- Email: [email protected]
On modeling teaching
ON IWODELIWG
TEACHI
ALANH.SCHOENFELD UNIVERSITYOF CALIFORNIA, BERKELEY
I would like to begin this response with an expression of sincere thanks to Jerry Carlson for inviting me to write Toward a Theory of Teaching-in-Context (T-I-C), and to my friends and colleagues for their thoughtful commentaries on the paper. It’s an absolute pleasure to have such fine thinkers scrutinize my work from multiple perspectives, pushing me to think more deeply about its foundations, strengths, and limitations. They have done their job admirably, raising deep and important questions. This is what scholarly collaboration is all about. The issues raised in the commentaries tend to cluster into three main areas: ??
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Context and background assumptions; Examining and pushing the boundaries of the model; Pragmatics (theory into practice).
This response is organized along those lines. In each of the three areas I will try to deal with the main points made in the commentaries. It is interesting to note that the responses to T-I-C, though all written independently, carry on a dialogue of sorts: questions and issues raised in one commentary are often addressed directly in the others. I shall enter into that dialogue, at times pointing to exchanges across papers and at times adding my opinion and/or ,AA:I.Z,.,,, Uc(Lcl. 1^L^ CIUUIU”ILdl
CONTEXT AND BACKGROUND ASSUMPTIONS The commentaries raise many questions. Schott, for example, raises issues of epistemology. Why do I separate beliefs and knowledge; indeed, what is knowledge? Why did I choose the three components I did? He argues, as does Lehrer, that I need to say more about mechanism, and about my underlying assumptions-e.g., am I grounding my work in the standard assumptions of cognitive psychology? And, what about methods? These themes are echoed in some of the other essays as well. Dim! all correspondence lo: Alan H. Schoenfeld, University of California, Berkeley, Elizabeth and Edward Conner Professor of Education, Graduate School of Education, Berkeley, CA 94720-1670. Issues In Education, Volume 4, Number 1, 1998,pages 149-162 All rights of repmduction in any form reserved.
Copyright D 1998 by JAI Press Inc. ISSN: 1080-9724
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Responses to these questions are found, in part, in the papers by Green0 and Leinhardt. Greeno’s essay begins with the direct assertion “Schoenfeld and the Teacher Model Group . . . used standard framing assumptions of cognitive theory to develop a framework for modeling episodes of teaching.” As I mention in T-I-C, I see the work described here as part of a decades-long effort on the part of the field to develop understandings of complex human behavior in highly interactive ---?-I __I..___ T _z_,__~3. -1-1__ _I__ __‘.I. _._ _*r:--_-I social serungs. Lemnarat eiaoorares wun signincanr and helpful detail, locating our work squarely in the middle of a number of well established traditions, the most central of which is the cognitive science approach to the study of thinking and learning. This is the arena within which my earlier work on problem solving resides. Indeed, in my 1985 book Mathematical Problem Solving one finds many of the antecedents to the current work-specifically the claim that to understand an individual’s mathematical problem solving behavior, one needs to attend to that person’s knowledge and strategies, metacognition (specifically with regard to decision-making), and beliefs. I think of teaching as a problem-solving activity, so it is not surprising that there are significant correspondences between my analyEPEvnf y’ nrnhlmn cnlxrino and toarhino Thic mdhdc ac ~~11 SE rcxrrltc~ “b” ““sb”’ ““““‘ b Ull.. “““““b. LAY”hnlrla I.“&_” fnr _“I I&.~C1L”UY U” ..CY UY L..Y..‘b”, CPP “CL below. Given the multiple audiences for this volume, I have not gone into extensive detail regarding the workings of the model or the correspondences between the results of the modeling process and behaviors that are being modeled. As suggested by Green0 and Leinhardt, the general sense of mechanism invoked for the model is drawn from the cognitive/artificial intelligence literature-think “spreading activation networks” if you will. There are, of course, much more data and detail to the work could be presented in T-I-C. Additional data and analysis can be found in Schoenfeld, Minstrel& and van Zee (1996). A general discussion of methods, which were given relatively short shrift in T-1-C may be found in Schoenfeld (1992). Specifics regarding one of our methodological tools, competitive argumentation, are given in Schoenfeld, Smith, and Arcavi (1993). [At a metatheoretical level, I note that there is an interesting tension in the modeling enterprise. “Going all the way”-implementing a computational model via a computer program-not only demands an extraordinary amount of time and energy, but it is often very constraining. When one implements computational models one is limited to the state of the art with regard to the entities that can be modeled, and this may mean that some crucially important cognitive phenomena cannot be attended to. In the 1970s and 198Os, for example, artificial intelligence models of problem solving were not capable of coping adequately with issues of metacognition or beliefs. If you thought such things were important and wanted to focus on them (which I did), then you opted not to build the actual computational model. Instead you got “close” intellectually by working through such models as gedanken-experiments. And, of course, you used the standard techniques of multiple data sources, triangulation, and replication to try to insure that you weren’t selling yourself a bill of goods. Much the same applies here.]
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As Leinhardt notes, the Teacher Model Group’s (TMG’s) work fits squarely in the tradition of fine-grained analyses of teaching-a tradition that extends over some decades and to which Leinhardt herself has been a major contributor (see, e.g., Leinhardt, 1993). We have profited immensely from that work. For somewhat technical reasons we have used a somewhat different vocabulary and an alternative set of representations. These differences should not obscure the significant _---..1_.- 1. _a_~___ -.__. _ -_-.I- _ _I rl-_L _..-.--.-1. __-1_ _._ 3 overlap uerween our work anu mar which preceded it. For example, goals ana action plans play many of the roles of agendas as Leinhardt and Borko and Peressini use the term. Borko and Peressini are right on the mark when they suggest that a common feature of the three lessons discussed in some detail in T-I-C is that the lesson segments are all agenda-driven, and that the model is capable of handling lessons of that type. Dijkstra points out, rightly, that I have made some tacit assumptions about teacher and students sharing a workspace, having access to common board space, etc. He chooses to represent the task of instruction as a design problem, asking questions such as: “Which learning situation should be made? Which questions to chnrrld the ctrrrlcmtc U” An en malcc. ‘ tnnurlm-lo~ rnnctrrrriinn nnccihl~?” a&? What UY_.. ..ILUL UILVUI.. CA&_ YLUULIL.” L” S111U1.L ...“‘.““6’ ~“-.YUU~.I”I. y”““‘ “‘b. This formulation is entirely consistent with the assumptions of T-I-C. There may be some differences between conceptualizing teaching as a design problem or as problem solving, but the two conceptualizations are fundamentally compatible. When one asks what the teacher brings to the design task, one must ask what knowledge the teacher has at his or her disposal, what the teacher’s goals are, and what the teacher believes is important. In sum, a case can be made that Dijkstra and I are working in much the same problem space. For different reasons and in different ways, most of the other commentators can also be situated in the same space. For example, Borko, Peressini, and Leinhardt raise some very deep questions about the scope of the model. Their questions, which I shall address directly in the next section of this response, are of the following type: “Does the model live up to the claims Schoenfeld makes for it? Indeed, can it live up to such claims?” This way of posing the questions situates us on common ground, which Green0 calls the “standard framing assumptions of cognitive theory.” Green0 labels my framing assumptions clearly because he wants to challenge them. It is a fair challenge, and it raises some very interesting issues. In brief, I have chosen to examine “teaching” from the teacher’s point of view. But teaching and learning are highly interactive, and the teacher is but one member of a community in which all of the members participate and change. What of the other members, and what of the community itself as an evolving entity? My working metaphor for this issue is the motion picture Rashomon, in which contrasting views of the same events are seen through the perspectives of different individuals. Where does the truth or “reality” reside? Clearly, each individual has his or her own version of what happened, and it is of great interest to understand those. About the larger issue of “what really happened,” reviewer James Berardinelli says the following:
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Many people watch Rashomonwith the intent of piecing together a picture of what really occurred. However, the accounts are so divergent that such an approach seems doomed to futility. Rashomon isn’t about determining a chronology of what happened in the woods. It’s not about culpability or innocence.Instead,it focuses on somethingfar more profound and thought-provoking: the inability of any one man to know the truth, no matter how clearly he thinks he sees things. Perspective distorts reality and makes the absolute truth unknowable.
