## (One Of Many Reasons) Why Students Hate Algebra

A youth group with 26 members is going to the beach. There will also be 5 chaperones that will each drive a van or a car. Each van seats 7 persons, including the driver. Each car seats 5 persons, including the driver. How many vans and cars will be needed?

Background

Tuesday was an all-school professional development day. The math departments joined from two campuses to learn about the Gradual Release of Responsibility from a couple of math coaches from the next county over.

One coach modeled a GRR lesson and opened with the problem above.

I leaned into another teacher and whispered, "I'm trying to decide which would be more socially acceptable right now, letting out a loud fart or saying what I really think about this problem."

We broke for lunch and came back to debrief. No one had commented on the problem by the end so I did.

"I see problems like this and I feel myself becoming less of a human and more of a math teacher. And I feel very lucky to teach our neediest students, students who punish me daily for problems like this one, students who are often very hard on me but who in return have helped me hold onto some of that humanity.

"I have three questions about this problem and we can discuss any of them or none of them."

Three Questions

"One, is the problem realistic? Would a real person need to solve this problem?

"Two, is the solution realistic? Would a real person solve the problem using a system of two equations?

"Three, in what ways does this problem help our students become better problem solvers?"

Elaboration

I didn't elaborate. I thought my questions were self-evident and their answers self-explanatory. I was wrong. The coach shrugged me off, saying, "Well, it's in your textbook." and I couldn't disagree. None of my colleagues seemed disturbed by the opening exercise of this quote model lesson unquote, so I didn't belabor the point. In hindsight, I wish I had soapboxed:

1. This is a problem you will only find in a math textbook. It's bizarre to me how many different ways just fifty words can fail to square with reality. Why does each chaperone have to drive? Why can't we take five vans? Why do our vehicles have to seat the exact number of people in our group and no more?
2. No youth group leader would ever solve this problem with a system of equations. I'd wager that no math teacher, if somehow faced with this completely fantastic scenario, would solve this problem with a system of equations. With 31 people, we'd just shuffle them around until they fit. Even if we insisted on the contrivances in #1, there are only [0, 5] possibilities for the vans so we'd use a table or just guess and check. ¶ I asked the coach why we were forcing the issue of systems when the easiest solution by a long shot was tables. She replied that we learned tables last class and this is the new skill we're learning.
3. This kind of algebra makes our students dumb, unimaginative, and scared of real problems. At the end of the model lesson, the coach put up our homework, which was a carbon copy of the original problem, new numbers swapped in for the old. ¶ I can't describe my contempt for this arrangement. ¶ This is how we make kids stupid and impatient with irresolution, eager for contrived problems that look just like the last contrived problem, completely lost if we so much as switch around the order of a few words. "We don't teach them problem solving skills anymore," my department head said to me. "We teach them problem types."

Algebra teachers sell students a cheap distortion of the real world while insisting at the same time that it really is the real world. The cognitive dissonance is obvious and terrible. Students know the difference. It cheapens my relationship to them and their relationship to mathematics when you ask me to lie to them.

It's like offering someone lust or manipulation while insisting that it's love. Not only are the short-term consequences devastating but it makes that person distrustful or wary of the real thing. Make no mistake. We are making an alien of algebra. We are doing real damage here.

## Final Exam Question #51

Who is better at Doodle Jump? Mike or Dan? Why?

The first semester ended, not with a bang, but with two days of canceled class1 and two days of hasty final exams. My remedial Algebra class spent a lot of time this semester on what California calls computational fluency and what I would rather call the awesome descriptive power of numbers.

Which has meant, thus far, everything from times tables to proportions to infographics all leading to the motivation for the question above: when your friend is being kind of insufferable about how good he is at Doodle Jump, you can use numbers to shut him up!

It is a feature not a bug, in my opinion, that Mike and Dan can draw their own self-serving conclusions from the same set of numbers.

1. … because you can't be too careful with those Santa Cruz tornadoes.

## How Do You Turn Something Interesting Into Something Challenging?

[Correction: an oil barrel contains 158,987.295 ml.]

Nat Torkington writes the Four Short Links column for O'Reilly's Radar, highlighting interesting articles around the web on a daily (or near-daily) basis. Recently, he's pitched me a few links via e-mail under the heading "WCYDWT?" which, due to my fallen nature, I have taken as a challenge to my sacred honor.

