Toh, Tin Lam

In this paper, several fallacies about the extension of the formula \frac{d}{dx} (\sin x) = \cos x to the erroneous formula \frac{d}{dx} (\sin x^\circ) = \cos x^\circ are discussed. In a Commognitive Theory Framework, misconceptions by ‘newcomers’ can be traced to the use of the word “unit”.

This paper examines the calculus curriculum in the current Singapore secondary and pre-university levels. Two concepts, (1) increasing and decreasing functions and their derivatives, and (2) the second derivative test for the nature of stationary points, are elaborated. An example of an incorrect calculus item in a national examination is brought up in relation to conditional reasoning involving calculus concepts. We reckon that the current emphasis on procedural knowledge in calculus is useful. However, we argue that formal conditional reasoning should not be introduced prematurely for school students.

In this paper we describe a problem solving course for first year undergraduate mathematics students who would be future school teachers.