ArticlePDF Available

The role of associative interference in learning and retrieving arithmetic facts

Authors:
A preview of the PDF is not available
... In cognitive arithmetic, it has been accepted for decades that people solve simple arithmetic problems (e.g., additions and multiplications with one-digit operands) through automatic retrieval of arithmetic facts from memory (e.g., network interference model, Campbell, 1987;Campbell et al., 2016;Chen and Campbell, 2016). Thus, in order to solve a simple math problem (e.g., 4 + 2, 3 × 2), an individual would retrieve the solution (e.g., 6) from semantic memory. ...
... Specifically, if medium small problems are resolved by retrieval and additions 1-4 by procedures (Barrouillet and Thevenot, 2013;Uittenhove et al., 2016), N400 amplitudes would be smaller for the first type of additions relative to additions 1-4. On the contrary, if all simple additions were resolved by memory retrieval (e.g., Campbell, 1987), there would be no differences in the N400 amplitude between these types of additions because all of them would imply access to semantic information (the retrieval of arithmetic facts). However, from the retrieval from memory view, LP modulations would be predicted with a larger positivity for medium small additions (control additions) than for small additions (additions 1-4). ...
... Specifically, if additions 1-4 involve counting and control additions memory retrieval (Barrouillet and Thevenot, 2013;Uittenhove et al., 2016), N400 modulations would be expected with smaller amplitudes for control additions vs. additions 1-4 because the semantic processing would be easier with medium small additions (retrieval of arithmetic facts) than with very small additions (counting procedures). Conversely, if all simple additions involve semantic access by retrieving arithmetic facts (Campbell, 1987), no N400 but LP modulations would be obtained. Thus, the type of problem would modulate the LP component with greater amplitude for control additions compared to additions 1-4, because retrieval from memory is difficult with medium small problems (control additions) than with small problems (additions 1-4). ...
Article
Full-text available
There is current debate about the way adult individuals solve simple additions composed of one-digit operands. There are two opposing views. The first view assumes that people retrieve the result of additions from memory, whilst the second view states that individuals use automatized counting procedures. Our study aimed to dissociate between these two hypotheses. To this end, we analysed the type of problem effect when participants resolved simple additions by comparing additions with operands between 1 and 4 and control additions with at least one operand larger than 4. Brain-waves activity of a group of 30 adult individuals were recorded with 64 scalp electrodes mounted on an elastic cap, referenced against an electrode between Cz and CPz and re-referenced to an average reference offline. We considered two electrophysiological indexes, event-related potentials, ERPs, time-locked to the addition problems to distinguish between retrieval from memory and the use of procedures: A late positivity component (LP, 500-650 time window) over posterior regions associated to memory retrieval difficulty with higher LP positivity when participants resolve difficult vs. easy additions, and a negative component (N400, 250-450 ms time window) over fronto-central regions related to the use memory retrieval vs. procedures with more pronounced N400 amplitudes when the difficulty in the retrieval of semantic information increased. LP modulations were observed depending on the type of problem over posterior regions, P3 and Pz electrodes, whilst the N400 component was not affected. This pattern of results suggests that adult individuals use retrieval from memory to solve simple additions.
... Although robust evidence indicates that non-human animals share a capacity for numerical representation and comparison with humans, less is known about the evolutionary foundations of arithmetic (but see Beran 2004;Beran and Beran 2004;Cantlon and Brannon 2007a, b;Pica et al. 2004). In particular, there are three classic psychological signatures of human arithmetic that have sometimes been described as unique to human arithmetic: the problem size effect, the tie effect, and practice effects (e.g., Ashcraft and Battaglia 1978;Campbell 1987;Campbell and Graham 1985;Geary 1996;Siegler 1987;Zbrodoff and Logan 2005). The problem size effect represents a systematic decline in accuracy and response time as the magnitude of the operands in an arithmetic problem increase (e.g., 5 ? ...
