Children's relational knowledge of addition and subtraction

Knowledge of addition combinations has long been thought to facilitate the learning of subtraction combinations (e.g., 8 - 5 = ? can be answered by thinking 5 + ? = 8). Indeed, it follows from Siegler's (1987) model that an associative facilitating effect should make the correct answer the most common response to a subtraction combination, even in the earliest phase of mental-subtraction development. Children in the initial or the early phase of development were examined in 2 studies. Study 1 involved 25 kindergartners and 15 first graders in a gifted program; Study 2 involved 21 first graders in a regular program. Participants were presented with pairs of items, such as 4 + 5 = 9 and 9 - 4 = ?, and asked if the first item helped them to answer the second. Many participants, particularly the less developmentally advanced ones, did not recognize they could use a related addition equation to determine a difference. Study 2 participants were also administered a subtraction timed test. Contrary to Siegler's model, developmentally less advanced children responded with the correct difference relatively infrequently on nearly all items, and even developmentally advanced children did so on more difficult items. The results of both studies are consistent with earlier findings that suggested the complementary relation between addition and subtraction is not obvious to children. They further indicate that an understanding of the complementary relation is not an all-or-nothing phenomenon. It often develops first with subtraction combinations related to the addition doubles, apparently because such addition combinations are memorized relatively early. Ready facility with related addition combinations may make it more likely that children will connect their knowledge of subtraction to their existing intuitive knowledge of part-whole relations. This process may also account for why Study 2 participants were able to master subtraction complements without computational practice. Methodological and educational implications are discussed.