Tuesday, January 10, 2012

Early morning maths will kill you - just ask Descartes!

Sorry - I couldn't resist. It's the hook I use to bring Descartes into my classroom. More below.

Do you remember the first time you encountered the idea that all those geometry concepts you had learnt for years could be expressed in algebra? How suddenly those two totally different topics in your maths classes could be viewed as one? I hope this was a special moment for you*.  At this special time, why not treat your class to a some time with  Descartes? There are many contemporary source materials on the internet just waiting to be woven together into a fascinating story of intrigue, intellectual curiosity and even the very question of our existence.

Portrait of Rene Descartes (1596 - 1650)
Louvre Museum - Richelieu - Level 2 - Room 27

Catch their attention with the story how Queen Christina killed Descartes.
I love telling the story how Queen Christina of Sweden killed Descartes by making him get up at six o'clock in the morning to teach her mathematics. Who can't relate to that story? Depending on your class and if you want to go there, they might also get a chuckle out of answering this question: which person in this picture is Queen Christina? You might like to explore the conspiracy theory Descartes was poisoned over religious turf wars. The Wikipedia pages on Descartes and Queen Christina offers some interesting starting points for further investigation.

Queen Christina of Sweden (on the left!) and Descartes. 
Pierre Louis Dumesnil (1698-1781); 1884 copy by Nils Forsberg 
See also http://commons.wikimedia.org/wiki/File:Dispute_of_Queen_Cristina_Vasa_and_Rene_Descartes.png

And now we have the class interested ...

Show the big picture: Descartes was interested in much more than just maths
The amazing thing is Descartes' work on analytic geometry first appeared as an appendix in a much larger book: The Discourse on Reason. I like to show students some of the images from the Geometry appendix.

A search on Google images for "Descartes La Geometrie" will turn up many photographs of early editions.
Personally I think students find this more interesting than just seeing reproductions of the diagrams.

and then take a look at the actual physical form of the original book. Here's a first edition auctioned at Christie's in 2010 for US$76,000. 

2010 auction at Christie's  http://www.christies.com/LotFinder/lot_details.aspx?intObjectID=5371784   
After we finish gasping at the price, I show the students how they can get a free copy of the book in digital form at Project Guttenberg.

 A student in my class who was studying French helped us read the title and explained it to us (much to the cheering of her peers).  Discours de la méthode pour bien conduire sa raison, &  chercher la vérité dans les sciences. Plus la Dioptrique. Les Meteores. Et la Geometrie. Qui sont des essais de cete Methode. I love the intonation and literal sequence of words  French - pour bien conduire - to well conduct reason, and to search for the truth in the sciences.  It's a lofty aim. 

Descartes the scientist.

Other appendices in the Discourse are on optics and astronomy. My students were fascinated how Descartes extended his observation how boats moved in a river to an explanation for the movement of astronomical bodies in a 'sea' of vortices.

His ideas on vision and how the brain interprets the senses provides a rich opportunity to engage with students on questions of knowledge**.

The mind-body interface in the pineal gland watches the image
formed on the retina (just like watching TV), and then orders the hand to move.
Descartes the philosopher : Je pense donc je suis.
Descartes said it in French first (1637) - unusually for the period - the Latin Cogito Ergo Sum came later (1644).  "I think therefore I am" is the usual English translation. Can you explain the idea in your own words for your maths class? Check out the Wikipedia page on Mind-Body dualism and Cogito Ergo Sum for starters. For me the most interesting aspect is that it's not actually an argument that we exist, but that something we call our mind exists - the rest of our body and the world around us might not be real.

I was pleasantly surprised how much my mathematics class took to even a surface discussion on the question of how do we know that 'we', or 'the mind' exists. I'm not a philosophy teacher and we did need to move on to the mathematics, so it was only a brief exploration,  but I could see it was an important and possibly rare opportunity for students to go there. I suspect we seriously underestimate the need and interest young people have for engaging with deeper philosophical questions. And mathematics offers a window, or at least one introductory path, to this place.

Is Descartes responsible for vivisection?
This little twist is interesting and engaging: the Wikipedia article on Descartes suggests his view that animals don't have have a mind, and thus no feelings, was responsible for centuries of believing it was fine to do live animal dissections. I guess he never had a cat or a dog?

Do pay a visit to Descartes next time you are in Paris.
It turns out the famous portrait of Descartes (see above) is in the Louvre Museum (Richelieu 2nd floor, Room 27).  I told the class I was very disappointed to discover this only this year, because in all my trips to the Louvre I had never seen the picture. I asked the students that when they went to the Louvre (and I assured them they would all eventually visit Paris!), could they please go and have a look at the picture for me and send a postcard.

So what's the point of this Descartes' distraction? I spread this material over several lessons as a short element in each lesson during our first week of coordinate geometry. I have absolutely no evidence or make any claim it helped the class learn the content better - but I do hope it helped them see some of the  people and worlds that produced the mathematics they are studying - and maybe encouraged them to explore further. Mathematics is not just concepts and skills - it's part of our culture - our never ending  exploration of the intellectual and natural world, which that rightly belongs in our classroom.

More resources:
"Frogs and Birds" - the great Freeman Dyson muses on the differences and synergies of Francis Bacon and Rene Descartes.

* There is a down side to this algebraic seduction - something I'll explore in a later post when I look at the ideas of another amazing mathematician, Michael Atiyah.
** Can't help it - just love that 'e' word (epistemology)

1 comment:

  1. Hello Nordin,

    I really enjoyed this post! I brought up Descartes not long ago in my geometry class. I was a science teacher for 12 years, then 8 years so far teaching middle school math, so I appreciate high quality blogs about teaching math. I just started blogging myself, posting math puzzles, recipes, and anecdotes about teaching. Hope you'll have some time to stop by: http://fawnnguyen.com
    Keep up the good work!