Ten-year-old children strategies in mental addition: A counting model account

What the Science of Learning Teaches Us About Arithmetic Fluency

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.

Unraveling the small tie problem mystery: Size effects from finger counting to mental strategies in addition

Determining how children solve arithmetic problems when they stop using their fingers is a real challenge. To take it up, the evolution of problem-size effects for tie and non-tie problems was observed when 6-year-olds (N = 65) shift from finger counting to mental strategies. These observations revealed that the problem-size effect remained the same for non-tie problems, whereas it drastically decreased for tie problems. Moreover, the solving strategy for tie problems switched directly from the representation of both operands on fingers to retrieval without transition through the representation of only one operand on fingers. This direct switch could be made possible by the relative ease to commit symmetrical representations to memory (in the case of tie problems) rather than non-symmetrical ones (in the case of non-tie problems). This would explain why, early during development, small tie problems are solved quickly and present null or negligible size effects. All in all, our results and interpretations provide an answer to the long-standing question as to why tie problems have a special cognitive status. Our results also nuance the classical description of the developmental pattern reported in all textbook chapters devoted to numerical cognition according to which a finger strategy where only one operand is represented on fingers constitutes a developmental stage between the representation of two operands on fingers and retrieval. We demonstrate here that it is true only for non-tie problems.

Finger counting training enhances addition performance in kindergarteners

Our study on 328 five- to six-year-old kindergarteners (mainly White European living in France, 152 girls) shows that children who do not count on their fingers and undergo finger counting training exhibit drastic improvement in their addition skills from pre-test to post-test (i.e., accuracy from 37.3% to 77.1%) compared to a passive control group (39.6% to 47.8%) (p < .001,η p 2  = .15). This result was replicated on a much smaller scale (37 five- to six-year-olds, mainly White European, 22 girls) but in more controlled setup and was further replicated with an active control group (84 five- to six-year-olds, mainly White European, 37 girls). Therefore, we demonstrate here for the first time that training finger counting constitutes a highly effective method to improve kindergarteners' arithmetic performance.

Age-related changes in how 5- to 8-year-old children use and execute finger-based strategies in arithmetic

To determine how young children use and execute finger-based strategies, 5- to 8-year-olds were asked to solve simple addition problems under a choice condition (i.e., they could choose finger-based or non-finger strategies on each problem) and under two no-choice conditions (one in which they needed to use finger-based strategies on all problems and one in which they could not use finger-based strategies). Results showed that children (a) used both finger-based and non-finger strategies to solve simple addition problems in all age groups, (b) used fingers less and less often as they grew older, especially while solving smaller problems, (c) calibrated their use of finger-based strategies to both problem features and strategy performance, and (d) improved efficiency of both finger-based and non-finger strategy execution. Moreover, (e) strategy performance was the best predictor of strategy selection in all age groups, and (f) when they had the possibility to use fingers, children of all age groups obtained better performance relative to when they could not use fingers, especially on larger problems.

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.

Differential Switch Costs in Typically Achieving Children and Children With Mathematical Difficulties

Children with mathematical difficulties need to spend more time than typically achieving children on solving even simple equations. Since these tasks already require a larger share of their cognitive resources, additional demands imposed by the need to switch between tasks may lead to a greater decline of performance in children with mathematical difficulties. We explored differential task switch costs with respect to switching between addition versus subtraction with a tablet-based arithmetic verification task and additional standardized tests in German elementary school children in Grades 1 to 4. Two independent studies were conducted. In Study 1, we assessed the validity of a newly constructed tablet-based arithmetic verification task in a controlled classroom-setting (n = 165). Then, effects of switching between different types of arithmetic operations on accuracy and response latency were analyzed through generalized linear mixed models in an online-based testing (Study 2; n = 3,409). Children with mathematical difficulties needed more time and worked less accurately overall. They also exhibited a stronger performance decline when working in a task-switching condition, when working on subtraction (vs. addition) items and in operations with two-digit (vs. one-digit) operations. These results underline the value of process data in the context of assessing mathematical difficulties.

