Counter-examples in Calculus - ISBN:0476012155

Counter-examples in Calculus

Date: 27 April 2011, 08:55

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This book is a welcome and refreshing antidote to the descending spiral of instrumentality. It is offered to those students and those teachers who know that there is more to learning mathematics than completing homework mechanically. It is consistent with the view put forward by A. Watson and J. Mason in “Mathematics as a Constructive Activity: The Role of Learner- Generated Examples” (Mahwah: Erlbaum, 2005) that athematics is a constructive activity, and that a central aspect of learning mathematics is enriching the space of examples which come to mind and to which you have access when you encounter a technical term.
An excellent way to do this is to become familiar with a wide range of examples, sometimes called ‘pathological’, but only because they are unfamiliar and even unexpected. The care and precision needed to do and to use mathematics depends upon and requires people to extend their range of ‘familiar’ examples. One of the classic behaviours of students trying to use mathematics in another discipline is a cavalier attitude to conditions and constraints.
Desperate to get a task completed, scant regard is paid to conditions which are necessary in order to apply a theorem or technique. By making the search for counter-examples an integral part of the way they expose students to mathematics, teachers can imbue all students with a more mathematical way of approaching and using mathematics. This book provides the groundwork on which to ascend the spiral of instrumentality towards appreciation and understanding of the mathematicsbehind the calculus.

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