# Inverse Reference in Adults-Elementary Arithmetic

## Article excerpt

Mauro, LeFevre, and Morris (2003) and Campbell (2008) manipulated problem format to assess university students' simple division and subtraction. Large division problems (dividend > 25; e.g., 42 ÷ 6 = _) and large subtraction problems (minuend > 10; e.g., 13 - 6 = _), but not small problems, were solved more quickly when presented in inverse operation format (e.g., 6 × _ = 42 for division; 6 + _ = 13 for subtraction). They concluded that adults often solve large simple division and subtraction problems by reference to the inverse operation but rely on direct memory retrieval for smaller problems. Their findings, however, might have resulted from unequal practice or mixing of the inverse operations. Here, in Experiment 1 (division) and Experiment 2 (subtraction) normal and inverse formats received equal practice and only one operation was practiced (i.e., division or subtraction). Large divisions and subtractions were solved substantially faster when presented in inverse format, but there was also evidence that subtraction ties (e.g., 12 - 6 = 6) and small subtractions (minuend ≤ 10) benefited from inverse format. The results affirm that inverse reference is an important element in adult's performance of elementary subtraction and division.

Keywords: elementary arithmetic, procedural strategies, problem format

As children learning elementary mathematics, we spend much time committing the basic facts of arithmetic to memory (e.g., 2 + 3 = 5, 7 - 3 = 4, 6 × 6 = 36, etc.). This surely is time well spent because most of us will need to refer to the basic math facts thousands of times during our lifetime. Furthermore, although alternative, conceptually based procedural strategies to solve basic arithmetic are taught, memory retrieval is generally, faster, more accurate, and requires less of our attentional resources (Imbo & Vandierendonk, 2008). Consequently, memory-based performance of elementary arithmetic is desirable and is generally associated with a higher level of skill (Campbell & Xue, 2001; LeFevre, DeStefano, Penner-Wilger, & Daley, 2006).

Nonetheless, it has become clear that educated adults do not rely only on direct memory retrieval for basic addition (Geary & Wiley, 1991; LeFevre, Sadesky, & Bisanz, 1996), multiplication (Hecht, 1999; LeFevre et al, 1996), subtraction (Seyler, Kirk, & Ashcraft, 2003), or division (Campbell, 1999; LeFevre & Morris, 1999). Instead, adults often report using procedural strategies based on counting, decomposition, or transformation. Procedure use is reported more frequently for large number problems (7 + 9 = 16) than for small problems (3 + 2 = 5) (Campbell & Xue, 2001; LeFevre et al., 1996). Small problems are often defined as those with a product/dividend ≤25 (Campbell & Xue, 2001) or sum/ minuend ≤ 10 (Seyler et al., 2003). Problem frequency is one important factor in this problem size effect (Zbrodoff & Logan, 2005). The probability of encountering any specific numeral decreases as numerical value increases (Dehaene & Mehler, 1992); consequently, small-number arithmetic problems are encountered more frequently than larger problems. Therefore, the latter develop relatively low memory strength and are more likely to be solved by procedural strategies, which are slower and relatively error prone compared to retrieval. This problem size effect on strategy use leads to longer response times (RTs) and more errors for larger problems, although a substantial problem size effect on RT and errors remains when only retrieval trials are analyzed (Campbell, Fuchs-Lacelle, & Phénix, 2006; Campbell & Xue, 2001).

Use of procedural strategies by educated adults is reported much more often for simple division and subtraction than the inverse operations of multiplication and addition (e.g., Campbell & Xue, 2001). One reason for this is that adults may adopt strategies for division and subtraction based on knowledge of the inverse addition fact (e. …

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