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Proportional reasoning is a key topic both at school and in everyday life. However, students are often misled by their preconceptions regarding proportions. Our hypothesis is that these limitations can be mitigated by working on alternative ways of categorizing situations that enable more adequate inferences. Multiple categorization triggers flexibility, which enables reinterpreting a problem statement and adopting a more relevant point of view. The present study aims to show the improvements in proportional reasoning after an intervention focusing on such a multiple categorization. Twenty-eight 4th and 5th grade classes participated in the study during one school year. Schools were classified by the SES of their neighborhood. The experimental group received 12 math lessons focusing on flexibly envisioning a situation involving proportional reasoning from different points of view. At the end of the school year, compared to a control group, the experimental group had better results on the posttest when solving proportion word problems and proposed more diverse solving strategies. The analyses also show that the performance gap linked to the school's SES classification was reduced. This offers promising perspectives regarding multiple categorization as a path to overtake preconceptions and develop cognitive flexibility at school.

Intuitive conceptions are prevalent among young learners and can impose constraints to knowledge acquisition. Even though the data suggests that instruction does not eradicate them, this phenomenon has rarely been quantified. In this study we raise the question of how robust intuitive conceptions are. Moreover, we look at their persistence long after instruction of the studied notions. Production tasks concerning the four elementary arithmetic operations were used for measuring the degree to which they prevail and impose constraints among adults, 131 bachelor students as well as 168 high-school teachers and 57 mathematics teachers. The findings revealed that in most cases (88.93%) participants evoked examples that are congruent with an intuitive conception. This was observed for all the arithmetic operations and populations involved in the study. Even when explicitly prompted to find incongruent cases, they failed on two thirds of the cases. The educational entailments of these findings are discussed.

A previous research by Raynal et al. (2021) has demonstrated that unconventional verb extensions based on abstract similarities (i.e. semantic approximations) elicited a reduced N400 compared to pseudoverbs in preschoolers. Whereas this result reveals young children’s analogical abilities, the current study aims to shed light on the specificity of mature metaphorical processing. To do so, the design of the previous research was replicated with adults. Participants saw images of actions (e.g., peeling an orange) followed by an auditory presentation of sentences containing either a conventional verb (e.g., peeling), an approximate verb (e.g., undressing), a superficially related verb (e.g., pressing) or a pseudoverb (e.g., rauging). We found that approximate verbs elicited a strong N400 response, followed by a reduced Late Negative Component (LNC) compared to pseudoverbs. These findings suggest that mature metaphorical reasoning involves a category violation followed by a reinterpretation process allowing one to recategorize the source and the target concepts through a common abstract concept.

Biodiversity is a complex concept entailing scientific and political aspects. The usage of analogies, especially metaphors, that have positive influences on the understanding of complex concepts, on attitudes and behaviors, seems an interesting strategy to achieve this goal. Based on biodiversity analogies elaborated by 259 participants, a first study aims to identify two important protective approaches: preservationism that encourages humankind to limit their intervention on nature and conservationism that allows humankind to exploit nature with parsimony. We analyzed their analogies and results highlight three major groups: a scientific, a conservationist and a preservationist dimension. A second study investigates the effects of metaphorical framing on environmental attitudes and behaviors. 277 University students read a short text framing biodiversity with a preservationist or conservationist metaphor or without metaphor framing. A decision-making task and an environmental concern scale were completed. Results showed an effect of the conservationist metaphor on the decision-making task.

This study investigates whether the ERPs of 4-years-olds in response to verbal overextensions reflect the encoding of actions through abstract categories. Participants were presented with images of actions (e.g. peeling an orange) while hearing a sentence containing a conventional verb (e.g. peeling), an approximative verb (e.g. undressing), a superficially related verb (e.g. pressing) or a pseudoverb (e.g. rauging). The N400 for approximative verbs significantly differed from the pseudoverb condition, but not from the conventional verb condition. In contrast, the N400 for superficially related verbs was significantly greater than for conventional verbs, but no significant difference was found with the pseudoverb condition. These results confirm our hypothesis that encoding mainly focuses on general categories (e.g. taking of an envelope). The implications of the findings are discussed regarding the conceptual organization of preschoolers and their analogical abilities.

