Monday, January 23, 2012

First Day of Class: Calculus II and Linear Algebra

I gave myself the gift this semester of no lower-level classes. I think I'll have more mature students, and I hope to have a great time with them.

Calculus II
I've taught this the previous two semesters, so it should be a breeze. We've switched to a new text, but I don't think that will have much effect on my teaching. I have a big class this time (36 or more), unlike the past two semesters. But putting students in groups of 4 makes it so much easier to learn their names.

We did the axes exercise (mentioned here), and I once again loved how it got them talking to one another and reminded me how weak they are on things I think of as pretty basic. (How would you label the axes?)

We also worked on the Calculus Review sheet I've used each term. Part of their homework is to make a list of 5 Calculus I problems that they can't do, and share it with their group tomorrow.

Linear Algebra
It's been over a decade since I've taught Linear, and I knew I needed lots of work on the material over the holidays to be well-prepared to teach it. That task is done.

We're using the text by David Lay (Linear Algebra and Its Applications, 4th edition), and it starts out tougher than many texts. He says, "I think that students' opinions of the course are set somewhere in the first two weeks, and they need to feel the conceptual emphasis early." (page xv, Notes to the Instructor) My colleagues have told me that our students have done especially well with this text. One thing that threw me at first was (from page 35, in section 1.4):
"If A is an mxn matrix, with columns a1, ..., an, and if x is in Rn, then the product of A and x, denoted by Ax, is the linear combination of the columns of A using the corresponding entries in x as weights; that is,
I don't think I had ever seen that before. I've been talking with Owen Thomas (aka vlorbik) about this course, and he grabbed right onto that when I mentioned it - he loved it. I think I will too, once I get used to thinking this way.

After doing the axes exercise, I gave my students a warmup sheet which was a review of what they've seen before regarding systems of equations.  Of course it wasn't the breeze I thought it would be, so we're already 'behind'. That's ok. Class went well. I had fun showing them three axes on the edges of my desk and in the air.

We did the axes exercise, and compared the different forms the equation of a line can take.

I'm tired - headed home...

Saturday, January 21, 2012

Math Adventures: Thinking about Spot it, and Learning Python

My brother lives in Minneapolis, and we visited him last January and this. Both times we used the light-rail train to get to the airport. Last year he had to drop us off pretty early, so we stopped at the Mall of America. We'd arrived too early and most of the stores were still closed, but we wandered around anyway. We found a toy/game store called Marbles - The Brain Store, and I knocked on their doors to ask if I could just come in and browse. They had a good collection of my favorite sorts of toys and games, and they ended up letting me buy a few things, Spot it among them.

It's not a math game at all. It's a contest to see who can find a matching picture first. My son is better at it than I am, so he enjoys winning. Every time I played, I pondered the math question that stared me in the face: How did they make sure every possible pair of cards would have just one match?! But I didn't know how to get started, so I never really pursued the question until recently.

Spot it is a good travel game, so I packed it this year as we headed off to visit family and friends in 4 Michigan cities, a Chicago suburb, and Minneapolis. My old friends Chris and Paul have twin girls my son's age, and our visits to their backwoods home are always a rich experience of outdoor play, good healthy food, and deep conversation. Chris and I decided one evening to look at how Spot it is put together. Chris doesn't think of herself as a mathematician, but she organized the information in a way that helped me solve my problem. It was also a great motivator to think through it together. We used numbers to represent the different pictures, and drew our cards in different layouts, looking for patterns.

We worked on it for a few hours, and got some good insights, but didn't solve it. That was enough to get me hooked. I got my mom to work on it with me. We counted all the pictures, and found 57 different pictures. Then I worked on it on the train to Chicago, and got some good stuff figured out.

One thing I did was to make a mini-version of the game. My 'cards' (circles with numbers in them), used only 4 numbers each. I figured out that I had to use 13 different pictures, and could have a deck of 13 cards - all different. That gave me enough push to find a scheme for the bigger deck. But I wasn't at all sure it was right. There are an awful lot of possible pairs when you have 55 cards. (How many?) How could I be sure I had exactly one match for every pair of cards? I thought I did, but I sure didn't know how to prove it.

That's where I was with the problem when I visited Prairie Creek Community School, about 40 minutes south of Minneapolis. Michelle Martin (who has a chapter in Playing With Math) let me show the kids (4th and 5th graders) the problem, and they all dug in. They pulled out all the cards with a heart - there were 8 - and thought about what they now knew. 8 pictures per card, one of which was the heart, along with 7 others, all different... They decided there had to be at least 57 different pictures (7 unique pictures/card*8 cards + the heart). I liked that they figured that out a better way than I had. The kids worked on the problem so diligently, but had to leave it and go on to other things. (And it didn't occur to me to leave my game there, so they couldn't keep working on it after I left.)

I went back to trying to find ways to prove I had a valid solution. My new burning question was 'Why did Spot it only have 55 cards, when it could have had 57 cards?'  I put my 'cards' on Excel. But I still didn't know how to test each card against each other card. I understand that the macros in Excel use visual Basic, and I hoped I could do something with that, but I was once again pretty stuck.

I asked a colleague to help me get started on the programming, and he recommended Python. I couldn't get started on my own. I found the Python documentation very confusing, and got stuck every step of the way. Yesterday we had a bit over an hour between meetings, and he showed me how to get started with Python. He's learning it too (and he's a computer science teacher). So the best thing he showed me (probably pretty obvious) was to google 'python example x' every time I had a question. This morning I got my program working, and it verified that my method worked!!! Maybe next year I'll generalize this to cards with n pictures on them.

The biggest lesson for me in this adventure was that working with others is a huge motivation for me. But I knew that.

[You may have noticed that I haven't given many details of my solution. I'm hoping you'll all play with it yourselves, of course. If you'd like to discuss all the gory details, please email me at mathanthologyeditor on gmail with your solution ideas.]

Friday, January 13, 2012

Playing With Math

If you just found my blog after reading my comment on Valerie Strauss' The Answer Sheet, you can start by watching this video of a math salon I conducted at my home. Or check out these posts:
 There's lots more to discover here, but that should get you started.

Thursday, January 12, 2012

Preparing to Teach Linear Algebra

I'll be teaching Linear Algebra this coming semester. It's been a decade since I last taught it. And the text is very different from the one I used before. We're using David C. Lay's, Linear Algebra and Its Applications.

I'd like to know if anyone reading this would like to mentor me when I have questions. Owen Thomas has already agreed. I'd like one or two more mentors.

I'd also like to know:
  • What you've done right when you've taught Linear Algebra,
  • What resources online you think are really useful,
  • Where students get the most stuck,
  • Anything else you think might help me make this a great course.
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