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.     

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.     

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.     

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.     

Individual differences in cognitive arithmetic

Unities in the processes involved in solving arithmetic problems of varying operations have been suggested by studies that have used both factor-analytic and information-processing methods. We designed the present study to investigate the convergence of mental processes assessed by paper-and-pencil measures defining the Numerical Facility factor and component processes for cognitive arithmetic identified by using chronometric techniques. A sample of 100 undergraduate students responded to 320 arithmetic problems in a true-false reaction-time (RT) verification paradigm and were administered a battery of ability measures spanning Numerical Facility, Perceptual Speed, and Spatial Relations factors. The 320 cognitive arithmetic problems comprised 80 problems of each of four types: simple addition, complex addition, simple multiplication, and complex multiplication. The information-processing results indicated that regression models that included a structural variable consistent with memory network retrieval of arithmetic facts were the best predictors of RT to each of the four types of arithmetic problems. The results also verified the effects of other elementary processes that are involved in the mental solving of arithmetic problems, including encoding of single digits and carrying to the next column for complex problems. The relation between process components and ability measures was examined by means of structural equation modeling. The final structural model revealed a strong direct relation between a factor subsuming efficiency of retrieval of arithmetic facts and of executing the carry operation and the traditional Numerical Facility factor. Furthermore, a moderate direct relation between a factor subsuming speed of encoding digits and decision and response times and the traditional Perceptual Speed factor was also found. No relation between structural variables representing cognitive arithmetic component processes and ability measures spanning the Spatial Relations factor was found. Results of the structural modeling support the conclusion that information retrieval from a network of arithmetic facts and execution of the carry operation are elementary component processes involved uniquely in the mental solving of arithmetic problems. Furthermore, individual differences in the speed of executing these two elementary component processes appear to underlie individual differences on ability measures that traditionally span the Numerical Facility factor.(ABSTRACT TRUNCATED AT 400 WORDS)     

Strategy use,the development of automaticity,and working memory involvement in complex multiplication

Participants practiced a set of complex multiplication problems (e.g., 3 x 18) in a pre-/postpractice design. Before, during, and after practice, the participants gave self-reports of problem-solving strategies. At prepractice, the most common strategy was a mental version of the standard multidigit algorithm, and dual tasks revealed that working memory load was high and heavier for problems solved via nonretrieval strategies. After practice, retrieval was used almost exclusively, and participant variability, automaticity level of problems (proportion of trials on which retrieval was used over the entire experiment), and error rates were significant predictors of problem-solving latencies. Practice reduced working memory involvement to minimal levels, and there was no relationship between automaticity level and working memory load. The commonalities between the present findings and findings related to automaticity development in simple arithmetic are discussed.     

Do multiplication and division strategies rely on executive and phonological working memory resources?

The role of executive and phonological working memory resources in simple arithmetic was investigated in two experiments. Participants had to solve simple multiplication problems (e.g., 4 x 8; Experiment 1) or simple division problems (e.g., 42 / 7; Experiment 2) under no-load, phonological-load, and executive-load conditions. The choice/no-choice method was used to investigate strategy execution and strategy selection independently. Results for strategy execution showed that executive working memory resources were involved in direct memory retrieval of both multiplication and division facts. Executive working memory resources were also involved in the use of nonretrieval strategies. Phonological working memory resources, on the other hand, tended to be involved in nonretrieval strategies only. Results for strategy selection showed no effects of working memory load. Finally, correlation analyses showed that both strategy execution and strategy selection correlated with individual-difference variables, such as gender, math anxiety, associative strength, calculator use, arithmetic skill, and math experience.     

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.     

