Trying to find math inside everything else

A few weeks ago I went on an interview, and I was trying to think about the part at the end when they always ask ,”Do you have any questions for us?” I used to never ask questions then, but I realize now that I am interviewing them as much as they are interviewing me. (This is especially true when you already have a job, as opposed to first getting one.) Most of the time my questions come from previous experiences with things that were lacking – much like my questions during apartment viewings while home-hunting. But since my experiences are not universal, I reached out to the #MTBoS for some suggestions, and got a lot of good ones back. Here’s a bunch of the ones I liked, courtesy of Kate Nowak, David Wees, Tina Cardone, Shannon Houghton, Anna Blinstein, Jonathan Claydon, and Brian Palacios.

  • Describe your students. (And take note of what kind of language they use.)
  • What are class sizes like?
  • What is your school/department working on improving?
  • What math curricula have you adopted?
  • What is your approach to students who failed previous math courses?
  • What would my schedule look like? (Prep time/number of courses/number of sections/length of classes)
  • What is [math] PD like here? What is the school’s PD priority?
  • How do important decisions get made?
  • Tell me more about the parent community.
  • What kind of technology is available for teacher use? For student use? How reliable is it?
  • Do you believe that all students can meet the standards?
  • What is the school struggling with right now? What is it excelling at?
  • What is discipline like at the school?
  • How is lateness/attendance? What policies are in place to handle it?
  • How much autonomy do I have regarding lesson plans?
  • What’s one thing you would change about the school?
  • What do you love about working here?

My math coach gave me this idea as we were planning my Circles unit. I think it went fairly well, so I’ll share it here. The idea is that we have, essentially, three basic objects that we’ve combined in different ways in geometry: circles, lines (including segments), and angles. So, as an opening activity to the unit, the task was this:

“Think of as many ways as possible to combine those three objects.”

First they brainstormed individually, as I reminded them that they can use multiple lines or angles or circles if they wanted. Then they went up to groups and made a master list per pair or group, eliminating ones that were “pretty  much” the same. I gave them some vocabulary based on what I saw they drew, and they had to use that vocabulary to describe what each drawing had. Finally, they chose one example and created one neat, fully correct example, in color that we combined into class posters. (I approved what they chose, to ensure a variety of possible layouts.)

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Between my two classes, they came up with almost every scenario I could think of that we would learn in the unit, with the exception of Tangent Line & Radius, which I drew and put in myself. Now they are hanging in the classroom, acting as a guide for our journey into circles.

Natural Circle Measures

Yesterday I introduce radians to my students for the first time. I started out by asking why they thought a circle had 360°. There were a few good answers – four right angles makes a circle, so 4*90 is 360; a degree is some object they measured in ancient Greece, and so a circle was made of 360 of them; something to do with the number of days in the year. All good answered, but I told them it was completely arbitrary based on the Babylonian number system.

Once we decided that it was arbitrary, I asked them to come up with their own method of measuring a circle. I would classify their responses into three categories

  1. Divide the circle up into 200 “degrees” (most common)
  2. Divide the circle up into 100 “degrees”
  3. Divide the circle up into 2 “degrees” (least common)

I was expecting 100 “degrees” to be the most common, so I was very surprised to see that most of the students want to split the triangle into two sections, each with 100 parts.

I have been a proponent of tau for a while, as I thought it was natural to think of radians as pieces of a whole circle, but my students were clearly thinking of the circle as two semicircles right off the bat.

I pushed the students who came up with the third way in a whole class discussion. If this whole semicircle is one student-name-degree, what would you call this section? And so we got to using fractions of those degrees.

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That made a pretty easy transition into radians. I went a little into the history; instead of using a degree, some mathematicians decided to use names based on the arc length – and so that semicircle’s angle was 1 π radians, instead of 1 student-name-degree. And the fractions we used were the same.

This almost made me doubt my tau ways – maybe π was more natural. But then, as we started converting angles from degrees to radians, some students kept complaining that, for example, 90° was 1/2 π instead of 1/4, since it was clearly a quarter-circle – so maybe I can stay a tau-ist.

I was listening to the episode of Wait Wait…Don’t Tell Me! yesterday with Neil deGrasse Tyson yesterday (I’m almost as behind on podcasts as I am on blog reading), and I noticed the following exchange between Peter Sagal and Tyson after he loses the game.

SAGAL: I have to say, this is a strange moment because, as listeners know, I usually like people to win and I often give hints, but I was so pleased by the idea of fooling you that – Neil DeGrasse Tyson – so, like, by doing this, I therefore, by the rule of succession, become the smartest person in the world.

(LAUGHTER)

SAGAL: I have slain the dragon – you know, it’s like…

TYSON: So I look – I look at it differently. I look at had I gotten all three right…

SAGAL: Yeah.

TYSON: …I would’ve learned nothing. But having gotten two wrong, I learned two things today.

JOBRANI: Wow.

SAGAL: There you go.

KURTIS: There’s a lesson.

