Running head: Children’s Game Play

Transforming Cultural Practices: Illustrations from Children’s Game Play

Steven R. Guberman, Jrene Rahm, and Debra W. Menk

University of Colorado at Boulder

Steven R. Guberman E-mail: steven.guberman@colorado.edu

School of Education Telephone: 303-492-8391

Campus Box 249

Boulder, CO 80309-0249

ABSTRACT

Analyses of children’s participation in cultural practices typically focus on the ways in which dimensions of activities shape the nature of children’s participation and learning. In contrast, our concern in this paper is to understand how children, in their participation, transform cultural practices. We use Saxe’s (1991) Emergent Goals Framework to illustrate how the mathematical problems that emerge in children’s play of Monopoly are interwoven with children’s developing competencies and social interactions.

Transforming Cultural Practices: Illustrations from Children’s Game Play

As researchers increasingly incorporate sociocultural context into analyses of children’s learning and development, they are faced with the challenge of devising methods for describing the interplay between actors and the social and cultural settings they inhabit. In general, relations between children and their environments tend to be conceived of in one-way terms: the focus has been on how the social world influences the behavior of individuals. John-Steiner and Mahn (1996) point out that this unidirectional model distorts sociocultural theory and reduces it to a social transmission model. Engeström (1993:65-66) notes that, although it is tempting to conceive of contexts "as containers of behavior, untouched in themselves by human actions," contexts emerge and are shaped by individuals acting in specific settings. As people take part in cultural practices over time, a single task often evolves in particular ways and has different meanings for distinct groups of participants (Holland and Reeves 1994; Nicolopoulou and Cole 1993). According to Minick (1987:47), Vygotsky

In order to understand learning and development in context, we need ways of studying how activities are interpreted, transformed, and enacted by participants, and the implications of these processes for learning. In this paper, we use analyses of children engaged in a common cultural activity—playing a board game—to illustrate one promising approach to the study of children’s learning as they participate in and contribute to everyday activities. Our focus in this paper is on mathematical environments: the types of mathematical problems and solutions that children are engaged in as they participate in collaborative endeavors.

The acquisition of mathematical understanding is a rich domain for examining relations between child development and culture. Increasingly, mathematical problem solving is viewed as a cultural practice "in which people become more proficient as they learn and understand particular ways of representing numbers and quantity and operating on them" (Carraher 1989:320). From this perspective, cultural practices—including mathematical problem solving—both reflect the historical achievements of social groups and are contexts that promote children’s acquisition of culturally-valued knowledge and behavior (Goodnow et al. 1995). Yet, despite consistent evidence that children acquire mathematical understanding and skill through participation in a variety of cultural practices—including commercial activities (Posner 1982; Saxe 1991), parent-child interactions (Saxe et al. 1987), and game play (Ainley 1991)—there have been few analyses that examine how mathematical activities are enacted and shaped by the actors who participate in them. In order to understand how participation in cultural practices may facilitate learning, we need to understand the many forms that participation may take. Such investigations are especially warranted if recommendations to base classroom instruction on everyday cultural practices—including the use of games to teach mathematics (National Council of Teachers of Mathematics 1991)—are to be implemented effectively.

Miller and Goodnow (1995:6) note that "cultural practices are not neutral; they come packaged with values about what is natural, mature, morally right, or aesthetically pleasing." At the same time, though, practices are transformed as they become inhabited by actors. We use the term transformation to describe the process by which cultural practices and activities--with their social and cultural norms and expectations—are altered by actors in ways that yield new or emergent properties. In children’s game play, transformations may be minor, such as altering a rule about the order of turn-taking, or may have a major impact on the nature of the activity, as when players decide to participate cooperatively rather than competitively. The outcome of transformations may take many forms, including the mathematical work that gets accomplished in game play.

Activities are often transformed when a more knowledgeable participant provides assistance—such as scaffolding (Wood et al. 1976) or completing key components or subtasks (Greenfield and Lave 1982)—so that less knowledgeable partners are able to participate with greater success than would be possible alone. Transformations of this type, which are well-documented in recent writings (Guberman 1996; Rogoff 1990; Saxe et al. 1987), are central to accounts of learning in the zone of proximal development (Rogoff and Wertsch 1984; Vygotsky 1978). These analyses have focused primarily on activities in which an uneven distribution of expertise leads one person (e.g., an adult) to transform a task in order to assist someone with less competence (e.g., a child). In contrast, we know little about how activities are enacted and transformed by groups in which expertise is more evenly distributed, such as peer groups. As Piaget (1965) noted in his discussion of children playing marbles, peer groups provide greater opportunities than do adult-child interactions for children to negotiate and jointly construct the character of their activities.

