The efficacy of manipulatives versus fingers in supporting young children's addition skills

Recent empirical investigations have revealed that finger counting is a strategy associated with good arithmetic performance in young children. Fingers could have a special status during development because they operate as external support that provide sensory-motor and kinesthetic affordances in addition to visual input. However, it was unknown whether fingers are more helpful than manipulatives such as tokens during arithmetic problem solving. To address this question, we conducted a study with 93 Vietnamese children (48 girls) aged 4 and 5 years (mean = 58 months, range = 47-63) with high arithmetic and counting skills from families with relatively high socioeconomic status. Their behaviors were observed as they solved addition problems with manipulatives at their disposal. We found that children spontaneously used both manipulatives and fingers to solve the problems. Crucially, their performance was not higher when fingers rather than manipulatives were used (i.e., 70% vs. 81% correct answers, respectively). Therefore, at the beginning of learning, it is possible that, at least for children with high numerical skills, fingers are not the only gateway to efficient arithmetic development and manipulatives might also lead to proficient arithmetic.

Learning basic arithmetic: A comparison between rote and procedural learning based on an artificial sequence

It is commonly accepted that repeatedly using mental procedures results in a transition to memory retrieval, but the determinant of this process is still unclear. In a 3-week experiment, we compared two different learning situations involving basic additions, one based on counting and the other based on arithmetic fact memorization. Two groups of participants learned to verify additions such as "G + 2 = Q?" built on an artificial sequence (e.g., "XGRQD…"). The first group learned the sequence beforehand and could therefore count to solve the problems, whereas the second group was not aware of the sequence and had to learn the equations by rote. With practice, solution times of both groups reached a plateau, indicating a certain level of automatization. However, a more fine-grained comparison indicated that participants relied on fundamentally different learning mechanisms. In the counting condition, most participants showed a persistent linear effect of the numerical operand on solution times, suggesting that fluency was reached through an acceleration of counting procedures. However, some participants began memorizing the problems involving the largest addends: Their solution times were very similar to those of participants in the rote learning group, suggesting that they resulted from a memory retrieval process. These findings show that repeated mental procedures do not systematically lead to memory retrieval but that fluency can also be reached through the acceleration of these procedures. Moreover, these results challenge associationist models, which cannot currently predict that the process of memorization begins with problems involving the largest addends. (PsycInfo Database Record (c) 2024 APA, all rights reserved).

Neural evidence for procedural automatization during cognitive development: Intraparietal response to changes in very-small addition problem-size increases with age

Cognitive development is often thought to depend on qualitative changes in problem-solving strategies, with early developing algorithmic procedures (e.g., counting when adding numbers) considered being replaced by retrieval of associations (e.g., between operands and answers of addition problems) in adults. However, algorithmic procedures might also become automatized with practice. In a large cross-sectional fMRI study from age 8 to adulthood (n = 128), we evaluate this hypothesis by measuring neural changes associated with age-related reductions in a behavioral hallmark of mental addition, the problem-size effect (an increase in solving time as problem sum increases). We found that age-related decreases in problem-size effect were paralleled by age-related increases of activity in a region of the intraparietal sulcus that already supported the problem-size effect in 8- to 9-year-olds, at an age the effect is at least partly due to explicit counting. This developmental effect, which was also observed in the basal ganglia and prefrontal cortex, was restricted to problems with operands ≤ 4. These findings are consistent with a model positing that very-small arithmetic problems-and not larger problems-might rely on an automatization of counting procedures rather than a shift towards retrieval, and suggest a neural automatization of procedural knowledge during cognitive development.

