Academic journal article Memory & Cognition

Identical Elements Model of Arithmetic Memory: Extension to Addition and Subtraction

Academic journal article Memory & Cognition

Identical Elements Model of Arithmetic Memory: Extension to Addition and Subtraction

Article excerpt

The identical elements (IE) model of arithmetic fact representation (Rickard, 2005; Rickard & Bourne, 1996) was developed and tested with multiplication and division. In Experiment 1, we demonstrated that the model also applies to addition and subtraction by examining transfer of response time (RT) savings between prime and probe problems tested in the same block of trials. As is predicted by the IE model, there were equivalent probe RT savings for addition with identical repetition (prime 6 + 9 [arrow right] probe 6 + 9) or an order change (9 + 6 [arrow right] 6 + 9), but much greater savings for subtraction with identical repetition (15 - 6 [arrow right] 15 - 6) than with an order change (15 - 9 [arrow right] 15 - 6), and no savings with an operation change (15 - 9 [arrow right] 6 + 9 or 6 + 9 [arrow right] 15 - 6). In Experiment 2, we examined transfer in simple multiplication and division and demonstrated symmetrical transfer between operations. Cross-operation RT savings were eliminated, however, when the RT analysis included only trials on which both the prime and the probe problems were reportedly solved by direct retrieval. An IE model extended to accommodate savings associated with procedural strategies provides a coherent account of facilitative transfer effects in simple arithmetic.

The cognitive processes that mediate simple arithmetic skills have been the focus of extensive experimental research over the last 30 years (Ashcraft, 1992, 1995; Zbrodoff & Logan, 2005). This research was an attempt to identify the memory processes and procedural strategies that underlie performance of basic arithmetic problems, such as 2 + 3 = 5 and 6 × 7 = 42. The work is motivated both by the importance of understanding the nature of this fundamental intellectual skill and by the fact that cognitive arithmetic provides a rich experimental domain for studying more general theoretical issues in cognitive science (Butterworth, 1999; Dehaene, 1997).

This article reports two experiments that were performed to test the predictions of a prominent theory of arithmetic fact representation: the identical elements (IE) model (Rickard, 2005; Rickard & Bourne, 1996; Rickard, Healy, & Bourne, 1994). According to the IE model, there is a single long-term memory node for problems consisting of the same numerical elements (i.e., operands and answer), regardless of operand order. For example, each multiplication node specifies the two operands, the operation, and the product (e.g., 6, 8, × [arrow right] 48) and is accessed by either operand order (6 × 8 or 8 × 6). In contrast, inverse problems that present different operands and have different answers access different nodes. Thus, inverse division problems (e.g., 48 ÷ 8 and 48 ÷ 6) are represented by different nodes [(48, 8, ÷ [arrow right] 6) and (48, 6, ÷ [arrow right] 8)].

Given these assumptions, practice should strengthen only the node corresponding to the practiced problem. Consequently, positive transfer of savings should occur equally between the two orders of multiplication problems, but there should be no transfer between the two orders of division problems and no transfer between inverse multiplication and division problems. Rickard and Bourne (1996) gave adults extensive practice on subsets of simple multiplication and division problems and then tested performance on the same problems (identical repetition), problems with the operand order reversed (order change), inverse problems with the other operation (operation change), and unpracticed control problems. Transfer of practice (response time [RT] savings) in the order change condition for multiplication (e.g., practice, 4 × 3; test, 3 × 4) was practically equivalent to identical repetition (practice, 3 × 4; test, 3 × 4), which supports the assumption that a common representation mediates both orders (see Verguts & Fias, 2005). They found no transfer of savings between the two orders of division problems (practice, 56 ÷ 7 = 8; test, 56 ÷ 8 = 7) and little evidence that practice transferred in either direction between corresponding division and multiplication problems (practice, 7 × 8 = 56, and test, 56 ÷ 8 = 7; or practice, 56 ÷ 8 = 7, and test, 7 × 8 = 56). …

Search by... Author
Show... All Results Primary Sources Peer-reviewed

Oops!

An unknown error has occurred. Please click the button below to reload the page. If the problem persists, please try again in a little while.