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Cultural Artifacts, Self Regulation, andLearning: Commentary on Neuman's "Canthe Baron von Münchausen Phenomenonbe Solved?"Rogers HallPublished online: 17 Nov 2009.

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Cultural Artifacts, Self Regulation, and Learning:Commentary on Neuman’s “Can the Baron von

Münchausen Phenomenon be Solved?”

Rogers HallGraduate School of Education

University of California, Berkeley

Because the focus article and commentaries of this symposium are concerned with history andmind, I feel compelled to start with a historical note. Neuman’s (this issue) article draws heavilyfrom a recently published article (Prawat, 1999) and series of commentaries in theAmerican Edu-cational Research Journal(AERJ,Spring, 1999, Vol. 36, No. 1). Prawat’s article proposed that a“learning paradox”, dating back to Socrates’ dialogue of the Meno, throws contemporary learningtheories into a crisis.

On my comparative reading of theAERJseries and the focus article in this issue, Neuman pro-poses that the “self-explanation effect” (discussed more fully in this article) is a special case ofthis learning paradox. He then draws directly from James Gee’s critical commentary on Prawat’s(1999) article, in which Gee rejected (literally, as part of the review cycle) that article for making“no substantive contribution to current debates about teaching and learning” (p. 87). Specifically,Gee argued (a) that there is no learning paradox, only the problem of studying how learning oc-curs, and (b) that attempts to resurrect the paradox neglect what sociohistorical theories have con-tributed to detailed studies of learning. As Gee put it:

Neo-Vygotskian approaches argue that the mind is furnished by social interactions (embodied-mindapproaches add the material world, as well). Thus, as social interactions change, the mind is furnishedand refurnished in various ways. Furthermore, neo-Vygotskian approaches have specific views on therelationship between intramental and extramental activity, as well as their own version of how cogni-tion goes beyond the skin. In particular, the way in which tools (artifacts, symbols, schemas, technolo-gies, participant frameworks, etc.) serve as mediating devices for human thinking, acting, and learningis crucial to the neo-Vygotskian account (see Wertsch’s work, in particular, especially his recent book,Mind as Action, 1998). Such mediating devices offer (as Wertsch points out) a very useful way to talkabout learning and change at the individual, social, cultural, and historical levels […] None of this iseasily discussable in broad, philosophical terms. One has to get down to discussions of specific theoret-ical frameworks and empirical approaches. (pp. 92–93)

MIND, CULTURE, AND ACTIVITY, 8(1), 98–108Copyright © 2001, Regents of the University of California on behalf of the Laboratory of Comparative Human Cognition

Requests for reprints should be sent to Rogers Hall, Graduate School of Education, University of California, Berkeley,4641 Tolman Hall #1670, Berkeley, CA 94720–1670. E-mail: rhall@socrates.berkeley.edu

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This is exactly what Neuman has set out to do in his article. I find myself in the strange position ofsimultaneously being critical of Neuman’s specific claims (as an anonymous reviewer, I rejectedhis article) and enthusiastic about the general project of using aspects of activity theory to reworkproblems of learning and development.

Neuman’s article proposes a radical break with traditional cognitive accounts of self-explana-tion (or, more generally, any sort of self-reference during problem solving and learning). He setsout to show how dominant philosophical traditions that inform psychological theory1 have createda false learning paradox, a situation in which the mind, after being cut off from others and theworld, needs either to contain already what could be learned (Rationalism) or to deploy mentalfaculties that build new, higher-order knowledge out of old, lower-order components (Empiri-cism). This is the paradox—the idea that the mind needs to extend beyond its own boundaries dur-ing learning—that Neuman seeks to dissolve.

Activity theory, Neuman argues, provides the solution by putting the learner back into theworld, and moreover a world filled with cultural artifacts that provide the means for transforminghuman thinking. Under this view, there is no learning paradox in phenomena of self-reference. Byparticipating in cultural systems of activity, which themselves reflect the construction of humanconsciousness through history, the learner acquires not only knowledge, but mind itself.

