Below is a page I wrote 15 years ago for the Exploring Calculus with the Geometer Sketchpad Summer Institute. Was thinking about it before going to Exeter in June. In thinking about starting calculus again next year, I am reflecting on the many tools available for illustrating calculus (and math) today; the question remains, I think, how to use the technology to spark students’ noticing and wondering, and inspire students to explore the terrain in pursuit of their own questions.
See here for one such example. Having learned about tangent lines to graphs, what if you are given only a tangent line, and no graph? In the sketch, we have a function f(x), and a tangent line through a point (a,b) with slope f(b). Move point A around – what do you notice? What do you wonder? What students ask when they see the tangent line is free to move around is enlightening and inspiring for me as a teacher. Students really do re-invent calculus in their own way through their own curiosity.
Understanding connections between graphical, numerical, symbolic, and verbal representations is essential in all areas of mathematics. In the study of calculus, symbolic language, in particular, is a key to opening doors of understanding. Without connections, however, it may be difficult to know what lock the key fits.
Understanding also depends upon a foundation of experience. If learners’ experiences are limited, they may not be able to “see” past the next question. When learners can “see” more of the terrain of ideas from which a particular idea arises, they are in a better position to construct meanings that will allow them to travel further across the terrain.
For example, students can shed light on the fundamental ideas of calculus by investigating limits, derivatives, antiderivatives, and integrals during the first week of calculus. When students develop a vision for calculus, they seek out the connections that enable them to develop deep understandings.
Conducting investigations as a means of developing precise definitions puts students into the same shoes as Newton and Leibniz. While these two mathematicians are credited with inventing the calculus, they did not use the definition of limit that we use today. The precise definitions contained in textbooks today arose, over years of study, from the intriguing questions that Newton and Leibniz asked but did not fully answer.
The students of today face the same questions and cognitive obstacles that faced those who developed the calculus, but we ask students to face them over the course of two semesters. The temptation for the teacher may be to “make things clear” for the students. Sometimes a shortcut, however, is not worth the sights the students will miss along the way.
With Sketchpad, students can experience calculus through direct action on dynamically defined mathematical objects. Using custom-built construction tools, students’ ideas become actions, and their actions become ideas. By experiencing calculus in this way, students can begin to make things clear through their own activity.
In this institute, you will take your own journey of exploration of the terrain of calculus. On the way, you will likely meet some familiar friends, but see them in a new light. You will walk in the steps of your students, to whom the terrain is unfamiliar, while using the shoes of Newton and Leibniz.
In this institute, you will see how using Sketchpad gives students a map for their journey through calculus, and a sense of where they are going and why. The point of their journey should not simply be “to arrive” – but to learn from the journey itself. We hope your journey this week is that kind of journey, and that you will be motivated to return to see the sights in new ways again and again.