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.

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.

The Evolution of Finger Counting between Kindergarten and Grade 2

In this longitudinal study, we aimed at determining whether children who efficiently use finger counting are more likely to develop internalized arithmetic strategies than children who are less efficient. More precisely, we analyzed the behavior of 24 kindergarteners aged between 5 and 6 years who used their fingers to solve addition problems, and we were interested in determining the evolution of their finger counting strategies towards mental strategies after 2 years (Grade 2). Our results show that kindergarteners who were the most proficient in calculating on fingers were the more likely to have abandoned this strategy in Grade 2. This shows that the use of efficient finger counting strategies early during development optimizes the shift to mental strategies later on during school years. Moreover, children who still use their fingers to solve additions in Grade 2 present lower working memory capacities than children who had already abandoned this strategy.

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.

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.

Are small additions solved by direct retrieval from memory or automated counting procedures? A rejoinder to Chen and Campbell (2018)

Contrary to the longstanding and consensual hypothesis that adults mainly solve small single-digit additions by directly retrieving their answer from long-term memory, it has been recently argued that adults could solve small additions through fast automated counting procedures. In a recent article, Chen and Campbell (Psychonomic Bulletin & Review, 25, 739-753, 2018) reviewed the main empirical evidence on which this alternative hypothesis is based, and concluded that there is no reason to jettison the retrieval hypothesis. In the present paper, we pinpoint the fact that Chen and Campbell reached some of their conclusions by excluding some of the problems that need to be considered for a proper argumentation against the automated counting procedure theory. We also explain why, contrary to Chen and Campbell's assumption, the network interference model proposed by Campbell (Mathematical Cognition, 1, 121-164, 1995) cannot account for our data. Finally, we clarify a theoretical point of our model.

Early Engagement of Parietal Cortex for Subtraction Solving Predicts Longitudinal Gains in Behavioral Fluency in Children

There is debate in the literature regarding how single-digit arithmetic fluency is achieved over development. While the Fact-retrieval hypothesis suggests that with practice, children shift from quantity-based procedures to verbally retrieving arithmetic problems from long-term memory, the Schema-based hypothesis claims that problems are solved through quantity-based procedures and that practice leads to these procedures becoming more automatic. To test these hypotheses, a sample of 46 typically developing children underwent functional magnetic resonance imaging (fMRI) when they were 11 years old (time 1), and 2 years later (time 2). We independently defined regions of interest (ROIs) involved in verbal and quantity processing using rhyming and numerosity judgment localizer tasks, respectively. The verbal ROIs consisted of left middle/superior temporal gyri (MTG/STG) and left inferior frontal gyrus (IFG), whereas the quantity ROIs consisted of bilateral inferior/superior parietal lobules (IPL/SPL) and bilateral middle frontal gyri (MFG)/right IFG. Participants also solved a single-digit subtraction task in the scanner. We defined the extent to which children relied on verbal vs. quantity mechanisms by selecting the 100 voxels showing maximal activation at time 1 from each ROI, separately for small and large subtractions. We studied the brain mechanisms at time 1 that predicted gains in subtraction fluency and how these mechanisms changed over time with improvement. When looking at brain activation at time 1, we found that improvers showed a larger neural problem size effect in bilateral parietal cortex, whereas no effects were found in verbal regions. Results also revealed that children who showed improvement in behavioral fluency for large subtraction problems showed decreased activation over time for large subtractions in both parietal and frontal regions implicated in quantity, whereas non-improvers maintained similar levels of activation. All children, regardless of improvement, showed decreased activation over time for large subtraction problems in verbal regions. The greater parietal problem size effect at time 1 and the reduction in activation over time for the improvers in parietal and frontal regions implicated in quantity processing is consistent with the Schema-based hypothesis arguing for more automatic procedures with increasing skill. The lack of a problem size effect at time 1 and the overall decrease in verbal regions, regardless of improvement, is inconsistent with the Fact-retrieval hypothesis.

