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Interactive Effects of Numerical Surface Form and Operand Parity in Cognitive Arithmetic
Journal of Experimental Psychology: Learning, Memory, and Cognition
Abstract
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.
... To study encoding and calculation, we relied upon two specific empirical effects found throughout the literature. First, a common signature of the calculation process is the problem size effect, which is the finding that response times and errors increase as problem operands grow in magnitude (Ashcraft, 1992;Campbell, Parker, & Doetzel, 2004;Groen & Parkman, 1972). For example, the simple addition problem 1+1 is typically solved more quickly and accurately than 8+9. ...
... A second class of models is characterized by an interaction between the encoding and calculation processes. That is, the processes involved in calculation directly depend upon the format in which stimuli are encoded (Campbell & Alberts, 2009;Campbell, Parker, & Doetzel, 2004). To illustrate, think back to the lunch bill example presented earlier. ...
... Given that the effects of problem size and format are strong in the literature (Ashcraft, 1992;Campbell, 1994;Campbell, Parker, & Doetzel, 2004;Dehaene & Cohen, 1995;Groen & Parkman, 1972;Noël, Fias, & Figure 1. Mean RT as a function of problem size (small, large), format (digits, words), and truth value (true, false). ...
Being diagnosed with a mental illness is quite common in the United States (Pratt & Brody, 2014), and the diagnosis itself can be challenging. When those same people experience stigma or social distance for simply having their mental illnesses, the pressure and embarrassment can seem unbearable (Corrigan, 2007). This study explored what circumstances increase or decrease the amount of mental health stigma and social distance participants place on a fictional character suffering from depression. After manipulating (1) origin of depression [biological versus psychological], (2) controllability of depression, and (3) target sex in a vignette, results showed that participant sex, target sex, and personal experience with mental health are predictors of social distance from mentally ill individuals. Surprisingly, mental health stigma was not affected by any of the independent variables. Results shed light on the circumstances in which social distance against mentally ill individuals increases and decreases. This information can aid clinicians in creating effective treatment plans for clients.
... To study encoding and calculation, we relied upon two specific empirical effects found throughout the literature. First, a common signature of the calculation process is the problem size effect, which is the finding that response times and errors increase as problem operands grow in magnitude (Ashcraft, 1992;Campbell, Parker, & Doetzel, 2004;Groen & Parkman, 1972). For example, the simple addition problem 1+1 is typically solved more quickly and accurately than 8+9. ...
... A second class of models is characterized by an interaction between the encoding and calculation processes. That is, the processes involved in calculation directly depend upon the format in which stimuli are encoded (Campbell & Alberts, 2009;Campbell, Parker, & Doetzel, 2004). To illustrate, think back to the lunch bill example presented earlier. ...
... Given that the effects of problem size and format are strong in the literature (Ashcraft, 1992;Campbell, 1994;Campbell, Parker, & Doetzel, 2004;Dehaene & Cohen, 1995;Groen & Parkman, 1972;Noël, Fias, & Brysbaert, 1997), we expected to find significant main effects of both. This expectation was confirmed. ...
Previous work has shown that the cognitive processes involved in mental arithmetic can bedecomposed into three stages: encoding, calculation, and production. Models of mental arithmetic hypothesize varying degrees of independence between these processes of encoding and calculation. In the present study, we tested whether encoding and calculation are independent by having participants complete an addition verification task. We manipulated problem size (small, large) as well as problem format, having participants verify equations presented either as Arabic digits (e.g., “3 + 7 = 10”) or using words (e.g., “three + seven = ten”). In addition, we collected trial-by-trial strategy reports. Though we found main effects of both problem size and format on response times, we found no interaction between the two factors, supporting the hypothesis that encoding and calculation function independently. However, strategy reports indicated that manipulating format caused a shift from retrieval based strategies to procedural strategies, particularly on large problems. We discuss these results in light of two competing models of mental arithmetic.
... To study encoding and calculation, we relied upon two specific empirical effects found throughout the literature. First, a common signature of the calculation process is the problem size effect, which is the finding that response times and errors increase as problem operands grow in magnitude (Ashcraft, 1992;Campbell, Parker, & Doetzel, 2004;Groen & Parkman, 1972). For example, the simple addition problem 1+1 is typically solved more quickly and accurately than 8+9. ...
... A second class of models is characterized by an interaction between the encoding and calculation processes. That is, the processes involved in calculation directly depend upon the format in which stimuli are encoded (Campbell & Alberts, 2009;Campbell, Parker, & Doetzel, 2004). To illustrate, think back to the lunch bill example presented earlier. ...
... Given that the effects of problem size and format are strong in the literature (Ashcraft, 1992;Campbell, 1994;Campbell, Parker, & Doetzel, 2004;Dehaene & Cohen, 1995;Groen & Parkman, 1972;Noël, Fias, & Figure 1. Mean RT as a function of problem size (small, large), format (digits, words), and truth value (true, false). ...
As part of a larger study, 116 adolescents and their parents (52 mothers and 32 fathers) were asked to (a) estimate the average amount of time each day the adolescents spend using five categories of social technology (i.e., Text/Instant Messaging, Talking on the Phone, Facebook, Twitter, and Picture/Video Messaging) and playing three categories of video games (i.e., Everyone 10 and Older, Teen, and Mature) and (b) rate the extent to which the parents monitor their adolescents' involvement in these activities. The adolescents reported spending more time Text/Instant Messaging than any other type of social technology, and they reported playing Mature video games more than any other category of video game. Despite the considerable amount of time the adolescents reported spending on the social technologies and playing video games, (1) the adolescents', mothers', and fathers' ratings of the parents' monitoring of the adolescents' involvement in these activities were generally quite low, and (2) the mothers and fathers rated themselves as monitoring their adolescents' involvement in these activities significantly more than did the adolescents. Potential explanations for the discrepancy between the parents' and adolescents' monitoring ratings are discussed.
... To study encoding and calculation, we relied upon two specific empirical effects found throughout the literature. First, a common signature of the calculation process is the problem size effect, which is the finding that response times and errors increase as problem operands grow in magnitude (Ashcraft, 1992;Campbell, Parker, & Doetzel, 2004;Groen & Parkman, 1972). For example, the simple addition problem 1+1 is typically solved more quickly and accurately than 8+9. ...
... A second class of models is characterized by an interaction between the encoding and calculation processes. That is, the processes involved in calculation directly depend upon the format in which stimuli are encoded (Campbell & Alberts, 2009;Campbell, Parker, & Doetzel, 2004). To illustrate, think back to the lunch bill example presented earlier. ...
... Given that the effects of problem size and format are strong in the literature (Ashcraft, 1992;Campbell, 1994;Campbell, Parker, & Doetzel, 2004;Dehaene & Cohen, 1995;Groen & Parkman, 1972;Noël, Fias, & Brysbaert, 1997), we expected to find significant main effects of both. This expectation was confirmed. ...
Previous work has shown that the cognitive processes involved in mental arithmetic can be
decomposed into three stages: encoding, calculation, and production. Models of mental arithmetic hypothesize varying degrees of independence between these processes of encoding and calculation. In the present study, we tested whether encoding and calculation are independent by having participants complete an addition verification task. We manipulated problem size (small, large) as well as problem format, having participants verify equations presented either as Arabic digits (e.g., “3 + 7 = 10”) or using words (e.g., “three + seven = ten”). In addition, we collected trial-by-trial strategy reports. Though we found main effects of both problem size and format on response times, we found no interaction between the two factors, supporting the hypothesis that encoding and calculation function independently. However, strategy reports indicated that manipulating format caused a shift from retrieval based strategies to procedural strategies, particularly on large problems. We discuss these results in light of two competing models of mental arithmetic.
... Concerning number notation, some have proposed that numerical processing is relatively abstract in nature McCloskey and Macaruso, 1995), whereas others emphasize the influence of number notation (Campbell and Alberts, 2009;Campbell and Fugelsang, 2001;Campbell et al., 2004). The former idea stems from several number processing phenomena that are not influenced by number notation. ...
... Moreover, this notation by size effect was replicated in self reported strategy use, with more calculation for number words than for Arabic numbers, especially for the larger sums (cf. Campbell and Alberts, 2009;Campbell et al., 2004). These latter findings emphasize a possible difference in solution method according to the number notation of the problem, even if both notations are symbolic in nature. ...
... More involvement of the IPS for simple addition of non-symbolic notations IPS activity during simple mental arithmetic was greater for non-symbolic (Dots) than for symbolic number notations (Arabic, Words), which suggests a greater use of magnitude calculations for the non-symbolic notation. This finding complements behavioral data that suggest that the route used for solving sums is dependent on number notation (Campbell and Alberts, 2009;Campbell and Fugelsang, 2001;Campbell et al., 2004), but it appears to be in contrast with the fMRI study by Venkatraman et al. (2005). Venkatraman and colleagues stressed their finding on shared IPS activity for non-symbolic and symbolic addition, which could lead to the idea that IPS activity reflecting arithmetic is notation independent. ...
Addition problems can be solved by mentally manipulating quantities for which the bilateral intraparietal sulcus (IPS) is likely recruited, or by retrieving the answer directly from fact memory in which the left angular gyrus (AG) and perisylvian areas may play a role. Mental addition is usually studied with problems presented in the Arabic notation (4+2), and less so with number words (four + two) or dots (:: + ·.). In the present study, we investigated how the notation of numbers influences processing during simple mental arithmetic. Twenty-five highly educated participants performed simple arithmetic while their brain activity was recorded with functional magnetic resonance imaging. To reveal the effect of number notation, arithmetic problems were presented in a non-symbolic (Dots) or symbolic (Arabic; Words) notation. Furthermore, we asked whether IPS processing during mental arithmetic is magnitude specific or of a more general, visuospatial nature. To this end, we included perception and manipulation of non-magnitude formats (Colors; unfamiliar Japanese Characters). Increased IPS activity was observed, suggesting magnitude calculations during addition of non-symbolic numbers. In contrast, there was greater activity in the AG and perisylvian areas for symbolic compared to non-symbolic addition, suggesting increased verbal fact retrieval. Furthermore, IPS activity was not specific to processing of numerical magnitude but also present for non-magnitude stimuli that required mental visuospatial processing (Color-mixing; Character-memory measured by a delayed match-to-sample task). Together, our data suggest that simple non-symbolic sums are calculated using visual imagery, whereas answers for simple symbolic sums are retrieved from verbal memory.
... This finding would be difficult to explain in terms of encoding processes since the same operands were used for both operations. More recently, Campbell et al. investigated the interaction between format and problem size by analyzing format effects in strategy choice to solve simple arithmetic problems (Campbell & Alberts, 2009;Campbell & Fugelsang, 2001;Campbell, Parker, & Doetzel, 2004). In these studies, participants reported their solution strategy (e.g. ...
... Campbell and colleagues have repeatedly suggested that since simple arithmetic is presented rarely as number words, then, the visual familiarity of this format is lower compared to the familiarity of the Arabic format. Hence, the strength of association between operands and answer will be weaker (e.g., Campbell, 1994;Campbell et al., 2004) and will influence the use of strategies. Less familiarity promotes greater use of calculation strategies against direct memory retrieval (Campbell & Fugelsang, 2001). ...
... The only significant effect related to the format condition was the size by format interaction due to greater wordformat costs for subtraction relative to addition operation. Campbell and colleagues (Campbell et al., 2004;Campbell & Alberts, 2009) also found no effects of format on errors. According to these authors, format effects on RT, errors, or both would depend on participants' emphasis on speed versus accuracy. ...
Adults' simple arithmetic performance is more efficient when operands are presented in Arabic digit (3 + 5) than in number word (three + five) formats. An explanation provided is that visual familiarity with digits is higher respect to number words. However, most studies have been limited to single-digit addition and multiplication problems. In this article, we examine to what extent format effects can be found in the context of arithmetic word problems, in which visual familiarity is reduced (Manuel had 3 marbles, and then Pedro gave to him 5). Participants with high and low arithmetic fluency solved addition and subtraction word problems in which operands were presented in both formats. The overall results showed an advantage for digits operands relative to words operands. In addition, the format effects were more evident for subtraction and low-skilled participants. These results were interpreted in terms of more rapid access of digits to numerical magnitude.
... The continued use of non-retrieval strategies by adults, as shown by the work of Campbell, et al. (2004) , supports the idea that the lack of a parity eff ect for the multiplication problems in the Batson and Combellick (1925) paper is due to a more common use of rote retrieval on these problems by their adult participants. Thus, the eff ect of parity on vari- 06-PR_Hines_130042.indd ...
... There is considerable evidence that marked concepts take longer to pro- cess ( Clark, 1973 ). In fact Campbell, et al. (2004) suggested that the eff ects found for odd digits, as well as other eff ects they reported on problem for- mat (e.g., word versus digit; number size), were due to "relatively poor memory strength" (p. 61) for problems involving odd digits. ...
... In this view, if the responses individuals make do not have a polarity, one would pre- sumably not fi nd any eff ect of markedness of the stimuli. But as the pres- ent results and those of Campbell, et al. (2004) show, this is not the case. ...