Evidence suggests that general, non-mathematical knowledgeabout the entities described in an arithmetic word problem mayinterfere with its encoding. We used behavioral and eye-tracking measures to investigate how the use of specificquantities may foster a cardinal representation of the numbersmentioned in a problem, whereas other quantities may favor anordinal representation instead. We asked 50 pre-serviceteachers to complete a solution validity assessment task. Wecompared participants’ gaze patterns on isomorphic problemsto gather insights into their encoded representations. Onproblems featuring cardinal quantities, we found that specificsentences describing elements relevant in a cardinalunderstanding of the problems but irrelevant otherwise werelooked at longer and were the focus of a higher number ofbackward eye movements. Additionally, an increase in pupildilation on correctly solved cardinal problems supported theidea that participants need to engage in a recoding processwhen facing semantic incongruence.

Informal strategies reflecting the representation of a situation described in an arithmetic word problem mediate students' solving processes. When the informal strategies are inefficient, teaching students to make way for more efficient ways to find the solution is an important educational issue in mathematics. The current paper presents a pedagogical design for arithmetic word problem solving, which is part of a first-grade arithmetic intervention (ACE). The curriculum was designed to promote adaptive expertise among students through semantic analysis and recoding, which would lead students to favor the more adequate solving strategy when several options are available. We describe the ways in which students were taught to engage in a semantic analysis of the problem, and the representational tools used to favor this conceptual change. Within the word problem solving curriculum, informal and formal solving strategies corresponding to the different formats of the same arithmetic operation, were comparatively studied. The performance and strategies used by students were assessed, revealing a greater use of formal arithmetic strategies among ACE classes. Our findings illustrate a promising path for going past informal strategies on arithmetic word problem solving.

We propose to assess conceptual knowledge of mathematical notions by having recourse to isomorphic word problems. We assumed that failing to solve isomorphic problems is an indicator of lack of conceptual knowledge. To reach these conclusions, two experiments were conducted among 4th and 5th grade students. In experiment 1, each student had to solve one isomorphic word problem allowing two solving strategies. Results from experiment 1 showed that semantic context promoted a one-sided perspective, constraining the strategies used. The second experiment questioned if the strategy choices by students reflected a lack of procedural knowledge or difficulty in conceptual knowledge. Students solved several problems with the explicit task of proposing two strategies. Results showed that students struggled to identify the conceptual structure of the problems: they changed their strategy depending on semantic contexts, and did not succeed in adopting two encodings for the same problem. This lack of flexibility does not reflect weak procedural knowledge but difficulties in conceptual knowledge. Once conceptual knowledge is characterized as the ability to get over dependence on semantic context, using word problems as assessment tools for conceptual development opens up educational perspectives.

In this study, we test the possibility that real-life events induce anabstract category activation in a way that permits structurally-basedretrievals. We used a free-recall reminding paradigm whereparticipants had to report any memory that come to mind when facedwith a target cue embodying a familiar concept. This methodallowed us to consider the retrievals of any analog that shares ameaningful structural similarity in the participants’ own eyes.Results revealed that most participants predominantly retrievedSuperficially Dissimilar Analogs (SDAs) rather than SuperficiallySimilar Disanalogs (SSDs). Interestingly, retrievals of SDAs werepreponderant over retrievals of Superficially Similar Analogs(SSAs). These data suggest that familiar abstract knowledge mayhave a more important role in promoting abstract encoding andstructurally-based retrievals than it was supposed to.

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.

Arithmetic problem solving is a crucial part of mathematics education. However, existing problem solving theories do not fully account for the semantic constraints partaking in the encoding and recoding of arithmetic word problems. In this respect, the limitations of the main existing models in the literature are discussed. We then introduce the Semantic Congruence (SECO) model, a theoretical model depicting how world and mathematical semantics interact in the encoding, recoding, and solving of arithmetic word problems. The SECO model's ability to account for emblematic results in educational psychology is scrutinized through six case studies encompassing a wide range of effects observed in previous works. The influence of world semantics on learners' problem representations and solving strategies is put forward, as well as the difficulties arising from semantic incongruence between representations and algorithms. Special attention is given to the recoding of semantically incongruent representations, a crucial step that learners struggle with.

