On Tue, 30 May 1995 Harvey Becker wrote: But do you agree that it is better to define sine as y/r before having students try to find the sine of 30 degrees?
Steve Means replied: Indeed. We place definitions up front.
What an interesting coincidence! One of my friends is working on her undergrad degree and often comes to me for help with her math courses. This summer she is taking precalculus at the university. Last week her class started on trig and she asked for help with problems in which they were asked to compute the trig ratios for right triangles with varying lengths given. After dealing with her questions, there was a rather long silence. Finally, she asked, "Why am I computing these ratios anyway?" I burst out laughing! What a great question! There was absolutely no reason to do any of those problems, other than to drill the definition which was placed up front! WHO CARES INDEED! It was useless!
In contrast, consider the approach we use in the Geometry Learning Project, a r/d project for high school geometry. We begin by developing the constancy of the tangent ratio (which we actually called slope) for right triangles with the same angles, tying in similarity, and its usefulness to determining missing lengths. Motivate the desirability of having a tabulation of slopes going with various angle measures. As a class project, have them make a table experimentally using grid paper. THEN give them a chart with enhanced accuracy and discuss the tan button on their calculators. They are pathetically grateful. "We'll never have to graph another triangle! Thanks!"
In other words, there was (a) external motivation for wanting to develop the idea (wanting to save time) AND (b) some internal mathematical structure for developing the idea (preservation of ratios of lengths in similar figures).
I fear that either of these two are easily ignored. By placing definitions up front, where is the motivation, either externally or internally?
On the other hand, as several people have commented, many new programs are great with external motivation but weak on providing the internal structure. We need both!
============================================================ W. Gary Martin 1776 University Ave. University of Hawaii Honolulu, HI 96822 Curriculum R & D Group (808)956-9956; FAX (808)956-4984