Those are chastening words for those of us engaged in the modeling enterprise. It is essential to recognize, up front, that we can never really get “inside the head” of another; and moreover, that the visions inside the heads of those involved in any event may not be commensurable. Nonetheless, it seems worth the effort to try, both at the individual and collective level. As Green0 points out, the teacher is just one participant in the classroom. The Teacher Model Group has chosen to focus (for now) on modeling the teacher as a central participant. In some circumstances, focusing on the teacher may seem to explain a large part of what happens in a classroom-even though, as we must always remember, the story is being seen through the teacher’s eyes. In others (e.g., where the agenda is co-constructed by teacher and students), what the teacher sees and thinks may seem to explain much less. In any case, understanding the view from each student’s eyes is a critical part of the overall story; equally critical is obtaining a description of the community as an organic entity. As I see it, it is an essential part of the theoretical enterprise to develop increasingly rich descriptions of individual participants, of the community, and of the relationships between and among them. What is not clear to me at this point is whether doing so will only require evolution (including changes of emphasis in what receives focal attention) or whether it will require something more dramatic. Two competing frameworks, cognitive and situative, are on the tabie. One comparative discussion of these frameworks may be found in Greeno, Pearson, and Schoenfeld (1997). Green0 et al. explore the two perspectives as constituting alternative framing assumptions for thinking about assessing people’s knowledge. From the cognitive perspective, the issues they identify as central for assessment include elementary skills, facts, and concepts; problem-solving strategies and schemata; metacognitive processes; beliefs and dispositions; engaging in mathematical practices. From the situative perspective, issues seen as central include basic aspects of participation in mathematical practice; personal identity and community membership; formulating and evaluating problems and solutions; making and checking meaning; and fluency with mathematical methods and representations. When one compares the two perspectives, one sees great overlap. A case can be made that the differences between them (though significant!) are largely differences in emphasis. Much teaching
the same seems to be the case here. For example, issue of establishing (co-constructing with students)
Green0 refers to the a discourse commu-
nity that has certain desired properties. I note that doing so is an explicit teaching goal for Ball and Minstrell, and for me-and that understanding how that hap-
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pens is one of my primary research goals. For reasons of grain size (we examined short instructional segments in detail) and focus (the big question for the T-I-C paper was, can we describe the model and show that it works?), we did not devote much detail to that particular topic. But we can-indeed, we have. Among the details in Schoenfeld, Minstrell, and van Zee (1996), there is an exegesis of lines l-35 of the Minstrel1 lesson segment discussed in T-I-C. That analysis identifies the ways in which specific questions and even wait time (a fuii thirteen seconds after asking, “Any questions that you have about course expectations or anything you’re wondering about the course?“) are devices used by Minstrell to help establish a desired classroom climate. In a similar analysis, Arcavi, Kessel, Meira, and Smith (1998) discuss various devices that I use in my problem solving course to build a particular kind of classroom community. At a larger level of grain size, in Schoenfeld (1994) I discuss the ways in which my problem solving class is structured
to foster mathematical
ownership
and agency on the part of my stu-
dents. I see these as part and parcel of the same thing, though different issues may come into focus at different levels of grain size. I. .* nt me same time, the &ishomon probiem remains. As a fieid, we have yet to develop the kinds of perspectives that provide meaningful syntheses of the individual and the collective. That problem needs attention.
EXAMINING AND PUSHING THE BOUNDARIES OF THE MODEL Leinhardt observes that “Schoenfeld needs to be careful about the specific relationship between the data he chooses, the claims he makes, and the assumptions about the power and use of modeling he carries over from its use in physics and chemistry” (p. 3). She notes, correctly, that I must be wary of introducing spherical --e-r-1 : ^^__^^ -L -1--e:--:_a... _^_._^I-L^1^^^_^^ - T -:-l..--rlL _-I_,.L”Wb llll” LI-^ Lllr Cld331”“lll. LUILILd1UL IdlbtD L.-A lU1lUdlll~l1Ldl ISSLlCb Ul ULJllldlll IZ~lSLlZmology-e.g., was the modeling enterprise successful in the three cases described because the subject matter is at the secondary level or above (so that developmental concerns re the students are not centrally important) and because mathematically related content is cut-and-dried enough to yield lessons that can be parsed neatly into chunks? Would a history lesson be more difficult to model, or perhaps not even amenable to modeling because of its fundamentally different nature? Borko and Peressini raise equally tough questions-some directly, some less so. They too raise developmental issues, conjecturing that [The theory
of teaching-in-context]
might do a better job modeling teachg
that occurs in secondary and post-secondary classrooms rather than elementary classrooms. The multiple, often contradictory goals central to [Magdalene] Lampert’s and [Deborah] Ball’s dilemmas and choices-goals which led Schoenfeld to acknowledge his difficulty with “the attribution and prioritization of goals and action plans”-did not seem to play a comparable role in the deliberations and decisions made by Nelson, Minstrell, and Schoenfeld (p. 5).
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Lurking behind that question, and behind Borko and Peressini’s invocation of dilemmas as discussed by Ball and Lampert, is the issue of whether the TMG’s framing of teacher behavior might be off the mark in the case of such dilemma-driven teaching. Beyond these issues, there are issues of scope and/or grain size-e.g., the questions Green0 raises about things that develop over time, and Lehrer, Borko and Peressini’s invocation of history as a factor in teachers’ decision-making. These are exactiy the right questions to ask. ruioreover, given that the evidence offered in T-I-C consisted of three cases described in partial detail, the commentators are right to be dubious about the answers. My response will be in two parts. First, I shall discuss one more (very small) instructional segment in rather finegrained detail. The segment of instruction comes from the January 19,199O “Shea Numbers” discussion in Deborah Balls classroom. I shall focus on one decision made by Ball in the midst of teaching. The analysis of this decision, which was done after the draft of the T-I-C paper was sent to the commentators, helps clarify some aspects of the model and the underlying theory. It also sets the stage for the second part of this section, a discussion of the issues summarized in the previous paragraph.
ONE FINAL EXAMPLE The context for this example was established in Section 7.2 of T-I-C, which the reader may wish to review. Ball’s class had been discussing the properties of even and odd numbers. The previous day they had held a special meeting with the previous year’s third grade class, in which such issues, and the question of whether the number zero is even, odd, or “special,” had been considered. As noted in Section 7.2, “one of Ball’s primary goals that day was to have students reflect on the previous day’s activities-not just on the mathematics but, more importantiy at the beginning of the lesson, on the state of their knowledge and learning.” She began the lesson with by announcing that agenda. And, she acted on it. Comments from students that focused on the nature of their thinking and learning were encouraged and elaborated upon, as in the following exchange: Ball: Student 1:
Ball:
Was there an example of something yesterday that you understood a little bit more during the meeting? Well, I didn’t think that zero was-zero, urn-even or odd until yesterday they said that it could be even because of the ones on each side is odd, so that couldn’t be odd. So that helped me understand it. Hmm. So you thought about something that came up in the meeting that you hadn’t thought about before? Okay.
Indeed, when two students began to dispute a particular mathematical point, Ball re-focused the class’s attention back to the issues of thinking and learning:
On Modeling Teaching
Ball:
Before we take this up again, I underst-I-I understand that this is still a probiem and that we didn’t a-we didn’t settle it, we’re probabiy not going to settle it. Urn, there’s a lot of disagreement about this issue, right? And you saw that the fourth graders who have been thinking about this for a long time also disagree about it, don’t they? I’m still kind of interested urn, in hearing some more comments about the meeting itself [Student 21 commented that it was good to have the two classes together because she heard an idea that she hadn‘t thought about and it made her think about and even revise her own idea when she was in the meeting yesterday. What other comments do other people have about the meeting and what happened yesterday? [Student 31, do you have a comment?
The conversation Ball: Student 4:
Ball:
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continued
in this vein, until the following exchange:
Other people? [Student 4]? Urn, first I said that urn, zero was even but then I guess I revised so that zero, I think, is special because urn, I-urn, even numbers, like they make even numbers; like 2, urn, 2 makes 4, and 4 is an even number; and 4 makes 4; 4 is an even number; and urn, like that. And, and go on like that and like 1 plus 1 and go on adding the same numbers with the same numbers. And so I, I think zero’s special. Can I ask you a question about what you just said? And then I’ll ask people for more comments about the meeting. Were you saying that when you put even numbers together, you get another even number-or were you saying that all even numbers are made up of even numbers?
it is this iast action i wish to expiore. Ball had announced her agenda, and she had stuck to it despite the tendency of some students’ comments to lead away from it. Yet at this point she deliberately took the conversation “off track.” Moreover, she did not do so to pursue the main mathematical issue under dispute, the question of whether zero is even, odd, or special. Instead she opted to pursue a somewhat peripheral issue, whether all even numbers are made up of even numbers. [For purposes of easy reference, I shall call this “the evenness question.“] The issue: why did Deborah Ball decide to pursue the evenness question at that point? The explanation ho-n nhlC,??r\...m.L “C-zll CzYlF L” “““IR
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salt. I include it because it serves to convey the flavor of the Teacher Model Group’s recent analyses, and to clarify some issues about the nature of the correspondence between goals and action plans. The argument, in brief, is that Ball undertook her (announced) detour in response to the presence of an emergent constellation of goals and constraints. None of the goals, in and of itself, was of high enough priority to warrant the detour. However, the combined priorities of the various goals was above threshold. That is, following up on the student’s comment allowed Ball to satisfy a num-
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ber of goals whose combined activation rendered important enough to pursue. In particular:
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the evenness question
1. Generally speaking, Ball is predisposed to pursue issues of mathematical substance when they emerge in classroom discourse-and when circumstances warrant their pursuit. The question of whether all even numbers are made up of /-_:-_ or) -A even ^____rmmuerb -__-l__-_ 15 :_ purentia~ly -_L_-L-l,__ one ^__ bucll ____LISSUE. :__.__ (yam I believe, however, that if there were no other factors favoring it, the “content value” of the evenness question would not have been enough to warrant its pursuit. After all, Ball had just declined to follow up on a more central content concern, the nature of zero. 2. Generally speaking, Ball is predisposed to explore issues related to her students’ understanding of various pieces of mathematics. That is, an important part of her teaching is developing a substantive understanding of what each of her students understands. Not only this student, but another as well had made a conjecture akin to the evenness question. Hence she might have wished to pursue their nnrl~rctnndinocnfit -"-~"'..^‘-~'b'""'
As in the case of (l), I do not think that this motivation would “win” on its own-or even in combination with (1). Note again that in the preceding conversation, Ball had bypassed the opportunity to pursue many students’ understandings of the nature of zero. 3. The second student mentioned in (2) had not been actively engaged in the discussion up to this point in the lesson. Discussing the evenness question would provide an opportunity to rope her directly into the conversation. This, too, was a non-negligible concern-but not of high enough priority to warrant action on its own. 4. In previous days the class had been engaged in making a range of conjecVV tures having to do with combinations of even and odd numbers. Ball’s intention was to return to these conjectures. Clarifying the nature of the evenness question, and possibly resolving it, could thus have established a platform for the work she intended to do later on. Much like (3), this is important but it is not important enough to warrant action by itself. 5. Since the nature of the student’s conjecture could be clarified in short order, only a small amount of time was necessary to pursue this detour-so Ball could return to the top item on her agenda, discussing the meeting. In contrast, there was the risk of a very substantial detour had the class pursued the discussion of whether the number zero, is even, odd, or special. In sum, the situation was as follows. Ball had announced and had been pursuing a high priority goal-discussing “what you thought about the meeting, what you noticed about the meeting, what you learned at the meeting.” A student made a comment closely related to the evenness question. Was the evenness question worth pursuing?
