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Mind, Culture, and Activity, 18: 237–256, 2011Copyright © Regents of the University of California

on behalf of the Laboratory of Comparative Human CognitionISSN 1074-9039 print / 1532-7884 onlineDOI: 10.1080/10749031003790128

Getting Unstuck: Learning and Histories of Engagementin Classrooms

Indigo Esmonde, Miwa Takeuchi, and Nenad RadakovicUniversity of Toronto

This article focuses on the role of history in shaping learning interactions in a high school mathematicsclass, in which we argue that students participate in two key activity systems: Learning mathematicsand doing school. Within the context of these two activity systems, we highlight the nature of socio-genesis, the patterns of shift in communities as people build on one another’s accomplishments, jointlysolve problems, and disseminate new and old ways of solving problems. Drawing on a yearlong studyof group work in a high school mathematics classroom in California, we discuss how mathematicalinscriptions in the classroom and the group’s mathematical interactions were influenced by and alsoinfluenced the group’s shared history. With this article we contribute to cultural-historical activitytheory by providing insights into the study of history in classroom interactions.

According to cultural-historical activity theory (CHAT), development should always be under-stood through an examination of its multilayered history. This is one of the key differencesbetween cultural-historical views of learning and other current theories of cognitive develop-ment. In school contexts, history—writ small—is absolutely built in to the structure of teachingand learning. What was learned in the past (last week, last year) is assumed to influence what canbe learned today, and the participation structures and social relationships that are built early on inthe year are assumed to shape those that occur later in the year. It is therefore surprising that thehistorical dimension of CHAT has so rarely played a central role in analyses of student interactionand mathematics learning in schools.

In this article, we investigate the seemingly commonsense claim that history does make a dif-ference for teaching and learning, and elaborate on how it does so. We focus on a collaborativegroup of four students and their teacher within a larger classroom and develop a detailed under-standing of how the group’s history in that school and in that classroom makes a difference fortheir joint engagement. This study should be seen against a backdrop of largely ahistorical stud-ies of classroom collaboration (for a review, see O’Donnell, 2006). The predominant researchquestions in this area have been, What kinds of collaborative tasks or activities can best supportlearning? Or, alternatively, what kinds of group interactions can best support learning? Research

Correspondence should be sent to Indigo Esmonde, Ontario Institute for Studies in Education, University of Toronto,252 Bloor Street West, Toronto, Ontario M5S 1V6, Canada. E-mail: [email protected]

238 ESMONDE, TAKEUCHI, AND RADAKOVIC

addressing either question tends to be short term and runs the danger of implying that any collab-orative interaction on any day of the year should have the same characteristics. If we view historyon a large scale, it is indeed accurate to assume that in most cases there would be little historicalchange over the course of a single academic year in a student’s life. However, if we examine thesmall-scale history of the cooperative group, it becomes clear that much is lost when history is notconsidered. In this article, we intend to demonstrate the importance of studying the small-scalehistory of interaction for an analysis of classroom mathematics learning. To do so, we first brieflyreview theoretical and empirical discussions that relate to history and learning, define some keyterms, and then provide a brief overview of the goals of the article.

A HISTORY OF “HISTORY” IN CHAT

Cultural-historical research draws from the foundational work of Vygotsky (1978) and in par-ticular on the importance of multiple levels of historical analysis for understanding individualcognition. These levels of historical analysis include analysis of the ways in which phylogeny(history as it concerns the evolution of the human species) and cultural history are both impli-cated in human cognitive development. That is, evolution (through physical changes in the body)makes particular forms of activity and forms of cognition possible, just as cultural history doesthrough the creation of contexts that support particular activities and forms of cognition. Further,each individual human has a personal history of development (commonly called ontogeny). Eachof these levels is interconnected with, and influences, the others.

Many have built on Vygotsky’s work, taking the concept of history in slightly different direc-tions. For example, proponents of CHAT have argued that the appropriate unit of analysis forstudying the influence of history is the activity system, because studying history in an individ-ual life amounts to simple biography, and studying history writ large is too complex and vast toidentify the relation to human activity (Engeström, 1999). Instead, focusing on an activity systemallows for a manageable and meaningful analysis of historical change. The history of an activitysystem provides a context that shapes the kinds of practices that individuals engage in; similarly,the accumulated actions of individuals shape and change the activity system itself, thus changinghistory.

The question for analysis becomes, What constitutes a distinctive activity system? In the orig-inal work in this area, activity systems were defined by a particular motive (Leont’ev, 1978). Allhuman activity was considered fundamentally object oriented, although the object may some-times be implicit or hidden. In practice, distinguishing between different activity systems can becomplicated. We draw on a recent study of workplace activity systems (Worthen, 2008), in whichat least two activity systems were shown to coexist in the workplace. These activity systems werenot entirely distinct from one another. They included many of the same people, artifacts, andso on, but with different motives. The first activity system’s motive was the productive work ofthe workplace, with the second’s motive being to keep one’s job and to maintain or improve theconditions of labor.

One could argue that these are, in fact, the same activity system and that doing one’s jobwell and navigating the conditions of labor are intimately connected within a larger motive.From our perspective, such debates must be grounded in efforts to answer a particular ques-tion about the activity system or systems in question. In our case, we draw a parallel to Worthen’s

HISTORIES OF ENGAGEMENT 239

workplace study and focus on two activity systems in the classroom: The activity system of learn-ing mathematics, and the activity system of doing school. Although we believe, and observed inour data, that these two activity systems overlap substantially, we found it useful to separate thesetwo motives in our analysis of small-scale historical change in the classroom. We elaborate onthis point throughout our methods and results.

