What affects strategy selection in arithmetic? The example of parity and five effects on product verification

Lemaire, Patrick; Reder, Lynne M.

doi:10.3758/bf03211420

Cited by 194 publications

(184 citation statements)

References 49 publications

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“…The two competing productions are influenced by their prior history of success -that is, how often the production correctly identified a target versus how often it required that the other production apply to confirm that the target was in the other location. 5 Empirically, the notion that procedures will adapt to previous successes and failures has been supported in higher-level domains: for example, deciding whether to search memory or work out the answer (Reder, 1987;Reder & Ritter, 1992), choosing between two problem-solving procedures (Lovett & Anderson, 1996), verifying an arithmetic statement by calculating or using a heuristic (Lemaire & Reder, 1999), or selecting a type of runway in an air traffic controller task (Reder & Schunn, 1999). This account is not completely sufficient, however, in that it would predict that repeated targets would be faster than control targets, and the evidence does not support this.…”

Section: Discussionmentioning

confidence: 99%

The adaptive character of the attentional system: Statistical sensitivity in a target localization task.

Reder¹,

Weber²,

Shang³

et al. 2003

Journal of Experimental Psychology: Human Perception and Perfor

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A localization task required participants to indicate which of 4 locations contained a briefly displayed target. Most displays also contained a distractor that was not equally probable in these locations, affecting performance dramatically. Responses were faster when a display had no distractor and almost as fast when the distractor was in its frequent location. Conversely, responses were slower when targets appeared in frequent-distractor locations, even though targets were equally likely in each location. Negative-priming effects were reliably smaller when targets followed distractors in the frequent-distractor location compared to the rare-distractor location, challenging the episodicretrieval account. Experiment 2 added a 5th location that rarely displayed distractors and never targets, yet responses slowed most when distractors appeared there. The results confirmed that the attentional system is sensitive to first-and higher-order statistical patterns and can make short-and long-term adjustments in preferences based on prior history of inspecting unsuccessful locations.The limited capacity of the human attentional system, combined with the complexity of the environment, necessitates a mechanism for effective selecting and responding to goal-relevant stimuli while disregarding irrelevant ones. The cost of splitting attention or of being distracted can even be seen in simple experimental tasks such as spatial localization. An example of this type of task (Tipper, Brehaut, & Driver, 1990) required participants to press a key corresponding to the spatial location of a target flashed on a computer display. Performance was slower when two stimuli were flashed and one was a distractor. It is not only extraneous information that interferes with the processing of the current display: Research in attention has also documented that when a stimulus (location or object, depending on the task) that should be ignored (i.e., is the distractor) on the prime display becomes the target on the subsequent probe display, performance suffers compared with trials in which this switch does not occur. This type of interference is commonly referred to as negative priming (Tipper, 1985).The fact that irrelevant stimuli interfere with performance implicates a selective attention mechanism that directly allocates processing resources to goal-relevant information (e.g., Neill, 1997). Although there is considerable agreement concerning the necessity of a selective attention mechanism, there is less agreement concerning how this selection occurs, that is, how the process of filtering or ignoring the irrelevant is achieved.

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Section: Discussionmentioning

confidence: 99%

The adaptive character of the attentional system: Statistical sensitivity in a target localization task.

Reder¹,

Weber²,

Shang³

et al. 2003

Journal of Experimental Psychology: Human Perception and Perfor

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“…People rejected violations of the parity relationship (i.e., 3 3 4 = 13) more quickly than nonviolations (i.e., 3 3 4 = 14). Other relationships within multiplication, such as the case of five rule (i.e., if either multiplicand is 5, the product must end in 5 or 0; Lemaire & Reder, 1999) and the effect of zero as a multiplicand (e.g., Miller, Perlmutter, & Keating, 1984), also appear to be represented. Second, developmental research on children's strategy selection within an operation has shown that children have rich representations of each operation.…”

Section: Principle Representation Of Mathematical Operationsmentioning

confidence: 99%

The representations of the arithmetic operations include functional relationships

2001

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Current theories of mathematical problem solving propose that people select a mathematical operation as the solution to a problem on the basis of a structure mapping between their problem representation and the representation of the mathematical operations. The structure-mapping hypothesis requires that the problem and the mathematical representations contain analogous relations. Past research has demonstrated that the problem representation consists of functional relationships, or principles. The present study tested whether people represent analogous principles for each arithmetic operation (i.e., addition, subtraction, multiplication, and division). For each operation, college (Experiments 1 and 2) and 8th grade (Experiment 2) participants were asked to rate the degree to which a set of completed problems was a good attempt at the operation. The pattern of presented answers either violated one of four principles or did not violate any principles. The distance of the presented answers from the correct answers was independently manipulated. Consistent with the hypothesis that people represent the principles, (1) violations of the principles were rated as poorer attempts at the operation, (2) operations that are learned first (e.g., addition) had more extensive principle representations than did operations learned later (multiplication), and (3) principles that are more frequently in evidence developed more quickly. In Experiment 3, college participants rated the degree to which statements were indicative of each operation. The statements were either consistent or inconsistent with one of two principles. The participants' ratings showed that operations with longer developmental histories had strong principle representations. The implications for a structure-mapping approach to mathematical problem solving are discussed.