These will be familiar themes to many readers ofMind, Culture, and Activity, but to help focusmy commentary on the role of cultural artifacts in self-reference, I include two quotations that Iunderstand2 to be central statements in a cultural-historical approach to mind. First, setting out aradical alternative to explaining the origin of consciousness in terms of consciousness itself (i.e.,the resurrected paradox that lead’s Neuman’s article), Vygotsky (1981) argued,

Any function in the child’s cultural development appears twice, or on two planes. First it appears on thesocial plane, and then on the psychological plane. First it appears between people as aninterpsychological category, and then within the child as an intrapsychological category. This isequally true with regard to voluntary attention, logical memory, the formation of concepts, and the de-velopment of volition. We may consider this position as a law in the full sense of the word, but it goeswithout saying that internalization transforms the process itself and changes it structure and functions.Social relations or relations among people genetically underlie all higher functions and their relation-ships. (p. 163)

A second statement, specifically concerned with the role of cultural tools or artifacts in activitywhere subject and object are formed, comes from Leont’ev (1981):

The tool mediates activity and thus connects humans not only with the world of objects but also withother people. Because of this, humans’ activityassimilates the experience of humankind.This meansthat humans’ mental processes (their “higher psychological functions”) acquire a structure necessarily

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1See Case (1992) for an integrative analysis of the contributions of these same two philosophical traditions with a cul-

tural–historical view, including a programmatic outline for research on human development. A similarly careful, integra-tive treatment can be found in Saxe (1991). There is, of course, considerable variety in how historical material (i.e.,sociogenesis) is said to play a role in (or be articulated with) analyses of individual development (ontogenesis).

2Quoting from central texts in activity theory strikes me as a risky business, particularly since I consider myself an out-

sider to this tradition. My purpose here is only to bring the terms of the argument into relief, and not to sort out the intellec-tual history of the tradition itself (e.g., Kozulin, 1986).

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tied to the sociohistorically formed means and methods transmitted to them by others in the process ofcooperative labor and social interaction. (p. 56)

Neuman adopts these (still radical, I think) ideas about the social or cultural origins of mind todispatch the learning paradox. Then, in the second half of his article, he sets out a detailed encod-ing of mental operations that people are said to employ while learning to solve algebra story prob-lems through self-explanation. The empirical material he examines most closely is the use of atable during a single subject’s attempt to solve a mixture problem. On this table rests both the phe-nomenon of self-explanation and Neuman’s use of activity theory to dissolve any apparent learn-ing paradox.

By my reading, Neuman’s empirical analysis looks quite like traditional cognitive accounts ofproblem solving in which “think aloud” protocols are segmented and coded as sequences of oper-ations that transform representational states in a given “problem space” (Newell & Simon’sHu-man Problem Solving, 1972, is cited). Neuman, however, argues that his detailed encoding occurswithin a “totally different context” for studying mediated activity.

In this commentary, I take up and extend the idea that activity theory could provide fresh theo-retical language for analyzing problem solving and learning in this particular area of mathematics(i.e., learning to solve “applied” problems typical of contemporary algebra texts). I take exceptionto some of Neuman’s claims, but overall, I think his article provides an opportunity to push issuesof empirical analysis into contact with theory in interesting ways. My comments invert the pro-gression of Neuman’s article, starting with the empirical case he analyzes, asking how Beth’s ut-terances constitute “explanation” or “self-reference” in some compelling way, and then lookingmore closely at what might be involved in a cultural–historical analysis of the particular table sheis using.

WHAT IS BETH EXPLAINING TO HERSELF?

I originally approached Neuman’s article with great interest, both because much of my own re-search has been about how people use and learn mathematics, but also because self-explanation hasbeen a hot topic in cognitive science over the past decade. I was eager to see what self-explanationconsists of in mathematical problem solving from the perspective of activity theory. In the tran-script fragment from Beth’s solution attempt on a typical mixture problem (Problem 2, given inFigure 1 and the Appendix of Neuman’s article), a table is drawn, column labels are written that or-ganize types of quantities (e.g., % sugar vs. amount of jam), rows are used to represent values forparticular jams, an algebraic variable and expression are introduced as cell entries, and trouble withthis expression (i.e., “No!” at Line 9) is apparently repaired by reworking the structure of the tableso that rows reflect three distinct jams (i.e., Lines 10–12).

