in whichthe framework can further support equitableinstruction will need to be elaborated as theframework is developed and refined.

A final reason why this framework haspromise is that the four skills can be applied toimprove learning with respect to all types oflearning goals. During their teaching careers,prospective teachers are likely to be asked tohelp their students achieve a variety of learninggoals. Earlier, we identified goals associated


with conceptual understanding and goals asso-ciated with procedural efficiency, goals thatare common in school mathematics. Othersubject areas also pose a range of learning goalsfor students. There is an advantage, we think, inacquiring skills for improving teaching that canbe applied to all goals rather than to a limited setof goals.

Readers might have noticed that the frame-work does not adopt a particular stance on pre-ferred theories of learning and teaching (e.g.,constructivism, behaviorism). In our view, therole for theories of learning and teaching is toprovide principles that can generate hypothesesthat link instructional activities with students’achievement of particular kinds of learning goals.They suggest predictions that streamline whatotherwise would be a lengthy and chaotic trial-and-error process of blindly producing and test-ing hypotheses about more effective instructionalactivities.

A related issue that we have not yetaddressed is what theories, or principles, oflearning and teaching underlie our efforts tohelp prospective teachers acquire the four skills.Just as teachers need to invoke principles oflearning and teaching to analyze teaching andimprove their students’ learning, teacher edu-cators must apply principles of learning andteaching to help prospective teachers achievethe program learning goals. Put simply, wesee the acquisition of the four skills in ourframework as conceptual learning. Using thefour skills effectively requires understanding therelationships among elements within the skillsand among the skills. Consequently, our analy-ses of our own instruction and our hypothesesdriving continuing revisions in our courses drawheavily on the two principles identified earlier:making the critical relationships explicit andencouraging prospective teachers to wrestlewith the key ideas contained in these skills.

CAN PROSPECTIVE TEACHERSDEVELOP THESE SKILLS?

Is the framework we have sketched feasibleas a set of learning goals for prospectiveteachers? Can they acquire the four skills wedescribed during their preparation program?

As we indicated earlier, there are no data thattest the success of this kind of program directly.However, the relevant data that exist areencouraging.

Van Es and Sherin (2002), using a somewhatdifferent framework and with beginning, ratherthan prospective, teachers found that teacherscould engage in analyzing videos of classroomlessons and, over time, move from descriptiveresponses to analytic responses. Of particularinterest is that with practice and discussion,teachers’ analytic responses gradually acquiredthe characteristics that would have yielded highratings on the criteria we suggested for measur-ing the growth of each skill. With a sample ofprospective teachers, albeit in Italy rather thanthe United States, Santagata et al. (in press) foundthat attention to teaching analysis skills of thekind we propose resulted in significant growth,over a few months, in participants’ analysisperformance. Finally, some evidence suggeststhat U.S. prospective elementary teachers haveacquired components of some of these skillsbefore entering their preparation programs; how-ever, they access and apply the skills only undercertain conditions (Morris, in press). This sug-gests that prospective teachers bring some ana-lytic competencies with them when they enterthe program and that appropriate conditionsmight enhance and even accelerate the acquisi-tion of intended skills. Furthermore, the prospec-tive teachers in the current study who diddisplay good analytic skills generated recom-mendations for revising the lesson that werejudged to improve students’ learning opportuni-ties. Although the data are too scarce to confirmthat prospective teachers can acquire the skills toanalyze teaching, the early results argue for con-tinued exploration and testing.

Complicating the assessment of the frame-work’s feasibility is the fact that the skills wepropose constitute a new curriculum for teacherpreparation programs. Little knowledge for howto implement such a curriculum currently exists.If one adopts the framework we propose, thelearning goals for prospective teachers are thefour skills that compose the framework. If onefurther accepts the view that meaningful acqui-sition of the skills is a conceptual task, thenappropriate principles that guide instruction of


prospective teachers can be suggested. However,teacher educators still face the same challenge asclassroom teachers—to engage in the difficult,ongoing work of improving teaching by helpingstudents (prospective teachers) more effectivelyachieve the learning goals.

A first step in this work is suggested by theframework itself—to specify the learning goalsmore precisely. Only by defining learning goalsprecisely and explicitly will it be possible collectthe relevant data—to assess whether and howthe preparation program facilitated prospectiveteachers’ achievement of the goals. This meansunpacking the four skills into more detailed andprecise component skills and examining howthe component skills can be developed andrecomposed. Clearly, there is much work to do;however, in the spirit of improving teachingthrough the deliberate and systematic study ofteaching, we believe that the framework we pro-pose provides a path toward improvement forteacher educators and classroom teachers.

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James Hiebert is the Robert J. Barkley Professor ofEducation at the University of Delaware, where heteaches in programs of teacher preparation, professionaldevelopment, and doctoral studies. His professionalinterests focus on mathematics teaching and learning inclassrooms. He has coauthored Making Sense: Teach-ing and Learning Mathematics with Understand-ing (1997) and The Teaching Gap (1999). He recentlyserved as the director of the mathematics portion ofthe TIMSS-R Video Study and currently is a primary

investigator on the National Science Foundation–funded Mid-Atlantic Center for Teaching and LearningMathematics.

Anne K. Morris is an associate professor in theSchool of Education at the University of Delaware, whereshe teaches mathematics content courses for prospectiveelementary and middle school mathematics teachers anddoctoral courses in mathematics education. Her profes-sional interests focus on preservice teachers’ analysis ofteaching skills, and K-16 students’ understanding ofmathematical proof.

Dawn Berk is an assistant professor in the School ofEducation at the University of Delaware, where sheteaches mathematics content courses for prospective ele-mentary and middle school mathematics teachers anddoctoral courses in mathematics education. Her profes-sional interests focus on the relationship between preser-vice teachers’ mathematics content knowledge andpedagogical skills.

Amanda Jansen is an assistant professor in the School ofEducation at the University of Delaware, where she teachesmathematics content and methods courses for prospectiveelementary and middle school mathematics teachers as wellas doctoral courses in mathematics education. Her profes-sional interests focus on motivational processes amongstudents and teachers in mathematics classrooms.


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