Developmental Trends in Arithmetic

Ilg, Frances L.; Ames, Louise Bates

doi:10.1080/08856559.1951.10533584

Cited by 46 publications

(32 citation statements)

References 19 publications

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“…In effect, subjects are less successful in counting n steps backward than in counting n steps forward (Fuson, Richards, & Briars, 1982). The errors made by subjects using counting strategies are of the same type as those observed in the case of addition (Ilg & Ames, 1951).…”

mentioning

confidence: 58%

From algorithmic computing to direct retrieval: Evidence from number and alphabetic arithmetic in children and adults

Barrouillet

Fayol

1998

Mem Cogn

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A number of theories of mental arithmetic suggest that the ability to solve simple addition and subtraction problems develops from an algorithmic strategy toward a strategy based on the direct retrieval of the result from memory. In the experiment presented here, 2nd and 12th graders were asked to solve two tasks of number and alphabet arithmetic. The subjects transformed series of 1 to 4 numbers or letters (item span) by adding or subtracting an operand varying from 1 to 4 (operation span). Although both the item and operation span were associated with major and identical effects in the case of both numbers and letters at 2nd grade, such effects were clearly observable only in the case of letters for the adult subjects. This suggests the use of an algorithmic strategy for both types of material in the case of the children and for the letters only in the case of the adults, who retrieved numerical results directly from memory.

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mentioning

confidence: 58%

From algorithmic computing to direct retrieval: Evidence from number and alphabetic arithmetic in children and adults

Barrouillet

Fayol

1998

Mem Cogn

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“…Eventually, with practice and the strengthening of memory traces for the declarative knowledge of addends and their sums, retrieval of the correct answer for a given problem becomes the more efficient way of solving simple addition problems. Ashcraft (1982;Ashcraft & Fierman, 1982) documented just such a developmental progression from use of procedural (e.g., digital) processes to use of declarative, retrieval processes, a trend noted in earlier research (e.g., Brownell, 1935;Ilg & Ames, 1951).…”

Section: Rule-based Procedural Processesmentioning

confidence: 95%

A componential model for mental addition.

Widaman¹,

Geary²,

Cormier³

et al. 1989

Journal of Experimental Psychology: Learning, Memory, and Cogni

180

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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

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“…Children solved 58% of these small problems using a strategy of retrieval of the answer from memory, and 12% of them used retrieval exclusively. Several other studies have reported a mix of algorithmic and retrieval strategies in children when solving elementary subtractions (Ilg & Ames, 1951;Lankford, 1974;Starkey & Gelman, 1982). Accordingly, studies on subtraction in adults have also reported frequent use of retrieval (Barrouillet & Fayol, 1998;Geary, Frensh, & Wiley, 1993).…”

Section: Mental Subtraction and Additionmentioning

confidence: 99%

Strategies in subtraction problem solving in children

Barrouillet

Mignon

Thévenot

2008

Journal of Experimental Child Psychology

112

111

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The aim of this study was to investigate the strategies used by third graders in solving the 81 elementary subtractions that are the inverses of the one-digit additions with addends from 1 to 9 recently studied by Barrouillet and Lépine. Although the pattern of relationship between individual differences in working memory, on the one hand, and strategy choices and response times, on the other, was the same in both operations, subtraction and addition differed in two important ways. First, the strategy of direct retrieval was less frequent in subtraction than in addition and was even less frequent in subtraction solving than the recourse to the corresponding additive fact. Second, contrary to addition, the retrieval of subtractive answers is confined to some peculiar problems involving 1 as the subtrahend or the remainder. The implications of these findings for developmental theories of mental arithmetic are discussed.

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Developmental Trends in Arithmetic

Cited by 46 publications

References 19 publications

From algorithmic computing to direct retrieval: Evidence from number and alphabetic arithmetic in children and adults

From algorithmic computing to direct retrieval: Evidence from number and alphabetic arithmetic in children and adults

A componential model for mental addition.

Strategies in subtraction problem solving in children

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