## Thursday, March 29, 2012

### What to Do with All the Technologies

Among people who know me professionally I have a bit of a reputation as a technology person. It might be because when people walk by my room, there's usually laptops and calculators all over the place. Or it might be because I put it all over my CV because people like that sort of thing. Or it might be my third robotic arm. Who knows.

But when people talk to me about the technology I have to constantly Reframe the Issue and explain how I'm not all pro any technology for its own sake. You don't go, "Oh here's this cool technology let me shoehorn it into my classroom." Instead you go, "I think I have thought of the best way to teach this, and it would be impossible in an analog world, but I know enough about the technologies to realize this idea." You don't go to a twenty-minute inservice about xyz.com and go "I'm going to make an xyz.com lesson." You use xyz.com for your own purposes, or you suspect its utility and put it in your back pocket, until your awesome instruction idea needs xyz.com in order to exist. Your lesson is the fuel and xyz.com is the oxygen.

So here is a lesson that would not exist without dynamic geometry software and classroom polling. It does not matter what sort of dynamic geometry software. I've done it with Sketchpad and I've done it with Nspire and next year I'll probably do it with Geogebra (damn, that sentence makes me sound disreputablllllle.) But I don't think you could get the same effect without the technology. Maybe you could give them diagrams on paper and rulers and protractors, but there's no way to make those not static, even if there's a lot of them.

We are discovering additional properties of special kinds of parallelograms. So everyone starts with a sketch of a parallelogram (that you give them, or in our case, that we constructed the day before.) And the children are in groups. Each group gets a different set of questions to explore. Writing these exploration-y questions is a bit of a dark art. You don't want to send them on a chase of the wild-goose variety but you don't want to set them down too much of a pre-defined path either.

Example: Group 1. Start with your sketch of a parallelogram. Construct both diagonals. Drag points around until the diagonals are perpendicular to each other. You will have to decide what to measure so you can be sure. What is the name of the special kind of parallelogram you get as a result of perpendicular diagonals? Now you need to find at least two NEW properties of this shape. They must be NEW properties that are NOT properties of any old parallelogram. You will have to measure some stuff. If you can't find two new properties, keep staring at it until you get a good idea. Write them down. Verify them with measurements. See if your group members agree. Drag vertices to make a different parallelogram where the diagonals are still perpendicular. Are your new properties still true? Challenge: It is possible to construct a quadrilateral with perpendicular diagonals that is NOT a parallelogram. Open a new page and construct such a shape. What other properties does it have?

They get ten or so minutes to play around. It is helpful to give them some verbal marginally-hysterical (at least in my case, they always feel slightly deranged) instructions like "It is not a square! Nobody has squares! The correct answer to any question is not "square!"" and you also need to run around like a crazy person and make sure everyone knows how to grab points and drag them around (you might as well admit that screaming about rhombuses to a roomful of 15 year olds makes you a little bit of a crazy person.) Because you KNOW there are at least three maybe four kids who try one thing for half a second and it doesn't work and then they will sit there and stare at their desk for twenty minutes unless you interrupt that little party.

Once they have had ample time to explore, and the faster workers are getting bogged down in the challenge questions you put at the end, you ask them to respond. I have queried each group verbally in front of the class, one by one, and that doesn't work so hot. Usually in a class of 30+ hardly anybody likes to talk in front of everybody. Better... send them a link to a Google Form where they can type an answer to each question. Or come up with your own response system that your existing tech will support. This year I used TI Navigator polls. They are annoying because the TI's don't have a qwerty keyboard. (Lesson number 9,125,698 learned the hard way.)

Once all the groups have had a chance to report by whatever method, then you write down your notes of properties of rhombuses and rectangles. And then you give them a bunch of problems to find missing measurements in rhombuses and rectangles. They can reason it out now. You don't have to show them example problems first. It feels like kind of a magic trick.

THAT's what the technology is for.

