This study aims at elaborating a well-established theoretical framework that distinguishes three modes of thinking in linear algebra: the analytic-arithmetic, the synthetic-geometric, and the analytic-structural mode. It describes and analyzes the bundle of signs produced by an engineering student during an interview, where she was asked to recall…

To understand the ways that manipulatives might support changes in students' reasoning about algebraic generalizations, a constructivist teaching experiment was conducted with two sixth-grade students. The students interpreted numerical situations with units of one and could construct units of units in mental activity. Initially, the students'…

The mathematics education literature indicates a consensus regarding the importance of developing algebraic thinking in elementary school mathematics. However, the approaches used to implement this concept vary around the world. This study examines how a popular standards-based elementary mathematics textbook series in China provides opportunities…

When a number sentence includes more than one operation, students are taught to follow the rules for the order of operations to get the correct result. In this context, brackets are used to determine the operations that should be calculated first. However, it seems that the written format of an arithmetical expression has an impact on the way…

Teachers' understanding of the concepts they teach affects the quality of instruction and students' learning. This study used a sample of 303 teachers from across the USA to examine elementary school mathematics teachers' knowledge of key concepts underlying fraction arithmetic. Teachers' explanations were coded based on the accuracy of their…

This article concerns the purpose, function, and mechanisms of students' rhythmic behaviors as they solve embodied-interaction problems, specifically problems that require assimilating quantitative information structures embedded into the environment. Analyzing multimodal data of one student tackling a bimanual interaction design for proportion,…

In this paper, we report how 5-year-olds' arithmetic skills developed through participation in an 8-month-long intervention. The intervention program aimed to enhance the children's ways of experiencing numbers' part-part-whole relations as a basis for arithmetic skills and was built on principles from the variation theory of learning. The report…

The use of artifacts to introduce the distributive law of multiplication over addition in primary school is a diffused approach: it is possible to find pre-constructed learning trajectories in instructional materials. However, it is still unclear how the teacher might support his/her students in transitioning from concrete to symbolic…

This study investigated 111 pre-service teachers' (PSTs') flexibility with referent units in solving a fraction division problem using a length model. Participants' written solutions to a measurement fraction division problem were analyzed in terms of strategies and types of errors, using an inductive content analysis approach. Findings suggest…

This paper examines the existence and impact of theoretical mathematical foundations on the teaching and learning of whole numbers in elementary school in France. It shows that the study of the New Math reform--which was eventually itself replaced in the longer term--provides some keys to understanding the influence of mathematical theories on…

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…

This paper deals with the competence of Cultural (epistemological, historical and anthropological) Analysis of the Content (CAC), which is important for teachers' awareness and autonomy when dealing with educational choices in a changing cultural and institutional context. We report on an 18-hour intervention in a teacher education course at the…

Children who experience difficulties with learning mathematics should be taught by teachers who focus on the child's best way of learning. Analyses of the mathematical difficulties are necessary for fine-tuning mathematics education to the needs of these children. For this reason, an instrument for Observing and Analyzing children's Mathematical…

Our aim here was to explore, via a specific instance, the potential for learners to develop mathematically as a consequence of the interplay between intuition and indirect classroom experience rather than through explicit tuition. A significant aspect of this study is the recognition of the possibility for learners to be able to thematize schemata…

This study tested the hypothesis that intuitions about the effect of operations, e.g., "addition makes bigger" and "division makes smaller", are still present in educated adults, even after years of instruction. To establish the intuitive character, we applied a reaction time methodology, grounded in dual process theories of reasoning. Educated…