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.


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

I'm tired - headed home...

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.]

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:

• Tin Ceilings, Triangles, and Loving Math
• Gift Ideas (for fun math games and books)
• Dots on a Circle 
• September 25th is Math Storytelling Day
• A Card Game for Integers
• Pentominoes

 There's lots more to discover here, but that should get you started.

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.

Thanks!

Grading...

I think that working in groups made a difference for my students. I can't be sure that the groups were the deciding factor, but all 3 of my classes did better than what I usually see. My pre-calculus had 13 A's. 27 passed, and only 13 dropped or failed. I'm impressed. (The other 2 classes were smaller groups, and so I wouldn't expect the same sort of statistical significance.)

But I figured I should look up the old grades to see if this is significantly better, and found a class that did lots better than this one (18 A's, 35 people passing). I remember that class - they asked so many good questions, we got slowed down and didn't finish our trig unit. That hurt them later. So the grade doesn't reflect learning the required material.

That class was asking questions from the assigned homework. I'm still willing to answer those questions, but students seldom ask lately. (Why?) I used to make sure to answer every question. Now I watch the time more.

I hate grading. It is a way in which teachers have power over students. That transcript is asked for over and over. (I've had to show mine for every job I apply for, but I'm in academia. Maybe other employers don't ask for transcripts?) I try to make it transparent, fair, flexible, and an accurate representation of how much students have learned.

• Transparent: I use percentages, and I explain my process on the syllabus (could I do more?).
• Flexible: I have more than one way to calculate the grade, and use the max function in Excel to give each student the formula that works out best for them. (One thing I do that's not transparent is to have a grade option where the final exam is the whole grade - if they learned it, they're set. But I don't want to tell overly optimistic students about that option until a few weeks before the end.)
• Fair: This semester I took to heart some of the blog posts I'd read that explain how a 0 affects the grade too much. (A and F should average to C. But 100% and 0% average to 50%, which is still an F in most classes.) I changed every 0 to a 40 before averaging. I chose 40 because my D goes down to 50% (and my C to 65%). I also agonize over making sure my bad feelings about troublesome students haven't affected their grade.
• Accurate: I let them re-test, and I let them take the final exam twice. Do some people get a grade that's better than it should be? Probably some, but not many. The more important thing to me is that no one be punished for learning things a bit later than they were supposed to.

I'm almost done with grading. Once I'm truly done, and the relief has washed over me, I expect I'll start blogging about all sorts of other things: tutoring, Linear Algebra (I'll be teaching it next semester, for the first time in 10 years), and Playing With Math (the book, and the concept) among them.

Happy holidays!

Testing...

I've arranged for all 3 of my courses to have two chances to take the final exam. Most of my students will not benefit from this, because they try to cram too much into too little time. But some students will really solidify their understanding, and so it's worth it.

The final exam is one grade, but that's not how I do my other tests. Those are broken up into subtests, each on one topic. (See the example below.) The student gets a grade for each subtest, and I allow retakes of any subtest they didn't score well on.  In Intermediate Algebra and Pre-calculus, they can come in whenever they want, show me that they've learned the material, and I'll make them a new version of the test (the whole thing, or just one or a few subtests).

I can't make new test versions so quickly in Calc II, so we work out a day that everyone who wants to retest can come. We scheduled a day this week or next for each of the previous tests. I hope this helps them!

The one new thing I've done this semester in relation to tests is to add a problem-solving subtest onto each test in Pre-calculus. My tests used to be way too hard, and I've made them easier over the years. I've worried that they were too easy (even though students don't do as well as I'd like). I want to test on their thinking skills. But I knew that was too stressful. This semester I came up with the idea of having one problem on each test that would require some real problem-solving. They only have to get this problem right once during the semester. There are still lots of students who haven't gotten one right, so during the final, I'll have a separate problem-solving test available, with 3 problems to choose from. Next semester I'll gather together the problems I've used this semester into a problem-solving handout, and we'll work more in class on how to problem solve.




My colleagues worry that doing this would take too much time. But I think I work less than they do, because they put more time into grading homework, and they probably agonize over partial credit like I used to. My tests are very short now, and both making and grading them is usually pretty quick.

I like seeing my students take more responsibility for their learning. It's really changing how some of them deal with math class. I know I have a long way to go to catch up with some of the people whose classes are blossoming with SBG; it's a great journey to be on.

Math Girls: A Novel Way to Learn Some Deep Math

I asked for a review copy, but I can't even wait until I finish it to tell you about this marvelous book. Math Girls was published just yesterday. (How I love the internet, let me count the ways!) My thanks to Robert Talbert for his blog post on the book, and to Bento Books for sending me a review copy so I can satisfy my desire for immediate gratification. You can download a sample (first two chapters) from Bento Books here.


Math Girls has gone through 18 printings in Japan, and the English translation has just been released. There are lots more books in the series, but those of us who don't read Japanese will have to wait for those.




Here's a bit for flavor:

When you’re doing math, you’re the one holding the pencil, but that doesn’t mean you can write just anything. There are rules. And where there are rules, there’s a game to play—the same game played by all the great mathematicians of old. All you need is some fresh paper and your mind. I was hooked. 

I had assumed it was a game I would always play alone, even in high school. It turned out I was wrong.

Our protagonist, a high school student, is intrigued by Miruka, an elusive girl at his school who gives him challenging math problems to ponder:

“Forget about the matrices for now,” she said. “Here’s a problem for you.”

Problem 3-1
Give a general term a_n in terms of n for the following sequence:
n     0 1  2 3 4 5   6  7···
a_n    1 0 −1 0 1 0 −1 0 ···

“Think you can you do it?” she asked.
“Sure, that’s easy. All you’re doing is going back and forth between 1, 0, and −1. Sort of. . . oscillating between them.”
“That’s all you see?”
“Am I wrong?”
“Not wrong, exactly. Go ahead and give me a generalization.”

And our protagonist (I don't know his name yet) helps another student, Tetra, with her math. So you get to see the same ideas played out at higher and lower levels. When Miruka kicked Tetra's chair out from under her, I had to skip the math to find out what would happen next between the characters. I'm not sure the storyline will make complete sense to me, but I am so loving it!

Tony, the rep from Bento books made a request:

If you blog about Math Girls, please be sure to let your readers know that this is a pretty advanced book. We’ve had many inquiries from parents looking for fun books for their middle school and younger children who love math, but Math Girls is probably best suited to, at a minimum, talented high school juniors and seniors who want to go beyond what they’re likely to be exposed to in a high school curriculum. The “sweet spot” for our readership will probably be first or second year college math majors who are looking for a more relaxed treatment of some of the stuff that they’re plunging into.

I'm counting the math lovers on my holiday gift list, and planning to buy each of them a copy. I'll be reading the rest of the book on the plane to Seattle this evening. And I'll let you know soon whether the adult content extends past the math.

===

It's Friday now. I just finished. My review copy is a pdf, and I need a paper copy to study the math. I just bought 4 copies; one for me, and 3 for the math lovers in my life. (Oops! I just thought of someone else I need to get it for.)

The topic I'm most interested in studying more closely is called the "Basel problem". It asks for an exact sum (in closed form) of the series . I knew the answer, but had no idea, until I read Math Girls, how it was derived. I've started to see it, and I love what I'm seeing through the mist. The coolest thing is how connected it is to what I'm teaching in Calculus II.

The only complaint I have about the mathematical exposition is when derivatives are given with no real explanation in chapter 9. I think that could have been fleshed out a bit more.

I would highly recommend this book for anyone at the level of pre-calculus and above who enjoys math. (There is no adult content besides the math.)