French preschool and primary teachers' attitude towards finger counting

Teachers' beliefs and attitudes are known to guide the type of activities they implement in their classrooms. A traditional conception that finger counting is merely a back-up when children fail to use more sophisticated and efficient strategies could therefore prevent teachers from encouraging children's use of fingers in arithmetic tasks. However, the potential benefit of finger counting for young learners has been recently documented and setting aside its practice within classrooms may hinder children's mathematical skill development. It is therefore important to establish whether there is a discrepancy between teacher's beliefs regarding finger counting and the latest discoveries in this field of research. To this aim, we interrogated 413 teachers from preschool to Grade 5. We found that, despite being generally positive towards finger counting, teachers think that finger counting is typical of children who present math difficulties or lack of confidence, even during the first years of learning. These results are discussed considering what is known and what remains to be determined in the current scientific literature.

The development of simple addition problem solving in children: Reliance on automatized counting or memory retrieval depends on both expertise and problem size

In an experiment, 98 children aged 8 to 9, 10 to 12, and 13 to 15 years solved addition problems with a sum up to 10. In another experiment, the same children solved the same calculations within a sign priming paradigm where half the additions were displayed with the "+" sign 150 ms before the addends. Therefore, size effects and priming effects could be considered conjointly within the same populations. Our analyses revealed that small problems, constructed with addends from 1 to 4, presented a linear increase of solution times as a function of problem sums (i.e., size effect) in all age groups. However, an operator priming effect (i.e., facilitation of the solving process with the anticipated presentation of the "+" sign) was observed only in the group of oldest children. These results support the idea that children use a counting procedure that becomes automatized (as revealed by the priming effect) around 13 years of age. For larger problems and whatever the age group, no size or priming effects were observed, suggesting that the answers to these problems were already retrieved from memory at 8 to 9 years of age. For this specific category of large problems, negative slopes in solution times demonstrate that retrieval starts from the largest problems during development. These results are discussed in light of a horse race model in which procedures can win over retrieval.

Acquisition of new arithmetic skills based on prior arithmetic skills: A cross-sectional study in primary school from grade 2 to grade 5

BACKGROUND

In several countries, children's math skills have been declining at an alarming rate in recent years and decades, and one of the explanations for this alarming situation is that children have difficulties in establishing the relations between arithmetical operations.

AIM

In order to address this question, our goal was to determine the predictive power of previously taught operations on newly taught ones above general cognitive skills and basic numerical skills.

SAMPLES

More than one hundred children in each school level from Grades 2 to 5 from various socio-cultural environments (N = 435, 229 girls) were tested.

METHODS

Children were assessed on their abilities to solve the four basic arithmetic operations. They were also tested on their general cognitive abilities, including working memory, executive functions (i.e., inhibition and flexibility), visual attention and language. Finally, their basic numerical skills were measured through a matching task between symbolic and nonsymbolic numerosity representations. Additions and subtractions were presented to children from Grade 2, multiplications from Grade 3 and divisions from Grade 4.

RESULTS AND CONCLUSIONS

We show that addition predicts subtraction and multiplication performance in all grades. Moreover, multiplication predicts division performance in both Grades 4 and 5. Finally, addition predicts division in Grade 4 but not in Grade 5 and subtraction and division are not related whatever the school grade. These results are examined considering the existing literature, and their implications in terms of instruction are discussed.

Raw eye tracking data of healthy adults reading aloud words, pseudowords and numerals

This paper describes data from de Chambrier et al. (2023). The dataset [2] contains raw eye tracking data of 36 healthy adults, collected using an EyeLink 1000 (SR Research Ltd., ON, Canada) during an on-screen reading task. Participants read 96 items including words, pseudowords and numerals. Each item was presented at the center of the screen until the participant produced an oral response and pressed the keyboard's space bar. Part of the data were analyzed to extract key metrics such as fixation number, fixation duration, saccade number, and saccade amplitude identified by the EyeLink 1000 [1]. Reuse potential includes (but is not limited to) pupil diameter data analysis, identification of fixations and saccades using custom algorithms, and secondary analyses using participant demographics (age, gender) as independent variables.