In the present paper, the results of early 20th century studies of children's difficulties with basic addition, subtraction, and multiplication problems are analyzed for parity effects. Parity influenced the difficulty of addition and subtraction problems, tasks in which any parity influence must be implicit, but did not influence difficulty of multiplication problems. Solution of multiplication problems relies to a greater extent on retrieval of rote-memorized answers. Addition and subtraction problems rely more on non-retrieval strategies. It is these latter strategies that depend on internal representations sensitive to the parity status of the numbers being processed. That parity affected responding in tasks where parity was irrelevant and no overt motor responses were made poses problems for the Markedness of Response Codes (MARC) and polarity explanations of parity effects in reaction time. Both these explanations require that an explicit parity judgment indicated by a binary motor response be made for a parity effect to be seen.
... Note that word (vs. digit) format produces delays in various tasks besides arithmetic (e.g., number comparison, Noel et al., 1997; parity judgments; Campbell et al., 2004). Another possibility is that arithmetic knowledge is represented exclusively in digit format (rather than an abstract format). ...
... encoding-based) explanation for the increased problem size effect in word format is that the less familiar word format increases the use of computation versus retrieval. In some studies on addition, participants self-reported more procedure use for problems in word versus digit format (Campbell et al., 2004; Campbell & Penner-Wilger, 2006). Such a strategy shift towards computation would indeed cause an increase in the problem size effect, because even within digit format problems, the problem size effect (i.e., slope) is larger among problems solved via computation then among problems solved via retrieval (LeFevre et al., 1996a; 1996b). ...
... Consequently, our account of the format by size interaction(s) is somewhat distinct and more general. Overall, however, our pattern of results seems incompatible with Campbell et al.'s (2004) access-based account in which the increase in the problem size effect (in word format) arises because low format familiarity produces a shift from retrieval to computation. If low familiarity with a problem's surface form induces a bias towards a computation strategy, then computation (vs. ...
We explored the impact of operand format (digit, word, pseudo-homophone) on single-digit addition and multiplication. Format manipulations are of theoretical interest because models of arithmetic knowledge differ with respect to predicted format effects. Latencies were shortest when operands were digits, and longest when they were pseudo-homophones. However, there was also an interaction of problem size with format: problem size effects were smaller in the pseudo-homophone condition relative to the digit condition, and were larger in the word format condition relative to the digit condition. We discuss our results with respect to the following question: Can this interaction be attributed (solely) to an encoding phase of processing, or might it (also) arise from a solution phase?
... Given the evidence that cue familiarity can affect strategy choice in other domains, it is worthwhile to determine if operand familiarity can similarly influence retrieval usage for elementary arithmetic facts. Differences in familiarity could explain the finding of Campbell et al. (2004) that direct retrieval was much more likely when simple addition problems appeared in digit format (4'/8) compared to written-word format (four'/eight). Arithmetic problems are frequently encountered in Arabic format, but rarely encountered in written word format. ...
... In the following experiment, we tested Canadian university students on simple addition problems (two'/five, seven '/eight, etc.), and asked them to report their strategy after each trial by selecting from ''remember '', ''count'', ''transform'', or ''other''. Theoretically, selection of the remember category corresponds to direct memory retrieval, whereas selection of one of the other categories corresponds to procedure use (Campbell & Austin, 2002;Campbell & Fugelsang, 2001;Campbell et al., 2004;Campbell & Xue, 2001). Participants received a practice phase in which they repeatedly solved a subset of addition problems, which familiarised four of the eight addends between 2 and 9. ...
... Nonetheless, operand familiarity is potentially a component of problem size and format effects on addition strategy choice. Campbell et al. (2004) speculated that lower rates of retrieval for addition in written word compared to digit format might reflect lower familiarity of arithmetic problems in the word format. The present results are consistent with the possibility that familiarity of the written word operands does contribute to this effect. ...
Are adults’ decisions to use direct memory retrieval for simple addition influenced by the familiarity of problem operands? We manipulated the familiarity of a subset of operands by having adults repeatedly practise specific additions (two+five=?; Experiment 1) or magnitude comparisons (two five, choose the larger; Experiment 2). Both experiments provided evidence that pre-exposure to single-digit operands increased reported use of direct retrieval for new combinations of the familiarised operands. RT and error patterns across experiments also supported the conclusion that increased use of retrieval facilitated performance. These results show that operand familiarity potentially plays a significant role in adults’ strategy choices for simple addition.
... To pursue these predictions, performance of the zero and one problems tested in Experiment 1 of Campbell et al. (2004) was analyzed in detail. In this experiment, participants received all the single-digit addition (0 + 0 to 9 + 9) and multiplication problems (0 × 0 to 9 × 9) as well as a parity-comparison task based on the combinations of the numbers 2 through 9. Campbell et al. (2004) excluded zero and one problems in order to match the parity task with the arithmetic tasks, and presented no analyses of the zero and one problems. ...
... To pursue these predictions, performance of the zero and one problems tested in Experiment 1 of Campbell et al. (2004) was analyzed in detail. In this experiment, participants received all the single-digit addition (0 + 0 to 9 + 9) and multiplication problems (0 × 0 to 9 × 9) as well as a parity-comparison task based on the combinations of the numbers 2 through 9. Campbell et al. (2004) excluded zero and one problems in order to match the parity task with the arithmetic tasks, and presented no analyses of the zero and one problems. Here the focus was on the zero and one problems specifically. ...
... Each of the 8 arithmetic blocks involved 110 trials, and each of the 4 parity blocks in involved 72, for a total of 1168 trials. No further details of the parity task are provided here (see Campbell et al., 2004). ...
This research examined adults’ performance of simple addition and multiplication involving rule-based zero and one problems (N+0=N, N×1=N, N×0=0) presented as Arabic digits (5×1; 0+4) or written English number words (five×one; zero+four). The results showed that digits and number words differed in their capacity to activate the zero and one rules: With Arabic numerals, N×0=0 was relatively difficult compared to the other zero and one problems. This may be understood as an interference effect from the competing N+0=N and N×1=N rules, an interpretation supported by the high rate of N×0=N errors with Arabic stimuli. In contrast, with word stimuli, N×0 items were among the fastest items and yielded a relatively low rate of N×0=N errors. The results provide evidence that problem encoding mechanisms and central calculation mechanisms are interactive rather than strictly additive processes.
... The source of these effects has been the focus of considerable debate. Some researchers argue that format effects in arithmetic arise only during problem encoding or response stages (e.g., McCloskey & Macaruso, 1995;Noël, Fias, & Brysbaert, 1997;Sokol, McCloskey, Cohen & Aliminosa, 1991), whereas others propose format can directly affect retrieval or calculation processes (e.g., Bernardo, 2001;Blankenberger & Vorberg, 1997;Campbell, 1994;Campbell & Alberts, 2009;Campbell, Parker, & Doetzel, 2004). ...
... There were 36 pairings of the numbers 2 through 9 ignoring order (e.g., 3 ϩ 8 vs. 8 ϩ 3), including eight ties (e.g., 2 ϩ 2, 8 ϩ 8). Tie problems were tested but excluded from analysis because of their unique encoding characteristics (Campbell et al., 2004;Noël et al., 1997). Participants received eight blocks of 72 trials in which all 36 problems were tested twice. ...
... whereas counting tended increase with word relative to digit format ϩ 6.8% for small problems and ϩ 5.2% for larger problems, F(1, 82) ϭ 2.7, mean square error (MSE) ϭ 34.7, p ϭ .11. The observation that written-word operands promote counting more than transformation, especially for small problems here, replicated previous research (Campbell & Alberts, 2009;Campbell et al., 2004). ...
Adults' simple addition performance (e.g., 3 + 4 = ?) is faster, more accurate, and more often based on direct memory retrieval (rather than a procedural method, such as counting) when problems are presented in digit format (3 + 4) than written-word format (three + four). A possible explanation is that the mathematical symbol + is more compatible to memory retrieval with Arabic numerals than word numerals. To investigate this, two groups of 42 participants received eight blocks of 72 simple addition problems. For one group, operand format (digits or words) switched across trials within each block and operator (the symbol + or the word plus) alternated between blocks. For the other group, operator switched across trials, whereas operand format alternated between blocks. In the switch-format condition, compatible formats (e.g., 3 + 4, three plus four) were solved by direct memory retrieval more often than were incompatible formats (3 plus 4, three + four). There was no compatibility effect on use of direct memory retrieval when operand format was fixed within blocks and operator format switched across trials. There was also a reaction time (RT) advantage only for digit operands with + relative to plus when format switched, but + facilitated only word problems when operand format was blocked. The results indicate that operand-operator compatibility and format switching had previously unsuspected effects that qualify previous research examining format effects in arithmetic.
... The interactive viewpoint assumes that the encoding conditions could affect subsequent retrieval or calculation. The main proponent of this perspective is Campbell, who proposed the encodingcomplex model (Campbell, 1992(Campbell, , 1994(Campbell, , 1999Campbell & Alberts, 2009;Campbell & Clark, 1989Campbell, Parker, & Doetzel, 2004;Clark & Campbell, 1991;Metcalfe & Campbell, 2008). Evidence for this model came from Campbell and colleagues' studies that showed interactions between problem format and arithmetic processing (e.g., type of operation, problem size). ...
... All rights reserved. doi:10.1016/j.bandc.2011.03.018 plication than for number comparisons (Campbell, 1999;Campbell et al., 2004), more salient for addition than subtraction, and more salient for division than multiplication (Campbell & Alberts, 2009). According to Campbell and Epp (2005), this word cost cannot be attributed to encoding of operands, but rather it occurs at the retrieval/calculation stage. ...
... According to Campbell and Epp (2005), this word cost cannot be attributed to encoding of operands, but rather it occurs at the retrieval/calculation stage. That is, the retrieval of the format-specific representations (e.g., verbal representations of problems expressed in word format) lead to interactions between problem format and arithmetic processing (Campbell, 1994;Campbell et al., 2004). Recently, Metcalfe and Campbell (2008) further found a three-way interaction among modality, operation, and problem size. ...
... Tie problems in both operations were tested but excluded from analysis because of unique encoding characteristics (cf. Campbell et al., 2004;Noël et al. 1997). For each operation, one operand order of each of the 28 non-tie pairs was selected at random for each participant and used throughout the experiment. ...
... Overall, less retrieval for large than small problems primarily reflected more use of reference and other transformation strategies for large than small problems (cf. Campbell et al., 2004;. ...
... This interaction mainly reflected more counting for addition problems in word format (20%) relative to digit format (11%) (cf. Campbell et al., 2004) whereas counting rates for subtraction were similar for word and digit formats (15% vs. 13%) [F(1, 43) = 14.1, MSE = 62.2, p = .001 for the Operation × Format interaction on reported counting]. ...
Educated adults solve simple addition problems primarily by direct memory retrieval, as opposed to by counting or other procedural strategies, but they report using retrieval substantially less often with problems in written-word format (four + eight) compared with digit format (4 + 8). It was hypothesized that retrieval efficiency is relatively low with word operands compared with digits and that this promotes a shift to procedural backup strategies. Consistent with this hypothesis, Experiment 1 demonstrated greater word-format costs on retrieval usage for addition than subtraction, which was due to increased counting for addition but not subtraction. Experiment 2 demonstrated greater word-format costs on retrieval for division than multiplication, which was due to increased use of multiplication-fact reference to solve division problems. Format-related strategy shifts away from retrieval reflected both the efficiency of retrieval for a given operation and the availability of viable alternative strategies. The results demonstrate that calculation processes are not abstracted away from problem surface form. The authors propose that retrieval efficiency for arithmetic connects diverse performance and strategy-related effects across key arithmetic factors, including arithmetic operation, numerical size, and numeral format.
... Alternatively, according to the encoding complex model proposed by Campbell and Clark (1988), the processes involved in number cognition are influenced by the format that the number information is presented. Empirical research has been found to support both the format-independent model (e.g., Noël et al. 1997;Noël and Seron 1992;Rickard et al. 1994) and the format-specific model (e.g., Bernardo 2001;Campbell 1994Campbell , 2005Campbell et al. 1999;Campbell et al. 2004;Frenck-Mestre and Vaid 1993). Research on bilinguals has largely supported the latter model. ...
... Rights reserved. format-specific model of number cognition (e.g., Bernardo 2001;Campbell 1994;Campbell et al. 2004;Frenck-Maestre and Vaid 1993). ...
Numbers are particularly interesting as they can be presented in different notations, for example, they can be represented as numerical digits or words. Moreover, many cultures around the world have different writing systems for representing number. Thai uses a more traditional Thai number system in conjunction with Arabic numbers. In the current study, we investigated the processing of numerical digits and words in unbalanced Thai-English bilinguals using a numerical parity judgment task. The flankers occurring on either side of the target were either congruent or incongruent with the target digit or word. In Experiment 1, we investigated the effects of Arabic digit and Thai digit flankers on English and Thai target number words and in Experiment 2, the effects of English and Thai number word flankers on Arabic and Thai digit targets. In Experiment 1, we found an interference effect from Thai digit flankers on Thai numerical words and in Experiment 2, an interference effect for Arabic digits from Thai word flankers. These results suggest that the first language is playing a greater contributing role than the second language and that numerical notation format contributes to the effect. Proficiency in the second language is likely to moderate this effect.
... Another interesting surface form study was conducted in 2004 by Campbell, Parker, and Doetzel [27]. In experiment 1 of their study, participants were presented with simple addition, multiplication, and parity tasks (the numbers were either Arabic digits or written number words). ...
... Another point to consider is that Arabic digits and written number word surface forms have been shown to elicit different arithmetical strategies (retrieval or calculation) from participants [27,28]. These different strategies which are used (and not just numerical processing) can affect RTs. ...
