Month: March 2008

Paris, or bust

I’m going to be heading off to Paris on Sunday, taking a much-needed break from the daily grind. I might post something before I leave, but I’ll be gone until March 26th.There’s a good chance I’ll be bringing my lappy toppy with me, so you might get an update or two from the French capital, but in case you don’t hear from me, I’m (probably) a-okay

Yahoo! for teachers

Update: Bill Fitzgerald and Dan Meyer are now on a similar quest.

Yahoo has started a new web 2.0 site for helping teachers create, manage, and share “projects.” I’ve dreamed of a site where teachers collaborate (as well as beg, borrow, and steal) online in a massive community. Yeah, bloggers read and comment, but that’s not what I dream of. I want a huge archive and discussions on projects.

This could be that, if we’re speaking with all the idealism and naivety of a ten year old. But for a site like this to work, people have to use it. Without having played with it, my initial spidey sense is telling me that instead of the website being adaptable to us, instead we’re going to have to adapt to it. Constrained by what the website constitutes a “project,” teachers are likely to think this site isn’t as natural as it could be. Instead of technology adapting to our needs, we might reconfigure our needs to adapt to this technology. And I’d like to have some… not promise… but strong indicator that it’s worth it before heading off into the technological blue.

It’s unclear to me how useful this is going to be (if at all), but I’m going to keep an open mind. One thing I’ve often noted is how hard it is to find smartboard presentations online. I create mine from scratch, but I also imagine that others would find them useful, as I would find looking at (and stealing parts of) their presentation of the same material useful. I secretly have a hope that one day I will be the facilitator to this giant city wide project which will get teachers who actually make lessons (smartboard, handwritten, typed, group projects, etc.) to upload them to some site to share.

In any case, the website is still in beta form: http://beta.teachers.yahoo.com/

I signed up for an invite forever ago, and just got invited to join today, but maybe anyone can get one now that they’ve opened it up for testing.

We love radians

Introducing trigonometry has become even more of a challenge than I thought. I think about each part of the lesson really hard; I want to give it a flow and focus on teaching the concepts. What I don’t want trigonometry to be is a huge mess of ad hoc rules.

Today was my introduction to radians. Looking back, my presentation was a bit more complicated than it needed to be to get the idea across. And what wasn’t clear (although that was one of my objectives) was why we use radians instead of degrees. So I’m going to start off class tomorrow with a little silent slideshow, replete with my own histrionics to make extraordinarily clear why we! love! radians!

Without further ado: why radians? (PDF file) [Unfortunately, SlideShare is only showing 18 of 23 pages for some reason. The PDF is complete.]

I’ve also come to realize that more is going on with kids than this whole forest for the trees crap that I wrote about before.

There’s a second reason things are getting mucked up, and that doesn’t have anything to do with my concept behind each lesson, or the flow, or anything like that. I realized today that a lot of the things that kids get tripped up on are (surprise surprise) basic facts about numbers. Is “1/2 times pi” the same thing as “pi over 2”? Yes. Do they know that? Possibly.

Or, for example, there’s the issue of manipulating visual and fractional information in their heads. We’re learning about radians, and we learn that there are “2 pi” radians in a circle. Then I ask them to draw an angle of “3 pi over 2” radians. It was as if I asked them to dance around like a chicken while singing Ave Maria. And since the “pi” was there, they thought that using the calculator wasn’t really going to help them.

I think we’re slowly getting it, but I’m not sure. I’m going slowly, but I have now started identifying key skills and concepts that need to be honed before we move on with radians. For example, because of what I noticed in my lesson on radians, we’re going to be practicing working with fractions (the eternal scourge of math teachers) and pi. (See my thrown-together worksheet here.)

Imagine (for surely, gentle reader, this has never happened to you before) that you’re at a mini golf course and you’re putting at the infamous and dreaded windmill hole. By mandate from the PuttPutt gods, you are not allowed to leave until you get the ball into the windmill. There’s a mini golf coach there, trying to give you advice and show you how to hold the club and how to swing. However, after 20 tries you aren’t getting it. And then you try another 2o times. No luck. Now tell me how you think you’d feel at your mini golf coach who has been standing there trying to help you.

There will be whining, complaining, anger, and frustration — anxiety — all directed to this coach.

The analogy isn’t quite right, and I hope that my students don’t direct those feelings to me (this was an extended allegory, duh), but I can’t help but notice that the anxiety level has shot up in my room in the past two weeks, when I feel that one of my teaching talents is keeping a totally relaxed atmosphere.

UPDATE: My presentation (see above) on”Why Radians?” took 5 minutes and I think did the trick. I did it in both classes, and both seemed to get it. And the levity of it all made the classroom less tense. And with Spring Break descending upon us, we’re going to have a much needed break.

I lost, but it’s the journey that matters.

So I didn’t *win* Monday Math Madness. For those of you not in the know, the blog WildAboutMath is having a rotating math problem contest with another website, and I put my hat in the ring. And I got the answer right. Sadly, the random number generator did not love me (click link for a few different solutions).

