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.

Parent Responses to SBG

We had Parent Night last night. The parents get a copy of their child’s schedule and go through it, spending about 10 minutes in each class.

I spent the majority of the time talking about my grading system. I’d sent home a description of it at the start of the year, but as we all know, those handouts don’t usually get a careful reading from parents. Plus, it’s much easier to explain something like that verbally; the written version is an overview.

So I told them that I’m using standards-based grading, and I explained what that means in my classroom – I told them that scores are based on Learning Targets, showed them my 4-point scoring scale, talked about reassessments, and showed them a sample gradebook with made-up students and scores.

They LOVED it, across the board. I got reactions and comments like these:

  • “I wish my teachers had graded like this!”
  • “I love that the focus is on making sure they learn!”
  • “So anybody can get an A in the class, as long as they’re tenacious about learning.”
  • “So when you give them a reassessment, you can show them where they have weaknesses so that they KNOW what they need to work on in order to understand it.”
  • (after I pointed out that my goal is for them to learn, even if it takes them longer than someone else) “Hallelujah!” :)

Some of the parents already had an idea of how my grading system works, and some didn’t but said they would be talking with their kids about coming in for help and then scheduling reassessments.

I’m not sure what I was expecting, but the response really blew me away. I mean, *I* think SBG is a fabulous idea with a philosophy that makes sense, but I’ve been reading about it for months and practicing it for several weeks now. For some of these parents, this was really their first time to hear anything about it. So the fact that they were so completely on board with it after my brief spiel was pretty amazing to me…and makes me think I did all right explaining it. :)

Water Tower Exploration

There’s a water tower right next to the building I teach in. Naturally, I had my trig students figure out how tall it is.

They were lying on the ground and measuring angles.

They were borrowing tools from other teachers (the science teacher has something she uses to see how high her 7th graders’ rockets go; the PE coach has a long tape measure for when he makes lines on the field).

They were mad that I wouldn’t let them climb the fence so they could get to the base of the water tower. (Dude, that’s not our property!)

They were making estimates before they took measurements.

They were recognizing when an answer they came up with wasn’t reasonable, working to figure out what went wrong, then trying again to correct it.

They were enjoying the nice weather.

They were noting that the ground isn’t completely level and trying to compensate for that in their measurements.

They were drawing pictures to represent their work.

After they’d gathered measurements and performed their calculations, I let a student call the city to find out the actual height of the water tower. Most of the students were within 8 feet of the right answer (which was 216 ft). One group was way off, but they realized that they hadn’t done a good job of determining the angles of elevation, so we got to see how much accuracy matters.

I love doing things like this. It seems to me like the students really feel like they own the mathematics when they tackle a problem like this and reach a solution.

However, I’m thinking about Dan Meyer’s recent post on pseudocontext. If we can just call the city to find out how tall the water tower is, what’s the point?

Well, it’s fun. It’s a chance to go outside. I think it’s significantly more engaging than the example in the textbook where you have to figure out how long the rope is that’s holding the tent up (the example is labeled as “Real World Application: Entertainment” – really? entertainment, because it’s a tent? yeesh).

But is the water tower activity flawed because there was an easier way to get the answer? My gut tells me no, but I’m still working on why.

Cool Stuff!

E was confused about marginal distribution on her quiz. She came in for a reassessment a couple of days ago…and was still confused, so her score didn’t change. I explained the concept to her again, and she seemed to get it.

We have homeroom at the end of the day. As soon as homeroom was over today, E came running into my classroom.

“Mrs. Dean!” she said. “I made a contingency table in homeroom!”

I said, “Great! Is it on the board in Mr. C’s room, then?”

“Yes – come see!” I was supposed to be going to a meeting, but I figured it could wait a minute or two, so I walked down to Mr. C’s room with her. As we walked, she continued: “Mr. C took a poll – somebody had this toy thing, and we were trying to decide whether it’s an evil fairy or an alien. So I said that we could break down the results by girls and boys. And I got the marginal distribution part and everything!”

We got to the classroom where, sure enough, she’d drawn this:

She pointed out that while there were some votes for alien, ALL of the girls voted for evil fairy. She also pointed out the marginal distribution that she’d written at the bottom. I asked her what percent of the people who voted for evil fairy were boys, and although she couldn’t calculate the percentage in her head, she knew that it was 2/12.

So I changed her score. Because she knows it, and I know she knows it. This wasn’t a scheduled reassessment that I generated for her; she saw an opportunity to use what she’d learned, and then she drew my attention to it because she knew it was a demonstration of her understanding. And that? Is awesome.

