Tag Archives: student thinking

Summertime: Recharging

School’s been out for over 3 weeks now, and I’m starting to feel like thinking about it again. :) So I’m ready to post now.

Next year will be different for me. Two of my classes will be middle school courses, which will be new to me, and one of the high school courses will change from trig/analyt to precalculus, so not new material exactly but a different emphasis and course layout. Geometry and calculus will remain in my schedule (with a new text for geometry), and statistics will not.

Also, I’m expecting baby #2 in late December or early January, so I’ll basically be taking the 3rd quarter off. That means preparing for a long-term sub. I’m a bit worried about the SBG aspect of things there.

Speaking of SBG, here are some thoughts I’ve had relating to assessment…

  • I need to get better about spiraling feedback for all students on all topics. I ended up with a “keep trying until you get it right, but then it’s okay to forget it” attitude among my students, and that’s not my goal.
  • I’m going to be using ActiveGrade this year, which I think will be great for my students. My school is going to online grade reporting as well (I know, we’re waaaay behind the times there!), but I think I can reasonably just keep the ActiveGrade updated and then enter scores in the official gradebook when I need to for report cards. ActiveGrade will give the kids (and their parents!) access to so much more information about their scores than a traditional gradebook.
  • A big problem I had last year was that as midterms or end-of-quarter approached, I would have large numbers of students in my classroom wanting to reassess. The rest of the time, not so much. It was a last-minute effort for most of the kids. (There were exceptions, naturally – this is just the general pattern.) I would also have kids who would say, “I want to reassess these three learning targets tomorrow. Can you tell me what they were over?” It was really, really painful to hear that, because it showed me clearly that it was still all about the grades. The kids were looking at their scores, finding the ones that were the lowest, and not even going back to their marked-up work to figure out what those were about, just asking me to tell them. Then they’d come in and, surprise!, not do well on the reassessment. Lather, rinse, repeat. I am working on how to remedy this, though I do think the previous two bullet points will help, but I also need to implement a better process for allowing reassessments, along the lines of Sam’s form email.
  • In the fourth quarter, I stopped keeping track of student homework completion. This was not a good idea. While I still think homework should not factor into the students’ grades, I also think that knowing they have to report what they’ve done is good accountability. And I think I need to start taking up homework for feedback purposes at least some of the time.
  • My students need to get better at knowing how to help themselves understand concepts. With some of them, I don’t know if they never bother to read my comments, or if they read them and think “okay, I’ve got it now” when they really don’t, or if they read them and think “I still have no clue what she’s talking about, or what to do from here.” I need to help them learn how to evaluate their own understanding better.

Other thoughts, not related to SBG…

  • I started out last year with some great, fun activities for the students, plenty of active uses of the mathematics they were learning. Then I got bogged down and into a rut of mostly class discussions. I want to get back to the way I started out. Discussions are great things, but not when they’re the only thing happening.
  • I want to continue working on incorporating the history of mathematics into my teaching.
  • I had a “math fair” as part of my students’ final exam grades for the second semester last year. I believe it was a worthwhile endeavor, and I plan to do it again, though it will not be part of the exam grade in the future. However, with my maternity leave planned for the 3rd quarter this year, I’m not sure if it’s something that can happen, as I don’t want to place any part of that burden on a sub. So it may be 2012-2013 before I do it again.

Seems like I had more things rolling around in my head than that, but some of them must have escaped. ;) Anyway, those are things I’m thinking about with regard to next year. Hopefully, I’ll be back with further thoughts on how I plan to improve as the summer goes on.

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Warm-up Follow-up

We’re in week 2 of doing daily warm-ups like I mentioned in my last post. One class had 4 days of them last week, and the other four classes had 3 (we spent one day going over the exam and one day reviewing what we’d been doing just before the break). What I told the kids is that my New Year’s resolution is to give them better and more frequent feedback, so that they don’t end up surprised on the quiz when they don’t know something as well as they thought. I also told them that I want them to improve their ability to self-assess, and the way to do that is by practicing it.

Here’s an example of what they look like, the final version (split into two images – they just have blank space in the middle to work the problem):

I have it set up so that I’ve got a Word document mail-merging an Excel file where I enter the variable information. (I <3 mail merge.) The only problem I have so far is that I can't enter both text and an equation into the same cell in Excel, so I sometimes have to write an equation on the worksheet before making copies. The same thing would apply if I wanted to add a figure. But that's not a huge deal.

I feel like it's working well. I've been able to get the papers marked from the previous day with no problem (I know that's just an issue of discipline, but marking papers quickly is something I have struggled with in the past). I am seeing where the kids are making errors and addressing them sooner rather than later. I get to see how well they feel like they're doing, and they've naturally started adding comments by the faces to explain why they're confused or don't feel completely confident with something. That is GREAT, because they are thinking about their thinking, which is a skill they need to continue practicing.

