Open Access11. Boundaries in Visualizing Mathematical Behaviour
Author(s) -
Andrew Francis Hare
Publication year - 2013
Publication title -
collected essays on learning and teaching
Language(s) - English
Resource type - Journals
ISSN - 2368-4526
DOI - 10.22329/celt.v6i0.3770
Subject(s) - boundary (topology) , sketch , witness , function (biology) , counterexample , calculus (dental) , simple (philosophy) , visualization , epistemology , computer science , mathematics , artificial intelligence , philosophy , mathematical analysis , discrete mathematics , programming language , algorithm , medicine , dentistry , evolutionary biology , biology
It is surprising to students to learn that a natural combination of simple functions, the function sin(1/x), exhibits behaviour that is a great challenge to visualize. When x is large the function is relatively easy to draw; as x gets smaller the function begins to behave in an increasingly wild manner. The sin(1/x) function can serve as one of their first counterexamples, helping them to appreciate better the tamer functions that they normally encounter. I see three boundaries here. First, a boundary erected by mathematicians between ‘nice’ versus ‘wild’ functions - captured for example by the concept of continuity. Second, a boundary between those functions that are most often studied in calculus and pre-calculus classrooms, and those that are more rarely looked at. Third, the boundary between the drawable and the undrawable. In this example, we can witness this last boundary first-hand even as we attempt to sketch the curve. Yet we can also continue the visualization in our mind’s eye beyond what we can represent on paper.