In cultural-historical analyses, history is usually considered in one of two ways. The predomi-nant method is to consider artifacts as representations of history in the present, because tools andsigns are created by human beings for particular purposes at particular points in history, and thosehistories cling to the artifacts themselves. Artifacts have both a material and an ideal aspect. Thematerial aspect is shaped by the ideal, and vice versa. Because artifacts are considered as histori-cally constructed and used, an analysis of the ways in which artifacts mediate human activity canbe considered a historical analysis.

Another way to include history in the study of activity systems has focused on what wecall “history writ large.” For example, a study of mathematical language in Oksapmin, PapuaNew Guinea, discussed the transformation of the word form fu from a nonquantitative meaning(complete) to a quantitative meaning (to double) in the context of historical change, primar-ily in the areas of economics and education (Saxe & Esmonde, 2005). In this and similarwork, large-scale trends across geographical regions and across decades or centuries are exam-ined to consider how these histories have shaped particular activity systems, cultural forms, orartifacts.

Both of these approaches certainly capture important aspects of a cultural-historical approach,but we feel that both approaches are inadequate if we want to understand classroom life. Althoughboth material and ideal aspects of artifacts do develop over long stretches of history, these lengthytrajectories are less salient in classroom life than the students’ and teachers’ shared experiencewith these objects. We wonder how the use of a textbook, a calculator, or a mathematical formulaor term develops materially and ideally within the small-scale history of the classroom and howthis history makes a difference for classroom learning.

For our purposes, a tri-level model of development, including microgenesis, ontogenesis, and(most relevant for this discussion) sociogenesis, is perhaps the most useful to understand thesmall-scale history of activity systems (Saxe, 1999). Microgenesis describes the ways in whichindividuals construct and use cultural forms in moment-by-moment interaction. As an example,a student faced with the task of graphing a line on a set of axes, given an equation, would usecultural forms such as a calculator, the number system, and prelearned algorithms, for specificfunctions such as finding the intercepts of the line and graphing it on the axes.

Ontogenesis describes patterns of change over time in the ways people solve recurrent problemtypes. As an example, if the student just discussed were to repeatedly graph lines correspondingto algebraic equations, the student might shift solution strategies over time, to make use of newinsights gained.

Finally, sociogenesis describes patterns of shift in communities as people build on oneanother’s accomplishments, jointly solve problems, and disseminate new and old ways of solvingproblems. Two students working together to graph a line may come up with a solution processdifferent from the process of either one alone; sociogenesis refers to the developmental shifts inthe way that people work together, and patterns of shift at the community or group level that areinfluenced by and may influence in turn individual development.


Sociogenesis has been used conceptually to describe history writ large—for example, thedevelopment of new meanings for the quantitative term fu in Oksapmin, Papua New Guinea, over40 years of use. It has also been used to describe history writ small, in a classroom communitywhere ideas are shared and developed between students and teacher. The key to understand-ing sociogenesis is that it focuses fundamentally on purposeful human activity, and the waysin which individual people shape collective history, whereas collective history also shapes indi-vidual activity. In the case of Oksapmin communities in which a new economic system wasintroduced, people unintentionally started to use the term fu in new ways to communicate logico-mathematical relations. As demonstrated by this example, the process of sociogenesis is “thereproduction and alteration of prior forms of representation as individuals engage as interlocutorsin collective practices, making efforts to communicate meanings” (Saxe, 2008, p. 88).

In this article, we take a slightly different approach to a study of sociogenesis. The discussionof the development and propagation of new meanings for fu was constructed to explain a set ofinterview data and, because the period in question covered decades, it was impossible to locatereal-time interactional data that might support this argument. We focus here on the details ofsocial interaction with a cooperative group to understand processes of sociogenesis specificallyas they relate to small-scale historical change.

Because we have access to the messy details of interaction in classrooms, we can also argueagainst naïve assumptions about history, which are all too easy to make in classroom life: Wepoint out that history is not a straightforward trajectory of continuous improvement. Althoughthere is perhaps nothing surprising about this statement, we believe that too often, educators andeducation researchers make the opposite assumption and believe learning and history will beclean, straightforward, and conflict-free. Rather, within the complex interlocking activity systemsthat make up classroom life, a variety of motives interact, prompting a variety of participationstructures, interactional styles, and even mathematical forms.

Given this theoretical overview, we can now state the objectives of the article more clearly.Within the complexity of classroom life, in which multiple activity systems intersect, we aim todemonstrate the importance of studying small-scale history if we wish to understand the natureand process of learning. In our case, we focus specifically on mathematics learning for one coop-erative group in a high school classroom, but we feel that the tools we use will be useful forthe study of other similarly complex learning environments, given that any classroom learning ishistorical.

A MULTILEVELED HISTORICAL CONTEXT

To understand the historical context of the project, we provide a sketch of several interconnectinglevels of history. Broadly speaking, the study took place in the early 21st century in California,in the United States. Throughout their existence in the United States, public schools have beencharacterized by a relatively high student–teacher ratio (in the early 21st century, usually about20 to 30 students per teacher at the secondary level). Pedagogically speaking, U.S. mathematicsteaching is generally fairly teacher directed, with the teacher taking on the role of impartinginformation and evaluating student learning. More recently, there has been considerable politicalconflict over whether schooling should be more teacher directed or more student centered, and


reform efforts have encouraged educators to create more space for students to construct and solveproblems for themselves. The students in this particular study had probably experienced a numberof different pedagogical types—from very teacher directed to very student centered—both withintheir mathematics education and across their other educational experiences.

