...used only one strategy that reflected the written procedure for each of the addition and subtraction algorithms taught in the classroom. Interviews were used to identify students' knowledge and ability with respect to number sense (including number facts, estimation, numeration, and effect of operation on number), metacognition and affects. Two conceptual frameworks were developed, one representing the "flexible" mental computers, and the other representing the inflexible mental computers. These frameworks identified factors and relationships between factors that influence mental computation. The frameworks were compared with an ideal framework that had been developed from a study of proficient mental computers. These frameworks showed that inaccuracy resulted from disconnected and deficient cognitive, metacognitive, and affective factors; and in some cases might have been affected by deficient short-term memory. It appeared that students' choices of mental strategies resulted from different forms of compensation for varying levels of deficiencies.

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Inaccurate Mental Addition and Subtraction: Two Case Studies

Researchers and educators have stressed the importance of including mental computation in number strands of mathematics curricula (e.g., Cobb & Merkel, 1989; McIntosh, 1996; Reys & Barger, 1994; Sowder, 1990; Treffers & de Moor, 1990; Willis, 1990). Reasons for its inclusion are that mental computation: (1) enables children to learn how numbers work, make decisions about procedures, and create strategies (e.g., Reys, 1985; Sowder, 1990); (2) promotes greater understanding of the structure of number and its properties (Reys, 1984); and (3) can be used as a "vehicle for promoting thinking, conjecturing, and generalizing based on conceptual understanding" (Reys & Barger, 1994, p. 31). In effect, mental computation promotes number sense (National Council of Teachers of Mathematics, 1989; Sowder, 1990). In fact, Willis (1992) suggested that mental computation should be the main form of computation, with written computation to serve as memory support.

Mental computation involves a wider range of strategies than traditional written procedures. A wide variety of mental addition and subtraction strategies has been identified in the literature (e.g., Beishuizen, 1993; Blote, Klein, & Beishuizen, 2000; Cooper, Heirdsfield, & Irons, 1996; Reys, Reys, Nohda, & Emori, 1995; Thompson & Smith, 1999). These strategies are summarized in Table 1.

The terms 1010 and u-1010 are used for separation strategies in the Dutch literature, N10 and u-N10 are used for the aggregation strategies, and N10C is used for the compensation strategy which is described here as wholistic (e.g., Blote, Klein, & Beishuizen, 2000). The strategy mental image of pen and paper algorithm is included in the table because of its presence in the literature (Reys, Reys, Nohda, & Emori, 1995). However, most literature considers mental image of pen and paper algorithm to be an inefficient strategy (Carraher, Carraher, & Schliemann, 1987; Ginsberg, Posner, & Russell, 1981; Hope, 1985; Kamii, 1989; Maier, 1977; Plunkett, 1979; Reys, Reys, Nohda, & Emori, 1995).

In terms of efficiency, Thompson and Smith (1999) classified the strategies so that aggregation and wholistic were the most sophisticated. Similarly, Heirdsfield and Cooper (1997) argued that separation right to left, separation left to right, aggregation and wholistic represented increasing levels of strategy sophistication.

While it has been posited in the literature that different strategy choice is effected by the semantic structure of word problems (e.g., Riley, Greeno, & Heller, 1983; Verschaffel & DeCorte, 1990), Blote, Klein, and Beishuizen (2000) also found that the number characteristics of problems can affect which strategy is chosen. However, some students do not consider either semantic structure of the word problem or the number characteristics; they employ a single strategy continuously. As mentioned before, this strategy is usually mental image of pen and paper algorithm.

Proficiency in mental computation has been the focus of several research projects (e.g., Beishuizen, 1993; Heirdsfield, 1996; Hope & Sherrill, 1987; McIntosh & Dole, 2000; Reys, Reys, Nohda, & Emori, 1995). In The Netherlands, where mental computation is taught before written computation, mathematics programs emphasize the use of aggregation (N10) as a more efficient mental strategy. However, weaker students tended to use less efficient separation strategies (Beishuizen, 1993). Hope and Sherrill (1987) reported that unskilled mental computers used strategies that reflected pen and paper algorithms. In contrast, skilled mental computers employed a variety of strategies that reflected understanding of number and operations. Reys, Reys, Nohda, and Emori (1995) also found that accuracy in mental computation was associated with strategies other than mental image of pen and paper algorithm. In contrast to these findings, McIntosh and Dole (2000) reported higher accuracy when students employed mental image of pen and paper algorithm than when they employed alternative mental strategies (although these alternative strategies revealed number sense). Heirdsfield (1996) also found that accuracy in mental computation did not need to be accompanied by employment of a variety of efficient mental strategies. Therefore, while some research appears to indicate that accuracy in mental computation is a result of efficient mental strategies; other research has reported accuracy as a result of employment of strategies that reflect pen and paper algorithms.

Research reported by the authors investigated mental computers and the factors that supported accuracy (Heirdsfield, 1998, 2001a, Heirdsfield & Cooper, 2002). This study investigated the part played by number sense knowledge (e.g., number facts, estimation, numeration, and effect of operation on number), metacognition (metacognitive knowledge, strategies and beliefs), affects (e.g., beliefs, attitudes), and memory (working memory and long-term memory) in mental computation. Flexibility in mental computation was defined as employment of efficient mental strategies, taking into account the number combinations to inform the mental strategy choice. The research showed that students proficient in mental computation (accurate and flexible) possessed integrated understandings of number facts (speed, accuracy, and efficient number facts strategies), numeration, and effect of operation on number. These proficient students also exhibited some metacognitive strategies and beliefs, and affects (e.g., beliefs about self and teaching) that supported their mental computation. Further, proficient mental computers had reasonable short-term recall to hold interim calculations and recall number facts (phonological loop--see Baddeley, 1986), and well developed central executive (Baddeley, 1986) to attend to the demanding task of mental computation and retrieve strategies and facts from a well-connected knowledge base in long-term memory. Proficient mental computers chose alternative and efficient strategies, as they possessed extensive and connected knowledge bases to support these strategies. Thus, there was evidence of the importance of connected knowledge, including domain specific knowledge, and metacognitive strategies, affects and memory for proficient mental computation. As a result of this study, a conceptual framework identifying associated factors involved in proficient mental computation (see Figure 1).

[FIGURE 1 OMITTED]

This leads to the question as to what are the effects on mental computation of less knowledge and fewer connections? It would be expected that one effect would be less accuracy. The purpose of this paper is to report on seven students who were inaccurate in mental computation. These students were part of a study that investigated addition and subtraction mental computation in seven and eight year old students. Conceptual frameworks for these students are developed and compared with a framework for the "ideal" mental computer (flexible and accurate) that was developed in the large study from which these students were drawn (Heirdsfield, 2001b).

Method

Participants

The participants were seven students, selected from a population of sixty Year 3 children from three classrooms (see Figure 2). The students were selected on the basis of an interview that probed for accuracy and flexibility (employment of variety of strategies) in mental computation. Emma, Jane and...

NOTE: All illustrations and photos have been removed from this article.



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Correction.(Correction Notice), June 22, 2004

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