Inaccurate Mental Addition and Subtraction: Causes and Compensation
Heirdsfield, Ann M., Cooper, Tom J., Focus on Learning Problems in Mathematics
This paper reports on a study of seven Year 3 students' diminished performance in mental computation, and compares their mental architecture. Although all students were identified as being inaccurate, three students used some variety of mental strategies, while the other students 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.
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. …