We now narrow our historical focus to the school itself. The students attended the small schoolMedia Academy within a large comprehensive high school, Bay Area High School (all namesare pseudonyms). The small school had begun operation less than a decade earlier, in response toa growing movement across the U.S. advocating small schools, and in an attempt to construct amore equitable school environment for Bay Area High School students. Media Academy had twofoci: Social justice education, and media production and literacy. The small group of teacherswho ran the school collaborated frequently to construct deep learning experiences across theircourses and to make sure struggling students got the support they needed. This school was dif-ferent than many large urban high schools because of this consistent focus on social justice inboth the curriculum (e.g., studying racism and sexism) and the process of learning (e.g., moreinclusive and culturally responsive pedagogies). The school emphasized community building andorganized students into cohorts.

We narrow still further to examine the history of the mathematics classroom. In this study,students were all 11th and 12th graders, and most had been taught by the same math teacher for the2004–5, the 2005–6, and the 2006–7 academic years. Throughout these three years, students wereexpected to collaborate, and the teacher had provided participation structures to support students’cooperative work. This practice was markedly different from historical precedents of schoolingin the United States, and was indeed unfamiliar to many of the students in her classroom.

During the year, there were five thematic mathematics units, each running for six to eightweeks. During each unit, the teacher usually switched group compositions two or three times.Each student was placed into a “home group” that he or she worked with for several weeksduring each unit. In between these sessions with their home groups, the teacher changed groupcompositions and students were grouped with other peers. In this study, we focused on onehome group as our focal group. Throughout the article, we present the data primarily througha series of descriptive episodes, and we provide further ethnographic detail as needed with eachepisode.

Brief Biographies of the Focal Group and Teacher

The four students in the focal group were chosen because they had participated in interviewsin an earlier ethnographic study at the school (two years earlier) and because their mathematicsteacher, Ms. Delack, agreed that they would form a reasonably functional group. The focal groupwas composed of the following people:

Tony, a senior: He was Latino and an English language learner who often visited family inMexico and Southern California. He was relatively proficient in academic and informal English,and Spanish was his first language. Tony was fairly quiet in class and rarely spoke up in whole-class discussions.

May, a senior: She had a strong biracial identity, with one White parent and one Latina parent.Her first language was English, and she spoke some Spanish. She participated as a facilitator in alocal program that provided multiculturalism and diversity workshops to young children.


Tariq, a junior: He was Latino who spoke English as his first language and little Spanish. Hewas easygoing and well-liked in the class, seeming to know just about everyone. He volunteeredat a local youth radio station.

Namaya, a junior: She was African American and spoke English as her first language. Shewas a high achiever in many of her classes but began to struggle that year to keep up with a highworkload while working at a part-time job after school.

The teacher, Cassie Delack, was in her third year of teaching at this school. Ms. Delack wasa young White woman who divided her time between teaching and political activism in thecommunity. She had taught these four students for the two previous years as well. She was com-mitted to equity in the classroom and believed that group work was important to her students’success.

I (Indigo) was a familiar face in the classroom. I was well known to students, having conductedresearch in their classroom two years prior, and volunteered in their classroom and the after-school program the previous year. When behind the video camera, I usually did not interact withthe students, but I sometimes stepped in to conduct Ms. Delack’s classes when she needed asubstitute teacher. My relationships with the students, developed over three years, were a part ofthe history of the classroom and its activity systems: Students sometimes called me over to helpthem with their work or talked to me about their peers or their teachers. Students were used to mypresence in the classroom and to the presence of the video camera.

Data and Methods

This study draws primarily from ethnographic methods. From October 2006 to June 2007, datawere collected in Ms. Delack’s classroom. Major data sources were video recordings of groupwork, with supporting data including student work and photos of class materials. The focal groupwas videotaped several times a week, whenever they returned to their home group. A total of 16days of video were collected for the focal group: Three in October, four in November/December,four in January/February, four in April, and one in June.

From activity theory, conflicts in interaction can serve as analytically significant moments tounderstand activity systems (Engeström, 1993). In particular, these moments of mathematicalconfusion or disagreement are significant in both activity systems of our focus (doing school andlearning mathematics) and are therefore likely to provide insights into processes of sociogenesis.We collected a set of clips in which the group was trying to “get unstuck” (i.e., resolve somedisagreement or difficulty they faced with the mathematical work; we also call these clips “diffi-culty episodes”). Over the 16 days we coded, there were 129 such episodes. Each episode beganwith one or more students expressing confusion or uncertainty and ended when the subject wasdropped or resolved in the group. This collection of episodes allowed us to see not only howconfusion or uncertainty were expressed but also how the group worked to move through theconfusion, to come to some mathematical understanding.

Once we had identified all difficulty episodes for our analysis, we viewed them repeatedly.We created detailed narrative descriptions of each episode and wrote analytic memos to describeways in which the group’s interactions referenced or relied on their shared past experiences.We searched explicitly for the influence of history and the process of sociogenesis in theseinteractions. In this article, we present some episodes as examples of the influence of history


on the development of the two classroom activity systems. These episodes were not chosenbecause they are typical. Rather, we selected rich episodes that highlight several of the the-oretical and methodological points we make about sociogenesis in doing school and learningmathematics.

TWO ACTIVITY SYSTEMS: DOING SCHOOL AND LEARNING MATHEMATICS

Before we describe the ways in which the group’s history shaped their interactions, we discussin more detail the two activity systems that we observed in the classroom: Doing school andlearning mathematics. Consider the following episode, which highlights the complexity of therelationship between these two motives. In this episode, the group was working on a “warm-up,”a typical activity included at the beginning of almost every day’s agenda. Warm-ups were usuallya set of mathematical problems posted on the overhead (so, created by the teacher rather thanfound in a textbook), and students were expected to begin work on these problems as soon asthe starting bell rang. While the students worked, the teacher circulated around the room, takingattendance, checking homework, and answering any questions students might have.

