A componential model for mental addition.

Abstract

A componential model capable of representing simple and complex forms of mental addition was proposed and then tested by using chronometric techniques. A sample of 23 undergraduate students responded to 800 addition problems in a true-false reaction time paradigm. The 800 problems comprised 200 problems of each of four types: two single-digit addends, one singleand one double-digit addend, two double-digit addends, and three single-digit addends. The results revealed that the columnwise product of addends, a structural variable consistent with a memory network retrieval process, was the best predictor of mental addition for each of the four types of problem. Importantly, the componential model allowed estimation of effects of several other structural variables, e.g., carrying to the next column and speed of encoding of digits. High levels of explained variance verified the power of the model to represent the reaction time data, and the stability of estimates across types of problem implied consistent component use by subjects. Implications for research on mental addition are discussed.Over the past 20 years, several types of models for mental addition have been proposed--for example, models hypothesizing that analog (Restle, 1970), counting (Groen & Parkman, 1972), or memory network retrieval (Ashcrafl & Battaglia, 1978) processes are invoked to arrive at the solution for a given problem. Although a great deal has been learned about the manner in which persons respond to addition problems, a comprehensive model identifying the several elementary processes underlying problem solution has not been developed. The primary aim of the present study is to propose and, by use of chronometric techniques, to evaluate a general processing model specifying the processes required to solve mental addition problems of any magnitude. Sternberg (1977) outlined the componential analysis approach for isolating the elementary information processes involved in solving ability problems. Chronometric, or reaction time (RT), tasks are typically used to validate proposed componential models. In the componential analysis framework, internal validation refers to the determination that RT to problems of a given domain is affected by the hypothesized elementary information processes. Internal validation may take two forms: intensive and extensive. Intensive validation

A componential analysis of an early learning deficit in mathematics

“…This result stands in contrast to the strong relationship between solution times for comparable addition problem sets and the prod variable found for adults (Gear-y et al, 1986;Miller et al, 1984;Widaman, Geary, Cormier, & Little, 1989). Moreover, the parameter estimates for the prod variable for both groups indicate a retrieval rate of more than double the rate estimated for adults (Geary et al, 1986;Widaman et al, 1989). The pattern of results described thus far suggests an immature memory representation of addition facts for the normal and LD-improved groups (Ashcraft, 1982;Brown, 1975;Hamann & Ashcraft, 1985) and perhaps an anomalous representation for the LD-no-change group.…”

“…(In fact, for these two groups retrieval trial RTs showed only modest correlations with each,of the previously described variables.) This result stands in contrast to the strong relationship between solution times for comparable addition problem sets and the prod variable found for adults (Gear-y et al, 1986;Miller et al, 1984;Widaman, Geary, Cormier, & Little, 1989). Moreover, the parameter estimates for the prod variable for both groups indicate a retrieval rate of more than double the rate estimated for adults (Geary et al, 1986;Widaman et al, 1989).…”

Cognitive addition: Strategy choice and speed-of-processing differences in gifted, normal, and mathematically disabled children.

“…The results for the young subjects do not support strong assumptions about the exclusive use of memory retrieval processes for the solution of mental arithmetic problems in adulthood (Ashcraft, 1982;Campbell, 1987a;Geary et al, 1986;Miller et al, 1984;Widaman et al, 1989). For the young adults, 12% of the rather simple problems used in the experimental task were apparently solved by means of either the verbal counting or decomposition backup strategies.…”

“…In fact, it is assumed that memory retrieval is the exclusive problem-solving strategy for simple arithmetic problems by the end of the elementary school years and certainly in adulthood (Ashcraft & Battaglia, 1978;Ashcraft & Fierman, 1982;Campbell, 1987a;Campbell & Graham, 1985;Geary, Widaman, & Little, 1986;Gears Widaman, Little, & Cormier, 1987;Miller, Perlmutter, & Keating, 1984;Widaman et al, 1989). Despite this strong assumption, the use of backup strategies by adults for solving simple arithmetic problems has not been empirically tested in terms of developmental models (but see Svenson, 1985).…”

“…The second purpose was to compare the performance of young and elderly adults on a relatively well-understood cognitive task: the mental solution of simple addition problems (Ashcraft, 1982;Widaman, Geary, Cormier, & Little, 1989). The cognitive addition task provides a useful vehicle for examining age-related performance differences for several reasons.…”