In addition to transformations that come about as a result of assistance, participants often transform activities unintentionally. For instance, in a study of college students, Holland and Reeves (1994) found that computer programming teams from a single class differed in their motives for engaging in an assigned project and, as a result, varied in their accomplishments. Forman and Larreamendy-Joerns (1995) found comparable results among fourth and seventh grade children working collaboratively on a projection of shadows tasks. Similar to Engeström and Vygotsky, Forman and Larreamendy-Joerns (1995:550) concluded that "the context of a collaborative learning situation is not static but is constantly being recreated by the participants in the group."

A similar concern with understanding how children’s activities are transformed in practice is evident in recent analyses of children’s game play. According to Goodwin (1995:262), "a major problem of research on games has been that much research has concentrated on the forms of games...rather than the interaction through which a game is accomplished in situ" (emphasis in the original). She, and others who research games (e.g., Fine 1986; Goldstein 1971; Hughes 1991), emphasize that studying children’s gaming requires an analysis of how games are interpreted and transformed in play. The focus of such analyses should be on how participants "actively collaborate in constructing the game of the moment" (Goodwin 1995:262). Similar concerns have been raised in education: The task children solve may be very different from the task intended by the teacher (Meloth 1991).

The variety of ways that activities may be transformed has yielded disparate approaches to the study of learning in collaborative activity. As Hoyles and Forman (1995:481) point out, the emergent nature of context in peer interactions alerts us to "the need for a new paradigm of research," a systematic approach to understanding how activities are constituted and reconstituted through children’s participation—that is, an analysis of emergent environments—and how activity transformations are interwoven with children’s abilities and opportunities for learning. In this paper, we use Saxe’s (1991) Emergent Goals Framework to examine the constitution and transformation of mathematical environments in children’s play of a popular board game, Monopoly. Similar to other researchers who have studied children’s games (e.g., Avedon and Sutton-Smith 1971; Greenfield 1994; Piaget 1965; Trevarthan and Grant 1980), we view games as cultural practices that both reflect the values and achievements of culture and provide opportunities for children to practice and acquire culturally-valued attitudes, behaviors, and knowledge. We agree with Hughes’ (1991:287) statement that children’s games are more than a listing of their rules; rather, "they are richly textured and highly situated instances of social life." As such, we believe the approach illustrated here is applicable to a wide range of children’s activities.

Our analysis is guided by a model developed by Saxe (1991) for understanding children’s mathematics learning in cultural practices. Saxe’s Emergent Goals Framework is well suited for studying how a given or intended structure of an activity is transformed in practice. The guiding tenet of Saxe’s model is that cognitive achievements are the outcome of children’s attempts to make sense of and accomplish the mathematical problems they encounter in routine cultural practices. Four parameters are key to understanding how problems take form in practice: the activity structure of the tasks comprising the practice, the prior knowledge children bring to the practice, the artifacts and conventions used in the practice, and the social interactions participants have with each other while engaged in the practice. In this paper, we use these parameters to understand how one activity, playing Monopoly, is constituted and transformed in children’s participation. Our focus is on the different mathematical environments that emerge in the play of peer groups.

As an example of how the four parameters provide a framework for understanding the nature of the mathematics that emerges in play, consider a common aspect of many board games, one that entails quite simple mathematics: moving one’s token from one space to another. In the standard version of Monopoly, the activity structure specifies that players take turns rolling two dice and then moving their token a number of spaces equal to the sum of the dice, giving way to mathematical problems of numerical representation, enumeration, and addition. The dice, an artifact of the game, engages players in particular mathematical problems, such as adding values through six plus six and counting to twelve. In contrast, in Monopoly Junior, a version of the game designed for children from 5- to 8-years of age, players use only a single die, which simplifies the mathematics of the game: no addition is needed to determine how far to move one’s token, and players never need to count more than six spaces as they move their token around the game board. Similarly, children’s prior knowledge influences the mathematical problems that emerge in play. For instance, in the standard version, a child may roll a five and a six on the dice. The child then may sum the values of the dice and move her piece eleven spaces; in contrast, a child with less mathematical competence may move her piece first the value of one die (six spaces) and then the value of the other die (five more spaces), not bothering to sum the die values. Finally, another participant may interrupt the player’s count and tell her to move eleven spaces, perhaps leading the player to recognize mental addition as a new strategy--an example of how social interaction may lead children to new ways of engaging in mathematical problem solving.

