Abstract
There appears to be a widespread assumption that deductive geometry is inappropriate for most learners and that they are incapable of engaging with the abstract and rule-governed intellectual processes that became the world’s first fully developed and comprehensive formalised system of thought. This article discusses a curriculum initiative that aims to ‘bring to life’ the major transformative (primary) events in the history of Greek geometry, aims to encourage a meta-discourse that can develop a reflective consciousness and aims to provide an opportunity for the induction into the formalities of proof and to engage with the abstract. The results of a pilot study to see whether 14–15 year old ‘mixed ability’ and 15–16 year old ‘gifted and talented’ students can be meaningfully engaged with two such transformative events are discussed.
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Notes
The Elements have been criticised for its lack of rigour (e.g. Russell 1971; Wilder 1981) and may not compare with the rigours of the axiomatics of today (volumes 1–6 of the Elements relies on the diagram for proof which compares rather unfavourable with a formalism that does not rely on diagrams), but pedagogically it serves as an exercise in deductive reasoning. The historical reason for teaching the Elements was that it taught people how to reason. It is interesting to note that Hartshorne (2000) teaches Greek deductive geometry to his mathematics undergraduates before teaching the axiomatics of university mathematics.
Much of this is still speculation as we can only go by Eudemus’ and Proclus’s account of the origins of Greek mathematics. Indeed, Netz (1999) goes so far as to state that there is no hard evidence that Greek mathematics as we know it pre-dates Socrates. Given the destruction of the library at Alexandra and the possible destruction of 99% of all Greek literature, this issue may never be resolved. However, the burning question that remains to be answered is how the Greeks transformed an empirical craft into a science of reason? The period of that transformation is certainly pre-Socratic and at least we have Proclus and Eudemus to go by. Although historically spurious, they can serve to justify the speculation in an educational context.
The following is speculation, but there is reason to suppose Solon, the elected head of state of Athens, actually met and conversed with Thales. If so, then the rule of law as governed by principle (as opposed to the dictate of kings) to which the law applies equally to all subjects, has been influenced by the principles of geometry. That Solon may have met Thales and in Egypt should be mentioned in the classroom in order to show the impact of geometry to other domains such as law (however, whether Thales was actually alone in creating abstract geometry is an historical point best left for the university rather than a point of heritage for the secondary classroom).
Despite the ancient Greek historians attributing proof to Thales, it is unlikely that he developed formal proof and perhaps went no further than cutting out templates to demonstrate opposite angles are equal. Experience from school visits have shown that novice 12 year-old-students may well generate the same kind of demonstration, which means they are on the right path to understanding how the method of proof developed. The essential point is if we took them back to that historical moment and immerse them in the relevant problem space then we can guide the transformation of their understanding as that primary event unfolds.
Such a blanket claim presupposes a contextual domain, in this case plane geometry, but the point here is that, for Plato, a ‘necessary truth’ is fundamentally different to opinion or the dictate of authority.
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Rowlands, S. A Pilot Study of a Cultural-Historical Approach to Teaching Geometry. Sci & Educ 19, 55–73 (2010). https://doi.org/10.1007/s11191-008-9181-3
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DOI: https://doi.org/10.1007/s11191-008-9181-3