In regards to numerical cognition and working memory, it is an open question as to whether numbers are stored into and retrieved from a central abstract representation or from separate notation-specific representations. This study seeks to help answer this by utilizing the numeral modality effect (NME) in three experiments to explore how numbers are processed by the human brain. The participants were presented with numbers (1-9) as either Arabic digits or written number words (Arabic digits and dot matrices in Experiment 2) at the first (S1) and second (S2) stimuli. The participant's task was to add the first two stimuli together and verify whether the answer (S3), presented simultaneously with S2, was correct. We hypothesized that if reaction time (RT) at S2/S3 depends on the modality of S1 then numbers are retrieved from modality specific memory stores. Indeed, RT depended on the modality of S1 whenever S2 was an Arabic digit which argues against the concept of numbers being stored and retrieved from a central, abstract representation.
... The irregular non-unit fractions in task 2 were less familiar than those in task 1 (Ganor-Stern, 2012), although their holistic values were equal. Previous studies have indicated that the stimuli format can affect numerical processing (e.g., Campbell et al., 2004;Campbell and Penner-Wilger, 2006). For example, Campbell et al. (2004) showed that an unfamiliar stimuli format disrupted number-fact retrieval and promoted the use of more effortful procedural strategies. ...
... Previous studies have indicated that the stimuli format can affect numerical processing (e.g., Campbell et al., 2004;Campbell and Penner-Wilger, 2006). For example, Campbell et al. (2004) showed that an unfamiliar stimuli format disrupted number-fact retrieval and promoted the use of more effortful procedural strategies. It is therefore understandable that holistic processing was adopted when fractions were relatively irregular and unfamiliar since this processing strategy is both more stable and more likely to produce a correct answer in spite of complexity. ...
Recent studies have indicated that people have a strong tendency to compare fractions based on constituent numerators or denominators. This is called componential processing. This study explored whether componential processing was preferred in tasks involving high stimuli variability and high contextual interference, when fractions could be compared based either on the holistic values of fractions or on their denominators. Here, stimuli variability referred to the fact that fractions were not monotonous but diversiform. Contextual interference referred to the fact that the processing of fractions was interfered by other stimuli. To our ends, three tasks were used. In Task 1, participants compared a standard fraction 1/5 to unit fractions. This task was used as a low stimuli variability and low contextual interference task. In Task 2 stimuli variability was increased by mixing unit and non-unit fractions. In Task 3, high contextual interference was created by incorporating decimals into fractions. The RT results showed that the processing patterns of fractions were very similar for adults and children. In task 1 and task 3, only componential processing was utilzied. In contrast, both holistic processing and componential processing were utilized in task 2. These results suggest that, if individuals are presented with the opportunity to perform componential processing, both adults and children will tend to do so, even if they are faced with high variability of fractions or high contextual interference.
... Numerous studies indicate that both of these factors affect use of procedural strategies (see Campbell & Epp, 2005, for a review). Specifically, procedure use increases with problem size, and self-reported use of procedures is much more common with written word operands (four ϩ eight) than with Arabic digits (4 ϩ 8), especially for simple addition (Campbell & Fugelsang, 2001;Campbell et al., 2004). Consequently, we expected verbal self-reports to replicate previous research with respect to the effect of these factors. ...
... For the Operation ϫ Format interaction, the wordformat cost (word -digits) was larger for addition (ϩ12% procedures) than for multiplication (ϩ5%; cf. Campbell et al., 2004). The Format ϫ Size and the Operation ϫ Format ϫ Size interactions were not significant. ...
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.
... Therefore, the conclusion that numbers are abstract may be due to a lack of statistical power, or the insensitivity of the paradigms used. Indeed, some studies have found differences or a tendency towards a difference between notations Cohen Kadosh & Walsh: Numerical representation in the parietal lobes BEHAVIORAL AND BRAIN SCIENCES (2009) 32:3/4 (e.g., digits, verbal numbers, numerosity, Mandarin numerals) (Campbell & Epp 2004; Dehaene 1996; Dehaene & Akhavein 1995; Droit-Volet et al. 2008; Ganor-Stern & Tzelgov 2008; Koechlin et al. 1999; Reynvoet & Ratinckx 2004) or modalities (i.e., visual or auditory) (Barth et al. 2003), but the implications of most of these results have either been ignored, or alternative explanations have been given that leave the idea of non-abstract representations unchallenged. ...
... The more expertise we have acquired for a specific type of numerical symbol, the more likely its presentation leads to an automatic activation of the underlying semantic magnitude representation. A similar account was put forward by Campbell and colleagues in their encoding-complex hypothesis: " task-specific retrieval processes will be more efficient when numerical stimuli appear in a familiar, well-practiced format, relative to the retrieval processes activated by an unfamiliar surface form " (Campbell & Epp 2004, p. 231). If differences in symbol-referent mapping expertise are not considered in tasks drawing on automatic number processing, notation-related effects are not conclusive with regard to the question of non-abstract representations . ...
Abstraction is instrumental for our understanding of how numbers are cognitively represented. We propose that the notion of abstraction becomes testable from within the framework of simulated cognition. We describe mental simulation as embodied, grounded, and situated cognition, and report evidence for number representation at each of these levels of abstraction.
... Therefore, the conclusion that numbers are abstract may be due to a lack of statistical power, or the insensitivity of the paradigms used. Indeed, some studies have found differences or a tendency towards a difference between notations Cohen Kadosh & Walsh: Numerical representation in the parietal lobes BEHAVIORAL AND BRAIN SCIENCES (2009) 32:3/4 (e.g., digits, verbal numbers, numerosity, Mandarin numerals) (Campbell & Epp 2004; Dehaene 1996; Dehaene & Akhavein 1995; Droit-Volet et al. 2008; Ganor-Stern & Tzelgov 2008; Koechlin et al. 1999; Reynvoet & Ratinckx 2004) or modalities (i.e., visual or auditory) (Barth et al. 2003), but the implications of most of these results have either been ignored, or alternative explanations have been given that leave the idea of non-abstract representations unchallenged. ...
... The more expertise we have acquired for a specific type of numerical symbol, the more likely its presentation leads to an automatic activation of the underlying semantic magnitude representation. A similar account was put forward by Campbell and colleagues in their encoding-complex hypothesis: " task-specific retrieval processes will be more efficient when numerical stimuli appear in a familiar, well-practiced format, relative to the retrieval processes activated by an unfamiliar surface form " (Campbell & Epp 2004, p. 231). If differences in symbol-referent mapping expertise are not considered in tasks drawing on automatic number processing, notation-related effects are not conclusive with regard to the question of non-abstract representations . ...
We contrapose computational models using representations of numbers in parietal cortical activity patterns (abstract or not) with dynamic models, whereby prefrontal cortex (PFC) orchestrates neural operators. The neural operators under PFC control are activity patterns that mobilize synaptic matrices formed by learning into textured oscillations we observe through the electroencephalogram from the scalp (EEG) and the electrocorticogram from the cortical surface (ECoG). We postulate that specialized operators produce symbolic representations existing only outside of brains.
... This emphasis stands in contrast to the models proposed by McCloskey (1992) and Dehaene and Cohen (1995), which do not provide mechanisms for format-specific number judgments or calculation. Our view is motivated by the C, K, & X study and others that have provided diverse evidence for format-specific retrieval in number processing (e.g., Bernardo, 2001;Blankenberger & Vorberg, 1997;Campbell, 1994;Campbell & Clark, 1992;Campbell & Fugelsang, 2001;Campbell, Parker, & Doetzel, 2004;Cipolotti, Warrington, & Butterworth, 1995;Frenck-Mestre & Vaid, 1993;Koechlin et al., 1999;McNeil & Warrington, 1994;Sciama, Semenza, & Butterworth, 1999;Szú´cs & Csépe, 2004;see Campbell & Epp, in press, for a review of research on format effects in simple arithmetic). ...
... Performance costs associated with an unfamiliar numeral format generally are expected to increase with item difficulty (cf. Campbell et al., 2004). This is expected because difficult discriminations are more susceptible to reductions in relevant associative or semantic information, or to increased interference from irrelevant associations. ...
We present a model of the cognitive architecture of basic numerical skills in adult Chinese-English bilinguals. The model is based on data reported by Campbell, Kanz, and Xue (1999) and combines Dehaene and Cohen's triple-code theory with Campbell and Clark's encoding-complex approach to modeling number processing. Participants were required to name, add or multiply Arabic or Mandarin numerals and to respond in English or Chinese. They also performed magnitude comparisons on pairs of Arabic or Mandarin numerals. The proposed model of their performance on this set of tasks assumes 1) that number processing is modular with respect to representational code (e.g., visual, visuo-spatial, verbal) rather than with respect to numerical function, 2) task-specific communication between representational codes is interactive rather than additive, and 3) memory for arithmetic facts is at least partially language-based and our Chinese-English bilinguals possessed both Chinese and English-language number-fact representations. We provide new analyses of the arithmetic data and a review of research on the role of language in simple arithmetic to substantiate our claims about linguistic codes for number-fact memory.
... Secondly, incorrect solutions for each addition were obtained by adding and subtracting one unit to and from their true solution (i.e., small-split solutions, which are solved less accurately than large-split solutions; Ashcraft & Battaglia, 1978; e.g., we used 8 + 3 = 12 instead of 8 + 3 = 21). Finally, in order to prevent participants from using the most efficient parity rule (i.e., the result of adding two even numbers is always an even number; otherwise the answer is incorrect; Campbell et al., 2004), for those additions with both operands being even numbers, two units were added or subtracted instead of one (e.g., "6 + 8 = 12″ instead of "6 + 8 = 13″). For the "2 + 4″ and "2 + 6″ additions in both directions, their form resulting from adding two units to the true solution was used twice since their form resulting from subtracting two units produces a non-plausible solution (e.g., "2 + 4 = 4″). ...
... Ce phénomène s'accommode mieux, aux yeux de Campbell et Clark (1988), d'un modèle supposant que les inputs numériques sont traités, ou font l'objet de calculs, selon la spécificité de leur format numérique.Si le modèle de Campbell est étayé par des cas cliniques et par des résultats expérimentaux (e.g. :Bernardo, 2001 ;Campbell, Parker, & Doetzel, 2004 ; Sczücs & Csépé, 2004 ; Myers & Sczücs, 2015), il convient de souligner que McCloskey (1992) s'est également attaché à l'analyse critique des arguments en faveur du modèlede Campbell. Campbell et Clark (1988) mettent ainsi au crédit de leur modèle les études deGonzales et Kolers (1982 ; avérant des temps de réponse fonction du format numérique, dans une tâche de validation de résultats d'additions avec chiffres arabes, romains ou à la fois arabes et romains (e.g. ...
Les nombres et les opérations sur les nombres contribuent à structurer notre rapport au monde. Il est ainsi logique que plusieurs études aient tenté de clarifier les mécanismes sous-tendant la cognition numérique ou les mécanismes cérébraux responsables de la dyscalculie. Ces études ont suggéré que les représentations mathématiques pourraient s’ancrer dans des expériences corporelles et/ou que la cognition numérique et la préparation du mouvement pourraient être sous-tendues par des mécanismes cérébraux similaires. Plusieurs études ont suggéré que des mouvements segmentaires ou du corps dans sa globalité peuvent influer sur la performance de tâches numériques ou arithmétiques. L’effet de telles taches sur la performance motrice demeurait en revanche à examiner, en particulier dans le cas des mouvements à haute intensité. Cet éventuel effet a été examiné via deux études impliquant au total 206 étudiants de sexe masculin, en Licence à la faculté de santé publique de l’Université Libanaise (Beyrouth, Liban). Une première étude (deux séries de deux expérimentations) a examiné les effets de la lecture d’un nombre et de la soustraction mentale complexe sur la hauteur de saut en squat jump vertical (SJV) et sur le temps de réponse d’un mouvement de pointage manuel (MPM). Dans chaque série, ces effets ont été examinés dans le cas de nombres en chiffres arabes et de nombres écrits en toutes lettres. Une seconde étude a examiné l’effet de tâches d’arithmétique mentale sur le temps de réponse d’un MPM. Trois expérimentations (1-3) ont étudié l’effet de la soustraction (complexe) et, respectivement, de : (1) l’addition (simple ou complexe), (2) la multiplication (simple ou complexe) et (3) la comparaison d’ensembles de points et la comparaison de nombres. Tout nombre était écrit en chiffres arabes. Dans ces deux études, les données ont été analysées en recourant à un modèle linéaire multiniveaux à effets mixtes. Les résultats de la première étude ont avéré une amélioration modérée de la performance en SJV (statistiquement significative, p < 0,05) suite à la lecture d’un nombre écrit en toutes lettres et un net effet de la performance en SJV et en MPM après une soustraction mentale (complexe) avec nombres en chiffres arabes (p < 0,001). Les résultats de la seconde étude ont avéré une amélioration statistiquement significative de la performance en MPM suite aux seuls calculs complexes (p < 0,001) et à la seule comparaison de nombres (p < 0,003). Ces résultats suggèrent que la relation entre une tâche arithmétique et la performance d’un mouvement à haute intensité est influencée par le format numérique, le recours à des nombres en chiffres arabes (à la différence de celui à des nombres écrits en toutes lettres ou à des ensembles de points) s’avérant conditionner un effet positif sur la performance motrice. Ces résultats ont cependant montré que cette condition n’est pas suffisante, la performance motrice étant améliorée après les tâches arithmétiques (avec chiffres arabes) favorisant le recours à des stratégies procédurales plutôt que le recours à des stratégies par recouvrement (en mémoire) de faits arithmétiques. Au regard de la littérature, l’effet des calculs mentaux complexes (soustraction, addition et multiplication) et de la comparaison de nombres, en notation arabique, sur la performance motrice peut s’expliquer par différents mécanismes. Cet effet peut être lié à la mobilisation de mécanismes d’encodage et/ou de mémorisation spécifiques des chiffres arabes. L’addition et la multiplication complexes et, éventuellement, la comparaison de nombres, peuvent en outre avoir favorisé une attention à la trajectoire optimale du mouvement subséquent. L’influence des calculs complexes et de la comparaison de nombres, avec notation arabique, sur la performance motrice pourrait enfin tenir à l’implication de régions cérébrales motrices, mobilisées durant une activité effective de calcul ou de comparaison.