Episode 1. “I don’t feel like ...”On November 27, 2006, the group was engaged in solving a few problems related to the area

of a triangle. They planned to use trigonometry to find the height of the triangle, but to do so, theyneeded to decide which trigonometric ratio was appropriate (sine, cosine, tangent). They labeled theirtriangles and Namaya wondered aloud which trigonometric ratio to use. Just then, the teacher walkedpast and Namaya said loudly, “I don’t get it! How do I know which one?” Ms. Delack approached andasked Namaya what could be done to find the area, and what information she would need. Namayaacknowledged that she needed the height. The teacher then told her to “pull out the triangle” tofind which trigonometric ratio was needed, and walked away to assist other groups. As Ms. Delackdeparted, Namaya loudly told her group that she had already done that (looked at the triangle), andcomplained “I don’t feel like . . . damn!,” as she returned to her work.

In this episode (see Figure 1), Namaya took advantage of the teacher’s proximity to ask for helpon a problem she was solving. The students had access to a representation of a right trianglewith the three primary trigonometric ratios labeled. This was a useful tool to help them withthis problem. Ms. Delack suggested that Namaya use this triangle, but it turned out that Namaya“just didn’t feel like” using the picture of the triangle to find the information she needed. In thisepisode, there is evidence that Namaya and her group could have solved the problem on theirown, but Namaya’s actions imply that she perhaps felt she could solve the problem more quickly,or with less effort, by enlisting the teacher’s help.

The two motives of doing school and learning mathematics are apparent in the episode. As theyworked to solve the nonroutine problem provided in the warm-up (doing school), Namaya andher peers had to work through some questions they had regarding trigonometric ratios (learningmathematics). In what was perhaps a shortcut, Namaya asked for the teacher’s help, even thoughshe says she already knew which strategy to employ. Especially in a context where the group hadlimited time to accomplish mathematical tasks, and in which completing the task correctly andquickly would influence both their grades in the class and their ability to understand the material,


FIGURE 1 Ms. Delack tells Namaya to “pull out the triangle” (colorfigure available online).

it is unsurprising that the group would call on the teacher to help them complete the task correctlyand efficiently. We claim that doing school was the motive in the provided interaction betweenNamaya and the teacher because of the focus on getting the task done quickly, regardless of one’slevel of understanding.

The excerpt illustrates our methods for identifying the activity systems that were at play inany given moment. When students were engaged in satisfying explicit or implied teacher expec-tations (whether these were mathematical or not), we considered their actions to be part of thedoing school activity system. When students were engaged in discussing or debating mathemat-ical content, we considered their actions to be part of the learning mathematics activity system.It was possible for students to be engaged in both activities simultaneously but also possible forthem to engage in only doing school (e.g., copying another student’s work without looking at it)or only learning mathematics (e.g., discussing some mathematical topic that did not form part ofthe class material or that was not appropriate at that point in time).

The very nature of classroom life demanded that students attend to these two different motivesin their work. They were not free to pursue mathematical understanding in any way they chose;although there was substantial overlap between doing school and learning mathematics, the twoactivity systems were sometimes in conflict. Episode 1 reminded us that in our analysis of thegroup’s collaborative activity, we would have to attend to both motives and recognize that thesegoals might not always match the teacher’s. We were also reminded that the group’s interactionswere goal directed, focused toward at least two motives, and highly dependent on the contextof the particular task they had been set. Therefore, we do not tell a story about the history ofgroup work as a continuous improvement in mathematical skills and in self-reliance. The story isconsiderably more complex.


HISTORY AND MATHEMATICAL INSCRIPTIONS

As we stated in our review of historically focused research in CHAT, artifacts have been one ofthe primary ways in which history has been taken into account. They are the logical place to startin our own analysis of classroom interaction. We focus in particular on mathematical inscriptions.In traditional mathematics classrooms, most of the definitions and formulas are already writtenin the textbook, which is positioned as an ultimate source of information and knowledge. In thisreform-oriented classroom, the textbook pages did not contain definitions, theorems, or workedexamples. Instead, the textbook contained only sets of problems to be solved. Based on theseproblems, the teacher orchestrated whole-class discussions in which any pertinent definitions,theorems, or formulas were developed and recorded on posters that were then displayed on thewall for the remainder of the unit. One such poster is displayed in Figure 2. The poster listedthe formulas developed by the class and was hung on the wall throughout the “High Dive” unit.Another poster was referenced in our discussion of Episode 1, in which Namaya consulted aposter that demonstrated a triangle with the primary trigonometric ratios.

Other inscriptions that were generally available to students included any materials written intheir notebooks, tests, or other assignments. Much of the talk in mathematics classrooms is cen-tered on inscriptions. These concrete inscriptions encoded past histories: Problems they solved,techniques they tried, formulas they developed, and so forth. Thus, these inscriptions encoded notonly the mathematical results but also the experiences the students had in solving the problems

FIGURE 2 Poster with formulas developed in class (color figure availableonline).


and participating in whole-class discussions to generate the posters. In other words, these class-room artifacts had both a material and an ideal aspect, and as we show, students routinely orientedto both aspects. In interaction, students sometimes referred to the overt content of these inscrip-tions (e.g., the formulas from the poster in Figure 2), and sometimes referred to their sharedexperiences when constructing or using these inscriptions. In one case, Tariq encountered a prob-lem that was similar to one he had recently tried in a quiz. He looked through his notes for thequiz, even while mentioning that he had done it incorrectly. The way that he consulted the quizto assist his current work was informed by his past experience and his knowledge that his writtenwork was incorrect.