In the analyses presented here, we provide evidence of transformations by comparing the types and complexity of mathematical problems and solutions that emerged in play across groups of children who were selected to highlight particular parameters of the Emergent Goals Framework. According to the Emergent Goals Framework, we expected that differences in the artifacts, conventions, and task structures of an activity would lead players to construct distinct mathematical problems and solutions. Therefore, we compared two groups of 8-year-old girls engaged in two versions of Monopoly that differ in their artifacts, conventions, and task structures: One group played Monopoly Junior and one group played the standard version. Also based on the Emergent Goals Framework, we expected that participants’ prior knowledge would influence the mathematics that emerged in play. Using age as an index of mathematical knowledge, we expected that children of different ages would transform the same activity in distinct ways; younger and less able players would be engaged in less frequent and less complex mathematical problem solving compared to older and more competent players. To examine this, we present analyses from two types of comparisons: In one analysis, we compare 8-, 11-, and 14-year-old girls playing the standard version, and in another analysis we compare high- and low-achieving 14-year-old boys and girls playing an abbreviated version of the standard game. For each comparison, we also examine how social interaction contributes to and shapes the emergent mathematics of play.¹

Methods of the Study

In this paper, we contrast the mathematics that emerged over the course of play in six games, each played by a different group of four children.² One group of 8-year-old girls played Monopoly Junior, and three groups of girls—8-, 11-, and 14-year-olds—played the standard version. Two additional groups of 14-year-olds, each composed of two boys and two girls, were selected from a single ninth-grade mathematics class to play an abbreviated version of the standard game: One group consisted of students selected by their mathematics teacher as low-achievers in mathematics, and the other group consisted of students selected as high-achievers by the same teacher. Participants were not told the basis for their selection. The players in all groups were either friends or acquaintances and all players reported that they were familiar with the game, having played it often. The girls playing Monopoly Junior reported that they were familiar with both the Junior and standard versions.

Each group was videotaped as they played the game. The observers provided no instructions or assistance other than telling the participants to play in their usual manner and to end the game when they wanted to stop.³

Analysis. We used the videotapes to reconstruct each game and recorded every instance of a mathematical problem. These ranged from the relatively simple addition of two die and counting spaces as players moved their tokens around the board, to the complex mathematics involved in purchasing multiple units of housing, determining the change due from rent, and negotiating trades and sales. When players were presented with options, such as whether to pay $200 or 10% of one’s net worth, we coded the option chosen by the players. We also recorded how problems were solved and the accuracy of players’ solutions. In this paper, we present the results of analyses of payment problems, instances in which the player whose turn it is needs to pay money to the bank (e.g., to purchase property), to another location (e.g., to FREE PARKING, a common unofficial rule), or to one or more players (e.g., to pay rent). Table 1 includes a listing and description of the types of problems and solutions included in the analyses that follow.

Interrater reliability was established by two coders independently applying the coding scheme to videotapes of two games: one Junior version and one standard game. Out of a total of 122 payment problems in the two games, agreement for problem complexity type was 97%, amount of payments was 95%, payment type was 98%, number of bills in payment was 97%, accuracy of problem formation was 95%, and accuracy of change payments was 100%.

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In addition to the quantitative analyses, we also conducted a case study analysis of the games in order to examine more closely the processes by which mathematical environments were transformed in game play. Of particular interest were the dynamics between the task structure, artifacts, children’s mathematical understandings, and social interactions that are interwoven with the emergent mathematical problems of play. At least two researchers independently watched videotapes of all the games and identified segments that illustrated the four components of the Emergent Goals Framework. Once an instance was identified, the entire game was examined to ensure that the incident was representative of the case. In transcribing the selected segments, we attempted to capture as accurately as possible the players’ words in conjunction with their nonverbal actions.

Emergent Mathematical Environments in Children’s Game Play

The games that emerged in the play of the six groups differed in several ways. Games ranged in duration from about 50 minutes (8-year-olds playing Monopoly Junior) to about 1½ hours (14-year-olds), and the average number of turns per player ranged from about 15 (8-year-olds playing Monopoly Junior) to more than 40 (14-year-olds). The mathematics that emerged in play also differed considerably between game versions and groups of players. Differences included the types of problems that emerged in play, the magnitude of the values entailed in players’ problems, and the strategies children used in their solutions. In our analyses, we focused on how the parameters described above—task structures, artifacts and conventions, prior knowledge, and social interactions—provide insight into the distinct mathematical environments constructed by each group.

The first set of analyses examined how activity structures, artifacts, and conventions influenced the mathematics that emerged in play. These parameters are important components of cultural practices, and variations in them provide insight into linkages between participation in cultural practices and learning outcomes. For these analyses, we compared 8-year-old girls playing either the standard or Junior versions of the game.

Table 2 contains a summary of the major similarities and differences between the two game versions. They share many features: Players begin with a specified amount of play money, take turns rolling a die (Junior version) or two dice (standard version), and move tokens around a game board. As players land on spaces they may make purchases or, if another player already bought the space, they may owe money to the other player. They both use die, play money, and game boards divided into spaces representing properties that players may purchase. The player who accumulates the most property and money is the winner.