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Not on the grounds of reason (1) alone-or any of the other reasons discussed immediately above. But in combination, reasons (1) through (5) did provide substantial motivation to pursue it. Ball did so, but without abandoning her initial agenda, which still had high priority. Indeed, she announced that what she was about to do represented a short detour. And, when she had clarified the evenness question, she did indeed return to the topic of the meeting.
DISCUSSION If this analysis (or something close to it) is right, it is important for at least two reasons. First, it clarifies what may have been an all-to-easy-to-develop mis-impression from the examples in the T-I-C paper. It is possible, though inaccurate, to get the impression from the examples given that there is a simple correspondence between goals and action plans. (See, e.g., the discussion of overarching gouls in Section 6.) That is, from the parsing scheme one might infer that whenever a high priority goal pops to the top of the stack, a new action plan is implemented; and whenever that action plan is completed, the corresponding goal is satisfied and a new goal (along with concomitant action plan) comes to the top of the stack. There are, to be sure, many times when this happens. However, the instance of Deborah Ball’s decision-making discussed immediately above provides a clear example of when it does not. More importantly, it provides a nice indication of the complex relationship between goals and action plans. On the one hand, a particular action plan may fully or partially satisfy a number of different goals. (Indeed, when selecting a course of action in class, the teacher often chooses one that meets multiple objectives.) On the other hand, a constellation of goals and constraints may lead the teacher to move in a particular direction, even if no one of those goals has high enough priority by itself to move the teacher that way. S~mn,-l 1 ““y’ hnnp thic ~~amnl~ chnwc that lancn~su~ nf e”“’ onal_Arimm “LLV’LU, L .I.*” “.““‘ y’L .,I.“,.” .llUL the .IIL TMC’ Il.__ c Y ‘ ““~C‘“b’ “I UII1L11 analyses and the language of dilemmas employed by Lampert, Ball, Borko and Peressini are not really very far apart from each other in substance. If you strip the language of goals from the example described above, you have what I think is a classic “dilemma story,” which might be told as follows: Throughout the opening minutes of the lesson Ball had to juggle between conflicting possibilities, e.g., whether she should pursue content considerations at the cost of her goal of having the students reflect on the meeting, or vice-versa. For some time she kept to her original path, but then she veered off, because pursuing a new direction for a short while served multiple purposes. Having that hridlv rn~~rs~red --1_-1 --.-- d&mm -_--_I_* -__,, the --.- retllrnd ._.I _ _.__ tn ._ hm _.__ nriuinal ““b . ..-. aoonda “b” ._-. Tt _L ic .y worth ,,“lL1. noting that, as the class unfolded, Ball’s choices were made opportunisticallytaking into account her original intentions (agendas) but also responding to emerging opportunities and constraints.
In short: the language of “goals and action plans” may sound cold, simplistic, and mechanistic, but their use doesn’t have to be any of those. Decision-making
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during teaching can be extraordinarily complex, but it appears that the architecture of the model can deal with that complexity. To put it another way: a goaldriven architecture can represent dilemmas quite comfortably. Let me now turn to the issue of domain epistemology. Leinhardt conjectures that the teaching of (especially secondary and post-secondary) mathematics might be easy to model, while the teaching of social studies might prove extremely diffi. . . . ..Lll :L ll”L -_^Lllny”aal”lr. :--..,,:~1, ‘llK PI-, 1tZcXl”I, _^^^^_ >lllZ “l-,, glvea - __^_1s :, &L-L A-_.- I”1111S L---- “I -L lusL”ry t:-r--. CLUL, LlldL*I-^ UK ueey instruction are grounded in the fundamental sense-making aspects of the discipline: Good history lessons braid together one question inside of and next to another. The class discourse moves around a core construct; there are no neat boundaries. Activity boundaries marking movement from small group to large group activities do not necessarily coincide with intellectual boundaries marking movement from one concept to another. Most discussions are both embedded and overlapping. Thematic concerns resonate through an entire semester of discussion and writing. . . . TL, ^‘cl,lllrlLLtz ^^..I ..^_^^ ClllU ..-A I~“ -^^^-_.,.L,.-.._.-:-I:I. ^ L:“ r--..l_%__llltz “ ccuIle‘^_^^ Lce “ I S”UIClIlt; 111d lllbl” ry Kbb”,, _..-..,A W”UlU -_I... u,anr a mess of tidy parsing and sequencing of teaching such as that represented columns 2 and 5 in Figure 4. So the critique here is that this model of teaching must either expand its content boundaries or admit that it is seriously limited by them. If it chooses to expand, then it must alter the suggestion of cleanly iterative embedding (Leinhardt, p. 6).
To this I say, “amen.” Once again, my exposition may have suffered from the crispness of the examples given. I note with all seriousness that if you replace the word “history” with the word mathematics in the first paragraph quoted above, you produce what I think is an accurate description of my problem solving course. The main goai of my probiem soiving instruction is to have students emerge from the course “thinking mathematically.” This includes having them develop the predilections to symbolize and analyze; to solve problems in multiple ways, to generalize; to consider a solution to a problem a beginning rather than an end. Issues of strategies, metacognition, and beliefs are braided inside of and next to another from the first day of the course to the last. These are leit motifs of my course, which resonate through an entire semester of discussion and writing. In the TMG’s language, these themes are represented as overarching goals. They are always potentially present, in that they may be triggered at any time. They are sometimes addressed in a collateral way when the main focus of attention is on another goal which at the time has higher priority. And sometimes they are the major foci of classroom interactions. Consider, for example, the first two goals of Figure 7: GoalA: Goal B:
Have the class interact as a community of inquiry with the freedom to explore, conjecture, and reason things through. Have students experience physics as a way of making sense of the world.
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These are perfect examples of overarching goals. They recur throughout Minstrell’s instruction. Their presence will become even more apparent as the unit of analysis is expanded to the week, month, or year of instruction. Thus the neat recursive structure discussed by Leinhardt will be seen less frequently than in the examples in the T-I-C paper. In addition, as we get better at goal analysis, I expect to see many more episodes of multiple and overlapping goal satisfaction such as +hnco cahnxro Yl in tho rlirrrracinn nC tho UL”OL olcahnrzatorl LIKAVVICILLU U”““L bI,L UIO~UO.DI”IL “I L11b“DXTP~~OEC L”Q.ILILLOO nr~aatinn yIULc7u”‘L. o
Can the model handle such things? At this point I have to admit that I have little to go on beyond gut feeling. I do believe that mathematics taught for understanding is every bit as complex as history taught for understanding-1 think it is no accident, for example, that there are such strong parallels between Sam Wineburg’s and my respective analyses of historical and mathematical thinking. And I do believe (largely on the basis of having lived with my problem solving course for 20+ years, and having done prospective analyses of long-term issues such as the development of a “mathematical culture” in the course) that most of the teaching in the problem solving course-even when I am exploring new territory, in ways similar to Ball’s exploration of the evenness question--can be modeled. I shall make a few additional comments in closing this section. First, I think the conjecture by Borko, Peressini, and Leinhardt that (good) elementary school teaching will tend to be more difficult to model than (good) secondary teaching is probably true. One of the main factors affecting the difficulty of the modeling task is the complexity of the decision-making undertaken by the teacher. The fact is that once the appropriate classroom culture has been established at the secondary level, things can often go smoothly while the teacher focuses primarily on content-individual student concerns aren’t necessarily that much of an issue, though perhaps they should be. But the elementary teacher who gives short shrift to developmental concerns does so at his or her peril. There is much more of an ;mr&..;A..~l ch*rln*\tc r.mc-l +h&s. *\onrle I-.vr;nrr en e.ecnmA +r\ ;m~or~t;~ra +n t-n.., thn IIILy~IcacI”~ L” NL”“” LllC IILUIVluual JLUUCllLJ c&llll LllFll ILFFUJ. ILylr,fj L” caLLFILL L” such issues, as well as to what is to be learned, will certainly give rise to multiple and conflicting goals. The resulting complex decision-making is likely to be difficult to model. Second, Borko and Peressini also conjecture that it might be easier in general to model novice teachers rather than experienced ones. Here, I think the answer is “it depends.” Complexity is one important variable, consistency another. True, novices may have “less to model”-their knowledge may be less differentiated, their action plans simpler. But, their behavior may be more random. And, being experienced doesn’t necessarily mean that a teacher will take on issues in a more complex manner. Some teachers become highly skilled at maintaining order and following a very simple lesson structure. Modeling this kind of “experienced” teaching is easy. Third, history. Borko and Peressini note that “some explanatory and predictive power-and by default, some scope-will be sacrificed if a model of Minstrell’s teaching does not take into account prior class experiences” (both with these students and with previous groups of students). Lehrer argues similarly. I agree. This
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is where the notion of lessonimageplays a central role. On the basis of prior experience, Minstrell knows what kinds of things to expect from his students at the beginning of the course. Later in the course, as he thinks about a day’s lesson, he knows where he wants the current group of students to get; he knows what knowledge and experiences from earlier in the course they bring to the lesson he is planning; and, he knows how other groups of students with similar backornllndc ..A... nnrl “.y”‘ ~vn~rienre hawe rcurtd tn hP r*..“’ nlanc tn (y”.a”“” u’“.. A.U. G _CU_.II .” what ..I..._ A._ C” r-in U”. Thlrc AA._“, mrhile ..A/__ it IC rnnv “‘“J not appear salient, history is taken seriously into account. More of a challenge will be modeling change-long-term modifications of a teacher’s knowledge, goals, and beliefs as a result of extended interactions with students.
