I recently attended the Ontario Math Coordinator’s Association (OMCA) 2011 coference, Math More Than Magic and had the pleasure of listening to Dr. Nathalie Sinclair, assistant professor at Simon Fraser University’s Faculty of Education and author of Mathematics and Beauty: Aesthetic Approaches to Teaching Children. She began her presentation by sharing a math doodle video clip by Vi Hart. Following the very cool video clip, Nathalie went on to speak about math in a way that I’m not accustomed to hearing. She called math elegant and beautiful and asked us when the last time we heard or used those words in the same sentence with the word “math” and also challenged the audience (a room full of math coaches and coordinators) to tell students that math is elegant and present them with opportunities to learn math not only visually but dynamically. This really got me thinking about math in education in general.
One question that math educators across North America probably hear most frequently from their students is, “When are we ever going to use this in real life?” How do you usually respond to this million dollar question? Sure, math can be practical and help us in our everyday lives (making change, estimating amounts, calculating HST, understanding the probabilities of poker hands when watching it on T.V. etc.) However, the truth is that some of the things that we teach in math will not help students in everyday practical situations. Try explaining to an intermediate student that they will need to know the Pythagorean Theorem so that they can know exactly how far a 5 metre ladder is from a wall if it is resting at a height of 4 metres. Is this the real reason why we teach the Pythagorean Theorem? I love the Pythagorean Theorem because of it’s simplicity and the fact it can be proved in so many ways other than using the formula. If you have a right triangle, the square on the biggest side of the triangle is the same as the sum of the squares of the other two sides. I love the theorem even more when students discover it through investigation and get excited to share how they proved it. Here are some beautiful examples of math proofs of the Pythagorean Theorem:
Paul Lockhart wrote an incredibly thought provoking math essay titled, “A Mathematician’s Lament” (a must read for any math educator). He refers to math as an art and the “art of explanation”. In fact, he states, “Mathematics is the purest of the arts, as well as the most misunderstood”. Lockhart is very much an advocate of bringing the beauty of mathematics into the curriculum. The following excerpt is a playful dialogue that was taken from Lockhart’s Lament:
SIMPLICIO: Are you really trying to claim that mathematics offers no useful or practical applications to society?
SALVIATI: Of course not. I’m merely suggesting that just because something happens to have practical consequences, doesn’t mean that’s what it is about. Music can lead armies into battle, but that’s not why people write symphonies. Michelangelo decorated a ceiling, but I’m sure he had loftier things on his mind.
SIMPLICIO: But don’t we need people to learn those useful consequences of math? Don’t we need accountants and carpenters and such?
SALVIATI: How many people actually use any of this “practical math” they supposedly learn in school? Do you think carpenters are out there using trigonometry? How many adults remember how to divide fractions, or solve a quadratic equation? Obviously the current practical training program isn’t working, and for good reason: it is excruciatingly boring, and nobody ever uses it anyway. So why do people think it’s so important? I don’t see how it’s doing society any good to have its members walking around with vague memories of algebraic formulas and geometric diagrams, and clear memories of hating them. It might do some good, though, to show them something beautiful and give them an opportunity to enjoy being creative, flexible, open-minded thinkers— the kind of thing a real mathematical education might provide.
SIMPLICIO: But people need to be able to balance their checkbooks, don’t they?
SALVIATI: I’m sure most people use a calculator for everyday arithmetic. And why not? It’s certainly easier and more reliable. But my point is not just that the current system is so terribly bad, it’s that what it’s missing is so wonderfully good! Mathematics should be taught as art for art’s sake. These mundane “useful” aspects would follow naturally as a trivial by-product. Beethoven could easily write an advertising jingle, but his motivation for learning music was to create something beautiful.
Lockhart makes some very bold statements but I think he definitely get’s his point across. Sometimes we have to be honest with our students and when they ask us when they will ever use the Pythagorean Theorem in their lives, we tell them, “You probably will never use the Pythagorean Theorem in your everyday life but it’s simple and beautiful and we’re going to have some fun trying to prove it.