... For three decades, the main interpretations of the problem size, practice, and tie effects have hinged on the assumption that basic arithmetic problems and outcomes are memorized in a symbolic format in humans (e.g., Ashcraft and Battaglia 1978;Campbell 1987;Geary 1996;Siegler 1987;Zbrodoff and Logan 2005). One interpretation of the problem size effect, for example, is that it emerges from the spreading of activation in a semantic network of precisely memorized arithmetic operands and outcomes in which small numbers are represented more strongly than large numbers (e.g., Widaman et al. 1992). ...
Article
Full-text available
Non-human primates compare quantities in a crude manner, by approximating their values. Less is known about the mental transformations that non-humans can perform over approximate quantities, such as arithmetic transformations. There is evidence that human symbolic arithmetic has a deep psychological connection with the primitive, approximate forms of quantification of non-human animals. Here, we ask whether the subtle performance signatures that humans exhibit during symbolic arithmetic also bear a connection to primitive arithmetic. Specifically, we examined the problem size effect, the tie effect, and the practice effect-effects which are commonly observed in children's math performance in school. We show that, like humans, monkeys exhibited the problem size and tie effects, indicating commonalities in arithmetic algorithms with humans. Unlike humans, however, monkeys did not exhibit a practice effect. Together, these findings provide new evidence for a cognitive relation between non-symbolic and symbolic arithmetic.
... The strength with which nodes were stored and interconnected is assumed to be a function of frequency of occurrence and practice. Campbell (1987;Campbell & Craham, 1985) focused on interference as a critical part of the retrieval process. According to Campbell individual multiplication problems activate a network substructure of candidate responses. ...
... In error priming, response by a prior retrieval promotes errors and slows the correct reaction time of subsequent problems that have a relatively high probability of generating that product as an error. According to this interference model the problem-size effect is also due substantially to a process of associative interference (Campbell, 1987). Small problems (sums smaller than 10) are practised more frequently and thus the strength of the association is greater and less susceptible to interference. ...
... Second, the PSE might be caused by differences in memory retrieval. Campbell's network interference hypothesis (Campbell, 1987;Campbell & Graham, 1985) suggests that retrieval of large problems involves increased interference from competing associations, resulting from weaker associations between operands and correct solutions. Third, individuals tend to rely on more effortful decomposition or counting strategies for solving large problems compared to small problems that typically rely on direct retrieval (Campbell & Fugelsang, 2001;Campbell & Timm, 2000;Campbell & Xue, 2001;Hecht, 1999Hecht, , 2002Siegler, 1987). ...
Article
Full-text available
The problem size effect (PSE) is defined by better performance solving small problems (e.g., 2 × 4) than large problems (e.g., 8 × 9). For monolinguals, the PSE is larger when problems are presented in unfamiliar formats (e.g., written words), reflecting increased processing difficulty. Bilinguals are typically faster and more accurate at retrieving multiplication facts in the language of learning (LA+) than in their other language (LA−). We hypothesized that the less familiar arithmetic language (i.e., LA−) would elicit larger PSEs than LA+. Here, fluent Spanish–English bilingual adults verified spoken multiplication problems presented in LA+ and LA− while event-related potentials (ERPs) were recorded (Experiment 1A). To further promote language differences, we increased task difficulty by presenting problems at a faster pace (Experiment 1B) and requiring bilinguals to verbally produce solutions (Experiment 2). Language differences in performance were only observed for Experiment 2, where solutions were produced more slowly in LA− than LA+. In the ERPs, a PSE was driven by larger P300s for small than large solutions. A language effect was only observed under time pressure where LA− elicited a PSE at the second operand. Additionally, the PSE was smaller for LA− at the solution. This suggests that categorizing multiplication facts is more effortful in LA−. In sum, very subtle language differences arise in fluent bilinguals when problems are more difficult, such as larger problems presented under time pressure in a weaker language. Critically, the effect of LA+ is at the level of response production and not access to the facts from memory.
... Furthermore, cognitive control processes are also important for retrieving arithmetic facts, as in the arithmetic fact network not only the correct but also associated wrong answers are activated and need to be inhibited (e.g. Barrouillet et al., 1997;Campbell, 1987;Campbell and Graham, 1985). Thus, cognitive control processes are involved in both, knowledge acquisition and retrieval. ...