PRAGMATICS(THEORYINTOPRACTICE) Schott raises the question of whether complex descriptive work of the type undertaken in T-I-C can improve instruction, and concludes that it cannot. My response: Read van Zee and Minstrell! Read Leinhardt! I want to start by stressing a strong personal belief in teacher professionalism. I do not like prescriptive models of teaching as suggested by Schott, for I think they are demeaning; moreover, they are inadequate preparation for the contingencies of the classroom, which require knowledge and judgment. In my opinion the goal of teacher preparation and professional development should be to help teachers develop the kind of knowledge and judgment that enables them to make wise, flexible, and well grounded decisions in the classroom. More to the point here, I believe that a primary mechanism for teachers to develop such knowledge is for them to reflect deeply on theirs and others’knowledge and actions. Leinhardt makes the case in general. Unclaimed,but I think even more helpful, is the role such an analysis can play in supporting new teachers. New teachers need to be able to unpack and annotate the lessons of more experienced “model” or mentor teachers. . . . The modeling activity itself may well be a useful tool for supporting new teachers’ emergent understandings. Schoenfeld argues convincingly that systematically pulling apart a lesson is revealing for researchers. It can be for teachers as well. Van Zee and Minstrell takes the same general stance and make the case via specifics, describing a series of activities by which the abstract analyses in T-I-C can be made real to teachers. It must be stressed that hers is not pie-in-the-sky speculation: she is intimately familiar with Minstrell’s work over the years, and has collaborated extensively with him in taking Minstrell’s abstract ideas and translating them into practice. They offer a series of concrete suggestions for providing tools and contexts that teachers can use in meaningful ways. I cannot add much to these, but I do want to make one more concrete suggestion and one more argument for having teachers engage in such analyses. The concrete suggestion involves the use of “video clubs,” in which groups of teachers (sometimes with the help of a researcher/facilitator, at least when they are getting
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started) gather to analyze and reflect on videotapes. As described in Frederiksen, Sipusic, Gamoran, and Wolfe (1992), and Sherin (1996), these can be valuable sites for professional development. The general argument concerns the feasibility and utility of having pre-service or in-service teachers engage in fine-grained analyses of the type found in T-I-C. First, I note that over the past few years, while half of the students in the Teacher Mm-lo1 I”I”ULI
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working toward a credential in our combined credential-plus-Master’s program. So, pre-service teachers can do such work! Should they? As Borko, Peressini, van Zee, and Minstrell point out, it is a lot of work to analyze a lesson in the way we do. Why do so, especially since they are not planning to be researchers? I shall make this final argument by analogy. Nearly 25 years ago, when I first moved into education from mathematics, I took a course in instructional design. It was a fascinating course. On average, members of the class spent more than 100 hours designing, creating, testing, and re-designing one hour’s worth of instruction on a topic of their choice. Their work started with the analysis of what it really meant to understand the topic. Believe it or not, it was mid-semester before they were allowed to start outlining the actual instruction! Now, you can argue that that kind of work was nuts. I was not planning to be an instructional designer; nor were any of the others in the class. Indeed, none of us were ever likely to spend more than a few hours planning for any particular class session. Yet, taking the course turned out to be a transformative experience for me. The net effect was that from then on I thought differently about instruction and the ways in which students engaged with it. Having thought so deeply about what it meant to understand one particular topic, I had a much better sense of what to look for and think about when I thought about teaching other topics. I mlannd anrl tallrrht Aiff oront1v uu 2.2 u 2 roc,,lt n-lx, yL~l,L~Lu Aifforontlxr UIIIUL1LUJ Cal&U ‘““6”’ UUILIL-ILUJ IL.JUIL nf “I “ ‘J nornr IIC.. llrw-lP?-PtcmvlinOE UI~UbIOC”I.UAIL&7, even though I may not have spent more time in the planning process. I hope and believe that teachers who develop a comparably deep understanding of even one episode of teaching will come to conceptualize their own teaching differently, and will teach differently as a result.
IN CONCLUSION Once again, my sincere thanks to all those who contributed to the work on modeling teaching, and to this exchange. What fun! I have learned a great deal. We have made great progress in the study of teaching. The kind of work done in the T-I-C paper waspossible only because of the quarter-century of work on thinking, learning, teaching and problem solving that preceded it. Only a few decades ago, researchers first embarked on (rather simplistic and basic) cognitive studies of problem solving. Now, we are starting to reap substantial benefits from that basic work Despite all the progress in general, we are still at the beginning of the learning curve when it comes to research on teaching-we are, after all, grap-
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pling with an extraordinarily difficult and complex task! It will take a while before we see the tangible benefits of this work, but we will. As van Zee and Minstrell note, “the promise is there but much work needs to be done.” So, let us have at it.
REFERENCES Atrnrri A Kocx.1 fMob.> 1 R, Smith 1 P Tocachinrr msthomntic~l nmhlom enl~~_ .Yx.U.L,‘..,a.Cuucr, L.,"'cuu,u.,u YUUL'L,,. I.1190+2\ \"/",. 'u.""'.(j IAL"ULCAn,UULLLI y&""luL, .7"1*analysis of an emergent classroom community. Research in Collegiate MathematVolume III (pp. l-70).Washington, DC: Conference Board of the Mathematical Sciences. Borko, H., & Peressini, D. (1998). Commentary on “Toward a theory of teaching-in-context.” Issues in Education, 4(l): 95-104. Dijkstra, S. (1998). The many variables that influence classroom teaching. Issues in Education, ing:
An
ics Education,
4(l): 105-110. Frederiksen, J., Sipusic, M., Gamoran, M., & Wolfe, E. (1992). Video portfolio assessment. Emeryville, CA: Educational Testing Service. Greeno, J, G. (1998). Where is teaching? Issues in Education, 4(l): 111-119. Greeno, J. G., Pearson, I’. D., & Schoenfeld, A. H. (1997). Implications for the National Assessment of Educational Progress of research on learning and cognition. In Assessment in transition: Monitoring the nation’s educational progress, background studies, (pp. 152215). Stanford, CA: National Academy of Education. Lehrer, R. (1998). Models as explanations. Issues in Education, 4(l): 121-123. Leinhardt, G. (1993). On teaching. In Robert Glaser (Ed.), Advances in Instructional psychology (Volume 4, pp. l-54). Hillsdale, NJ: Erlbaum. . (1998). On the messiness of overlapping goals and real settings. Issues in Education, 4(l): 125-132. Schoenfeld, A. H. (1985). Mathematical problem solving. Orlando, FL: Academic Press. (1992). On paradigms and methods: What do you do when the ones you know -. don’t do what you want them to? Issues in the analysis of data in the form of videotapes. journal of the Learning Sciences, Z(2): 179-214. (1994). Reflections on doing and teaching mathematics. In A. Schoenfeld (Ed.), -. Mathematical thinking and problem solving (pp. 53-70). Hillsdale, NJ: Erlbaum. (1998). Toward a theory of teaching-in-context. Issues in Education, 4(l): l-94. -. Schoenfeld, A. H., Minstrell, J., & van Zee, E. (1996). The detailed analysis of an established teacher carrying out a non-traditional lesson. Paper presented at the annual meeting of the American Educational Research Association, New York, April g-12,1996. Schoenfeld, A. H., Smith, J. I’., & Arcavi, A. A. (1993). Learning. In R. Glaser (Ed.), Advances in instructional psychology (Volume 4, pp. 55-175). Hillsdale, NJ: Erlbaum. Schott, F. (1998). On complexity and relevance in education research. Issues in Education, 4(l): 133-139. Sherin, M. G. (1996). The nature and dynamics of teachers’content knowledge. Unpublished docL-_-l -I:____L^ti^- TT^:-*^-^:~-L rrl:lz.-:^ -A 0^..1~..1,.. L”ldl UlbXlldll”1,, u‘uvnol~y “I ~cIIll”I‘IICIar ur1nnry. van Zee, E. M., & Minstrell, J. A. (1998). The promise of a theory of teaching-in-context. Issues in Education, 4(l): 141-147.