Edited because apparently writing a post quickly makes me leave verbs out of my sentences…sorry ’bout that.

Wait, what?

I hate it when kids seem to really get something, it works great for them, they can do the problems…and then they ask me a question that shows me they missed the point of the whole thing.

My calc kids are taking a quiz over the precalc review stuff. One of the questions asks them to find a natural logarithm regression equation for a set of data. Should be no problem – they’ve been doing great with that.

But one of them just came up and said, “I found the equation, but when I look back at the x-values from the data we were given, it doesn’t have the right y-values.”

Now, I know it’s not the stats class, but still, I didn’t realize that they didn’t know what a regression equation is all about. I’m glad to realize it now, but I hate that I was just having them find regression equations without understanding what they were doing. Sigh.

So, what to do about it? I think I need to be more careful, more deliberate, about making sure they understand concepts that I think should be prior knowledge for them. I need to stop assuming that they know something because they can execute an algorithm; that doesn’t help them learn, and it will end up causing me frustration down the line when I want them to build on a concept they never had to start with.

Two Weeks In: SBG Thoughts

I’m liking this whole SBG thing. Here are a couple of reasons why I like how it’s working so far.

1. I feel like I can expect clarity from my students.
When grading in the past, I often found myself interpreting students’ answers. “Well,” I would say to myself when reading a response that could have been clearer, “he’s saying this, but I’m pretty sure he means this, so I’ll give him credit, or only take off one point.” That was something I didn’t like about myself as a teacher…but at the same time, if the question was worth 5 points, marking it completely wrong would hurt their grade. What if they really did get it, and they just didn’t make that clear? Not good for me to take points away, necessarily. But on the other side, what if I decided they got it, but they really didn’t? I hated those times when a kid would say, “Really? I got some points on this question? How’d THAT happen? I just put down a random guess!”

Now, though, I don’t have to worry about the points. A geometry student wrote down that the pattern for a sequence of numbers was “divide the number and its quotient by two, then two again.” I was pretty sure he understood that the pattern was to divide a term by two in order to get the next term, but that wasn’t quite what he communicated. So I gave him a score of 3 on that Learning Target (there were other aspects of the problem wherein he demonstrated better understanding, but he wasn’t all the way there). He came back for a reassessment and showed me that he did understand it clearly. He wasn’t stuck with a bad grade, and I wasn’t stuck having to guess whether he got it or not. Taking the focus away from the points lets me demand excellence in their communication skills, and so far, they’re rising to the occasion.

2. I can see clear relationships between students’ homework effort and their understanding.
I am keeping track of homework completion as a gradebook category that I’ve set to 0%, and I just mark each assignment as 0 (less than 25% done), 1 (25-75% done), or 2 (more than 75% done). With the scores on quizzes separated into Learning Targets, it is so clear that there’s a connection between doing the homework and understanding the material. I love how this system lets me see that the kids who scored low on Concept X are the same kids who got 0’s or 1’s on the homework for Concept X. Much clearer than a score on Test #3.

3. Students are taking charge of their grades.
They aren’t asking for extra credit or how they can bring their grade up. They’re coming to me and saying, “I want to have a reassessment for Learning Target 2, the one about domain and range.” Some of my high achievers are shocked to realize that they got a 75% on something (if they scored 3 on my 4-point scale), but I just say, “You know what to do if you’re not satisfied with that,” and they say, “Right. Reassess. Can I come in at lunch on Tuesday?”

—–

So, yeah, I think the SBG kool-aid tastes better and better. Some things I need to work on:

A. Broader Learning Targets.
I knew when I was writing them that I was probably focusing too narrowly, but now that I’m walking it out I’m seeing how I can make LTs that are more broadly defined. This may be something I just make notes to myself about and then change next year, since I already made up all the LTs for the year. I have decided at assessment time to skip a Learning Target here and there, but I think the overall restructuring is something I’ll just do next year.

B. More frequent assessments.
I have never been good at remembering to give frequent quizzes. I’ll plan them, and then forget to announce them, and then it ends up being too close to the test over the whole unit, so…yeah. With SBG, though, I really want to give more frequent but shorter assessments. The biggest thing really is remembering to announce it to the students. I don’t know why that’s always been such a challenge for me. Right now it isn’t helped by the fact that I’m still figuring out how to pace things with math.

Those aren’t the only things I need to work on, but they’re the most glaring in my mind right now.