Ensuring students correctly understand the notion of equality is a fundamental problem in teaching mathematics. Is it possible to teach arithmetic in such a way that students do not misinterpret the equal sign "=" as indicating the result of an arithmetic operation and, consequently, viewing the arithmetic operation symbols (+, -, or x) as systematically meaning perform a computation? The present paper describes the implementation of an experimental arithmetic teaching program (called ACE) drawn up in conjunction with teachers and focusing on these issues. We assessed the program's impact via an experiment involving 1140 experimental group students and 1155 control group students, and using a pretest/posttest design. The experimental group students achieved higher composite arithmetic scores, combining all four subdomains (arithmetic writing, mental computation, word-problem solving, and estimation), than control group students. The effect size, computed using individuals' posttest scores adjusted for pretest scores was d = 0.559. The ACE program was effective in all four subdomains tested; however, it was particularly successful in the arithmetic writing subdomain, especially in writing equalities. The program's effect not only held one year later, it was cumulative, as the benefits produced by following the program in both first and second grade were almost double those of following the program in just one grade.

When solving arithmetic problems, semantic factors influencethe representations built (Gamo, Sander & Richard, 2010). Inorder to specify such interpretative processes, we createdstructurally isomorphic word problems that could be solvedwith two distinct algorithms. We tested whether a distinctionbetween cardinal and ordinal quantities would lead solvers, dueto their daily-life knowledge, to build different representations,influencing their strategies as well as the nature of theirdrawings. We compared 5th grade children and adults in orderto assess the validity of this hypothesis with participants ofvarying arithmetic proficiency. The results confirmed that thedistinction between cardinal and ordinal situations led todifferent solving strategies and to different drawings amongboth age groups. This study supports the ontological distinctionof cardinal versus ordinal quantities and calls for theconsideration of the role of daily-life semantics whenaccounting for arithmetic problem solving processes.

The predominant view concerning determinants ofanalogical retrieval is that it is preferentially guided by superficialcues. In order to test the cognitive plausibility of a structuralsimilarities-based retrieval, we constructed a story-recall task inwhich life-like scenarios shared structural correspondences. InExperiment 1, we showed that such structural similarities induceretrievals when the participant had several source candidatesituations sharing superficial similarities with the target cue.Experiment 2 was designed to test whether the encoding wassufficiently oriented on structural similarities to drive retrievals,even if the participants possess only one source candidate situationwith superficial matches in memory. The results of the two presentexperiments lead us to conclude that in some contexts, abstractencoding induces a superiority of structural similarities oversuperficial ones in retrieval. Further implications for analogicalretrieval approaches are discussed.

In problem solving situation, cognitive flexibility appears tobe a major skill. Fostering cognitive flexibility is therefore aspecific stake in mathematics education. This researchintroduces a learning method to develop mathematicalconcepts when solving word arithmetic problems. The studywas conducted with 8 classes (4 th -5 th Grades) from high-priority education schools in the Paris area following thisprotocol: pre-tests, 5 learning sessions for experimental andcontrol groups, post-tests. During learning sessions, studentsstudied arithmetic word problems that can be solved in twodifferent ways: an expansion strategy and a factorization one.The experimental teaching method, based on arecategorization principle, allowed experimental students toimprove more than the control students in ability to use thefactorization strategy even in contexts where it is the lessintuitive and to consider the two successful strategies.Educational entailments of our finding are discussed.

Young children use informal strategies to solve arithmeticword problems. The Situation Strategy First (SSF) frameworkclaims that these strategies prevail even after instruction. Thepresent study was conducted with second grade students inorder to investigate the persistence of intuitive, situation-based strategies, on word problems that do not involvedynamic temporal changes. This is challenging for the SSFframework, since the lack of this dimension might bypassintuitive strategies. The results revealed that intuitivestrategies persist, are valid for these types of problems, andimpact the problems' difficulty. Indeed problems that requirethe application of arithmetic principles remain hard, eventhough they have been practiced at school. These findingsprovide complementary evidence to how mental calculationstrategies articulate with arithmetic word problem solving andcall for the extension of the SSF framework.