## Sunday, March 18, 2012

### Nerding Out with the Dictionary

In my Spanish studies, I recently came across radical-changing verbs. When Mark, the teacher (I'm using, among other sources, the excellent Coffee Break Spanish) first said "radical-changing" I first thought WHOA, RADICAL, they must be really extremely different. But here's how it works. In conjugating a standard verb, the stem stays the same, and the ending changes. For example, bailar - to dance: bailo, bailas, baila, bailamos, bailáis, bailan mean I dance, you dance, he dances, we dance, you-all dance, they dance. The stem stays the same and the ending changes. But in a radical-changing verb, there are spelling changes to the stem as well. For example pensar - to think, the "e" changes to "ie" sometimes: pienso, piensas, piensa, pensamos, pensáis, piensan mean I think, you think, he thinks, we think, you-all think, they think.

Why this is interesting to me, is it's another clear example of "radical" referring to a root. "Radical-changing verb" is referring to changes in the root of the word, as well as the ending. With numbers, an example is the square root of a number, like how "radical 9" means "the square root of 9." The symbol for which is $\sqrt{9}$, and that symbol is derived from a stretched out "r." The rationale I've heard for this word choice is, picture an upright square resting on the ground:
If the square has an area of 9, the root, the part resting on the ground, has a length of 3. The square root of 9 is 3.

So I went and looked up "radical," and behold the first definition:

(esp. of change or action) Relating to or affecting the fundamental nature of something; far-reaching or thorough

not, as I might have guessed, extreme or far-fetched, which are included in the definition, but farther down.

Anyway, you all probably knew that already. But I thought it was a neat connection.

## Saturday, March 3, 2012

### This Instructor's Guide is SO GOOD

I just want to draw your attention to, should you happen to be teaching calculus using Calculus: Early Transcendentals (Stewart's Calculus Series), the existence of the Instructor's Guide. This text is different from the others I have used, in that the Instructor's Edition of the textbook is not that much different from the Student Edition. They are roughly the same size, and you would have to make a close comparison to find any differences. Most other textbooks I've seen, the Teacher's Edition is a crazy behemoth of a thing with marginal notes of highly variable usefulness that take up 3/4 of the page.

So ANYWAY, with the Stewart book, I got this whole other huge 3-ring binder that I barely glanced at all year until maybe four weeks ago when I started idly wondering what was in there. And holy crap, you guys, it is really full of some amazing stuff. It does have suggestions for how to organize and emphasize lectures, and questions to ask, which is nice. But most sections have two or three handouts called "Group Work" or "Find the Error in Reasoning" or something like it - and these are just very, very well done. They are thoughtfully-structured, with good questions and writing components. Some are simple and straightforward, but they're still nicely organized and typeset with pretty graphs and nice, realistically big spaces to write. So now when I'm tempted to sit and write an investigation or problem set or activity for Calculus, I check the binder first.

For example, today we worked through Group Work 1 for Section 5.2, "The Area Function." Students found expressions for areas under a constant function
Given f(t) = 4, find  $\int_{1}^{2} f(t)dt, \int_{1}^{3}f(t)dt, \int_{1}^{4} f(t)dt, \int_{1}^{x}f(t)dt$

and another linear function f(t) = 2t + 2, find  $\int_{0}^{2} f(t)dt, \int_{0}^{4}f(t)dt, \int_{0}^{x} f(t)dt$, and then the same function again but starting at -1 instead of 0. They do all this just by finding areas geometrically. Then they look at the first derivatives of those, and notice hey wait a minute, the first derivative of the area function is just the original function. (Then they go, and this is a direct quote, "That's so stupid!" (The area is just the antiderivative?! Why didn't you just tell us this before instead of making us do all those limits!) and then I get sad because this is so neat and beautiful and I jump around and get very enthusiastic about how they were just a minute ago finding areas by ADDING UP TRIANGLES FOR CRYING OUT LOUD, but now they have the power to find the area under ANY CRAZY CURVY FUNCTION THEY FEEL LIKE, and sometimes it's really pretty easy, way easier than summing infinite rectangles.)

So, yeah. The binder. Check the binder.