## [Confab] Snow Day

Earlier this week, Matt Reinhold tweeted:

Fearing our buses wouldn’t start due to cold, our district let them idle overnight. The first student question this morning: “How much did that cost?”

1. That’s kind of amazing.
2. There’s a local, personally relevant, real-world math problem somewhere in there for students to work on and learn from. But one of my theses with fake-world math is that relevance and the “real world” aren’t necessary or sufficient. They don’t guarantee interest and they don’t guarantee learning.

So tell me about an effective treatment of this situation in math class. (Draw on research on curiosity, abstraction, and the CCSS modeling framework if they’re helpful.) Also tell me about an ineffective treatment of this situation in math class.

BTW. “Curriculum Confab” will be a recurring feature around here, similar to our early “What Can You Do With This?” days only with more design and theory attached.

2014 Feb 02. Molly helps out enormously with this confab:

Ineffective: If gas costs 3.38 per gallon, and the bus burns 1.1 gallons per hour idling, what is the cost of the fuel burned by 32 buses over a period of 13 hours?

Effective: 1. What questions do we need to ask in order to answer this question?

The first treatment offers no “information gap” of the kind that’s generative of student curiosity. Moreover, curious or incurious, the first treatment doesn’t have students doing modeling of the sort promoted by the CCSS, where students set themselves to “ identifying variables in the situation and selecting those that represent essential features.”

I’d only add one question to Molly’s effective treatment: “How much would you guess it cost the district to keep the buses idling overnight?”

I'm Dan and this is my blog. I'm a former high school math teacher and current head of teaching at Desmos. More here.

1. #### Kenneth Tilton

January 29, 2014 - 2:55 pm -

Google says \$1-\$4 an hour depending on engine size. Even with childhood obesity I suspect school bus engines are on the low side, let’s say \$30 to idle from 4pm one day to 7am the next. So far it is fun but trivial arithmetic.

What gets interesting is estimating the cost of getting the things started if we let them drop to ambient temperature — diesels are not like gas engines, so we prolly cannot get them started without heating them externally which will take so long that (a) the cost of the mechanic getting them started blows away the fuel cost and (b) we are adding the risk cost of kids standing in the cold waiting for buses that are not coming.

At this point \$30 a bus (a dollar a kid) looks like — well, let;s just say the district and its fleet mechanics knew what they were about.

Here’s a fun one: some Europeans I met said the law required them to turn their cars off at a red light. I asked if anyone had worked out how much extra fuel it cost to restart an engine compared to idling — starting takes quite a gulp. They all yelled out the answer together. Left as an exercise.

-kt

2. #### Grant Wiggins

January 29, 2014 - 4:33 pm -

Have you seen my blog post on the meaning and history of ‘authenticity’? It supports your point and expands on it, based on my original definition of the concept 25 years ago.

grantwiggins.wordpress.com

3. #### M Ruppel

January 30, 2014 - 5:32 am -

The question burning in my mind is ‘was it worth it,’ but the modeling gets hopelessly complex to the point where you would have to give kids too much information. They would have to account for the cost of a mechanic to fix a bus that won’t start, find some way to measure the cost of a kid not getting to school because the bus won’t start, etc, and then take the probability of that happening into account…not the kinds of calculations that are readily approachable

It seems like this could lose ‘authenticity’ really quick with kids that might not have engaged that question. You could take a great context and blow it pretty quickly that way

As a lower middle school task (Grade 6 or 7), we could get pretty hands-on with the question ‘how much did that cost’

Take the kids outside, let them start the bus and monitor the gas level for thirty minutes of idling (or do it in a video yourself), and then scale that for the overnight and the size of the fleet. Multiply by the cost of gas and you’re there. I would want to let kids make assumptions about the size of the fleet, the hours each bus was left on, and the cost of gas. Discuss how the assumptions affect the model.

Sounds like gwe are hitting SMP.4 there as well as some pretty in-depth ratio analysis…

4. #### Sammy

January 30, 2014 - 7:01 am -

Perhaps there are several ratios worth examining when asking “was it worth it?” For high school students, particularly those also in Chemistry, estimating how many pounds of carbon dioxide were emitted from idling buses all night is also an interesting question. Many municipalities have laws and regulations about how long buses can idle and how close to buildings because of the health implications. (Most of the needed information for solving this problem is easily found online, or in curricula such as Chemistry in the Community.)

What was the cost of this idling bus decision – economically, environmentally, and socially?

5. #### Kenneth Tilton

January 30, 2014 - 7:12 am -

“What was the cost of this idling bus decision – economically, environmentally, and socially?”

If students can prove that the optimal choice would be to declare a snow day it would answer why they should learn algebra.

6. #### Dan Meyer

January 30, 2014 - 7:45 am -

M Ruppel identifies two important dilemmas here:

One, it’s important to sand this problem down to a size that is manageable for students but not so smooth that it feels like the context exists just to disguise some math.

Two, how should students get the information necessary to answer their question?

7. #### wwndtd

January 30, 2014 - 8:28 am -

I jokingly submitted “The amount of black ice formed from the exhaust pipes when water is generated from combustion.” on Twitter as a terrible question. It’s bad for the reasons Dan just described (basically a poor camouflage for calculations).

I don’t have a problem with students needing to search online for info, as long as they’ve been training to sort through all of the bad info they get. I know Language Arts teachers have to teach kids to do this now, and maybe it’d be good for some math/science training too.

Maybe it’s better (like Sammy said) to leave the question at “Is it worth it?” so that the kids can decide how to define the question. But then, it still could be a poor mask for doing calculations.

8. #### Molly

January 30, 2014 - 10:52 am -

Ineffective: If gas costs 3.38 per gallon, and the bus burns 1.1 gallons per hour idling, what is the cost of the fuel burned by 32 buses over a period of 13 hours?

Effective: 1. What questions do we need to ask in order to answer this question? 2. What conditions in the district, state, national or global economy/environment would make it worthwhile to answer this question?

I’m not sure calculating the cost would actually be the point!

9. #### William Carey

January 30, 2014 - 1:31 pm -

An interesting question is this: how do you know whether you’re right? The more interesting question is this: did the *adults* do the math?

Somewhere in the bowels of the the school (county?) office, is there an excel spreadsheet? It takes a few variables – number of busses in the fleet, gallons per hour, cost of diesel, cost of a snow day, and spits out a yes or no.

An intriguing (if high risk?) exercise would be to divide the students into groups. