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Introducing equations in early algebra

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Abstract

The overwhelming presence of a procedural meaning of equality and equations reported in previous research has led to a call for suitable pedagogical interventions to nurture a relational meaning of these concepts. This paper is a response to that call. Drawing on the theory of objectification, the first part deals with the configuration of a Grade 3 (8–9-year-old students) teaching–learning activity that seeks to create the classroom conditions for the formation of the mathematical operations and operation-based rules that underpin the algebraic simplification of linear equations. Instead of using problems involving abstract open arithmetic sentences or alphanumeric equations (e.g., \(5 + \_\_\) = 16; \(2n + 3 = 11\)), the teaching–learning activity resorts to story-problems. Two visual semiotic systems serving to model and solve the story-problems were devised. The story-problems were framed in narratives that allowed the teacher and the students to infuse equations, their equating parts, and the mathematical operations with contextual meanings. The first part of the paper includes the theoretical assumptions about the teaching–learning activity and its configuration, and a rationale behind the devisal of the semiotic systems. The second part presents a Vygotskian multimodal genetic analysis of the teaching–learning activity; that is, an analysis that shows the formation of concepts in motion, in the process of their genesis. The genetic analysis sheds some light on the way students, in their work with the teacher, encountered and refined the cultural-historical algebraic meanings of the equal sign and equations, and the concepts required in solving equations.

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Notes

  1. The tenth century mathematician Ikhwan al-Ṣafā’ wrote, “So I say that the product of the ten by seven is equal to (musāwin) the product of seven by itself and three by seven” (Oaks, 2010, p. 267).

  2. A story-problem is a sub-category of the category of word-problems. The difference is that a story-problem includes an experiential context and a narrative in which agents participate in some way or another. As described in the following sections, our story-problems include children (agents) having and/or receiving things (narrative) such as candies or cards (experiential context). A problem like “Divide 10 into two parts such that the division of one by the other gives 4” is a word-problem, not a story-problem.

  3. Our CSS is not far from balance models used sometimes in research and instruction (see, e.g., Blanton et al., 2018; Stephens et al., 2022; Vlassis, 2002; for a review, see Otten et al., 2020). In our case, the balance is replaced by story-problems that, through narratives, bring to life the equality of the equating parts.

  4. The first contact with equations in Grade 2 was described by Radford (2017).

  5. Figures are numbered from left to right and top to bottom. In Fig. 1 they are numbered as 1.1, 1.2, 1.3, 1.4.

  6. The six problems discussed in this paper constituted the task and were chosen following the theoretical guidelines of our approach to task design. One distinctive idea of task design in the theory of objectification is that the problems of the task follow “an organization... according to an increasing conceptual complexity” (Radford, 2021, p. 134). Simpler problems come first (e.g., x + 2 = 8), complex problems come later (e.g., 3x + 1 = 5 + x), while what is learned in one problem is put in the service of solving the next problems.

  7. For an interesting discussion of equality and similarity in Greek mathematics, see the account by Fried (2009).

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Acknowledgements

This paper is a result of a research program funded by the Social Sciences and Humanities Research Council of Canada. I am grateful to the reviewers and editors for their comments and critique.

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    Luis Radford

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Radford, L. Introducing equations in early algebra. ZDM Mathematics Education 54, 1151–1167 (2022). https://doi.org/10.1007/s11858-022-01422-x

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