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# Algebraic approach to the derivative and continuity

Colignatus, Thomas

Continuity is relevant for the real numbers and functions, namely to understand singularities and jumps. The standard approach first defines the notion of a limit and then defines continuity using limits. Surprisingly, Vredenduin (1969), Van der Blij (1970) and Van Dormolen (1970), in main Dutch texts about didactics of mathematics (journal Euclides and Wansink (1970, volume III)), work reversely for highschool students: they assume continuity and define the limit in terms of the notion of continuity. Vredenduin (1969) also prefers to set the value at the limit point (x = a) instead of getting close to it (xa). Their approach fits the algebraic approach to the derivative, presented since 2007. Conclusions are: (1) The didactic discussions by Vredenduin (1969), Van der Blij (1970) and Van Dormolen (1970) provide support for the algebraic approach to the derivative. (2) For education, it is best and feasible to start with continuity, first for the reals, and then show how this transfers to functions. (3) The notion of a limit can be defined using continuity. The main reason to mention the notion of a limit at all is to link up with the discussion about limits elsewhere (say on the internet). Later, students may see the standard approach. (4) Education has not much use for limits since one will look at continuity. The relevant use of limits is for infinity.

A website on re-engineering mathematics education is: http://thomascool.eu/Papers/Math/Index.html
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