Monthly Archives: October 2010

Math Department Identity

Our principal has challenged us (me and the other two math teachers) to articulate who we are as a math department. I’m trying to consider what that means in order to work on answering it. So far, here are the questions I think we need to address:

  1. What are our overall mathematical goals for our students?
  2. In what ways does our department work toward the goals of the school as a whole?
  3. What is distinctive about our math program?
  4. What is our perspective on the nature of mathematics, and how does this perspective influence our instruction?
  5. What is our perspective on the purpose of mathematics, and how does this perspective influence our instruction?

What other questions should I add to the list?

Have you gone through an exercise like this in your own department? Any additional thoughts?

Thesis Defense

I defended my master’s thesis this afternoon, and…I passed! And there was much rejoicing! And a little editing to be done to the paper before the final sign-off from my advisor, but that’s to be expected. My committee had great questions and suggestions for my work, and it was really a cool experience (now that I don’t have the “what if they hate it” worry).

So I’ll make my edits and graduate in December (on 12/11/10, to be exact), with a master of science in mathematics education. Yay! :)

For anyone who’s interested, the title of my thesis is “Identification and analysis of pedagogical techniques in Descartes’ La Géométrie.”

“Could we have an oral quiz?”

What would you do (or have you done) if your students asked you that?

I said sure, and tried to figure out a way to make it happen.

This was my 1st period geometry class, but I decided to use the same method for 5th period as well. The unit was on angles, and to be honest, I was feeling very very bored with the whole chapter. I understand the necessity of having students grasp the basic vocabulary and concepts, but it’s so tedious to have to go through it.

So for their quiz, I had the students do an angle scavenger hunt. They worked with a partner, and each pair had a list of terms (same for everyone) and a picture of a quilt block (varied by group).* They had to label points on their pictures and then identify on the list what examples they’d found & labeled for each item. The items on the list were things like “pair of adjacent angles,” “segment that bisects an angle,” etc.

When they’d finished labeling these things with their partners, I called them up individually and asked them questions. “Okay, you wrote that angle ABC and angle CBD are complementary. What do you mean by that? How do you know they’re complementary? You drew them so that they share a side – do they have to share a side in order to be complementary?” And so on. I had my list of Learning Targets beside me and was able to mark their scores as they showed me their examples and explained them to me.

The Good:

  1. I was able to see and correct errors in understanding much more immediately than when students take completely written assessments.
  2. I was able to see some really awesome things about my students’ thinking – one student’s spatial reasoning is really strong, and I know he wouldn’t have written out all the words he said to me about how “if you flip the angle over like this and then slide it over here, it will fit exactly on top of the other one.” Another student tends to rush through written work but was taking his time and thinking carefully so he could communicate using extremely precise language.
  3. I had worried that the working together part might result in a poor measure of individual students’ understandings, but it didn’t – the kids that had different levels of understanding still demonstrated that when they spoke with me one on one, even if they had the exact same things written on their papers.
  4. The “scavenger hunt” aspect made it a little more interesting, at least in my opinion. Don’t know if the kids agree or not – I need to ask them.

The Bad:

  1. This is a classroom management nightmare. Once the kids were done with their own written work, they had to sit around and wait for their turn to talk to me (or wait after they talked to me). Not a productive use of their time, and I need to come up with something else for them to do while they wait if I’m going to use this idea again. It was poor planning.
  2. It took a long time. A traditional assessment would have taken one class period. This took two, and in 1st period where I have kids who take longer to think through things, I still need to talk to a couple more kids. Combined with the previous note especially, I need to work on this. At the same time, though, I don’t want to skip any questions, because I need to make sure they truly have understanding on all the Learning Targets.

I’m sure there were more drawbacks, but those were the really big ones. The kids felt nervous about talking to me individually and getting a grade for it, but hey, I’m defending my thesis on Wednesday and am right there with them on the nervousness thing. Doing this more often would help them feel more confident in their ability to communicate their understanding verbally, I think.

Anyway, just reviewing and reflecting on how this little experiment went. I’m not going to do something like this for every unit, but I may well try something like it again later in the year and/or with other classes.

* The pictures I used were this, this, and this, all of which came from this quilt. Hey, if you have a hobby with mathematical tie-ins, use it, right?

Apples and Oranges

We compared apples and oranges in statistics today.

The teacher’s resource manual for the textbook comments that standard deviation lets us compare apples and oranges, because we can use it as a “ruler” to figure out how a particular data value compares with the mean. I took that idea and ran with it – to the grocery store, during my 4th period prep. I got 9 apples and a bag of clementines (there were 20 of them in the bag).

