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).

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.

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.

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.

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.

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.

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).

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.