In many accounts of the ideal nature of artifacts, authors describe the ways in which inscrip-tions are built up over historical time and are constructed within sociohistorical contexts forparticular purposes (e.g., Roth & McGinn, 1998). Such accounts highlight the ways in whichthe social practices of inscription users imbue these inscriptions with meaning. Although it istrue that the artifacts can be linked to the particular historical moment of their creation or mod-ification, this historical progression is often largely invisible to the students and teacher in theclassroom. More relevant is the smaller scale history of their past experiences with these sym-bols and conventions in and outside of school, and their joint engagement with these symbolstogether. Inscriptions change as they are used in the classroom because their ideal aspects weretightly connected to use.

Consider the following episode, which illustrates how inscriptions were used in practice, whilestudents worked toward the motives of doing school and learning mathematics. Namaya wasabsent on the day and Tariq, Tony, and May were working together. The three students had onetextbook to share between them, as two group members had left their books at home or in theirlockers. The students are shown in Figure 3, hard at work on the problem. The textbook problemwas typical in that it was quite lengthy, involved quite a bit of reading and writing, and led

FIGURE 3 May, Tony, and Tariq work to discover how to multiplymatrices (color figure available online).


students to “discover” a mathematical principle on their own. In this case, that principle wasthe multiplication of matrices.

Episode 2. “I remember all this . . . ”On December 5, 2006, the group’s task was to organize some information about a pilot delivering

two kinds of products into two matrices and then create a third matrix that would provide the resultof multiplying the matrices together. (For an explanation of the mathematics of this example, see theappendix.) Tariq read the problem aloud, commenting, “I remember all this shit,” as he read, sincethe same problem context (a pilot delivering cases of materials for several different clients) had beenused the previous year.

As Tariq and Tony started to describe what should go into the “first row” of the matrix, theyrealized they couldn’t remember which were rows, and which were columns. Tony noted that hehad written this information on an earlier page of the textbook, so he flipped back a few pages tohomework 9.

He pointed to one of the examples, and said, “These are the columns, and these are the rows.” Thethree students then created their matrices, with both May and Tariq verbally confirming the numbersthey were entering into the various positions, and all three students huddled around the single copy ofthe text they were sharing.

Once they had created the first two matrices, the group paused. They could compute the totalweight and volume that the pilot would have to deliver, but they were unsure how to do this as matrixmultiplication. Tariq wondered if they should add the two matrices together. “Do you multiply?” Tonyasked.

The teacher came over and inspected their notebooks, reminding them to label their work (thiswould be important later, during the whole-class discussion and explanation of why multiplyingmatrices makes sense). Ms. Delack urged them to figure out the result “logically,” and explain itto their peers.

In the episode, students coordinated a variety of mathematical inscriptions (found in their text-books, notebooks, and calculators) in their work. In large-scale history, students were accessingthe body of mathematics knowledge about matrices that had developed over time and for partic-ular purposes. Because this type of inscription was initially unfamiliar to students, they struggledto learn the meanings and the use of the artifact. The process of acquiring a new historical artifactwas supported by the small-scale histories as students gradually imbued these strange objects withmeaning. When they first read the problem, Tariq remarked that he remembered doing similarproblems about the pilot the previous year. In developing new meanings for these mathematicalobjects, they drew on their familiarity with other mathematical artifacts and processes, as theywondered whether they should add or multiply the terms in the matrices. In Episode 2, the use ofsmall-scale histories is seen in the way Tariq and Tony flip back and forth in their textbook—likegoing back and forth in time—in search of how they had used the matrices in the recent past.

Ideal aspects of these mathematical inscriptions were collectively developed. In the episode,students created the first two matrices together, and then with the help of that inscription, theystarted to develop further meanings for matrices as they discussed how to create a third matrixby multiplying the first two. Both motives were visible in their work. In Episode 2, the teacherwas trying to navigate students to a deeper understanding of matrices by encouraging them tofocus on a logical explanation for their intuitive understanding. However, from the students’ side,completing the task assigned to them was a primary goal. Students started to create matriceswithout having a clear vision of why they would need the inscription and how they might use it


(doing school). After coming up with a mathematical inscription, students started to unpack themeaning or the use of that inscription (learning mathematics).

As the students created these inscriptions in their notebooks, the teacher monitored what theywere writing and responded to their questions based on what they had written. Notebooks wereoften treated as reflections of students’ mathematical thinking, but in the context of the two class-room activity systems, the interpretation of notebook writing was not entirely straightforward.Because of the two motives for classroom work, there were multiple distinct reasons for writingsomething down in a notebook. One predominant goal of doing school was to appear productiveand to complete assignments. Completing these assignments correctly (and learning mathematics)was sometimes a secondary goal, given that students’ notebooks were not evaluated for correct-ness. Thus, students sometimes wrote computations down in their notebooks so that the teacherwould see that they had “started the work,” or they wrote down nonsense answers to homeworkassignments because they knew the teacher was marking for completeness, not for correctness.Although educators and researchers often assume that students’ written work reflects only theirmathematical thinking, we emphasize that notebooks encoded partial histories of engagement intwo activity systems. In other words, this written work cannot be interpreted as a straightforwardreflection of what students “know” or “don’t know.”

The importance of classroom inscriptions for learning mathematics in this classroom contrastswith ethnographic descriptions of the creation and use of inscriptions in a biology laboratory(Latour & Woolgar, 1986). This ethnographic study demonstrated that inscriptions were generated(relatively) sequentially, one replacing the other until the final “product” (e.g., a journal article)was generated. One important feature of the inscription-generating process was that once the endproduct (the final inscription) was available, all the intermediary steps that made its productionpossible were hidden from view. The scientists typically referred to the earlier inscriptions onlywhen the information presented in the final inscription was contested. In that case, the scientistscould trace backward through the sequence of transformations to resolve the problem.