Here's one: the relative price of different liquids which illustrates the disturbing fact that HP printer ink is several orders of magnitude more expensive than crude oil.

So I opened our first day back from winter break with a learning moment built around Nat's link and then recorded video of the moment which you'll find below. My apologies in advance for the pitiful production value. Initially, I was going to forward this only to Nat as some kind of retort but I found the experience so difficult, messy, and exhilarating, I had to debrief myself here. Notwithstanding the video quality, you're welcome to pummel me for anything you see.

Classroom Video

Color Commentary

Synonymous with "What Can You Do With This?" is "How Do You Turn Something Interesting Into Something Challenging?" I have asked educators that question on this blog, in online classes, and in several conference presentations over several years. Here is — by far — the most common answer:

"I'd put it on the wall and we'd talk about it."

Which is a weak start. A certain kind of student inevitably dominates these pseudo-Socratic discussions and then invites another kind of student to disengage. But Nat has dealt us a strong hand. If we play those cards right, we can retain and empower a lot of those (mathematically and conversationally) reticent students.

1. Calm down with the math for a moment. Invite their intuition.

At one point in my career, I would have led this off by giving them all the data and asking them to compute the ratio of cost to volume. but my blue students are poorly-served by that approach. So many of them have been burned so badly by math that if I open the conversation with terms like "ratio" and "volume," pushing numbers and structure right at them, I'll lose the students I want to keep. Moreover, this confuses master with slave. We use math to make sense of the world around us more often than the reverse.

So I put seven liquids on the wall and asked them to rank them from most expensive to least. Simple speculation. Nothing more mathematical than that. Please imagine, here, how much more fun it is to walk around and talk about the question, "Which do you think is the most expensive?" rather than the lead balloon "Which has the highest ratio of cost to volume?"

Ask a student to come up and share her ranking with the class. Argue a bit. Entertain opposing opinions. Ask a student if he'd trade a can of Red Bull for a can of his own blood. Student investment at this point is very nearly 100%. It's mine to lose.

2. Slowly lower mathematical structure onto their intuition.

"Here's the answer," I told them, but students know at this point to triple-check me. Several went straight for Red Bull, which totes does not cost \$51.15.

"So you're saying that how much you get matters as much as how much it costs."

Fine.

We used cell phones to text Google and ask for unit conversion. This always strikes my students as magical and suspicious.

And here, finally, we talked about the ratio of the cost of blood to how much blood you get. I asked them to visualize one milliliter of blood. "What does .40 mean?" We talked about the cost of one milliliter and how it's useful to compare that cost across liquids.

The rest (hopefully) writes itself, though, for the record, I kind of hate how explain-y I get in the last third of the video.

The Virtuous Cycle Specific To Our Line Of Work

1. Find an interesting thing1.
2. Transform that interesting thing into a classroom challenge.
5. Repeat.

The feeds in your reader then spiral upwards and out of your control. WCYDWT ideas begin to pile up faster than you can capture them. It'll freak you out and you'll wish you could turn it off for just a few hours while you're watching TV but you realize this a rare ancillary benefit in an occasionally tortuous job and you accept it gratefully.

[BTW: Mr. K rightly points out that this problem is of a piece with the nickel thieves from a few years back.]

[BTW: You should read Burt's commentary on the lack of real-world meaning of these statistics.]

[BTW: Great list of liquids and prices here.]

1. It's sad how often the conversation with other teachers ends here, after it becomes obvious that they just aren't interested in all that much.

## Put THAT On The Fridge

There aren't a lot of firsts left in your life when you're twenty-seven so imagine my exhilaration last week at Google when I encountered an annoying technical problem and rather than grind the solution out over several hours of pointing, clicking, and transcribing, for the first time ever, I wrote twenty lines of code that solved the problem in several minutes.

I created something from nothing. And that something did something else, which is such a weird, superhuman feeling. I've got to chase this. I'm on a very dark path right now.

At the same time, I'm going to resist to the bitter end the urge to write up some grand prescription for education-writ-large based on something as flimsy as my own personally satisfying learning experience. There's way too much of that in the edublogosphere without my own contribution.

## The Blue Students

These are my people, my students this year. They're averaging just a bit above a 1.5 GPA.

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.