... Furthermore, cognitive control processes are also important for retrieving arithmetic facts, as in the arithmetic fact network not only the correct but also associated wrong answers are activated and need to be inhibited (e.g. Barrouillet et al., 1997;Campbell, 1987;Campbell and Graham, 1985). Thus, cognitive control processes are involved in both, knowledge acquisition and retrieval. ...
Article
Full-text available
Over the last decades, interest in transcranial electrical stimulation (tES) has grown, as it might allow for causal investigations of the associations between cortical activity and cognition as well as to directly influence cognitive performance. The main objectives of the present work were to assess whether tES can enhance the acquisition and application of arithmetic abilities, and whether it enables a better assessment of underlying neurophysiological processes. To this end, the present, double-blind, sham-controlled study assessed the effects of six active stimulations (three tES protocols: anodal transcranial direct current stimulation (tDCS), alpha band transcranial alternating current stimulation (tACS), and theta band tACS; targeting the left dorsolateral prefrontal cortex or the left posterior parietal cortex) on the acquisition of an arithmetic procedure, arithmetic facts, and event-related synchronization/desynchronization (ERS/ERD) patterns. 137 healthy adults were randomly assigned to one of seven groups, each receiving one of the tES-protocols during learning. Results showed that frontal theta band tACS reduced the repetitions needed to learn novel facts and both, frontal and parietal theta band tACS accelerated the decrease in calculation times in fact learning problems. The beneficial effect of frontal theta band tACS may reflect enhanced executive functions, allowing for better control and inhibition processes and hence, a faster acquisition and integration of novel fact knowledge. However, there were no significant effects of the stimulations on procedural learning or ERS/ERD patterns. Overall, theta band tACS appears promising as a support for arithmetic fact training, but effects on procedural calculations and neurophysiological processes remain ambiguous.
... With ready access to basic math facts, students are less likely to be overwhelmed by the amount of cognitive engagement problems often require and will be better positioned to solve more complex problems (Sweller, 1994). Several other studies (Ashcraft, 1992;Campbell, 1987;Compton & Logan, 1991) assert that remembering memorized math facts quickly will free cognitive capacity and result in greater success with more complex problems. ...
Article
Full-text available
While conceptual understanding of properties, operations, and the base-ten number system is certainly associated with the ability to access math facts fluently, the role of math fact memorization to promote conceptual understanding remains contested. In order to gain insight into this question, this study looks at the results when one of three elementary schools in a school district implements mandatory automaticity drills for 10 minutes each day while the remaining two elementary schools, with the same curriculum and very similar demographics, do not. This study looks at (a) the impact that schoolwide implementation of automaticity drills has on schoolwide computational math skills as measured by the ITBS and (b) the relationship between automaticity and conceptual understanding as measured by statewide standardized testing. The results suggest that while there may be an association between automaticity and higher performance on standardized tests, caution should be taken before assuming there are benefits to promoting automaticity drills. These results are consistent with those that support a process-driven approach to automaticity based on familiarity with properties and strategies associated with the base-ten number system; they are not consistent with those that support an answer-driven approach to automaticity based on memorization of answers.
... Direct-facts and indirect-facts approaches are based on the assumption that memorizing a basic fact entails the relatively simple process of forming and strengthening an association between an expression and its answer and that practice is the basis for increasing associative strength (Ashcraft, 1992), or at least, the most important factor in this process (Siegler, 1987;Torgesen & Young, 1983). Large doses of practice are justified by Thorndike's (1922) law of frequency: The more two stimuli are presented together, the stronger their association (see also Ashcraft, 1992;Campbell, 1987;Logan, 1991;Siegler & Shipley, 1995). For example, according to the distribution-of-associations model (Siegler & Jenkins, 1989) and its successors (Shrager & Siegler, 1998;Siegler & Araya, 2005), a memory trace is laid down each time an item is practiced, and thousands of such traces are necessary to achieve efficient fact recall. ...
