Mathematical Problems and Goals in Children’s Play of an Educational Game

Steven R. Guberman
University of Colorado at Boulder

Geoffrey B. Saxe
University of California at Berkeley

Abstract

In his development of activity theory, Leontiev explained that the emergence of divisions of labor in society necessarily leads to collaborative activity in which individuals, with their own goals and actions, contribute to collective achievements. In this paper, we describe emergent "divisions of labor" that are common in children’s collective problem solving. Parallel to Leontiev’s argument, we show that when labor becomes divided, children often become engaged in accomplishing different goals leading to different learning outcomes. We illustrate the utility of this analytic tack in analyses of 64 third and fourth graders playing an educational game, Treasure Hunt.

Mathematical Problems and Goals in Children’s Play of an Educational Game

Though geared to the emergence of divisions of labor in the history of societies, Leontiev’s remarks bear on current issues concerning children’s learning in collective activity. Consider a group of friends playing the board game Monopoly (Guberman, Rahm, & Menk, 1998). In the course of game play, the children become engaged in meeting a variety of game-linked "needs." Such needs include purchasing properties and paying and collecting rent. To satisfy these needs, children may become engaged with mathematical work—such as calculating the cost of a purchase and using play money to pay for it (Leontiev’s "intermediate results")—work that is often distributed across several players. One player, for instance, lacking "exact change," may determine that the $500 bill is the appropriate bill to offer for the purchase of Atlantic Avenue, a property that costs $260. Another player, who has the role of "banker," engages in computing the amount due the purchaser, 500 minus 260, and returns $240 in change. Such divisions of labor are the rule rather than the exception in many games and activities with which children are engaged.

Recent sociocultural analyses have highlighted the importance of collective activities as contexts for children's learning and development , and researchers are looking for new ways of conceptualizing relations between individuals and groups in children’s learning. In this paper, we extend a research framework developed in prior work on cultural practices and cognitive development to analyses of children’s joint play of an educational game that we developed, Treasure Hunt.¹ We focus our efforts here on the emergence of divisions of labor in the children's collective activity. Following Leontiev, we note that "divisions of labor" emerge frequently in children’s group problem-solving efforts. One result of such divisions is that although all participants may be engaged in the same nominal activity, the cognitive work that gets accomplished collectively often is distributed among group members. Thus, while participants may be engaged in accomplishing a common problem (Leontiev’s "product"), the construction of goals and the means for accomplishing them (Leontiev’s "intermediate results") are often quite distinct across individual participants. Sometimes, goals may complement one another; other times, goals may conflict; and still other times, goals may be similar. In order to distinguish between collective and individual processes in our analyses, we differentiate between the problems that groups collectively generate and solve in their interactions, and the goals that individuals construct as they participate with others in group activity.

The Play of Treasure Hunt

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Treasure Hunt

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Supply purchases are perhaps the key activity in Treasure Hunt because they are an occasion for players to acquire supplies that will subsequently enable them to attain greater quantities of doubloons. When purchasing supplies, a player reads prices on a supply menu, printed prominently on each of the six islands on the game board. For example, a player who lands on Snake Island, depicted in Figure 2, may decide to purchase two shovels and two spyglasses. As indicated in the supply price menu in Figure 2, the price to the player would be 14 doubloons if the shovels and spyglasses are purchased in pairs. One of two types of arithmetical problems issues from this decision depending on the configuration of doubloons in the player's Treasure Chest.

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Snake Island

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Problem and Strategy Types

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We have described the strategies used to solve payment problems as though players accomplished them on their own. In fact, solutions were often accomplished collaboratively. Regardless of strategy type—exact payment, trade-pay, or remove-replace—players often gave and received varied forms of assistance in carrying out the strategy. As we discuss below, the different ways that participants assisted each other to accomplish supply purchases had consequences for the goals individual players formulated and the mathematics learning that took place during play.

We begin our analyses with a focus on the collaborative solutions produced by dyads. We expected that success in solving payment problems would vary as a function of problem complexity: Direct payment problems would be easier to solve than indirect payment problems. To test this expectation, we computed rates of accurate problem solving for each problem type.

