Matrices!

Matrices! In Algebra! Whoa! As you may be able to tell, I was surprised and delighted to find we were learning about matrices in class today. Not baby matrix math either. We were adding, subtracting, and multiplying. Multiplying! Multiplying matrices! Matrices! We even covered the idea of a transformation matrix and used one to scale the points of a triangle. The few students I was able to sit down with today got to hear about how matrices are used a lot in computer graphics. A couple students asked about programming games for their calculators. When I said "you could download them, or you could make them yourself" they seemed amazed at the thought that a person could actually program a video game.

"How do you think video games are made? Of course people program them!" Incidentally, my college roommate, Matt*, recently left his game development job to start a game development company. Check out a demo of their new game here! I wish I could bring up youtube links in real life. Until then, I guess retroactively posting on my blog will have to do.

Despite how difficult I find matrix math to be (there are lots of additions, which I think is the hardest part of math), matrix multiplication is easy to algorithm-ize. Give students a recipe, and BAM, they get it. Initially I was surprised how after 2 examples, everyone I talked to was able to multiply 2x2 by 2x3 matrices and 2x3 by 3x3 matrices without error. Perhaps it was the straightforwardness of the work. Compare this to solving 2x+4=y for x, which can be done many ways and still confuses some students. Maybe there's a lot to be said for not having choices.

This partnership program has inspired me to write a math book. I imagine anyone that has had a similar experience has thought similarly. So, how do you write a math book that's fun, useful, and free?

Happy Halloween!

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When I got up this morning, I realized that I hadn't prepared a Halloween costume, and felt a little out-of-sorts.  Luckily, I was able to find my bear costume from two years ago and put that on before grabbing my camera and dashing out the door.  I was pretty excited to see all of the kids at Ypsi High dressed up in crazy costumes...

...but then I got there and realized maybe 10 people had Halloween spirit.  Nobody was wearing costumes!  Disappointed and a little embarrassed that I was the only person in the room with a fluffy tail, I proceeded to grade tests from the previous hours while our students took today's big test.  Unfortunately, this caused me to forget to get pictures of 3rd hour, but above you can catch a glimpse of 4th hour when they're being good.

Regarding the test, there weren't any big surprises.  Your usual set of questions about functions and your usual set of answers.  Grading is nice for a lot of reasons.  It's objective, so you can get fast at it and do it without much thought.  Mrs. Colwell always mentions it is a big help to her, so I like that.  Pretty much the only thing that isn't great about grading is when a student misses points that you know they could've gotten.  If only there were some way to help them improve... 

baby steps

Remember Bill Murray in What About Bob?

Today in class when I was helping students with a problem, I noticed everyone just wanted to rush to the answer.  They'd read the question.  They'd see the empty blank on the right side of the page.  Then they'd freak.  Dizzy spells.  Nausea.  Cold sweats.  Hot sweats.  Fever.  Blisters.  Difficulty breathing, difficulty swallowing, blurred vision, involuntary trembling, dead hands.   Numb lips.  Fingernail sensitivity.  Pelvic discomfort. 

Math anxiety isn't any different from Bob anxiety and solution for both is baby steps.  Instead of getting frustrated, panicked, or pelvis discomforted when trying to throw an answer in the blank, take a baby step.  What is the problem asking for?  Baby step.  What information does the problem give us?  Baby step.  What's the first step in getting from here to there?  Baby step.

Teaching someone to take baby steps is hard.  Even if you do, oftentimes  it will still only get a student halfway.  Still, I think this is a is a good model.  What's a teacher for, besides helping a student find the next baby step when they can't find it themselves?  

One class-wide baby step I've been happy about is the willingness to ask for help.  Of the students that need the most baby-stepping, some of the shy ones are now asking for help and some of the loud ones are quieting down and doing the same.  At the same time, some students are unfortunately stepping down a class- a few faces I'd grown accustomed to helping are no longer there.  

Not much I can do but take it one step at a time.

after a break

You probably didn't notice, but I didn't go in last week. Today I'm back, so you get a new blog post.

Grading quizzes today, the one salient observation that popped out of the student responses was that they still don't know what a Range is. What is the range of the function y=-7? Answer? -7. What did students put? Everything from 0, to all reals, to x is in [-7,7], to my favorite: -7>0. Maybe 5 out of the 40 or so tests I saw answered -7. Everyone probably already knows my attitude towards math vocabulary, but given that I had to double check what the domain and range were again today I couldn't help but mention this once more Let it be known, all students who got that wrong, I didn't feel good taking the red pen to your paper. I feel for ya.