According to Neuman, there are 10 codable instances of “transformatory-explanation” in thetranscript.3 These come in threetypes of activity: object transformations, chains of action, and sit-uated activity. These activities can be implemented through fourtypes of action: mapping values

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3Line numbers for transcribed utterances do not correspond with numbered “explanations” in Figure 1, though both pre-

serve the sequence of utterances and actions. I am not sure what the arrows in Figure 1 represent, other than to indicate typesof coded actions.

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from one representational form to another, inferring new semantic categories, deducing valuesfrom known quantitative relations, and articulating new variables. Each type of action is said to bean explanation.4

After several readings of Neuman’s current article, I cannot find a direct account of what Bethis explaining to herself. This is a serious omission, and it remains despite repeated efforts to elicitthis account during the review cycle. To pursue this question, I focus first on what is meant by“self-explanation” in the existing literature (i.e., what is being explained and why) and second onwhat aspects of this transcript might correspond to this or some other form of self-reference.

Beth is doing something in this excerpt from a solution attempt, but she is not, I think, doingwhat Chi et al., (1989, cited in Neuman’s article) originally described as a self-explanation:

[A] good student5 “understands” an example solution and will succeed in generalizing because he orshe makes a conscious effort to ascertain the conditions of application of the solution steps beyondwhat is explicitly stated. To do so, the student must “explain”how the example instantiates the princi-ple [italics added] which it exemplifies. (p. 149)

In Chi et al.’s (1989) study and in a stream of follow on research seeking to replicate and extend the“self-explanation effect” (some cited in Neuman’s article), learners were said to “self-explain” inone of two senses. First, they fill in or justify steps in a worked example; second, they elaborate onthe reason for taking a particular step while solving a new problem. The idea, by my reading (seepreceding italics) is that a learner asked herself about the point of the current activity and orients insome fashion toward the future of this activity (i.e., toward re-using some principle relevant in thecurrent activity).

For example, participants in (Chi et al., 1989) were asked to “talk aloud” while studying aworked example to a physics problem involving an inclined plane. After reading the followingtextual instruction,

It is convenient to choose the x-axis of our reference frame to be along the incline and the y-axis to benormal to the incline

a study participant reported,

and it is very, umm, wise to choose a reference frame that’s parallel to the incline, parallel and normal tothe incline, because that way, you’ll only have to split up mg, the other forces are already, componentvectors[italics added] for you. (p. 164)

The textual instruction does not explicitly state why it is “convenient” to overlay a referenceframe along the incline, but the learner fills in this instructional gap by noticing that forces will be

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4It is possible that I misunderstood Neuman's argument for these analytic categories, but I think I am right about the idea

that action categories implement types of activity. These would correspond, I assume, to Leont’ev’s (1981) distinction be-tween activity, action, and operation as different levels of analysis.

5Contrasting “good” and “poor” students is central to research on the self-explanation effect, since “poor” problem solv-

ers are said to produce fewer and more superficial self-explanations. This distinction is not used in Neuman's article. Howone goes about being a “good” or “poor” participant in some cultural activity is, of course, theoretically important (Lave,1996).

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split in ways that lead to a simpler solution (i.e., in italics). In Neuman’s example, Beth is solving anew mixture problem, having (presumably, we are not told) studied worked examples to similarproblems in some prior phase of the study. But by my reading of the transcript, none of her utter-ances contain the kind of deliberate, future-oriented reflection reported as self-explanation by Chiet al. (1989).

Even if Neuman has chosen a poor empirical example to illustrate his coding scheme, he mightstill be right when he claims that an “orthodox cognitive approach” offers little to an analysis ofself-explanation or learning. This is, of course, his reason for turning to activity theory. But look-ing over the existing literatures on self-explanation and solving algebra story problems, there aregood reasons to disagree with Neuman’s assessment. Part of the published literature on self-ex-planation contains detailed analyses of the content of participants’ explanations (e.g., Chi et al.’s,1989, original study does this, as do others). In an ambitious related project (VanLehn, Jones, &Chi, 1992), the content of these explanations was even used to construct and test computationalmodels of problem solving and learning. Specifically, VanLehn et al.’s rule learning system(CASCADE) is said to acquire new content rules and search heuristics by applying methods of in-duction and analogical inference over fragments of solution derivations. At least at the level ofgoal-directed action implemented in rule-based operators (i.e., analytic categories very similar toNeuman’s), the self-explanation effect has been extensively investigated.

