Cognitive arithmetic: Evidence for obligatory activation of arithmetic facts

In two experiments, obligatory activation of arithmetic facts resulted in interference on a simple number-matching task. Subjects were required to verify the presence of a probe number (e.g., 5) in a previously presented pair (e.g., 5+1). Items for which the probe was the sum of the initial pair (e.g., 5+1 and 6) were rejected more slowly than items for which the probe was not the sum (e.g., 5+1 and 3), and this effect was largest at stimulus onset asynchronies of less than 180 msec between the number pair and the probe. The results are consistent with the notion that arithmetic knowledge is represented in an associative network and accessed by means of spreading activation.     

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).     

Simple arithmetic processing: Individual differences in automaticity

This study investigated individual differences in the ability to automatically access simple addition and multiplication facts from memory. It employed a target-naming task and a priming procedure similar to that utilised in the single word semantic-priming paradigm. In each trial, participants were first presented with a single digit arithmetic problem (e.g., 6+8) and were then presented with a target that was either congruent (e.g., 14) or incongruent (e.g., 17) with this prime. Response times for congruent and incongruent conditions were then compared to a neutral condition (e.g., X+Y, with target 14). For the high skilled group, significant facilitation in naming congruent multiplication and addition targets was found at SOAs of 300 and 1000?ms. In contrast, for the low skilled group, facilitation in naming congruent targets was only observed at 1000?ms. Significant inhibition in naming incongruent multiplication and addition targets at 300?ms, and addition targets at 1000?ms, was found for the high skilled group alone. This advantage in access to simple facts for the high skilled group was then further supported in a problem size analysis that revealed individual differences in access to small and large problems that varied by operation. These findings support the notion that individual differences in arithmetic skill stem from automaticity in solution retrieval and additionally, that they also derive from strategic access to multiplication solutions.     

When it hurts to be misled: A Stroop-like effect in a simple addition production task

In four experiments, subjects saw simple addition equations (e.g., 3 + 4 = 9) and produced the sums while ignoring the presented answer. If the presented answer was false, subjects took longer to produce the sum, as compared with when the presented answer was true (Experiment 1), when there was no answer presented (blanks; Experiment 2), when a letter was presented (Experiment 3), and when a symbol was presented (Experiment 4). The results suggest that subjects were unable to ignore the presented answers, which raises problems for theories of arithmetic verification (i.e., deciding whether 3 + 4 = 9 is true or false) that claim that subjects verify equations by first producing the sum and then comparing the produced sum with the presented answer. Our results are more compatible with theories that claim that in verification and production, an arithmetic knowledge base is used in different ways.     

Capacity and contextual constraints on product activation: Evidence from task-irrelevant fact retrieval

Three experiments tested the limiting conditions of multiplication facts retrieval in a number-matching task (LeFevre, Bisanz, & Mrkonjic, 1988). By presenting two digits as cue and by requiring participants to decide whether a subsequent numerical target had been present in the pair, we found interference when the target coincided with the product of the cue digits. This was evidence for obligatory activation of multiplication facts. Also, we showed that multiplication facts retrieval occurred even in the absence of any arithmetic context (i.e., a multiplication sign between the cue digits) and did not require processing resources (i.e., the process met the capacity criterion of automaticity; Jonides, 1981), whereas manipulation of the spatial relation between the two operands (cue digits) negatively affected retrieval. The present work appears to be unique in the context of previous similar studies on mental calculation, which invariably adopted an arithmetic task as the primary demand. We identify this difference as the reason for the failure of all previous studies in revealing independence of multiplication facts from attentional resources. Furthermore, we suggest the application of a contextual definition of automaticity to this kind of retrieval, given the fact that it might depend both on association strength and on contextual setting variables.     

Processes underlying dimensional interactions: Correspondences between linguistic and nonlinguistic dimensions

In six experiments, we examined speeded classification when one dimension was linguistic and the other was nonlinguistic. In five of these, attributes on the dimensions corresponded meaningfully, having in common the concepts "high" and "low." For example, in Experiment 1, the visually presented words HI and LO were paired with high- or low-pitched tones; in Experiment 2, the dimensions were visual words and vertical position, in Experiment 3, they were spoken words and position, and in Experiments 4 and 5, spoken words and pitch. For each dimension in each pair, subjects suffered Garner interference when dimensions were varied orthogonally. Garner interference remained constant across 15 blocks of trials (Experiment 5). Subjects also showed significant congruity effects in all experiments, with attributes from congruent stimuli (e.g., HI/high pitch) classified faster than attributes from incongruent stimuli (e.g., HI/low pitch). These results differ from those obtained previously with noncorresponding pairs of linguistic-nonlinguistic dimensions. The results also differ from those obtained with traditional Stroop dimensions (colors and color words; Experiment 6), which showed minimal Garner interference and diminishing congruity effects across blocks of trials. We conclude that the interactions found here represent cross-talk between channels within a semantic level of processing. We contrast our view with current models of dimensional interaction.     

