Think Aloud: Planning Inequality

From the always fun Zappa Blamma

In my preassessment for preservice teacher (PST) high school mathematics course, I had a new topic grab their attention - inequalities.  In previous times teaching this course, that hasn't been an issue to which we've given much attention. My bad, as I've seen plenty of struggles with it in secondary classrooms. It's clearly an error where we retreat to instrumental understanding (using Skemp's terminology); rules, rules and more rules.

So what are the key ideas? It's an extension of the algebraic methods we're already using. So I want to focus more are the conceptual components of inequalities... which I haven't ever really deeply considered before.

The heart of inequalities is comparison, which is a key component of number sense. A key and often omitted component. Usually we introduce numbers (multi-digit, fractions, decimals, signed) and then are lucky if we even get representation work before we get to computation rules and rote. So that makes me think of a couple of my favorite games - Fraction Catch and Decimal Pickle. Probably not appropriate for a high school class, though I use them in the preservice middle school class to good effect. What would a high school version look like?

Hmm. It's the rules for changing the sign that get people all bothered. What if the players/teams started with a number, then multiplied or added to both players numbers... the goal could be to get the larger quantity in the end.  You could make special cards... or (I love being able to use regular dice, cards, dominoes, etc.) have the suits imply the operations. I'd have to think of how many turns, how many cards to have, some of the game mechanics. But that idea might be worth the work of developing something new.

Before being in math ed, I worked in index theory, that includes plenty of analysis. Analysis, as a mathematical field, involves a lot of estimates. Think nitty-gritty epsilon-delta proofs. This is less that that, which is less than or equal to the other, which means the whole thing is less than... I loved that stuff. So is there a way to get students making those kind of estimates?

That makes me think about experimental error. Error analysis is nice because of the connections to measurement. That gets us into volume formulas. Could even do a guess how many kind of competition. Hmm. I took a picture of a neat set up like that at the ND-U of M football game over the weekend, but it came out too blurry. And I don't know the result! (I guessed 1337. [LEET.] The bottom pyramid was 10 balls wide and about 20 balls tall.) Lots of good images for that on 101qs.com. Could use in class or make a good HW assignment out of that, and get the PSTs looking at that great resource. It would also let us talk a bit about measuring technique - important to get in somewhere. Volume ties into some of the higher degree polynomial stuff we've been doing...

I think that would make for good foundational experiences, and then we could do a follow up day to see how the ideas of inequalities we see apply in traditional algebra and algebra 2 problems.

Now to get working on that game...

ps. What I didn't do here that I usually do when planning, is look around what other people have about inequalities. That usually involves a visit to Sam's virtual cabinet, searching reader, raiding Kate's archive, etc. Where do you look for inspiration, adaptation and out and out robbery? (Accredited, of course.)

This story continues in the post on coaching. With video of an actual dialogue!

Storytelling

Not me, sadly.
(by Travel Aficiando @ Flickr)

We just finished up our Vacation Bible School, which had Joshua as a theme.  I'm the storyteller, and my thoughts were much on teaching throughout the whole process.  Dave Coffey and I often talk about teaching as story telling (EDIT: turns out he was writing about storytelling at the same time as I was - shocking), and love drawing lessons from Harry Potter as we both love the books.  (So much so that when my otherwise-entirely-admirable summer class revealed they had not read them I was at a bit of a loss when it came to examples for questioning in literacy. Stunned, I was.)

The process started with learning about the subject.  I'm not a bible scholar, but I am willing to write bible studies.  Mostly I just share the questions I wonder about as I try to make sense of the readings. I use BibleGateway.com as a tool, as it has many translations and a robust search feature.  I copied the relevant bits of scripture out, and made it into a study.  (Shared here.) I got to discuss it with three groups of people before writing the story.  In particular, with a men's group where (rather unbelievably) I'm a junior member.  Finally it was time to write the story. (Shared here.) I wrote the narrative, following the facts of the story. I went over it again, thinking about the teaching point(s) of each of the three days.  I'm not always explicit with those, but it's going to be harder for kids to notice if I don't put it in there to notice.  I went over it again, strengthening and adding connections. Things like: if the ark is going to be used on day 2, make sure it's mentioned where I can on day 1. Go over it again - does it include what's important and is what's important emphasized?  Finally, practice the telling.

My daughter performed with me to help with the illustration and comic relief, and we'd go through the story right before telling it. Originally she was going to be Joshua the first night when young, then Israelites later, but we wound up keeping her as Joshua because the kids seemed to identify her really strongly with the part. As we told the story, we responded to the kids' response. We monitored both their general engagement and asked questions about what they would do, or what thought might happen or what they knew about things in the story.  We told one version to the K-4 crowd, and another to the age 3 and 4 group (with lots more marching around). We used the props and set pieces we had and figured out how to do things with limited time and resources.

Sounds a lot like teaching, right?

I've been really interested as some teachers share on Twitter their summer planning process.  It seems some are frustrated by trying to plan without their students, and I agree that this is questionable. But I also got to thinking that it offers an opportunity to think about your objectives in another way.  As a story.  Your plans won't be able to be set until you know your students, and assess what they're bringing to the story.  But you can think about what's important and look for the narrative.  To me it's quite like what Dan Meyer's been writing about when he's considering how to set, describe and pitch the problems you find.  I also think that's the kind of work we need to find better and better ways to share with each other, as it's very portable amongst schools and students.

For your listening and viewing pleasure

One of the many interesting ceiling panels.

Goals of Math Ed

Arthur Benjamin, self-proclaimed mathemagician and a very popular mathematician among teaching mathematicians, at TED on why calculus is not an appropriate pinnacle for math education. To be replaced by: discrete mathematics (statistics and probability).

Teacher Props
Taylor Mali, teacher/slam poet on What Teachers Make. One of my calc students shared this. As Mike warned me, I'll warn you: profanity.

Interesting glass floor and 6 or 7 story dome of the main rotunda.

Planning
An audio link from the new teacher resource center: an audio interview with Suzanne Lieurance, a teacher trainer and children's author, about planning in threes. There's a lot here that's compatible with workshop teaching, emphasis on assessment and evaluation and teaching for engagement. A form for note taking is in the post here.

The graphics are from the beautifully restored Michigan State Capitol that we visited this weekend, which include a couple of nice mathematical designs . Lots of nice 4-, 8- and 16-fold rotational symmetry. And much nice frieze-type translational symmetry, too. The architect was Elijah Meyers, and this was his crowning work.