From ‘proofs without words’ to ‘proofs that explain’ in secondary mathematics

Gierdien, M. F. (2007-06)

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From the mid 1970s onwards in almost every issue of the undergraduate mathematics journals Mathematics Magazine and College Mathematics Journal there is at least one ‘proof without words’ (Nelsen, 1993). A proof without words can be thought of as a ‘proof’ that makes use of visual representations, that is, pictures or other visual means to show a mathematical idea, equation or theorem (Casselman, 2000). It does not contain any words other than literal or numerical symbols and geometrical drawings, for example. There is debate around whether a proof without words really qualifies as a proof. It helps the observer see why a particular mathematical statement may be true, and also to see how one might begin to go about proving it true. It may also have an equation or two, arrows or shading in order to guide the reader in this process. In it there is a clear emphasis on providing visual clues to the reader in order to stimulate thinking with the eventual goal of writing a proof. Many proofs without words in the referred journals are directly related to the secondary Mathematics curriculum in South African schools although not exclusively so.

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