We put each piece of fruit on a paper plate with a letter on it (we had to use three Greek letters). We borrowed a balance from a science teacher. Before weighing the fruit, the class made conjectures about which apples would be biggest and smallest, and which oranges would be biggest and smallest. (They did know that our method of measurement would be using the balance.) I told them that in making those decisions, they were comparing apples with apples and oranges with oranges, but what if they wanted to determine which was relatively bigger, the biggest apple or the biggest orange? That, I said, was what they were going to figure out how to do today.

After they made their conjectures, they wrote the name of each piece of fruit on the board. (They decided the fruits needed names, so instead of boring old A, B, and C, we had Amy, Billy, and Caroline as our first three apples. I’ve studied Greek, so I was able to come up with some Greek names for those extra letters.) Then they recorded the mass of each piece of fruit.

The next part was easy: For each type of fruit, which piece was the biggest, and which the smallest? It turned out that the biggest apple was Francis (212.1 g), which had been their guess, but that was the only one they guessed correctly. Caroline was the smallest apple (144.4 g), Wilhelm the biggest orange (84.6 g), and Violet the smallest orange (52.1 g).

Then I told them to figure out the mean and standard deviation for each type of fruit. Doing this REALLY helped some of the students to better understand what standard deviation is (a measure of spread); I was able to point out that the standard deviation of apple masses (19.72) was more than twice that of orange masses (8.26), and the students were able to look at the actual fruits and see that yeah, there’s a lot more variation in size in the apples than in the oranges. They had been pretty confused by it before, so I was really glad to have that visual for them.

Once we had that concept a little more firmly understood, I said that we should look at Francis (biggest apple) and Wilhelm (biggest orange). I asked how we could figure out how many standard deviations above the mean Francis is, and they knew right away how we could do it. We did the same for Wilhelm, and we discovered that Francis was 0.981 standard deviations above, but Wilhelm was 1.746 standard deviations above. So even though Wilhelm is SMALLER than Francis when we look at the measurements themselves, Wilhelm is bigger as a big orange than Francis is as a big apple. We did the same thing with the smallest ones, and I pointed out that since we were subtracting the mean from smaller data values, we were ending up with negative answers…which just show us that we’re that many standard deviations below the mean. They got it.

And then I told them that these “how many standard deviations away” things they were coming up with are called z-scores. They were excited that “it has a cool name.” :) And I’m excited that I think they will actually remember it because of how we got there – they figured it out rather than having me throw a formula at them.

And then? We had a healthy snack to conclude the class. :) (My homeroom may or may not have juggled the leftover clementines right after the statistics class left.)

Updated link for quilt solution

Thanks to Elizabeth S for informing me that I had a bad link in my post about my solution to the quilt problem. The correct link for my solution is this one. I’ve updated the original post with the right link as well.

Historical Intro to the Tangent Line

I mentioned when I started this blog that this is my first year teaching math, but that I spent the last seven years teaching history. So it’s probably not a surprise that I find the history of mathematics interesting, though I’ll admit I have a LOT to learn in that area. I’ve pulled a few lessons from this CD-ROM, called Historical Modules for the Teaching and Learning of Mathematics, which I highly recommend. And then I developed a lesson for my calculus students as an introduction to The Derivative.

For full disclosure, you guys are actually getting the edited version. What I gave the students was overly ambitious and overlooked the fact that ellipses and hyperbolas have more complicated derivatives than parabolas, and I was really just going for something simple to ease them into things. I made some on-the-spot changes during the presentation of the lesson, and now I’ve come back to the lesson and changed it to what I really ended up doing.

All that said, here’s the lesson.

Some things to note about the execution of it…

  • I didn’t actually say the word “derivative” until they got to the very end. Their textbook doesn’t give them the word until the next chapter, but hey, they found it! Why not tell them that’s what they were doing?
  • Your students need to be familiar with GeoGebra in order to make this thing work. Mine ended up having to use the Van Schooten worksheet to find their tangent line slopes because they couldn’t actually do what I’d asked them to – this was the first time I’d told them to do something other than graph a function on GeoGebra, and it was over their heads. At the same time, they did learn a lot about what it can do and what doesn’t work! :) I’m hoping to have next year’s group prepared by working with GeoGebra a good bit while I have them in trig/analyt this year.
  • Don’t tell your students you just ordered a book on the history of calculus unless you want them to look at you like you’re insane.
  • My students didn’t know what “analytic” meant or how it was different from “geometric.” My own understanding of geometric solutions to problems has been greatly increased as I’ve done my thesis (on Descartes’ La Geometrie), so hopefully I was able to explain it to them well. I am much more of an analytic thinker than a geometric one.

My goal in using this was to have the historical setting provide a context for studying the mathematical concept. I think it worked pretty well. In any case, staring at the board where they’d done a lot of the work together and where I’d circled an expression and written “derivative,” one student said,

You know, I really do think I get this. It looks complicated, but it’s really not that hard – it’s just slope.