I met with an older student this evening, who’s story is classic. He excelled in math effortlessly; everything was obvious. This until he hit college level abstract algebra, and then… failure.

This is a ridiculously common story among students with natural talent for math. There’s a bravado in not writing things down, not needing to work out the details aloud. You know because you’re smart, not because you had to work for it. And then one day, you hit some mathematical ideas that are too hard and too deep to “get.” Time to learn how to work.

What’s the solution? Step 1: write it down! Putting ideas on paper is a way to turn an intractable ten step problem into ten easy one step problems, and you can’t do it unless you write. You need to start before you know what the end is, and allow yourself to make mistakes that will become steps towards the end.

I heard Malcolm Gladwell speak about this tussle between genius and sweat on Radiolab recently.

His conclusion? Love is the critical factor (Robert Krulwich tries to ask how critical, but doesn’t get too far). Malcolm Gladwell, we here at Math for Love agree. Love is a necessary (but not sufficient) condition to brilliance.

Why are people so hostile to the notion that what genius is is an extraordinary love for a particular thing?

–Malcolm Gladwell, about 8′ in above

Speaking of sweating through big projects, my girlfriend and I signed up to participate in NaNoWriMo, which means that we’ll each be writing novels this November. I actually plan to be a NaNoWriMo rebel (there’s a forum for us) and write math related nonfiction (or fiction), but at the end of day one, I’ve got a weird fairy tale-ish thing that I can’t really control. Is it good? Will I end up with a real novel?

Who cares. The goal here is to write it down. In novel writing as in mathematics, there’s a time to just put pen to paper and see what you can build. You have to make something ugly before you make something beautiful. Be proud of the ugly, I say.

So today I wrote 2074 words, which puts me ahead of schedule for 50,000 by the end of the month. Will I make it? Not sure. But damned if I won’t try to put quantity ahead–way ahead–of quality for a while.

I can’t wait to see what comes of your novel writing attempt.

Math got hard for me when I took my first college math course, a (double) honors level calculus course. All of the other students had taken calculus in high school. (My school didn’t offer it.) I worked 4 hours a night, and still got a B-. I still liked it ok, but sure couldn’t muster up the energy to work that hard the next term (which I failed).

I got out of the University of Michigan thinking I didn’t like math. But I loved teaching community college, and needed a masters degree for fulltime. I warily started at Eastern Michigan University, and fell back in love. My masters degree is full of love and sweat.

Math was never exactly easy for me, but I came to it so late that I just expected it to be hard. Where I went to college, we were partially evaluated on how much we wrote (seriously) on homework, so I fell early and deeply into the habit of writing extensively on math problems. Similar to work of writing a novel, NaNoWriMo style, sometimes I’d be at a loss for what to write so I would just write down anything that seemed even remotely relevant to problem (and very occasionally, I’d give in to my aesthetic impulses and draw comics instead). And sometimes, by giving myself permission to spend the time flopping around aimlessly, I’d stumble onto a path that would take me where I hoped to go. The beauty of this approach to working on math is that it takes talent out of the equation. It almost even takes the destination out of the equation, allowing instead for a purity of focus on the journey.

Then at graduate school I got marks taken off for writing too much. Go figure.

Teaching geometry at the secondary level I have noticed that many students who do not have a problem understanding algebra 1 sometimes have problems with the concepts that are introduced in geometry. One of the main reasons for this is because the students do not like to write down every step, let alone the reasons involved to solving problems. This becomes a major problem when I teach the students how to complete a geometric proof. A lot of times students try to look at a proof and solve it without starting the proof. They will read the problem and try and solve it in their head without writing any steps down. Most of the time they do not succeed in getting the right answer. Just a few days ago I was helping a group of students with a proof and the first thing I asked them was “What are you trying to prove?” The interesting part was that all three of the students did not know what they were trying to prove, but of course they were trying to complete the proof. When I introduce proofs to students I tell them I always want to see four things when they start. The first is to state the given information for the proof. Second, they are to state what they are trying to prove. Next is to draw the picture that describes the proof. Finally, make the two columns and write the given as the first statement and reason. It is amazing how many students who excelled at algebra do not see the reasons for these steps. I explain to them that all you are doing is rewriting the problem and that this may help you understand the logical reasons that are needed to complete the proof. Also, this will give the student a better understanding of how to solve mathematical problems that require more than one step to solve. Even with this information students are still reluctant to write things down.

One thing I’ve thought about a lot is how to motivate different parts of math. Writing explanations is one of the things I’d like to motivate: it’s crucial to understanding and communication, but there tends to be a lot of student resistance to doing it.

To this end, I’ve played with having students write their explanations for each other, and exchange proofs or explanations, often providing critiques and suggestions for improvement (and compliments on what was good). When a student explains something and another student says, “This helped me understand what was going on,” or “I couldn’t follow what you were trying to say,” it can have a powerful effect on motivating clear writing. Sadly, the student’s own understanding doesn’t seem to be as powerful a motivator.

I also have the students work together when going over math problems that require more than one step. With proofs I find it easier for me to have the students write out the problems on the board in small groups of two or three. This way the students and I can cover more problems during class time. Two things happen when I call a student to try and complete a proof on the board. First, many of them are afraid to be in front of the class when they may not know how to solve the problem. Usually I will call on them and if they are hesitant to try I tell them to “bring a friend with you”. This helps to relieve some of the stress students might feel when trying to complete a problem by themselves. Second, students have an opportunity to talk about the problem and the steps it takes to complete the problem correctly.