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The Linearity Prototype for Graphs: Cognitive and Sociocultural Perspectives

Linear models are frequently inappropriately applied to various situations. When misapplied, linearity is traditionally referred to as a tendency, a misconception, or an error. Cognitive linguistic, situated cognition, and sociocultural or cultural-historical activity theory, on the other hand, suggest that linear models are adopted by learners meaningfully as “prototypes,” “cultural models,” or “instruments” that mediate discourse and activity in a particular context of situation. This paper explores these theoretical frameworks as possible explanatory frameworks for some of our data on pupils' and teachers' performance on two test items that were designed to diagnose linearity: pupils' tendency to draw graphs of the form y  =  ax + b (or joined segments of this form). A high percentage of 14- to 15-year-old pupils tended toward linearity on these items, and so do their teachers, who seem generally oblivious to “linearity” as a phenomenon. We argue that this can be explained by the context of “schooling” as a social practice, or activity, in which tasks take on schooling meanings that may therefore relate only weakly to the contexts and conditions presented in the items themselves. This leads us to rethink the relationship of “modelling” with “context” in mathematics pedagogy.