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Mean Percentage of Direct and Indirect…

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In her analyses of girls playing hopscotch, Goodwin notes that "moves in a turn-taking system (e.g., turns at talk) are not single-party activities but instead are constructed through the ongoing mutual orientation and collaborative action of multiple participants" (pp. 265-266). Like Goodwin, we find in Treasure Hunt that although the player taking his or her turn appears to be the most salient actor, the opponent also was central to our understanding the mathematics that emerged in play. To understand the effect of the opponent on children’s problem solving accuracy, we turn to analyses of social interaction during play, especially how individuals—players and their opponents—contributed to problem solutions. Our focus is on group differences in forms of social interaction and individuals' construction of mathematical goals that emerged in accomplishing indirect supply purchase problems.

In our observations of children playing Treasure Hunt, we noted that players and their opponents engaged in two primary forms of social interaction as they attempted to solve payment problems. We refer to them as direct and thematically-organized assistance. These interactional types support different kinds of mathematical goals that players structure in their activities.

In this excerpt, Kevin appears to not understand the denominational values of the doubloons—that some doubloons are worth one unit, others are worth ten units, and so forth. In placing the 10-doubloon piece in the bank, Kevin may merely be structuring and accomplishing a goal of moving the piece denoted by Erica, something he can do even though he may not understand the logic of Erica’s solution. With Erica's assistance, though, Kevin is able to complete his purchase (the collective problem).

Many children also provided direct assistance when players were faced with indirect payment problems. In the following excerpt, Erica and Kevin attempt to accomplish an indirect payment problem that results from Kevin's decision to purchase supplies at a value of 13 doubloons, for which he cannot make an exact payment. In his choice of supplies to purchase, Kevin often generated complex arithmetical problems that required multiple doubloon trades for adequate completion. But, as in the above excerpt, it was Erica who structured and accomplished the mathematical goals that led to an accurate solution to the purchase problem.

As in the first excerpt, Kevin appears to not understand the denominational values of the doubloons and, when faced with another payment problem, he does not generate the mathematical goals involved in either a trade-pay or remove-replace strategy. Erica, who appears to possess a more adequate understanding of the doubloon denominations, formulates the goals of summing supply costs, selecting doubloons for payments, and returning change, goals that, in concert, are the basis for the remove-replace strategy.

Similar to Kevin and Erica, in many of the mixed ability dyads the more able child provided direct assistance that made it possible for the less able player to accomplish purchases and payments. This finding explains, at least in part, why third graders playing against fourth graders had higher rates of adequate solutions to payment problems than did third graders playing against other third graders (as discussed above; see Figure 4). In mixed ability dyads, the more able fourth grader accomplished much of the mathematical work, providing assistance that was not available to third graders paired with other third graders. Through direct assistance, both players’ and opponents’ mathematical goals often took form and were modified as a function of the activity of the other member of the group. Indeed across dyads, we observed that the mathematical goals entailed in purchasing and paying for supplies was sometimes formulated and accomplished by the player, sometimes by the opponent, and sometimes distributed between them.

Emergence of goals in 'thematically-organized' interactions. The second form of social interaction that was common in children's play was thematically-organized assistance. The distinction between problems and goals is especially helpful for understanding the learning that occurred in such interactions, and they are the focus of the remainder of this paper. In thematically-organized interactions, the collective arithmetical problems are similar to those that emerge with direct assistance. However, the goals that lead to problem accomplishment become distributed in thematic play, as one child assumes the role of customer and the other child assumes the role of storekeeper. For example, in thematically organized assistance interactions, the player-as-customer selects supplies to purchase. In response, the opponent-as-storekeeper usually determines the cost of supplies, tells the customer the purchase price, receives doubloons from the customer, and produces change.

Similar to direct assistance, thematically-organized assistance has the effect of engaging each of the players in accomplishing distinct mathematical goals as he or she attempts to solve joint problems. But in contrast to direct assistance, thematically-organized assistance emerges in play as a function of the roles players assume, rather than in response to a player's need for assistance.