Neural representations of transitive relations predict current and future math calculation skills in children

A large body of evidence suggests that math learning in children is built upon innate mechanisms for representing numerical quantities in the intraparietal sulcus (IPS). Learning math, however, is about more than processing quantitative information. It is also about understanding relations between quantities and making inferences based on these relations. Consistent with this idea, recent behavioral studies suggest that the ability to process transitive relations (A > B, B > C, therefore A > C) may contribute to math skills in children. Here we used fMRI coupled with a longitudinal design to determine whether the neural processing of transitive relations in children could predict their current and future math skills. At baseline (T₁), children (n = 31) processed transitive relations in an MRI scanner. Math skills were measured at T₁ and again 1.5 years later (T₂). Using a machine learning approach with cross-validation, we found that activity associated with the representation of transitive relations in the IPS predicted math calculation skills at both T₁ and T₂. Our study highlights the potential of neurobiological measures of transitive reasoning for forecasting math skills in children, providing additional evidence for a link between this type of reasoning and math learning.

Effects of working memory updating on children's arithmetic performance and strategy use: A study in computational estimation

The current study investigated how children's working memory updating processes influence arithmetic performance and strategy use. Large samples of third and fourth graders were asked to find estimates of two-digit addition problems (e.g., 42 + 76). On each problem, children could choose between the rounding-down strategy (i.e., rounding both operands down to the closest decades) or the rounding-up strategy (i.e., rounding both operands up to the closest decades). Four tasks were used to assess updating. Analyses of strategy use revealed that children with more efficient updating showed higher levels of (a) strategy flexibility (i.e., they were less likely to use a single strategy on all or nearly all problems within a test block), (b) strategy adaptivity (i.e., they selected the better strategy overall more often and were more adaptive specifically on homogeneous and rounding-up problems), and (c) strategy performance (i.e., they tended to execute strategies more quickly, especially on homogeneous and larger problems). Finally, updating exerted a more important role for problem type effects in younger children than in older children. These findings have important implications for further understanding how working memory updating processes influence children's arithmetic performance and age-related differences therein.

Mathematical thinking in children with developmental language disorder: The roles of pattern skills and verbal working memory

Previous research suggests that children with language disorders often have difficulties in mathematical tasks. In the current study, we investigated two relevant factors - working memory and pattern skills - that may underlie children's poor mathematics performance. Children with developmental language disorder (DLD, n = 18, ages 6-13) and age-matched typically-developing children (n = 18) completed three math tasks that tapped calculation skill and knowledge of concepts. Children also completed a visual pattern extension task and a verbal working memory task. There were four key findings: (1) children with DLD exhibited poorer mathematical knowledge than typically-developing children, both in calculation and on key math concepts, (2) children with DLD performed similarly to typically-developing children on the visual pattern extension task, (3) children with DLD had lower verbal working memory scores than typically-developing children, and these differences in working memory accounted in part for their poorer calculation performance, and (4) children's pattern extension scores predicted their arithmetic calculation scores, but not their concept scores.

Is a fact retrieval deficit the main characteristic of children with mathematical learning disabilities?

Although a fact retrieval deficit is widely considered to be the hallmark of children with mathematical learning disabilities (MLD), recent studies suggest that even adults use procedural strategies to solve small additions, except for ties that are unanimously considered to be solved by retrieval. Our study, based on how MLD children process ties and non-ties compared to typically developing (TD) children, sheds new light on their retrieval and procedural difficulties. Our results show that, by the end of the second grade, MLD children do not differ in their ability to solve the tie problems that are certainly solved by retrieval, but they do struggle with both small and large non-ties. These findings emphasize the extend of the difficulties that MLD children exhibit in procedural strategies relatively to retrieval ones.

