A Call For Answers!

I sent out a couple of tweets yesterday asking for your assistance:

You see, I was asked to be the high school teacher-speaker at an MSRI workshop.

The 2014 CIME workshop will focus on the role played by mathematics departments in preparing future teachers.  As part of this focus, the workshop will consider two broad questions: What mathematics should teachers know, and how should they come to know this mathematics?

Harvey Mudd professor, creator of morning problem sets at PCMI, and all-around awesome person,  Darryl, asked me to think about the following questions,

“Do you have any thoughts on the connection between the mathematical preparation that you had in college and the mathematical skills and knowledge that you need now as a teacher?  In what ways were you prepared well, and in what ways not? What could we do better?”

and I was intrigued enough by these questions to say “yes” and commit to giving a little presentation.

I’ve been thinking over the past couple of weeks about the “lessons” I learned in my college math classes and how they impacted me in my work as a math teacher. So far I’ve come up with this list of things I learned:

  1. I don’t hate math.
  2. Math is supposed to make sense–even when math doesn’t make sense, it makes sense.
  3. Seemingly different ideas in math are connected, often strongly & deeply.

Which I will flesh out more fully when I write my talky-thing, of course.

So what I need from you is your answer(s) to the question:

How did your math courses/major prepare you for teaching?

Comment here, send me a DM (or a regularM) on twitter, email me using the comment form below, hire a sky-writer, send off a carrier pigeon, whatever. Just get me the goods (pretty please).


Over the weekend, in Pacific Grove at the CMC-North conference, at a late dinner with a bunch of my former colleagues and some twitter-folk I was reminded of a moment from Palm Springs last month that I had planned on blogging about. For whatever reason it fell off my radar, but Dan Meyer was kind enough to bring it back to the forefront of my mind. Dan asked Avery whether he ever got push-back from people at his workshops who claimed that Avery’s ideas “would never work with my kids” when they learned that he teaches at a small, independent (private) school that primarily serves high SES students.

Dan’s question reminded me of this moment when I was at Palm Springs in November. I was at a workshop—the first workshop in fact—in which the presenter came from the Phillips Exeter Academy. If you don’t already know, Exeter is a private boarding school. Small class sizes, privileged population…perhaps you’ve heard of it? The presenter introduced himself and his school at the beginning of the talk, so I assumed we were all on board. He spoke about how the classes worked at Exeter—the Harkness table mythos—and how the curriculum developed over students’ time at the school. At some point during the talk, the presenter showed us a blog from a teacher in a typical public school situation who was using the Exeter problem sets in his classroom: Harkness for Thirty. Shortly after this very part of the talk, a woman seated at one of the back tables asked our presenter about his classroom setting. He told her the information I outlined above. Said he had classes of about 14 students. She immediately replied

“We can’t talk”

and followed it up with

“You’ve just lost all credibility.”

I remember writing down on my paper “Wow!” I couldn’t believe the rudeness. More importantly I couldn’t believe that her attitude towards his classroom situation—which was very unfamiliar to her own—meant that she shut out all possible benefit of the ideas being discussed because her idea of a classroom didn’t match up to the classroom situation at Exeter.

Every classroom situation is different. At the same school, even moving from one section of the same course to another section of that course—taught by the same teacher, no less—can require different approaches. Obviously, not everything we hear about in a conference, even in an individual workshop, will work for us without adaptation—whether it’s to better suit our student body, to better suit our teaching style, or to reduce cost, or de-technify something, we make changes to things we learn about all the time. I can’t remember the last time I took something someone handed me, and handed them to students unchanged. So of course we are not going to find the perfect match for our needs at a workshop at a conference, no matter how well we select our speakers. It just doesn’t exist. We have to take what we can use, discard or postpone the rest, and make new ideas work together with our old ones in the environment we have.

But to dismiss everything a teacher says because they teach at a different type of school that you?—Wow.