... The problem size effect occurs at this stage. When the numeric values are larger, the response time is longer and accuracy rate is lower (e.g., 1 þ 1 is easier than 8 þ 9; Ashcraft, 1992;Campbell et al., 2004). Researchers postulated that individuals might encounter more difficulties retrieving arithmetic facts with larger values and tend to use more complex operation procedures in situations with larger numeric values (Campbell & Alberts, 2009;Campbell & Xue, 2001;Zbrodoff & Logan, 2005). ...
The aim of this paper was to examine the role of phonological working memory in specific mental arithmetic difficulties and general arithmetic learning difficulties (ALD; difficulties presenting in both mental arithmetic and written arithmetic). In Study 1, we categorized 53 sixth graders into a control group, a group with specific mental arithmetic difficulties, and a group with general ALD. The findings indicated the group with specific mental arithmetic difficulties performed significantly worse on the task involving phonological working memory than did the control group. However, a significant difference was not found between the group with general ALD and the control group. In Study 2 involving 54 sixth graders, we decreased the load of phonological working memory by changing the format of the problems from horizontal (more reliance on phonological codes) to vertical (more reliance on visual resources). We found that the group with specific mental arithmetic difficulties performed comparably to the control group. In other words, when the working memory load is reduced, they no longer lag significantly behind on mental arithmetic. However, the group with general ALD still performed significantly worse than the control group when the problems were presented vertically, indicating that reduced phonological working memory load did not alleviate their arithmetic difficulties. The findings in both studies suggested that poor phonological working memory might contribute to the underlying mechanism for specific mental arithmetic difficulties but not as much for general ALD.
... In other words, after stimulate the coding, problems in discrete numerical surface form remain its specificity and process in different paths (e.g., Campbell & Clark, 1988). A substantial body of research supports the separate-pathway hypothesis (e.g., Campbell, 1994;Campbell, 1999;Campbell & Clark, 1992;Campbell & Fugelsang, 2001;Campbell, Parker, & Doetzel, 2004;. The core hypothesis of interacting neighbors model is that "7 × 4" and "4 × 7" have only one representational unit, Butterworth et al. (2001) proposed the core assumptions of the COMP model took a similar view, However, Robert and Campbell's (2008) study found, "7 × 4" and "4 × 7" problems didn't have significant difference in reaction time in addition and multiplication tasks, thus demonstrated that "7 × 4" and "4 × 7" problems didn't have inherent comparison and have not been converted into a common internal representational unit, which also indicated that the "7 × 4" and "4 × 7" may have a different kind of problem representation. ...
In the present study, under the spoken Mandarin number words format, we employed verification tasks to investigate the neighborhood effects in single-digit multiplication. The results revealed that, in the Arabic digits format condition, the neighborhood effects like as the former studies discovered is natural, however, the unexpected reversed neighborhood effects were found in the spoken Mandarin number words format. Specifically, RTs of higher neighborhood effects multiplication problems were longer than lower neigh-borhood effects.
... The parity e®ect as noticed by Krueger (Krueger & Hallford, 1984) though a®ecting the results was weaker than the \distance" e®ect. Moreover, the strongest \parity e®ect" was for even þ even numbers which in further studies were interpreted not as result of parity but rather of more familiarity with even numbers (i.e., we count by \two's") (Campbell et al., 2004; Lochy et al., 2000). In our experiment, only two cases were of the type even þ even numbers. ...
... That is, numeric stimuli maintain their surface properties as specific codes throughout processing by various separate pathways (Campbell & Clark, 1988). A large body of research supports the separate pathway hypothesis (e.g., Campbell & Clark, 1992;Campbell & Fugelsang, 2001;Campbell, Parker, & Doetzel, 2004;Metcalfe & Campbell, 2008). ...
Two perspectives compete to explain how the surface form of digits affects cognitive processing of numerical magnitude; one argues for a common pathway and the other for separate pathways. This study examined the operand-related error effect in simple multiplication operations using different combinations of visually presented Arabic digits and auditorily presented Mandarin number words. The study suggested two conclusions, both consistent with the separate pathway perspective. First, the numerical surface form (Arabic digits, spoken Mandarin number words) affected retrieval. That is, surface properties were maintained as specific codes throughout processing. Second, the phonological code activated by spoken Mandarin number words interfered with activation of answers during retrieval.
... On the other hand, it is mathematically impossible to match the carry condition with the conditions 5 break and no break with regard to the factor unit sum because unit sum needs to be larger than 10 in the carry and smaller than 10 in the other conditions per definition. Deschuyteneer et al. (2005) Square of the sum Ashcraft and Battaglia (1978) Product of the summands Widaman et al. (1989) Sum of the square of the addends (SSA) Widaman et al. (1989) Parity of the two summands Campbell et al. (2004) Lemaire and Siegler (1995) Vandorpe et al. (2005) For the latter, the In line with our hypothesis we indeed observed consistent sub-base-five effects: when summing up the unit digits crossed the sub-base-five boundary, this was associated with a significant increase of response times. First, the comparison of addition problems involving a 5 break with no break problems revealed reliably prolonged reaction times for the former. ...
Recently, a strong functional relationship between finger counting and number processing has been suggested. It has been argued that bodily experiences such as finger counting may influence the structure of the basic mental representations of numbers even in adults. However, to date it remains unclear whether the structure of finger counting systems also influences educated adults’ performance in mental arithmetic. In the present study, we pursued this question by examining finger-based sub-base-five effects in an addition production task. With the standard effect of a carry operation (i.e., base-10 crossing) being replicated, we observed an additional sub-base-five effect such that crossing a sub-base-five boundary led to a relative response time increase. For the case of mental arithmetic sub-base-five effects have previously been reported only in children. However, it remains unclear whether finger-based numerical effects in mental arithmetic reflect an important but transitory step in the development of arithmetical skills. The current findings suggest that even in adults embodied representations such as finger counting patterns modulate arithmetic performance. Thus, they support the general idea that even seemingly abstract cognition in adults may at least partly be rooted in our bodily experiences.
... In addition, when children learn to make parity judgments, they do so with Arabic numerals rather than with their alphabetic counterparts. Second, in the mathematics literature, the Arabic format is considered the familiar format for arithmetic (including parity) but not for reading aloud (e.g., Campbell, Parker, & Doetzel, 2004; Dehaene, Bossini, & Giraux, 1993). Therefore, when a person is reading aloud, the strength of the connections between levels of lexical representations should differ little, if at all, between Arabic numerals and their alphabetic counterparts. ...
Frequency-of-occurrence effects (e.g., effects of word frequency or familiarity) are widely thought to arise through differences in resting levels of activation in localist input-output modules. A different account posits that these effects at least partially reflect the strength of connections between various localist modules. Given that Arabic numerals appear more frequently than their alphabetic counterparts, we contrasted reaction times to stimuli in both formats in a naming/reading-aloud task and a parity-judgment task. The script effect (the difference between reaction times to Arabic and to alphabetic formats) was large in the parity-judgment task but absent in the naming/reading-aloud task. This script-by-task interaction follows naturally from the idea that at least part of the effect of frequency of occurrence of a printed word or digit (and other instances of familiarity) resides in the strength of connections between specialized localist input-output modules and a localist semantic module. This conclusion is likely applicable across a variety of domains.
... The experimental within-participant 2 × 2 × 2 design comprised the three factors carry (carry vs. non-carry; e.g., 27 + 48 vs. 21 + 48), problem size (sum < 40 vs. > 60; e.g., 13 + 14 vs. 13 + 54), and distractor type. Half of the incorrect solution probes (distractors) differed from the correct result by ±2, whereas the distance between the correct result and the distractor was ±10 for the other half, in order to minimize parity based solution strategies [21-23]. Additionally, half of the distractors differed from the correct result only in the units position, while in the other half only the tens position was different from the correct result. ...
Recently it was suggested that the carry effect observed in addition involves both categorical and continuous processing characteristics.
In the present study, we aimed at identifying the specific neural correlates associated with processing either categorical or continuous aspects of the carry effect in an fMRI study on multi-digit addition.
In line with our expectations, we observed two distinct parts of the fronto-parietal network subserving numerical cognition to be associated with either one of these two characteristics. On the one hand, the categorical aspect of the carry effect was associated with left-hemispheric language areas and the basal ganglia probably reflecting increased demands on procedural and problem solving processes. Complementarily, the continuous aspect of the carry effect was associated with increased intraparietal activation indicating increasing demands on magnitude processing as well as place-value integration with increasing unit sum.
In summary, the findings suggest representations and processes underlying the carry effect in multi-digit addition to be more complex and interactive than assumed previously.
... Half of the solution probes in incorrect equations differed from the correct result by ±2, whereas the split for the other half was ±10. These splits were chosen to minimize parity based solution strategies (Krueger, 1986;Lemaire & Siegler, 1995;Campbell, Parker & Doetzel, 2004). Additionally, this choice implies that half of the probes in incorrect equations differed from the correct result only in the tens position, while in the other half only the unit position was different. ...
Recent research has suggested addition performance to be determined by both the need for a carry operation and problem size. Nevertheless, it has remained debatable, how these two factors are interrelated. In the current study, this question was pursued by orthogonally manipulating carry and problem size in two-digit addition verification. As the two factors interacted reliably, our results indicate that the carry effect is moderated by number magnitude processing rather than representing a purely procedural, asemantic sequence of processing steps. Moreover, it was found that the carry effect may not be a purely categorical effect but may be driven by continuous characteristics of the sum of the unit digits as well. Since the correct result of a carry problem can only be derived by integrating and updating the magnitudes of tens and units within the place-value structure of the Arabic number system, the present study provides evidence for the idea that decomposed processing of tens and units also transfers to mental arithmetic.
... The problem-size effect reflects, in part, differences in solvers' solution approaches. In particular, the problem-size effect is greater when adults use procedural solutions (such as counting) than when they use memory retrieval to solve problems (Campbell & Fugelsang, 2001;Campbell, Parker, & Doetzel, 2004;Campbell & Penner-Wilger, 2006;LeFevre et al., 1996). Accordingly, the problem-size effect in subtraction is typically much larger than that in addition, because solvers use procedural solutions more frequently (Campbell & Xue, 2001;LeFevre et al., 2006;Seyler et al., 2003). ...
Are negative numbers processed differently from positive numbers in arithmetic problems? In two experiments, adults (N = 66) solved standard addition and subtraction problems such as 3 + 4 and 7 - 4 and recasted versions that included explicit negative signs-that is, 3 - (-4), 7 + (-4), and (-4) + 7. Solution times on the recasted problems were slower than those on standard problems, but the effect was much larger for addition than subtraction. The negative sign may prime subtraction in both kinds of recasted problem. Problem size effects were the same or smaller in recasted than in standard problems, suggesting that the recasted formats did not interfere with mental calculation. These results suggest that the underlying conceptual structure of the problem (i.e., addition vs. subtraction) is more important for solution processes than the presence of negative numbers.
... As in other domains, the acquisition of arithmetic expertise is reflected by a shift from slow and effortful processing to skilled and fast retrieval. While young children use timeconsuming step-by-step procedures (Ashcraft, 1992;Fuson, 1982Fuson, , 1988, school age children show an increasing degree of optimization (Barrouillet and Fayol, 1998;Lemaire and Siegler, 1995) and finally retrieve a set of arithmetic problems from memory (note, however, that adults still often use strategies for solving arithmetic problems; Campbell et al., 2004;Campbell and Timm, 2000;Campbell and Xue, 2001;Geary et al., 1993;Geary and Wiley, 1991;Kirk and Ashcraft, 2001;LeFevre et al., 1996a,b;LeFevre and Morris, 1999). ...
It is widely accepted that the human brain is remarkably adaptive not only in child development, but also during adulthood. Aim of this work is to offer an overview and a systematic analysis of neuroimaging studies on the acquisition of arithmetic expertise. In normally developing children and adults, the gain of arithmetic competence is reflected by a shift of activation from frontal brain areas to parietal areas relevant for arithmetic processing. A shift of activation is also observed within the parietal lobe from the intraparietal sulci to the left angular gyrus. Increases in angular gyrus activation with gaining of expertise have also been documented in other cognitive domains. It appears that the left angular gyrus activation is modulated by inter-individual differences in arithmetic performance. The comparison of normal individuals with exceptionally performing individuals (e.g., calculating prodigies) suggests that the experts' arithmetic proficiency relies on a more extended activation network than the network found in non-experts. In expert individuals with long-lasting, extensive mathematical training, specific structural brain modifications are also evident.