Like the scientists, the students in our study were also involved in the transformation of inscrip-tions. In this case, they transformed the inscription of a lengthy problem description into twomatrices (first having to decode what was meant by “row,” “column,” and “row matrix”). Theythen had to transform a set of intuitive computations they had done into a written procedure for“matrix multiplication.” As the students were learning mathematics, their work resembled thework of the scientists in Latour and Woolgar’s (1986) study. Students could resolve disputes byback-tracing the chain of inscriptions, as when Tony remembered that he had written informationabout matrix rows and columns on an earlier page of the textbook. On the other hand, because stu-dents were also involved in “doing school,” their work with inscriptions differed from that of thescientists. Because the students were graded on completion of their work, all inscriptions neededto be equally visible. In this episode, the two previous matrices were as important as the newmatrix because they were graded on the assignment and they were required to show their work.

HISTORY AND MATHEMATICAL TALK

Inscriptions and physical resources around the classroom were perhaps the most visible artifactsthat were used to support learning mathematics and doing school, but mathematical talk andgestures can also be considered cultural-historical artifacts—though these artifacts are weighted


perhaps more heavily toward their ideal aspects and less towards the physical. In this section, wediscuss some ways that the group’s shared history was evident in their talk and shaped the courseof their interactions.

Large-scale history and the group’s past experience in a particular sociocultural milieu wasmade evident in the shared language and gestures that the group developed to facilitate the dis-cussion of mathematical objects. For example, when the group worked on a project about settheory, they learned about the concept “intersection of sets” that frequently uses the symbol ∩ torepresent an intersection. Because they were learning about this topic on their own through read-ing a variety of texts, they had never heard any oral name for this symbol. The group began to usethe word hat to denote this symbol. The term was first introduced by one group member and waspicked up by other group members until the teacher intervened to tell them an accepted mathe-matical term for the symbol, intersection. This incident resembles Saxe’s (2008) description of theprocess of sociogenesis: “In their efforts to get across intended meanings, people unintentionallydrew on prior representational forms, using them in new ways” (p. 88). Although representationalforms were used in innovative ways in the classroom, the uptake of these innovations were influ-enced by clear power differentials. The teacher, in this context, had more power than the studentsdid to determine proper names for mathematical objects.

Almost every mathematical conversation revealed some shared history that supported thegroup’s activity. Consider Episode 3, in which the students were trying to find a mathemati-cal function that corresponded to a table of values they had been given. The table of values isreproduced in Table 1.

As part of their solution to this problem, in Episode 3 the group worked with the numbers inthe table to calculate the stopping distance for a car that was moving at a speed of 0 miles per hour.

Episode 3. “Zero is point two”On April 16, 2007, while working on this problem, May and Tariq had agreed that the table repre-

sented a quadratic function of the form ax2+bx+c. Tony was quietly writing and using his calculator,apparently working on his own and not attending to his peers. Namaya was distracted and doing workfor another class. To find c, Tariq commented that they needed to find the stopping distance “at zero”(for a speed of 0 miles per hour). In order to find this stopping distance, they did a computation inwhich they repeatedly subtracted numbers from the right-hand column, eventually coming up with0.2. May turned to the group and said, “So zero is point two.” Tony asked, “So, point two gives uswhat?” and Tariq and May responded in unison, “c.”

TABLE 1Values for Better Braking Problem

Speed (in Miles per Hour)Stopping Distance (to the Nearest Tenth of a Foot)

Including Distance During Reaction Time

20 44.225 62.230 83.035 106.540 132.845 161.950 193.8


FIGURE 4 The group works on the Better Braking problem (color figureavailable online).

In Figure 4, the group is pictured while at work on this problem. Tariq and Tony are bothpunching buttons in their calculators, May is writing down Tariq’s results, and Namaya has somework for another class out on her desk while she flips through her binder (another example of theconflict between doing school and learning mathematics).

In this episode, when May found the answer “point two,” she explained its significance inthe shorthand phrase “zero is point two,” which meant something like, “When the speed is zeromiles per hour, the stopping distance is two tenths of a foot.” Because of their shared historyof engagement on this and similar problems, she was able to use this shorthand—which wouldprobably be nonsensical to anyone else but which seemed to make perfect sense to Tariq. Anothergroup member, Tony, seemed to face more difficulty in interpreting her comment. When he askedfor the significance of the “point two,” Tariq and May were both able to respond quickly to tellhim that “point two” was the “c” they were looking for in the expression ax2+bx+c.

This episode has interesting implications for researching classroom activity systems. As in thisepisode, when people have a history of joint engagement, they often leave many things unsaid inthe interest of quick, efficient communication. Thus, in the group’s efforts to find a function thatcorresponded to the points in the table, they did not articulate every aspect of their conceptualunderstanding of their task. This is reminiscent of a multisited ethnography that studied scientificengagement inside and outside of school with a group of middle school students (Bricker &Bell, 2007). In this study, participants drew on a shared history of engagement when makinginformal scientific arguments. To an outsider with no access to this shared history, the students’arguments would appear to be much weaker than they were understood to be by other participants.Although our study draws on data from only one site and one group of students, our findingsalso demonstrate that students often bolstered their mathematical argumentation by references toshared past experiences.

Detailed explanations were simply unnecessary for their task as May and Tariq computed the“zero” of the function. In other words, although mathematics educators tend to value detailed


explanations in problem solving, for the purpose of doing school such explanations were notalways necessary for students with shared histories. Yet when one group member expressed con-fusion at the telegraphic nature of their talk, May and Tariq were able to quickly expand on it. It ispossible that if Tony still had questions about their one-word response, “c,” they would have beenable to offer further background. Yet we speculate that many educators or researchers would bedismayed at what is missing from May and Tariq’s discussion and would expect detailed accountsof mathematical thinking at every turn.