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In addition to thematic differences in the game versions—buying real estate and earning rent in the standard version versus buying ticket booths and charging admission to amusement park rides in Monopoly Junior—they also vary in their artifacts and activity structures. For instance, embedded in the structure of the standard version, but absent from the Junior version, are problems of multiplication, percentage, and complex addition and subtraction of large values. Also, as indicated in Table 2, currency denominations range from $1 to $500 in the standard version but only from $1 to $5 in the Junior version; players are instructed to begin with $1500 in the standard version and $31 in the Junior version. Differences in currency parallel differences in the cost of properties and rents (see Table 2). Below, we document the consequences of these variations for the mathematics that emerged in play.

Activity structure. The activity structure of Monopoly led players to participate frequently in mathematical problem solving. Players in both versions engaged in summing units of currency to make purchases and pay rents (or admissions), and, when necessary, subtracted in order to determine change. Yet, differences in the rules and organization of play yielded distinct patterns of emergent mathematical problems and solutions across game versions. Table 3 summarizes the primary differences in the mathematical problems that emerged in children’s play.

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The problems that emerged in play were influenced by the opportunities for mathematical problem solving afforded by the structures of the two game versions. For instance, in the Junior version, when players land on an unowned ride they are (according to the rules) required to buy it and no further "improvements" are possible. Unlike the standard version, there are no rules or opportunities in Monopoly Junior that would lead players to multiply, determine percentages, or calculate the cost of multiple items (e.g., to buy several houses). Indeed, the prescribed activity structure of the Junior version yields only a limited range of possible mathematical problems: to make and receive payments of $1 to $6, and, potentially, to double admission charges (which also range from $1 to $6). One outcome of these variations in activity structure concerns the types of problems players encountered when making payments to the bank or other players. (See "Problem complexity" in Table 3.) Although players rarely needed to calculate the amount of their payments in either version—96% and 83% of problems entailed only a single term (e.g., PAY $2) in the standard and Junior versions, respectively—when they did calculate, players in the Junior version were engaged with problems of doubling small quantities (17% of their payment problems), whereas players in the standard version were engaged with more complex mathematical problems entailing the multiplication of two-digit values (4% of their payment problems). On the rare occasions that players needed to calculate the amount of a payment—six problems in the Junior version and two problems in the standard version (see "Accuracy: Problem formation" in Table 3)—they did so with 100% accuracy. This indicates that the problems that emerged in both versions were matched well to the children’s mathematical abilities.

In addition to variation in problem types, the mathematical complexity of problems that emerged in play differed as a result of the numerical values with which players were engaged. Due to differences in the cost of properties and rents (admissions), players in the standard version were involved in problems with larger numerical values compared to players in the Junior version: Purchases in the standard version averaged almost $200 and rents almost $30, engaging children with values entailing two-and three-digits; in the Junior version purchases and admissions both averaged around $3, and problems never entailed values of more than one-digit. Clearly, the mathematical work that was accomplished—and opportunities for players to develop new problem solving skills—varied across game versions.

Artifacts. In addition to dealing with mathematical values of different magnitudes, variation in the currency provided with the game versions also generated distinct problem types. For instance, of the 36 payment problems in the Junior game, all but one (97%) were accomplished by giving the exact amount of money that was owed. (See "Payment type" in Table 3.) In contrast, 35% of the payment problems in the standard version were accomplished by paying more than the amount owed, a procedure that required players to calculate and tender change. In the following excerpt we consider the mathematical complexity that results from a player’s exact payment in the standard game. Carla has landed on KENTUCKY AVENUE and, after some hesitation, decides to buy it for $220.

The low incidence of overpayments was a fortunate adaptation because, as this excerpt illustrates, 8-year-old players’ ability to give accurate change was limited: As shown in Table 3 ("Accuracy: Change"), more than 20% of change payments were incorrect across the two game versions.

When players used exact amounts to make purchases and pay their debts, different levels of mathematical complexity emerged in the two versions. The average number of bills tendered in payments—an index of the number of currency units players needed to add together—was 2.1 in the standard version versus 1.3 in the Junior version. Indeed, 75% of all payments in the Junior version entailed only one bill (which required players to recognize numerical symbols but not to operate on them), in contrast to 40% in the standard game.

Conventions. Conventions that were not included in the official rules of the game but were adopted by groups of players also were interwoven with the mathematics that emerged in play. Some conventions—apparently passed from one generation of players to another—have become so common that players often were unaware that their play did not conform to official game rules. These conventions included what happens when players do not want to purchase the properties they land on, whether or not players in jail continue to receive rent payments, and whether the proceeds from certain types of payments are put into the bank or put on a space (Free Parking) for players to win.