Article
Full-text available
How best to promote fluency with basic sums and differences is still not entirely clear. Some advocate a direct approach-using drill to foster memorization of basic facts by rote. Others recommend an indirect approach that first involves learning reasoning strategies. The purpose of the present study was to evaluate the efficacy of 2 computer-based interventions that embody an indirect approach by highlighting the conceptual bases for 2 relatively difficult reasoning strategies: subtraction as addition (e.g., for 8 - 5, think: "What plus 5 equals 8?") and use-10 (e.g., "If 10 + 5 = 15 and 9 is 1 less than 10, then 9 + 5 is 1 less than 15"). After pretest, 85 Grade K-3 students were randomly assigned to subtraction, use-10, or drill conditions. The subtraction and use-10 conditions served as an active control and represented regular classroom instruction for each other; the drill condition controlled for the effect of extra practice and represented the direct approach. Each intervention involved 30-min sessions, twice weekly, for 12 weeks. Using pretest scores, mathematics achievement, and age as covariates, mixed-model analyses of covariance revealed that, at posttest at least 2 weeks later, the subtraction group outperformed both comparison groups on progress toward fluency and fluency rate with unpracticed subtraction items. The use-10 group achieved analogous results with unpracticed add-with-8 or -9 combinations. This transfer suggests that the conceptually based indirect programs were efficacious and more successful than regular classroom instruction or the direct approach in promoting progress toward fluency and fluently itself. (PsycINFO Database Record
... This model has received strong support from several studies (Barrouillet & Fayol, 1998;Campbell & Timm, 2000;Geary & Brown, 1991;Geary & Burlingham-Dubree, 1989;Hamann & Ashcraft, 1986;Imbo & Vandierendonck, 2007;Imbo & Vandierendonck, 2008;Reder, 1988) and has provided a theoretical basis to the recurrent observation that adults retrieve from memory the answer of small additions instead of having to calculate it (Ashcraft, 1982;Ashcraft, 1987;Ashcraft & Battaglia, 1978;Ashcraft & Stazyk, 1981;Barrouillet & Fayol, 1998;Campbell, 1987a;Campbell, 1987b;LeFevre, Sadesky, & Bisanz, 1996;Miller, Perlmutter, & Keating, 1984). Thus, it is almost universally admitted that small additions have so often been encountered that their answer is necessarily retrieved from memory in adults (see Zbrodoff & Logan, 2005, for a review). ...
Article
Full-text available
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 × 4 = 36), whereas other facts are stored in the form of a general rule (e.g., 0 × 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.
Article
Full-text available
The development of mental arithmetic is approached from a mathematical perspective, focusing on several process models of arithmetic performance which have grown out of the chronometric methods of cognitive psychology. These models, based on hypotheses about the nature of underlying mental operations and structures in arithmetic, generate quantitative predictions about reaction time performance. A review of the research suggests a developmental trend in the mastery of arithmetic knowledge—there is an initial reliance on procedural knowledge and methods such as counting which is followed by a gradual shift to retrieval from a network representation of arithmetic facts. A descriptive model of these mental structures and processes is presented, and quantitative predictions about children's arithmetic performance at various stages of mastery are considered.
Article
Full-text available
Considers a number of models that specify how children and adults solve single-digit addition problems. It is shown that the most adequate of these for children's response latencies is a model that assumes the existence of a counter with 2 operations: setting and incrementing. The child adds 2 digits, m and n, by setting this counter to max (m,n) and then incrementing it min (m,n) times. This model also accounts for adults' latencies, though with a drastically reduced incrementing time. Some theoretical issues raised by this reduced time are considered, and an alternative model is suggested which assumes that adults usually use a memory look-up process with homogeneous retrieval times, but occasionally revert back to the counting process used by children. (2l ref.) (PsycINFO Database Record (c) 2012 APA, all rights reserved)
Article
Full-text available
Collected response time (RT) and error data on multiplication problems up to 9 × 9 from 86 Ss in Grades 2–5 and from 60 undergraduate and graduate students. Results show that most errors involved correct products to other problems and that a developmental trend emerged in which the specific errors made by children mirrored adult errors by Grade 5. The error patterns indicate that an associative network evolves in which problem operands become linked to specific sets of candidate answers. Retrieval is governed by a process that activates candidates, and accessibility of correct answers is impeded by competing associations: At all skill levels, both problem-error rates and product-error rates (i.e., how often a problem's correct product occurs as an incorrect response to other problems) contributed to predicting correct problem RT in multiple regression analyses. These interference variables yielded higher correlations than did structural variables (e.g., the numerical size of problem operands), the latter having provided the basis for previous models of arithmetic memory. A network-interference account is proposed that explains the slow course of acquisition and differential problem difficulty in terms of interference by false associations. (French abstract) (37 ref) (PsycINFO Database Record (c) 2012 APA, all rights reserved)