TEACHI
ALANH.SCHOENFELD UNIVERSITYOF CALIFORNIA, BERKELEY
I would like to begin this response with an expression of sincere thanks to Jerry Carlson for inviting me to write Toward a Theory of Teaching-in-Context (T-I-C), and to my friends and colleagues for their thoughtful commentaries on the paper. It’s an absolute pleasure to have such fine thinkers scrutinize my work from multiple perspectives, pushing me to think more deeply about its foundations, strengths, and limitations. They have done their job admirably, raising deep and important questions. This is what scholarly collaboration is all about. The issues raised in the commentaries tend to cluster into three main areas: ??
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Context and background assumptions; Examining and pushing the boundaries of the model; Pragmatics (theory into practice).
This response is organized along those lines. In each of the three areas I will try to deal with the main points made in the commentaries. It is interesting to note that the responses to T-I-C, though all written independently, carry on a dialogue of sorts: questions and issues raised in one commentary are often addressed directly in the others. I shall enter into that dialogue, at times pointing to exchanges across papers and at times adding my opinion and/or ,AA:I.Z,.,,, Uc(Lcl. 1^L^ CIUUIU”ILdl
CONTEXT AND BACKGROUND ASSUMPTIONS The commentaries raise many questions. Schott, for example, raises issues of epistemology. Why do I separate beliefs and knowledge; indeed, what is knowledge? Why did I choose the three components I did? He argues, as does Lehrer, that I need to say more about mechanism, and about my underlying assumptions-e.g., am I grounding my work in the standard assumptions of cognitive psychology? And, what about methods? These themes are echoed in some of the other essays as well. Dim! all correspondence lo: Alan H. Schoenfeld, University of California, Berkeley, Elizabeth and Edward Conner Professor of Education, Graduate School of Education, Berkeley, CA 94720-1670. Issues In Education, Volume 4, Number 1, 1998,pages 149-162 All rights of repmduction in any form reserved.
Copyright D 1998 by JAI Press Inc. ISSN: 1080-9724
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Responses to these questions are found, in part, in the papers by Green0 and Leinhardt. Greeno’s essay begins with the direct assertion “Schoenfeld and the Teacher Model Group . . . used standard framing assumptions of cognitive theory to develop a framework for modeling episodes of teaching.” As I mention in T-I-C, I see the work described here as part of a decades-long effort on the part of the field to develop understandings of complex human behavior in highly interactive ---?-I __I..___ T _z_,__~3. -1-1__ _I__ __‘.I. _._ _*r:--_-I social serungs. Lemnarat eiaoorares wun signincanr and helpful detail, locating our work squarely in the middle of a number of well established traditions, the most central of which is the cognitive science approach to the study of thinking and learning. This is the arena within which my earlier work on problem solving resides. Indeed, in my 1985 book Mathematical Problem Solving one finds many of the antecedents to the current work-specifically the claim that to understand an individual’s mathematical problem solving behavior, one needs to attend to that person’s knowledge and strategies, metacognition (specifically with regard to decision-making), and beliefs. I think of teaching as a problem-solving activity, so it is not surprising that there are significant correspondences between my analyEPEvnf y’ nrnhlmn cnlxrino and toarhino Thic mdhdc ac ~~11 SE rcxrrltc~ “b” ““sb”’ ““““‘ b Ull.. “““““b. LAY”hnlrla I.“&_” fnr _“I I&.~C1L”UY U” ..CY UY L..Y..‘b”, CPP “CL below. Given the multiple audiences for this volume, I have not gone into extensive detail regarding the workings of the model or the correspondences between the results of the modeling process and behaviors that are being modeled. As suggested by Green0 and Leinhardt, the general sense of mechanism invoked for the model is drawn from the cognitive/artificial intelligence literature-think “spreading activation networks” if you will. There are, of course, much more data and detail to the work could be presented in T-I-C. Additional data and analysis can be found in Schoenfeld, Minstrel& and van Zee (1996). A general discussion of methods, which were given relatively short shrift in T-1-C may be found in Schoenfeld (1992). Specifics regarding one of our methodological tools, competitive argumentation, are given in Schoenfeld, Smith, and Arcavi (1993). [At a metatheoretical level, I note that there is an interesting tension in the modeling enterprise. “Going all the way”-implementing a computational model via a computer program-not only demands an extraordinary amount of time and energy, but it is often very constraining. When one implements computational models one is limited to the state of the art with regard to the entities that can be modeled, and this may mean that some crucially important cognitive phenomena cannot be attended to. In the 1970s and 198Os, for example, artificial intelligence models of problem solving were not capable of coping adequately with issues of metacognition or beliefs. If you thought such things were important and wanted to focus on them (which I did), then you opted not to build the actual computational model. Instead you got “close” intellectually by working through such models as gedanken-experiments. And, of course, you used the standard techniques of multiple data sources, triangulation, and replication to try to insure that you weren’t selling yourself a bill of goods. Much the same applies here.]
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As Leinhardt notes, the Teacher Model Group’s (TMG’s) work fits squarely in the tradition of fine-grained analyses of teaching-a tradition that extends over some decades and to which Leinhardt herself has been a major contributor (see, e.g., Leinhardt, 1993). We have profited immensely from that work. For somewhat technical reasons we have used a somewhat different vocabulary and an alternative set of representations. These differences should not obscure the significant _---..1_.- 1. _a_~___ -.__. _ -_-.I- _ _I rl-_L _..-.--.-1. __-1_ _._ 3 overlap uerween our work anu mar which preceded it. For example, goals ana action plans play many of the roles of agendas as Leinhardt and Borko and Peressini use the term. Borko and Peressini are right on the mark when they suggest that a common feature of the three lessons discussed in some detail in T-I-C is that the lesson segments are all agenda-driven, and that the model is capable of handling lessons of that type. Dijkstra points out, rightly, that I have made some tacit assumptions about teacher and students sharing a workspace, having access to common board space, etc. He chooses to represent the task of instruction as a design problem, asking questions such as: “Which learning situation should be made? Which questions to chnrrld the ctrrrlcmtc U” An en malcc. ‘ tnnurlm-lo~ rnnctrrrriinn nnccihl~?” a&? What UY_.. ..ILUL UILVUI.. CA&_ YLUULIL.” L” S111U1.L ...“‘.““6’ ~“-.YUU~.I”I. y”““‘ “‘b. This formulation is entirely consistent with the assumptions of T-I-C. There may be some differences between conceptualizing teaching as a design problem or as problem solving, but the two conceptualizations are fundamentally compatible. When one asks what the teacher brings to the design task, one must ask what knowledge the teacher has at his or her disposal, what the teacher’s goals are, and what the teacher believes is important. In sum, a case can be made that Dijkstra and I are working in much the same problem space. For different reasons and in different ways, most of the other commentators can also be situated in the same space. For example, Borko, Peressini, and Leinhardt raise some very deep questions about the scope of the model. Their questions, which I shall address directly in the next section of this response, are of the following type: “Does the model live up to the claims Schoenfeld makes for it? Indeed, can it live up to such claims?” This way of posing the questions situates us on common ground, which Green0 calls the “standard framing assumptions of cognitive theory.” Green0 labels my framing assumptions clearly because he wants to challenge them. It is a fair challenge, and it raises some very interesting issues. In brief, I have chosen to examine “teaching” from the teacher’s point of view. But teaching and learning are highly interactive, and the teacher is but one member of a community in which all of the members participate and change. What of the other members, and what of the community itself as an evolving entity? My working metaphor for this issue is the motion picture Rashomon, in which contrasting views of the same events are seen through the perspectives of different individuals. Where does the truth or “reality” reside? Clearly, each individual has his or her own version of what happened, and it is of great interest to understand those. About the larger issue of “what really happened,” reviewer James Berardinelli says the following:
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Many people watch Rashomonwith the intent of piecing together a picture of what really occurred. However, the accounts are so divergent that such an approach seems doomed to futility. Rashomon isn’t about determining a chronology of what happened in the woods. It’s not about culpability or innocence.Instead,it focuses on somethingfar more profound and thought-provoking: the inability of any one man to know the truth, no matter how clearly he thinks he sees things. Perspective distorts reality and makes the absolute truth unknowable.
