Constructing the Concept Image of a Tangent

When a learner meets a new mathematical concept, it may be invested with implicit properties arising from the context, producing an idiosyncratic concept image which may cause cognitive conflict at a later stage. The purpose of this empirical research is to test the hypothesis that interactive computer programs, encouraging teacher demonstration and pupil investigation of a wide variety of examples and non-examples, may be used to help students develop a richer concept image capable of responding more appropriately to new situations. Three experimental classes of sixteen year-olds were taught using computer packages capable of magnifying graphs to see if they “looked straight”, and to draw a line through two close points on a graph. These formed the basis of class discussion and small group investigation to encourage the formation of a coherent relationship between the concepts of gradient and tangent. For comparison, five other classes were taught by more traditional methods. Two questionnaires administered during the course confirmed that the experimental students were able to respond more appropriately in new situations, for example in the case where a function is given by a different formula to the left and right. However, the notion of a “generic tangent” – an imagined line touching the graph at only one point (even where this is inappropriate) – persisted in both groups, though significantly less amongst the experimental students.