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).

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.