We investigated what happens when the spontaneousencoding of a problem is incongruent with its solvingstrategy. We created word problems from which two distinctsemantic representations could be abstracted. Only one ofthese representations was consistent with the solving strategy.We tested whether participants could recode a semanticallyincongruent representation in order to access another, lesssalient, solving strategy. In experiment 1, participants had tosolve arithmetic problems and to indicate which problemswere unsolvable. In experiment 2, participants receivedsolved problems and had to decide whether the solution wasappropriate or not. In both experiments, participants had moredifficulties acknowledging that problems inducing anincongruent representation could be solved than they had forproblems inducing a congruent representation. This wasconfirmed by response times. These results highlight howsemantic aspects can lead even adults to fail or succeed in thesolving of arithmetic problems requiring basic mathematicalknowledge.

Intuitive conceptions in mathematics guide the interpretation of mathematical concepts. We investigated if they bias teachers' conceptions of student arithmetic word problem solving strategies, which should be part of their pedagogical content knowledge (PCK). In individual interviews, teachers and non-teaching adults were asked to describe students' strategies in situational contexts within or outside the scope of the intuitive conception. The results revealed that teachers relied on their PCK and identified student strategies; however, in the presence of the intuitive conception, their PCK was overshadowed and they ceased to differ significantly from non-teachers. This brings the attention to certain biases that can have a strong impact on teachers' efficient use of PCK.

The nature of the quantities involved in arithmetic problemspromotes semantic encodings that affect the strategy chosento solve them (Gamo, Sander, & Richard, 2010). Suchencoding effects might prevent positive transfer to problemssharing the same formal mathematical structure (Bassok, Wu,& Olseth, 1995). In this study with 5th and 6th graders, weinvestigated the conditions promoting positive and negativetransfer in arithmetic problems that could be solved with twodistinct strategies. We showed that basic training do notovercome the initial impact of semantic encodings, and weprovided evidence that a poor semantic encoding of thetraining problems leads to transfer errors. This suggests theexistence of ontological restrictions on the representationmechanisms involved in word arithmetic problem solving

"Natural syllogisms" are arguments formally identifiable with categorical syllogisms that have an implicit universal affirmative premise retrieved from semantic memory rather than explicitly stated. Previous studies with adult participants (Politzer, 2011) have shown that the rate of success is remarkably high. Because their resolution requires only the use of a simple strategy (known as "ecthesis" in classic logic) and an operational use of the concept of inclusion (the recognition that an element that belongs to a subset must belong to the set but not vice versa), it was hypothesized that these syllogisms would be within the grasp of non-adult participants, provided they have acquired the notion of deductive validity. Here, 11-year-old children were presented with natural syllogisms embedded in short dialogs. The first experiment showed that their performance was equivalent to adults' highest level of performance in standard experiments on syllogisms. The second experiment, while confirming children's proficiency in solving natural syllogisms, showed that they outperformed children who solved non-natural matched syllogisms in the same experimental setting. The results are also in agreement with the argumentation theory of reasoning.

Transfer of strategies between problems sharing the same formal structure is facilitated by a semantic recoding that makes evident the structural similarities between the problems. Two experiments were carried out among 4th and 5th grade pupils, with an experimental group trained to compare strategies in order to reinterpret an arithmetic word problem so that the calculations are consistent with the interpretation, and a control group. The experimental group in Experiment 1 improved significantly by choosing the alternative strategy in problems different from those used in training but no significant progress was observed in the control group. Experiment 2 showed that the improvement observed in the experimental group could not be attributed to the fact that children only learned to use a superficial rule. These results support the idea that the activity of problem re-representation may be a crucial step in mathematization. (Contains 1 figure and 3 tables.)

Before instruction, children solve many arithmetic word problems with informal strategies based on the situation described in the problem. A Situation Strategy First framework is introduced that posits that initial representation of the problem activates a situation-based strategy even after instruction: only when it is not efficient for providing the numerical solution is the representation of the problem modified so that the relevant arithmetic knowledge might be used. Three experiments were conducted with Year 3 and Year 4 children. Subtraction, multiplication and division problems were created in two versions involving the same wording but different numerical values. The first version could be mentally solved with a Situation strategy (Si version) and the second with a Mental Arithmetic strategy (MA version). Results show that Si-problems are easier than MA-problems even after instruction, and, when children were asked to report their strategy by writing a number sentence, equations that directly model the situation were predominant for Si-problems but not for MA ones. Implications of the Situation Strategy First framework regarding the relation between conceptual and procedural knowledge and the development of arithmetic knowledge are discussed.