Hello, reader! If you intend to post a link to this blog on Twitter, be aware that for utterly mysterious reasons, Twitter thinks this blog is spam, and will prevent you from linking to it. Here's a workaround: change the .com in the address to .ca. I call it the "Maple Leaf Loophole." And thanks for sharing!

Tuesday, October 28, 2008

Nova Nerdplosion

Check out "Hunting the Hidden Dimension", with a basic explanation of fractals, the Mandelbrot set, and a large collection of applications (blood vessels! forest health! textiles! eyeballs! heartbeats!). The website has neat stuff now, and you can watch the whole thing online starting Wednesday, October 29.

I love Nova. Especially Nova episodes about math. I show "The Proof", about Andrew Wiles' quest to prove Fermat's Last Theorem, to my classes when we have a down day. It gives them a taste of what professional mathematicians actually do all day. Also when they complain that a problem takes too long, I can remind them that Wiles kept at the same problem for seven years.

Monday, October 27, 2008

I <3 Green Globs

If you have access to this game by Sunburst software called "Green Globs", use it. One of my more smartass freshmen exclaimed today "I'm learning! I'm learning!", but they were mostly having fun. It's a game that gives you a bunch of globs on a cartesian plane, and you knock out globs by typing in the equations whose graphs intersect them. The more you hit with one graph the higher your score.

I like it because they *need* to know how to write the equations for graphs they want for some other purpose (an artificial purpose, but one to which they are accustomed to submitting)...I haven't run across anything better for clarifying for them positive vs. negative slopes, rise vs. run, how to slide the graph up and down, how to adjust the steepness. I drag it out again when they graph quadratics.

Anyone know of a free version of something like it? I'd love to hear about it.

Wednesday, October 22, 2008

The Professional Development I Need

Secret message to School District Administrators responsible for Staff Development:

I have many good ideas, and shamelessly adopt other people's good ideas, but I'm not great at executing them. Some people seem to be born with a gift for it, but I'm not. I need practice, feedback from people who know what they are talking about, and chances to observe people who know what they are doing.

The scene: 8th period. My miraculously small, at 11 students, Algebra 1 class. I decide that I would rather stab myself in the eye than do the notes-practice-repeat routine about the slope formula.

I close the smartboard file and have them arrange their desks in a horseshoe. I lie and say that last period, before they were even aware, they were in pitched competition with 7th period. A competition to see which class can do "the wave" fastest. We talk about how we can compare which class is fastest, since we have about half the students they do.

And when I say "talk", I mean it's me and about 3 kids, the rest look listless, like they are waiting for me to just tell them what they need to know. We "decide" we need to measure how many kids can cycle through doing the wave at various timed intervals.

They have fun with this. There is open but good-natured rebellion when I won't let them do the wave for a full 5 minutes.

I decide how to record and organize everything on the board. I don't even know how I'd get them to do that part. I sketch a hasty graph.

There is lively debate about whether it would be better to add up all the people and all the seconds from all our trials and divide, or rather divide people by seconds for each trial and average the rates. These calculations come out differently, but they don't see why. I do a crappy job of explaining why the first option is better.

Then I fumble through a discussion of how "somethings per one something" is called "slope". They remembered the word, they remembered "rise over run". Briefly talk about extrapolation and interpolation, and an equation for the graph. A good teacher would have had just the right sequence of questioning prepared.

5 minutes before the bell, I panic and have them write m = (y2 - y1)/x2 - x1) in their notes and copy a few examples.

It's not easy for me to write that I'm not better at this by now. I hate sucking at stuff. I've seen how they need concrete models to hang their learning on. I don't know how to give it to them.

On Friday, all the high school teachers will come to school for a Staff Development Day. I am leading a "Smartboard Work/Share" session in my room. Not that I won't get some good work done, but honestly I am already excellent at using a computer. I need some one or ones to help me improve the execution of this lesson.

Tuesday, October 21, 2008

Leading a Math Circle Proves at Least as Difficult as Expected

I finally had a first math circle meeting today. I invited any and all Math League students to come. Around 25 showed up at the beginning...8 or 10 had to leave in the middle and 2 more stopped in toward the end. I didn't know most of them but I think they were mostly juniors and seniors (in precalc and calculus). 6 or so students completely dominated the conversation.

I started with "what's the biggest number?" which quickly morphed into "are there different sizes of infinity?" and stalled at "is infinity a number?". They never agreed on whether or not infinity was a number, and by the end decided that they didn't even know what a number was. That was kind of neat. They also kept looking at me imploringly and saying "IS THERE AN ANSWER?!" and you all would have been proud of my aikido-like deflections.

For my part, I tried to steer them to focusing on the countable sets first, and broached the idea of one to one correspondence, but no one picked up that particular ball and ran with it.

So I guess I need help with -

Managing such a big group? More than 20 seems too big to guarantee everyone gets involved. I have 2 other teachers who are interested in "helping" but one of them could only come for about half the time and the other just stopped in said hi and left.

Encouraging dominant students to be more inclusive and collaborative?