Reading numbers is harder than reading words: An eye-tracking study

We recorded the eye movements of adults reading aloud short (four digit) and long (eight to 11 digit) Arabic numerals compared to matched-in-length words and pseudowords. We presented each item in isolation, at the center of the screen. Participants read each item aloud at their pace, and then pressed the spacebar to display the next item. Reading accuracy was 99 %. Results showed that adults make 2.5 times more fixations when reading short numerals compared to short words, and up to 7 times more fixations when reading long numerals with respect to long words. Similarly, adults make 3 times more saccades when reading short numerals compared to short words, and up to 9 times more saccades when reading long numerals with respect to long words. Fixation duration and saccade amplitude stay almost the same when reading short numerals with respect to short words. However, fixation duration increases by ∼50 ms when reading long numerals (∼300 ms) with respect to long words (∼250 ms), and saccade amplitude decreases up to 0.83 characters when reading long numerals with respect to long words. The pattern of findings for long numerals-more and shorter saccades as well as more and longer fixations-shows the extent to which reading long Arabic numerals is a cognitively costly task. Within the phonographic writing system, this pattern of eye movements stands for the use of the sublexical print-to-sound correspondence rules. The data highlight that reading large numerals is an unautomatized activity and that Arabic numerals must be converted into their oral form by a step-by-step process even by expert readers.

Individual Differences in the Evolution of Counting

The alphabet-arithmetic paradigm, in which adults are asked to add a numeral addend to a letter augend (e.g., D + 3 = G), was conceived to mimic the way children learn addition. Studies using this paradigm often conclude that procedural learning leads to the memorization of associations between operands and answers. However, as recently suggested, memorization might only be used by a minority of participants and only for problems with the largest addend. In the present paper, we aim at investigating these individual differences through transfer effects from trained problems to new ones. Participants were trained over 12 learning sessions, followed by 3 transfer sessions. A group of participants, that we called the nonbreakers, showed a linear function associating solution times and addends throughout the experiment. In this group, transfer was observed during the first transfer session, suggesting that a procedural strategy, transferable to new items, was still used at the end of training. In another group of participants, that we called the breakers, we observed a decrease in solution times for problems with the largest addend. In this group, transfer was only observed after two transfer sessions, suggesting that procedural strategies were not used as often in this group than in the other group. This was especially true for problems with the largest addend because transfer effects were stronger when they were excluded. Therefore, during learning and for breakers, the answers to problems with larger addends are retrieved first and, as for non-breakers, the answers to problems with very small operands remain computed.

Automatization through Practice: The Opportunistic-Stopping Phenomenon Called into Question

As a theory of skill acquisition, the instance theory of automatization posits that, after a period of training, algorithm‐based performance is replaced by retrieval‐based performance. This theory has been tested using alphabet‐arithmetic verification tasks (e.g., is A + 4  = E?), in which the equations are necessarily solved by counting at the beginning of practice but can be solved by memory retrieval after practice. A way to infer individuals’ strategies in this task was supposedly provided by the opportunistic‐stopping phenomenon, according to which, if individuals use counting, they can take the opportunity to stop counting when a false equation associated with a letter preceding the true answer has to be verified (e.g., A + 4  = D). In this case, such within‐count equations would be rejected faster than false equations associated with letters following the true answers (e.g., A + 4  = F, i.e., outside‐of‐count equations). Conversely, the absence of opportunistic stopping would be the sign of retrieval. However, through a training experiment involving 19 adults, we show that opportunistic stopping is not a phenomenon that can be observed in the context of an alphabet‐arithmetic verification task. Moreover, we provide an explanation of how and why it was wrongly inferred in the past. These results and conclusions have important implications for learning theories because they demonstrate that a shift from counting to retrieval over training cannot be deduced from verification time differences between outside and within‐count equations in an alphabet‐arithmetic task.