In the area of learning to solve algebra story problems, there is a long line of research on howpeople construct and use “situation models” to bridge between problem texts and normative de-mands for algebraic solutions (Kintsch & Greeno, 1985; Hall, Kibler, Wenger, & Truxaw, 1989;Nathan et al., 1992, cited in Neuman’s article). Whereas Neuman mentions the study by Nathan etal. as an example of the “orthodox” approach, he completely excludes any mention of the role ofsituation models as intermediary supports for mathematical problem solving and learning. This isstriking omission, since one design goal of Nathan et al.’s ANIMATE system was to providelearners with a yoked collection of representational systems that facilitate the coordination of situ-ation and problem models. These representations included tables exactly like the artifact Neumanpicks out of Beth’s solution attempt.

As part of my effort to make sense of Neuman’s empirical claims when writing this commen-tary, I sought out one of his earlier studies of the self-explanation effect, published in theBritishJournal of Educational Psychology(Neuman & Schwarz, 1998, cited in his current article). Bymy reading of this earlier study, which focuses on “analytical reasoning problems” typical of col-lege entrance examinations, the authors work in exactly the cognitive stance that Neuman nowcriticizes:

We suggest that self-explanations during problem solving serve the same function as self-explanationswhile learning from worked-out examples, namely the construction of a complete representation (Chiet al., 1989, 1994). This function may be fulfilled while solving problems by justifying a solution stepor clarifying the problem. Justifying a solution step may support theselection of operators and expli-cate the rules to be “fired” in a given condition. Clarifying the problem may explicate the initial prob-lem-space and the legal operators to be used. (p. 17)

I am left to wonder about the relation between cognitive orthodoxy and Neuman’s use of activitytheory, particularly his adoption of Leont’ev’s three level model of an activity system (i.e., activity,action, and operation). Users of Leont’ev’s model have frequently been criticized for inadequately

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analyzing the social or historical basis of activity (Engeström, 1999a), and this is one place whereNeuman’s “totally different” alternative to cognitivism seems barely to have started.

The complete study from which Beth’s transcript was excerpted is listed as being in press, so Ifocus more closely on what Beth is doing that might be considered a form of self-reflection. Shecertainly encounters a “snag”6 and makes an effort to “repair” it within the transcript, and these ac-tivities could sensibly be called a kind of self-reflection. At Lines 8–12, Beth begins to write out acluster of quantities related through an algebraic expression (i.e., X, 80, and 80–X). But she stopsherself (i.e., “No!” at Line 9) and reorganizes her table to show three distinct jams as rows of thetable (i.e., the given mixture is labeled “new” in Line 10). What results is a shift from a column la-beled “Mixture” (Line 4) to a row labeled “New,” and this could be interpreted as a dramaticchange in Beth’s use of the table.

As these tables usually appear in textbooks, column labels describe a three-part multiplicativerelation between the types of quantities making up a mixture (i.e., amount of sugar, rate of concen-tration, and amount of mixture). Rows describe an additive relation between different mixtures(e.g., two jams with different concentrations, added to make a third jam). If Beth originally in-tended the column labeled “Mixture” to represent different kinds of jams (Line 4), then she hasnoticed (“No!”) and recovered from a conceptual error that mixes together multiplicative and ad-ditive relations in this problem. One would, of course, need a transcript of the full solution attemptto confirm this interpretation.

Beth’s use of self-explanation is in doubt, but she clearly struggles to coordinate different sys-tems of representation while working on this mixture problem. At Lines 8 and 9, she attempts tolocate algebraic expressions involving a variable in the cells formed by a table of typed quantities.This appears to be what Neuman means by “articulatory transformatory explanation” as an actioncategory that introduces a new variable (i.e., what Beth calls the “unknown” at Line 8). As I readthe transcript, however, the moment of self-reference (“No!”) appears to coincide with Beth’s at-tempt to place one form of representation inside another (i.e., writing expressions into cells of thetypological array formed by the table).

