Ben Blum-Smith inspires a new anti-jargon edu-slogan:

… what happens is that when kids reach a point in their mathematical education where they are asked to prove things, they find

1. that they have no idea how to accomplish what is being asked of them, and
2. that they don’t really get why they’re being asked to do it in the first place.

The way out of this is to give them a crisis. We need to give them problems where the obvious pattern is not the real pattern. What you see is not the whole story! Then, there is a reason to prove something.

Create crisis.

I have become convinced, overwhelmingly, that my value to my students as an educator is bound up less in what I know (though knowing things doesn't hurt) than in my ability to engineer a safe, controlled crisis in their learning process (which, I think, requires knowing a whole lot about something very little).

Blum-Smith's post connects a lot of my recent disconnected writing — in particular, my intention to be less helpful and my interest in a learning process that positions new knowledge as the solution to the limitations of old knowledge. As far as teacher mug slogans go, "create crisis" should start some good conversation in the faculty lounge.

Aside: I'm reviewing some of O'Reilly's Head First material (for some small profit, in full disclosure) and their authors do a fantastic job creating crises. There are two chapters in the Head First Programming, in which the reader learns searching, strings, and branching. Watch how they do this:

• First, you learn how to pull down the text of a webpage for your "employer," the CEO of Starbuzz.
• But then your employer gets sick of looking at raw HTML and you have to search strings to find the price information he's looking for.
• But then he needs an emergency option so he can place an order if the price drops below a certain point.
• But then he needs a notification system, so your program will Tweet him that information immediately.

And so on. It's extremely satisfying.

2.

Nina Simon, a Santa Cruz neighbor, designs participatory museum experiences which she has dubbed "Museum 2.0." She compiled an excellent list of questions that support and suppress participation:

Here are some of the wrong questions:

1. What is the girl in the painting doing? (too teacherly)
2. What does freedom mean to you? (too abstract)
3. How would you define nanotechnology? (too impersonal)
4. What's the best song you've ever heard? (avoid superlatives – they make some people anxious)
5. What do you think? (too general)

Let's set aside the fact that she uses "teacherly" as a pejorative. The applicability of this post to anyone who spends the majority of her work day asking students questions should be obvious.

Furthermore, I haven't set aside the puzzle I posted in You Have No Life. I'm ripping pages out of the playbooks of people who spend their working lives "engaging the imagination so that people don't feel threatened by it": Nina Simon (museums), Jane McGonigal (games), and David Milch (television, quoted). The projects highlighted at the Kickstarter blog (miscellaneous) are essential reading also.

3.

Robert Cringely hops on the higher-education-is-screwed meme in Burn Baby Burn [via]:

Education, which — along with health care — seems to exist in an alternate economic universe, ought to be subject to the same economic realities as anything else. We should have a marketplace for insight. Take a variety of experts (both professors and lay specialists) and make them available over the Internet by video conference. Each expert charges by the minute with those charges adjusting over time until a real market value is reached. The whole setup would run like iTunes and sessions would be recorded for later review.

I'm not in the prognostication business. I draft off more serious thinkers on the open question of the effect of the Internet on higher education. I don't know if I've missed a dumptruck-sized hole in Cringely's reasoning but I found the piece extremely provocative and I'll be revisiting it later in this space.

4.

I want to buy a coffee table book packed with anecdotes just like this one from Ian Garrovillas, illustrating the healthy, cool interaction between a student and teacher.

I tried to graft a structure onto this post but nothing stuck. Topical bullet points from the failed drafts:

1. a description of what happens to the blue students next, of their regrettable slides further leftward and their occasional, triumphant slides rightward.
2. tortured musings about correlation and causation. (ie. "if I take some credit for their progress, must I then accept some blame for dot dot dot et cetera.")
3. a description of effective motivators for my blue students, none of which include teacher approval, parent approval, disciplinary consequences, or perfect attendance badges at the end-of-year assembly.
4. the economies of scale I can't seem to access as a part-time teacher, two of which, however tacky the terms may seem in this context, are "word of mouth" and (even tackier) "branding."
5. really, how irresponsible and inaccurate it is to compare one class to the next and yet, wow, that was some group last year, the first and last group for whom I'll ever take a summer school bullet.

The only draft that mattered was this:

The blue students indulge none of my laziness. They tolerate none of my bad habits. There are all kinds of students at this school — gray students, we'll call them — who will let me slide on all kinds of carelessness so long as I keep them moving toward graduation, college, and career.

