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

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. We suggest 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).     

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

The parity effect in arithmetic problem verification tasks refers to faster and more accurate judgments for false equations when the odd/even status of the proposed answer mismatches that of the correct answer. In two experiments, we examined whether the proportion of incorrect answers that violated parity or the number of even operands in the problem affected the magnitude of these effects. Experiment 1 showed larger parity effects for problems with two even operands and larger parity effects during the second half of the experiment. Experiment 2 replicated the results of Experiment 1 and varied the proportion of problems violating parity. Larger parity effects were obtained when more of the false problems violated parity. Moreover, all three effects combined to show the greatest parity effects in conditions with a high proportion of parity violations in problems containing two even operands that were solved during the second half of the experiment. Experiment 3 generalized the findings to the case of five rule (i.e., checking whether a false product ends in 5 or 0), another procedure for solving and verifying multiplication problems quickly. These results (1) delineate further constraints for inclusion in models of arithmetic processing when thinking about how people select among verification strategies, (2) show combined effects of variables that traditionally have been shown to have separate effects on people's strategy selection, and (3) are consistent with a view of strategy selection that suggests a bias either in the allocation of cognitive resources in the execution of strategies or in the order of execution of these strategies; they argue against a simple, unbiased competition among strategies.     

Interactive effects of numerical surface form and operand parity in cognitive arithmetic

In Experiment 1, adults (n = 48) performed simple addition, multiplication, and parity (i.e., odd-even) comparisons on pairs of Arabic digits or English number words. For addition and comparison, but not multiplication, response time increased with the number of odd operands. For addition, but not comparison, this parity effect was greater for words than for digits. In Experiment 2, adults (n = 50) solved simple addition problems in digit and word format and reported their strategies (i.e., retrieval or procedures). Procedural strategies were used more for odd than even addends and much more for word than digit problems. The results indicate that problem encoding and answer retrieval processes for cognitive arithmetic are interactive rather than strictly additive stages.     

Why 2×2=5 looks so wrong: On the odd-even rule in product verification

Subjects judged whether the proposed product for two multipliers was true or false. Each equation could be judged as plausible or implausible because a product must be even if any of its multipliers is even; otherwise, it must be odd. A proposed product that violates the required odd-even status of the product—that is, deviates from the correct product, whether odd or even, by an odd value (e.g., splits of ±l, ±3, ±5)—can be rejected as false before normal processing is completed (i.e., before the correct product is retrieved and compared with the proposed product). Subjects were indeed faster and more accurate in rejecting a split of +1 or +3 than a split of +2 or +4, and this effect increased as the number of even multipliers in the pair increased. Subjects did not use the odd-even rule when either multiplier was 0 or 1 (Experiment 2), perhaps because other rules are available to bypass normal processing in those cases. A similar odd-even rule is used in sum verification (Krueger &Hallford), and use of the odd-even rules may help to explain why oddness and evenness are such salient features of numbers as abstract concepts (Shepard, Kilpatric, & Cunningham).     

Why 2 + 2 = 5 looks so wrong: On the odd-even rule in sum verification

The odd-even status of a sum depends on the odd-even status of its addends. A sum must be odd if an odd number of its addends are odd; else it must be even. A proposed sum that violates the required odd-even status of the sum—that is, deviates from the correct sum, whether odd or even, by an odd value (e.g., splits of ±1, ±3, ±5)—can be rejected immediately as false. Subjects in the present study did indeed use the odd-even rule in sum verification, because they were as fast and accurate in rejecting a split of ±1 as one of ±2, and a split of ±3 as one of ±4, even though a larger split generally is easier to reject (symbolic distance effect), and splits of ±3 and ±4 were rejected faster and more accurately than those of ±1 and ±2. Performance on separate odd-even tasks indicated that the odd-even properties of numbers and sums are readily available for use by adults, and that persons who do well on such tasks are especially likely to use the odd-even rule in sum verification.     

Production,verification, and priming of multiplication facts

In the arithmetic-verification procedure, subjects are presented with a simple equation (e.g., 4 × 8 = 24) and must decide quickly whether it is true or false. The prevailing model of arithmetic verification holds that the presented answer (e.g., 24) has no direct effect on the speed and accuracy of retrieving an answer to the problem. It follows that models of the retrieval stage based on verification are also valid models of retrieval in the production task, in which subjects simply retrieve and state the answer to a given problem. Results of two experiments using singledigit multiplication problems challenge these assumptions. It is argued that the presented answer in verification functions as a priming stimulus and that on “true” verification trials the effects of priming are sufficient to distort estimates of problem difficulty and to mask important evidence about the nature of the retrieval process. It is also argued that the priming of false answers that have associative links to a presented problem induces interference that disrupts both speed and accuracy of retrieval. The results raise questions about the interpretation of verification data and offer support for a network-interference theory of the mental processes underlying simple multiplication.     