Do production and verification tasks in arithmetic rely on the same cognitive mechanisms? A test using alphabet arithmetic

In this study, 17 adult participants were trained to solve alphabet-arithmetic problems using a production task (e.g., C + 3 = ?). The evolution of their performance across 12 practice sessions was compared with the results obtained in past studies using verification tasks (e.g., is C + 3 = F correct?). We show that, irrespective of the experimental paradigm used, there is no evidence for a shift from counting to retrieval during training. However, and again regardless of the paradigm, problems with the largest addend constitute an exception to the general pattern of results obtained. Contrary to other problems, their answers seem to be deliberately memorised by participants relatively early during training. All in all, we conclude that verification and production tasks lead to similar patterns of results, which can therefore both confidently be used to discuss current theories of learning. Still, deliberate memorization of problems with the largest addend appears earlier and more often in a production than a verification task. This last result is discussed in light of retrieval models.

Differences in event-related potential (ERP) responses to small tie, non-tie and 1-problems in addition and multiplication

Using ERP, we investigated the cause of the tie advantage according to which problems with repeated operands are solved faster and more accurately than non-tie problems. We found no differences in early or N400 ERP components between problems, suggesting that tie problems are not encoded faster or suffer from less interference than non-tie problems. However, a lesser negative amplitude of the N2 component was found for tie than non-tie problems. This suggests more working-memory and attentional resource requirements for non-tie problems and therefore more frequent use of retrieval for tie than non-tie problems. The possible peculiarity of problems involving a 1 was also investigated. We showed less negative N2 amplitudes for these problems than for other non-tie problems, suggesting less working-memory resources for 1-problems than other non-tie problems. This could be explained either by higher reliance on memory retrieval for 1-problems than non-1 problems or by the application of non-arithmetical rules for 1-problems.

Priming effects of arithmetic signs in 10- to 15-year-old children

In this research, 10- to 12- and 13- to 15-year-old children were presented with very simple addition and multiplication problems involving operands from 1 to 4. Critically, the arithmetic sign was presented before the operands in half of the trials, whereas it was presented at the same time as the operands in the other half. Our results indicate that presenting the 'x' sign before the operands of a multiplication problem does not speed up the solving process, irrespective of the age of children. In contrast, presenting the '+' sign before the operands of an addition problem facilitates the solving process, but only in 13 to 15-year-old children. Such priming effects of the arithmetic sign have been previously interpreted as the result of a pre-activation of an automated counting procedure, which can be applied as soon as the operands are presented. Therefore, our results echo previous conclusions of the literature that simple additions but not multiplications can be solved by fast counting procedures. More importantly, we show here that these procedures are possibly convoked automatically by children after the age of 13 years. At a more theoretical level, our results do not support the theory that simple additions are solved through retrieval of the answers from long-term memory by experts. Rather, the development of expertise for mental addition would consist in an acceleration of procedures until automatization.

The use of automated procedures by older adults with high arithmetic skills during addition problem solving

In contrast to other cognitive abilities, arithmetic skills are known to be preserved in healthy elderly adults. In fact, they would even outperform young adults because they more often retrieve arithmetic facts from long-term memory. Nevertheless, we suggest here that the superiority of older over younger adults could also stem from the use of more efficient automated and unconscious counting procedures. We tested 35 older participants using the sign priming paradigm and selected the 18 most efficient ones, aged from 60 to 77. Sign priming is interpreted as the indicator of the preactivation of an abstract procedure as soon as the arithmetic sign is presented. We showed that expert elderly arithmeticians behaved exactly as 26 young participants presenting the same level of arithmetic proficiency. More precisely, we showed that presenting the "+" sign 150 ms before the operands speeds up the solving process compared to a situation wherein the problem is classically presented in its whole on the screen. Only tie problems and problems involving 0 were not subjected to these priming effects, and we concluded that only these problems were solved by retrieval, either of the answer for tie problems or of a rule for + 0 problems. These results could provide new insights for the conception of training programs aiming at preserving older individuals' arithmetical skills and, in a longer-term perspective, at maintaining their financial autonomy, which is decisive for keeping them in charge of their daily life. (PsycInfo Database Record (c) 2020 APA, all rights reserved).

