Mathematics is not a series of Statement-Reason proofs punctuated by the occasional “QED”.
Mathematics, as Paul Lockhart writes in A Mathematician’s Lament, is “wondering, playing, amusing yourself with your imagination.”
If you haven’t read Lockhart’s piece, please do so. It’s way more interesting than anything I am about to say. It begins with first a musician and then a painter waking up in a cold sweat after their respective nightmare’s on how their field has come to be taught.
Lockhart compares Math education to these nightmare scenes of music and art because he sees math as being the same sort of endeavor. Math is a playful, creative art.
Only later after we have discovered something new and worth sharing do we go back and formalize it. We make sure that what we stumbled on is really true. Then we present it to others in a way that they can see its truth.
Unfortunately, this means that we often present new findings to others in a way that they don’t see its beauty.
That’s a shame.
There have been a bunch of blogs recently that have talked about what should be taught in math classes. Most of them focus on answering the question “when will I ever use this.”
My daughter has been taking band since fifth grade. At her school they have music every day. In the high school they begin the year with marching band and then audition and split into five smaller bands. They play an array of interesting music and, in my opinion, play well beyond their years. Their top band was the only band in regionals to receive the top marking from the judges.
Of the three hundred kids taking band this year, a handful will continue in music or music education beyond college. Most of the rest will never pick up their instrument again. Most will say “I used to play trumpet (or flute, or …).”
The marching band plays at half-time of the football games. The football program has well over a hundred kids on the teams if you count varsity, junior varsity, and the freshman team. The team is not very good. Two years ago we only saw one victory. Last year they won a couple of games and almost won a couple more.
Some of the students will play in college. It would be very unusual for any of them to play beyond college.
The band kids and the football kids never ask their teachers and coaches, “when will I ever use this”.
Both sets of kids have to show up in August two weeks before the school year begins to prepare for the season. Families cut their vacations short so that their kids can spend all day out in the heat of the summer working hard at something they’ll never use in life.
And yet they come into my math class and raise their hand half-way through my demonstration of the mean value theorem to ask me when they will ever use this. They want me to justify that I’m not wasting their time.
I never mind answering this question.
It allows me to give them a glimpse into the beauty that I see when I play with mathematics. To me math is a game. We have a set of rules and we take turns playing this game. When the rules change — say we relax that fifth postulate so that a line can have no parallel lines through a given point — we are playing a different game.
The reason for learning mathematics should never be restricted to how it will help us do something else.
The hard part is that math is so darned useful. There is math everywhere. It’s easy for us to think about learning the math we need to do science or economics.
A recent blog post questioned why so much of the math we learn is continuous as opposed to discrete. We should learn more discrete math, but we shouldn’t ignore the continuous. The continuous describes a lot of the world we live in.
We pour a cup of coffee and it is 180 degrees F. When we remember to drink it is is cold. Blech. It dropped 120 degrees while we weren’t looking. It didn’t drop suddenly. It followed Newton’s Law of Cooling. It’s kind of cool that the temperature of the coffee at any given time is roughly governed by a law that only describes the rate at which the liquid cools.
Here’s the email I received that set me off on this post. Several people thought that I would be interest in this mathematical oddity and sent me this:
It made me sad.
The email described this amazing phenomenon of having five Fridays, Saturdays, and Sundays in the month of Dec. The text of the email said that this only happens every 823 years.
The people who sent it to me were impressed that they will about to live through an event that wouldn’t happen again for more than 800 years.
Look again at the calendar.
You can see that this happens any time the month of Dec begins on a Friday. You can reason that this happens about as often as Dec beginning on any other day. It would be shocking to think that this won’t happen for another 823 years. In fact, if Dec 1, 2023 is Friday this year then it will be Sunday next year. The day on which Dec 1st falls advances by one day each non-leap year and by two each leap year. On average every seven years Dec 1 falls on a Friday. In reality, it will sometimes take more than seven years to return to Friday and sometimes take more – but these numbers are close to seven and not close to 823.
In order to confirm that Dec 1 advances by one day in a non-leap year and two days in a leap year I had to look at the actual calendar. There I discovered that the premise of the email was wrong.
In other words, the claim that this is the only time in your life that this will occur is wrong on two counts. First, it’s not going to occur next year. Second, if you’re reading this blog post it is likely to have already occurred in your life and it will happen again in 2028.
This, to me, is mathematics. Playing, confirming, questioning — even in my real life. It gave me a much sadder answer for people when they ask me “when am I ever going to use this”.