Inverse reference in subtraction performance: An analysis from arithmetic word problems

Abstract

Studies of elementary calculation have shown that adults solve basic subtraction problems faster with problems presented in addition format (e.g., 6 + _ = 13) than in standard subtraction format (e.g., 13 – 6 = _). Therefore, it is considered that adults solve subtraction problems by reference to the inverse operation (e.g., for 13 – 6 = 7, “I know that 13 is 6 + 7”) because presenting the subtraction problem in addition format does not require the mental rearrangement of the problem elements into the addition format. In two experiments, we examine whether adults' use of addition to solve subtractions is modulated by the arrangement of minuend and subtrahend, regardless of format. To this end, we used arithmetic word problems since single-digit problems in subtraction format would not allow the subtrahend to appear before the minuend. In Experiment 1, subtractions were presented by arranging minuend and subtrahend according to previous research. In Experiment 2, operands were reversed. The overall results showed that participants benefited from word problems where the subtrahend appears before the minuend, including subtractions in standard subtraction format. These findings add to a growing body of literature that emphasizes the role of inverse reference in adults' performance on subtractions.

Keywords:

• Strategy use
• Subtraction by addition
• Arithmetic word problems

Acknowledgments

This research was supported by Grant PSI2011-27737 to J. Orrantia from the Ministerio de Ciencia e Innovación in Spain.

Notes

¹ Ending a sentence with a number is grammatically correct in the Spanish language.

² The problem size effect does not derive solely from procedure use; other factors have been proposed to account for this effect, such as differences in the associative strength of small versus large problems or interference from competing associations (see Zbrodoff & Logan, 2005, for a discussion of the problem size effect). For instance, the strength of the association between a problem and its answer is stronger for small problems, so that these problems are primarily solved by direct retrieval. Recently, Klein and colleagues have suggested that problem size may be confounded with the need for a carry operation when using only single-digit problems (Klein et al., 2010; Klein, Nuerk, Wood, Knops, & Willmes, 2009), because problems with a carry (i.e., with a sum or minuend ≥10) are always larger than problems without a carry (i.e., with a sum or minuend ≤10). Since problems requiring a carry are usually associated with longer response times than problems not requiring a carry, it seems plausible that differences observed between small and large problems were also influenced by the need for a carry in the present study (we thank an anonymous reviewer for this suggestion). Nevertheless, this possibility needs to be investigated in future studies using two-digit problems, since problem size and carry are inevitably confounded when using single-digit problems.