There are lots of qualities that make someone “good” at math. Knowledge and skills are important, and so is creativity. But perhaps the most important qualities are patience and persistence.
I was inspired to write this because of this post by Alexander Bogomolny.
So many of my students shut down as soon as they see a problem, especially those students who have had a bad relationship with math over the years. Some students even give up before really understanding (or reading!) a question. This is even more true when it comes to ‘word problems.’ If you haven’t yet seen this now famous TED talk from Dan Meyer, I encourage you to watch it now (and visit his great blog!). I think he might be the first person I’ve heard use the phrase “patient problem solving” so I’ll give him the credit for that :-).
The importance of patient problem solving has broader application than just math, of course. In so many areas of life, we give up too easily when faced with a problem. We don’t realize, that if we just looked a the problem a little longer, if we came back to it a few more times, if we dove a little deeper, the problem would crack.
When I do my graduate class homework, I find great value in starting the homework problems as soon as possible. Sometimes the problems just need to percolate in my brain!
For those of us who teach, it’s important to keep ourselves fresh and engaged in mathematical problem solving on a regular basis so that we can (1) remain familiar with what real mathematicians actually do, and (2) relate to (and empathize with) our students who are being faced constantly with fresh problems.
The folks over at Math Fail recently encouraged us in that direction, saying “If you didn’t focus on mathematics today, you should at least keep the gears in motion by trying to solve this mathematical puzzle” and then they proceeded to give us a good puzzle. Go try it now! (I solved it this morning with my wife as we were driving to church!)
I want my students to experience the immense satisfaction that comes from having solved a stubborn problem. I want them to know that if you bang your head enough against the problem, eventually it will crack! Alexander Bogomolny made this point in the the inspiring blog post I mentioned above, saying
Still, there is great satisfaction in having solved a problem – even a simple one, and extra satisfaction in being able to appreciate an elegant proof; this kind of satisfaction is multiplied manifold after you devised a solution on your own. Yes, it all may start with inspiration, but to keep the flame burning involves hard work…the more you sweat, the greater the satisfaction.
Some of my students are starting a big math paper this week, in which they choose their own topic. One of my hopes for them is that they get to experience the deep satisfaction that comes from actually doing mathematical thinking and solving hard problems. There’s also great satisfaction in coming up with good mathematical questions! And they’ll have a chance to do that too.
On a related note, when reading math textbooks, students sometimes don’t understand that reading math is very different than reading other subjects. In other subjects you might be willing to devote 5 minutes per page. But in math, a reader shouldn’t be discouraged if it takes 20 minutes or more to understand a page of text. Math is dense!