Verification of multiplication facts: an investigation using retrospective protocols

Retrospective verbal protocols collected throughout participants' performance of a multiplication verification task (e.g., "7 x 3 = 28, true or false?") documented a number of different strategies and changes in strategy use across different problem categories used for this common experimental task. Correct answer retrieval and comparison to the candidate answer was the modal but not the only strategy reported. Experiment 1 results supported the use of a calculation algorithm on some trials and the use of the difference between the candidate and correct answers (i.e., split) on others. Experiment 2 clearly demonstrated that participants sometimes bypassed retrieval by relying on the split information. Implications for mental arithmetic theories and the general efficacy of retrospective protocols are discussed.     

Do people combine the parity- and five-rule checking strategies in product verification?

The basic question of the present experiment was whether people use a combination of arithmetic problem solving strategies to reject false products to multiplication problems or whether they simply use the single most efficient strategy. People had to verify true and false, five and non-five arithmetic problems. Compared with no-rule violation problems, people were faster with (a) five problems that violated the five rule (i.e., N×5=number with 5 or 0 as the final digit; e.g., 15 × 4=62), (b) problems that violated the parity rule (i.e., to be true, a product must be even if either or both of its multipliers is even; otherwise, it must be odd; 4 × 38=149), and (c) problems that violated both the parity and five rules (e.g., 29 × 5=142). Finally, people were equally fast and accurate when they solved two-rule violation problems than when they solved five-rule violation problems, and faster for those two types of problems than for parity-rule violation problems. Clearly, people use the single most efficient strategy when they reject false product to multiplication problems. This result has implications for our understanding of strategy selection in both arithmetic in particular and human cognition in general. Received: 18 October 1999 / Accepted: 27 January 2000     

Simple and complex mental addition across development

Students in Grades 1, 4, 7, and 10 were timed as they solved simple and complex addition problems, then were presented similar problems in an untimed interview. A manipulation of confusion between addition and multiplication, in which multiplication answers were given to addition problems (3 + 4 = 12), revealed evidence for the hypothesized interrelatedness of these operations in memory only in 10th graders. The overall pattern of results suggests a strong reliance on memory retrieval, even in the first-grade group, with discernible time differences when “procedural” knowledge of carrying is required for problem solution. The results were judged consistent with a fact retrieval model which invokes explicit procedural information when problem difficulty is high or when processes like carrying and estimating magnitudes are required. In agreement with several other reports, the overall slowing of performance to larger problems is best explained in terms of normatively defined problem difficulty or associative strength in memory.     

The odd-even effect in Sudoku puzzles: effects of working memory, aging, and experience

The odd-even effect in numerical processing has been explained as the easier processing of even numbers compared with odd numbers. We investigated this effect in Sudoku puzzles, a reasoning problem that uses numbers but does not require arithmetic operations. Specifically, we asked whether the odd-even effect occurred with Sudoku puzzles and whether individual differences in working memory (WM), aging, and experience with Sudoku modulated this effect. We manipulated the presence of odd and even numbers in Sudoku puzzles, measured WM with the Wisconsin Card Sorting Test and backward digit span task, tested older and younger adults, and collected Sudoku experience frequency. Performance on Sudoku was more accurate for even puzzles than odd ones. Younger, experienced, and higher-WM participants were more accurate on Sudoku, but these individual difference variables did not interact with the odd-even effect. Odd numbers may impose more cognitive load than even numbers, but future research is needed to examine how age, experience, or WM may influence the odd-even effect.     

On mental multiplication and age.

In 2 experiments, younger and older adults were presented with simple multiplication problems (e.g., 4 x 7 = 28 and 5 x 3 = 10) for their timed, true or false judgments. All of the effects typically obtained in basic research on mental arithmetic were obtained, that is, reaction time (a) increased with the size of the problem, (b) was slowed for answers deviating only a small amount from the correct value, and (c) was slowed when related (e.g., 7 x 4 = 21) versus unrelated (e.g., 7 x 4 = 18) answers were presented. Older adults were slower in their judgments. Most important, age did not interact significantly with problem size or split size. The authors suggest that elderly adults' central processes, such as memory retrieval and decision making, did not demonstrate the typical age deficit because of the skilled nature of these processes in simple arithmetic.     