Musical space synesthesia: Automatic,explicit and conceptual connections between musical stimuli and space

In musical–space synesthesia, musical pitches are perceived as having a spatially defined array. Previous studies showed that symbolic inducers (e.g., numbers, months) can modulate response according to the inducer’s relative position on the synesthetic spatial form. In the current study we tested two musical–space synesthetes and a group of matched controls on three different tasks: musical–space mapping, spatial cue detection and a spatial Stroop-like task. In the free mapping task, both synesthetes exhibited a diagonal organization of musical pitch tones rising from bottom left to the top right. This organization was found to be consistent over time. In the subsequent tasks, synesthetes were asked to ignore an auditory or visually presented musical pitch (irrelevant information) and respond to a visual target (i.e., an asterisk) on the screen (relevant information). Compatibility between musical pitch and the target’s spatial location was manipulated to be compatible or incompatible with the synesthetes’ spatial representations. In the spatial cue detection task participants had to press the space key immediately upon detecting the target. In the Stroop-like task, they had to reach the target by using a mouse cursor. In both tasks, synesthetes’ performance was modulated by the compatibility between irrelevant and relevant spatial information. Specifically, the target’s spatial location conflicted with the spatial information triggered by the irrelevant musical stimulus. These results reveal that for musical–space synesthetes, musical information automatically orients attention according to their specific spatial musical-forms. The present study demonstrates the genuineness of musical–space synesthesia by revealing its two hallmarks—automaticity and consistency. In addition, our results challenge previous findings regarding an implicit vertical representation for pitch tones in non-synesthete musicians.     

Cue Combinatorics in Memory Retrieval for Anaphora

Many studies have shown that memory retrieval for real‐time language processing relies on a cue‐based access mechanism, which allows the cues available at the retrieval site to directly access the target representation in memory. An open question is how different types of cues are combined at retrieval to create a single retrieval probe (“cue combinatorics”). This study addresses this question by testing whether retrieval for antecedent‐reflexive dependencies combines cues in a linear (i.e., additive) or nonlinear (i.e., multiplicative) fashion. Results from computational simulations and a reading time experiment show that target items that match all the cues of the reflexive are favored more than target items that mismatch these cues, and that different degrees of mismatches slow reading times in comparable amounts. This profile is consistent with the predictions of a nonlinear cue combination and provides evidence against models in which all cues combine in a linear fashion. A follow‐up set of simulations shows that a nonlinear rule also captures previous demonstrations of interference from nontarget items during retrieval for reflexive licensing. Taken together, these results shed new light on how different types of cues combine at the retrieval site and reveal how the method of cue combination impacts the accessibility of linguistic information in memory.     

Bidirectional links in the network of multiplication facts

In three experiments, we tested the hypothesis that activation of multiplication operand nodes (e.g., 3 and 8) can occur through presentation of their product (e.g., 24). In Experiments 1 and 2 we found activation of the operands when the product was presented as a cue in a number-matching task. In Experiment 3, activation also occurred in a parity-matching task, where the product (24) was not relevant to the parity matching on its operands (3 and 8). We concluded that bidirectional links exist among the operands and their product for multiplication problems and these links can be activated in a purely stimulus-driven manner. We suggest this may constitute the basis for the solution of simple divisions by mediation through the complementary multiplication facts.     

Making remembering more memorable*

Memory retrieval is a cognitive operation that itself can be remembered or forgotten, with potentially important consequences. To study memory for prior remembering, we had participants first study target words (e.g., bark) alongside semantically related cue words (e.g., dog). Then, on Test 1, participants retrieved targets in response to either the study cue or a changed cue that was semantically related to a homograph of the target (e.g., birch). Finally, on Test 2, participants retrieved all targets in response to the original study cues, and participants judged whether targets were previously retrieved on Test 1. As in previous research, cue change on Test 1 rendered target retrievals less memorable, suggesting context changes harm memory for prior remembering. We hypothesised that the negative effect of context change could be ameliorated by reminding participants of the original study cues during Test 1. We had participants either retrieve (Experiments 1 and 3, Ns?=?46 and 62) or view (Experiment 2, N?=?118) the study cue following each target retrieval. Reminding significantly reduced the negative effect of cue change, with self-generation being especially potent. This indicates that reminding can make remembering more memorable in the face of context change.     