This leads me to a second puzzle in Neuman’s analysis. Why not analyze algebraic expressionsas mediating artifacts, just as he proposes to do with the table? In my own studies (Hall, 1990,1996; Hall et al., 1989), people’s attempts to solve algebra story problems are often supported by aheterogeneous collection of representational forms. Some of these are part of standard algebrapedagogy (e.g., algebraic expressions and “role tables” of the sort analyzed by Neuman), but otherforms are less standard and could even be said to be “invented”7 as problem solvers struggle toalign their understanding of the problem situation with demands for a precise algebraic solution.Participants in these studies are clearly (painfully) aware of the struggle, as I show here, and theysometimes are able to repair conceptual errors by using nonstandard forms to work around diffi-culties associated with the tables or algebraic expressions they are taught.

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6I borrow the terms “snag” and “repair” from Olivia de la Rocha's (1986) study of Weight Watchers struggling with the

demands of precise measurement in their kitchens. She argued that snags and repairs where central to a form of gap-closingarithmetic that was organized (culturally and historically) in ways quite different from school or psychological experi-ments.

7The idea that ordinary learners might “invent” representational forms in ongoing activity is intriguing. DiSessa, Ham-

mer, Sherin, & Kolpakowski (1991) investigated this in the context of representing motion. Most directly relevant toNeuman’s project, Meira (1995) uses Leont’ev’s three level scheme to analyze inventions in the microgenesis of quantita-tive reasoning about related rates.

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WHICH CULTURAL ARTIFACTS MATTER IN BETH’S WORK, ANDHOW DO THEY FIGURE IN A BROADER SYSTEM OF ACTIVITY?

As Neuman uses Leont’ev’s (1981) theoretical framework, the object of Beth’s brief struggle is to“bridge the gap” between the story problem (called a “primary object”) and an algebraic equation(the “goal object”). The table simply appears as a “material artifact” for this purpose. Neuman’sanalysis tells us nothing about where the table comes from, how it might provoke some form ofself-reference for Beth, or how its use as a “symbolic tool” might matter for her participation orlearning in a broader activity system.

To dig further into these questions, I do two things. First, I consider briefly what kind of activ-ity system Beth is facing, and this also involves looking into the history of some of the material ar-tifacts in play. Second, I contrast fragments of “talk aloud” protocol excerpted from one of myown studies of how people with different backgrounds solve algebra story problems (Hall, 1990).This contrast, between a student and a teacher of algebra, may be helpful for understanding thehistorical origin and conflicted use of Beth’s table.

By my reading, Beth is participating in a very specific activity system—the undergraduate psy-chology experiment—and this deserves some analysis from a theory of activity.8 Because I some-times participate in this kind of system, I make some guesses. Beth is probably encountering theexperimenters for the first time, or if not, she is bartering with them to exchange her participationfor course credit. In this sense, her participation has a very short history (and future) in terms of theactual social relations involved and the material or institutional setting in which these activitiesappear. For example, related to my earlier question about self-explanation as a future-oriented ex-perience, the motive force behind Beth’s objective may not be to improve her performance onsome later occasion. This kind of guessing could go on, but much has already been written aboutthe psychological experiment as an institutional site for constructing human rationality (e.g.,Lave, 1980/1997, 1988, 1996; Cole, Hood, & McDermott, 1978/1997). I simply want to index theidea that Beth is participating in something quite a lot more complicated—and within activity the-ory, more analyzable—than translating words into equations.

From Beth’s perspective (as I continue to imagine it), a variety of cultural artifacts mightseem very important. These include the text of the story problem, the table she recalls, and thealgebraic expressions she constructs and attempts to distribute over the cells of the table. Fol-lowing Gee’s advice (quoted earlier), we might also want to consider her ongoing conversationwith the interviewer (a durable participation structure, even if a strange one) and her own em-bodied efforts to coordinate one inscribed form with another under the lidless gaze of a videocamera. As the list of cultural artifacts gets longer, it becomes clear that Beth inhabits a complexscene in which present time actions (i.e., “thinking aloud” or solving a problem) are sedimentedtogether and constructed out of artifacts and structures of interaction that have a substantial (andanalyzable) history.