Those are chastening words for those of us engaged in the modeling enterprise. It is essential to recognize, up front, that we can never really get “inside the head” of another; and moreover, that the visions inside the heads of those involved in any event may not be commensurable. Nonetheless, it seems worth the effort to try, both at the individual and collective level. As Green0 points out, the teacher is just one participant in the classroom. The Teacher Model Group has chosen to focus (for now) on modeling the teacher as a central participant. In some circumstances, focusing on the teacher may seem to explain a large part of what happens in a classroom-even though, as we must always remember, the story is being seen through the teacher’s eyes. In others (e.g., where the agenda is co-constructed by teacher and students), what the teacher sees and thinks may seem to explain much less. In any case, understanding the view from each student’s eyes is a critical part of the overall story; equally critical is obtaining a description of the community as an organic entity. As I see it, it is an essential part of the theoretical enterprise to develop increasingly rich descriptions of individual participants, of the community, and of the relationships between and among them. What is not clear to me at this point is whether doing so will only require evolution (including changes of emphasis in what receives focal attention) or whether it will require something more dramatic. Two competing frameworks, cognitive and situative, are on the tabie. One comparative discussion of these frameworks may be found in Greeno, Pearson, and Schoenfeld (1997). Green0 et al. explore the two perspectives as constituting alternative framing assumptions for thinking about assessing people’s knowledge. From the cognitive perspective, the issues they identify as central for assessment include elementary skills, facts, and concepts; problem-solving strategies and schemata; metacognitive processes; beliefs and dispositions; engaging in mathematical practices. From the situative perspective, issues seen as central include basic aspects of participation in mathematical practice; personal identity and community membership; formulating and evaluating problems and solutions; making and checking meaning; and fluency with mathematical methods and representations. When one compares the two perspectives, one sees great overlap. A case can be made that the differences between them (though significant!) are largely differences in emphasis. Much teaching
the same seems to be the case here. For example, issue of establishing (co-constructing with students)
Green0 refers to the a discourse commu-
nity that has certain desired properties. I note that doing so is an explicit teaching goal for Ball and Minstrell, and for me-and that understanding how that hap-
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pens is one of my primary research goals. For reasons of grain size (we examined short instructional segments in detail) and focus (the big question for the T-I-C paper was, can we describe the model and show that it works?), we did not devote much detail to that particular topic. But we can-indeed, we have. Among the details in Schoenfeld, Minstrell, and van Zee (1996), there is an exegesis of lines l-35 of the Minstrel1 lesson segment discussed in T-I-C. That analysis identifies the ways in which specific questions and even wait time (a fuii thirteen seconds after asking, “Any questions that you have about course expectations or anything you’re wondering about the course?“) are devices used by Minstrell to help establish a desired classroom climate. In a similar analysis, Arcavi, Kessel, Meira, and Smith (1998) discuss various devices that I use in my problem solving course to build a particular kind of classroom community. At a larger level of grain size, in Schoenfeld (1994) I discuss the ways in which my problem solving class is structured
to foster mathematical
ownership
and agency on the part of my stu-
dents. I see these as part and parcel of the same thing, though different issues may come into focus at different levels of grain size. I. .* nt me same time, the &ishomon probiem remains. As a fieid, we have yet to develop the kinds of perspectives that provide meaningful syntheses of the individual and the collective. That problem needs attention.
EXAMINING AND PUSHING THE BOUNDARIES OF THE MODEL Leinhardt observes that “Schoenfeld needs to be careful about the specific relationship between the data he chooses, the claims he makes, and the assumptions about the power and use of modeling he carries over from its use in physics and chemistry” (p. 3). She notes, correctly, that I must be wary of introducing spherical --e-r-1 : ^^__^^ -L -1--e:--:_a... _^_._^I-L^1^^^_^^ - T -:-l..--rlL _-I_,.L”Wb llll” LI-^ Lllr Cld331”“lll. LUILILd1UL IdlbtD L.-A lU1lUdlll~l1Ldl ISSLlCb Ul ULJllldlll IZ~lSLlZmology-e.g., was the modeling enterprise successful in the three cases described because the subject matter is at the secondary level or above (so that developmental concerns re the students are not centrally important) and because mathematically related content is cut-and-dried enough to yield lessons that can be parsed neatly into chunks? Would a history lesson be more difficult to model, or perhaps not even amenable to modeling because of its fundamentally different nature? Borko and Peressini raise equally tough questions-some directly, some less so. They too raise developmental issues, conjecturing that [The theory
of teaching-in-context]
might do a better job modeling teachg
that occurs in secondary and post-secondary classrooms rather than elementary classrooms. The multiple, often contradictory goals central to [Magdalene] Lampert’s and [Deborah] Ball’s dilemmas and choices-goals which led Schoenfeld to acknowledge his difficulty with “the attribution and prioritization of goals and action plans”-did not seem to play a comparable role in the deliberations and decisions made by Nelson, Minstrell, and Schoenfeld (p. 5).
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Lurking behind that question, and behind Borko and Peressini’s invocation of dilemmas as discussed by Ball and Lampert, is the issue of whether the TMG’s framing of teacher behavior might be off the mark in the case of such dilemma-driven teaching. Beyond these issues, there are issues of scope and/or grain size-e.g., the questions Green0 raises about things that develop over time, and Lehrer, Borko and Peressini’s invocation of history as a factor in teachers’ decision-making. These are exactiy the right questions to ask. ruioreover, given that the evidence offered in T-I-C consisted of three cases described in partial detail, the commentators are right to be dubious about the answers. My response will be in two parts. First, I shall discuss one more (very small) instructional segment in rather finegrained detail. The segment of instruction comes from the January 19,199O “Shea Numbers” discussion in Deborah Balls classroom. I shall focus on one decision made by Ball in the midst of teaching. The analysis of this decision, which was done after the draft of the T-I-C paper was sent to the commentators, helps clarify some aspects of the model and the underlying theory. It also sets the stage for the second part of this section, a discussion of the issues summarized in the previous paragraph.
ONE FINAL EXAMPLE The context for this example was established in Section 7.2 of T-I-C, which the reader may wish to review. Ball’s class had been discussing the properties of even and odd numbers. The previous day they had held a special meeting with the previous year’s third grade class, in which such issues, and the question of whether the number zero is even, odd, or “special,” had been considered. As noted in Section 7.2, “one of Ball’s primary goals that day was to have students reflect on the previous day’s activities-not just on the mathematics but, more importantiy at the beginning of the lesson, on the state of their knowledge and learning.” She began the lesson with by announcing that agenda. And, she acted on it. Comments from students that focused on the nature of their thinking and learning were encouraged and elaborated upon, as in the following exchange: Ball: Student 1:
Ball:
Was there an example of something yesterday that you understood a little bit more during the meeting? Well, I didn’t think that zero was-zero, urn-even or odd until yesterday they said that it could be even because of the ones on each side is odd, so that couldn’t be odd. So that helped me understand it. Hmm. So you thought about something that came up in the meeting that you hadn’t thought about before? Okay.
Indeed, when two students began to dispute a particular mathematical point, Ball re-focused the class’s attention back to the issues of thinking and learning:
On Modeling Teaching
Ball:
Before we take this up again, I underst-I-I understand that this is still a probiem and that we didn’t a-we didn’t settle it, we’re probabiy not going to settle it. Urn, there’s a lot of disagreement about this issue, right? And you saw that the fourth graders who have been thinking about this for a long time also disagree about it, don’t they? I’m still kind of interested urn, in hearing some more comments about the meeting itself [Student 21 commented that it was good to have the two classes together because she heard an idea that she hadn‘t thought about and it made her think about and even revise her own idea when she was in the meeting yesterday. What other comments do other people have about the meeting and what happened yesterday? [Student 31, do you have a comment?
The conversation Ball: Student 4:
Ball:
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continued
in this vein, until the following exchange:
Other people? [Student 4]? Urn, first I said that urn, zero was even but then I guess I revised so that zero, I think, is special because urn, I-urn, even numbers, like they make even numbers; like 2, urn, 2 makes 4, and 4 is an even number; and 4 makes 4; 4 is an even number; and urn, like that. And, and go on like that and like 1 plus 1 and go on adding the same numbers with the same numbers. And so I, I think zero’s special. Can I ask you a question about what you just said? And then I’ll ask people for more comments about the meeting. Were you saying that when you put even numbers together, you get another even number-or were you saying that all even numbers are made up of even numbers?
it is this iast action i wish to expiore. Ball had announced her agenda, and she had stuck to it despite the tendency of some students’ comments to lead away from it. Yet at this point she deliberately took the conversation “off track.” Moreover, she did not do so to pursue the main mathematical issue under dispute, the question of whether zero is even, odd, or special. Instead she opted to pursue a somewhat peripheral issue, whether all even numbers are made up of even numbers. [For purposes of easy reference, I shall call this “the evenness question.“] The issue: why did Deborah Ball decide to pursue the evenness question at that point? The explanation ho-n nhlC,??r\...m.L “C-zll CzYlF L” “““IR
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salt. I include it because it serves to convey the flavor of the Teacher Model Group’s recent analyses, and to clarify some issues about the nature of the correspondence between goals and action plans. The argument, in brief, is that Ball undertook her (announced) detour in response to the presence of an emergent constellation of goals and constraints. None of the goals, in and of itself, was of high enough priority to warrant the detour. However, the combined priorities of the various goals was above threshold. That is, following up on the student’s comment allowed Ball to satisfy a num-
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ber of goals whose combined activation rendered important enough to pursue. In particular:
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the evenness question
1. Generally speaking, Ball is predisposed to pursue issues of mathematical substance when they emerge in classroom discourse-and when circumstances warrant their pursuit. The question of whether all even numbers are made up of /-_:-_ or) -A even ^____rmmuerb -__-l__-_ 15 :_ purentia~ly -_L_-L-l,__ one ^__ bucll ____LISSUE. :__.__ (yam I believe, however, that if there were no other factors favoring it, the “content value” of the evenness question would not have been enough to warrant its pursuit. After all, Ball had just declined to follow up on a more central content concern, the nature of zero. 2. Generally speaking, Ball is predisposed to explore issues related to her students’ understanding of various pieces of mathematics. That is, an important part of her teaching is developing a substantive understanding of what each of her students understands. Not only this student, but another as well had made a conjecture akin to the evenness question. Hence she might have wished to pursue their nnrl~rctnndinocnfit -"-~"'..^‘-~'b'""'
As in the case of (l), I do not think that this motivation would “win” on its own-or even in combination with (1). Note again that in the preceding conversation, Ball had bypassed the opportunity to pursue many students’ understandings of the nature of zero. 3. The second student mentioned in (2) had not been actively engaged in the discussion up to this point in the lesson. Discussing the evenness question would provide an opportunity to rope her directly into the conversation. This, too, was a non-negligible concern-but not of high enough priority to warrant action on its own. 4. In previous days the class had been engaged in making a range of conjecVV tures having to do with combinations of even and odd numbers. Ball’s intention was to return to these conjectures. Clarifying the nature of the evenness question, and possibly resolving it, could thus have established a platform for the work she intended to do later on. Much like (3), this is important but it is not important enough to warrant action by itself. 5. Since the nature of the student’s conjecture could be clarified in short order, only a small amount of time was necessary to pursue this detour-so Ball could return to the top item on her agenda, discussing the meeting. In contrast, there was the risk of a very substantial detour had the class pursued the discussion of whether the number zero, is even, odd, or special. In sum, the situation was as follows. Ball had announced and had been pursuing a high priority goal-discussing “what you thought about the meeting, what you noticed about the meeting, what you learned at the meeting.” A student made a comment closely related to the evenness question. Was the evenness question worth pursuing?