Other conventions arose spontaneously during play. For instance, in the standard version, players decided that they would not buy properties if someone already owned one from the same color group; this convention led to fewer purchase and rent problems, no negotiation of trades, and only one monopoly (owning all the properties of a color group), which limited players’ ability to purchase houses and hotels and kept rents low. As the subsequent comparisons with older children will indicate, the 8-year-olds’ convention restricting purchases led to relatively few opportunities for them to engage in complex mathematics.

A third type of convention, which one group referred to as "house rules," was noted in the standard game played by 14-year-olds. In the following excerpt, an alteration to the intended activity structure of play is suggested by one player and, with some modification, adopted by the group:

The establishment and transmission of conventions—either among a small group of friends or from one generation of players to another—illustrates that although situated activities are dynamic contexts whose form is continually negotiated, there also is the potential for establishing enduring and stable features. The creation and establishment of conventions is one way that "children re-create culture as they learn to participate in the practice" (Miller and Goodnow 1995:12).

In summary, the comparison of two closely-related versions of a single game points to how mathematical problems that emerge in play are interwoven with the activity structure, artifacts, and conventions of the game. Distinct mathematical environments, in terms of the diversity, complexity, and types of mathematical problems, occurred in each game: Children playing the standard version had more complex problems that contained more and larger quantities on which they had to operate. The emergent mathematics in both games appeared to be well suited to the players’ abilities. But, whereas the Junior version was transformed for children by the game’s designers, the children themselves transformed the structure of the standard version through their play by, for instance, limiting the amount of their purchases and engaging in a restricted range of problem types. The following sections provide further evidence of how children’s transformations influenced the mathematical problems that emerged in play.

In the following analyses, the focus is on relations between children’s prior knowledge and the mathematical environments that emerge in their play. We present data from two sets of comparisons. In one set we used age as an index of children’s mathematical knowledge, comparing the emergent mathematics in the standard version of games played by 8-, 11-, and 14-year-old girls. In the second comparison, we varied players’ mathematical knowledge by comparing ninth-grade students selected as either high-or low-achievers by their mathematics teacher.⁴ As in the previous analyses, we are concerned with transformations that affect the frequency, type, and complexity of the mathematical problems that emerged in children’s play. But, whereas the analyses above focused primarily on the way features of the game (e.g., activity structure, artifacts) are interwoven with the emergent mathematics, now our concern shifts to how players themselves transformed the game in ways that have consequences for the mathematical problems that emerge in play. We focus on two components of the Emergent Goals Framework: the knowledge that players bring with them to the game, and the social interactions that take place during play.

Although they were engaged in the same nominal activity (the standard version of Monopoly) the mathematics that emerged in play differed across games played by groups of 8-, 11-, and 14-year-old girls. In general, older children—especially the 14-year-olds—were more likely to buy property, acquire monopolies, and build houses and hotels. As a result, older children’s mathematical problem solving was more frequent—14 year-olds engaged in more than 100 payment problems in their game compared to fewer than 50 for the 8- and 11-year-olds (see frequencies of "Problem complexity" in Table 4). The more properties that were purchased led to a greater likelihood that players would land on one and have to pay rent: There were 54 rent payments for the 14-year-olds compared to 18 for the 11-year-olds and 15 for the 8-year-olds.

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Some aspects of the mathematics that emerged in play were more susceptible to players’ influence than were others. For instance, as shown in Table 4, the average payment for purchases increased just a little across age groups, from $196 to $205 to $229. In contrast to purchases, whose cost is fixed, rents are more responsive to players’ actions—such as whether or not they acquire monopolies and buy houses and hotels. Consequently, the average rent payment increased substantially with age, from a low of $30 for the 8-year-olds to a high of $98 for the 14-year-olds.

We again found that most payment problems required no calculation, although there were differences in problems across the games. The 11- and 14-year-old players were more likely than 8-year-old players to structure problems that required calculation, and 14-year-olds engaged in a wider variety of problem types and problems of greater mathematical complexity than did younger players, including multiplication and percentage problems. Despite the differences, players in all three age groups had high rates of accuracy in forming problems and calculating change.

Children’s prior knowledge and options for play. Differences in the mathematics that emerged in play arose in several ways. Sometimes, children of different ages chose distinct options provided as part of the game’s activity structure. Consider, for instance, problems entailing the calculation of percentages. Percentage problems were most likely to arise when a player landed on the Income Tax space: Players have the choice of forfeiting $200 or 10% of their assets. In the following excerpts, we see how children’s mathematical understandings gave shape to the problems that emerged. The first excerpt is from the 8-year-olds’ game:

In this excerpt, there was no discussion that Carla could have paid 10%, even though she has just a few hundred dollars and doing so would have saved her money. In contrast, in the following excerpt we see that 11-year-old players struggled with the meaning of percentage.

The following excerpt, from the 14-year-olds’ game, illustrates a shift in the players’ understanding of percentage and, as a consequence, in the emergent mathematics of the game. Faye and her partner Carla have passed GO and landed on INCOME TAX.