For example, texts such as Beth’s mixture problem have a rich history in cognitive psychology,where for over 30 years (e.g., Paige & Simon, 1966) they have provided a flexible technology ofgraded “task domains” for studies of problem solving, learning, the development of mathematicalthinking, and instructional design (Reed, 1999). As parts of the mathematics curriculum, these

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8See Saljo & Wyndhamn (1993) for a comparative analysis of how different contexts of instruction affect students’ so-

lutions to word problems.

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texts have an even longer history stretching back to “practical arithmetics” put into wide circula-tion by the development of movable type printing technologies in the 15th century (Swetz, 1987).These texts were studied by young men aspiring to work as “computers” along an expanding net-work of European trade routes. The mixture problems used by Neuman are not much differentfrom “alligation” problems like the following:

A merchant has 46 marks, 7 ounces of silver, alloyed at 7 ounces and 1/4 per mark. He wishes to cointhis so that it shall contain 3 ounces and 1/2 of fine silver per mark. Required is to know the amount inthe mixture and how much brass he must add. (Treviso Arithmetic, 1478, R. 52, v., translated in Swetz,1987)

I am not sure how (or whether) to follow a historical trajectory from minting money, to makingjam, to learning mathematics, to solving algebra problems under the watchful eye of psycholo-gists. But some kind of bracketing or periodization seems necessary for understanding how mindforms out of participation in cultural activity. As described in a recent overview of the field byEngeström (1999a; see also Engeström, 1999b; Hall, 1999b), these issues are a going concern foractivity theory:

Historical analysis must be focused on units of manageable size. If the unit is the individual or the indi-vidually constructed situation, history is reduced to ontogeny or biography. If the unit is the culture orthe society, history becomes very general or endlessly complex. If a collective system is taken as theunit, history may become manageable, and yet it steps beyond the confines of individual biography.(Engeström, 1999a, p. 26)

What kind of “collective system” could help to open up Neuman’s analysis of a table that, as amediating cultural artifact, appears from nowhere and leads Beth into an episode of self reflection(if not explanation)? To make a provisional start on this question, I contrast transcript fragmentsfrom a student and teacher asked (in separate interviews) to “think aloud” while solving the fol-lowing algebra story problem (Hall, 1990):

Two trains leave the same station at the same time. They travel in opposite directions. One train travels60 km/hr and the other 100 km/hr. In how many hours will they be 880 km apart?

After reading this problem, Karen (an algebra student attending a community college) commentson failing to remember relevant formulas and then sets out to construct a table similar to the one thatBeth uses (her completed table appears on the right).

Ok, well let me set up my little dirt table … I call it dirt … R T(drawsa 2 × 3table, labels d, r, t for columns). So we’ve got train A and trainB (labels A and B for rows). Ok, the rate is … for train A is 60. Is thatright? Yeah, rate … so I’m trying to figure out the time. The distance… is 880 for both of them. Well, yeah, it’s A plus B equals 880.That’s right … well, yeah. (writes 880 in each cell under d)Thatseems wrongto put 880 in there, though (pp. 115–116, italics added).

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What “seems wrong” to Karen is, I think, similar to what led Beth to announce “No!” inNeuman’s transcript. Although the table is a cultural artifact mediating meaningful activity inboth examples, it can be filled with all sorts of quantities. Both Karen and Beth are struggling tocoordinate the structure of a recalled representational form (i.e., what Karen routinely calls a “dirttable”) with algebraic expressions that are also in play (e.g., Karen’s statement that “A plus Bequals 880”).

Karen’s snag leads her to generate incompatible algebraic expressions (60t = 100t and 60t +100t = 880), then she returns to her “dirt” table and focuses on the column for time, where she haswritten “t” in both cells for trains A and B (i.e., rows of her table). After briefly updating an earlierdrawing (not shown) to indicate that the trains would move apart by 160 km in one hour, Karen be-gins writing out the rows of a very different sort of table.

All right, so in one hour, we’ve got 160 kilometers (writes 1 = 160).Well, I could just keep doubling it (laughs). That seems like aweirdway to do it[italics added]. Or not doubling, but …[…]

So in five and half hours, it would take them five and half hours toreach 880 kilometers apart. Now why can’t I figure out how to putthat into the formula? Or the damn table?That’s weird[italics added](writes 5 1/2 = 880). (pp. 117–118)

Karen iterates through this table of intermediate values until exceeding the given 880 km. She thenbacktracks to find 80 km covered in one half hour and again reviews her solution approach as“weird” (i.e., italics).

