I’ve been thinking about Elizabeth Green’s wonderful NY Times article “Why do Americans stink at math?” and wondering how my own math education has affected the way I teach math. So what was my math education?

I really struggled with math as a kid. I couldn’t count money or tell time (and haughtily told my 3rd grade teacher that I’d just get a digital watch) or subtract (to do 12 – 7, I had no strategy other than 7 + ? = 12).

In middle school, I was in the honors class, but the teacher felt there were too many kids in the class and set out to eliminate as many as possible. I failed the first term (which unfortunately included percents). I started going to extra help every week and when I got my first 100 on an exam, I said, “Mr. Epler, aren’t you proud of me?” He replied, “No.” He was not a warm and fuzzy kind of guy.

When I got to high school, numbers disappeared and suddenly everything made sense and I loved math. But please don’t think the tenor of my education changed — one of my high school math teachers was a nun who wore full habit. She is the only teacher I’ve ever had who hit me (when she caught me checking what sin 60 was, which I was supposed to have memorized) and once took off 5 points from my test because I had dropped the edges of my paper on the floor and forgotten to pick them up.

So, I didn’t really have the sort of progressive math education Green posits is best for learning. But since I started teaching 7 years ago, I tried to go in that direction even though I had little experience with it as a learner. Each year I’ve pushed myself more and, especially with my international classes, I feel good about a lot of the activities and investigations my classes do.

But then two nights ago I couldn’t sleep; I got really bothered by something and felt ashamed of myself: circles. When doing circumference with my Dutch 2nd graders (8th grade), I’d always shown the students that the circumference was always a bit more than 3x the diameter.

Ugh how I cringe to write that. *Shown*. Why didn’t I let them find out themselves? And then I didn’t even explain where the formula for area came from, just gave it to them. Then worst of all, I taught them something that was taught to me by a colleague in my first year of teaching: Cherry Pies — Delicious! Apple Pies Are Too!

How simple! Of course! Now it’d be so easy for them to remember the formulas! Except it wasn’t. They’d constantly forget whether it was A = or A = . This year I finally (after three years! How could I have let this go on for THREE YEARS??) started saying things like, “well, what should the unit be for area?” hoping that if they could remember centimeters squared, they could also remember radius squared.

Tricks just do not work. I couldn’t sleep thinking about it. I’d been doing it so wrong.

How could I get the students to discover the formula for area themselves, just like they should’ve discovered circumference for themselves? I am embarrassed to admit that I had no idea how one could even come up with this formula — after all, I’d never been taught any way but memorizing it!

Finally I hit upon this:

Christopher Danielson helped me refine my thinking as well to include another idea:

Before today I literally did not truly know this myself.

I think that before we can teach in the progressive way that Green advocates, we need to teach ourselves this way, to pull apart the math and at every turn ask ourselves how a student could come up with this themselves, and then have them do it. Even though I’d been paying lip service to this idea for years and actually implementing it quite a bit, there’s still so many holes in my own practice.

So I’m filling up some holes. This year, in addition to actually having my students measure circumferences and diameters, I’m going to have them draw a circle of a certain radius and ask them to come up with an estimate of the area of it, using what they already know about how to find area. I want them to give me a reasonable guess that’s too low and one that’s too high, articulate how they came up with it and then write the damn formula themselves.

If they’ve done areas of triangles by the “stick another like it next to it and upsidedown, chop a bit off and stick it on the other end” method then a bit of prodding may get them to consider chopping the circle up into little triangles (sectors) and rearranging them into a knobbly rectangle. This gives you area = radius x half the circumference, and this certainly deals with the “is it r or r squared” decision.

best wishes with your new way, and have fun.

That’s a great idea as well, Howard. My plan of attack is to leave it open to them how they want to calculate/approximate it to encourage their problem solving skills, but this is definitely going in my arsenal of how to think about it. I love that there are multiple approaches that will all get you to the formula!

Thanks for all the encouraging words. ðŸ™‚

I agree with the thought of gaps in my math knowledge. I used tricks all through middle and high school and while I it came easier for me than others, I don’t feel like I have built a good foundation. I want to go and reteach myself all kinds of areas of math but I don’t even know how to categorize what I do and don’t know. I wouldn’t even know where to begin.

All I need to do to find my gaps is read any blog post by @samjshah! ðŸ˜‰

I think if we’re vigilant as we plan our lessons, we can find more and more places where the gaps lie. I hope?