Welcome to Recipes for Pi!

I’d like to introduce myself to writing about teaching math by giving some history about me and math while explaining part of why I’m starting to write this. This blog was in part born out of a mixture of enthusiasm and anger about the ways that the world learns and shares math, triggered and brought to a head by an event in my own math education. For me, writing is the best way to process my feelings about and reactions to events. I was motivated to share my writing, and to begin a blog, when one of my students asked me about the event in question. So, read on and enjoy!

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I was recently reminded of how much I dislike math class – and how I learn math best.

Last semester, I registered to take a graduate-level math class. I hadn’t taken a math class since college, and had begun to miss being on the learning – rather than teaching and learning-for-teaching end – of the classroom. I figured that taking the class would be difficult – both because it was a graduate-level class and because I’d be taking it while teaching – but was motivated to face the challenge because, well, I like doing math.

Before about 5 ½ years ago, I hated math. That’s surprising to those who’ve gotten to know me best in the past 5 ½ years, but the fact that I love math now is a total shocker to those who know me from before. Throughout elementary, middle, and high school, I was one of those students who often baffle math teachers most – the extremely conscientious kid who does all of the homework, stays after class to ask questions, and aces tests, but is convinced she’s terrible at math. I always felt as though while I could eventually do anything with enough memorization, I never really understood what was going on. If you’d asked me then what it meant to do math, I would have said, “Math is learning complicated, nonsensical procedures and algorithms, and using them again and again and again until you can do it with your eyes closed (or at least without your notes).”

But when I got to college, something changed. Here’s what I think happened:

- I stopped solving problems by memorizing everything the teacher said to do, because the problems weren’t like that. Most of the problems weren’t like the examples – you had to build your own methods from the tools you were given, not copy a procedure.
- I stopped worrying about the answer and started paying attention to the process, because the problems weren’t about the “answers.” The problems we were doing were proofs – and when you’re doing a proof, you already have the answer. Math wasn’t working the machine properly, but building the machine you needed.
- I started working with other people. My classmates and I figured out how to do the problems together, and the professors came to our informal homework meetings to talk to us, too. Math was social, cooperative, a conversation among people – students and teacher – learning together.

As I started really thinking about math and bringing my tastes to my mathematical work, I began to see how many ways there are to interact with math. I also began to see how very beautiful the work of finding, simplifying, proving, and changing patterns can be.

And when I began teaching math at Saint Ann’s, I grew to love math even more. I had to really think about interesting things to do with math, and do them – mostly for the first time, because these were precisely the activities I avoided in elementary and high school. My peers’ love of doing math was infectious, and their patience with and appreciation of my pace of thinking were blessings. I got to try to build classes where, whether by conscious or sub-conscious effort on my part, my students learned math the way I learn math best – by talking with each other, figuring things out for themselves, and certainly not just listening to the teacher.

I probably thought that taking a graduate math class was a good idea because the images of math class I had in my head were of math classes at Haverford and the classes I try to run. And the feelings I had after one week in this math class were probably so very negative for those reasons as well.

After about a half an hour into doing the homework assignment, I could feel my old math anxiety returning. Here was math class at its near-worst for me – math as competition, math as listening to a professor spout information, math as a race to accumulate facts. I was working alone. My professor quoted from a textbook for two hours and did nothing to create an environment in which a bunch of people really interested in math (himself included) could work and learn together. There were a TON of problems for homework, because we’d covered an entire chapter – itself covering all of introductory group theory – in two hours.

And I began to feel math anxiety all over again.

This made me angry. Math anxiety?? Me?? I like math! I know that, in working on math with others, I can contribute ideas, ways of thinking, and ways of structuring thoughts that are valuable. I also know that I can solve and find the resources to help me solve problems that I come across and find interesting.

There are many, many ways to learn and participate in math (and any subject of study, for that matter). One of those ways is listening to a professor in lecture and working through chapters-worth of math problems every week – and maybe that accelerated, private, listening-based way of learning is great for some people. But it certainly isn’t good for me. I can learn more modern algebra by working through a good book on my own, at my own pace, by sharing interesting problems with friends and peers, and, most importantly, teaching it.

So, I dropped the class. And I know, better than I did before, how I learn math and how I want to teach it.

Solid first post! It’s quite a good way to introduce yourself, because we get a sense of who you are as a teacher, as a mathematician, as a learner, but not the whole picture. I definitely want to learn more about your story after reading it, so keep it coming!

Anna,

What an awesome foray into the world of teacher blogging! So great to have another math blogger out there, especially one who comes with such high recommendations from Paul and Sam.

I love your vision of math teaching, and wish more students got to see it. I especially like this line:

So many students find math tied up in it’s competitive aspects—being the first to complete the test, math competitions, and more. I wonder what we can do to emphasize the creative, playful side of math. This certainly seems like something St. Ann’s does well. I wonder how one can spread it to other schools?

John,

Thank you for the wonderful comment! I feel very honored to be recommended by Paul and Sam.

I agree with what you’re saying about many students’ encounters with math being mostly tied-up in competition. I know that I never participated in outside-of-class math activities in high school because they were all competitions – math team, chess team, etc. One thing we’ve tried to do at Saint Ann’s to show math’s non-competitive side is to offer electives. The mathematical art seminar that Paul, Justin Lanier, and another teacher and I are running this semester might be the best encounter with math I’ve ever had. I’ll certainly write more about it later, but, basically, the class is run as half sharing time, where students and teachers share work they’ve done or found, and half creating art time. As far as I can tell, it isn’t competitive at all – the students are full of praise for each other’s work, help each other out, and take inspiration from the work being done around them. Having teachers actually participating in the class, rather than simply leading it and acting as experts, also helps to reduce the competitive feel. I’ve also taught two other high school electives this semester.