Suggestions for steering them toward Cantorian set theory and the proofs I have in mind? I want to get them at least through the diagonal proof of the nondenumerability of the continuum, so that they can see that there is more than one distinct cardinality. At the end I postulated that there were as many positive even integers as there were positive integers but I didn't want to do too much. How should I start next week? I tried to pick a topic where I already knew "how it went", maybe that was a mistake, maybe I should have picked something unfamiliar.

Monday, October 20, 2008

When They Are Going to Use This

A graduate of my school, currently a freshman studying fine arts, sent these pictures to my colleague. She had to make a sculpture from a single piece of poster board. Check out all the geometry she used!

Buried Treasure

In Out of the Labyrinth: Setting Mathematics Free
, a book I have turned to a dozen times, I found this passage about "teaching by indirection" that informs all of my lesson planning. I don't remember ever seeing it before, though I suspect I must have:
Means to an end seem to be more readily absorbed than the end itself...So if you want your students to learn some topic A, find another topic B, which A entails - and have them work on B. You'll see that A is effortlessly taken in stride (although B itself may escape).

Let's say for some reason or other you thought it important for your young students to learn their times tables. Set them instead to puzzling over which of the first 100 or so integers are prime. This is always a heady pursuit for our pattern-hungry minds, and in the course of it such "facts" as that 51 is (disappointingly) 17 x 3 are hit on -- and stick. Once they dope out how the times tables are made ("once you know 3 x 4 your get 4 x 3 for free"), a very few significant encounters tamp down the actual values. You could also have had them figure out how to add fractions: finding a common denominator is tense and rewarding, and puts times tables in their right (ancillary) place. Or if you want them to master deductive reasoning, put in front of them some intriguing topic in Euclid. If the subject is calculus, embody it in a particular physical problem. It takes some thought to find a good B for each A - fortunately no system of facile entailments is ready at hand.

(Emphasis mine.) It's that "fortunately" that undoes me. That the Kaplans expect me to know my subject with such intimacy that I can set my kids on a path where they will necessarily overcome the thing I want them to learn anyway, without even being aware of it. And I can't look it up which B goes with my A. And I shouldn't even try to compile a list. And the idea we can hand the uninitiated a script, or a binder, and expect real mathematics to be learned, would be laughable if it weren't so sad and terrifying.

Friday, October 17, 2008

A fun little limit problem

You drive the first half of a car trip at 30 mph. How fast would you have to travel during the second half of your trip so that the average speed of the entire trip was 60 mph?

Wednesday, October 15, 2008

Direct and Inverse Variation

Planning a lesson about direct and inverse variation, I thought I'd share the hook I use to introduce the concept. This is the script I use in class, but it was heavily cribbed from

If 5 cats can catch 5 mice in 5 days, how many days does it take 3 cats to catch 3 mice?

The "obvious" answer is three. Three cats catch three mice in three days.

However, this answer is not correct!

As the number of cats increases the number of mice they can catch increases. "Direct variation" means that as one quantity increases, the other quantity will also increase.

On the other hand, as the number of cats increases, the number of hours required to catch a certain number of mice decreases. "Inverse variation" means that as one quantity increases, the other quantity decreases.

If 5 cats can catch 5 mice in 5 days, how many mice can they catch in one day? (discuss)

Now, if 5 cats catch 1 mouse in 1 day, How many days would it take 1 cat to catch 1 mouse? (discuss)

And finally, since 1 cat catches 1 mouse in 5 days, if you have three times as cats, they'll catch three times as many mice in the same amount of time. So 3 cats catch ____ mice in 5 days.

photo credit: Denis Defreyne via flickr

Monday, October 13, 2008

Lesson Reflection: Measuring Error

New to our Integrated Algebra standards last year was percent error in measurements. Last year I did a miserable job of it. "Copy notes followed by practice problems" miserable. This year was better.

Things to keep in mind:
  1. They already have learned how to calculate "percent error" in science labs.
  2. This concept is a hybrid of "greatest possible error" and "percent error".
I started this year with a slide and a discussion:

They all said "5.4 cm". To which I replied, "Are you sure it's not really 5.38 cm? 5.41 cm? 5.40236 cm? We discussed the mechanics of measuring something, how you end up visually rounding to the nearest marking on the measuring device. They talked about their digital scales in science class, and how you can set the readout to the nearest tenth or hundredth, and we discussed that the little onboard computer is doing the rounding, even though you can't see it. (For next year: at this point, have them measure several different items to the nearest _______th, to ensure everyone groks the "visual rounding" idea.) Then, I got them to conclude verbally, stated several different ways, that any measurement could really be up to half a unit smaller or larger than the stated value.

Then a few problems:

Then brought in area:

The next day, we tie greatest possible error to percent error, and calculate measuring error (greatest possible area divided by measurement).

On the assessment, 83% of the students scored a 4 or 5 on the question compared to 26% last year. I think the difference was in the introduction.

For next year: take 5 minutes and have them measure some stuff to the nearest whatever. Measure your desk/textbook/etc to the nearest inch, nearest centimeter, nearest 16th of an inch, nearest millimeter, etc. Perhaps borrow a few of those digital scales from science.

Also, it's still pretty dry. Any suggestions to make it more engaging? Other NY teachers (and anywhere else you cover this topic), what are you doing?