So (as to not assume across the board – just in the particular example), not only does Tyson demonstrate a growth mindset, as is obvious, but we see Sagal talking about the typical fixed mindset – wanting to show his superiority and prove his “worth,” so to say. It was a pretty clear constrast. And if you’re looking for growth mindset role models, it’s hard to do better than Neil deGrasse Tyson.

Angle Chasing

On Friday our school was supposed to have a Quality Review, but it was canceled at the last minute. (That’s a whole ‘nother story.) But that pushed me to do a lesson that I probably wouldn’t’ve done otherwise, so that’s good. I actually think it went pretty well.

I noticed in our last exam that I should probably explicitly teach angle chasing as a problem solving strategy, so I asked the MTBoS for some good problems. Justin Lanier came through in the most wonderful way. So I picked out some problems into a nice sequence that would use a bunch of the theorems we’ve already done.

I wanted the students to work as a group up on the whiteboards, so I gave each person in each group a different color marker. I then had the students write a key in the corner. Each student’s color represented 1-3 of the theorems that they would have to use to solve the problems. Then they would draw up the diagram of the problem. As they went through, each person was only allowed to write when their theorem was used to deduce the measure of the angle. That way, with the colors, I could actually trace through the thought processes they used to solve the problem, which was really nice. (I wonder if I can use that as an assessment some how, having students trace through the same process. Maybe as a warm-up, once I get my smartboard working again.)

Here’s some pics of their great work.

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The Silent Treatment

Sometimes I just have one of those classes. (Well, we all do.) The behavior’s not been that bad, really, not at first. But it slowly slips away from me. I need to do something to get things back on track, because none of the other little course corrections I’ve been making have been working. So I turned to something I’ve done less than a handful of times before – I gave the whole class the silent treatment.

I first did it my first year of teaching, out of actual despair at how I felt I was being treated. That day, with that class, I wrote them a letter explaining how I felt and what I was doing and projected it onto the board. They felt bad, but were only marginally better. The next day it continued, they realized I meant it, and it got better.

Now I don’t really take it to heart, but I still think it is an effective thing to do. When students talk over for you, most often they take it for granted that someone else heard and can explain it, or that I’ll come over and explain it to them personally, or various other reasons. Those all come to relief when I stop talking.

They came in today and I handed out their cards for their seats, but no high fives today, which was the first omen. Then I went around and serenely place a written task in front of each of them. One student, at this point, says “Why are you so calm?!? It’s making me mad!” That was unexpected.

I got through the rest of the class with a mixture of gestures, pointing, and writing on the desks. Often a student would ask me the same question another student already did, so I would point them to what I wrote on the other student’s desk. Some of the students tried to take charge and guide the class through getting on task, but only with moderate success. Many students begged me to talk to them. Then, at the end of the period, I verbally wished them a good day, which they all took with a breath of relief.

One thing that sticks with me, though, was how this made it clear that I talk way too much in class. And I didn’t even think I talked that much! But left without my guiding words, students had to struggle with the task on their own, knowing that I would be of only limited help. It made me realize that maybe I’ve been too quick to help recently, and I need to pare it back (though my students would certainly argue the opposite). But I took that feeling to heart and, in the subsequent class, I decided I would still keep my talking to a minimum (though I did talk occasionally).

So maybe it’s something to keep in mind, even without the classroom management angle – when my words were few, each one had more meaning.

Quadrilateral Congruence

Stressful as it is, I am loving teaching new courses. When I first start teaching, I felt like I was learning new stuff all the time, stuff about algebra (and how it connects to other courses) that I didn’t know I didn’t know, and now it keeps happening with geometry, especially with the more transformational tinge CC geometry has.

One of the things that struck me was, last week, when I used this Illustrative Mathematics task as a follow-up to my lesson about the diagonals of quadrilaterals. I feel like the understanding I had internalized that you can prove triangles congruent with less information because they are rigid structures, but quadrilaterals are not, so there are no quadrilateral congruence theorems. But I realized that’s not true.

Last time, we constructed all of the special quadrilaterals by taking a triangle and applying a rigid motion transformation. That meant that every special quadrilateral can be split into two congruent triangles. Therefore, if you had enough information to prove one pair of triangles is congruent, you could prove the whole quadrilaterals are congruent.

Parallelogram SSSS

So if we’re looking at SSSS in terms of the triangles, we really only know two sides of the triangles. Since that’s not information to prove the triangles congruent, then it’s not enough for the parallelograms. But SAS is enough for the triangles, so it’s enough for the parallelograms.

Isosceles Trapezoid SSA

Here’s a non-parallelogram example. Here are two isosceles trapezoids with the same diagonals, same legs, and the same angle between the diagonals and one of the bases, but the trapezoids are not congruent. But that’s because, when you look at the triangles, we have Angle-Side-Side, which we all know is not a congruence theorem. If, instead, we had had SSS (a leg, a base, and a diagonal), then they would be congruent.

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