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“…An exception is the Lemaire and Reder (1999) study, which compared types of problems and found a greater parity effect when both operands were even than when one operand was even, and a reverse, but not significant, effect on 0 X 0 problems. The authors proposed EVENNESS FAMILIARITY EFFECT 359 that noting evenness as a feature invoking the parity rule is twice as likely when there are two even operands; however, no explanation was proposed for the reverse trend noted for two-odd problems.…”

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confidence: 99%

“…Besides the issues concerning the representation format in which parity information is coded and the way this information is retrieved (Campbell & Clark, 1992;Clark & Campbell, 1991;Dehaene, Bossini, & Giraux, 1993;McCloskey, Aliminosa, & Sokol, 1991;McCloskey, Caramazza, & Basili, 1985), the role of parity in arithmetic has also been studied (Krueger, 1986;Krueger & Hallford, 1984;Lemaire & Fayol, 1995;Lemaire & Reder, 1999). Infact, several studies have proposed that subjects have access and sometimes use parity information without explicit awareness when doing mental calculation.…”

mentioning

confidence: 99%

“…Furthermore, Lemaire and Reder (1999) found that the use of the parity rule depended on stimuli and contextual characteristics. They observed that when subjects were verifying complex problems (49 X 8 = 394), the incongruence effect was greater when both operands were even and when the efficiency ofthat strategy was highthat is, when the proportion of problems violating the rule was high.…”

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confidence: 99%

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The odd-even effect in multiplication: Parity rule or familiarity with even numbers?

et al. 2000

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This study questions the evidence that a parity rule is used during the verification of multiplication. Previous studies reported that products are rejected faster when they violate the expected parity, which was attributed to the use of a rule (Krueger, 1986;Lemaire & Fayol, 1995). This experiment tested an alternative explanation of this effect: the familiarity hypothesis. Fifty subjects participated in a verification task with contrasting types of problems (even X even, odd X odd, mixed). Some aspects of our results constitute evidence against the use of the parity rule: False even answers were rejected slowly, even when the two operands were odd. Wesuggest that the odd-even effect in verification of multiplication could not be due to the use of the parity rule, but rather to a familiarity with even numbers (three quarters of products are indeed even).Besides the issues concerning the representation format in which parity information is coded and the way this information is retrieved (Campbell & Clark, 1992;Clark & Campbell, 1991;Dehaene, Bossini, & Giraux, 1993;McCloskey, Aliminosa, & Sokol, 1991;McCloskey, Caramazza, & Basili, 1985), the role of parity in arithmetic has also been studied (Krueger, 1986;Krueger & Hallford, 1984;Lemaire & Fayol, 1995;Lemaire & Reder, 1999). Infact, several studies have proposed that subjects have access and sometimes use parity information without explicit awareness when doing mental calculation. For instance, it seems that in some circumstances, when subjects are verifying simple arithmetic operations (e.g., 2 X 3 = 6 or 2 + 3 = 5), they do not retrieve the result but use a shorthand strategy for checking for the plausibility of the proposed result. Among the cues used for checking that plausibility, its parity has proven to be relevant. It has been found that adults reject a false result more rapidly when its parity is incongruent with the parity of the correct answer (e.g., 2 X 3 = 7) than when it is congruent with it (e.g., 2 X 3 = 8). This incongruence efThis work was supported by the Belgian National Scientific Research Funds (FNRS) and by the Belgian Federal government (I.U.A.P. on temporal control of dynamic tasks situations and the nature of knowledge representation). We thank Marie-Pascale Noel and Agnesa Pillon for their helpful comments on this work. Correspondence should be addressed to A. Lochy,

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What affects strategy selection in arithmetic? The example of parity and five effects on product verification

Cited by 194 publications

References 49 publications

The adaptive character of the attentional system: Statistical sensitivity in a target localization task.

The adaptive character of the attentional system: Statistical sensitivity in a target localization task.

The representations of the arithmetic operations include functional relationships

The odd-even effect in multiplication: Parity rule or familiarity with even numbers?

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