That Karen finds her successful solution weird by comparison to the “damn table” she startedwith makes more sense when compared with the “think aloud” protocol of a community collegealgebra teacher, Richard. After reading the same motion problem, he announces:

I love this kind of problem because there are two ways to look at it, and the algebraic way is not the easi-est way, in my opinion. [I: Ok.] But I’m going to do it the algebraic way, cause that’s how I’d be doing itin front of a class. (p. 123)

First he draws a labeled “picture” showing directed motion segments, given values, and the writtenphrase “time is ??” as a representation of the unknown. Reflecting briefly on this latter bit of in-scription (shown in the following), he then produces a table that matches Karen’s exactly in struc-ture and labels (not shown).

Otherwise you don’t know what you’re after. Andthe standard technique that I think most of us teacharound here[italics added], is for distance rate time problems we … take a box, sort of a window, anddivide it into three columns with as many rows as there are trips that are taking place (draws role tablebelow diagram). (p. 125)

Richard’s approach makes explicit an interesting contradiction. On the one hand, he feels thisis “not the easiest way” to solve these kinds of problems. But on the other hand, he marks this asthe “standard technique” that he and his fellow teachers present to students. Later in the interview,Richard returns to what he considers a more sensible way to solve these problems, giving an elab-

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orate narrative from the perspective of two travelers, observing each other from the back of themoving trains after synchronizing their watches (i.e., a florid example of a situation model). Re-flecting this same contradiction (i.e., what “makes sense” is at odds with what “counts” in manyversions of the mathematics curriculum), Karen struggles with a “damn table” (a standard form orartifact) and marks her successful, hour-by-hour simulation of train travel (a nonstandard form) asa “weird way to do it.”

This contrast between “standard” and “weird” ways to solve algebra problems, marked in a re-flective way by participants to either side of an asymmetric social relation (student/teacher), pro-vides one line of analysis into a broader system of activity in which algebraic problem solving islearned and taught. Whether (or how) the traces of self reflection that I can find in Neuman’s tran-script take their motive force within this broader system is open for analysis. From my perspec-tive, this would be a particularly powerful use of activity theory, and it could help us to understandbetter how cultural artifacts not only structure but are structured in activity (i.e., what Engeström,1999a, described as an “expansive cycle” of internalization and externalization).

In closing, I point out that this kind of research is already underway and growing (e.g., Cobb,Yackel, & McClain, 2000; Boaler, 1998; Hall, 1999a). A synthetic review of these developmentsis beyond the scope of my commentary, but I feel this body of work provides a welcome alterna-tive in research on mathematical activity and learning. Neuman’s article is a move in this directionas well, and I hope that the commentary series in this issue brings these problems to the attentionof the broaderMind, Culture, and Activitycommunity.

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Cobb, P., Yackel, E., & McClain, K. (2000).Symbolizing and communicating in mathematics classrooms. Mahwah, NJ:Lawrence Erlbaum Associates, Inc.

Cole, M., Hood, L., & McDermott, R. (1997). Concepts of ecological validity: Their differing implications for comparativecognitive research. In M. Cole, Y. Engeström, & O. Vasquez (Eds.), Mind, culture, and activity: Seminal articles fromthe Laboratory of Comparative Human Cognition(pp. 49–56). Cambridge, England: Cambridge University Press.(Original work published 1978)

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diSessa, A. A., Hammer, D., Sherin, B., & Kolpakowski, T. (1991) Inventing graphing: meta-representational expertise inchildren.Journal of Mathematical Behavior, 10,117–160.

Engeström, Y. (1999a). Activity theory and individual and social transformation. In Y. Engeström, R. Miettinen, & U.Punamaki (Eds.),Perspectives on activity theory.Cambridge, England: Cambridge University Press.

Engeström, Y. (1999b). Communication, discourse and activity. Communication Review, 3(1–2), 165–185.Hall, R. (1990).Making mathematics on article: constructing representations of stories about related linear functions.

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