THE APPLICATION AND DEVELOPMENT OF AN ADDITION GOAL SKETCH

Arthur J. Baroody University of Illinois at Urbana-Champaign Baroody@uiuc.edu

Sirpa H. Tillikainen University of Illinois at Urbana-Champaign tiilikai@students.uiuc.edu

Yu-chi Tai University of Illinois at Urbana-Champaign yuchitai@students.uiuc.edu

Abstract: Siegler’s latest strategy-choice model includes an addition goal sketch, which is presumed to develop as a child practices concrete counting-all and to subsequently affect strategy choice by identifying legal and illegal strategies and suppressing the latter. A study of 20 kindergartners was undertaken to examine key assumptions of this model. Participants were individually interviewed to determine their own strategy use and their ability to evaluate legal and illegal strategies. Consistent with Siegler’s model, children who had just learned a concrete counting-all procedure were prone to use illegal strategies. Inconsistent with the model, more advanced adders did also. Furthermore, children who had not yet invented counting-on did not view this strategy as legal as Siegler has suggested. Implications for computer simulations of development are discussed.

Rationale

A key aspect of the latest version of Siegler’s (e.g., Shrager & Siegler, 1998) strategy-choice model is an addition goal sketch.

Recognition of Novel Legal Strategies

According to Siegler (e.g., Crowley, Shrager, & Siegler, 1997), a goal sketch permits children to evaluate new versions of strategies they invent or even novel strategies they have never used themselves. Consistent with this claim, Siegler and Crowley (1994) found that 5-year-olds who themselves had not yet invented counting-on from the larger addend (COL; e.g., for 3 + 5, counting: 5; 6 [is one more], 7 [is two more], 8 [is three more]) rated this strategy and the familiar, concrete counting-all strategy (CCA; e.g., for 3 + 5, counting out three items to represent the first addend, then five more items to represent the second addend, and finally counting all the items put out to determine the sum) as "smart."

Siegler and Crowley's (1994) finding contradicts Baroody's (1984) case study observations of Felicia. This 5-year-old typically used a verbal counting-all strategy—either beginning with the first addend (CAF ; e.g., for 3 + 5, counting: "1, 2, 3, 4 [is one more], 5 [is two more], 6 [is three more], 7 [is four more], 8 [is five more]") or beginning with the larger addend (CAL; e.g., for 3 + 5, counting: "1, 2, 3, 4, 5; 6 [is one more], 7 [is two more], 8 [is three more]") for single-digit combinations. However, when presented with larger challenge items, she immediately and consistently used either a COL strategy (e.g., 5 + 22: "23, 24, 25, 26, 27") or a COL-like strategy (e.g., 32 + 6: "31, 32, 33, 34, 35, 36, 37, 38"). When single-digit combinations were reintroduced, Felicia reverted to using CAF or CAL. Moreover, when COL was modeled for her with these smaller combinations and she was asked to evaluate the strategy, the girl declared, "You can't do it that way." These results were replicated on several more occasions.

The reason for this discrepancy may be that Siegler and Crowley (1994) modeled a conceptually less-advanced form of COL than did Baroody (1984). More specifically, the former demonstrated an indirect-modeling version of counting-on (IM), a strategy only somewhat more sophisticated or (in the case of CAF or CAL users) less advanced than their own. (Indeed, Fuson and Secada,1986, found that learning an IM strategy did not promote the learning of COL.) In contrast, Felicia (Baroody, 1984) evaluated a COL strategy conceptually more advanced than her own. A main purpose of the present study, then, was to determine if other children with less-advanced addition strategies can typically recognize COL as a legal strategy or not.

Recognition of Illegal Strategies

Siegler and Crowley (1994) also found that their 5-year-old participants, whether they had already invented COL or not, rated an illegal strategy (representing one addend twice) as “not smart.” Crowley et al. (1997) have further noted that “children never discover illegitimate addition” (p. 469). Shrager and Siegler (1998) attribute the ability to distinguish between legal and illegal strategies and choose only the former to an addition goal sketch. Unfortunately, this generalization was based on a small and select sample (eight above-average calculators). Our previous research indicates that children of all abilities do sometimes invent and use illegal addition strategies. One such strategy is to create a nondistinct representation of the two addends. For 2 + 4, for instance, Jonna (almost 8-years old and diagnosed as having learning difficulties) represented the first addend by counting, "One, two" (accompanied by raising two fingers), looked at the expression and represented the second addend by counting, "three, four" (accompanied by raising two fingers), and then determined the sum by counting the four extended fingers. In effect, the "one, two" was used to represent the first addend and the first two counts needed to represent the second addend.

The second primary goal of the present study was to gauge whether children could identify a genuine and more subtle error (Jonna’s error) than the artificial and obvious error of counting the same addend twice. A secondary goal was to find additional evidence of illegal strategy use.

The Development of an Addition Goal Sketch

Crowley et al. (1997) argued that an additive goal sketch is constructed from the repeated application of the most basic informal counting strategy, concrete counting-all (CCA) which is learned by rote by imitating parents, siblings, or peers. As a child practices CCA, a metacognitive component is increasingly relieved of the burden of micromanaging the execution of this strategy and increasingly takes on a monitoring function. As monitoring requires using only the key elements of CCA, only these are reinforced. The other steps (elements) fade from memory leaving a goal sketch of the strategy. A key implication of this view is that children who have just learned CCA do not have a goal-sketch and, thus, are likely to invent illegal strategies as they are legal strategies. Another secondary goal of the present study was to check whether this was true.

Method

Participants

Twenty kindergartners, ranging in age from 5 years and 4 months (5-4) to 7-2 (median age = 6-1) participated in the study. Gender was equally represented; 14 participants were Caucasian; 3, Asian; 2, African-American; and 1, Indian.