The following excerpt is an illustration of how thematic roles led two players, Veronica and Toni, to form and accomplish distinct mathematical goals as they worked jointly toward solving an indirect payment problem.

Although Veronica’s decision to buy a map and a parrot initiated the joint problem, Toni, in her role of storekeeper, formulated and accomplished the mathematical goals of adding the cost of the two items and returning the appropriate change. Veronica also contributed to accomplishing the purchase problem: Once Toni said that the total cost of her purchase would be 11 doubloons, Veronica, as customer, formulated and accomplished the mathematical goals entailed in completing an adequate payment.

As Toni and Veronica’s interaction illustrates, when payments are organized by thematic roles, the mathematical goals and the means for accomplishing them usually are less complex for the player than for the opponent. In thematically-organized assistance, the player’s goals principally involve counting doubloons while respecting their denominational structure to produce an appropriate overpayment. In contrast, the opponent is engaged with more complex goals. An opponent-as-storekeeper may be asked to add together the cost of several supplies and, if handed a 100-unit doubloon, be required by virtue of the thematic role to determine and return the appropriate change. Such a task involves not only the representation of doubloons in quantitative terms, but also an arithmetical transformation such as subtraction (often accomplished by means of a complex counting strategy).

Thus, through thematic roles of customer and storekeeper, mathematical work was divided among the players. Thematic roles were not included in the instructions for playing Treasure Hunt. Rather, in cases like Veronica and Toni, they emerged spontaneously in play. In the next section, we analyze how general such thematic roles were and how they varied across our targeted groups.

To analyze the frequency of thematic roles in play, we coded each players’ supply purchases as either ‘using’ or ‘not using’ thematic roles. As shown in Figure 5, when third graders played third graders, the incidence of thematically-organized assistance was low, and when fourth graders played fourth graders, its incidence was high. Importantly, though, the data presented in Figure 5 also indicate that thematic roles emerged as a function of the dyad, not of the individual player: Third graders who played against fourth graders were more likely to use thematic roles during their turns than were third graders who played against other third graders; and fourth graders who played against fourth graders were more likely to use thematic roles during their turns than were fourth graders who played against third graders. Thus, like Veronica and Toni, many dyads used thematically-organized assistance in play, though the frequency of such assistance varied across groups.

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Percent of turns...

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Our results indicate that thematically-organized assistance was implicated in children’s construction of arithmetical goals in interesting ways. Since third and fourth graders playing in mixed-ability dyads were playing against each other, we note that thematically-organized assistance was used more often during the third graders’ turns than during the fourth graders’ turns. For example, the above excerpt used to illustrate thematically-organized assistance was typical of play during Veronica's turns (the player less competent in mathematics), but when play shifted to Toni's turn, Toni (the more competent player) usually calculated the cost of her own purchases, paid for them, and took the appropriate change with little input from Veronica. Thus, while Veronica was engaged with structuring and accomplishing arithmetical goals involved in overpayments, Toni was engaged with structuring and accomplishing arithmetical goals in calculating supply costs and producing change.

We have argued that both across groups and within dyads, children were structuring and accomplishing different goals in play. As a consequence, we expected that children would differ in the mathematics they acquired through play. For illustration purposes, we limit our discussion to children’s solution to a single posttest problem.

The posttest problem is similar to the indirect payment problems that emerged in play, although its minuend is greater than the values children typically encountered in Treasure Hunt: A child has a set of blocks and must remove (or pay to the bank) a specified quantity, as if purchasing supplies. We identified three types of solutions to the posttest problem that were similar to the collective solutions to indirect payment problems used in game play: inadequate solutions, adequate solutions that made use of the trade-pay strategy, and adequate solutions that made use of the remove-replace strategy. (The two adequate solutions were discussed above and are depicted in Figure 3.)

