The underground secant: a tour in search of a definition. Italian pupils and students interpretation
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The mathematical concept of secant has an intrinsic double nature: formal and informal, the second, often, is a result of a visual approach. We can apply to the secant what Sfard (2008, p. 39) states, i.e. that “there are some specialized keywords we use without ever being exposed to their explicit definitions”. We consider the word and the mathematical meaning of secant as an example of what Sfard notices and we can name ‘underground curriculum’. As to secant, the assumption is that informal, visual, content should be enough for to appreciate the formal definition but this assertion clashes with the result of a questionnaire administered to pupils of grade 11–13 (16–18 year old) and first enrolled in university students. A widespread didactic practice strictly connects the mathematical concept of secant with the derivative through the construction of the tangent to the curve. Therefore misunderstanding the concept of ‘secant’ could negatively affect the comprehension of Calculus.