The locus of self-inhibition in sequential retrieval

Self-inhibition refers to the suppression of a representational node following its use in a sequential task. Two general models of self-inhibition exist in the literature: one that models suppression following activation of a node (Houghton & Tipper, 1994), and one that models suppression following motor production of a node (MacKay, 1986). These two models make opposite predictions for units that are activated but not produced, with the post-activation model predicting interference and the post-production model predicting facilitation. These predictions were tested using operand--answer priming in two mental addition experiments. Addition problems with sums <10 were primed with addition problems that contained the correct sum as one of the operands (e.g., 5 + 8 preceding 3 + 2). Experiment 2 included an additional manipulation of prime-target discriminability to rule out episodic retrieval accounts of the interference effect. Both experiments indicated interference following operand-answer primes relative to control primes, consistent with the predictions of the post-activation model of self-inhibition.     

Negative priming from activation of counting and addition knowledge

In two experiments, subjects were presented with digit pairs (e.g., 32) and asked to respond to the rightmost number. Negative priming, that is, slowed processing, was evident when the rightmost number was a counting-string (e.g., 43 following 12) or addition-sum (e.g., 65 following 32) associate of the number pair from the preceding trial. The studies are the first to demonstrate negative priming with counting and arithmetical memory representations and suggest the obligatory activation of these representations with the presentation of number pairs. The results are also consistent with the view that negative priming often occurs at the semantic level. Received: 15 February 2000 / Accepted: 8 June 2000     

Retrieval-induced forgetting of arithmetic facts but not rules

Retrieval of a multiplication fact (2×6 =12) can disrupt retrieval of its addition counterpart (2+6=8). We investigated whether this retrieval-induced forgetting effect applies to rule-governed arithmetic facts (i.e., 0×N=0, 1×N=N). Participants (n=40) practised rule-governed multiplication problems (e.g., 1×4, 0×5) and multiplication facts (e.g., 2×3, 4×5) for four blocks and then were tested on the addition counterparts (e.g., 1+4, 0+5, 2+3, 4+5) and control additions. Increased addition response times and errors relative to controls occurred only for problems corresponding to multiplication facts, with no problem-specific effects on addition counterparts of rule-governed multiplications. In contrast, the rule-governed 0+N problems provided evidence of generalisation of practice across items, whereas the fact-based 1+N problems did not. These findings support the theory that elementary arithmetic rules and facts involve distinct memory processes, and confirmed that previous, seemly inconsistent findings of RIF in arithmetic owed to the inclusion or exclusion of rule-governed problems.     

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.     

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.     

Automatic activation of phonological information in reading: Evidence from the semantic relatedness decision task

A semantic relatedness decision task was used to investigate whether phonological recoding occurs automatically and whether it mediates lexical access in visual word recognition and reading. In this task, subjects read a pair of words and decided whether they were related or unrelated in meaning. In Experiment 1, unrelated word-homophone pairs (e.g., LION-BARE) and their visual controls (e.g., LION-BEAN) as well as related word pairs (e.g., FISH-NET) were presented. Homophone pairs were more likely to be judged as related or more slowly rejected as unrelated than their control pairs, suggesting phonological access of word meanings. In Experiment 2, word-pseudohomophone pairs (e.g., TABLE-CHARE) and their visual controls (e.g., TABLE-CHARK) as well as related and unrelated word pairs were presented. Pseudohomophone pairs were more likely to be judged as related or more slowly rejected as unrelated than their control pairs, again suggesting automatic phonological recoding in reading.     

Semantic Competition as the Basis of Stroop Interference:Evidence From Color-Word Matching Tasks

Abstract—A color-word matching task was used to investigate the basis of Stroop interference. Subjects were shown a pair of stimuli: an ink color (e.g., a red bar) and a colored word (e.g., RED printed in red or blue) and decided whether the two items had the same meaning (meaning decisions) or whether they had the same surface color (visual decisions). In Experiment 1, the two stimuli were shown simultaneously, and conflicting visual information of the word (e.g., RED printed in blue, against a red bar) led to interference in meaning decisions, whereas conflicting verbal information (e.g., BLUE printed in red, against a red bar) produced no interference in visual decisions. In Experiment 2, as an increasing time interval was imposed between presentation of the color bar and the colored word, interference in meaning decisions diminished, whereas interference in visual decisions was established. These results suggest that semantic competition, not response competition, is the major source of Stroop and Stroop-like interference.     

Simple arithmetic processing: the question of automaticity

In adult simple arithmetic performance, it is commonly held that retrieval of solutions occurs automatically from a network of stored facts in memory. However, such an account of performance necessarily predicts a uniform reaction time for solution retrieval and is therefore not consistent with the robust finding that reaction time increases with problem size and difficulty. Additionally, past research into arithmetic performance has relied on tasks that may have actually induced and measured attentional processing, thereby possibly confounding previous results and conclusions pertaining to automaticity. The present study therefore, attempted to more reliably assess the influence of automatic processing in arithmetic performance by utilizing a variant of the well-established semantic word-priming procedure with a target-naming task. The overall results revealed significant facilitation in naming times at SOAs of 240 and 1000 ms for congruent targets i.e., targets that represented the correct solutions to problems presented as primes (e.g., 6+8 and 14). Significant inhibition in comparison to a neutral condition (0+0 and 17) was also observed at 120 and 240 ms SOAs in naming incongruent targets (e.g., 6+8 and 17). Furthermore, response times were found to vary as a function of both arithmetic fluency and problem size. Differences in performance to addition and multiplication operations and implications for cognitive research and education are considered.     