The expected sequence of strategy use—younger, less-able children using trade-pay strategies and older, more-able children using remove-replace strategies—also was supported by an analysis of strategy use on the posttest for the children who had not played Treasure Hunt. The performances of third and fourth grade nonplayers (represented in Figure 6), provide "baseline" information that supports our argument for a developmental ordering of strategies. At third grade, nonplayers who produce adequate solutions tended to use trade-pay strategies, whereas at fourth grade, nonplayers who produced adequate solutions tended to use remove-replace strategies. Thus, both a logical analysis based on the mathematical complexity of direct and indirect payment problems, and an empirical analysis based on nonplayers’ performances in third and fourth grades, indicate that the trade-pay strategy is a precursor to the remove-replace strategy.

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Percent of Players and Nonplayers Using Each Strategy

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In addition to facilitating children’s accuracy on the posttest, game play also had an impact on children’s strategic knowledge. Here we return to the importance of analyzing emergent mathematical goals as a basis for understanding children’s learning in collective activity. We expected that the mathematical goals that emerged during children’s play of Treasure Hunt, which varied as a result of children's participation in distinct forms of social interaction, might alter the progression from trade-pay to remove-replace strategies that we found among nonplayers.

We expected that players who used thematic roles to solve their purchase problems during game play were likely to have acquired an understanding of the mathematical goals entailed in the remove-replace strategy. Thematic roles typically engaged the player in a remove-replace strategy, albeit a form of the strategy that is distributed across players: Rather than the player making an overpayment and determining his or her own change, in thematically-organized solutions the player makes an overpayment, but responsibility for calculating change shifts to the opponent. In this way, the joint accomplishment of the payment problem constitutes a remove-replace strategy "stretched over" or "distributed" among the two members of the dyad. Players who used these distributed remove-replace strategies had many opportunities to construct the goals associated with overpayments by making sense of the change they received in relation to both the cost of their purchase and their payment. In contrast, children who did not use thematic roles in their purchases, tended to use more trade-pay strategies during play and, consequently, had fewer opportunities to construct the mathematical goals entailed in the remove-replace strategy.

This analytic tack led us to the expectations presented in Table 1. Third graders who played fourth graders had regularly engaged in the distributed remove-replace strategy during play, and we expected that they would use remove-replace strategies on the posttest. In contrast, third graders who played against other third graders rarely used thematically-organized solutions during play, but were more likely to construct mathematical goals associated with exact payments, including trading denominational blocks to accomplish them. We expected, therefore, that on the posttest they would use the trade-pay strategy more often then the remove-replace strategy.

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To analyze children's posttest performance, we examined how many children in each group used trade-pay versus remove-replace strategies. As expected, third graders who played fourth graders were more likely to use remove-replace than trade-pay strategies on the posttest, and third graders who played third graders more frequently used trade-pay than remove-replace strategies (see Figure 6a).

It is important to note that for both grade levels, the strategy children used on the posttest was consistent with the mathematical goals we hypothesized they would construct during play. Depending on the social organization of their problem solving during play, some third graders used remove-replace strategies and others used trade-pay strategies on the posttest. Similarly, some fourth graders used remove-replace strategies and others used trade-pay strategies on the posttest consistent with the forms of social interaction they had engaged in during play. The expected normative sequence of strategies was not consistent with the posttest performances of children who had played Treasure Hunt.

Concluding Remarks

As a first pass at understanding the dynamics of play that led to learning, we turn to Vygotsky’s "Genetic Law of Cultural Development":

Vygotsky’s law well describes our findings in certain respects.

Consider first the contrast between the findings for non-players and players on our arithmetical posttest task. With an increase in grade level, children who had not played Treasure Hunt tended to shift from using the trade-pay strategy to using the remove-replace strategy. This was an expected developmental sequence based upon a logical analysis of task requirements. For players, however, we observed developmental trajectories that differed from the "normative sequence." In the case of players, solutions that were initially distributed over the dyad in play led to solution strategies that were used by individual players in the posttest. Players who had used distributed remove-replace strategies in play (whether third graders playing fourth graders or fourth graders playing fourth graders) were likely to use such strategies in their solitary problem solving in the posttest. Similarly, players who used distributed trade-pay strategies in play more frequently used trade-pay strategies in the posttest (third graders playing third graders and fourth graders playing third graders).

