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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.

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.

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.

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.

"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.

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. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]