Conceptual Knowledge of Decimal Arithmetic

In two studies (N's = 55 and 54), we examined a basic form of conceptual understanding of rational number arithmetic, the direction of effect of decimal arithmetic operations, at a level of detail useful for informing instruction. Middle school students were presented tasks examining knowledge of the direction of effects (e.g., "True or false: 0.77 * 0.63 > 0.77), number line estimation of decimal magnitudes, and knowledge of decimal arithmetic procedures. Their confidence in their direction of effect judgments was also assessed. We found (1) most students incorrectly predicted the direction of effect of multiplication and division with decimals below one; (2) this pattern held for students who accurately estimated the magnitudes of individual decimals and correctly executed decimal arithmetic operations; (3) explanations of direction of effect judgments that cited both the arithmetic operation and the numbers' magnitudes were strongly associated with accurate judgments; and (4) judgments were more accurate when multiplication problems involved a whole number and a decimal below one than with two decimals below one. Implications of the findings for instruction are discussed. [At time of submission to ERIC this article was in press with the "Journal of Educational Psychology."]