Article
Full-text available
Evaluated 40 undergraduates' performance on a simple mental multiplication task and the adequacy of several different models of mental addition as extended to multiplication. Exps I and II revealed that the performance of multiplication resembles simple mental addition, showing similar effects of problem size and of split (the numerical difference between a stated and correct answer). Exp III included a special manipulation of "confusion products," incorrect answers that were multiples of one of the problem's digits. Consistent with the assumption of interrelated network storage, confusion products slowed RT significantly, even when the problem was presented 600 msec in advance of the answer. Results are discussed in terms of a network-retrieval approach to mental arithmetic, the commonalities between addition and multiplication, and rule- vs retrieval-based performance. (26 ref) (PsycINFO Database Record (c) 2012 APA, all rights reserved)
Article
Full-text available
Showed 1 female and 5 male undergraduates a series of simple multiplication problems of the form p * q = r for 6 sessions. Ss were asked to respond rapidly whether the solution given with each problem (r) was correct or not. On 1/2 the trials, the presented solution equaled the true product, and on the remaining trials, the presented solution differed from the true product in a systematic way. Additive multiplication- and comparison-stage RT effects were found for correct responses. For the multiplication stage, latencies increased monotonically as a function of the sum of p and q and as a function of min (p,q), excluding problems where p or q equaled zero. For the comparison stage, negative responses had longer latencies than did positive responses. Results are very similar to those from an earlier study of simple addition. Some theoretical implications of this similarity are explored. (PsycINFO Database Record (c) 2012 APA, all rights reserved)
Article
Full-text available
Performed 2 experiments involving a recall task and a recognition test in classic paired-associate paradigms. 80 undergraduates were Ss. It was shown that, even when interference conditions were equated to control conditions in percent recall by extra study trials, there was an RT deficit in the interference condition. The only theories compatible with these data are those that assume probability of recall is affected by encoding and retrieval factors, but that RT is only affected by retrieval factors. Further, interference must affect retrieval more than encoding. The author's (1976) theory is shown to be compatible with these results at both qualitative and quantitative levels. That theory proposes that traces are formed in an all-or-none manner but that their retrieval depends on a continuous strength that is subject to interference. Probability correctly reflects both the all-or-none encoding and strength-determined retrieval, whereas latency is affected only by retrieval. (32 ref) (PsycINFO Database Record (c) 2012 APA, all rights reserved)
Article
Full-text available
Defines the notion of strength in several alternative ways for chains of associations connected in series and in parallel. Network strength theory is extended to handle retrieval dynamics for a network of associations, in a manner that permits various degrees of serial vs parallel processing through a chain of serially ordered associations. In the parallel-processing version, a speedy activation pulse passes through the chain and initiates a relatively slower retrieval process virtually simultaneously at each link. It is demonstrated that under many conditions, the theory yields the same storage and retrieval dynamics for a network as for any component association. The theory is applied to recall and recognition, semantic memory, speech recognition, and reading. (44 ref) (PsycINFO Database Record (c) 2012 APA, all rights reserved)
Article
We respond to A. Baroody's comment (1984, Developmental Review, 4, 148–156) with an empirical comparison of the production and verification tasks. With the exception of performance at the first grade level, the two tasks yield essentially identical conclusions. The results of an adjunct task, in which the rate of mental counting was assessed, suggest that children as young as second grade are relying on memory retrieval to a significant degree. In contrast to Baroody's speculation, there appear to be no widespread difficulties associated with results from the verification task. Furthermore, the task permits a more analytic examination of performance and underlying mental process than is afforded by the production task. We conclude by reiterating the empirical support for a model of fact retrieval, and suggesting that accessibility of the arithmetic facts is the basic factor which underlies both fact retrieval and procedural knowledge performance.