... Developmental studies (Barrouillet and Fayol, 1998; Lemaire and Siegler, 1995; Siegler, 1988) as well as experimental studies with adults (Anderson et al., 1999; Logan, 1988; Logan and Klapp, 1991; Rickard, 2004) converge on the view that the acquisition of arithmetic expertise is reflected by a shift from slow and effortful back-up strategies to skilled and fast retrieval from memory (but see for a different view Baroody, 1983 Baroody, , 1994 Baroody, , 1999). However, there is evidence that adults do not systematically retrieve answers to all simple addition or multiplication problems from long-term memory, but still apply a variety of back-up strategies even in problems with one-digit operands (Campbell and Timm, 2000; Campbell and Xue, 2001; Campbell et al., 2004; Geary and Wiley, 1991; Geary et al., 1993; Kirk and Ashcraft, 2001; LeFevre and Morris, 1999; LeFevre et al., 1996a,b). Whether memory retrieval and arithmetic back-up strategies are applied in parallel on a particular item or whether they exclude each other is under debate. ...
The present fMRI study investigates, first, whether learning new arithmetic operations is reflected by changing cerebral activation patterns, and second, whether different learning methods lead to differential modifications of brain activation. In a controlled design, subjects were trained over a week on two new complex arithmetic operations, one operation trained by the application of back-up strategies, i.e., a sequence of arithmetic operations, the other by drill, i.e., by learning the association between the operands and the result. In the following fMRI session, new untrained items, items trained by strategy and items trained by drill, were assessed using an event-related design. Untrained items as compared to trained showed large bilateral parietal activations, with the focus of activation along the right intraparietal sulcus. Further foci of activation were found in both inferior frontal gyri. The reverse contrast, trained vs. untrained, showed a more focused activation pattern with activation in both angular gyri. As suggested by the specific activation patterns, newly acquired expertise was implemented in previously existing networks of arithmetic processing and memory. Comparisons between drill and strategy conditions suggest that successful retrieval was associated with different brain activation patterns reflecting the underlying learning methods. While the drill condition more strongly activated medial parietal regions extending to the left angular gyrus, the strategy condition was associated to the activation of the precuneus which may be accounted for by visual imagery in memory retrieval.
High-quality mathematics education not only improves life outcomes for individuals but also drives innovation and progress across society. But what exactly constitutes high-quality mathematics education? In this article, we contribute to this discussion by focusing on arithmetic fluency. The debate over how best to teach arithmetic has been long and fierce. Should we emphasize memorization techniques such as flashcards and timed drills or promote “thinking strategies” via play and authentic problem solving? Too often, recommendations for a “balanced” approach lack the depth and specificity needed to effectively guide educators or inform public understanding. Here, we draw on developmental cognitive science, particularly Sfard’s process–object duality and Karmiloff-Smith’s implicit–explicit knowledge continuum, to present memorization and thinking strategies not as opposing methods but as complementary forces. This framework enables us to offer specific recommendations for fostering arithmetic fluency based on the science of learning. We define arithmetic fluency, provide evidence on its importance, describe the cognitive structures and processes supporting it, and share evidence-based guidance for promoting it. Our recommendations include progress monitoring for early numeracy, providing explicit instruction to teach important strategies and concepts, implementing well-structured retrieval practice, introducing time-limited practice only after students demonstrate accuracy, and allocating sufficient time for discussion and cognitive reflection. By blending theory, evidence, and practical advice, we equip educators and policymakers with the knowledge needed to ensure all children have access to the opportunities needed to achieve arithmetic fluency.
The ability to select the correct dose when using pen-injectors is both a prerequisite for accurate delivery of medicinal products and a high priority for many users when choosing their treatment. A large body of published work has explored how different pen-injector designs affect safety and effectiveness. However, despite the importance of dose selection, publications on the design of dose dials remain scarce. Many pen-injectors enable users to vary the dose by twisting a dial and the selected dose is shown on an analogue scale drum. Thus, visual inspection is necessary to confirm that the correct dose has been selected, and it is crucial that the visual design of the scale drum accommodates selection of all possible doses. However, it is not always possible to represent all doses numerically on the scale, and users might need to rely on markings between numbers. This paper details an online user study (n=200) exploring how the readability of doses displayed on scale drums is affected by different scale drum designs. Specifically, we explored designs where (1) the scale is based on even or odd tens in increments of 20 (i.e., either 0, 20, 40, 60, etc. or 10, 30, 50, 70, etc.), and (2) the dose is represented by a numerical value or a line marking between two values. The study did not reveal differences between scale drums displaying odd and even tens, which might indicate that the two number representations are similar in terms of readability. However, the results suggest that it might be harder to read the current dose from a line marking than a number. Taken together, these results indicate that it is relevant to consider whether common doses are administered in an even or odd number of units when designing scale drums. Moreover, the results indicate that even-numbered endpoints on odd scales might be harder to read, and none of the compared designs were able to alleviate this problem completely. Finally, even though the online format limits ecological validity, the study yielded valuable insights about the relative readability of the compared design. Thus, the study indicates that surveys may be used as an inexpensive and effective supplement to physical studies.
Some models of memory for arithmetic facts (e.g., 5+2=7, 6×7=42) assume that only the max-left order is stored in memory (e.g., 5+2=7 is stored but not 2+5=7). These models further assume an initial comparison of the two operands so that either operand order (5+2 or 2+5) can be mapped to the common internal representation. We sought evidence of number comparison in simple addition and multiplication by manipulating size congruity. In number comparison tasks, performance costs occur when the physical and numerical size of numerals are incongruent (8 3) relative to when they are congruent (8 3). Sixty-four volunteers completed a number comparison task, an addition task, and a multiplication task with both size congruent and size incongruent stimuli. The comparison task demonstrated that our stimuli were capable of producing robust size congruity and split effects. In the addition and multiplication task, however, we were unable to detect any of the RT signatures of comparison or reordering processes despite ample statistical power: Specifically, there was no evidence of size congruity, split, or order effects in either the addition or multiplication data. We conclude that our participants did not routinely engage a comparison operation and did not consistently reorder the operands to a preferred orientation.
The commentators have raised many pertinent points that allow us to refine and clarify our view. We classify our response comments into seven sections: automaticity; developmental and educational questions; priming; multiple representations or multiple access(?); terminology; methodological advances; and simulated cognition and numerical cognition. We conclude that the default numerical representations are not abstract.
Some current models of mathematical cognition (Dehaene, 1992; Campbell & Clark, 1992) make strong claims about the code in which arithmetical operations are solved, basing themselves on how these operations were originally acquired or are most frequently employed. However, data on acquisition and use are often derived from anecdotic reports and no systematic figures have ever been collected. In this study a questionnaire was devised to investigate how participants learned multiplication tables, as well as the code in which one-digit and multi-digit operations are usually solved. The questionnaire was administered to two groups of university students, one Spanish (Study 1) and the other Belgian (Study 2). The results show that multiplication tables are mainly learnt by oral rehearsal, but adults solve multiplications more frequently by visualizing Arabic digits. This is also their preferred code for calculating additions, subtractions, and divisions. The preference for the Arabic code increases when subjects have to solve multidigit operations. When dealing with numbers, many different formats can be used to convey the same meaning: Arabic digits, number words, Roman numbers or dots, among others, are all valid ways of representing a precise numerosity. The question, however, is whether some formats are particularly good at performing a given numerical task. On this matter the three main current models on numerical cognition differ (see Figure 1).
Korean deaf signers performed a number comparison task on pairs of Arabic digits. In their response times profiles, the expected magnitude effect was systematically modified by properties of number signs in Korean sign language in a culture-specific way (not observed in hearing and deaf Germans or hearing Chinese). We conclude that finger-based quantity representations are automatically activated even in simple tasks with symbolic input although this may be irrelevant and even detrimental for task performance. These finger-based numerical representations are accessed in addition to another, more basic quantity system which is evidenced by the magnitude effect. In sum, these results are inconsistent with models assuming only one single amodal representation of numerical quantity.
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.
Die Arbeit beschäftigt sich mit der Frage, inwiefern sich die medialen Eigenschaften von Zeichensystemen auf die mentale Verarbeitung solcher Systeme auswirken. Die Zahlenverarbeitung von deutschen Gehörlosen eignet sich besonders für die Untersuchung dieser Fragestellung, da sich das Zahlensystem der Deutschen Gebärdensprache (DGS) nicht nur hinsichtlich der visuell-räumlichen Modalität von dem gesprochener Zahlen unterscheidet, sondern auch im Hinblick auf die zugrunde liegende Struktur von gesprochenen und arabischen Zahlen verschieden ist. Es wird gezeigt, dass es sich bei DGS-Zahlen um ein Dekaden-basiertes Zahlensystem mit einer Sub-Basis-5-Struktur handelt. Das gilt nicht für alle Gebärdensprachen; z.B. hat das Zahlensystem der Amerikanischen Gebärdensprache (American Sign Language; ASL) eine reine 10er-Basis. Dadurch, dass deutsche Gehörlose auch sehr vertraut im Umgang mit arabischen Zahlen und geschriebenen deutschen Zahlwörtern sind, nutzen sie also nicht nur drei verschiedene Zahlensymbolsysteme, sondern auch zwei verschiedene Zahlensysteme. In sieben empirischen Studien zu wichtigen Effekten der Zahlenverarbeitung (SNARC-Effekt, MARC-Effekt und Distanz-Effekt) wird in dieser Arbeit die mentale Verarbeitung der verschiedenen Zahlensysteme bei deutschen Gehörlosen systematisch untersucht. Mit dem Vergleich der Ergebnisse der deutschen gehörlosen Probanden mit denen von US-amerikanischen Gehörlosen und deutschen Hörenden sollten Hinweise darüber gewonnen werden, welche Effekte der Zahlenverarbeitung an eine bestimmte sprachliche Modalität und welche an sprachspezifische Zahlensysteme gebunden sind. In den Ergebnissen zeigten sich formatspezifische Unterschiede zwischen der Verarbeitung von den 10er-basierten Zahlensystemen und dem DGS-Zahlensystem auf der Sub-Basis-5. In einem Experiment, in dem deutsche gehörlose Probanden Paritätsentscheidungen mit DGS-Zahlen zu fällen hatten, zeigte sich, dass der Paritätsstatus der dominanten Hand relevanter für die Reaktionsgeschwindigkeiten war, als der Paritätsstatus der gesamten Zahl. Dies wurde für die Beurteilung von sprachlichen Zahlzeichen beobachtet und bei einigen Probanden auch für arabische Zahlen, aber nicht bei der Darbietung von Punktmengen. Darüber hinaus wurde ein DGS-spezifischer Distanz-Effekt beobachtet. In numerischen Größenvergleichen mit der festen Vergleichszahl fünf waren die Reaktionen auf die Zahlen vier und sechs im Mittel besonders langsam, während sich die mittleren Reaktionszeiten für alle anderen Zahlen nicht überzufällig voneinander unterschieden. Dieser Effekt wurde lediglich in den Ergebnismustern deutscher Gehörloser für DGS-Zahlen oder entsprechende geschriebene Zahlwörter beobachtet. Aufgrund der ähnlichen Ergebnismuster für DGS-Zahlen und geschriebene Zahlwörter wurde angenommen, dass sprachliche Zahlzeichen von Gehörlosen über ein inneres Gebärden enkodiert werden. Beide DGS-spezifischen Effekte wurden ausschließlich in den Ergebnissen der deutschen Gehörlosen, nicht aber in denen der US-amerikanischen Gehörlosen oder deutschen Hörenden beobachtet. Auch deshalb wurde angenommen, dass diese Effekte durch die Sub-Basis-5-Struktur des DGS-Zahlensystems bedingt sind. Aufgrund der formatspezifischen Effekte wurde weiterhin gefragt, ob DGS-Zahlen und arabische Zahlen abgesehen von einer formatspezifischen Enkodierungsphase durch dieselben kognitiven Strukturen verarbeitet werden, wie dies für Hörende angenommen wird (z.B. Dehaene, 1997; Nacchache & Dehaene, 2001; Nuerk, Wood & Willmes, 2005). Die Ergebnisse einer cross-notationalen Primingaufgabe stützten diese Annahme. Allerdings wurde in derselben Studie gezeigt, dass das zweimalige Zeigen derselben DGS-Zahl zu keiner Reaktionszeitbeschleunigung führte. Dieser fehlende Priming-Effekt innerhalb einer Zahlendarstellung wurde mit Interferenzen zwischen den motorischen Programmen zum Sprachverstehen und solchen zur Antwortseingabe erklärt. Weitere Hinweise auf solche Interferenzen ergaben sich aus den Ergebnissen beider gehörloser Gruppen: bei Größenvergleichsaufgaben zeigte sich eine Überlagerung der SNARC-Effekte durch ebendiese Interferenzen. Insgesamt wurden in den Experimenten deutliche Hinweise auf einen Einfluss der medialen Eigenschaften des DGS-Zahlensystems auf die mentale Zahlenverarbeitung beobachtet. Möglicherweise wurden vergleichbare formatspezifische Effekte in den Ergebnissen von hörenden Probanden aufgrund der fehlenden medialen Differenz in Bezug auf die Struktur der sprachlichen und der arabischen Zahlensysteme bisher nicht sichtbar. This dissertation addresses the question whether media-specific features of sign systems have an influence on the mental processing of these signs or not. This research question is investigated with the mental number processing of German deaf signers. The number system in the German Sign Language (Deutsche Gebärdensprache, DGS) differs from German spoken numbers in regard to the visual-spatial language modality. Besides that, the structure of the DGS number system is different from that of Arabic and spoken numbers. This paper shows that the DGS number system is a base-10 one with a sub-base-5. However, this system is language-specific and does not apply for all sign languages. The American Sign Language (ASL) is one example of a visual-spatial language with a pure base-10 number system. German signers are familiar with the processing of base-10 Arabic numbers as well as the processing of written German numbers. Thus, German signers are not only using three different number symbol systems but also two different number systems. In this dissertation, the processing of those different symbol systems in German deaf signers is examined systematically by means of seven empirical studies regarding several important number-processing effects (SNARC effect, MARC effect and distance effect). A comparison of the results of the German deaf group with those of US-American deaf as well as those of hearing German participants should give some answers to the question which aspects of number processing are linked to language modality and which are associated with a language-specific number system. The results indicated format-specific differences in the processing of base-10 number systems on the one hand and the DGS number system on the sub-base-5 on the other hand. In a parity decisions task with DGS numbers the parity status of the dominant handshape seemed to be more relevant for reaction time patterns than the parity status of the whole number sign. This was observed in the DGS and written German number judgments by German signers. It was also partly true for some participants judging Arabic numbers, but not for the judgment of dot patterns. Moreover a DGS-specific distance effect was observed. In magnitude comparisons with a fixed standard of five the reaction times for the numbers four and six (5 +- 1) were very slow, but the mean reaction times for all other number comparisons did not differ significantly from each other. This DGS-specific distance effect was exclusively observed when German signers judged DGS or written numbers, but not in the results patterns of Arabic numbers or dot patterns. Since the result patterns for the DGS numbers and the written numbers were very similar, it was assumed that written numbers are encoded by German deaf signers via an inner signing. Both effects were observed for German signers only, not for US-American signers or hearing German participants; this substantiated the assumption that those effects were related to the sub-base-5 structure of the DGS number system. These DGS-specific effects raised the question whether DGS and Arabic numbers are processed via the same cognitive structures as is assumed for the hearing population (e.g. Dehaene, 1997; Nacchache & Dehaene, 2001; Nuerk, Wood & Willmes, 2005). The results of a cross-notational priming study with German signers supported this assumption. However, the same study showed that there was no reaction facilitation when a DGS number was primed with exactly the same DGS number. This lack of a priming effect was explained with interferences between the motor programs involved in language perception and those motor programs controlling the bimanual key presses necessary for supplying an answer to the experiment. Further indications for this kind of interferences were observed in the results patterns of the number magnitude comparisons of both deaf groups. Since there were no visible SNARC effects, it was assumed that the SNARC effect can be overlaid by those interferences. On the whole, there were clear links to an influence of the media-specific features of the DGS number system on mental number processing. Probably similar format-specific effects have not yet been observed for the hearing population because there are no differences in the main structure of the Arabic number system and the number systems of the investigated languages.
