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| Raphael "School of Athens" - detail showing Euclid. |
I caught myself out again today - (see future post : "Assume : making an ass out of you and me") : I assumed my students, and indeed my colleagues, knew the wonderful story about Euclid and his five postulates. Assumed they knew how our high school geometry is built on the foundation of these five unprovable axioms. But I'm of course being unreasonable and unfair - Euclid isn't in our school syllabus any more.
While we still ask our top students to replicate and develop geometric proofs, Euclid and the idea of axioms is effectively removed from the content. There is just a single reference to the word 'axiom' in our syllabus - a background note tucked away in the NSW Board of Studies 7-10 Mathematics syllabus (p161): "The Elements of Euclid (c 325-265 BCE) gives an account of geometry written almost entirely as a sequence of axioms, definitions, theorems and proofs. Its methods have had an enormous influence on mathematics. Students could read some of Book 1 for a far more systematic account of the geometry of triangles and quadrilaterals." Fortunately these precious words survive on in the new 'Australian Curriculum' version of our syllabus.
But I wonder. What have we done? Why did we keep the formalism of doing proofs, keep our students busy with it for months of syllabus time while letting go of one of the most powerful ideas in mathematics: the idea of proof built on axioms? How will our students relish those mind-blowing moments in their future when they encounter parallel lines that do meet, or better yet, encounter Godel or sit in a philosophy class wondering how we know something*, if they don't first meet Euclid?
While we still ask our top students to replicate and develop geometric proofs, Euclid and the idea of axioms is effectively removed from the content. There is just a single reference to the word 'axiom' in our syllabus - a background note tucked away in the NSW Board of Studies 7-10 Mathematics syllabus (p161): "The Elements of Euclid (c 325-265 BCE) gives an account of geometry written almost entirely as a sequence of axioms, definitions, theorems and proofs. Its methods have had an enormous influence on mathematics. Students could read some of Book 1 for a far more systematic account of the geometry of triangles and quadrilaterals." Fortunately these precious words survive on in the new 'Australian Curriculum' version of our syllabus.
But I wonder. What have we done? Why did we keep the formalism of doing proofs, keep our students busy with it for months of syllabus time while letting go of one of the most powerful ideas in mathematics: the idea of proof built on axioms? How will our students relish those mind-blowing moments in their future when they encounter parallel lines that do meet, or better yet, encounter Godel or sit in a philosophy class wondering how we know something*, if they don't first meet Euclid?
And sad to say, I had to confess to myself I couldn't actually remember those five postulates. So it was time for a visit to Wikipedia.
And then I discovered something I hadn't seen before:
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| The Pons Asinorum aka "the isosceles triangle theorem" |
I smiled. It was if Euclid himself was winking at me across the millennia. So tomorrow my class is exploring the idea of axioms and getting a history lesson! And like students of many generations past, I think they will appreciate the humour of the Bridge of Assess.
Two interesting resources for The Elements:
A full digital copy of Oliver Bryne's 1847 famous pictorial version of the Elements (most of the proofs are done without words) is at http://www.math.ubc.ca/~cass/Euclid/byrne.html. This could make an interesting source document for students.
A good explanation and commentary from David E Joyce http://aleph0.clarku.edu/~djoyce/java/elements/bookI/propI5.html - who has an amazing website exploring the whole contents of the Elements.
* Say it quietly : epistemology.
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