Storage and retrieval of addition facts: The role of number comparison

Abstract

It is proposed that arithmetical facts are organized in memory in terms of a principle that is unique to numbers-the cardinal magnitudes of the addends. This implies that sums such as 4 + 2 and 2 + 4 are represented, and searched for, in terms of the maximum and minimum addends. This in turn implies that a critical stage in solving an addition problem is deciding which addend is the larger. The COMP model of addition fact retrieval incorporates a comparison stage, as well as a retrieval stage and a pronunciation stage. Three tasks, using the same subjects, were designed to assess the contribution of these three stages to retrieving the answers to single-digit addition problems. Task 3 was the addition task, which examined whether reaction times (RTs) were explained by the model; Task 1 was a number naming task to assess the contribution of the pronunciation stage; Task 2 was a magnitude comparison task to assess the contribution, if any, of the comparison stage. A regression equation that included just expressions of these three stages was found to account for 71% of the variance. It is argued that the COMP model fits not only the adult RT data better than do alternatives, but also the evidence from development of additional skills.The basic phenomena involved in single-digit addition performance are robust, widely replicated and well known, yet there has been much controversy as to the psychological processes involved. It is generally agreed that competent adults use some mixture of memory retrieval and procedures, but there is little agreement as to how the addition facts are represented and Requests for reprints should be sent to Brian Butterworth,

How abstract is symbolic thought?

“…Both formal models (Butterworth, Zorzi, Girelli, & Jonckheere, 2001;Anderson, 2005) and more theoretical conceptions (Stenning, 2002;Chandrasekaran, 2005) conceive of notation interpretation as a two-stage process: First, a visual scene is decomposed into a string of basic graphemes. Second, abstract representations of these basic graphemes are recomposed using the internally represented laws of an algebraic grammar.…”

Mental representations of arithmetic facts: Evidence from eye movement recordings supports the preferred operand-order-specific representation hypothesis

“…The representation does not contain any order information. In contrast, the COMP model for simple addition proposed by Butterworth, Zorzi, Girelli, and Jonckheere (2001) proposes operand-order-specific (max + min) single representations. In the following paragraphs, we provide some details about these two models.…”

“…As experience with addition increases, children and adults can directly retrieve answers from memory. As Butterworth et al (2001) claimed, the counting procedure could affect the organization of facts in memorythat is, the preferred form is likely to be 5 + 3 = 8 rather than 3 + 5 = 8. In addition to the min model, children and adults also rely on transformation strategies to acquire and practise addition facts, such as, 8 + 5 = 8 + 2 + 3, 5 + 8 = 3 + 2 + 8 (e.g., Dehaene & Cohen, 1997;Geary, Hoard, Byrd-Craven, & DeSoto, 2004;LeFevre, Sadesky, & Bisanz, 1996;LeFevre, Smith-Chant, Hiscock, Daley, & Morris, 2003;Roussel, Fayol, & Barrouillet, 2002;Siegler, 1987).…”

“…Then they use the max and min to access stored addition facts. Butterworth et al (2001) asked college students to perform a number-naming task, a magnitude-comparison task, and an addition task. They found that performance on the naming and magnitude-comparison tasks accounted for 71% of the variance of the performance on the addition task.…”

“…The finding that addition showed the operand order effect in eye movements, but not in the behavioural measurement, was probably due to the greater sensitivity of the HEOG measurements than of behavioural measurements. Previous research has shown that the operand order effect for addition was consistent but very weak (Butterworth et al, 2001;Zhou & Dong, 2003). Zhou and Dong found that smaller operand first problems took 8 ms longer to answer than did larger operand first problems.…”