Avery had a great answer for Dan’s question. Avery talked about how he sees the professional role of teachers as one where their job at a conference is to find the parts that work for them, with their students. I liked Avery’s response. And this is something I think about, as I too teach at a private school. Students who attend my classes are mostly there because their parents can pay the 37,000 dollars a year it costs to go there. My classes this trimester are 10, 10 and 13 students. Last trimester I had a “large” class of 18…teaching Topology to juniors and seniors. I know that I have it good.

Just because it was on my mind, I decided to track the types of comments regarding “my school” versus “your school” that I heard at Asilomar last weekend. I was surprised—and happy—that I didn’t hear that many. Both of these came out of the first workshop I attended:

8:14 am: “teaching at a demographic such as mine…75% free and reduced lunch…”

8:44 am: “for those of us teaching at school’s where students are mostly far below basic…”

CMC South Presentation


For those who are interested.


If Scribd doesn’t work for you:

Telling Stories Teaching Math CMC South 2013

MTBoS Taught My Class Today

Except for Topology. That was all me…and the student presenter who taught for the first half of the class. But other than that, it was all me. :)

So here’s what I did today.

Got to work, had a free block first thing. Parent association was doing a breakfast thing for the faculty, so I went up and got myself a plate. Came back to my office and played the daily Set game from nytimes while I ate. Did a little work on my presentation for CMC South and in doing so went through my Evernote files. I found a rather well-populated tag for Coordinate Geo, which is my current unit so I opened it up and out fell my lesson plan for the day.

I start the class off with a warm-up (okay, this I wrote). Have a student put up a solution for each problem. Thumbs up/thumbs down for “do you agree/disagree” and we discuss the documentation of work.

Then I throw up this image:

DD 59b

The grid lines aren’t visible enough on the projector screen so I walk the kids through how to get to the dailydesmos page. [daily desmos, not just desmos] They stare blankly at it for a few moments before I tell them they need to reconstruct the graph on desmos before they can proceed. Students get themselves some parallel lines. I interject to show them that desmos will give you the coordinates for points of intersection and remind them that they need to make sure the side lengths are the same. Many “ahas” about how points with the same distance from the origin will have reversed coordinates (when lines are perpendicular).

Then we proceed to the Best Square video. We watch it twice because students didn’t realize what was going on the first time–totally expected. Much discussion ensues about whether Timon pronounces his name TEE-mon or TIH-mon. Consensus is that he has the best name. We discuss what information would be helpful in deciding who drew the best square. Coordinates, lengths and angles are all mentioned in both classes. I give them the coordinates and the angles. I ask the class if they really need the lengths after I’ve given them the coordinates and they agree that they don’t need them. Students start calculating distances. In my second time around, I assign each group one square to speed things along. Much better flow that way.

While the conversation about what made a square better than the rest didn’t really go anywhere deep, I was happy with the class. The combination of activities allowed students to apply some of the skills they’d been working on with distance formula and parallel and perpendicular lines, and it drove home the idea that coordinates are actually really handy things to think about and use to help them solve problems.

Since I just did all of this on the fly, I didn’t think to add in the Squareness activity! Maybe next year.

One Problem Problem-Set

I’ve been thinking about assessments a lot in my Topology class this trimester. For the first few problem sets, I just copied the ones the previous teacher created…and they were a b**** to grade. They took me ages, and most of the questions we’d already gone over in class. Probably the worst part about grade these problem sets was that I didn’t care that much about the questions, and therefore I didn’t enjoy reading over students’ answers at all. I wasn’t learning enough about their thought process to keep me interested.

I came to my senses and revised the third problem set to just ask the questions that I thought were good ones–ones the students hadn’t already seen, that pushed them to think about new ideas, or about applying old ideas in new ways.

When it came time to do their last problem set, I realized I didn’t really have many good questions to ask students. The content of the chapters we’d gone over wasn’t as meaty as previous sections in the text. I didn’t want to ask the same numeric questions and just substitute different numbers into the problems. Not my style. So, I asked myself what I wanted the students to get out of the material.