But graduation, college, and career are all abstractions wrapped in scare quotes to my blue students. So they pummel my flabby pedagogy daily to the point that I'm burger. Lean burger. You can't believe the gratitude I have for such a challenging year.

Take-home lesson: never underestimate your ability to fool yourself into believing your students understand something when really what they are doing is watching you. To force them to engage the material it is often necessary to restrict their access to you or systematically confound the signals they get from you.

I think this is a central issue for modern math teachers. We need to explicitly develop ways of question-posing and interacting with our classes and individual students that hide or disguise our intentions for how they are supposed to respond. This needs to be part of the core training of math teachers, much more than it already is.

A late link to some really great writing. Forward, retweet, reblog, subscribe, etc.

Two reasons:

1. The early levels are ridiculously easy. Not a serious problem in and of itself. The same is true of Flight Plan, which you'll recall I rather liked.
2. But game play gets harder only over a series of completely nonsensical contrivances. You're dropping gears into a system, blitzing your way through easy. Then on level 21, as the game flips to medium, you're confronted with "no-drop zones." That's really it. Everything else is the same. You're arbitrarily excluded from routes you know would otherwise work for reasons that have nothing to do with the function of gears.

There's no good reason to criticize an iPhone game from this forum except for the robust metaphor it offers for conceptual growth in math. Few textbooks get this right — and I include here the ones that do a pretty good job of being less helpful:

whenever possible, introduce new skills and new knowledge as the solution to the limitations of old skills and old knowledge.

Typical:

Better:

Please argue with me here but I don't think my freshmen really care if career professionals use math in their jobs. This "career" concept is supremely abstract to most and therefore mostly useless to me as a motivator. I've found a much stronger motivator in a palpable sense of forward momentum, in a coherent skill set, in real, uncontrived challenges.

I'm teaching remedial Algebra for a fourth year now and the change I make to my curriculum far more than any other is to add this connective tissue.

You're comfortable with a dot plot? Fine. Let's put you in a place where a dot plot is tough to execute — say, a large data set with no mode and a huge range. That's annoying. Then bring in the box-and-whiskers, the histogram, or whatever. I try not to introduce the next concept simply because it's the next chapter in the book or the next bullet point on a list of standards or because it's "what we're learning today." In other words, I try to stay away from the no-drop zones.

Academic Green Circumference and Area Problem from Kyle Webb on Vimeo.

First, let's pay respect to how fast the video moves, how it sets a scene and establishes a problem in just 14 slides and 57 seconds. Webb knows his audience and its attention span. Also, none of this is stock photography. Every photo selected is of high bandwidth and relates directly to the problem. After 12 seconds, we have three different views of the lawn. After 15 seconds, a panoramic shot. I'll begin my redesign 23 seconds in, when he mentions the lawn is 75 steps across.

This is really, really close to my textbook's own installation of the problem. The text would ask a question like "how far is it around?" or something with a real-world spin like "how large would the ice rink be?" (standing in for "what is the area?") and then it would explicitly define the only variable we need: 75 steps. My students would identify the formula and then solve.

This kind of instructional design puts students in a strong position to resolve problems the textbook draws from the real world but in no position to draw up those problems for themselves. This kind of instructional design also yields predictably lopsided conversation between a teacher and his students.

The fix is simple but difficult: be less helpful.

Let's start here: is circle area just something math teachers talk about to amuse themselves or do other people care? If they care, why do they care? How do we convey that care to our students? Maybe someone needs to fertilize the lawn. Maybe someone wants to spray paint the dead lawn green in the winter. Without this component, the answer to the question "how far is it around?" is little more than mathematical trivia to many students.

So put them in a position to make a choice, a tough choice that's true to the context of the problem, a choice that math will eventually simplify.

For instance: "how many bags of fertilizer should I buy to cover the entire lawn?"

Or, a little weirder: "how many cans of spray paint should I buy to cover the entire lawn?"

In both cases, we're putting every student on, more or less, a level playing field. They are guessing at discrete numbers (ie. "fifty bags — no — sixty bags.") and drawing on their intuition, which, from my experience, is a stronger base coat of for mathematical reasoning than the usual lacquer of calculations, figures, and formula.