Today's warm-up says this:

We’ve been doing these warm-ups for a few days now. Do you feel like they are helping you understand the concepts more clearly? Why or why not?


Their feedback is almost entirely positive, and they’ve made some thoughtful comments that are helping me consider how I can improve things. If you’d like to see their responses, I’ve typed them up here (yes, I have small classes, though I was missing 3 kids in 1st period).

Unexpected Misconceptions

My trig students are now learning about the graphs of trig functions. (Yes, I’m behind.) I started off by talking about periodicity, and then I talked about how the values of sine and cosine repeat once we get all the way around the unit circle and start going around again. I then presented the idea of f(x)=sin(x) and started getting them to tell me what values I would plot.

They were confused. I figured out two major reasons for their confusion. The first one, I think I’ve addressed, but the second is, I think, still causing befuddlement.

Reason #1: I put a graph up there and used crazy things like pi/2, pi, 3pi/2, and 2pi as my notch marks on the x-axis. They’re used to integers there, and to think that I could divide my axis so that each notch represented pi/2…well, that was just insane to them. I think it relates to the way they don’t really like to think of pi as a constant, since they can’t express its value exactly in any way other than a weird Greek symbol. But I think I dealt with this one okay, though I might mention it again today just to make sure they caught on.

Reason #2: This is best expressed by giving a quote from a student.

“Why are you saying ‘sin x’ when sine is y?”

The student was confused because, on the unit circle, she had memorized that the y-value is the sine of the angle (which we’ve been calling theta). All that relating of the values on the unit circle to right triangles? Yeah, that didn’t work for her. She just memorized that “sine is y.” I tried to show her that we had been saying that sin(theta)=y on the unit circle, and that now instead of theta, x is the angle of which we’re taking the sine for our function…but I don’t think it clicked for her. Well, I thought it had, but then when I said that we could also do a function f(x)=cos(x), she said, “But then cos(x) would be y, and I thought sine was y?”

So I’m still working on that one. But I’m making myself notes, so that next year I’ll be able to anticipate these misconceptions and hopefully prevent them by changing my earlier instruction. I’m sure new ones will pop up to surprise me then. ;)

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.)

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.

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.

Quilt problem: My solution

I finally got my solution for yesterday’s quilt problem typed up and published as a pdf. It’s here. I used both composition and multiplication of functions. Would you do it differently?

I had to go and ask another teacher a question right before that class came in today, so they were already in the room when I got back. And they were at the board, putting up what they’d done on the problem last night, or looking over each other’s work. I hadn’t instructed them to do that; they just did. YAY! :)

I think they probably could have continued working on it all period today, but I did have other things to teach them as well. So first I let them share their thoughts, but I gave more telling feedback. “Hmm…I think I’d run out of fabric if I only got that much. Can you figure out why?” (That kid was very close; he just didn’t think about how I need to cut rectangles of a particular size, so I can’t just divide the area of all the rectangles by the width of the fabric – in other words, the necessity of the greatest integer function for this problem.) “So for a quilt that’ll only be 48 by 64 inches–” (she was giving me an example of a specific block size rather than a function for any block size) “–you want me to get thirty-four YARDS of fabric? It’ll probably cost $3.99 a yard…do I need to spend THAT much money?” (I think she was trying to make me cut one super-long strip of fabric, leaving about 37.5 inches of the 40-inch fabric width untouched.)

So I took a few of their ideas, and really, most of them were focusing on figuring it out for a particular size block. One pair of girls who tried to make it a function didn’t recognize that the width and the length of a fabric strip in a block can both be expressed in terms of the same variable, so they were working with x and y. But you know what? When I was organizing my thinking to start off, *I* was going through a specific example in my mind. So I don’t think there’s a problem with using a specific example to help orient yourself to a problem.

I think the key is that in my mind, I always knew that I’d identified x and was working to figure out what I wanted to do to it by rehearsing what I do when x=6. I’m not sure whether they did that or not; in fact, I’d be willing to bet that most of them were just planning to figure out what x should be after they’d solved the problem with their particular example, or else that they forgot they were supposed to be looking for a function and would just consider themselves done when they reached the solution for their example. So the next time I use this activity, I need to make that point more clear at the outset, and I need to emphasize it over and over while they’re working as well.

I just realized that I’m going to be late for our church supper tonight if I don’t get out of here, but I think that was pretty much what I wanted to say. Oh, and after letting them share, I walked them through my solution, asking them questions to get them to come up with the functions I had. I don’t know if any of them drew out the fabric with rectangles cut out of it…importance of drawing a picture to help you solve something!

A great day in trig

My trig students are reviewing some algebra concepts before we jump into trigonometry, and the most recent thing was composition of functions. So today, I put this picture up on the screen:

And I asked them to find a set of functions, the composition of which would show how much fabric I need to buy of each color. I told them to work with one another, and that I would answer any questions they have about quilting or fabric.