Cognitive representations and processes in arithmetic: inferences from the performance of brain-damaged subjects.

In this article, we present data from two brain-damaged patients with calculation impairments in support of claims about the cognitive mechanisms underlying simple arithmetic performance. We first present a model of the functional architecture of the cognitive calculation system based on previous research. We then elaborate this architecture through detailed examination of the patterns of spared and impaired performance of the two patients. From the patients' performance we make the following theoretical claims: that some arithmetic facts are stored in the form of individual fact representations (e.g., 9 x 4 = 36), whereas other facts are stored in the form of a general rule (e.g., 0 x N = 0); that arithmetic fact retrieval is mediated by abstract internal representations that are independent of the form in which problems are presented or responses are given; that arithmetic facts and calculation procedures are functionally independent; and that calculation algorithms may include special-case procedures that function to increase the speed or efficiency of problem solving. We conclude with a discussion of several more general issues relevant to the reported research.     

Retrieval-induced forgetting of arithmetic facts

Retrieval-induced forgetting (RIF) is a widely studied phenomenon of human memory, but RIF of arithmetic facts remains relatively unexplored. In 2 experiments, we investigated RIF of simple addition facts (2 + 3 = 5) from practice of their multiplication counterparts (2 × 3 = 6). In both experiments, robust RIF expressed in response times occurred only for high-strength small-number addition facts with sums ≤ 10, indicating that RIF from multiplication practice was interference dependent. RIF of addition-fact memory was produced by multiplication retrieval (2 × 3 = ?) but not multiplication study (2 × 3 = 6), supporting an inhibitory mechanism of RIF in arithmetic memory. Finally, RIF occurred with multiplication practiced in word format (three × four) and addition tested later in digit format (3 + 4), which provides evidence that digit and written-word formats for arithmetic accessed a common semantic retrieval network. The results support the view that addition and multiplication facts are stored in an interrelated semantic network and that RIF of competing addition facts is an intrinsic process of multiplication fact retrieval.     

Prospective memory: Are preparatory attentional processes necessary for a single focal cue?

The preparatory attentional and memory processes theory of prospective memory (PM) assumes that PM retrieval requires resource-demanding preparatory attentional processes, whereas the multiprocess theory assumes that retrieval can also occur spontaneously. On the basis of showing slowing on an ongoing task (i.e., task interference) —even when the PM cue was highly salient (i.e., the participant’s own name) —Smith, Hunt, McVay, and McConnell (2007) concluded that preparatory attentional processes are always necessary for PM retrieval. We argue that the presence of preparatory attentional processes cannot be used to rule out the existence of spontaneous retrieval processes, and the goal of the present research was to examine whether PM retrieval can occur in the absence of preparatory attentional processes. We varied whether we emphasized the importance of the PM task or the ongoing task, and we assessed task interference across quarters of the ongoing task. Our results showed no evidence of task interference and, hence, no evidence of preparatory attentional processes in the periods proximal to the target event, and yet participants showed high PM performance. Thus, the results suggest the existence of spontaneous retrieval processes and support the multiprocess theory.     

How the mind shapes action: Offline contexts modulate involuntary episodic retrieval

Involuntary retrieval of previous stimulus–response episodes is a centerpiece of many theories of priming, episodic binding, and action control. Typically it is assumed that by repeating a stimulus from trial n–1 to trial n, involuntary retrieval is triggered in a nearly automatic fashion, facilitating (or interfering with) the to-be-executed action. Here we argue that changes in the offline context weaken the involuntary retrieval of previous episodes (the offline context is defined to be the information presented before or after the focal stimulus). In four conditions differing in cue modality and target modality, retrieval was diminished if participants changed the target selection criterion (as indicated by a cue presented before the selection took place) while they still performed the same task. Thus, solely through changes in the offline context (cue or selection criterion), involuntary retrieval can be weakened in an effective way.     