Whereas Episode 3 highlighted the influence of shared history on the group’s talk aboutmathematical ideas, the next episode highlights that students were learning about more than math-ematical content. They also learned about one another as people and as mathematics students.Their conversations provided them with access to one another’s habitual ways of interacting (thenorms, or rules, of interaction, or perhaps the division of labor within the group).

For example, from our own observations, we found Tony was quieter than the other three focalstudents. He spoke less frequently and usually at a lower volume. His gaze was often directedat his own notebook, and he rarely engaged in the frequent social conversations between May,Namaya, and Tariq. Because he was usually working, Tony often had mathematical contributionsto make to group discussions, but he rarely volunteered them. We found several instances inour corpus of video in which another student made a general query for help, was met with nosatisfactory response, and then made a direct request to Tony for help. For example, in Episode4, the group was trying to find a function to match a given set of points.

Episode 4. “Are you getting this, Tony?”On April 17, 2007, the whole group was working on the assignment to identify a function to match

the points. The conversation had reached an impasse, and Tariq asked the group if they should get help(presumably, from the teacher). May agreed that they should, and Tariq raised his hand. At that point,May turned to Tony and asked, “Are you getting this, Tony?” Tony said no, but then began to explainwhat he had figured out so far. He and May continue talking about how to solve the problem but couldnot reach a conclusion. May decided to ask a question to Ms. Delack and raised her hand.

Figure 5 depicts the group at the point when May joined Tariq in raising her hand to call forthe teacher’s help.

The group’s shared history —both in the immediate past, when Tony had not participated intheir discussion, and in the extended past, when he had contributed ideas only if they were directlysolicited—may have allowed May to surmise that Tony would respond better to a direct requestthan the implicit request for help as Tariq raised his hand to call the teacher.

The group’s developing knowledge about one another as people helped them to navigate bothlearning mathematics and doing school. When she needed mathematical help, May knew thatasking Tony directly was more likely to get the help she needed than to wait for him to volunteerhis help. This was important for learning mathematical content as well as for doing school.

RECONSIDERING SMALL-SCALE HISTORY IN AND ACROSS ACTIVITY SYSTEMS

The historical dimension of CHAT has received less attention than it deserves. As we have dis-cussed in this article, history on a large scale and a small scale is a fundamental aspect of jointinteraction in classrooms. In particular, we focused on one group of students who worked together


FIGURE 5 May and Tariq raise their hands to call the teacher (colorfigure available online).

frequently over the course of a school year. In contrast with many classroom-based studies thatfocus on snapshots of moments in time, without considering the shared histories of participants,we wanted to demonstrate that small-scale history lies at the heart of classroom interaction, anddiscuss some ways in which researchers can take this history into account.

To theoretically and empirically examine multiple levels of history, we relied on two key con-cepts: Sociogenesis and scientific chains of inscriptions. Throughout, we continually noted twomajor motives that shaped the classroom activity systems in which these students and teacherengaged: Learning mathematics and doing school. Although some have argued that a single activ-ity system is an appropriate unit of analysis, in this particular case, the two activity systems oflearning mathematics and doing school were so intimately connected that our analysis had toconsider these interconnections. The development of each activity system depended in part onthe other.

Because inscriptions have been the focus of so much CHAT research that attends to the his-torical dimension, we began our analysis with inscriptions. In this classroom, the group pursuedmathematical learning and doing school through the construction of chains of inscriptions. As inmany mathematics curricula, students often began with informal problem-solving methods thatwere later crystallized into formulas and procedures. These procedures were inscribed in posters,textbooks, and notebooks, and they became a resource for future work. In a chain, one inscriptionbuilt on and distilled some aspect of the previous, and thus over time these chains of inscriptionstended to “black box” a set of procedures, assumptions, and concepts that were not always appar-ent when focusing on the last inscription in the chain. In this way, their work was perhaps similarto the scientists that Latour and Woolgar (1986) studied.

However, we found some distinct differences, which we attribute to the dual motives of school,and especially to the “doing school” motive. Perhaps for scientists, the dual motives of learningscience and doing the work of science are more closely aligned—at least, in the aspects of their


work that formed part of the study. A key aspect of our classroom context was the daily evaluationof students. The teacher often wanted access to more than the last inscription in a chain; shewanted students to demonstrate a complete understanding of procedures and concepts. Thus,while in science laboratories inscriptions often become invisible unless they are the endpoint ofthe chain, in this mathematics classroom (and, we suspect, in many others) multiple inscriptionswere often displayed side by side.

In our analysis of mathematical talk during group work, we found that students often “blackboxed” the concepts behind the procedures they discussed. They spoke telegraphically, withthe minimum necessary amount of detail. Although they could elaborate as necessary (e.g.,when the teacher or another student asked), they did not routinely describe their mathematicalthinking.

Finally, we presented a brief analysis of some of the people knowledge that groups developedthrough their work together. We speculated that the group members had learned through experi-ence that Tony was relatively quiet and would rarely volunteer information. They began to askTony directly when they wanted his help. We suspect that this type of knowledge was developednot only through their mathematical discussions but also through social talk, which we did notstudy for this article.

We speculate that this shared social history might also have supported their mathematicalcollaboration. In a recent study of collaboration on an engineering design task, pairs who engagedin social talk actually did better on this challenging task than pairs who did not (Mercier, 2008).One provocative implication of this study was that the group’s non-task-relevant talk might haveprovided an atmosphere in which students could take academic risks. Although we cannot provethis claim, we highlight the importance of considering both mathematical and social conversationsin analysis.

