Monthly Archives: August 2010

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. :)

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Going over homework?

How do you review homework with your classes? Right now I’m just having the students ask questions and then we go through solving the problems they were unsure of. (Sometimes I do it, sometimes I have another student do it, sometimes I have the student who asked do it so we can figure out where they made a mistake.)

But it feels like that takes up waaaay too much class time, and I’m running out of time for teaching whatever new material I’d planned on. So…how do you do it?

Thesis: Coming Along

Since my last post on this subject, I’ve narrowed the focus of my thesis to just one mathematician, René Descartes. My advisor and I realized that with the level of detail I’m using, doing more than one mathematician would be insane. With that change, I have now finished coding my data. That process involved pretty much retyping the whole of La Géométrie (in French* and in English) into a spreadsheet so that I could enter codes for each sentence.

So, with that done, I had the whole text in a spreadsheet, copy-and-paste-able…and I thought, hey, Wordle! So here it is.

“Expression” and “equation” are both probably a bit bigger than they should be, because I started just typing [expression] or [equation] rather than typing the whole things out, as the actual values are irrelevant for my research. However, there are a lot of expressions and equations in there, so I don’t think it’s that off to have those words enlarged.

*And can I just say, the French that I learned in school and the French that Descartes was writing? NOT the same thing. I can figure it out – the letters u and v are interchanged sometimes; i is used for j; non-final s’s look very similar to f’s; y is sometimes used for i; words that now have an accent over a vowel then had an s following the vowel; etc. But it took some adjusting, and it wasn’t until I was nearly done that I realized the adverb “desia” that I kept seeing was actually the word “déjà.” All that said…it’s very cool to see how the language has changed, and I really feel like my grasp of French has increased significantly as a result of this work.

Off to a good start

Regular classes started on Wednesday. Despite my having tried to condense my “this is what the class looks like” stuff so I could get into actual content, I only had a few minutes for math in 3 of the classes, and didn’t get to it at all in the other two. That was a bit sad for me.

The main reason for that, though, was that I went ahead and introduced standards-based grading to the kids, and it took longer than I expected. (Amazingly, despite MY having mulled this thing over for several weeks, it was totally new to them!) But they are very enthusiastic about the concept. I was reading through my description, and they were kind of staring at me glassy-eyed, and then I stopped reading and said, “That means that if I give a test on Thursday, and you don’t really get it on Thursday, but over the weekend you practice and then you come in on Monday and show me, I will change your grade to show that you really do understand it.” In every single class, that was what got their attention. I think they’re currently thinking that it’s too good to be true. :)

I do have a few kids who seem worried about my not grading their homework. I reassured them that (1) they will have the answers ahead of time, so they’ll be able to see if they’re doing it correctly, (2) we will go over problems that confused them in class the next day, and (3) I will look over any individual student’s homework and give more specific feedback if he or she wants me to do so. Also, I’m keeping track of whether they’re doing the homework or not in an “ungraded” category in my grade book, just on a 0-1-2 scale, so that if I see they’re having trouble understanding, I have a record of their practice and can say “you need to try doing your homework” or “you’re already doing your homework, so let’s figure out another strategy to help you.” With all those things in place, the kids who were worried seemed satisfied.

I told the other high school math teacher about SBG, and she is on board and using it in her classes as well (though her implementation is a little different, like she’s using a 5-point scale where I’m using a 4-point one). I’ve also talked about it with teachers in other subject areas, and while they’re not yet drinking the kool-aid, they’re talking about how tasty it looks. :) One English teacher in particular is really interested in learning more about it with the goal of possibly switching to SBG for the second semester. Do you know of any English edubloggers using SBG? If so, let me know so I can point her to them to read.

I have about a million thoughts on how my first couple of days of actual math lessons went, but I don’t have the energy to post them right now, as Little Precious was up sick all last night. So I’ll just leave you all a picture of the door to my classroom:

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. :)

Potpourri

Teachers at my school report back today – summer is officially over for me. Our kids don’t start until the 23rd, and normally we wouldn’t be back until Monday, but this year we’re having some additional professional development today through Friday. I expect it’ll probably be related to global things like our school improvement plan, but I’ll find out for sure in less than an hour.

I’m still having difficulty thinking of myself as a Math Teacher. I guess it’ll seem more real once I’m actually teaching kids math, but right now, I’ve been a History Teacher for seven years, and it just seems weird, even though it’s what I have been wanting to do. I chatted with a stranger in a fabric store yesterday, and when she asked what I teach, I said “math” and it sounded odd to have that come out of my mouth.

This next month is going to be pretty intense for me, with school starting up as I try to get my thesis completed. I had a great meeting with my advisor yesterday, and we decided that I’m going to narrow my focus to just Descartes rather than all four mathematicians. I’m feeling excited about getting the work completed, even though I know that the next few weeks are going to be a challenge to get through. (There’s also a quilt top I’ve committed to piecing soon, not to mention a husband and little girl who like to see me from time to time, and various other things that seem to pop up demanding my time. It’s that “life” thing, ya know?)

Anyway, I mostly just wanted to get a post up here. Since I’m starting back today, I’ll hopefully have more things to write about in the near future. :)

SBG Question: Keep Moving Along?

I went to the school today, met with the person who’s taking on 7th grade social studies to give her my resources, met with a student to try to determine math placement, moved some stuff from my old room to my new, told one of my principals that I’m going to be using standards-based grading.

He had some questions.

The biggest thing he was worried about was this (and this is me trying to reword the question):

What about the student who doesn’t get these first few concepts? In a context where you are allowing for reassessment on those learning targets, does it make sense to keep moving along in the curriculum?

One thing I said in response was that in my experience, later math can illuminate earlier math. Sometimes you don’t truly understand a concept until you’re a few steps down the line and you see how it fits with other concepts.

Another thing I thought about since I left his office was that this isn’t an issue particular to a grading system. You’ve got kids who get left behind conceptually in a traditionally-graded classroom, too – more so, I’d argue, because of the whole “you didn’t learn it by Thursday’s test, too bad for you” approach. I think SBG encourages kids to go back and learn, or to get help learning, much more than the traditional system.

What thoughts do you guys have on this question?