Interacting neighbors: A connectionist model of retrieval in single-digit multiplication

For most adults, retrieval is the most common way to solve a single-digit multiplication problem (Campbell & Xue, 2001). Many theories have been proposed to describe the underlying mechanism of arithmetical fact retrieval. Testing their validity hinges on evaluating how well they account for the basic findings in mental arithmetic. The most important findings are the problem size effect (small multiplication problems are easier than larger ones; cf. 3 x 2 and 7 x 8), the five effect (problems with 5 are easier than can be accounted for by their size), and the tie effect (problems with identical operands are easier than other problems; cf. 8 x 8 and 8 x 7). We show that all existing theories have difficulties in accounting for one or more of these phenomena A new theory is presented that avoids these difficulties. The basic assumption is that candidate answers to a particular problem are in cooperative/competitive interactions and these interactions favor small, five, and tie problems. The theory is implemented as a connectionist model, and simulation data are described that are in good accord with empirical data.     

When is an odd number not odd? Influence of task rule on the MARC effect for numeric classification

When classifying numbers as odd or even with left-right keypresses, performance is better with the mapping even-right/odd-left than with the opposite mapping. This linguistic markedness association of response codes (MARC) effect has been attributed to compatibility between the linguistic markedness of stimulus and response codes. In 2 experiments participants made keypresses to the Arabic numerals or number words 3, 4, 8, and 9 using the odd-even parity rule or a multiple-of-3 rule, which yield the same keypress response for each stimulus. For both stimulus modes, the MARC effect was obtained with the odd-even rule, but tended to reverse with the multiple-of-3 rule. The reversal was complete for the right response, but task rule had little influence on the left response. The results are consistent with the view that the MARC effect and its reversal are caused by correspondence of the stimulus code designated as positive by the task rule with the positive-polarity right response code.     

The representations of the arithmetic operations include functional relationships

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.     

Starting to add worse: effects of learning to multiply on children's addition

A major stumbling block in acquiring a new skill can be integrating it with old but related knowledge. Learning multiplication is a case in point, because it involves integrating new relations with previously acquired arithmetic knowledge (in particular, addition). Two studies explored developmental changes in the relations between single-digit addition and multiplication. In the first study, third-graders, fifth-graders, and adults performed simple addition or multiplication in mixed- and blocked-operations formats. Substantial interfering effects from related knowledge were found at all age levels, but were more pronounced for younger subjects. Thus in the early stages of learning multiplication, one consequence of learning a new operation is interference in performance of an earlier, related, but less recently studied skill. Consideration of error patterns supported the view that the problem of integrating operations is a prominent one even in the early stages of mastering multiplication. Patterns of errors were generally consistent across all age groups, and all groups were much more likely to give a correct multiplication response to an addition problem than the reverse. A second, longitudinal study confirmed this finding, showing evidence for impaired performance of addition over time within individual children (second-, third-, and fourth-graders) tested on simple addition and multiplication over a 5-month period. Analysis of reaction times for addition indicated that second-graders in advanced math classes and third-graders in regular math classes tended to slow down over the year in responses to addition problems. Fourth-graders, on the other hand, tended to increase their speed of addition over the course of the year. Multiplication showed a different pattern during this period, with no evidence for slowing among children who were able to perform this task. Disruption of previously learned knowledge in the course of acquiring new skills provides evidence that new knowledge and old knowledge are being integrated. This kind of non-monotonic development may provide an empirical method for determining the functional limits of a domain of knowledge.     

A methodological note about the measurement of the false‐consensus effect

Although the existence of the false consensus effect is beyond all doubt, its interpretation as a judgemental bias is still a matter of debate. Krueger recently proposed a new measure for this phenomenon, called the truly false consensus effect (TFCE). This measure consists of correlating the subject's endorsement of each item with the error he or she made in estimating the popularity of that same item. I question the validity of this measure, arguing that it could be negative even in cases where false consensus is at work. I present an inter‐group study where participants made percentage estimates for various behaviours. The results yielded a significantly negative TFCE, although other measures of false consensus demonstrate that the phenomenon is present. I discuss the differences between Krueger's studies and my own, and suggest that a better measure of false consensus might be the particle correlation between endorsement and estimated popularity, controlling for true popularity. Copyright © 2000 John Wiley & Sons, Ltd.     

