Neural correlates of unconventional verb extensions reveal preschoolers' analogical abilities.

In the current event-related potential (ERP) study, we assessed 4-year-olds' ability to extend verbs to new action events on the basis of abstract similarities. Participants were presented with images of actions (e.g., peeling an orange) while hearing sentences containing a conventional verb (e.g., peeling), a verb sharing an abstract relation (i.e., an analogical verb, e.g., undressing), a verb sharing an object type (i.e., an object-related verb, e.g., pressing) with the action, or a pseudoverb (e.g., kebraying). The amplitude of the N400 gradually increased as a function of verb type-from conventional verbs to analogical verbs to object-related verbs to pseudoverbs. These findings suggest that accessing the meaning of a verb is easier when it shares abstract relations with the expected verb. Our results illustrate that measuring brain signals in response to analogical word extensions provides a useful tool to investigate preschools' analogical abilities.

Uncovering the interplay between drawings, mental representations, and arithmetic problem-solving strategies in children and adults.

There is an ongoing debate in the scientific community regarding the nature and role of the mental representations involved in solving arithmetic word problems. In this study, we took a closer look at the interplay between mental representations, drawing production, and strategy choice. We used dual-strategy isomorphic word problems sharing the same mathematical structure, but differing in the entities they mentioned in their problem statement. Due to the non-mathematical knowledge attached to these entities, some problems were believed to lead to a specific (cardinal) encoding compatible with one solving strategy, whereas other problems were thought to foster a different (ordinal) encoding compatible with the other solving strategy. We asked 59 children and 52 adults to solve 12 of those arithmetic word problems and to make a diagram of each problem. We hypothesized that the diagrams of both groups would display prototypical features indicating either a cardinal representation or an ordinal representation, depending on the entities mentioned in the problem statement. Joint analysis of the drawing task and the problem-solving task showed that the cardinal and ordinal features of the diagrams are linked with the hypothesized semantic properties of the problems and, crucially, with the choice of one solving strategy over another. We showed that regardless of their experience, participants' strategy use depends on their problem representation, which is influenced by the non-mathematical information in the problem statement, as revealed in their diagrams. We discuss the relevance of drawing tasks for investigating mental representations and fostering mathematical development in school.

Reconsidering conceptual knowledge: Heterogeneity of its components.

Cognitive arithmetic classically distinguishes procedural and conceptual knowledge as two determinants of the acquisition of flexible expertise. Whereas procedural knowledge relates to algorithmic routines, conceptual knowledge is defined as the knowledge of core principles, referred to as fundamental structures of arithmetic. To date, there is no consensus regarding their number, list, or even their definition, partly because they are difficult to measure. Recent findings suggest that among the most complex of these principles, some might not be "fundamental structures" but rather may articulate several components of conceptual knowledge, each specific to the arithmetic operation involved. Here, we argue that most of the arithmetic principles similarly may rather articulate several core concepts specific to the operation involved. Data were collected during a national mathematics contest based on an arithmetic game involving a large sample of 9- to 11-year-old students (N = 11,243; 53.1% boys) over several weeks. The purpose of the game was to solve complex arithmetic problems using five numbers and the four operations. A principal component analysis (PCA) was performed. The results show that both conceptual and procedural knowledge were used by children. Moreover, the PCA sorted conceptual and procedural knowledge together, with dimensions being defined by the operation rather than by the concept. This implies that "fundamental structures" rather regroup different concepts that are learned separately. This opens the way to reconsider the very nature of conceptual knowledge and has direct pedagogical implications.

What we count dictates how we count: A tale of two encodings.

We argue that what we count has a crucial impact on how we count, to the extent that even adults may have difficulty using elementary mathematical notions in concrete situations. Specifically, we investigate how the use of certain types of quantities (durations, heights, number of floors) may emphasize the ordinality of the numbers featured in a problem, whereas other quantities (collections, weights, prices) may emphasize the cardinality of the depicted numerical situations. We suggest that this distinction leads to the construction of one of two possible encodings, either a cardinal or an ordinal representation. This difference should, in turn, constrain the way we approach problems, influencing our mathematical reasoning in multiple activities. This hypothesis is tested in six experiments (N = 916), using different versions of multiple-strategy arithmetic word problems. We show that the distinction between cardinal and ordinal quantities predicts problem sorting (Experiment 1), perception of similarity between problems (Experiment 2), direct problem comparison (Experiment 3), choice of a solving algorithm (Experiment 4), problem solvability estimation (Experiment 5) and solution validity assessment (Experiment 6). The results provide converging clues shedding light into the fundamental importance of the cardinal versus ordinal distinction on adults' reasoning about numerical situations. Overall, we report multiple evidence that general, non-mathematical knowledge associated with the use of different quantities shapes adults' encoding, recoding and solving of mathematical word problems. The implications regarding mathematical cognition and theories of arithmetic problem solving are discussed.