Arithmetic word problems describing discrete quantities: E.E.G evidence for the construction of a situation model

In this research, university students were asked to solve arithmetic word problems constructed either with discrete quantities, such as apples or marbles, or continuous quantities such as meters of rope or grams of sand. An analysis of their brain activity showed different alpha levels between the two types of problems with, in particular, a lower alpha power in the parieto-occipital area for problems describing discrete quantities. This suggests that processing discrete quantities during problem solving prompts more mental imagery than processing continuous quantities. These results are difficult to reconcile with the schema theory, according to which arithmetic problem solving depends on the activation of ready-made mental frames stored in long-term memory and triggered by the mathematical expression used in the texts. Within the schema framework, the nature of the objects described in the text should be quickly abstracted during problem solving because it cannot impact the semantic structure of the problem. On the contrary, our results support the situation model theory, which places greater emphasis on the problem context in order to account for individuals' behaviour. On a more methodological point of view, this study constitutes the first attempt to infer the characteristics of individual's mental representations of arithmetic text problems from EEG recordings. This opens the door for the application of brain activity measures in the field of arithmetic word problem.

Spatial-Numerical Associations Enhance the Short-Term Memorization of Digit Locations

Little is known about how spatial-numerical associations (SNAs) affect the way individuals process their environment, especially in terms of learning and memory. In this study, we investigated the potential effects of SNAs in a digit memory task in order to determine whether spatially organized mental representations of numbers can influence the short-term encoding of digits positioned on an external display. To this aim, we designed a memory game in which participants had to match pairs of identical digits in a 9 × 2 matrix of cards. The nine cards of the first row had to be turned face up and then face down, one by one, to reveal a digit from 1 to 9. When a card was turned face up in the second row, the position of the matching digit in the first row had to be recalled. Our results showed that performance was better when small numbers were placed on the left side of the row and large numbers on the right side (i.e., congruent) as compared to the inverse (i.e., incongruent) or a random configuration. Our findings suggests that SNAs can enhance the memorization of digit positions and therefore that spatial mental representations of numbers can play an important role on the way humans process and encode the information around them. To our knowledge, this study is the first that reaches this conclusion in a context where digits did not have to be processed as numerical values.

Dexterity and Finger Sense: A Possible Dissociation in Children With Cerebral Palsy

Both hand and finger sensory perception and motor abilities are essential for the development of skilled gestures and efficient bimanual coordination. While finger dexterity and finger sensory perception can be impaired in children with cerebral palsy (CP), the relationship between these two functions in this population is not clearly established. The common assumption that CP children with better sensory function also demonstrate better motor outcomes has been recently challenged. To study these questions further, we assessed both finger dexterity and finger gnosia, the ability to perceive one’s own fingers by touch, in groups of 11 children with unilateral (i.e., hemiplegic CP) and 11 children with bilateral spastic CP (i.e., diplegic CP) and compared them with typical children. In our sample, children with hemiplegia exhibited finger dexterity deficit in both hands and finger gnosia deficit only in their paretic hand. In contrast, children with diplegia exhibited finger gnosia deficits in both hands and finger dexterity deficit only in their dominant hand. Thus, our results indicated that children with spastic hemiplegia and diplegia present different sensory and motor profiles and suggest that these two subgroups of CP should be considered separately in future experimental and clinical research. We discuss the implications of our results for rehabilitation.