Adults' strategies for simple addition and multiplication: verbal self-reports and the operand recognition paradigm

Accurate measurement of cognitive strategies is important in diverse areas of psychological research. Strategy self-reports are a common measure, but C. Thevenot, M. Fanget, and M. Fayol (2007) proposed a more objective method to distinguish different strategies in the context of mental arithmetic. In their operand recognition paradigm, speed of recognition memory for problem operands after solving a problem indexes strategy (e.g., direct memory retrieval vs. a procedural strategy). Here, in 2 experiments, operand recognition time was the same following simple addition or multiplication, but, consistent with a wide variety of previous research, strategy reports indicated much greater use of procedures (e.g., counting) for addition than multiplication. Operation, problem size (e.g., 2 + 3 vs. 8 + 9), and operand format (digits vs. words) had interactive effects on reported procedure use that were not reflected in recognition performance. Regression analyses suggested that recognition time was influenced at least as much by the relative difficulty of the preceding problem as by the strategy used. The findings indicate that the operand recognition paradigm is not a reliable substitute for strategy reports and highlight the potential impact of difficulty-related carryover effects in sequential cognitive tasks.     

Cognitive load causes kinematic changes in both elite and non-elite rowers

The current motor literature suggests that extraneous cognitive load may affect performance and kinematics in a primary motor task. A common response to increased cognitive demand, as observed in past studies, might be to reduce movement complexity and revert to previously learned movement patterns, in line with the progression-regression hypothesis. However, according to several accounts of automaticity, motor experts should be able to cope with dual task demands without detriment to their performance and kinematics. To test this, we conducted an experiment asking elite and non-elite rowers to use a rowing ergometer under conditions of varying task load. We employed single-task conditions with low cognitive load (i.e., rowing only) and dual-task conditions with high cognitive load (i.e., rowing and solving arithmetic problems). The results of the cognitive load manipulations were mostly in line with our hypotheses. Overall, participants reduced movement complexity, for example by reverting towards tighter coupling of kinematic events, in their dual-task performance as compared to single-task performance. The between-group kinematic differences were less clear. In contradiction to our hypotheses, we found no significant interaction between skill level and cognitive load, suggesting that the rowers' kinematics were affected by cognitive load irrespective of skill level. Overall, our findings contradict several past findings and automaticity theories, and suggest that attentional resources are required for optimal sports performance.     

Facts, rules, and procedures in normal calculation: evidence from multiple single-patient studies of impaired arithmetic fact retrieval.

This article presents results from multiple single-case studies of brain-damaged patients with impairments in retrieval of arithmetic facts (i.e., "table" facts such as 8 x 7 = 56). The results provide a basis for exploring the types of knowledge implicated in simple arithmetic performance, the internal representations for the various knowledge types, the processes operating upon these representations, and the ways in which the representations or processes may be disrupted by brain damage.     

An evaluation of sex and cultural differences in arithmetic retrieval-induced forgetting

Retrieval practice of arithmetic facts (e.g. 2?×?3) can interfere with retrieval of other, closely related arithmetic facts (e.g. 2?+?3), increasing response time (RT) and errors for these problems. Here we examined potential sex and culture-related differences in arithmetic retrieval-induced forgetting (RIF). This was motivated by re-analyses of several published arithmetic RIF data sets that appeared to show that the effect occurred for women but not men. Experiment 1 (n?=?72) tested for possible sex differences in a diverse but predominantly Canadian university sample. Experiment 2 (n?=?48) examined potential sex differences in native Chinese participants, which previous research indicated may not be susceptible to the RIF effect for a particular subset of small addition problems (sum?≤?10). In Experiment 1, we found no evidence that the addition RIF effect differed between male and female adults. In Experiment 2, the Chinese adults showed RIF for tie problems (e.g. 2?+?2, 3?+?3, etc.) regardless of sex, but neither sex presented RIF for small non-tie addition problems. The results indicated that the RIF effect is not gender specific, and there might not be strong memory retrieval competition between addition and multiplication facts for non-tie problems in Chinese adults.