Academic journal article Australian Primary Mathematics Classroom

Eliciting Algebraic Reasoning with Hanging Mobiles

Academic journal article Australian Primary Mathematics Classroom

Eliciting Algebraic Reasoning with Hanging Mobiles

Article excerpt

How algebraic reasoning can be fostered within the important big idea of equivalence is demonstrated using hanging mobiles. A concrete-representational-abstract approach is used, without any formal algebraic symbolism, to elicit algebraic reasoning and higher-order thinking.

The importance of laying a foundation for algebraic reasoning at a young age is increasingly being emphasised. In this article, we report on an activity that elicits in a natural way algebraic strategies that in a later stage of learning algebra are crucial for solving equations. The activity brings about the students' spontaneous use of symbolic notations. It also makes students' reasoning visible both to themselves and to their teacher and helps build students' conceptual understanding and foster productive classroom discussion. This activity involves working with a hanging mobile.

In this article we describe the work of students aged 11-12, using a hanging mobile as shown in Figure 1. This mobile can be considered as a balance model representing an equation with unknowns. The chains on both sides of the mobile support coloured bags. The different bags each have a particular weight to make the balance 'workable', but the weight aspect is not explicitly mentioned to the students. Neither are the students told that the activity is about solving equations with unknowns. For them, this is a puzzle. They have to figure out what you can do with the bags in the hanging mobile while keeping the mobile in balance and then use these strategies to find relationships among the bags. Figuring out these puzzles does not require formal algebra, as students can physically add, remove and exchange bags and can see the result of their actions.

In this way, they develop strategies such as restructuring (e.g. moving bags with respect to each other but keeping them on one side), isolation (e.g. removing the same bags from both sides of the hanging mobile), and substitution (replacing bags with different coloured bags), strategies which are very important for solving equations. In fact, the informal strategies that come up now form a pre-stage of the most prominent strategies that are used when solving formal equations. Promoting such informal reasoning in the field of algebra is in line with the proficiency strands of the Australian Curriculum: Mathematics (ACARA, 2015; see also Hurrell, 2012).

The large-size physical hanging mobile that we used in this activity was inspired by mobile puzzles. The idea of these puzzles is old (Kroner, 1997) but their curricular use is newer (Goldenberg, Mark, Kang, Fries, Carter, & Cordner, 2015; see for an online version). Like the mobile puzzles on paper, the physical model can trigger algebraic reasoning in students. Both do this without requiring formal techniques or notations, but the advantage of the physical hanging mobile is that the students physically interact with the mobile and have an embodied experience in keeping the mobile in balance. Another positive aspect of the physical mobile is that the students can watch each other's actions, which can support classroom discussion.

A physical model of equivalence

A balance models the mathematical idea of equivalence, a crucial concept for understanding equations (Faulkner, Walkowiak, Cain, & Lee, 2016; Greenes & Findell, 1999; Hurrell, 2012). The mobile thus represents an equation that students can handle physically, using their sense and intuition to build the logical system of algebra. Part of this system is the meaning of the equal sign. Students often interpret the equal sign as a "to do"sign--as it is on a calculator--instead of an "is equal to"sign (Carraher & Schliemann, 2007). The hanging mobile presents this latter meaning without any notational hurdle and builds on the balancing experiences young children already have in everyday life (e.g., seesaws). The mobile gives students bodily experiences of the concept of equivalence, which can help them anchor this concept, in line with the theory of embodied cognition (cf. …

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