Successful aging: The role of cognitive gerontology

This commentary explores the relationships between the construct of successful aging and the experimental psychology of human aging-cognitive gerontology. What can or should cognitive gerontology contribute to understanding, defining, and assessing successful aging? Standards for successful aging reflect value judgments that are culturally and historically situated. Fundamentally, they address social policy; they are prescriptive. If individuals or groups are deemed to be aging successfully, then their characteristics or situations can be emulated. If an individual or a group is deemed to be aging unsuccessfully, then intervention should be considered. Although science is never culture-free or ahistorical, cognitive gerontology is primarily descriptive of age-related change. It is not prescriptive. It is argue that cognitive gerontology has little to contribute to setting standards for successful aging. If, however, better cognitive function is taken as a marker of more successful aging-something not universally accepted-then cognitive gerontology can play an important assessment role. It has a great deal to contribute in determining whether an individual or a group evidences better cognitive function than another. More importantly, cognitive gerontology can provide tools to evaluate the effects of interventions. It can provide targeted measures of perception, attention, memory, executive function, and other facets of cognition that are more sensitive to change than most clinical measures. From a deep understanding of factors affecting cognitive function, cognitive gerontology can also suggest possible interventions. A brief narrative review of interventions that have and have not led to improved cognitive function in older adults. Finally, the enormous range is addressed in the estimates of the proportion of the population that meets a standard for aging successfully, from less than 10% to more than 90%. For research purposes, it would be better to replace absolute cutoffs with correlational approaches (e.g., Freund & Baltes, 1998, Psychology and Aging, 13, 531-543). For policy purposes, cutoffs are necessary, but we propose that assessments of successful aging be based not on absolute cutoffs but on population proportions. An example of one possible standard is this: Those more than 1 standard deviation above the mean are aging successfully; those more than 1 standard deviation below the mean are aging unsuccessfully; those in between are aging usually. Adoption of such a standard may reduce the wide discrepancies in the incidence of successful aging reported in the literature.

Hippocampal spatial mechanisms relate to the development of arithmetic symbol processing in children

Understanding the meaning of abstract mathematical symbols is a cornerstone of arithmetic learning in children. Studies have long focused on the role of spatial intuitions in the processing of numerals. However, it has been argued that such intuitions may also underlie symbols that convey fundamental arithmetic concepts, such as arithmetic operators. In the present cross-sectional study, we used fMRI to investigate how and when associations between arithmetic operators and brain regions processing spatial information emerge in children from 3^rd to 10^th grade. We found that the mere perception of a ‘+’ sign elicited grade-related increases of spatial activity in the right hippocampus. That is, merely perceiving ‘+’ signs – without any operands – elicited enhanced hippocampal activity after around 7^th grade (12–13 years old). In these children, hippocampal activity in response to a ‘+’ sign was further correlated with the degree to which calculation performance was facilitated by the preview of that sign before an addition problem, an effect termed operator-priming. Grade-related increases of hippocampal spatial activity were operation-specific because they were not observed with ‘×’ signs, which might evoke rote retrieval rather than numerical manipulation. Our study raises the possibility that hippocampal spatial mechanisms help build associations between some arithmetic operators and space throughout age and/or education.

Identifying strategies in arithmetic with the operand recognition paradigm: a matter of switch cost?

Determining adults' and children's strategies in mental arithmetic constitutes a central issue in the domain of numerical cognition. However, despite the considerable amount of research on this topic, the conclusions in the literature are not always coherent. Therefore, there is a need to carry on the investigation, and this is the reason why we developed the operand recognition paradigm (ORP). It capitalizes on the fact that, contrary to retrieval, calculation procedures degrade the memory traces of the operands involved in a problem. As a consequence, the use of calculation procedures is inferred from relatively long recognition times of the operands. However, it has been suggested that recognition times within the ORP do not reflect strategies but the difficulty of switching from a difficult task (calculation) to a simpler one (recognition). In order to examine this possibility, in a series of 3 experiments we equalized switch-cost variations in all conditions through the introduction of intermediate tasks between problem solving and recognition. Despite this neutralization, we still obtained the classical effects of the ORP, namely longer recognition times after addition than after comparison. We conclude that the largest part of the ORP effects is related to different strategy use and not to difficulty-related switch costs. The possible applications and promising outcomes of the ORP in and outside the field of numerical cognition are discussed.

