The definition of cognitive flexibility remains debated in the literature. Some refer to flexibility as a general shifting ability. Others argue that it is a context-dependent skill. For others, flexibility is a dimensional construct including reactive flexibility and spontaneous flexibility. To help decide between these views, we analyzed the performances of first-graders across four flexibility tasks. Two tasks focused on reactive flexibility (rule-shifting and verbally-cued-shifting), and two targeted spontaneous flexibility (role-shifting and divergent thinking). Three additional executive tasks investigated the contribution of inhibition, working memory, and processing speed. Significant — though moderate — positive correlations emerged between all but one of the pairs of flexibility tasks. Besides, executive functions were not found to systematically correlate with any measure of flexibility. Furthermore, confirmatory factorial analyses failed to identify a unique or bi-dimensional structure of flexibility. Taken together, these results did not provide evidence for a domain-general ability. Implications and future directions are discussed.

Because of its importance in academic achievement, especially in mathematics, training cognitive flexibility at school is a major issue. The present research investigates the effectiveness of a school-based intervention to improve proportion arithmetic problem solving. The study was conducted with 5th graders of 10 classes from 5 high-priority education schools in the Paris region. Students of the control and experimental groups took part in 8 learning sessions about proportion problem solving. The experimental group’s training focused on comparing and flexibly categorizing the problems in the hopes to help students achieve a deeper understanding of proportion problems. Results show that training flexible categorization allowed the experimental group to progress more than the control group, in both categorization and solving tasks. The educational implications of our results are discussed.

Recent evidence suggests that non-mathematical world knowledge influences the semantic encoding of arithmetic wordproblems (Gamo, Sander & Richard, 2010; Gros, Thibaut & Sander, 2017). We used isomorphic problems that couldbe encoded in two distinct ways to investigate this issue. Depending on the world knowledge evoked by the elementsdescribed in the problem statement, we made the hypothesis that different mathematical relations would be made salient inthe encoded representation. We tested this hypothesis by presenting participants with an unexpected problem recognitiontask following a problem solving task. Participants tended to erroneously recognize modified problems in the recognitiontask when they had been rewritten so as to explicitly describe the relation that could have been inferred from worldknowledge, but not when the world knowledge evoked during the encoding did not make this relation salient. Thishighlights the crucial influence of world knowledge on arithmetic word problems representations.

Evidence suggests that general, non-mathematical knowledge about the entities described in an arithmetic word problem may interfere with its encoding. We used behavioral and eye- tracking measures to investigate how the use of specific quantities may foster a cardinal representation of the numbers mentioned in a problem, whereas other quantities may favor an ordinal representation instead. We asked 50 pre-service teachers to complete a solution validity assessment task. We compared participants’ gaze patterns on isomorphic problems to gather insights into their encoded representations. On problems featuring cardinal quantities, we found that specific sentences describing elements relevant in a cardinal understanding of the problems but irrelevant otherwise were looked at longer and were the focus of a higher number of backward eye movements. Additionally, an increase in pupil dilation on correctly solved cardinal problems supported the idea that participants need to engage in a recoding process when facing semantic incongruence.

We investigated what happens when the spontaneous encoding of a problem is incongruent with its solving strategy. We created word problems from which two distinct semantic representations could be abstracted. Only one of these representations was consistent with the solving strategy. We tested whether participants could recode a semantically incongruent representation in order to access another, less salient, solving strategy. In experiment 1, participants had to solve arithmetic problems and to indicate which problems were unsolvable. In experiment 2, participants received solved problems and had to decide whether the solution was appropriate or not. In both experiments, participants had more difficulties acknowledging that problems inducing an incongruent representation could be solved than they had for problems inducing a congruent representation. This was confirmed by response times. These results highlight how semantic aspects can lead even adults to fail or succeed in the solving of arithmetic problems requiring basic mathematical knowledge.

The nature of the quantities involved in arithmetic problems promotes semantic encodings that affect the strategy chosen to solve them (Gamo, Sander, & Richard, 2010). Such encoding effects might prevent positive transfer to problems sharing the same formal mathematical structure (Bassok, Wu, & Olseth, 1995). In this study with 5th and 6th graders, we investigated the conditions promoting positive and negative transfer in arithmetic problems that could be solved with two distinct strategies. We showed that basic training do not overcome the initial impact of semantic encodings, and we provided evidence that a poor semantic encoding of the training problems leads to transfer errors. This suggests the existence of ontological restrictions on the representation mechanisms involved in word arithmetic problem solving

When solving arithmetic problems, semantic factors influence the representations built (Gamo, Sander & Richard, 2010). In order to specify such interpretative processes, we created structurally isomorphic word problems that could be solved with two distinct algorithms. We tested whether a distinction between cardinal and ordinal quantities would lead solvers, due to their daily-life knowledge, to build different representations, influencing their strategies as well as the nature of their drawings. We compared 5th grade children and adults in order to assess the validity of this hypothesis with participants of varying arithmetic proficiency. The results confirmed that the distinction between cardinal and ordinal situations led to different solving strategies and to different drawings among both age groups. This study supports the ontological distinction of cardinal versus ordinal quantities and calls for the consideration of the role of daily-life semantics when accounting for arithmetic problem solving processes.