Following this excerpt, Brenda and her partner, Amanda, discuss that when they landed on INCOME TAX earlier in the game, they should have paid 10% of their money, rather than the $200 alternative. In fact, they had $1412 at the time and would have saved themselves about $60.

In the above excerpts, children’s mathematical knowledge led them to choose one or another of the options provided by the manufacturer. Other times, particular forms of mathematics emerged as a result of players’ own creative transformations. For instance, players sometimes structured problems that combined two or more mathematical computations into one step, rather than solving each computation sequentially. In the following excerpt from the 11-year-olds’ game, Bonnie has passed GO (for which she collects $200) and landed on a Railroad (cost is $200), which she decides to buy.

As the excerpt illustrates, one player’s decision to combine calculations into a single compound problem often confused other players. Sometimes, like Bonnie in this excerpt, the player needed to "unpack" or explain the compound problem. Whereas compound problems never emerged in the 8-year-olds’ play and often led to confusion among 11-year-olds, they were both more common and routine in the play of the 14-year-olds.

Children’s prior knowledge and social roles. Mathematical work also became distributed among players as a result of the social roles they took on. Most notably, the game rules provide for one player to be the banker. Sometimes, the banker took on special responsibilities that engaged her in frequent problems of equivalence trades—changing one denomination of currency for an equal value in another denomination. For instance, in the 11-year-olds’ game, when the banker noticed that the bank was running low on $100 bills, she asked the other players to exchange five $100 bills for one $500 bill.

In addition to engaging more frequently in mathematical problems than other players, the banker role also led to particular problem formats, such as returning change when players made overpayments. The following excerpt shows how the banker role led Bonnie, an 11-year-old player, to engage frequently in complex mathematics. At the start of the interaction, Laura has landed on VENTNOR AVENUE, which costs $260, and decided to buy it.

As the excerpt makes clear, the mathematical problems that emerged as a function of the banker’s role often challenged the younger players’ abilities. Indeed, similar to Laura in the above excerpt, it appeared that some of the players used overpayments to simplify the computation entailed in making an exact payment, thereby shifting the mathematical work to the banker, who then was responsible for returning the appropriate amount of change.

Social interaction. We noted several ways that social interaction influenced the mathematics that emerged in play. Some of these were related to the games’ activity structure, such as interactions that resulted from the need to pay rent to another player, or the competitive nature of play that led older players to be very concerned with purchasing properties and acquiring monopolies. As a result, the 14-year-olds were more likely than younger players to encourage each other to purchase properties and houses and often involved each other in deal-making. The nature of the artifacts, too, encouraged players to engage in certain forms of social interaction, as when players paid their rent with more than the exact amount and the recipient had to calculate and return the appropriate amount of change.

Especially noteworthy were forms of social interaction that distributed or shared the mathematical work that needed to be accomplished. Players sometimes assisted each other in calculating costs, payments, and change. These joint solutions were most common in the 8-year-olds’ game, in which some players had considerable difficulty with the mathematics entailed in play. As the following excerpt illustrates, some problems engaged all players in their solution. Nancy has landed on WATER WORKS, which is owned by Carla. According to the rules, Nancy owes Carla an amount equivalent to four times the quantity shown on the die, eleven.

By sharing the mathematical problem, all players were engaged in its solution. Nancy’s ability to articulate not just the answer but also the informal process she used to arrive at the solution provided a model of a problem solving strategy that other players could appropriate.

How mathematical work was distributed shifted over the course of play among the 8-year-old girls. Consider two mathematical problems that Carla encountered in play and the nature of the assistance she received from other players. On her first turn, Carla landed on CONNECTICUT AVENUE.

In this excerpt, Carla is provided with a great deal of assistance. Sarah shows her the denominations she needs to pay to the bank and tells her that she needs to give just one of each. In order to follow Sarah’s guidance, Carla does not need to recognize the value of the bills—she could just match colors—nor does she need to engage in any mathematical calculations. Indeed, all the mathematical work is accomplished by Sarah, and there is little opportunity for Carla to improve her understanding. Later in the game, though, a similar problem—purchasing a property—leads to interactions that require Carla to engage in slightly more mathematics. Carla has landed on and decided to buy VIRGINIA AVENUE, which costs $160.

Unlike the excerpt from her first turn, in this instance Carla is expected to use the numerical symbols on the currency to make her payment. Although Carla needs, and receives, a great deal of help throughout the game, other players often shift the amount of assistance they provide in ways that both challenge and support Carla’s understanding while making it possible for play to continue.

