Students' formal written communication

This work is part of Swedish Institute for Educational Research grant 2020-00066

Making Sense of Negative Numbers

Numbers are abstract objects that we conceptualize and make sense of through metaphors. When negative numbers appear in school mathematics, some properties of number sense related to natural numbers become contradictory. The metaphors seem to break down, making a transition from intuitive to formal mathematics necessary. The general aim of this research project is to investigate how students make sense of negative numbers, and more specifically what role models and metaphorical reasoning play in that process. The study is based on assumptions about mathematics as both a social and an abstract science and of metaphor as an important link between the social and the cognitive. It is an explorative study, illuminating the complexity of mathematical thinking and the richness of the concept of negative numbers. The empirical data were collected over a period of three years, following one Swedish school class being taught by the same teacher, using recurrent interviews, participant observations and video recordings. Conceptual metaphor theory was used to analyse teaching and learning about negative numbers. In addition to the four grounding metaphors for arithmetic described in the theory, a metaphor of Number as Relation is suggested as essential for the extension of the number domain. Different metaphors give different meanings to statements such as finding the difference between two numbers, and result in incoherent mappings onto mathematical symbols. The analyses show affordances but also many constraints of the metaphors in their role as tools for sense making. Stretching metaphors, from the domain of natural numbers to fit the domain of signed numbers, changes the metaphor, with unfamiliarity, inconsistency and limited applicability as a result. This study highlights the importance of understanding limitations and conditions of use for different metaphors, something that is not explicitly brought up during the lessons or in the textbook in the study. Findings also indicate that students are less apt to make explicit use of metaphorical reasoning than the teacher. Although metaphors initially help students to make sense of negative numbers, extended and inconsistent metaphors can create confusion. This suggests that the goal to give metaphorical meaning to specific tasks with negative numbers can be counteractive to the transition from intuitive to formal mathematics. Comparing and contrasting different metaphors could give more insight to the meaning embodied in mathematical structures than trying to fit the mathematical structure into any particular embodied metaphor. Participants in the study showed quite different learning trajectories concerning their development of number sense. Problems that students had were often related to similar problems in the historical evolution of negative numbers, suggesting that teachers and students could benefit from deeper knowledge of the history of mathematics. Students with a highly developed number sense for positive numbers seemed to incorporate negatives more easily than students with a poorly developed numbers sense, implying that more time should be spent on number sense issues in the earlier years, particularly with respect to subtraction and to the number zero

Tinkering in algebra - the case of John

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Algebraic thinking in the shadow of programming

International audienceThis paper calls attention to how the recent introduction of programming in schools interacts with the teaching and learning of algebra. The intersection between definitions of computational thinking and algebraic thinking is examined, and an example of a program activity suggested for school mathematics is discussed in detail. We argue that students who are taught computer programming with the aim of developing computational thinking will approach algebra with preconceptions about algebraic concepts and symbols that could both afford and constrain the learning of algebra

Algebraic thinking in the shadow of programming

International audienceThis paper calls attention to how the recent introduction of programming in schools interacts with the teaching and learning of algebra. The intersection between definitions of computational thinking and algebraic thinking is examined, and an example of a program activity suggested for school mathematics is discussed in detail. We argue that students who are taught computer programming with the aim of developing computational thinking will approach algebra with preconceptions about algebraic concepts and symbols that could both afford and constrain the learning of algebra

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We characterize the recently included programming content in Swedish mathematics textbooks for elementary school. Especially, the connection between programming content and traditional mathematical content has been considered. The analytical tools used are based on the so-called 5E’s, a theoretical framework of action, developed within the ScratchMath project, and Brennan and Resnick’s (2012) terms computational concepts and practices. The result uncovers “follow a procedure” as the dominating action, in which the concepts stepwise instruction and repeated pattern were frequent. Bridging between programming and mathematics is weak in the sense that the programming content does not enhance the possibility to explore mathematical concepts and ideas

Exploring the intersection of algebraic and computational thinking

This article investigates how the recent implementation of programming in school mathematics interacts with algebraic thinking and learning. Based on Duval’s theory of semiotic representations, we analyze in what ways syntax and semantics of programming languages are aligned with or divert from corresponding algebraic symbolism. Three examples of programming activities suggested for school mathematics are discussed in detail. We argue that although the semiotic representations of programming languages are similar to algebraic notation the meanings of several concepts in these two domains differ. In a learning perspective these differences must be taken into account, especially considering that students have to convert between registers with both overlapping and specific meanings