This week’s problem was:

A popular blog has just three categories: brilliant, insightful, and clever. Every blog post belongs to exactly one of the three categories and the category for each post is selected at random. What is the probability of reading at least one post from each category if a reader reads exactly five posts?

After solving it, I tried thinking of natural extensions. Clearly, one is: “if you have p posts and n tags: what is the probability that a reader reads at least one post from each category if a reader reads exactly p posts?” Or one could think about the more difficult question, where the frequency of each tag is different (not equal chance of stumbling across each post).

My submitted solution after the fold. You can see how hard it is to explain math clearly in an email.

(more…)

The Mathematician’s Lament

It’s 10:45pm and I just came across this compelling article called “The Mathematician’s Lament” that I’ve now read half of. I’m going to finish it off when I have more time, and post my reaction to it. From what I’ve gleaned thus far, this article is incredible provocative. I can’t say I wholly agree with everything the author says, but he also does hit a few points home.

Some juicy quotations to get you hooked:

Sadly, our present system of mathematics education is precisely this kind of nightmare. In fact, if I had to design a mechanism for the express purpose of destroying a child’s natural curiosity and love of pattern-making, I couldn’t possibly do as good a job as is currently being done— I simply wouldn’t have the imagination to come up with the crushing ideas that constitute contemporary mathematics education.

By concentrating on what, and leaving out why, mathematics is reduced to an empty shell. The art is not in the “truth” but in the explanation, the argument. It is the argument itself which gives the truth its context, and determines what is really being said and meant. Mathematics is the art of explanation. If you deny students the opportunity to engage in this activity— to pose their own problems, make their own conjectures and discoveries, to be wrong, to be creatively frustrated, to have an inspiration, and to cobble together their own explanations and proofs— you deny them mathematics itself. So no, I’m not complaining about the presence of facts and formulas in our mathematics classes, I’m complaining about the lack of mathematics in our mathematics classes.

But why?

Trigonometry is one of those topics that if you get the basics, the rest of it will make a heck of a lot of sense. But if you miss it, you’re going to be trying frantically to come up with ad hoc ways to understand each new concept.[1]

I have been teaching the beginning of trig, and I’ve noticed a few things that I have to watch out for next year:

  1. Have a really good reason prepared for explaining why we care about angles greater than 360. In general, the kids don’t have a good idea of why we’re doing what we’re doing. (“Why do we want to find csc(421)?”; “Why don’t we just say we have an angle of 1 instead of 361? When would we ever need 361 degrees if a circle has only 360?”)[My explanation didn’t hold over well, but it’s true. I said that often times we use angles to measure time, like in a clock. So if we have something repeating — like a spring with a mass in physics, or a ferris wheel going around and around, or a bike wheel spinning — we will be able to model how far it’s gone or how many oscillations its made by using this angle.]
  2. In fact, have a good reason to explain why we care about angles greater than 90. I started out teaching triangles and SOH CAH TOA, and they got it. Then I started teaching how to see angles on the coordinate plane, and I lost some of them. They can — I hope — calculate the sine of 210. But they don’t get why the sine of 210 is at all related to the sine of 30. They see the 210 angle, and they say “where’s our triangle”? And I show them the 30 angle, and they understand that we can form a right triangle with it, but they don’t get why we use the triangle with 30 to deal with the 210 angle.[My explanation dealt with looking at triangles made in the first quadrant, like the sine of 45. I showed them that the opposite side of the triangle was the y-coordinate, the adjacent side with the x-coordinate, and that the hypotenuse was the radius. Then I said the problem with the whole “opposite,” “adjacent,” “hypotenuse,” method of things was that it restricted our angles to lie between 0 and 90. So to expand the domain of these trig functions, we put them on the coordinate plane, and defined sine to be y/r, cosine to be x/r, and tangent to be y/x.But then they asked: “Why? Who cares about angles greater than 90?” Which then takes us back to #1.]
  3. I haven’t given them a big picture, which is part of the problem. Right now they’re learning smaller skills, but they don’t know what the whole point of it is. So what if you can find cot(260)? Why do we care? What does cotangent mean in the real world?

Next year… I might want to motivate trigonometry on the first day, and give a hard application problem that we’ll be able to solve by the end of the unit. Then I’ll give an schematic diagram of what we’ll be doing and try to motivate each step. That will give them the big picture. And hopefully the rest will fall into place once we have the big picture.

[1] It’s like that old adage… “I must’ve been absent that day.” In this case, it’s pretty disastrous.

UPDATE: A student asked me today, “Mr. Shah, will you promise me something?” “Not without knowing what it is.” “Well, will you promise me that you’ll explain why we’re doing all of this at some point?” So I was right on the ball, in terms of students just not knowing what’s going on because they can’t see the forest for the trees. And I went looking in the book to see what applications they have, and they’re awful (kites, ferris wheels, and bicycles). I’ve got to get on this right away!