Why is 9+7 harder than 2+3? Strength and interference as explanations of the problem-size effect

In four experiments, the problem-size effect was investigated, using an alphabet-arithmetic task in which subjects verified such problems as A + 2 = C. Problem size was manipulated by varying the magnitude of the digit addend (e.g., A + 2, A + 3, and A + 4). The frequency and similarity of problems was also manipulated to determine the contribution of strength and interference, respectively. Experiment 1 manipulated frequency at low levels of practice and found that strength could account for the problem-size effect. Experiment 2 manipulated frequency at higher levels of practice, and found that strength alone could not account for the problem-size effect at asymptote. Experiment 3 manipulated frequency and similarity and found a substantial problem-size effect at asymptote, suggesting that both strength and interference contribute to the problem-size effect. Experiment 4 manipulated similarity, keeping frequency constant, and found no problem-size effect at asymptote, suggesting that interference alone is not responsible for the problem-size effect. The results are related to findings with number arithmetic.     

Activation of number facts in bilinguals

Two experiments examined the effect of the presentation format of numbers—digits versus word format in the first and in the second languages of bilinguals—on mental arithmetic. Speed of number-fact retrieval and the presence of interference produced by numbers that were either numerically close to or associatively related to the correct answers of stored arithmetic problems (e.g., 2+5 and 7×8) were compared across formats. The verification of true problems was increasingly slower and less accurate from the digit condition to the second-language condition. Interference was produced by both types of incorrect answers in the digit and first-language conditions, whereas in the second-language condition, it was constrained to answers that were numerically close to correct answers. Together, the results suggest that the retrieval of arithmetic facts and the automatic spreading of activation within the network of numerical facts are not only language-sensitive, but format-sensitive in general.     

Perceptual processing of nontargets in an attention task

The capability of nontargets to qualitatively influence the semantic processing of coincident targets was investigated in three experiments. Subjects were aurally presented a series of word pairs and attempted to detect homonymic instances of a predesignated category (e.g., animals). The nontarget with which a target (e.g., ANT) was paired was appropriate (e.g., CRAWLING), inappropriate (e.g., UNCLE), or neutral (e.g., STRAW). Experiments 1 and 2 established that detection of targets can be facilitated by appropriate nontargets and inhibited by inappropriate ones. Thus, nontargets can influence the way in which targets are semantically represented. Experiment 3 showed that this effect is eliminated when subjects are precued as to the ear of entry of targets. Thus, precuing appears to curtail the perceptual processing of nontargets. The data run counter to theories that claim that focused attention does not entail the perceptual suppression of nontargets.     

Middle-School Students' Understanding of the Equal Sign: The Books They Read Can't Help

This study examined how 4 middle school textbook series (2 skills-based, 2 Standards-based) present equal signs. Equal signs were often presented in standard operations equals answer contexts (e.g., 3 + 4 = 7) and were rarely presented in nonstandard operations on both sides contexts (e.g., 3 + 4 = 5 + 2). They were, however, presented in other nonstandard contexts (e.g., 7 = 7). Two follow-up experiments showed that students' interpretations of the equal sign depend on the context. The other nonstandard contexts were better than the operations equals answer context at eliciting a relational understanding of the equal sign, but the operations on both sides context was best. Results suggest that textbooks rarely present equal signs in contexts most likely to elicit a relational interpretation—an interpretation critical to success in algebra.     

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.     

Arithmetic split effects reflect strategy selection: an adult age comparative study in addition comparison and verification tasks.

We tested whether split effects in arithmetic (i.e., better performance on large-split problems, like 3 + 8 = 16, than on small-split problems, like 3 + 8 = 12) reflect decision processing or strategy selection. To achieve this end, we tested performance of younger and older adults, matched on arithmetic skills, on two arithmetic tasks: the addition/number comparison task (e.g., 4 + 8, 13; which item is the larger?) and in the inequality verification task (e.g., 4 + 8 < 13; Yes/No?). In both tasks, split between additions and proposed numbers were manipulated. We also manipulated the difficulty of the additions, which represents an index of arithmetic fact calculation (i.e., hard problems, like 6 + 8 < 15, are solved more slowly than easy problems, like 2 + 4 < 07, suggesting that calculation takes longer). Analyses of latencies revealed three main results: First, split effects were of smaller magnitude in older adults compared to younger adults, whatever the type of arithmetic task; second, split effects were of smaller magnitude on easy problems; and third, calculation processes were well maintained in older adults with high level of arithmetic skills. This set of results improves our understanding of cognitive aging and strategy selection in arithmetic.