The neural realization of number in abstract form is implausible, but from this it doesn't follow that numbers are not abstract. Clear definitions of abstraction are needed so they can be applied homogeneously to numerical and non-numerical cognition. To achieve a better understanding of the neural substrate of abstraction, productive cognition--not just comprehension and perception--must be investigated.
Single-neuron recordings may help resolve the issue of abstract number representation in the parietal lobes. Two manipulations in particular - reversible inactivation and adaptation of apparent numerosity - could provide important insights into the causal influence of "numeron" activity. Taken together, these tests can significantly advance our understanding of number processing in the brain.
We delineate a developmental model of number representations. Notably, developmental dyscalculia (DD) is rarely associated with an all-or-none deficit in numerosity processing as would be expected if assuming abstract number representations. Finally, we suggest that the "generalist genes" view might be a plausible--though thus far speculative--explanatory framework for our model of how number representations develop.
Much evidence cited by Cohen Kadosh & Walsh (CK&W) in support of their notation-specific representation hypothesis is based on tasks requiring automatic number processing. Several of these findings can be alternatively explained by differential expertise in mapping numerical symbols onto semantic magnitude representations. The importance of considering symbol-referent mapping expertise in theories on numerical representations is highlighted.
Cohen Kadosh & Walsh (CK&W) present convincing evidence indicating the existence of notation-specific numerical representations in parietal cortex. We suggest that the same conclusions can be drawn for a particular type of numerical representation: the representation of time. Notation-dependent representations need not be limited to number but may also be extended to other magnitude-related contents processed in parietal cortex (Walsh 2003).
The dual-code proposal of number representation put forward by Cohen Kadosh & Walsh (CK&W) accounts for only a fraction of the many modes of numerical abstraction. Contrary to their proposal, robust data from human infants and nonhuman animals indicate that abstract numerical representations are psychologically primitive. Additionally, much of the behavioral and neural data cited to support CK&W's proposal is, in fact, neutral on the issue of numerical abstraction.
Like number representation, basic arithmetic seems to be a natural candidate for abstract instantiation in the brain. To investigate this, researchers have examined effects of numeral format on elementary arithmetic (e.g., 4+5 vs. four+five). Different numeral formats often recruit distinct processes for arithmetic, reinforcing the conclusion that number processing is not necessarily abstracted away from numeral format.
The study of neuronal specialisation in different cognitive and perceptual domains is important for our understanding of the human brain, its typical and atypical development, and the evolutionary precursors of cognition. Central to this understanding is the issue of numerical representation, and the question of whether numbers are represented in an abstract fashion. Here we discuss and challenge the claim that numerical representation is abstract. We discuss the principles of cortical organisation with special reference to number and also discuss methodological and theoretical limitations that apply to numerical cognition and also to the field of cognitive neuroscience in general. We argue that numerical representation is primarily non-abstract and is supported by different neuronal populations residing in the parietal cortex.
A fundamental question in the study of consciousness is the connection between subjective report and objective measures. We explored this question by testing NM, a grapheme-color synesthete, who experiences colors when viewing digits but not dot patterns. Synesthesia research has traditionally used variants of the Stroop paradigm as an objective correlate of these subjective synesthetic reports. We used both a classical synesthetic Digit Stroop task and a novel Numerosity Stroop task, in which random dot patterns were colored either congruently or incongruently with the colors NM reported for digits. We observed longer response times in the incongruent condition for both tasks, despite the fact that NM denied experiencing colors for random dot patterns, constituting a clear dissociation between subjective and objective measures of synesthetic experience. We argue that distinguishing synesthesia from learned synesthesia-like associations (pseudosynesthesia) should depend primarily on the presence of subjective reports, validated by objective measures. More generally, we suggest that consciously and unconsciously mediated interference may arise from qualitatively different mechanisms.
The problem size effect in adult arithmetic performance is generally attributed to direct retrieval processes operating on a network representation in long-term memory. J. LeFevre and her colleagues (J. LeFevre, J. Bisanz, et al., 1996; J. LeFevre, G. S. Sadesky, & J. Bisanz, 1996) challenged this explanation using verbal report evidence that adults also use time consuming nonretrieval strategies to solve simple addition and multiplication. The authors replicated J. LeFevre and colleagues' methods, but added instructional biasing and silent control conditions to test these methods. Both reaction time and report results suggest that LeFevre and colleagues' conclusions about nonretrieval frequency may have been influenced by instructions that revealed the experimental hypothesis and affected participants' strategy reports. Obtaining evidence about adult strategy use in simple arithmetic will require understanding instructional demand and appropriate report methodology.
This article presents a computational theory of retrieval of simple addition and multiplication facts (e.g. 9 x 6 = ?; 3 + 4 = ?). According to the network-interference model, a presented problem activates memory representations for a large number of related number facts, with strength of activation of specific facts determined by featural and magnitude similarity to the presented problem. Nonnative frequencies of confusion errors (e.g. 9 x 6 = 36) were u,sed to quantify similarity factors. Problem nodes continuously receive similarity-based excitatory input during retrieval and compete by way of mutual inhibition until one node reaches a critical activation threshold and triggers a response. The model demonstrates that similarity-based interference, in addition to accounting for many features of eJTors, also provides accurate prediction of variability in speed of correct retrieval among problems. The model also accounts for subtle features of inter-trial err.or priming, as well as changes in the rates and types of errors observed as a function of elapsed retrieval time. Most educated adults can quickly and accurately produce the answers to a large number of simple arithmetic problems (e.g. 4 + 2 = ?, 6 x 6 = ?). This knowledge of basic number facts is usually characterised as "simple arithmetic", but experimental analysis has uncovered many subtle phenomena reflecting a complex system of interacting memory representations and retrieval processes. Understanding these memory processes is important both because of the status of simple mental arithmetic as a fundamental intellectual skill and because simple arithmetic provides a unique opportunity to study elementary memory processes in a highly constrained domain. The article presents a revision of the network-inJeiference model of number-fact retrieval described by Campbell and Oliphant (1992). The model constitutes a detailed theory of the memory codes Requests for reprints should be sent to Jamie Campbell,
We tabulated the frequency with which simple addition and multiplication facts occur in elementary school arithmetic texts for grades 1-6. The results indicated a strong "small-fact bias" in both addition and multiplication. "Large" facts, with operands larger than 5, occurred up to half as frequently as those with operands in the 2-5 range. As was also found in an earlier tabulation for grades K-3, facts with operands of 0 and 1 occurred relatively infrequently; the ostensible exceptions to this pattern, high frequencies for combinations like 1+2 and 1× 3, were caused by the small-fact bias in multicolumn problems. The small-fact bias in the presentation of basic arithmetic, at least to the degree observed here, probably works against a basic pedagogical goal, mastery of simple arithmetic. It may also provide a partial explanation of the widely reported problem size or problem difficulty effect, that children's and adults' responses to larger basic facts are both slower and more error prone than their solutions to smaller facts. Theoretical and practical implications of the small-fact bias are discussed briefly.
Nine experiments of timed odd–even judgments examined how parity and number magnitude are accessed from Arabic and verbal numerals. With Arabic numerals, Ss used the rightmost digit to access a store of semantic number knowledge. Verbal numerals went through an additional stage of transcoding to base 10. Magnitude information was automatically accessed from Arabic numerals. Large numbers preferentially elicited a rightward response, and small numbers a leftward response. The Spatial–Numerical Association of Response Codes effect depended only on relative number magnitude and was weaker or absent with letters or verbal numerals. Direction did not vary with handedness or hemispheric dominance but was linked to the direction of writing, as it faded or even reversed in right-to-left writing Iranian Ss. The results supported a modular architecture for number processing, with distinct but interconnected Arabic, verbal, and magnitude representations. (PsycINFO Database Record (c) 2012 APA, all rights reserved)
Widely usedmathematics textbooks from kindergarten, first, second, and third grades were evaluated both for the frequency and the order of presentation of the basic 100 addition facts 0 + 0 through 9 + 9. A striking result was that the frequency of occurrence distributions for the basic facts were markedly skewed. There were many fewer presentations of large than small problems in the texts, and problems involving the addition of 0 were relatively infrequent at all grades; such was also the case in a similar study conducted in 1924. Correlations among the tabulated variables and a variety of published normative and performance measures are presented. We suggest several possible relationships between these results and the empirical literature on mental arithmetic performance. In particular, reaction time performance across the entire developmental span is closely related to both the frequency of presentation and the order of presentation of the basic facts in elementary texts. Furthermore, both the frequency and order of presentation variables have strong and possibly important relationships to adults' ratings of problem difficulty.