In 4th hour, I had an opportunity for more one-on-one tutoring with students. One had turned in a blank quiz after about 5 seconds. We absconded to the far corner of the room where we went through some examples. After about 10 minutes, I revealed to him that our examples were really the quiz questions and that he had just gotten all of them right. "Sure is a shame you gave up so fast..." He thought this was pretty funny and I'm not sure if he spent the rest of class paying attention, but at least spent he wasn't causing trouble.

Not a whole lot else to report, so I'll shut up. Later!

A class divided

I just found this story about Jane Elliot's classroom discrimination experiment from 1968:

• A Class Divided
  

What an extraordinary documentary.  Watch it.  The whole thing.

Update:  I just finished watching it and read an interview of her from 2002.  It's sad to hear that her exercise is still as enlightening today as it was in 1968- but it's also inspiring to see that racism is a learned behavior and that you can "teach an old dog new tricks."  The comments from the children, both in the classroom and years later, and the comments from people from her later trials were poignant.  Watch it.  The whole thing.

Dividing both sides

Today was the first of two review days for Monday's test: students formed groups and worked in teams on the review sheet.  Tomorrow, the review sheet will be reviewed and the students will get a chance to check their answers.  I mentioned before that I like this format and today was no exception. 

In 3rd hour, one group of girls was flying through the review assignment.  This group also happened to contain the girl who was elected to homecoming court.  See kids?  Math skills make you popular =)

In 4th hour, I had two noteworthy interactions.  In the first, I had a chance to talk with a student who is failing the course.  Citing the fact that "no matter how hard I try, I'll still fail, so why even try?" she refused to work on the worksheet.  I wasn't able to work any miracles here, she didn't work on the worksheet all class, but at least we had a chance to talk about it.

The second interaction was with an international student that is clearly frustrated, but has a great attitude.  We only made it through the 3rd question on the review sheet (there are around 20, i think), but even so I think we had some great progress today.  As we worked through a problem, we discovered that he didn't "get" dividing.  If he'd reduced a problem to 2k/4=40, he'd stop.  We worked through a few more examples, and he seemed really pleased.  He even asked for more examples of this sort of problem, so he wouldn't forget!!  I scribbled a couple more examples out as the bell rang.  I apologize if the numbers work out to be really ugly- but I didn't have time to check!  How's that for pro-activeness, though?  

This same student also mentioned that nobody was at tutoring last week.  I don't know if he didn't wait around long enough, didn't go to the right room, or what happened, but I told him I'd mention this to the tutoring folks tonight at dinner.  I also gave him my email address in case he had other questions.  I don't know if he'll email at all, and I don't know if I've just opened up Pandora's box.  We'll see!

Hung-Hsi Wu

Last night I found the website of a professor at Berkeley who has been quite prolific in the area of Mathematics education.  I've only had a chance to read one of them (here), but I found it very timely and interesting.  I look forward to reading the rest.

I bring this up today, because Wu's comments on "order of operations" and other vocabulary like domain and range meshed perfectly with some of my observations thus far.  The first sentence:

One of the flaws of the school mathematics curriculum is that it wastes fruitless exercises in notation, definitions, and conventions, when it should be spending time on the mathematics of substance

Wu goes on to explain order of operations, the Please Excuse My Dear Aunt Sally mnemonic, and how all conventions like this are artificial.  My favorite example was

Evaluate 4+5*6/10

"Now, one never gets a computation of this type in real life, for several reasons.  In math the / sign (in the paper it's the divide sign with two dots, but i'm not going to the trouble of encoding it properly here) basically disappears after grade 7.  Once fractions are taught, you almost always see 6 * 1/10.  Moreover, if anyone wants you to compute that, he would certainly make sure that you do what he wants done, putting parentheses in the right spot.  The original problem is therefore a kind of Gotcha! parlor game designed to trap an unsuspecting person by phrasing it in terms of a set of unreasonably convoluted rules."

Or, at least I thought it was my favorite example until page 9, when I ran into "most beginning algebra texts still devote pages to teaching the concepts of the domain and range of relations which are not functions, and teachers are known to proudly drill their students on learning htese terms.  But these concepts are not used in any part of mathematics except for rare circumstances such as the discussion of correspondances in algebraic geometry.  There is not much to gain by spending valuable classroom time on this topic."  Our friends the domain and range return!  

The point of all this was mostly just to make a record of these math education papers.  Check back later to see if I've reviewed any more.  Ciao for now!