The following excerpts illustrate different forms of assistance and distribution of mathematical work in players’ payments across the three age groups. The complexity of the mathematics that is accomplished increases with players’ ages. The first is an example of an assisted overpayment. Although 8-year-old players generally had little difficulty paying for purchases when they had the exact amount of money they owed, they often became confused when they could not make an exact payment. In this excerpt, Carla has landed on a property owned by Sarah and owes her $14 rent. Carla looks at her money hesitantly, apparently unsure what to do. Amy begins to provide assistance, touching Carla’s money.

In this excerpt, Carla is provided with explicit assistance about how to make her payment ("one ten and one five") and the logic of overpayments ("she’ll owe you one dollar"). Included in Nancy’s suggestion about how to complete the payment is additional information about the amount of change that Sarah should give Carla.

Overpayments often served to shift mathematical work from one player to another. As indicated in Table 4 (see "No. of bills in payments"), overpayments were accomplished using fewer bills than were exact payments for each age group. Exact payments typically required more than two bills, engaging players in adding together currency values, whereas most overpayments were accomplished with a single bill. Thus, some players, especially those who were challenged by the mathematics entailed in payments, routinely paid their debts using the largest bill they had available, a strategy that greatly simplified the mathematical work they needed to accomplish.

Although overpayments typically shifted mathematical work from the player to an opponent, we noted several forms of overpayments that added mathematical complexity to the work done by the player. Sometimes, as in this example from the 8-year-olds’ game, players needed to provide assistance to their opponent after making an overpayment. Nancy has landed on Water Works, which is owned by Carla. After determining that she owes Carla $44, Nancy gives Carla a $50 bill.

In a more advanced form of overpayments, players took into account the change they would receive when making their payments. Although these complex overpayments were rare and occurred only among 14-year-old players, they illustrate well the way that, through social interaction, more complex mathematics emerges in relation to players’ knowledge. In this excerpt, Faye has decided to purchase a property costing $260. She counts out $100 bills but, finding that she has only one of them, begins counting her $20 bills.

Unlike the overpayments of younger players, which simplified the player’s mathematics, these more complicated overpayments typically increased the mathematical work entailed in determining the payment, but reduced the number of bills required for change. In this manner, even the 14-year-old players often were challenged by the mathematics that emerged from their social interactions.

Our comparison of high- and low achieving 14-year-old students also revealed differences in the mathematical environments that emerged in play. But unlike the games discussed above, in which increasing ability was associated with the construction of more complex and more diverse mathematical problems, the high-achieving students constructed less complex mathematical problems during play than did their low-achieving peers. As seen in Table 5, the low-achieving students had a wider range of problem types, higher payments (especially for rent, which is more responsive to the nature of play than is other payment types), and were more likely than high-achieving students to make their payments with the exact amount of money they owed, although they used fewer bills to do so.

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Both groups of players had high accuracy rates for formulating payment problems and calculating change. It is likely that neither group of 14-year-old students was especially challenged by the mathematics that emerged in play. Rather, what seemed important in their games were differences in the way they approached the mathematics that did emerge. The high-achieving students often engaged in lengthy mental calculations, as can be seen in this excerpt from the start of play. Laura needs to pay $350 for her property. She gives the money slowly to Derek without saying anything as Derek keeps track for her.

In this excerpt, the choice of mental calculation suggests that the high-achieving students are comfortable with arithmetical problem solving, even though it led to two minor calculation errors that would have been unlikely had they used another method, such as creating piles worth $100 each. The use of mental calculation as a problem solving strategy may be related to the lesser range and complexity of problems engaged in by the high-achieving students: High-ability students structured only problems that could be solved readily using mental calculation—problems with numerical values that could be manipulated easily and without taxing limited memory resources—and avoided more complex (e.g., multiple term) problems. In contrast, the low-achieving students seemed to avoid arithmetical problem solving—especially mental arithmetic—as much as possible. Two ways they accomplished this were to decide at the start of play to round all values to the nearest five dollars and to not use the $1 bills, and to employ a calculator for more difficult problems. These strategies, which may represent more the players’ attitude toward mathematics than their problem solving ability, show how emergent mathematical environments may take many forms as a result of players’ actions.

In summary, the comparisons in this section indicate that the mathematics that emerges in children’s play of Monopoly is not simply a function of its activity structure and artifacts, but also is shaped by the participants and the knowledge and attitudes they bring with them. Comparing games played by children across a wide range of ages revealed ways that children’s mathematical understandings are interwoven with the problems that emerged in play: Through their selection of options incorporated into the game’s design (such as Income Tax), their decisions about buying properties and acquiring monopolies, and their social interactions, older players constructed a more complex mathematical environment than did younger players. The comparison of 14-year-old students who differ in their school mathematics achievement revealed additional ways that children transformed the activity, including their use of problem solving strategies (mental versus electronic calculation) and artifacts (eliminating the $1 bills). As we discuss in the next section, these transformations have important implications for understanding children’s learning through participation in activities.