The analyses we have presented illustrate how sociogenesis occurs in classrooms like thisone, in which the dual motives of doing school and learning mathematics are both present. Ina compelling example from a nonschool setting, we observed that Oksapmin people repurposedpreexisting cultural forms when faced with novel problems. As more and more people interactedtogether around these problems, these new uses of old forms began to spread. We certainly sawthis process of sociogenesis in our data, when the students used the preexisting word “hat” todescribe a new mathematical symbol, “intersection.” We also saw this process when students(with the teacher’s support) drew on familiar mathematical processes (e.g., arithmetic operationswith numbers, and addition and subtraction of matrices) to create a new process—multiplicationof matrices.

But the dual motives of the classroom introduced a different set of processes for sociogen-esis as well. In the hat/intersection example, when the teacher heard students using the word“hat,” she corrected them, and thereafter they mostly used the accepted mathematical term “inter-section.” The teacher, due to her role in the classroom and school hierarchy, had the power toredirect sociogenesis by strongly encouraging students to pick up particular words, procedures,and mathematical concepts. However, students were still responsible for making sense of theideas that the teacher encouraged, and at every step, they adapted their prior understandings andfamiliar cultural forms. For example, although the teacher was largely responsible for the distill-ing of classroom discussions into posters that lined the walls, students had to turn these postersinto resources through joint engagement. As they did so, their own prior histories and experienceswith the posters were often at the forefront of discussions.


The process of sociogenesis in the context of these dual motives has serious implications forresearch on learning. Teachers and researchers often focus on the motive “learning mathematics”and ignore “doing school”. With this in mind, it may be tempting to treat student talk and studentnotebooks as revealing only their mathematical thinking, yet we uncovered several examples inwhich students asked questions that they already knew the answer to (e.g., Episode 1) or studentswrote down answers that they either didn’t understand or knew to be incorrect. In the activitysystem of doing school, having something (anything!) written in a notebook was one way forstudents to show the teacher that they were focused on work. Notebook writing was a strategy tomaintain standing with the teacher as a competent student.

Further, student talk in groups was often telegraphic, relying on a shared history to explicateideas that had been fully fleshed out in the past. Thus, even when students did appear to understandthe work and to be discussing mathematics, their talk could be difficult to interpret. This has boththeoretical and methodological implications. Theoretically speaking, this phenomenon influencesthe processes of sociogenesis. Inscriptions and verbalizations become the link in the chain onwhich the next link is built. When chains of inscriptions become black boxed, or as forms ofmathematical talk or inscriptions become tools mainly to show the teacher that one has beenworking (and not necessarily linked to one’s ongoing mathematical learning), opportunities tocontinue to develop community understanding are diminished. We saw many opportunities whenstudents began to open up these black boxes and to more fully explicate the processes they hadelided in earlier talk. At these moments, the motives of doing school and learning mathematicswere closely aligned. This may be one reason why, as other analysts have pointed out, momentsof difficulty, confusion, or disagreement can be so productive for cooperative learning settings.

Methodologically, this finding reminds us as analysts that if we want to understand “what stu-dents think” about mathematics, we need to triangulate data from multiple sources and acrosstime. If the motive of doing school trumps the motive of mathematical learning, and if the expec-tations for “doing school” allow students to forego detailed explanations, to write down incorrectanswers, and so on, then we have limited windows into student thinking. The historical underpin-ning of these interactions was made visible only because of our sustained focus on this one group.Without such a focus, we would probably have misinterpreted many of the group’s mathematicaldiscussions, disregarding them as superficial, unhelpful, or otherwise problematic.

Through our analysis, we confirmed that multiple layers of histories and multiple motivesafford or constrain student learning in the classroom. Success of student mathematical learning inthe classroom cannot be discussed without considerations of these histories and motives. In con-texts marked by conflicting activity systems, the influence of history is far from straightforwardand is deserving of closer analysis within CHAT.

ACKNOWLEDGMENTS

This work is partially based on a presentation at the Canadian Society for the Study of Education’sAnnual Conference in Ottawa, Ontario, May 2009. The research was supported by the NationalScience Foundation under Grant No. SBE-0354453 to the Learning in Informal and FormalEnvironments Science of Learning Center, as well as by the Connaught Fund at the University ofToronto. Any opinions, findings, and conclusions or recommendations expressed in this material


are those of the authors and do not necessarily reflect the position, policy, or endorsement of theNational Science Foundation or the Connaught Fund.

We thank Baolong Fu for her contributions to data analysis, and Rubén Gaztambide-Fernández, Joseph Flessa, Lance T. McCready, and Roland S. Coloma, as well as anonymousreviewers, for their thoughtful critique of earlier versions of this article. We also thank the studentsand teacher at Bay Area High School who generously donated their time to this project.

REFERENCES

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Latour, B., & Woolgar, S. (1986). Laboratory life: The construction of scientific facts. Princeton, NJ: Princeton UniversityPress.

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APPENDIX

The matrix problem that the group had to solve is as follows.They were told that a pilot, Linda Sue, had to deliver packages to two clients. The first client,

Charley’s Chicken Feed, had packages that weighed 40 pounds each and had a volume of 2 cubicfeet. The second client, Careful Calculators, had packages that weighted 50 pounds each and hada volume of 3 cubic feet. They organized this information into a matrix, A.

A =[

40 250 3

]


Next, students were told that Linda Sue had to transport 500 containers of chicken feed and 200cartons of calculators on Monday, and they had to put this information into a matrix, B. They thenhad to create a matrix that displayed the total weight and total volume for Monday’s packages,and call this matrix C.

B = [ 500 200 ]C = [ 30000 2500 ]

The students were told that matrix C had been obtained by multiplying matrix B times matrix A.They had to inspect their work, and explain in words how to multiply matrices. They were thenprovided a few other scenarios involving the calculators, and had to repeat the procedure.