Week after next I will be giving a talk to 60 or so high school Heads of Math Departments from around NSW (that’s New South Wales, the state of Australia in which I live, containing prominently Sydney, Wollongong, Newcastle, and smaller towns) on a rather contentious educational issue currently here in Australia. The government has decided that we need a national curriculum in mathematics to replace the current hodge-podge of state curricula, which probably makes a lot of sense. Unfortunately the current Draft of the new maths curriculum for advanced maths (Years 11 and 12) is deeply inadequate, and I will be talking about that.
I won’t bore you with the details, [if you are interested, you can view our detailed report at http://www.maths.unsw.edu.au/news/2012-07/schools-response-draft-senior-mathematics-curriculum-acara ] but there is an issue here that is of more general interest, and that is the role of geometry in maths education. Geometry used to be the core of higher training in maths (here in Australia we use this abbreviated form often instead of “mathematics”, and I will be following that a lot of the time), with Euclid’s Elements the main text until the beginning of the 20th century. Around that time a steady decline in the amount of geometry taught in Schools began; with the hefty and careful sequence of Euclid’s Propositions slowly giving way to ever briefer descriptive approaches; the use of instruments and drawing diminished; and the emphasis on strict, perhaps overly rigid, logical presentation was replaced by a more flexible view. Linear algebra expanded to fill the void. These days Euclid has well and truly been relegated to historical interest, and the subject re-badged with titles like “Shape, space and measurement”. Students memorize terminology and facts, but problem solving and an appreciation for proofs and logical structure has diminished markedly.
In 2001 the British Royal Society proclaimed that “We believe that geometry has declined in status within the English mathematics curriculum and that this needs to be addressed. It should not be the ‘subject which dare not speak its name’.”
Ironically the decline of geometry in schools was accompanied by the development and rise of key geometrical mathematical subjects of the 20th century, such as differential geometry, algebraic geometry (which used to be called projective geometry), topology, and linear algebra/functional analysis. While maths students spend less time on pure geometry, the physics community has slowly but steadly, starting with the pivotal work of Einstein, come to appreciate the close synthesis between geometry and physics. This is a turning full circle, since much of geometry originally was motivated by astronomical interests.
Is any of this likely to change in our new millenium? In fact I think it is already doing so. Young people are visually oriented, and so geometry appeals to a large cohort of students who are bored by algebraic manipulation and large amounts of numerical data. Computer graphics, video games, 3D movies all clearly require spatial understanding. But I think the key drivers are, and will increasingly be, the remarkable new dynamic computer software programs such as Geometer’s Sketchpad, C.a.R., Cabri, GeoGebra and Cinderella, that allow you to create 2D, and even 3D, constructions on a computer and then manipulate inputs to see how relationships change and are maintained, and the newish construction sets that make it easier to physically make models of interesting geometrical objects and explore them, such as Zome, polydron and Frameworks, and many others (for a good list, see http://www.ics.uci.edu/~eppstein/junkyard/toys.html)
If you feel like experimenting with dynamic software packages, check out the free programs C.a.R. (a one-man creation: thanks to Rene Grothmann) which is called Z.u.R. in German, or GeoGebra. But all the programs I listed are really quite special!
In my thinking, there can be no better way to interest students in geometry, and perhaps mathematics, than playing around with such programs and construction sets. Perhaps mathematics education will turn around, and educators will come to realize that stripping geometry from mathematics renders it bloodless and potentially tedious.
In Australia, I hope that the people in charge of curriculum design (ACARA) wake up to the fact that a high school Year 11 and 12 core mathematics course with mostly calculus and statistics, but no geometry, is a disaster for prospective engineers and scientists, and will drive away a lot of students who otherwise might be attracted to the subject.
There was a good reason that geometry was the heart of mathematics for more than 2000 years!