Transfer of training in simple addition

In recent years, several researchers have proposed that skilled adults may solve single-digit addition problems (e.g., 3 + 1 = 4, 4 + 3 = 7) using a fast counting procedure. Practicing a procedure often leads to transfer of learning and faster performance of unpracticed items. Such transfer has been demonstrated using a counting-based alphabet arithmetic task (e.g., B + 4 = C D E F) that indicated robust generalization of practice (i.e., response time [RT] gains) when untrained transfer problems at test had been implicitly practiced (e.g., practice B + 3, test B + 2 or B + 1). Here, we constructed analogous simple addition problems (practice 4 + 3, test 4 + 2 or 4 + 1). In each of three experiments (total n = 108), participants received six practice blocks followed by two test blocks of new problems to examine generalization effects. Practice of addition identity rule problems (i.e., 0 +  N =  N) showed complete transfer of RT gains made during practice to unpracticed items at test. In contrast, the addition ties (2 + 2, 3 + 3, etc.) presented large RT costs for unpracticed problems at test, but sped up substantially in the second test block. This pattern is consistent with item-specific strengthening of associative memory. The critical items were small non-tie additions (sum ≤ 10) for which the test problems would be implicitly practiced if counting was employed during practice. In all three experiments (and collectively), there was no evidence of generalization for these items in the first test block, but there was robust speed up when the items were repeated in the second test block. Thus, there was no evidence of the generalization of practice that would be expected if counting procedures mediated our participants’ performance on small non-tie addition problems.

Longitudinal development of subtraction performance in elementary school

A major goal of education in elementary mathematics is the mastery of arithmetic operations. However, research on subtraction is rather scarce, probably because subtraction is often implicitly assumed to be cognitively similar to addition, its mathematical inverse. To evaluate this assumption, we examined the relation between the borrow effect in subtraction and the carry effect in addition, and the developmental trajectory of the borrow effect in children using a choice reaction paradigm in a longitudinal study. In contrast to the carry effect in adults, carry and borrow effects in children were found to be categorical rather than continuous. From grades 3 to 4, children became more proficient in two-digit subtraction in general, but not in performing the borrow operation in particular. Thus, we observed no specific developmental progress in place-value computation, but a general improvement in subtraction procedures. Statement of contribution What is already known on this subject? The borrow operation increases difficulty in two-digit subtraction in adults. The carry effect in addition, as the inverse operation of borrowing, comprises categorical and continuous processing characteristics. What does this study add? In contrast to the carry effect in adults, the borrow and carry effects are categorical in elementary school children. Children generally improve in subtraction performance from grades 3 to 4 but do not progress in place-value computation in particular.

Transfer of training in alphabet arithmetic

In recent years, several researchers have proposed that skilled adults may solve single-digit addition problems (e.g., 3 + 1 = 4, 4 + 3 = 7) using a fast counting procedure. Practicing a procedure, however, often leads to transfer of learning to unpracticed items; consequently, the fast counting theory was potentially challenged by subsequent studies that found no generalization of practice for simple addition. In two experiments reported here (Ns = 48), we examined generalization in an alphabet arithmetic task (e.g., B + 5 = C D E F G) to determine that counting-based procedures do produce generalization. Both experiments showed robust generalization (i.e., faster response times relative to control problems) when a test problem's letter augend and answer letter sequence overlapped with practiced problems (e.g., practice B + 5 = C D E F G, test B + 3 = C D E ). In Experiment 2, test items with an unpracticed letter but whose answer was in a practiced letter sequence (e.g., practice C + 3 = DEF, test D + 2 = E F) also displayed generalization. Reanalysis of previously published addition generalization experiments (combined n = 172) found no evidence of facilitation when problems were preceded by problems with a matching augend and counting sequence. The clear presence of generalization in counting-based alphabet arithmetic, and the absence of generalization of practice effects in genuine addition, represent a challenge to fast counting theories of skilled adults' simple addition.