And this was it:

Problem Set Question WordPress

I just finished grading their papers. And it was a breeze to read through them. I enjoyed myself. I enjoyed myself! I know: I can’t believe it myself.

You can’t hide in a problem like this; you can’t BS your way through; you can’t check your answer in the back of the book. And that’s what made it so enjoyable: I had hand-drawn diagrams that were beautiful, I had students describe spaces from a “Euclidean-eye view,” I had pictures of Pringles potato chips. My students explained their understanding of the material, in ways that were uniquely theirs.

Student Reflection

One change evolution I have gone through from the start of my career as a teacher up to now is with the use of student self-reflection. Giving time for it, and giving weight to it.

In “Teacher School” (as I like to refer to my certification program) we were told on many occasions about the value of reflection. I don’t really remember what people said, in what context they said it, or with what strategies they suggested using reflective tasks. Because I didn’t buy into the whole idea. Or rather, I agreed that it was good, but I didn’t see how to make it effective or useful to my teaching. After all, I needed to cover x amount of material in y amount of time (where x>y for all x and y). Who had the time? And, having students write about their work and their thinking meant that I then had to read it all. Who had the energy? I was resistant.

But I did it a bit. And sometimes it was really hard. When I asked students for feedback on my teaching, sometimes they wrote things that hurt. That made me confront weaknesses in my teaching that I didn’t want to acknowledge. But it was worth it. I got better. Both at teaching and in confronting my own challenges.

When people suggested that I ask students to comment on their own work and thinking, I was also resistant. I assumed that students would be dishonest, that they would see themselves through distorted lenses and only notice and comment on the good things, ignoring all the rest. But I was wrong. Students are sometimes incorrect in their self-assessments; sometimes they think things are fine when they aren’t, but more often they think things are going poorly when they are fine–or better than fine. The vast majority of the time, however, students know themselves really well–and they are willing to share to people who are willing to listen. They will tell you so much about themselves, with depth and complexity.

For the past four years I have written narrative comments for students three times a year. This is a major task, which requires me to know and speak to each of my students on an individual level. I comment on their work, their participation in class, their strengths and their challenges. I give each of them feedback on how they can grow as mathematical thinkers and as students in the classroom community. I have become more and more reliant on student reflections to help me with this task. Not because I want to get out of doing the work–but because the students’ reflections about what they need to work on are so insightful that without them, I would be groping in the dark, trying to make out the shapes of the objects around me. Student reflections are like flipping on the light switch.

Return of the Mistake Game

So last year I tried out the mistake game, and it kinda bombed.

This year I am trying it out again…on the exact same material.

But wait, Bree–didn’t you say last year that these problems weren’t great for making good, rich, intentional mistakes? Why, yes; I did. Now, I’m not exactly sure *where* I said that, but I’m sure that I did.

Mistake Game, version 2.0 is already going better than last year. Same course, same unit, same material. What’s different? Well, mostly just the way I set things up and introduced the idea, but also the format to some extent. Last year I got the idea to do the mistake game on this packet of problems when most students had finished (or nearly finished) the work. This time around we’re doing presentations in several rounds as we go. The added bonus is that while students are waiting for other groups to finish writing up their solutions on the whiteboard, they have something to work on!

Several of my students wrote on their midterm self-reflections that the Mistake Game was one of the most helpful and/or most enjoyable aspects of the class. Now, that may be due in part to being one of the more recent things we’ve done in class, but I still count this as a success.

One more small share: Earlier in the year I used the red/yellow/green cups. For ONE day. ONE activity. And I still have kids in one of my classes who will draw a red or green cup on the whiteboard when they want to get my attention. I chastise them when they draw the red cup with green markers, and vice versa. But they have convinced me that this, too, was a good idea.

The cups are now living full-time in the classroom.