Making Human Sense of Calculus - David Tall скачать в хорошем качестве

Making Human Sense of Calculus - David Tall

This video is a presentation given in Norway in August 2019 to offer new insights into the meanings of the calculus and its teaching and learning. It introduces ideas from mathematics and its visual and symbolic meaning, taking account of how our human perception and mental processing naturally inspire certain ways of thinking and how the differing cultural needs of different communities cause us to approach calculus in different ways. In particular, by zooming in on a graph on a retinal display, familiar graphs look less curved until a small magnified portion looks like a straight line, so the slope of the graph can be seen and the derivative is visibly the changing slope of the graph itself. This leads to a 'locally straight' approach to differentiation where the limit process is implicit and need not be introduced in a formal way. It does, however, allow all the standard derivatives to be given meaning. Continuous functions, drawn freely with a pencil on paper can be imagined digitally represented on a high resolution display. Maintaining the same vertical scale, but stretching the picture horizontally, causes a small portion of the graph to become a horizontal line of pixels. A continuous function 'pulls flat'. This leads naturally to the idea of integration and the fundamental theorem. The presentation questions the current approach to teaching calculus in a manner typified by US AP Calculus building on an informal version of the formal definition of limit which is known to cause significant difficulties for many students. It shows that dynamic visual ideas of variable quantities, including infinitesimals as 'arbitrarily small' variables used in applied mathematics have a natural counterpart that can be proved formally to reveal a number line where each finite point is either a real number or a real number plus an infinitesimal. This supports an approach to the calculus that builds from human perception and action UP to the formal idea of limit, not DOWN from the formal definition to practical operations. It suggests an entirely new meaningful approach to the calculus appropriate for the new digital age.