At first, they were just sitting there at their desks, making uncertain pencil marks on their papers. I said, “Work together. Talk to someone near you. Get up and talk to someone far away from you. Use one another to figure it out.”

Within a couple of minutes, they were all sitting on the floor at the front of the room “so we can powwow.” One girl didn’t want to sit on the floor, so she started writing on the board. Most of them ended up getting up and crowding around the board (or the pull-down graph, or the other board); a few of them continued to huddle on the floor and work on paper. But they were all engaged and trying to figure this thing out.

They first asked me how many blocks there are in the quilt, and I pointed to the image and counted. They asked me how big the blocks were. I said that for the one I made in the picture, I used blocks that were 6″ squares, but that the block size was what needed to vary. I did have to point out to them that quilting uses 1/4″ seam allowances, but one student who does some sewing knew what I meant and explained once I said the phrase, catching on immediately to why that was important.

They asked if they could just write a function of how much it would cost to buy a quilt from Wal-Mart instead. I denied that request.

There was a lot of argument over whether the size of the block or the size of the strips within a block was the most important. Once they remembered that they were looking at how much fabric per color to buy, they focused in on the strips instead of the block.

Then they pretty much all got hung up on the area of the block.

“If it’s gonna be six inches finished,” they told me, “then the strips will be six-and-a-half by two-and-a-half inches –”

Wait, I said. Where’d you get two and a half?

“We added the quarter inch seam allowance four times, once for each side.”

Didn’t you already give it a seam allowance on the top and bottom? Why are you using those quarter inches again? (This part actually caused more confusion than this summary indicates.)

“Oh! Right, so they’re six-and-a-half by two inches.”

Okay, got it.

“So then, 6.5 times 2 is…thirteen, and then we need 48 of those, so we need 624 inches.”

So…you multiplied a length times a width times a count, and you ended up with a length?

“Ugh! I knew there was something about that that didn’t quite work! (to friend) Come on, let’s try again.”

At one point a group asked me how long a bolt of fabric was. Hiding my surprise that a 16-year-old boy knew that fabric comes on bolts, I clarified, “Do you want to know how long or how wide?” He changed the question to how wide, and I told him, normally 40 to 44 inches, and I usually use 40 when I’m calculating how much to get. A girl drew on the board to show a classmate how the fabric is wrapped around the bolt and where the 40 would be. A boy asked if I would want to buy any extra fabric; I said that normally, yes, I would, but for the sake of our problem they could assume that I was buying exactly what I needed. (The amount of extra I get varies, so it’s not like a constant they could add.)

They weren’t done at the end of class. I told them their homework is to continue to struggle with it – it’s okay if they don’t come up with an answer, but I want them to work on it some more. One girl said, “But my work is on the board!” So I let her use her phone to take pictures of her work, and the rest of the class asked her to send the pictures to them.

All of them were engaged. All of them were thinking and working to figure out what information was important and what wasn’t. It was awesome, and I hope I can come up with more problems like it. :)

Day One, Sort Of

Today was the first day of school, but our high schoolers (it’s a K-12 school) don’t have regular classes for the first two days, so it was a little odd. They have Orientation, which is basically a series of seminars. I did lead two math seminars today, one for the juniors & seniors and one for the freshmen & sophomores.

I had a little more than an hour and a half with each group, which was way too long; I’m going to request that the times be reduced for next year. I split the session into two parts.

The first thing we did was based on George Woodbury’s post on doing a study skills inventory. I did a lot of asking, “How do you do that?” For example, a few kids listed “organization” as something that’s characteristic of a successful math student. We talked about what it means to be organized as a math student, and how one can make that happen. It was a pretty good discussion, but I think a lot of the kids were zoning out. That may have been related to their schedule for the day (lots of seminars, as I said), but I think even so I should work on making sure ALL students take part if I do this again in the future.

For the second part of the seminar, I took an idea from @Mythagon and decided to have them investigate spirolaterals. I chose this because it’s accessible to all of them, regardless of math course level, and because it was something I could do to get them thinking mathematically, looking for patterns and using mathematical terms to describe what they saw. Here is the worksheet I developed. (Buddy the Bunny is one of the stuffed animals who lives in my classroom, just as an fyi.) The kids really got into doing the spirolaterals, and they were engaged and working hard to find the patterns.

With the younger group, I didn’t get into questions 4 and 5, except to point them to this website where you can make those changes and generate more spirolaterals. But the students in that group were asking fabulous questions as they tried to articulate the rules they were developing about the kinds of patterns they were seeing – a lot of “what ifs” came from them. It was really awesome to have them so into what we were looking at, and it was great to say, “That’s a great question. Here’s another sheet of graph paper – why don’t you try to figure it out?” We didn’t have time during class to explore whether palindromes in spirolaterals make any particularly cool patterns, but I think it will be something to investigate!

Tomorrow I don’t have any classes, and then Wednesday will begin the real deal. I’m excited. :)