At the end of each school year, Manhattan Country School faculty and staff are given a “suggested reading list,” from which they can choose a book to read, to enhance their knowledge of some area of education, social justice or one of many issues relating to race and gender. During the first few years, I used to read one of these books, and thoroughly enjoyed them.
This year, I decided to choose my own book, or, as is often the case, the book “chooses me.” While listening to the local public radio station, I heard an interview with an engaging mathematician named Jordan Ellenberg, whose book, How Not To Be Wrong: The Power of Mathematical Thinking, had just been published in paperback.
During the course of the interview, he discussed all sorts of mathematical phenomena that related to “real life.” Intrigued, I ordered it, and have been slowly working my way through it.
Ellenberg is an engaging writer, and his popularity is based on his ability to apply mathematical concepts in surprising ways. My favorite chapter (so far) was entitled “Miss More Planes!”, in which Ellenberg provides a convincing argument that if you travel a lot, and don’t ever miss a plane, you’re probably wasting too much time at the airport. Using simple mathematics, Ellenberg shows that arriving at the airport early to make a plane means you have less time doing other things, and I think we can admit, waiting for a plane is among the least useful things that one can do.
Depending on how you value your time, arriving at the airport very early can be quite expensive: let’s say you estimate that one hour of airport time is worth 3 hours of “non-airport” time, when you can be doing much more productive things, like cooking, exercising or practicing the bassoon. If you make a habit of arriving at the airport an hour earlier than you need to (instead of arriving “just on time”) you’ll waste 3 hours of “useful” time, which we can refer to as “utils.” If you take 10 flights per year, you’ve lost 30 utils of time, which could have been put to more useful ends. All of this points to the idea that you should spend less time getting to the airport early, which may lead you to miss a flight once or twice.
While visiting the sixth graders recently, I had a similar moment: I went around the classroom and noticed that one of the students was trying to correct an incorrect answer, but her eraser had been worn out. I complimented her, noting that since she had used up much of her eraser, that she had fixed lots of mistakes, which is an important habit to develop while doing mathematics.
I then walked around the room to inspect erasers. Some students had lots of eraser left; I mentioned that they were probably working too carefully, or not
taking risks, and that’s why they had not been making enough mistakes. I advised them to not be too careful while doing their work, because it might be slowing them down, which means they don’t get to finish the entire activity, or it might be a sign that they’re too confident, and have overlooked an error somewhere. In general, I’m suspicious of mathematics assignments that are completed too neatly, and so should you.
One of the questions I like to ask my students is, “How do you know when you’re doing “real” mathematics?” The answer usually has to do with numbers and equations and solutions; I usually say the following: “I know I’m doing real mathematics when I’m a little lost, making mistakes, and going back to fix them. If you’re not, you’re probably just doing plain old mathematics, and that’s not all that interesting.”
Over the last week, I’ve been working on designing a series of puzzles to practice addition and subtraction strategies, while incorporating deductive reasoning skills. Here’s what my notebook looks like:
It’s messy work, and because lots of this usually takes place while sitting on the subway, I use a 4 color pen, so that I can keep track of all the mistakes and fixes. Through all this messiness, small ideas emerge, and, if I’m lucky, every once in a while, I’ll discover a big, fat truth.
Look over your child’s math assignments and notebooks once in a while; if you don’t see enough erasing and cross outs, encourage them to make more mistakes!