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Not on the grounds of reason (1) alone-or any of the other reasons discussed immediately above. But in combination, reasons (1) through (5) did provide substantial motivation to pursue it. Ball did so, but without abandoning her initial agenda, which still had high priority. Indeed, she announced that what she was about to do represented a short detour. And, when she had clarified the evenness question, she did indeed return to the topic of the meeting.
DISCUSSION If this analysis (or something close to it) is right, it is important for at least two reasons. First, it clarifies what may have been an all-to-easy-to-develop mis-impression from the examples in the T-I-C paper. It is possible, though inaccurate, to get the impression from the examples given that there is a simple correspondence between goals and action plans. (See, e.g., the discussion of overarching gouls in Section 6.) That is, from the parsing scheme one might infer that whenever a high priority goal pops to the top of the stack, a new action plan is implemented; and whenever that action plan is completed, the corresponding goal is satisfied and a new goal (along with concomitant action plan) comes to the top of the stack. There are, to be sure, many times when this happens. However, the instance of Deborah Ball’s decision-making discussed immediately above provides a clear example of when it does not. More importantly, it provides a nice indication of the complex relationship between goals and action plans. On the one hand, a particular action plan may fully or partially satisfy a number of different goals. (Indeed, when selecting a course of action in class, the teacher often chooses one that meets multiple objectives.) On the other hand, a constellation of goals and constraints may lead the teacher to move in a particular direction, even if no one of those goals has high enough priority by itself to move the teacher that way. S~mn,-l 1 ““y’ hnnp thic ~~amnl~ chnwc that lancn~su~ nf e”“’ onal_Arimm “LLV’LU, L .I.*” “.““‘ y’L .,I.“,.” .llUL the .IIL TMC’ Il.__ c Y ‘ ““~C‘“b’ “I UII1L11 analyses and the language of dilemmas employed by Lampert, Ball, Borko and Peressini are not really very far apart from each other in substance. If you strip the language of goals from the example described above, you have what I think is a classic “dilemma story,” which might be told as follows: Throughout the opening minutes of the lesson Ball had to juggle between conflicting possibilities, e.g., whether she should pursue content considerations at the cost of her goal of having the students reflect on the meeting, or vice-versa. For some time she kept to her original path, but then she veered off, because pursuing a new direction for a short while served multiple purposes. Having that hridlv rn~~rs~red --1_-1 --.-- d&mm -_--_I_* -__,, the --.- retllrnd ._.I _ _.__ tn ._ hm _.__ nriuinal ““b . ..-. aoonda “b” ._-. Tt _L ic .y worth ,,“lL1. noting that, as the class unfolded, Ball’s choices were made opportunisticallytaking into account her original intentions (agendas) but also responding to emerging opportunities and constraints.
In short: the language of “goals and action plans” may sound cold, simplistic, and mechanistic, but their use doesn’t have to be any of those. Decision-making
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during teaching can be extraordinarily complex, but it appears that the architecture of the model can deal with that complexity. To put it another way: a goaldriven architecture can represent dilemmas quite comfortably. Let me now turn to the issue of domain epistemology. Leinhardt conjectures that the teaching of (especially secondary and post-secondary) mathematics might be easy to model, while the teaching of social studies might prove extremely diffi. . . . ..Lll :L ll”L -_^Lllny”aal”lr. :--..,,:~1, ‘llK PI-, 1tZcXl”I, _^^^^_ >lllZ “l-,, glvea - __^_1s :, &L-L A-_.- I”1111S L---- “I -L lusL”ry t:-r--. CLUL, LlldL*I-^ UK ueey instruction are grounded in the fundamental sense-making aspects of the discipline: Good history lessons braid together one question inside of and next to another. The class discourse moves around a core construct; there are no neat boundaries. Activity boundaries marking movement from small group to large group activities do not necessarily coincide with intellectual boundaries marking movement from one concept to another. Most discussions are both embedded and overlapping. Thematic concerns resonate through an entire semester of discussion and writing. . . . TL, ^‘cl,lllrlLLtz ^^..I ..^_^^ ClllU ..-A I~“ -^^^-_.,.L,.-.._.-:-I:I. ^ L:“ r--..l_%__llltz “ ccuIle‘^_^^ Lce “ I S”UIClIlt; 111d lllbl” ry Kbb”,, _..-..,A W”UlU -_I... u,anr a mess of tidy parsing and sequencing of teaching such as that represented columns 2 and 5 in Figure 4. So the critique here is that this model of teaching must either expand its content boundaries or admit that it is seriously limited by them. If it chooses to expand, then it must alter the suggestion of cleanly iterative embedding (Leinhardt, p. 6).
To this I say, “amen.” Once again, my exposition may have suffered from the crispness of the examples given. I note with all seriousness that if you replace the word “history” with the word mathematics in the first paragraph quoted above, you produce what I think is an accurate description of my problem solving course. The main goai of my probiem soiving instruction is to have students emerge from the course “thinking mathematically.” This includes having them develop the predilections to symbolize and analyze; to solve problems in multiple ways, to generalize; to consider a solution to a problem a beginning rather than an end. Issues of strategies, metacognition, and beliefs are braided inside of and next to another from the first day of the course to the last. These are leit motifs of my course, which resonate through an entire semester of discussion and writing. In the TMG’s language, these themes are represented as overarching goals. They are always potentially present, in that they may be triggered at any time. They are sometimes addressed in a collateral way when the main focus of attention is on another goal which at the time has higher priority. And sometimes they are the major foci of classroom interactions. Consider, for example, the first two goals of Figure 7: GoalA: Goal B:
Have the class interact as a community of inquiry with the freedom to explore, conjecture, and reason things through. Have students experience physics as a way of making sense of the world.
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These are perfect examples of overarching goals. They recur throughout Minstrell’s instruction. Their presence will become even more apparent as the unit of analysis is expanded to the week, month, or year of instruction. Thus the neat recursive structure discussed by Leinhardt will be seen less frequently than in the examples in the T-I-C paper. In addition, as we get better at goal analysis, I expect to see many more episodes of multiple and overlapping goal satisfaction such as +hnco cahnxro Yl in tho rlirrrracinn nC tho UL”OL olcahnrzatorl LIKAVVICILLU U”““L bI,L UIO~UO.DI”IL “I L11b“DXTP~~OEC L”Q.ILILLOO nr~aatinn yIULc7u”‘L. o
Can the model handle such things? At this point I have to admit that I have little to go on beyond gut feeling. I do believe that mathematics taught for understanding is every bit as complex as history taught for understanding-1 think it is no accident, for example, that there are such strong parallels between Sam Wineburg’s and my respective analyses of historical and mathematical thinking. And I do believe (largely on the basis of having lived with my problem solving course for 20+ years, and having done prospective analyses of long-term issues such as the development of a “mathematical culture” in the course) that most of the teaching in the problem solving course-even when I am exploring new territory, in ways similar to Ball’s exploration of the evenness question--can be modeled. I shall make a few additional comments in closing this section. First, I think the conjecture by Borko, Peressini, and Leinhardt that (good) elementary school teaching will tend to be more difficult to model than (good) secondary teaching is probably true. One of the main factors affecting the difficulty of the modeling task is the complexity of the decision-making undertaken by the teacher. The fact is that once the appropriate classroom culture has been established at the secondary level, things can often go smoothly while the teacher focuses primarily on content-individual student concerns aren’t necessarily that much of an issue, though perhaps they should be. But the elementary teacher who gives short shrift to developmental concerns does so at his or her peril. There is much more of an ;mr&..;A..~l ch*rln*\tc r.mc-l +h&s. *\onrle I-.vr;nrr en e.ecnmA +r\ ;m~or~t;~ra +n t-n.., thn IIILy~IcacI”~ L” NL”“” LllC IILUIVluual JLUUCllLJ c&llll LllFll ILFFUJ. ILylr,fj L” caLLFILL L” such issues, as well as to what is to be learned, will certainly give rise to multiple and conflicting goals. The resulting complex decision-making is likely to be difficult to model. Second, Borko and Peressini also conjecture that it might be easier in general to model novice teachers rather than experienced ones. Here, I think the answer is “it depends.” Complexity is one important variable, consistency another. True, novices may have “less to model”-their knowledge may be less differentiated, their action plans simpler. But, their behavior may be more random. And, being experienced doesn’t necessarily mean that a teacher will take on issues in a more complex manner. Some teachers become highly skilled at maintaining order and following a very simple lesson structure. Modeling this kind of “experienced” teaching is easy. Third, history. Borko and Peressini note that “some explanatory and predictive power-and by default, some scope-will be sacrificed if a model of Minstrell’s teaching does not take into account prior class experiences” (both with these students and with previous groups of students). Lehrer argues similarly. I agree. This
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is where the notion of lessonimageplays a central role. On the basis of prior experience, Minstrell knows what kinds of things to expect from his students at the beginning of the course. Later in the course, as he thinks about a day’s lesson, he knows where he wants the current group of students to get; he knows what knowledge and experiences from earlier in the course they bring to the lesson he is planning; and, he knows how other groups of students with similar backornllndc ..A... nnrl “.y”‘ ~vn~rienre hawe rcurtd tn hP r*..“’ nlanc tn (y”.a”“” u’“.. A.U. G _CU_.II .” what ..I..._ A._ C” r-in U”. Thlrc AA._“, mrhile ..A/__ it IC rnnv “‘“J not appear salient, history is taken seriously into account. More of a challenge will be modeling change-long-term modifications of a teacher’s knowledge, goals, and beliefs as a result of extended interactions with students.