Strategy over operation: neural activation in subtraction and multiplication during fact retrieval and procedural strategy use in children

Arithmetic development is characterized by strategy shifts between procedural strategy use and fact retrieval. This study is the first to explicitly investigate children's neural activation associated with the use of these different strategies. Participants were 26 typically developing 4th graders (9- to 10-year-olds), who, in a behavioral session, were asked to verbally report on a trial-by-trial basis how they had solved 100 subtraction and multiplication items. These items were subsequently presented during functional magnetic resonance imaging. An event-related design allowed us to analyze the brain responses during retrieval and procedural trials, based on the children's verbal reports. During procedural strategy use, and more specifically for the decomposition of operands strategy, activation increases were observed in the inferior and superior parietal lobes (intraparietal sulci), inferior to superior frontal gyri, bilateral areas in the occipital lobe, and insular cortex. For retrieval, in comparison to procedural strategy use, we observed increased activity in the bilateral angular and supramarginal gyri, left middle to inferior temporal gyrus, right superior temporal gyrus, and superior medial frontal gyrus. No neural differences were found between the two operations under study. These results are the first in children to provide direct evidence for alternate neural activation when different arithmetic strategies are used and further unravel that previously found effects of operation on brain activity reflect differences in arithmetic strategy use. Hum Brain Mapp 38:4657-4670, 2017. © 2017 Wiley Periodicals, Inc.

An Extension of the Procedural Deficit Hypothesis from Developmental Language Disorders to Mathematical Disability

Mathematical disability (MD) is a neurodevelopmental disorder affecting math abilities. Here, we propose a new explanatory account of MD, the procedural deficit hypothesis (PDH), which may further our understanding of the disorder. According to the PDH of MD, abnormalities of brain structures subserving the procedural memory system can lead to difficulties with math skills learned in this system, as well as problems with other functions that depend on these brain structures. This brain-based account is motivated in part by the high comorbidity between MD and language disorders such as dyslexia that may be explained by the PDH, and in part by the likelihood that learning automatized math skills should depend on procedural memory. Here, we first lay out the PDH of MD, and present specific predictions. We then examine the existing literature for each prediction, while pointing out weaknesses and gaps to be addressed by future research. Although we do not claim that the PDH is likely to fully explain MD, we do suggest that the hypothesis could have substantial explanatory power, and that it provides a useful theoretical framework that may advance our understanding of the disorder.

Similarity interference in learning and retrieving arithmetic facts

Storing the solution of simple calculations in long-term memory is an important learning in primary school that is subsequently essential in adult daily living. While most children succeed in storing arithmetic facts to which they have been trained at school, huge individual differences are reported, particularly in children with developmental dyscalculia, who show a severe and persistent deficit in arithmetic facts learning. This chapter reports important advances in the understanding of the development of an arithmetic facts network and focuses on the detrimental effect of similarity interference. First, at the retrieval stage, connectionist models highlighted that the similarity of the neighbor problems in the arithmetic facts network creates interference. More recently, the similarity interference during the learning stage was pointed out in arithmetic facts learning. The interference parameter, that captures the proactive interference that a problem receives from previously learned problems, was shown as a substantial determinant of the performance across multiplication problems. This proactive interference was found both in children and adults and showed that when a problem is highly similar to previously learned ones, it is more difficult to remember it. Furthermore, the sensitivity to this similarity interference determined individual differences in the learning and retrieving of arithmetic facts, giving new insights for interindividual differences. Regarding the atypical development, hypersensitivity-to-interference in memory was related to arithmetic facts deficit in a single case of developmental dyscalculia and in a group of fourth-grade children with low arithmetic facts knowledge. In sum, the impact of similarity interference is shown in the learning stage of arithmetic facts and concerns the typical and atypical development.