Another thing we’ve tried to do is separate the pace of a class from “tracking.” We’ve stopped having tracked classes and, instead, have two or three “pacings” which, in high school at least, the students can choose when they sign up for classes.

A more complicated issue is how we balance competition and cooperation within our classes, however, no matter whether they are electives or regular classes, tracked or not tracked. Competition is woven into performance on tests and quizzes, but also who finishes what fastest, who makes the “best” comments, who asks or answers the most questions, whose ideas get the most attention, who needs help and who gives it… Competition in this sense can be positive – it can motivate and spur creativity – but it can also hold some kids down. How do you keep the tone of a class celebratory and encouraging of achievement, but not really competitive?

Thank you emphasizing the idea of competition in your comment – it’s very important to me and certainly worth thinking about. I hope we can discuss more in the future!

Hi Anna. Welcome to the blogosphere. I’ll be following along.

Indeed – welcome! Looking forward to more thoughtful posts! There really can never be too many of us math folks talking about teaching, can there?

Thanks for the welcome, both of you!

Hi Anna,

I’ve really enjoyed catching up on all your old blog posts! I have to disagree with this one, though. You write “There are many, many ways to learn and participate in math (and any subject of study, for that matter). One of those ways is listening to a professor in lecture and working through chapters-worth of math problems every week – and maybe that accelerated, private, listening-based way of learning is great for some people.”

I’m pretty convinced by the education research that shows that this is not a good strategy, at least for almost everyone (so your “some people” must be pretty small, and your “learn and participate” doesn’t involve much learning and participation). I’m pretty mystified why so much teaching is still in this style. I figure that once the printing press was invented, the lecture was no longer a very good way to teach, since people could read instead of listening and copying down the book.

I do a lot of work with math circles with kids and teachers, and the most shocking educational experience I’ve had recently was when I was a guest in a college class and I opened with a nice easy softball question to get things going, and was met with complete silence. It was as though I was violating some kind of social contract, that professors were supposed to lecture and students were supposed to take notes and actually asking the students to participate or, heaven forbid, think, was just out of the question.

I’ve done a lot of work bringing more mathematics and problem solving into school classrooms, by connecting mathematicians with teachers (particularly in the middle school), but despite moderate amounts of effort I’ve had almost no success bringing good pedagogy like yours into college classrooms. I’m also somewhat mystified why mathematicians don’t think they have anything to learn from professional teachers. This is one of the big differences I noticed in my visit to Russia: the mathematicians there view middle and high school teachers as professionals in a way that mathematicians here rarely do.

Hi John,

Thank you very much for your thoughtful comment! (And sorry for the delayed response.)

You write, “your ‘some people’ must be pretty small, and your ‘learn and participate’ doesn’t involve much learning and participation.” It’s true – I don’t know very many people who learn well from pure lecture. But, in my time as a student in a variety of classrooms and in different subjects, I have encountered people who thrive in that environment. How do they do it? Because, no, it’s not an environment that encourages genuine interaction with the material being “taught.”

The people I remember being very successful in this sort of environment were capable of thinking critically and posing and answering questions during lecture. They would often raise their hands to ask questions even when that wasn’t the norm – or they were capable of questioning and sharing ideas in other forums, such as homework and papers. You could sometimes even tell from the looks on their faces while they listened that they weren’t just listening like the rest of us – they were thinking and listening at the same time. They could come up with questions and interact with the material without being directly prompted or encouraged to do so.

This definitely wasn’t me in math class – though I have vivid memories of fellow students who were like this in math. It actually was me in other classes. How did I end up this way in, say, history class, and how did some of my peers end up this way in math? I can’t really say. Possibly some mixture of subject affinity, good teachers early in schooling, and family culture. The ability to self-engage almost certainly can be taught, because, while I couldn’t do it at all in math class during elementary and high school, I can now, after years of training to think inquisitively about mathematics.

Now, just because some people can engage and learn during lecture – and do it well – and because it’s a learnable skill certainly doesn’t mean that it’s a good way to teach. While I could keep up an interesting inner dialogue during a history lecture, I almost certainly would have learned more in a non-lecture environment.

So, while I stand by what I said, I also agree with you. Though I also think that learning to self-engage is something all kids should do. We can set up a framework for that by creating classrooms that constantly prompt kids to question, interact, make, and do. But maybe we also sometimes have to remove the training wheels and let them have a go at it on their own?

Again, thank you very much for your thoughtful and thought-provoking comment! I look forward to continuing to correspond with you about teaching!

Thanks for the thoughtful reply. I really appreciate the time you took to craft a careful response that made me think more about students’ different learning styles. Of course we agree much more than we disagree!

Still, I think that even those people you describe who have the skills to learn a lot from a lecture-type class would learn even more from a class that promoted even more engagement with the material. There are certainly plenty of studies that show big gains on average for interactive engagement questions as compared with lecture (particularly in the realm of physics education, where the existence of the Force Concept Inventory seems to make people more able to agree about whether classes were or weren’t effective).

On the other hand, you give a good reminder that the goal of a class isn’t only for kids to learn the most they can about mathematics this year — we’re also (perhaps more importantly) setting them up with skills to enable them to learn more effectively in future years. One way to do that is with mathematical habits of mind (or maybe I should capitalize that to refer to the relevant paper), but another way is to show them how to work on their own and carry out an internal dialogue to question the material when the classroom environment doesn’t contain that conversation. They will certainly encounter classes in the future where the lecturer expects them to already have those skills, so part of our job should be to prepare them to make the most of that type of class, too. Your penultimate paragraph there is a good reminder to me about including those kinds of skills and habits in my list of useful things for students to learn how to do.