Are Superficially Dissimilar Analogs better retrieved than Superficially Similar Disanalogs?

In the present study, we tested the assumption that structural similarity overcomes surface similarity in the retrieval of past events, by observing whether structural similarity alone is a better cue than surface similarity alone. To do so, in four story-recall experiments, we provided the participants with multiple source stories and then with a target cue story. This target cue only shared either surface or structural similarity with the source stories. In Experiment 1A, a Superficially Similar Disanalog source story (SSD) and a Superficially Dissimilar Analog source story (SDA) were presented among Superficially Dissimilar Disanalog source stories (SDDs). A soundness rating task was used in Experiment 1B to control the absence of structural similarity among the SSDs presented in Experiment 1A. In Experiment 2, the number of SSDs was increased in the aim to reproduce more ecological conditions. In two further experiments, a five minute (Experiment 3) and a 45 minute (Experiment 4) delay was introduced, and supplementary source stories were presented, in order to make the study more similar to previous story-recall paradigms. The results of the four story-recall experiments support the dominance of structural over surface similarities in analogical retrieval. The role of a structurally-based access regarding the retrieval of Superficially Similar Analogs (SSAs) and SDAs is discussed, as well as the factors underlying the rare occurrence of SDAs retrievals in previous experiments.

When masters of abstraction run into a concrete wall: Experts failing arithmetic word problems.

Can our knowledge about apples, cars, or smurfs hinder our ability to solve mathematical problems involving these entities? We argue that such daily-life knowledge interferes with arithmetic word problem solving, to the extent that experts can be led to failure in problems involving trivial mathematical notions. We created problems evoking different aspects of our non-mathematical, general knowledge. They were solvable by one single subtraction involving small quantities, such as 14 - 2 = 12. A first experiment studied how university-educated adults dealt with seemingly simple arithmetic problems evoking knowledge that was either congruent or incongruent with the problems' solving procedure. Results showed that in the latter case, the proportion of participants incorrectly deeming the problems "unsolvable" increased significantly, as did response times for correct answers. A second experiment showed that expert mathematicians were also subject to this bias. These results demonstrate that irrelevant non-mathematical knowledge interferes with the identification of basic, single-step solutions to arithmetic word problems, even among experts who have supposedly mastered abstract, context-independent reasoning.

Preadolescents Solve Natural Syllogisms Proficiently.

"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.

Refining the assessment of internal working models: the Attachment Multiple Model Interview.

The Attachment Multiple Model Interview (AMMI) was developed to assess internal working models (IWMs) of specific relationships in adulthood (e.g., with mother, father, and romantic partner). In an initial effort to validate the AMMI, the interview was administered to participants who were followed from age 4 to 23. ANOVA and contrast tests confirmed the AMMI's capacity to discriminate between mother, father, and partner IWMs. AMMI security with each parent was correlated with coherence according to the Adult Attachment Interview, and AMMI disorganization with mother with unresolved trauma (N = 53). AMMI dimensions of security, deactivation, and hyperactivation with the mother were associated with cumulative lifetime scores of security (N = 23), avoidance, and resistance (N = 34), respectively. Intercorrelations between these AMMI scales were also theory-consistent. Associations with the AAI and between AMMI security scores of different relationships are consistent with previous findings suggesting a contribution from both parents in the development of a state of mind, but a more important role of the mother for representations of the partner.

Arguments, more than confidence, explain the good performance of reasoning groups.

In many intellective tasks groups consistently outperform individuals. One factor is that the individual(s) with the best answer is able to convince the other group members using sound argumentation. Another factor is that the most confident group member imposes her answer whether it is right or wrong. In Experiments 1 and 2, individual participants were given arguments against their answer in intellective tasks. Demonstrating sound argumentative competence, many participants changed their minds to adopt the correct answer, even though the arguments had no confidence markers, and barely any participants changed their minds to adopt an incorrect answer. Confidence could not explain who changed their mind, as the least confident participants were as likely to change their minds as the most confident. In Experiments 3 (adults) and 4 (10-year-olds), participants solved intellective tasks individually and then in groups, before solving transfer problems individually. Demonstrating again sound argumentative competence, participants adopted the correct answer when it was present in the group, and many succeeded in transferring this understanding to novel problems. Moreover, the group member with the right answer nearly always managed to convince the group even when she was not the most confident. These results show that argument quality can overcome confidence among the factors influencing the discussion of intellective tasks. Explanations for apparent exceptions are discussed.

Arithmetic word problem solving: a Situation Strategy First framework.

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.