## Triangles

Pre Algebra is on my teaching list this year. The last time I taught it was five or six years ago, and I’ve changed quite a bit as a teacher since then, so I’m building everything again from scratch. (It feels appropriate to note that my philosophy of teaching/learning has shifted more significantly than my actual practice, at least in my other classes.)

For the past week or so we’ve been constructing, measuring, labeling, identifying, and discussing triangles (where the constructing has been mostly with non-digital tools). Given three sides, given three angles, given some combination, similarity, congruence, impossible triangles, etc. Our progress has been frustratingly slow, in large part because my routine of late has consisted of me going into class with an activity and some excitement only to discover that some key element of the activity is seriously flawed. I spend the next evening revising the activity (thereby rekindling my excitement) in order to try it again the next day. I imagine (hope?) our progress won’t be quite as slow next year, but I’m not sure if that’ll be the case.

This is the activity I wrote for today. Since it’s late, I’ll cut to the chase: The conclusions students drew at the end of the activity were on the disappointing side, both in terms of the depth of insight of their observations, as well as their ability to express what they noticed.

This (here and here and here) is what I have planned for tomorrow. I’m hoping that by giving them space to record their measurements (in an organized manner) our debriefing conversation will include more insightful comments from students. We’ll see.

If you’re game, have a look at the handouts and let me know in the comments what you like, what you don’t, and how you’d make it better.

## Estimation 180 Rocks

I learned about Andrew Stadel’s Estimation 180 a few weeks ago. I decided this morning it was time to stop watching and time to start playing along. We’re a bit behind the rest of the estimating world, but in first period (Pre Algebra) we worked through Days 1 and 2 and in third period (Honors Algebra 1) we worked through Day 1.

It. Was. Awesome.

I won’t spend a lot of time in this post talking about Estimation 180 in general or how I used it in class today. If you already use it, you’ve probably formed your own opinion and approach by now. If you don’t use it (or have never heard of it), get yourself over there as fast as you can.

What I will say is this: My students were more engaged today, even in those ten short minutes, than they have been in quite a while.

I imagine it had something to do with the accessibility (everyone can make a guess, even if it’s horribly, horribly wrong), the anticipation (“Am I right? Am I right? Am I right?”), and the way these estimation tasks present a natural, un-intimidating opportunity for students to defend their responses by explaining their reasoning. The reasoning—to students, at least—often doesn’t appear very technical or mathematical, but it’s great exercise anyway. And it’s a healthy start to making sure this is a regular part of my students’ experience.

So where to from here? Well, for one thing I’ll continue to use Estimation 180 with these students. But beyond that, I’ll try to incorporate those three elements (accessibility, anticipation, and ready-made opportunities for answer-defending) into other tasks and courses.

I just have one question: When this year’s Pre Algebra students have me again next year in Algebra 1, will there be an Estimation 181-360?

## Reason and Wonder

In my third year of teaching I taught AP Calculus AB for the first time. I had 10 students (I teach at a small school; more on that later). All of them passed. In fact, six of them earned 5’s. I felt like the king of the world. In reality, I had no idea what I was doing. (In many ways, I still have no idea what I’m doing; more on that later as well.)

A couple years later, four of my students (out of 18) failed to pass the exam. I had a minor crisis for at least a couple of reasons. One, as I continued investing exorbitant amounts of time and energy into my calculus course, as I thought I was getting better, my students’ results (by one measure) were declining. This was more than a little discouraging. Two, if my goal was to equip students to pass the AP exam in order to earn college credit, then I had failed.

I had to stop and think: Was the year of teaching and learning and struggling a complete waste for these four students (and for me in relation to these four students)? My gut told me no, it was not a waste, not even close. So if it wasn’t a waste, then I must have been deceived when I thought (in no uncertain terms) that the primary (only?) goal of the course was to pass an exam and earn some college credit. So what was the goal?

I sat down on September 17, 2009 and tried to sort out the thoughts in my head by writing in a private journal. After a few passionate, yet meandering paragraphs, I settled on this as the reason I taught calculus:

I teach calculus in order that students may reason more soundly and see the beauty of the created world more clearly.

The reason I share this story now, as the first post in what I hope will be more than a few over the days ahead, is that with some minor revisions it expresses why I teach not just calculus, but anything at all. It also explains why I chose the name for my blog. (Thanks to Justin Lanier and Michael Pershan for encouraging me to finally start blogging, and to consider naming the blog in a way that expresses what I’m most passionate about in teaching.)

So here goes:

I teach mathematics in order that my students may reason more soundly and that they may have the capacity to wonder more deeply and profoundly about the world.

That’s it. That’s why I teach. And while I still wait anxiously for AP results to show up online each July, I no longer count my year as a waste or a success based solely on the numbers that appear in those reports. If I’m helping my students to reason with skill and precision, and if they’re growing in their capacity to wonder about and be amazed by the world, then I think I’m doing alright.