12 males and 8 females made odd vs even classification judgments on numbers presented visually in 3 different formats: digits, number words, and dot patterns. Males showed longer decision times for odd numbers only when the numbers were presented in the dot pattern format. Females showed this effect only when the stimuli were presented in the word format. These results suggest that a differential speed of response to odd and even numbers is found most strongly when the stimuli are presented in a format which is processed more efficiently by the Ss. This finding may imply that the effect is being produced by some higher order cognitive process, based on higher order representations that do not necessarily involve linguistic or verbal coding. (PsycINFO Database Record (c) 2012 APA, all rights reserved)
McCloskey, Sokol, and Goodman (1986) presented a model of verbal-number production that was based on the Arabic-number-reading errors of several brain-damaged subjects. The model assumed that number processing entails separate comprehension, calculation, and production mechanisms interconnected by a single type of abstract quantity code. We propose instead that numbers activate multiple specific representations functionally integrated in an encoding complex. Further analyses of the number-word confusion matrix produced by one of their subjects, HY, showed that his Arabic-number-reading errors were predicted by the visual similarity of digits and by numerical relations (numerical proximity and odd–even agreement). These findings and other research on number processing suggest a more complex encoding architecture than McCloskey et al assumed and raise questions about their conclusion that HY's deficit is localized within a verbal-number production system, and also about the psychological distinctions among number comprehension, calculation, and production. (PsycINFO Database Record (c) 2012 APA, all rights reserved)
How do people decide whether to try to retrieve an answer to a problem or to compute the answer by some other means? The authors report 2 experiments showing that this decision is based on problem familiarity rather than on retrievability of some answer (correct or incorrect), even when problem familiarization occurred 24 hr earlier. These effects at the level of the individual problem solver and the results reported by L. M. Reder and F. E. Ritter (see record
1992-30097-001) are well fit with the same parameter values in a spreading-activation computational model of feeling of knowing in which decisions to retrieve or compute an answer are based on the familiarity or activation levels of the problem representation. The authors therefore argue that strategy selection is governed by a familiarity-based feeling-of-knowing process rather than by a process that uses the availability of the answer or some form of race between retrieving and computing the answer. (PsycINFO Database Record (c) 2012 APA, all rights reserved)
Researchers have assumed that adults solve simple arithmetic problems by retrieving answers from a network of stored facts. In 2 studies, undergraduates described their solutions of single-digit multiplication problems. They reported direct retrieval on approximately 80% of trials but also reported rules (e.g., anything times 0 is 0), repeated addition (e.g., 2 × 4 = 4 + 4), number series (e.g., 3 × 5 = 5, 10, 15), and derived facts (e.g., 6 × 7 = [6 × 6] + 6). Participants were slower to retrieve problems that were most likely to be solved by nonretrieval procedures and faster to retrieve problems that were usually solved by retrieval. These results indicate that direct-retrieval models are incomplete accounts of adults' performance and support a continuing influence of learning and experience on the mental representation of simple multiplication problems. (PsycINFO Database Record (c) 2012 APA, all rights reserved)
To assess the validity of verbal reports in children's subtraction, students in Grades 1, 3, and 5 were asked to solve a set of simple subtraction problems and were placed in the no-report, retrospective-report, or concurrent-report conditions. Two aspects of verbal report validity were assessed: reactivity, or whether providing a verbal report alters subsequent task performance, and veridicality, or whether the verbal reports are accurate reflections of solution strategies. Students in all grades and in both the retrospective-report and concurrent-report conditions were able to provide veridical strategy reports, and the instruction to verbally report had few effects on task performance. Informal findings indicate that students had less difficulty reporting retrospectively than concurrently. (PsycINFO Database Record (c) 2012 APA, all rights reserved)
Current models of mental arithmetic disagree on whether the same mental representation is activated by numbers in different forms (single-format assumption). Two experiments examined if the response times in arithmetic fact retrieval from arabic digits, number words, or dice are independent of the graphical representation of the numbers. S. Sternberg's (1969) additive factors method was used to analyze which response time interactions are compatible with the single-format assumption. In accord with the assumption, additive effects of form and operation were found in Experiment 1. In contrast, Experiment 2 showed additive effects only when an operand was held in memory, but a strong Form X Operation interaction otherwise. The authors discuss how phonological interference and form-specific retrieval could produce these interactions and how the single-format assumption can be extended to handle the findings. (PsycINFO Database Record (c) 2012 APA, all rights reserved)
Adults' solution times to simple addition problems typically increase with the sum of the problems (the problem size effect). Models of the solution process are based on the assumption that adults always directly retrieve answers to problems from an associative network. Accordingly, attempts to explain the problem size effect have focused either on structural explanations that relate latencies to numerical indices (e.g., the area of a tabular representation) or on explanations that are based on frequency of presentation or amount of practice. In this study, the authors have shown that the problem size effect in simple addition is mainly due to participants' selection of nonretrieval procedures on larger problems (i.e., problems with sums greater than 10). The implications of these results for extant models of addition performance are discussed. Twenty years of research on mental arithmetic has shown that problems involving larger numbers (e.g., 9 + 6) are solved more slowly than problems involving smaller numbers (e.g., 3 + 4). Surprisingly, in spite of the wealth of empirical data and the extensive theoretical development on mental arith-metic, the problem size effect has eluded satisfactory explana-tion (Ashcraft, 1992; McCloskey, Harley, & Sokol, 1991; Widaman & Little, 1992). The goal of the present research was to test an explanation of the problem size effect in adults that has been used to account for the arithmetic performance of children (Ashcraft, 1992; Siegler, 1987). We hypothesized that variability in the selection of procedures to solve simple addition problems has a major impact on solution latencies and may account for a substantial portion of the problem size effect.
The area of cognitive arithmetic is concerned with the mental representation of number and arithmetic, and the processes and procedures that access and use this knowledge. In this article, I provide a tutorial review of the area, first discussing the four basic empirical effects that characterize the evidence on cognitive arithmetic: the effects of problem size or difficulty, errors, relatedness, and strategies of processing. I then review three current models of simple arithmetic processing and the empirical reports that support or challenge their explanations. The third section of the review discusses the relationship between basic fact retrieval and a rule-based component or system, and considers current evidence and proposals on the overall architecture of the cognitive arithmetic system. The review concludes with a final set of speculations about the all-pervasive problem difficulty effect, still a central puzzle in the field.
This paper contrasts two views of the cognitive architecture underlying numerical skills and acalculia. According to the abstract-modular theory (e.g., McCloskey, Caramazza, & Basili, 1985), number processing is comprised of independent comprehension, calculation, and production subsystems that communicate via a single type of abstract quantity code. The alternative, specific-integrated theory (e.g., Campbell & Clark, 1988), proposes that visuospatial, verbal, and other modality-specific number codes are associatively connected as an encoding complex and that different facets of number processing generally involve common, rather than independent, processes. The hypothesis of specific number codes is supported by conceptual inadequacies of abstract codes, format-specific phenomena in calculation, the diversity of acalculias and individual differences in number processing, lateralization issues, and the role of format-specific codes in working memory. The integrated, associative view of number processing is supported by the dependence of modular views on abstract codes and other conceptual inadequacies, evidence for integrated associative networks in calculation tasks, acalculia phenomena, shortcomings in modular architectures for number-processing dissociations, close ties between semantic and verbal aspects of numbers, and continuities between number and nonnumber processing. These numerous logical and empirical considerations challenge the abstract-modular theory and support the encoding-complex view that number processing is effected by integrated associative networks of modality-specific number codes.
Are numerical stimuli represented in an abstract code or is information about surface format preserved in memory? Three experiments bearing on this issue are reported. All used an incidental recall procedure. Two experiments examined memory for numeral versus word format in monolinguals and the third examined memory for words in the first versus second language of Spanish-English bilinguals. Across all studies, memory for format of number presentation was above chance and was as high as 76% under certain conditions. Implications for number representation are discussed.
Eight experiments are reported that first establish and then explicate a serendipitous finding that judgments about whether digits are odd or even take longer for odd than for even digits. The slowing of judgments about odd digits is more pronounced when digit pairs or triples are used, but is still weakly present when a single digit must be classified. A similar effect is seen when judgments of nouns are based on whether the nouns are the names of living or dead objects. Nouns that name dead objects are judged more slowly than ones that name living objects. The concept "alive" is linguistically marked. Past research has shown that unmarked concepts are processed more rapidly than marked ones. The similarity in the pattern of results when digits and words are judged is used to argue that the slower judgments about odd digits are due to the fact that "odd" is a linguistically marked and "even" a linguistically unmarked concept.
In the present study, we examined the conditions that favor the plausibility judgment strategy over the retrieval strategy when we verify some statements. In particular, we examined the effect of odd-even status of numbers on subjects' verification of single-digit arithmetic problems. In Experiment 1, we explored how factors such as problem difficulty and stimulus onset asynchrony (SOA) influence this effect in adults. In Experiment 2, we present evidence that this odd-even effect is also present in elementary school children, although it varies with the age of the children, the difficulty of the problems, and the SOA. We argue that the odd-even information is helpful in arithmetic verification tasks with difficult problems early in the verification processes and that the time course of these effects differs across ages. The present results are consistent with the view that the plausibility strategy is preferred over the retrieval strategy at an early stage of processing and with information that is not quickly accessible. Finally, we discuss the implications of the present experiments for understanding of single-digit arithmetic and for understanding the more general issue of how people coordinate use of multiple strategies.
This study reports a longitudinal investigation of French 2nd graders' acquisition of single-digit multiplication skill. Speed, accuracy, and strategy use were assessed 3 times within the year when children learned multiplication. The data showed that improvements in speed and accuracy that generally accompany learning can reflect at least 4 types of specific strategic changes: introduction of new strategies, increasing use of the most efficient existing strategies, more efficient execution of each strategy, and more adaptive choices among strategies. The data also showed substantial continuities in learning: At all 3 points of measurement, children used multiple strategies, used retrieval most often on the same classes of problems, and used repeated addition on the most difficult problems. Stable individual differences were also apparent. The findings supported a number of predictions of Siegler and Shipley's (1995) adaptive strategy choice model. Implications for understanding learning, arithmetic, and strategy choice processes are discussed.
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 7x8) 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.
Do number-fact retrieval (4.6 =?) and numeral reading (e.g., transcoding 46 into "forty six") access the same retrieval structures? Data from the present experiment suggest that they do. Under instructions to respond quickly, adults performed simple multiplication problems oriented horizontally or vertically with operands presented simultaneously or with a 500 ms preview of one operand. Errors involving congruent operand intrusions (e.g., 2.8 = 24 or 9.6 = 36) were most frequent when conditions afforded the left-to-right encoding sequence that is standard for reading multi-digit numbers. Ninety percent of such intrusions involved the correct answer to another simple multiplication problem (e.g., 4.8 = 28) rather than a miscellaneous answer (e.g., 4.8 = 38). This high rate of arithmetically related intrusions suggests that numeral reading and verbal production of number facts involve a shared representational system.
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.
We measured cerebral activation with functional magnetic resonance imaging at 3 Tesla while eight healthy volunteers performed various number processing tasks known to be dissociable in brain-lesioned patients: naming, comparing, multiplying, or subtracting single digits. The results revealed the activation of a circuit comprising bilateral intraparietal, prefrontal, and anterior cingulate components. The extension and lateralization of this circuit was modulated by task demands. The intraparietal and prefrontal activation was more important in the right hemisphere during the comparison task and in the left hemisphere during the multiplication task and was intensely bilateral during the subtraction task. Thus, partially distinct cerebral circuits with the dorsal parietal pathway underlie distinct arithmetic operations.
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).
In this chapter we address the logical and empirical arguments offered by Campbell and Clark (this volume) against our model of numerical processing. We argue that several of their objections are based upon misinterpretations of our model; that the empirical results cited by Campbell and Clark do not constitute clear evidence against the model; and that the approach to theory construction embodied in our model is likely to be more productive than the approach exemplified by Campbell and Clark's encoding complex theory.
According to the encoding-complex approach (Campbell & Clark, 1988; Clark & Campbell, 1991), numerical skills are based on a variety of modality-specific representations (e.g., visuo-spatial and verbal-auditory codes), and diverse number-processing tasks (e.g., numerical comparisons, calculation, reading numbers, etc.) generally involve common, rather than independent, cognitive mechanisms. In contrast, the abstract-modular theory (e.g., McCloskey, Caramazza, & Basili, 1985) assumes that number processing is comprised of separate comprehension, calculation, and production subsystems that communicate via a single type of abstract quantity code. We review evidence supporting the specific-integrated (encoding-complex) view of number processing over the abstract-modular view, and report new experimental evidence that one aspect of number processing, retrieval of simple multiplication facts, involves non-abstract, format-specific representations and processes. We also consider implications of the encoding-complex hypothesis for the modularity of number skills.
In the preceding chapter, McCloskey, Macaruso, and Whetstone (henceforth MM&W) present a number of theoretical and empirical challenges to the encoding-complex view of number processing advanced by Campbell and Clark (this volume; henceforth C&C). In this reply I examine their explanations for the effects of number format on number-fact retrieval found by C&C. I argue that the main alternative accounts offered by MM&W – the “internal deadline” hypothesis and the “encoding efficiency” hypothesis – are not supported by the data and do not provide convincing alternatives to C&C's proposal that retrieval processes differ as a function of format. I also argue that aspects of the explanations offered by MM&W compromise the basic abstract-modular theory and, in fact, undermine MM&W's claim that the modular view is likely to be more productive than the encoding-complex approach. I propose further that the abstract-code theory of number meaning assumed within the modular framework is counterproductive, because it takes for granted complex, fundamental aspects of cognitive number processing. In contrast, it is a primary goal of the encoding-complex approach to provide explanatory mechanisms for these basic elements of numerical cognition.
To investigate effects of surface notation on basic numerical skills we examined number naming, magnitude selection, and simple arithmetic performed by adult Chinese-English bilinguals born and educated in China. Stimuli were presented using either arabic (7 + 8) or ''mandarin'' symbols and participants were cued to respond either in English or Chinese. The naming task demonstrated that the mandarin characters were as easy to identify as the arabic digits, but for both arithmetic and magnitude selection there were faster RTs and fewer errors overall with arabic notation. Arabic notation also produced smaller problem-size effects in arithmetic and a smaller split effect in magnitude selection relative to mandarin notation. These results suggest that retrieval processes in the arithmetic and selection tasks were more efficient with arabic than mandarin stimuli. Arithmetic RTs were substantially slower with English than Chinese responses given either arabic or mandarin stimuli, but the English-language cost was greater with mandarin stimuli. The form of the Notation Language RT interaction is consistent with language-specific Chinese and English number-fact stores (''arithmecons'') that were differentially accessible as a function of notation. Naming RTs also presented a significant Notation Language interaction due mainly to slow RTs to produce English number names for mandarin stimuli. These Notation Language interactions are not easily reconciled with the standard version of McCloskey's (1992) model of number processing, which holds that numeral reading and arithmetic performance are based on a single, abstract-semantic code regardless of input or output conditions. Instead, the results suggest that different Notation Language combinations were mediated by independent associative paths that varied in strength and efficiency as a function of prior experience.