Are Arithmetic Networks Interdependent In Number-Matching Task?

Addition and multiplication facts are retrieved from a network-like structure, as shown by data from number-matching tasks. Even if several evidences (e.g., cross-operation confusion effect) suggest that these networks are interrelated, the interdependency between addition and multiplication networks could be influenced by the type of task used (e.g., verification task). The present study aimed to investigate whether the addition and multiplication networks were interdependent or separate using a number-matching task. Eighty participants were divided in four groups. The Groups A (x, x, x) and B (+, +, +) performed the task in which only one arithmetic interference effect was implemented through three sessions (pure condition). The Groups C (x, x, +) and D (+, +, x) performed the same task in which the same arithmetic interference effect appeared in the first and second sessions, while a different arithmetic problem was presented in the last session (mixed condition). In the last session, the interference effect in the mixed condition was higher than that in the pure condition. The results argued more for an independency of addition and multiplication networks than for their interdependency.     

When you don't have to be exact: Investigating computational estimation skills with a comparison task

The present study is the first systematic investigation of computational estimation skills of multi-digit multiplication problems using an estimation comparison task. In two experiments, participants judged whether an estimated answer to a multi-digit multiplication problem was larger or smaller than a given reference number. Performance was superior in terms of speed and accuracy for smaller problem sizes, for trials in which the reference numbers were smaller vs. larger than the exact answers (consistent with the size effect) and for trials in which the reference numbers were numerically far compared to close to the exact answers (consistent with the distance effect). Strategy analysis showed that two main strategies were used to solve this task—approximate calculation and sense of magnitude. Most participants reported using the two strategies. Strategy choice was influenced by the distance between the reference number and exact answer, and by the interaction of problem size and reference number size. Theoretical implications as to the nature of numerical representations in the ANS (approximate number system) and to the estimation processes are suggested.     

Conditions of error priming in number-fact retrieval

Analysis of errors in simple multiplication has shown that answers retrieved on previous trials are initially inhibited (negative error priming) but later are promoted as errors to subsequent problems (positive error priming). Two experiments investigated whether error priming is associated either with problem-specific retrieval processes or with representations of answers that can be manipulated independently of problems. In Experiment 1, answers were primed by visually presenting products for 200 msec prior to problems. Correct-answer primes facilitated retrieval, related-incorrect primes interfered with retrieval more than unrelated primes, and both effects were greater for more difficult problems. Primes affected only the trial on which they were presented, however, whereas both negative and positive error priming from previous problems were observed across trials. In Experiment 2, subjects named and retrieved multiplication products on alternating trials. Just-named products were inhibited as errors to the following multiplication problem (i.e., negative error priming), but, compared to positive priming from previous retrieved products, positive error priming from previously named numbers was weak. The results indicate that positive error priming is due mainly to an encoding or retrieval bias produced by previous problems, whereas negative error priming entails suppression, or de-selection, of answer representations.     

Switch costs and the operand-recognition paradigm

Experimental research in cognitive arithmetic frequently relies on participants’ self-reports to discriminate solutions based on direct memory retrieval from use of procedural strategies. Given concerns about the validity and reliability of strategy reports, Thevenot et al. in Mem Cogn 35:1344–1352, (2007) developed the operand-recognition paradigm as an objective measure of arithmetic strategies. Participants performed addition or number comparison on two sequentially presented operands followed by a speeded operand-recognition task. Recognition times increased with problem size following addition but not comparison. Thevenot et al. argued that the complexity of addition strategies increases with problem size. A corresponding increase in operand-recognition time occurs because, as problem size increases, working memory contains more numerical distracters. However, because addition is substantially more difficult than comparison, and difficulty increases with problem size for addition but not comparison, their findings could be due to difficulty-related task-switching costs. We repeated Thevenot et al. (Experiment 1) but added a control condition wherein participants performed a parity (odd or even) task instead of operand recognition. We replicated their findings for operand recognition but found robust, albeit smaller, effects of addition problem size on parity judgements. The results indicate that effects of strategy complexity in the operand-recognition paradigm are confounded with task-switching effects, which complicates its application as a precise measure of strategy complexity in arithmetic.