The role of fingers in number processing in young children

The aim of the present study was to investigate the relationship between finger counting and numerical processing in 4-7-year-old children. Children were assessed on a variety of numerical tasks and we examined the correlations between their rates of success and their frequency of finger use in a counting task. We showed that children's performance on finger pattern comparison and identification tasks did not correlate with the frequency of finger use. However, this last variable correlated with the percentages of correct responses in an enumeration task (i.e., Give-N task), even when the age of children was entered as a covariate in the analysis. Despite this correlation, we showed that some children who never used their fingers in the counting task were able to perform optimally in the enumeration task. Overall, our results support the conclusion that finger counting is useful but not necessary to develop accurate symbolic numerical skills. Moreover, our results suggest that the use of fingers in a counting task is related to the ability of children in a dynamic enumeration task but not to static tasks involving recognition or comparison of finger patterns. Therefore, it could be that the link between fingers and numbers remain circumscribed to counting tasks and do not extent to static finger montring situations.

The use of the operand-recognition paradigm for the study of mental addition in older adults

OBJECTIVES

Determining how individuals solve arithmetic problems is crucial for our understanding of human cognitive architecture. Elderly adults are supposed to use memory retrieval more often than younger ones. However, they might backup their retrieval by reconstructive strategies. In order to investigate this issue, we used the operand-recognition paradigm, which capitalizes on the fact that algorithmic procedures degrade the memory traces of the operands.

METHOD

Twenty-three older adults (M = 70.4) and 23 younger adults (M = 20.0) solved easy, difficult, and medium-difficulty addition and comparison problems and were then presented with a recognition task of the operands.

RESULTS

When one-digit numbers with sums larger than 10 were involved (medium-difficulty problem), it was more difficult for younger adults to recognize the operands after addition than comparison. In contrast, in older adults, recognition times of the operands were the same after addition and comparison.

DISCUSSION

Older adults, in contrast with younger adults, are able to retrieve the results of addition problems of medium difficulty. Contrary to what was suggested, older participants do not seem to resort to backup strategies for such problems. Finally, older adults' reliance on the more efficient retrieval strategy allowed them to catch up to younger adults in terms of solution times.

Better Elementary Number Processing in Higher Skill Arithmetic Problem Solvers: Evidence from the Encoding Step

This paper addresses the relationship between basic numerical processes and higher level numerical abilities in normal achieving adults. In the first experiment we inferred the elementary numerical abilities of university students from the time they needed to encode numerical information involved in complex additions and subtractions. We interpreted the shorter encoding times in good arithmetic problem solvers as revealing clearer or more accessible representations of numbers. The second experiment shows that these results cannot be due to the fact that lower skilled individuals experience more maths anxiety or put more cognitive efforts into calculations than do higher skilled individuals. Moreover, the third experiment involving non-numerical information supports the hypothesis that these interindividual differences are specific to number processing. The possible causal relationships between basic and higher level numerical abilities are discussed.

Retrieval from Memory or Procedural Strategies for Addition Problems

In this study, we used a paradigm recently developed ( Thevenot, Fanget, & Fayol, 2007 ) to determine whether 10-year-old children solve simple addition problems by retrieval of the answer from long-term memory or by calculation procedures. Our paradigm is unique in that it does not rely on reaction times or verbal reports, which are known to potentially bias the results, especially in children. Rather, it takes advantage of the fact that calculation procedures degrade the memory traces of the operands, so that it is more difficult to recognize them when they have been involved in the solution of an addition problem by calculation rather than by retrieval. The present study sharpens the current conclusions in the literature and shows that, when the sum of addition problems is up to 10, children mainly use retrieval, but when it is greater than 10, they mainly use calculation procedures.