Exploring Calculus with The Geometer's Sketchpad
How the publisher describes it:
“Your students will directly experience the dynamic, geometric nature of calculus with the activities in this module.”
Review by Chris Little
“Techniques (executed correctly) mean ticks, and ticks (in an examination) mean marks!”
This book provides activities for developing calculus using the software package Geometer’s Sketchpad (GSP). At the start it is worth noting that, in order to run the Sketchpad files included on the CD which accompanies the book, GSP Version 4 is required.
The activities in the book are divided into five chapters - Exploring Change, Exploring Limits, Exploring Derivatives, Exploring Anti-derivatives and Exploring Integrals. A section of Activity Notes in the back of the book include suggested pre-requisites, answers, explanations of how the sketches were constructed, and extensions for experienced Sketchpad users.
One fascinating aspect inspired by the book is seeing how much a dynamic geometry package like Sketchpad can provide students with genuine kinaesthetic insights into the calculus. How much easier it would have been for Isaac Newton (who had to stockpile paper and write his ideas in very small writing in order to conserve his precious paper) if only he’d had GSP to play with!
The first chapter of the book is essentially pre-calculus, in that it lays the groundwork for calculating average rates of change, and exploring the relationship between distance-time and velocity-time graphs. Chapter 2 explores continuity - using the idea of a ‘crevasse’ that appears in the apparently smooth slopes of a ‘Mount Everest’ as a warning about assuming something is continuous!
The introduction to epsilon and delta analysis will probably be skipped by teachers, and students doing AS level as it’s likely to be way over their heads. Indeed, teachers may well want to gloss over these chapters and move rapidly to the tasks in chapter 3. Here, the derivative is introduced as the limit of the average rate function shown in Fig 1.
In the book, this limit is illustrated by dragging one point close to the other point and getting a trace of the average rate function, then repeating with an increment h. Once the derivative is established in this way, the results for the standard functions are very easily ‘reinvented’ using the efficient function plotter. Parameters can be adjusted easily using sliders, and the zoom slides are equally efficient in delving deeply into the neighbourhood of a point.
The approach to integration is to establish this through Riemann sums, adjusting the number of rectangles and seeing how the error bound shrinks. There is what can only be described as a bit of ‘waving of hands’ in ‘establishing’ the fundamental theory of calculus - even using Sketchpad can’t short-circuit the genius of Newton and Leibnitz which led to this extraordinary insight!
There is no doubt that this is an incredibly rich package which would enable students to gain genuine insights into calculus. However, there are practical caveats on how it would work with sixth formers in the UK. One is that this book is clearly designed for stand-alone U.S. calculus courses. There is, therefore, far too much material for the average AS level student to work through on their own, never mind in class. In an ideal world, it would be nice for students to know about epsilons and deltas, but it’s not in the already crowded UK syllabus (though the approach in the book might well be useful for university-level analysis courses).
The second caveat about the book is the relentless step-by-step instructions. These are meticulously and accurately written, but it is questionable whether students will be motivated to religiously plough their way through them. In practice, many teachers may well be tempted to delve into the ready-made files and use these as demonstration tools (albeit powerful ones), but omit the painstaking, blow-by-blow build-up of the concepts.
Such an approach, if adopted, is, no doubt, a pity, and is probably less a criticism of the material in the book than of UK A/AS syllabuses and the way in which these encourage teachers to gloss over the more difficult analytical foundations of calculus and teach the techniques - the product, quotient, chain rules, integration by parts, etc, etc. Because, as we all know, techniques (executed correctly) mean ticks, and ticks (in an examination) mean marks! The issue remains about whether or not A-level students should able to prove derivatives from first principles, and if not, whether they should be able to.
Chris Little • St Vincent's College, Gosport, Hampshire
Exploring Calculus with The Geometer’s Sketchpad
by Cindy Clements, Ralph Pantozzi, Scott Steketee
Key Curriculum Press