Discussion

In this paper, we have provided examples of the way in which the mathematics of game play varied across versions of a game and several groups of children. Our results indicate that the mathematical content of the activity—playing Monopoly—is neither fixed in the game itself nor solely a function of some underlying mathematical competence that participants bring with them to the game. Rather, the mathematics that emerged during play—including the frequency with which problems were posed, and their diversity, complexity, and solutions—was a result of the interweaving of many factors. The four parameters offered by Saxe—activity structures, artifacts and conventions, social interactions, and prior knowledge—were useful guideposts for understanding the interplay of cultural, social, and cognitive processes in children’s creation of mathematical environments.

Each game took on a distinct mathematical character--featuring particular types of problems, magnitudes, and solutions—as a result of players’ transforming the game as they negotiated how play should proceed and the role that mathematics should have in it. The variation across games reaffirms the need for process-oriented studies of children’s game play and, more generally, of collaborative interactions; analyses of learning must examine how learners contribute to the construction and transformation of their own learning environments.

Differences in the mathematics that emerged in the two versions of Monopoly highlight the role of activity structures and artifacts in constituting mathematical environments. Some differences were designed by the manufacturer with the intent of appealing to children of different ages, an example of how we, as a society, structure age-appropriate activities for children (Guberman in press; White and Siegel 1984). The comparison of age groups highlighted the role of children’s knowledge in creating their own mathematical environments, especially how they interpret and transform tasks in ways that challenge both themselves and others. With increasing age, players appear to have structured mathematical environments of greater complexity, creating contexts for them to use the mathematics they possess. Yet, as several excerpts indicate, diversity within age groups often led children to observe and question problem solving approaches that were new to them. Such interactions created frequent opportunities for learning, and the assistance children provided each other facilitated acquiring new ways of formulating and solving problems. Finally, the comparison between high- and low-achieving students revealed that children’s approaches to mathematics also contribute to emergent mathematical environments by transforming activity structures and conventions, and using artifacts imported from outside the activity itself, such as calculators.

In cultural practices such as game play, we see what Cole (1996:103) refers to as "the dual process of shaping and being shaped through culture": In their transformations children shape the emergent mathematics of play; they simultaneously have opportunities to identify new problem solving goals and to construct the strategies and knowledge needed for their resolution. We believe that our findings are applicable to many of children’s everyday activities and have implications for their mathematics learning, especially for how teachers structure classroom tasks. For instance, the Professional Standards for Teaching Mathematics from the National Council of Teachers of Mathematics (1991) includes suggestions for game-like activities as ways of engaging students in classroom mathematics. Importantly, we found that children in all the games that we observed almost never erred in their problem solving. Perhaps classroom tasks that also allow for participation at various levels of mathematical competence, including well-organized but flexible task structures, diverse artifacts, and meaningful social interactions, will provide children with both opportunities and motivation to build on and extend their mathematical abilities.

Acknowledgements

We thank the children who allowed us to videotape their game play and the parents and teachers who made it possible to do so. We also are grateful to the students in the seminar, Advanced Child Development and Educational Growth, during which these ideas took shape, and to two anonymous reviewers. Partial support for the research reported here was provided to the first author by the IMPART Program and the Committee on Research and Creative Work, University of Colorado at Boulder.

Notes

1. Although each player participated in only one game, in our analyses we use the 8-year-olds playing the standard version of the game in two sets of comparisons, first in contrast to other 8-year-olds playing Monopoly Junior, and then in contrast to 11- and 14-year-old girls playing the standard version.

2. We present analyses to illustrate the approach we take to understanding children’s learning in collective practices. Generalization of the findings would require a larger sample of games and players, although we note that the findings reported here are similar to the results of additional analyses we have completed with groups that vary in composition (Guberman, Menk, & Rahm, in preparation).

3. The 14-year-old girls and both groups of 14-year-old ninth graders selected to play in competing teams of two.

4. The ninth grade players were instructed to play an "abbreviated" version of the standard game in order to shorten the amount of time needed to complete a game. Following instructions supplied by the manufacturer, players randomly distributed two properties to each participant before beginning play.

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Table 1

Examples of Emergent Mathematical Problems and Solutions in the Play of Monopoly

Table 2

Similarities and Differences Between the Junior and Standard Versions of Monopoly

Table 3

Characteristics of Emergent Mathematical Problems and Solutions for 8-Year-Old Girls Playing the Standard and Junior Versions of Monopoly

Table 4

Characteristics of Emergent Mathematical Problems and Solutions for Three Age Groups of Girls Playing the Standard Version of Monopoly

Table 5

Characteristics of Emergent Mathematical Problems and Solutions for Low- and High-Achieving Ninth-Grade Students Playing the Standard (Abbreviated) Version of Monopoly