Proof barely exists in the Australian Curriculum. This is nuts, of course, but leave that be. Even when it is accepted that proof should – must – have a significant role in the curriculum, there are questions to be asked:
In which topics should proof be introduced and emphasised, and at what stage(s)?
Should “proof” be more a topic(s), or more an omnipresent concern?
How proofy should teacher-presented or students’ proofs be?
These questions, particularly in relation to Euclidean geometry, arose in comments to the post on the recent “trigonometric proof” of Pythagoras. (Also relevant are the excellent, thought-provoking articles by Hung-Hsi Wu, particularly this and this, deserving of their own post for discussion.)
It’s rare, at least for Australian students, to see anything proper of Euclidean geometry, except a piece here and a piece there. Which is exactly not the point. It is difficult to imagine anyone still using Jacobs’ beautiful Geometry, but perhaps there is some backwater where it has not been forgotten.
In their great (and free) book, The Essence of Mathematics Through Elementary Problems, Alenxandre Borovik and Tony Gardiner outline what they refer to as a “semi-formal” approach to school geometry (Chapter V). They introduce their outline with an extended discussion of the teaching of geometry (pp 169-173). A couple excerpts:
… in many instances, the “essence” that is captured by a problem requires that the problem be seen within an agreed logical hierarchy – a sequencing of properties, results, and methods, which establishes what is a consequence of what – and hence, what can be used as part of a solution. In particular, we need to construct proofs that avoid circular reasoning. …
… the cumulative architecture of [Borovik and Gardiner’s outline] conveys a rather different aspect of the “essence of mathematics”, deriving not just from the individual problems, but from the way a carefully crafted, systematic arrangement of simple “bricks” can create a much more significant mathematical structure.
There is no question that an extended logical hierarchy is more valuable than the sum of the individual bricks. But is there still value in presenting isolated bricks? What about the proof, or “proof”, of Pythagoras pictured below (Proof #9 in cut-the-knot’s list)? When should this proof be presented to students, and what hierarchy, if any, should accompany it?
Or, should it not be presented at all? Surely, that cannot be right.