Going Beyond “Calculus Reform”
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| Much effort has been expended on improving the teaching and learning of university calculus. Our work differs in two fundamental ways from this traditional calculus reform. |
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| First, the calculus reform has focused exclusively on the existing calculus course, leaving everything else in K-12 mathematics untouched. Our work concentrates on exactly what traditional calculus reform ignored. We are working in schools at every grade level, from grade one to first year college students. One critical consequence of beginning early is that instead of perhaps 10% of the population having a chance to learn these basic ideas, ALL students have a chance. |
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| Second, the reform calculus has concentrated on linking what are now known as “the big three” representations, numeric, graphic and symbolic, where these are used in combination either to represent each other, or, less directly, to model some situation described using text. This approach emphasizes symbolically defined functions as the primary objects that the other systems represent. Research reveals that student learning, while improved over traditional rule-based approaches, continues to be fragile and disconnected. The representations need to represent something other than each other! They need to be anchored in students' experience. Our approach puts phenomena at the center, and begins with directly manipulable graphs that both represent and create the phenomena. |
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| We begin with the phenomena of motion, where the mathematics of change and variation began historically with the Scholastics, more than 200 years before Newton and Leibniz—and more importantly, before modern algebraic symbolism had evolved. Later, we expand the phenomena to include fluid flow and connections to geometric shapes. While we do not attempt to parallel historical development, we draw from two historical facts, that all this mathematics was rooted in the mathematization of phenomena, and that symbolism co-evolved with this gradual mathematization over the centuries. |
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