PRAGMATICS(THEORYINTOPRACTICE) Schott raises the question of whether complex descriptive work of the type undertaken in T-I-C can improve instruction, and concludes that it cannot. My response: Read van Zee and Minstrell! Read Leinhardt! I want to start by stressing a strong personal belief in teacher professionalism. I do not like prescriptive models of teaching as suggested by Schott, for I think they are demeaning; moreover, they are inadequate preparation for the contingencies of the classroom, which require knowledge and judgment. In my opinion the goal of teacher preparation and professional development should be to help teachers develop the kind of knowledge and judgment that enables them to make wise, flexible, and well grounded decisions in the classroom. More to the point here, I believe that a primary mechanism for teachers to develop such knowledge is for them to reflect deeply on theirs and others’knowledge and actions. Leinhardt makes the case in general. Unclaimed,but I think even more helpful, is the role such an analysis can play in supporting new teachers. New teachers need to be able to unpack and annotate the lessons of more experienced “model” or mentor teachers. . . . The modeling activity itself may well be a useful tool for supporting new teachers’ emergent understandings. Schoenfeld argues convincingly that systematically pulling apart a lesson is revealing for researchers. It can be for teachers as well. Van Zee and Minstrell takes the same general stance and make the case via specifics, describing a series of activities by which the abstract analyses in T-I-C can be made real to teachers. It must be stressed that hers is not pie-in-the-sky speculation: she is intimately familiar with Minstrell’s work over the years, and has collaborated extensively with him in taking Minstrell’s abstract ideas and translating them into practice. They offer a series of concrete suggestions for providing tools and contexts that teachers can use in meaningful ways. I cannot add much to these, but I do want to make one more concrete suggestion and one more argument for having teachers engage in such analyses. The concrete suggestion involves the use of “video clubs,” in which groups of teachers (sometimes with the help of a researcher/facilitator, at least when they are getting
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started) gather to analyze and reflect on videotapes. As described in Frederiksen, Sipusic, Gamoran, and Wolfe (1992), and Sherin (1996), these can be valuable sites for professional development. The general argument concerns the feasibility and utility of having pre-service or in-service teachers engage in fine-grained analyses of the type found in T-I-C. First, I note that over the past few years, while half of the students in the Teacher Mm-lo1 I”I”ULI
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working toward a credential in our combined credential-plus-Master’s program. So, pre-service teachers can do such work! Should they? As Borko, Peressini, van Zee, and Minstrell point out, it is a lot of work to analyze a lesson in the way we do. Why do so, especially since they are not planning to be researchers? I shall make this final argument by analogy. Nearly 25 years ago, when I first moved into education from mathematics, I took a course in instructional design. It was a fascinating course. On average, members of the class spent more than 100 hours designing, creating, testing, and re-designing one hour’s worth of instruction on a topic of their choice. Their work started with the analysis of what it really meant to understand the topic. Believe it or not, it was mid-semester before they were allowed to start outlining the actual instruction! Now, you can argue that that kind of work was nuts. I was not planning to be an instructional designer; nor were any of the others in the class. Indeed, none of us were ever likely to spend more than a few hours planning for any particular class session. Yet, taking the course turned out to be a transformative experience for me. The net effect was that from then on I thought differently about instruction and the ways in which students engaged with it. Having thought so deeply about what it meant to understand one particular topic, I had a much better sense of what to look for and think about when I thought about teaching other topics. I mlannd anrl tallrrht Aiff oront1v uu 2.2 u 2 roc,,lt n-lx, yL~l,L~Lu Aifforontlxr UIIIUL1LUJ Cal&U ‘““6”’ UUILIL-ILUJ IL.JUIL nf “I “ ‘J nornr IIC.. llrw-lP?-PtcmvlinOE UI~UbIOC”I.UAIL&7, even though I may not have spent more time in the planning process. I hope and believe that teachers who develop a comparably deep understanding of even one episode of teaching will come to conceptualize their own teaching differently, and will teach differently as a result.
IN CONCLUSION Once again, my sincere thanks to all those who contributed to the work on modeling teaching, and to this exchange. What fun! I have learned a great deal. We have made great progress in the study of teaching. The kind of work done in the T-I-C paper waspossible only because of the quarter-century of work on thinking, learning, teaching and problem solving that preceded it. Only a few decades ago, researchers first embarked on (rather simplistic and basic) cognitive studies of problem solving. Now, we are starting to reap substantial benefits from that basic work Despite all the progress in general, we are still at the beginning of the learning curve when it comes to research on teaching-we are, after all, grap-
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pling with an extraordinarily difficult and complex task! It will take a while before we see the tangible benefits of this work, but we will. As van Zee and Minstrell note, “the promise is there but much work needs to be done.” So, let us have at it.
REFERENCES Atrnrri A Kocx.1 fMob.> 1 R, Smith 1 P Tocachinrr msthomntic~l nmhlom enl~~_ .Yx.U.L,‘..,a.Cuucr, L.,"'cuu,u.,u YUUL'L,,. I.1190+2\ \"/",. 'u.""'.(j IAL"ULCAn,UULLLI y&""luL, .7"1*analysis of an emergent classroom community. Research in Collegiate MathematVolume III (pp. l-70).Washington, DC: Conference Board of the Mathematical Sciences. Borko, H., & Peressini, D. (1998). Commentary on “Toward a theory of teaching-in-context.” Issues in Education, 4(l): 95-104. Dijkstra, S. (1998). The many variables that influence classroom teaching. Issues in Education, ing:
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4(l): 105-110. Frederiksen, J., Sipusic, M., Gamoran, M., & Wolfe, E. (1992). Video portfolio assessment. Emeryville, CA: Educational Testing Service. Greeno, J, G. (1998). Where is teaching? Issues in Education, 4(l): 111-119. Greeno, J. G., Pearson, I’. D., & Schoenfeld, A. H. (1997). Implications for the National Assessment of Educational Progress of research on learning and cognition. In Assessment in transition: Monitoring the nation’s educational progress, background studies, (pp. 152215). Stanford, CA: National Academy of Education. Lehrer, R. (1998). Models as explanations. Issues in Education, 4(l): 121-123. Leinhardt, G. (1993). On teaching. In Robert Glaser (Ed.), Advances in Instructional psychology (Volume 4, pp. l-54). Hillsdale, NJ: Erlbaum. . (1998). On the messiness of overlapping goals and real settings. Issues in Education, 4(l): 125-132. Schoenfeld, A. H. (1985). Mathematical problem solving. Orlando, FL: Academic Press. (1992). On paradigms and methods: What do you do when the ones you know -. don’t do what you want them to? Issues in the analysis of data in the form of videotapes. journal of the Learning Sciences, Z(2): 179-214. (1994). Reflections on doing and teaching mathematics. In A. Schoenfeld (Ed.), -. Mathematical thinking and problem solving (pp. 53-70). Hillsdale, NJ: Erlbaum. (1998). Toward a theory of teaching-in-context. Issues in Education, 4(l): l-94. -. Schoenfeld, A. H., Minstrell, J., & van Zee, E. (1996). The detailed analysis of an established teacher carrying out a non-traditional lesson. Paper presented at the annual meeting of the American Educational Research Association, New York, April g-12,1996. Schoenfeld, A. H., Smith, J. I’., & Arcavi, A. A. (1993). Learning. In R. Glaser (Ed.), Advances in instructional psychology (Volume 4, pp. 55-175). Hillsdale, NJ: Erlbaum. Schott, F. (1998). On complexity and relevance in education research. Issues in Education, 4(l): 133-139. Sherin, M. G. (1996). The nature and dynamics of teachers’content knowledge. Unpublished docL-_-l -I:____L^ti^- TT^:-*^-^:~-L rrl:lz.-:^ -A 0^..1~..1,.. L”ldl UlbXlldll”1,, u‘uvnol~y “I ~cIIll”I‘IICIar ur1nnry. van Zee, E. M., & Minstrell, J. A. (1998). The promise of a theory of teaching-in-context. Issues in Education, 4(l): 141-147.