Procedure

The participants were individually interviewed or tested over five sessions. Preliminary testing of prerequisite counting and number skills was done in Session 1. In Session 2, children were administered a nonverbal addition and subtraction task (for details, see Huttenlocher, Jordan, & Levine, 1994) and change add-to word problems to assess their informal understanding of arithmetic; to determine which children already knew informal, counting-based strategies; to teach those who did not have such a strategy CCA; and to gauge the general level of a child’s addition strategies. The word problems and a child’s answer to each were translated into a number sentence to familiarize participants with symbolic addition. In Sessions 3 and 4, children were administered the strategy-evaluation task (see Siegler & Crowley, 1994, for details). In each session, five strategies were modeled in the following order: CCA, IM (which Siegler & Crowley, 1994, incorrectly called COL), counting out an addend twice (E1), COL, and Jonna’s error (E2). For each strategy, a child was asked to rate it as “very smart,” “smart,” or “not smart.” To analyze the data, these responses were coded as 2, 1, and 0, respectively, and the average a child’s rating for each strategy was determined. In Session 5, the children’s strategy use was assessed by presenting them 10 written addition expressions, accompanied by a verbal description. These included small items (1 + 3, 4 + 2, 3 + 2), medium items (3 + 5, 6 + 4, 5 + 6, 8 + 4) and large items (16 + 1, 11 + 2, 3 + 12). The last type was included to give children who knew a COL strategy a particularly strong incentive to use it. The tasks were typically presented in the format of a game to maximize motivation.

Results

Eleven children used COL at least once or, in two cases, relied solely on strategies more advanced than COL (e.g., retrieval); nine children exhibited no use of COL. Both quantitative and qualitative analyses were performed.

Quantitative Data

Figure 1 summarizes the quantitative results of the study. A 2 (group: use: COL user or non-COL user) x 5 (demonstrated strategy: E1, E2, CCA, IM, COL) repeated-measures ANOVA revealed a significant main effect for demonstrated strategy F (4, 72) = 8.627, p < .001, no effect for group, and (unlike Siegler & Crowley, 1994) a significant interaction effect for the two factors, F (4, 72) = 3.291, p = 0.016. Tests for simple main effects were conducted to follow up the significant interaction. The COL users judged the legitimate strategies significantly more favorably than the illegitimate strategies (F [4, 80] = 1.969, p = 0.05), but the non-COL user’s judgments of the five strategies did not differ significantly. The non-COL users (incorrectly) judged the Jonna’s error strategy (E2) more favorably than did the COL users (F [1, 90] = 6.621; p = .05), but the judgments of other specific strategies were not significantly different between the groups.

Figure 1. Mean Evaluation Scores for Five Addition Strategies by Group (COL Users vs. Non-Col Users)

Qualitative Data

Like the quantitative analysis, qualitative analyses indicated that, consistent with the case of Felicia (Baroody, 1984), children who have not yet invented COL may have trouble recognizing this strategy as legal. In fact, five of these children favored CCA over COL; three rated the former as “very smart” and the latter as “not smart” in both sessions. For example, Brianna, who had to be taught the CCA strategy in Session 2 and who rated this strategy as “very smart,” rated the IM strategy as “kinda’ smart” and the COL strategy as “not smart.” Beth, who was an accomplished user of CCA or its shortcuts considered the CCA and the IM strategy as "smart" but Jonna's procedure (E2) and representing one addend twice (E1) as "not smart."

Consistent with Crowley et al.’s (1997) prediction, children who had just learned CCA were prone to invent illegal, as well as legal, strategies. Inconsistent with their predictions, children who adopted more advanced strategies and presumably had constructed a goal sketch also used illegal strategies. For example, in Session 5, Beth devised a relatively advanced strategy, which involved keeping track. For example, for 4 + 2, she counted, "one, two, three, four" (as she extended four fingers consecutively) and then "five, six" (as she extended two more fingers consecutively). Yet for 5 + 6 and 3 + 12, she devised a strategy similar to Jonna's. For example, for the former item, she counted, "one, two, three, four, five" (while extending five fingers in turn), next counted, "six" (as she extended one more finger), and then announced an answer of "six." In brief, she devised an illegal strategy to deal with relatively large and challenging problems, despite fairly clear evidence of conceptual or metacognitive knowledge of addition.

Conclusions

Some scholars argue that computer simulations based on information-processing theory have important advantages over verbal (constructivist and social-learning) theories in that they require explicitly delineating questions, assumptions, and predictions (e.g., Klahr & MacWhinney, 1998), and that this permits more precise theorizing and theory testing. In fact, the validity of a computer simulation such as SCADS depends on the completeness and accuracy of the developmental theory and data used to design it. The results of the present study indicate that even this latest and most sophisticated version of the strategy-choice model does not accurately represent the development of addition strategies. In particular, SCADS does not adequately account for how children construct an understanding of addition and how they this knowledge to invent addition strategies. For example, many, if not most, children invent CCA, because preschoolers begin to construct a concept of addition before they devise counting strategies for solving word problems or symbolic expressions (e.g., Huttenlocher et al., 1994). That is, contrary to Shrager and Siegler’s (1998) assumption, the development of addition does not begin when parents or others teach CCA to a child. Inconsistent with Shrager and Siegler’s (1998) assumption that the same goal sketch underlies the invention of all strategy improvement beyond CCA, the results of this study indicate that there is a qualitative or conceptual leap that must be made to move from counting-all to counting-on (e.g., Fuson,1992; Steffe, von Glasersfeld, Richards, & Cobb, 1983). Furthermore, non-COL users were less able to distinguish between legal and illegal strategies than Siegler and Crowley (1994) evidence suggests and, contrary to Shrager and Siegler's (1998) claim, some non-novices, do invent illegal strategies.

References

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