Mathematics: Visible & Tangible
3 Once these attempts to solve the above questions have led us to select some notions and ways of working, is it possible to look at these notions per se and discover properties that were not needed for the original problems? Indeed, there are a number of these properties, the consideration of which has become part of trigonometry courses. Some of these discoveries made it possible to select different introductions to this subject and, while keeping “trigonometry” as the label for the awarenesses which came from these discoveries, to propose very different ways of working. As a result, it became possible to examine which introductions are most beneficial to the learners. This we can now do, selecting what we think is the most primitive introduction, which we have found, over 30 years of testing, to be the most efficient. *** Just as we can run a finger along the edge of a coin and come back to where it started, we can conceive that a point on the circumference of a circle could move around, passing through all the points of the circumference before returning to its original position. We therefore invite students (and our readers here) to imagine this “running” point describing the circumference: 1
at various speeds;
2 alone, or with the ray which emanates from the center of the circle and passes through the point; 3 with the same ray, but not showing the point (which is now conceived as the intersection of the ray and the circumference).
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