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).

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.

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.

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.

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.

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.

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.

An insect brain organizes numbers on a left-to-right mental number line.

The "mental number line" (MNL) is a form of spatial numeric representation that associates small and large numbers with the left and right spaces, respectively. This spatio-numeric organization can be found in adult humans and has been related to cultural factors such as writing and reading habits. Yet, both human newborns and birds order numbers consistently with an MNL, thus raising the question of whether culture is a main explanation for MNL. Here, we explored the numeric sense of honey bees and show that after being trained to associate numbers with a sucrose reward, they order numbers not previously experienced from left to right according to their magnitude. Importantly, the location of a number on that scale varies with the reference number previously trained and does not depend on low-level cues present on numeric stimuli. We provide a series of neural explanations for this effect based on the extensive knowledge accumulated on the neural underpinnings of visual processing in honey bees and conclude that the MNL is a form of numeric representation that is evolutionarily conserved across nervous systems endowed with a sense of number, irrespective of their neural complexity.

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.

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.

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.

Developmental changes in size effects for simple tie and non-tie addition problems in 6- to 12-year-old children and adults.

In the domain of cognitive arithmetic, the size effect corresponds to an increase in solution times as a function of the size of the operands involved in the problems. In this study, we tracked the evolution of size effects associated with tie and non-tie addition problems across development. We scrutinized the progression of solution times for very small problems involving operands from 2 to 4, larger problems, and 1-problems (problems involving 1 as one of the operands) in children from Grade 1 to Grade 5 and adults. For the first time, we document the presence of a size effect for tie problems with a sum up to 8 in Grade 1 children. In contrast, from Grade 3 until adulthood, this size effect could not be evidenced. Crucially, for non-tie problems, whereas a general size effect is observed when contrasting small one-digit additions with large additions, we show that, from Grade 1 until adulthood, a continuous size effect as a function of the sum of the problems is not observed. In fact, for all age groups, medium problems with sums of 8, 9, and 10 do not present a size effect at all. Given that the problem size effect is sometimes referred to as one of the most robust and reliable effects in the numerical cognition literature, our results necessarily challenge its theoretical interpretation.

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.

Learning to run the number line: the development of attentional shifts during single-digit arithmetic.

Solving single-digit subtraction and addition problems is associated with left and right shifts of attention in adults. Here, we explored the development of these spatial shifts in children from the third to fifth grade. In two experiments, children solved single-digit addition (Experiments 1 and 2), subtraction (Experiment 1), and multiplication (Experiment 2) problems in which operands and the arithmetic sign were shown sequentially. Although the first operand and the arithmetic sign were presented on the center of a screen, the second operand was presented either in the left or the right visual field. In Experiment 1, we found that subtraction problems were increasingly associated with a leftward bias by the fifth grade, such that problem solving was facilitated when the second operand was in the left visual field. In Experiment 2, we found that children can also associate addition problems with the right side of space by the fourth grade. No developmental increase in either leftward or rightward bias was observed for multiplication problems. These attentional shifts might be due to the increasing reliance on calculation procedures that involve mental movements to the left or right of a sequential representation of numbers during subtraction and addition.

Scrutinizing patterns of solution times in alphabet-arithmetic tasks favors counting over retrieval models.

According to associationist models, initial sequential processing of algorithmic steps is replaced through learning by single-step access to a memory instance. In an alphabet-arithmetic task where equations such as C + 3 = F have to be verified, the shift from algorithmic procedures to retrieval would manifest in a transition from steep slopes relating solution times to addends at the beginning of learning to a flat function at the end (e.g., Logan & Klapp, 1991). Nevertheless, we argue that computation of the slopes at the end of training is biased by a systematic drop in solution times for the largest addend in the study set. In this paper, this drop is observed even when the longest training period in alphabet-arithmetic literature is doubled (Experiment 1) and even when the size of the largest addend is increased (Experiment 2). We demonstrate that this drop is partly due to end-term effects but remains observable even when end-term problems are not considered in the analyses. As Logan and Klapp suggested, we conclude that the drop is partly due to deliberate memorization of the problems with the largest addend. In contrast, departing from Logan and Klapp, we demonstrate that, when problems with the largest addend are excluded from the analyses, the possibility that counting is still used after learning cannot be discarded. This conclusion is reached because after this exclusion, the slopes were still significant. To conclude, our results advocate that practicing an algorithm leads to its acceleration and not to a shift from algorithmic procedures to retrieval.