This approach also forces students to reconcile the fact that the problem is impossible to solve as written. This is an essential moment. They need more information, but what? What defines a circle? Would it be easier to walk across the lawn's diameter or around the lawn's circumference? Which would be more accurate? Why is the radius difficult to measure? Did Kyle really walk through the center of the lawn or does he just think he did?

When you write "75 steps" on a photo, that conversation never happens.

My thanks to Kyle for jogging my thoughts here.

[previously]

Everyone out there seems so full of love for the students and the job that it carries them through the long hours, but it hasn't been enough for me to break out of the vicious cycle of frantic work and procrastination I've been stuck in since first grade.

This is as good a description of teaching's tumultuous first year as you'll find out there on the blogs. It also summarizes:

1. teaching's great deception — "love your students and the rest will follow."
2. teaching's jarring transition — from sleepwalking into your 08h00 MAT 180 class to teaching your own classroom of sleepwalkers where every bad work habit you've accumulated over your entire life pays off huge negative returns.

Let's table this post for a few years. It took me five years to feel even a little put-together in this job, to feel like I wasn't just scrambling to keep pace, but I give Alison half that.

Had I gone to grad school this year, I would have put some time into a collage of new teacher profiles. Not my kind of new teacher. Not the traditionally inducted teacher, two mentors assigned by his district over two years, mentors who in all likelihood teach an unrelated subject. The sort of new teacher aptly described by statistics like "50% attrition rate."

Rather, scan the list of commenters at Alison's blog, scan her Twitter crowd (Twitter account required, sorry), and tell me you don't think she's going to bend the induction curve upwards.

Let's assemble a control group. We'll have the experimental group spamming questions at jackieb, jdyer, dcox21, colleenk, samjshah, k8nowak, sweenwsweens, dgreenedcp, et al, while blogging the experience as time permits.

I don't know if it's any kind of model. I only know it would've made me a much happier teacher, much sooner.

The deceit here is vaguely mathematical so I asked my first class of students, "What is it selling and how does it try to sell it?" Most identified the product as some kind of weight loss formula and/or protein shake. No one identified the deceit.

So I asked my next class, "What is it selling and how does it lie?" Many suggested we were looking at two different people here, which I said was false (with about 99% certainty). One student thought the ears were the giveaway, which is true, though not the giveaway he thought they were.

I guess I'm curious a) if you notice the deceit and b) if there's any way to translate that deceit into an actionable math unit?

Update: Yeah, it's the distortion, which is about 17% too wide in the first image and 17% too tall in the second image. Should it worry me that none of my students caught it?

Here are my estimations of the undistorted images.

Graph: effectiveness of ad v. percent distortion.

[Update: Holy cow. They fixed the proportions in the latest ad buy.]

Appeal To Their Intuition

"How much cash is this?" Take guesses. The student risks nothing with a guess but that investment pays off huge for the teacher over the life of the exercise because the student wants to know who guessed the closest.

Build Slowly

Again, ask "how much cash?" but also ask "how heavy?" Show them the weight. (I zeroed out the jar from every weight measurement you'll see here. Don't worry about it.) Spitball some ideas for determining the value of those coins. You're trying to motivate the idea that the weight of the coins ties directly to how much the coins are worth. Pull up the relevant Treasury website.

Then mix in some nickels. Scoop out a small sample. Play with that. Set up a proportion between value and weight.

Iterate

Now you have pennies, dimes, nickels, and quarters. I took nine sample scoops, everything from small to big.

I formatted these at 4×6 so I could print them out at our local one-hour shop for a few bucks and put one in front of every student.

Throw A Curve Ball

Some will finish quickly. You tell them you have a jar of coins that weighs 5,500 grams. You reach in and pull out 14 nickels. How much is the jar of coins worth?

They'll run these calculations and come up with an estimate of $55. You tell them it was really $34, which is huge error. Ask for sources of error. Then toss this up and talk about it.

Confirm The Answer

$84.00, if you were curious.

It's essential to give some kind of visual confirmation of the answer, both so we can give credit to good initial guesses and so we can talk about sources of error. (ie. "who was off by the most? did sample size matter at all?")

Miscellaneous

1. Show them CoinCalc, the backend of which does exactly what we've done here.
2. This activity follows-up nicely on the goldfish activity, where we used a small sample of fish to determine the total population of a lake.
3. We yield the floor to Jason Dyer and anybody else who would like to debate the question, "why are we doing this digitally?"

Download

Here's the entire learning packet [62MB].