Spontaneous meta-arithmetic as a first step toward school algebra

Taking as the point of departure the vision of school algebra as a formalized meta-discourse of arithmetic, we have been following five pairs of 7th grade students as they progress in algebraic discourse during 24 months, from their informal algebraic talk to the formal algebraic discourse, as taught in school. Our analysis follows changes that occur along time in the discourse produced by the students when they deal with specific types of algebraic problems. In this paper we take a look at the earliest stage in this process. Our data have shown, unsurprisingly, that while reflecting on arithmetic processes and relations, the uninitiated 7th graders were employing colloquial means, which could not protect them against occasional ambiguities. More unexpectedly, this spontaneous meta-arithmetic, although not supported by any previous algebraic schooling, displayed some algebra-like features, not to be normally found in everyday discourses. This finding prompted us to extend the study and take a look at spontaneous meta-arithmetic of younger children. Our data collected among 5th graders have confirmed that informal algebraic discourse emerges early, and they shed light on the development of meta-arithmetic in the span of two years prior to the formal introduction of algebra in school.

► The older children's informal algebra was more reified than younger students.
► The discourses of all participants were still in the need of disambiguation.
► Spontaneous meta-arithmetic starts developing years before the “official” algebra.
► Teachers should engage children in gradual formalization of their meta-arithmetic.