Share your passion

Three links yesterday conspired to get me thinking of misunderstandings and creativity.

It began with Mark Chu-Carroll's article on a Crossword Guy who claimed correctly that Sudoku need not use numbers as its symbols. But then, the Crossword Guy concluded wrongly, that because it could use any nine symbols "it's not a mathematical puzzle at all."

I've used carefully selected Sudoku puzzles to illustrate different forms of mathematical reasoning for years. The different strategies used to fill in squares in Sudoku help illustrate different strategies mathematicians use in constructing proofs. Just as you might need different strategies to fill in different squares in a single Sudoku puzzle, you might need to apply different strategies to different stages of a proof.

Mark writes that "One of my pet peeves about people and math is that most people don't really have a clue of what math is."

In Nat Torkington's Four short links: 25 June 2009 he writes about the Wired article on "The Nike Experiment". Nat adds to his excerpt and link that "Mathematics gets by with just an 'equals' operator. The rest of us need a 'can equal' operator."

Now that's a cute comment from a very bright guy. It's a nice play on words but it isn't the mathematics that I know an love. Here's one of my favorite examples that starts with a formula you probably remember from High School or earlier: the area of a triangle equals half the product of the base and the height. 

A = (1/2) b * h

Take out a scrap of paper and draw a line segment that will represent the base of a triangle. Now draw a round point not on that line segment that will be the third vertex of the triangle. In other words, the three corners of the triangle are the two endpoints of your line segment and this third point you just drew. Lightly sketch in this triangle. 

Now here's just one "can equal" operator in mathematics. Keep the same base of the triangle, choose another point for the third vertex so that it has the same area as your existing triangle. Choose another. And another. 

Now -- what do you notice about all of these "third points"?

We didn't use any numbers but we're doing some pretty interesting mathematics. We've actually discovered something equivalent to Euclid's fifth postulate. 

I'll let you play around with this a bit and we'll look at the answer in a future post but I also want to recommend that you read Lockhart's Lament (pdf) when you get a chance.

The story begins with a musician who wakes up from a nightmare in which he has dreamt that music education is compulsory in school. As a result everyone needs to learn to write sheet music and work with symbols. They learn to memorize axioms of music and take classes in which they need to recite the circle of fifths and apply rules of harmony and progressions. The kids don't listen to or play any music, they just manipulate symbols. 

An artist has a similar dream in which children learn color theory for years before they ever paint anything. They prepare for the college courses in painting by taking Paint-by-numbers in high school and they prepare for this with a Pre-Paint-by-numbers class.

Think of how much blander your life would be if you'd never gotten to actually enjoy art and music because of all of the formalism that it was determined you would have to master before you could ever understand or participate.

That's what we have in so many of our subjects. The soul has been ripped out of mathematics and so I think it is completely understandable that bright folks like Nat and the Crossword Guy don't understand what math is.

I used to teach a section of a course called "Math and Creativity" at John Carroll University. One semester a student pulled me aside to say that the passion I feel for Mathematics is similar to the passion he feels for History. He saw the look on my face and said "History is not just dates and names and battles." He then painted a picture of History that was exciting and engaging. It was unlike anything I'd every gotten from any of my History teachers over the years as we trudged through a list of unconnected events. Actually "raced" is more the word for it--we had to finish the book by the end of the year.

David Love used to have a similar lament about courses in Shakespeare. The syllabus would require the students to read two to three plays a week. "Why?"  he asked. Mostly, I suppose, so they could say they had read all of those plays. "Why not read two or three plays over the entire term?" he asked. "Really read them. Understand them deeply. Foster such a love and respect for the work that you read or see others on your own time."

My point is that although I care about the way math is taught and perceived, the problems I see there are shared by other fields. With this triangle problem I've tried to share with you some of what makes math come alive for me. This weekend, think about your field and find something that excites you that you could share with someone who isn't in your field. Help someone outside of your area of expertise understand what it is that you love about what you do. Make it something specific.

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My point is that although I care about the way math is taught and perceived, the problems I see there are shared by other fields.