The agreement of our findings with Vygotsky’s Genetic Law is strong; however, we note two features that suggest a need to expand this account. First, as others (e.g., Stone, 1993; Wertsch & Stone, 1985) have noted, Vygotsky does not elaborate the mechanisms whereby activity that is distributed socially may lead to "intra-psychological" accomplishments. Without an account that focuses on mechanism, we are left with an observation elevated to the status of "Law"—what is distributed in play becomes a property of the individual in independent activity. We are concerned that the "how" or "why" of this process is left unexplained.

Second, our data indicate that movement from the social plane to the psychological plane entails more than the reproduction of social activity by individuals. Accounts of internalization that focus on reproduction at the expense of participants’ constructive activity lead to the expectation that both of the children engaged in an interpsychological process would develop similar intrapsychological understanding (Cobb et al., 1997). But, although we found that third graders who played fourth graders used remove-replace strategies on the posttest (these children used distributed remove-replace strategies in play), their fourth grade partners (engaged in the same interpsychological process) did not use such strategies on the posttest at the same rate. Thus, a simple internalization of distributed strategies into individual problem solving does not account for the posttest performance of both participants in the social interaction.

We find it useful to return to our treatment of problems and goals as a way to understand the shift from "interpsychological" to "intrapsychological" functioning in play—constructs that build on Leontiev’s remarks cited at the beginning of this essay. Although problem solutions were accomplished jointly, as thematic roles emerged in play children became engaged in routines in which the particular mathematical goals they were structuring and accomplishing became distinct as "labor" became divided. For instance, when a third grader playing a fourth grader took on the role of customer, the third grader was likely to form goals associated with making overpayments and receiving change. In contrast, in the same interaction, the fourth grader, taking on the role of storekeeper, structures different (but complementary) goals, ones associated with calculating the cost of the customer's purchase and the change that is due. Thus, although engaged in the same interpsychological activity, they are constructing distinct goals, and the arithmetic learning that each player accomplishes is linked to these goals. In this way, social activity and the individual construction of knowledge are inextricably linked. Thus, children's performances on the posttest are not mere copies of the strategies used in social interaction during game play, but represent constructions linked to the mathematical goals that children structured during play.

In closing, we find that Leontiev’s statement (in the introduction) about divisions of labor in emerging social groups is a useful stepping stone toward understanding the reflexivity between children’s joint activity and their individual accomplishments in Treasure Hunt. In our analyses, individuals’ goal-directed activities both sustained and were constitutive of the collective play in which children participated; reflexively, collective efforts to accomplish emergent problems valued in play had implications for participants’ goals. We find that emergent problems and goals provide a critical nexus for understanding children's collective practice and learning in Treasure Hunt, and we believe that the analytic tack presented here is applicable to understanding children’s learning in collective activities across a broad range of cultural and educational practices.

Footnotes

1. Treasure Hunt was developed by the UCLA Peer Interaction Group, including Joseph Becker, Teresita Bermudez, Lisa Butler, Kristin Droege, Randy Fall, Steven Guberman, John Iwanga, Marta Laupa, Scott Lewis, Anne McDonald, David Niemi, Mary Note, Pamela Paduano, Laura Romo, Geoffrey Saxe, Rachelle Seelinger, Tine Sloan, Christine Starczak, and Michael Weinstock.

2. We refer to the child whose turn it is as the "player," and to the other member of the dyad as the "opponent."

3. Due to attrition, our final sample was reduced.

References

Table 1. The Predominant Organizations of Play and Their Relation to Problem Solving Strategies During Play and on the Posttest

Figure Legends

Figure 1. The Treasure Hunt game.

Figure 2. Snake Island: One of six islands on the Treasure Hunt game board.

Figure 3. Problem and strategy types for paying 14 doubloons.

Figure 4. Percent of direct and indirect payment problems solved correctly as a function of players’ grade level, opponents’ grade level, and session.

Figure 5. Percent of turns in which dyads used thematic roles in indirect payment problems as a function of players’ grade level, opponents’ grade level, and session.

Figure 6. Percent of players and nonplayers using inadequate, trade-pay, and remove-replace strategies on the posttest: (a) third graders, (b) fourth graders.