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.)