Eliciting Algebraic Reasoning in Primary School Students From an Embodiment Perspective

• In Event: Higher Order Thinking in Primary School Mathematics Education

Abstract

Algebraic competence is seen as a gatekeeper for further studies and later occupational opportunities (Katz, 2007). Because algebra implies seeking and exploring patterns and relationships, and searching for generalizations (Caspi & Sfard, 2012), it is strongly related to mathematical higher-order thinking. Even though there is strong consensus that these activities can be a vital part of primary
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school mathematics (Kaput et al., 2008), early algebra is often lacking in the primary school mathematics curriculum. The goal of our study is gaining knowledge about how algebra, i.e. context-based equations, can be included.
Teaching primary school students to solve equations often means (cf. Smith & Thompson, 2008) considering algebra as an elaboration of arithmetic: from reasoning with numbers to reasoning with unknowns. In our study, we chose another approach. We started with context-based problems in which students have to reason with unknowns. This reasoning was elicited by offering students physical experiences with a hanging mobile functioning as a balance scale. With this mobile, students could explore relations between unknowns and spontaneously develop algebraic strategies of restructuring, isolation, and substitution.
This way of teaching early algebra is grounded in embodiment theory, which assumes the fundamental coupling between action and perception as a basis for mathematical reasoning (Abrahamson, 2017). This implies that perceptuo-motor experiences with particular (mathematical) concepts could be beneficial when learning new concepts (Lakoff & Núñez, 2000). As equivalence is a crucial concept for understanding equations (e.g., Knuth et al., 2005), learning activities in which children physically interact with this concept could provide a strong basis for developing understanding of equivalence and equation solving. When working with the hanging mobile, the bodily based experience of balance – a pervasive physical experience we have in everyday life (Gibbs, 2006) – is revitalized and can be used as the metaphorical representation of equivalence. The elicited algebraic strategies are consequently linked to this revitalized experience of maintaining balance. The current study aimed to map whether, and to what extent, algebraic reasoning of the students developed during these activities.
Method
The experimental intervention consisted of six one-hour lessons that were taught in six fifth-grade classes (N = 132). There was an embodiment and a non-embodiment condition. Each class received a mathematical higher-order thinking test before and after the intervention (macro-genetic development) and a domain-specific test after each lesson (micro-genetic development). Both tests consisted of open-ended questions, explicitly inviting students to explain their reasoning. Students’ work was analyzed based on the frequency of correct answers and their level of reasoning. The development over the domain-specific tests was analyzed and pre- and post-measures were compared.
Results and significance
Data are currently being gathered. We expect our results to reflect students’ development of algebraic strategies over the lesson series. This development would be a strong indicator of students’ potential for learning algebra. It will give insight in how algebraic reasoning and related higher-order thinking evolve, and in the role of embodiment as a learning facilitator. This knowledge will be valuable for teaching linear equations and designing future embodied learning activities.

Authors

• Mara Otten, Freudenthal Group FSW - Utrecht University

• Marja Van Den Heuvel-Panhuizen, Utrecht University, the Netherlands; Nord University, Norway

• Michiel Veldhuis, Utrecht University

• Aiso Heinze, Leibniz Institute for Science and Mathematics Education