In a verification task of simple additions composed of Arabic or Roman numerals, Gonzalez and Kolers (1982) reported data that were interpreted as supporting the idea that cognitive operations are not independent of the symbols that instigate them. We propose an alternative interpretation of these results and argue that the effects reported may have been produced by a peculiarity of the Roman code for which the encoding time would not be constant for all numerals. We hypothesize that three different “structures” can be distinguished in the Roman code, and that the time necessary to encode a numeral would vary according to its structure, with the analogical (numerals I, II, and HI) and the symbolic (V, X) structures being processed faster than the complex structures (IV, VI, VII, VIII, IX, XI,…). This structure effect is tested in two experiments: a verification of transcoded forms and a parity judgement. Data repeatedly showed support for this hypothesis. Moreover, a verification task for additions showed that the presentation format of the addends played a role in the encoding stage but did not interact with variables relative to the size of the addition problems. These data could thus sustain the hypothesis of a “translation model”, according to which numerals would be translated into a specific code to which the calculation process would be applied.
Across two experiments, the magnitude of the problem-size effect in mental addition was examined for kindergarten and elementary school children, as well as adults, from mainland China and the United States. In North American samples, the problem-size effect represents the finding that arithmetic problems consisting of larger-valued numbers (e.g. 8+7) take longer to solve and are more error prone than are problems consisting of smaller-valued numbers (e.g. 2+3). This standard finding was found for the kindergarten, elementary school, and adult samples from the United States. For the Chinese children, the problem size effect was evident in kindergarten and at the beginning of first grade. However, the effect had disappeared at the end of first grade and had reversed (i.e. larger- valued addition problems were solved more quickly than smaller-valued problems) by the end of third grade. However, the standard problem-size effect "reappeared" for the Chinese adults. The results are interpreted in terms of theoretical models of the nature of the memory representation for arithmetic facts and in terms of the mechanisms that govern the development of these representations. In the nearly 25 years since Groen and Parkman's (1972) seminal study of the mental processes underlying the solution of simple addition problems, cognitive arithmetic has emerged as a vibrant area of research. Scientists in this area have mapped the cognitive processes and neurological correlates that govern the mental solution of simple and complex arithmetic problems and have extended these basic findings to more applied issues, such as mathematical anxiety and mathematical disabilities (Ashcraft, 1992, 1995; Ashcraft & Faust, 1994; Ashcraft, Yamashita, & Aram, 1992; Campbell & Clark, 1988; Campbell & Graham, 1985; Dehaene & Cohen, 1991; Geary,
discuss some aspects of mental arithmetic and the cognitive operations that compose it and, in doing so, are especially concerned to describe the role that particular forms of symbolization or notation play therein
models of mental arithmetic / analog / counting / network models
coding by language / numbering systems (PsycINFO Database Record (c) 2012 APA, all rights reserved)
Adults educated either in the People's Republic of China or in Canada solved single-digit multiplication problems. Chinese adults were faster and made fewer errors than Canadian adults and showed smaller increases in latency and errors as problem size increased. Chinese adults transformed problems with the larger operand on the left (e.g. 6) by reversing the digits and solving the complementary problem (e.g. 9). Only the Canadian adults showed a substantial advantage in latencies and errors for problems with operands of 5. Although both groups showed a latency advantage on ties (e.g. 3) as compared to other problems, the advantage was much larger for the Canadian than for the Chinese adults. These findings were only partially attributable to overall differences in skill; patterns of differences persisted when groups were equated on multi-digit arithmetic performance. Chinese adults made more errors that reflect verbal-production processes that may occur after retrieval, whereas Canadian adults made more errors that reflect retrieval processes. The results are consistent with Siegler's (1988) experiential model of multiplication. Solution of simple arithmetic problems provides a fruitful area within which to examine issues of representation and process in cognitive psychology (Ashcraft, 1995; McCloskey & Macaruso, 1994). Understanding of the arithmetic domain has progressed such that a variety of empirical effects have been closely examined (e.g.
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).
We argue that to best comprehend many data sets, plotting judiciously selected sample statistics with associated confidence intervals can usefully supplement, or even replace, standard hypothesis-testing procedures. We note that most social science statistics textbooks limit discussion of confidence intervals to their use in between-subject designs. Our central purpose in this article is to describe how to compute an analogous confidence interval that can be used in within-subject designs. This confidence interval rests on the reasoning that because between-subject variance typically plays no role in statistical analyses of within-subject designs, it can legitimately be ignored; hence, an appropriate confidence interval can be based on the standard within-subject error term--that is, on the variability due to the subject condition interaction. Computation of such a confidence interval is simple and is embodied in Equation 2 on p. 482 of this article. This confidence interval has two useful properties. First, it is based on the same error term as is the corresponding analysis of variance, and hence leads to comparable conclusions. Second, it is related by a known factor 2 to a confidence interval of the difference between sample means; accordingly, it can be used to infer the faith one can put in some pattern of sample means as a reflection of the underlying pattern of population means. These two properties correspond to analogous properties of the more widely used between-subject confidence interval.
This article discusses cognitive neuropsychological research on acquired dyscalculia (i.e., impaired numerical processing resulting from brain damage), surveying issues of current interest, and illustrating the ways in which analyses of acquired deficits can contribute to an understanding of normal processing. I first review the logic whereby inferences concerning normal cognition are drawn from patterns of impaired performance. I then consider research exploring the general functional architecture of the cognitive numerical processing mechanisms, and finally turn to studies aimed at probing the internal structure and functioning of individual processing components.
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.
Since Japanese learn the multiplication tables by means of mnemonic rhymes called kuku, i.e., via the auditory-speaking route, it is hypothesized that aphasics who have a deficit in kuku might relearn it via the visual-writing route. After having ordinary language therapy for more than 3 months, eight patients were admitted to this special kuku training. The comparisons of scores on multiplication tests showed that the visual-writing route improved most after a 1- or 2-month training period while the auditory-speaking route improved least. These results, suggesting that the visual processor can take on the new role in conducting multiplication, can be considered a realization of functional reorganization in the brain.
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.
A patient is described who is affected by an inability to recall and use 'arithmetical facts' of one-digit multiplications and divisions. This loss contrasts with the preservation of a wide set of complex notions that the patient exploits in order to overcome his deficit and get the right result. This observation helps in isolating and describing an important component of arithmetical long-term memory that is not overlearnt and the functioning of which is not automatic or mechanistic. An account of such a component is lacking in models of arithmetic currently referred to in cognitive neuropsychology. In a remediation study, performed over several weeks, the effect of training was selective for each single arithmetical fact: not even skills with multiplication complements (e.g. 6 x 3, 3 x 6) fully benefited from the rehabilitation of a specific fact. This suggests that the storage format of each fact is independent from that of other facts.
We report the case of an anarithmetic patient with a selective deficit of memory for elementary arithmetic facts, who produced, for instance, "25" in answer to "4 x 5." The patient showed good language comprehension and production abilities, had minimal number transcoding difficulties, and mastered normally multidigit arithmetic procedures. She was submitted to a series of calculation, verification, and number classification tasks. The arithmetic deficit was evident in both recognition and recall tasks, was consistent across testing sessions, and did not vary as a function of the format used for presentation of the problems. The patient failed even when only implicit access to arithmetic facts was expected: In a timed addition verification task, she did not show a normal inhibition effect when rejecting addition problems in which the proposed result was the product of the two operands (e.g., "3 + 4 = 12"). We suggest that the deficit resulted from a specific and permanent degradation of some connections and nodes in arithmetic long-term memory.
Current theories of numerical cognition differ in assumptions about the componential architecture of number processing and about the extent of notation-specific processes. To investigate these issues, 64 adult subjects were tested on simple addition and multiplication problems presented in Arabic digit or English number-word format. Overall, response times and error rates were much higher with the word format, but more importantly, presentation format interacted with arithmetic operation and problem size. Operation errors (2 + 4 = 8), operand-naming errors (2 + 8 = 8), and operand-intrusion errors (9 x 6 = 36) were each characterized by a different format x operation interaction, and analysis of inter-trial error priming showed selective interference from preceding trials as a function of number format. These types of format-specific retrieval interference and operation-specific effects of format are problematic for models that hypothesize notation-independent memory processes for arithmetic. Furthermore, analyses of operand-naming errors, operand-intrusion errors, and other operand-priming effects, revealed strong interactions of number reading and number-fact retrieval processes; processes that are typically posited to be functionally independent. The results suggest a complex encoding architecture that incorporates notation-dependent activation of addition and multiplication facts, as well as interpenetration of number reading and number-fact retrieval processes.
A patient with a severe dyscalculia and a mild arabic number dyslexia is described. He could perform simple addition and subtraction sums with oral presentation. However with written arabic number sums he was impaired with addition but not with subtraction. These findings require modifications to current models of arithmetic processing which have suggested that numerical inputs are converted into abstract internal representations before arithmetical processing can occur.
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.
We describe two acalculic patients, one with a left subcortical lesion and the other with a right inferior parietal lesion and Gerstmann's syndrome. Both suffered from "pure anarithmetia": they could read arabic numerals and write them to dictation, but experienced a pronounced calculation deficit. On closer analysis, however, distinct deficits were found. The subcortical case suffered from a selective deficit of rote verbal knowledge, including but not limited to arithmetic tables, while her semantic knowledge of numerical quantities was intact. Conversely the inferior parietal case suffered from a category-specific impairment of quantitative numerical knowledge, particularly salient in subtraction and number bissection tasks, with preserved knowledge of rote arithmetic facts. This double dissociation suggests that numerical knowledge is processed in different formats within distinct cerebral pathways. We suggest that a left subcortical network contributes to the storage and retrieval of rote verbal arithmetic facts, while a bilateral inferior parietal network is dedicated to the mental manipulation of numerical quantities.
This article presents the results of two experiments. In Experiment 1, French-speaking participants were asked first to retrieve the product of two numbers presented in Arabic or verbal code, and then to perform a number-matching task on the same material to assess the encoding time difference between numerals in the two formats. Experiment 2 involved the same multiplication task with Dutch-speaking participants who name two-digit numbers in reverse order. The format effects obtained by Campbell and Clark (1992); Campbell (1994) for multiplication were replicated. However, several observations suggest that some of these effects may be due to encoding time differences between word and digit numerals. The same size-by-format interaction was found for the number-matching task as for the multiplication task, and the effect disappeared with practice in the multiplication task. Finally, despite the fact that the linguistic structure of number names differs between French and Dutch, the types of error produced in both groups were identical. The last result does not match with the hypothesis that operand intrusion errors are due to interference between reading processes and arithmetical-fact retrieval processes. Implications of these findings for the debate about the nature of arithmetical-fact retrieval are discussed.
Noël et al. (Noël, M.-P., Fias, W., Brsybaert, M., 1997. About the influence of the presentation format on arithmetical-fact retrieval processes. Cognition 63, 335-374) examined the simple-multiplication errors of 24 Dutch- and 24 French-speaking adults for evidence that number reading interferes with language-specific, number-fact retrieval processes. They concluded that arithmetic memory is not influenced by reading-based interference and is based on a notation and language-independent mental representation. Alternative analyses of their error data, however, provide strong evidence that arithmetic performance is subject to reading-based interference and provide some support for the language-specificity of number-fact memory.
Sixty-four university students received simple addition problems with operands presented as arabic digits (e.g. 2 + 3, 8 + 6) or as English number words (two + three, eight + six). Operands either were displayed simultaneously or sequentially with the left operand appearing 800 ms before the right operand. Consistent with previous findings, word problems were slower than digit problems and this word-format cost was larger for large- than small-number problems. The central question addressed by this experiment concerned whether this Format x Size interaction arises during problem encoding processes or in subsequent retrieval or production processes. In the simultaneous condition, both operands would contribute to format-related differences in encoding, whereas in the sequential condition encoding differences would arise only in connection with the second operand. Critically, however, the Format x Size interaction did not differ between the sequential and simultaneous conditions, although the experiment had ample power to detect such an effect. The results argue that the Format x Size interaction does not arise during encoding, but instead arises during calculation or production processes.
Adults (N = 32) solved simple multiplication (e.g., 8 x 7) and corresponding division problems (e.g., 56/8). Self-reports of solution processes were given by half of the participants. Latency patterns and error rates were closely related across operations and were similar in self-report and no-report conditions. Solution of division problems, however, facilitated solution of multiplication problems more than the reverse. On large division problems, participants reported that they "recast" problems as multiplication (e.g., 56/8 as 8 x = 56). These results support the hypothesis that multiplication and division are stored in separate mental representations but that solution of difficult division problems sometimes involves access to multiplication.
Contrary to predictions of current solution process models, adults used a variety of procedures other than retrieval to solve addition and multiplication math facts. Predictors assumed to capture retrieval processes posited by such models did account for a substantial proportion of variance in averaged retrieval solution times. But most of the variance in individual participants' retrieval times remained unaccounted for. Cross-operation associations in patterns of strategy use and retrieval latencies were obtained. Adults with stronger higher level math achievement were more likely to use retrieval, solved math facts faster and less variably, and executed retrieval processes posited by current solution process models faster than participants with less math attainment. The results are explained within the context of the adaptive strategy choice model.