Balancing Rigor and Whimsy

Do you think like an artist? Perhaps you think more like an engineer?  Do you make broad strokes in your arguments- sometimes glossing over the finer details? We know what this sounds like,

“Who cares if the constant is wrong? It’s just a constant.”

Or perhaps you are very concerned about the precision of your work.  You can complete computations correctly the first time and have confidence in your answer.  Does hyperbole bother you when used in scientific contexts? Is it more important that there are a LOT of chickens on that farm or is it important that there are 12,393 chickens?

I think mathematicians tend to think like artists.  This is why we believe our craft to be creative and challenging.  We consider a proof like an artist considers a canvas. Bob Rauschenberg will make rules for himself about how he is going to create a piece of art.  …then breaks them.  Haven’t we done that from time to time? We say, “let’s prove by contradiction” then we start our proof and realize that, in the end, we proved it directly.  Our artwork is sometimes hard to understand and is more beautiful the longer the audience studies the craft- just like fine art is. Mathematicians are the abstract artists of analytical thinking.  Our most beautiful theorems can require years of training before they can be appreciated.  We don’t just care about the finished product (the theorem) but we want to analyze the brush strokes of the proof. As though we were talking about an oil painting we ask,

“How did they layer the colors of the argument to provide a beautiful conclusion?”

Despite my waxing poetical treatment here, the line between the artistic whimsy and mathematical rigor is crisp.

Mathematicians must also think like engineers. We must be very concerned about precision and accuracy. Because we can’t BS our way through a proof. Our colleagues will see straight through us. Perhaps this is why I can’t sell my abstract paintings for millions. The trained fine art audience can see through my flimsy attempts at their art. We must care that there are 12,393 chickens on the farm. It may be important to our argument that the number of chickens isn’t prime. We must swap between rigor and whimsy to succeed.  I think this is part of what draws people into mathematics. I don’t know about you, but I can’t think of too many other fields were individual creativity is valued as highly as precision and accuracy.

Successful Tutoring

(A frizzy haired student sits down with a middle-aged professional tutor to do some mathematics.)

“Which problem would you like to work on first?,” asks the tutor.

“Number 52. It’s even so the answer isn’t in the back,” replies the student.

“Okay. Let’s take a look.”

(Five or 10 minutes pass while the tutor struggles to solve the problem.)

“You know, I think if I spent 10 minutes thinking about this problem I could probably figure it out too,” the student observes.

(Student leaves.)

 

I wish I could tell you I was the frizzy haired student in the above story. In some ways the story is true. I was tutored in mathematics when I was a high school student. I wasn’t a bad student in mathematics, but I wasn’t getting the A that I wanted. In some ways the story is untrue. I never actually walked out of a tutoring session. But I often wanted to! I always assumed it was the tutor’s job to know how to solve every question that I brought to them immediately. So I was always disappointed when it took them a long time to solve the problem.

Let’s take an aside here for a second.  I believed that 5-10 minutes on a hard problem was too long. I wanted instant gratification! Who doesn’t? My school teacher can solve things instantly, why can’t my tutor? This is a really common assumption for mathematics students. Thankfully, this assumption is challenged by going to tutoring.  As the student, you are more aware of how much time things take (cause you are paying for it!) and you start to notice that math takes time!  Okay, back to my childhood tutoring assumptions.

Sometimes I thought, and continue to think, that tutors are not as prepared as they should be for the work they are doing.  But, on the other hand, I realize that tutoring mathematics is really hard.  I should know- I’ve been doing it for the past five years. There are lots of little techniques that are specific to certain authors and complicated uses of the methods taught in each section. I solve this by being selective about my students so that I know that I can help them in the specific course they are taking. I wonder how many smart students have disappointing experiences with tutoring.  Is it because they expect instant answers or because their professional tutor is not sufficiently well trained?

Singing about Math

I vividly remember singing the quadratic formula in high school. I bet you learned a song to go with the formula also.  Hum a few bars of it.  I promise I won’t listen.  Now, when is the last time you had to think about the quadratic formula?  Do you still remember it?  I learned the quadratic formula to the tune of “Rose Rose.”  In fact, I think I learned the quadratic formula to several different tunes, though this is the one I remember best.  I included a lovely version of the original song in the video below.  Sadly, I could not find a version of the quadratic formula sung to this tune.

I think learning the song made a huge difference to my memorization.  If I could have learned a song for my multiplication tables, then I bet I would have enjoyed math a lot more in elementary school.  If you enjoy singing or mathematics then there is a fun website for you.  Sing about Science and Math is a lovely, searchable, clickable collection of songs.  You might be surprised at some of the fantastic pieces of mathematical artistry there are out there.  Though not everything the site leads you too is golden, I recommend searching for “finite simple group” and watching what comes up.   Enjoy!