This is my post for The Virtual Conference of Mathematical Flavors!
“How does your class move the needle on what your kids
think about the doing of math, or what counts as math,
or what math feels like, or who can do math.”
It’s difficult to explain experience or perspective to someone. That’s why teenagers of every generation will make the same mistakes, and it’s why the adults of every generation will complain about teenagers as though their generation had been any different.
Getting students to develop their sense of mathematics beyond basic operations, interpreting word problems, or working with equations and graphs is going to be a subtle process. You can’t just tell them what mathematics looks like, you need to show them and ideally give them some genuine experiences.
Anyone reading this knows that these metaphors abound in blogs and books, but the way we present math in school is like having a basketball team that practices fundamental drills every day, never connects those drills, and maybe gets to play one game at the end of the year. Someone on that team might be able to explain some accurate but oddly detailed aspect of a particular drill, but they aren’t a basketball player.
So what does math look or feel like? There are many possible good answers to this, but I’ll try to give my sense of it through a brief anecdote: I once had a physics student who, after a good deal of time and study, crossed a threshold beyond just going down a checklist of equations looking for “the answer.” I overheard her one day telling some friends that she had noticed something interesting in the way her pen rolled on the table, and that she woke up that morning with an idea for how to model it. Sure, she could probably look it up, she told them, but she didn’t have to. She had internalized concepts into intuition, and now her intuition was guiding her thoughts about quantifiable relationships. She played with ideas and passed everything she saw through a lense of that intuition. She was doing math.
These are three ideas I think about when I want to develop my students’ understanding of mathematics:
- Don’t present math as complete or static
- Share the stories of mathematics
- Be a person, not a math person
These are really cultural approaches, leaving out the obvious approach of trying to get them to do math.
DON’T PRESENT MATH AS COMPLETE OR STATIC
Math is happening right now, so take care to acknowledge that! The issue is that there are plenty of ideas in math that actually are complete, and it tends to be only these ideas that we present in class… I don’t think there are any unsolved problems mentioned in the Common Core. There’s some practicality in that, but it misses a valuable opportunity.
I love talking about unsolved problems, why they’re still unsolved, and who is working on solving them. Some of my favorites are
If you haven’t, familiarize with Unsolved K-12, a great set of unsolved problems with one problem chosen for every grade K to 12.
I start each year by presenting a small glimpse of what mathematics looks like. Here are my slides from last year, which was an interesting year as progress had just been made by Kallus and Romik on an upper bound in the moving sofa problem, and Blum had uploaded “A Solution of the P versus NP Problem” to arXiv just weeks before school started. It wasn’t long until Blum updated arXiv to acknowledge flaws in his proof. How dramatic! And what a great example of a professional mathematician trying and failing!
SHARE THE STORIES OF MATHEMATICS
History provides context, and people are fascinating. You surely know the story of George Dantzig, but have you ever heard the legendary tale of the final days of Évariste Galois? Much of the legend is probably untrue as traditionally told, but that doesn’t stop me from reciting it to my students! You should never ever discuss arithmetic series without prefacing it with the story of Gauss. If you can’t drop Maryam Mirzakhani’s name in casual conversation, you don’t deserve math. Andrew flip’n Wiles. The Pythagoreans were both amazing and cray cray. Did you realize how controversial the idea of zero was? Or negative numbers? Or irrational numbers? Don’t even get me started on complex numbers.
Read about mathematics. An assignment that I give to Algebra classes is to read a book from the Goodreads list of Best Books About Mathematics. No report, no test, just read one of these books and then have a conversation with me about it. It’s important to guide students towards appropriate books. For example, I would not suggest a student attempt to read GED or Euclid, but I would suggest they read anything by Bellos, Gleick, Singh, or Strogatz, or most any biography of a mathematician.
Will understanding a little mathematics history or some details about the dramatic lives of some mathematicians be the key difference in helping my students on standardized tests? I will say this, I don’t see any reason why it would be.
BE A PERSON, NOT A MATH PERSON
I want my students to know that I’m passionate about doing awesome stuff with math, but I don’t want them to think that you have to have a particular set of traits to do awesome stuff with math. I share interesting nerdy projects that I work on because it’s relevant and awesome and they get excited, but there are any number of interests anyone might develop and math can be just one of them. More than that, I want students to look beyond the math to see the awesome stuff, including when that awesome stuff is more sophisticated math! I want students to realize that what we can do with math is so diverse that there’s a place for everyone.
I don’t want to be a part of perpetuating the stereotype of the “math person.” I’m not in any way suggesting that I change who I am, I’m saying that I don’t hide who I am just to hype math. My students know that I was a dance instructor and that I write bad poetry. They know that math is just one of the things I like to do well.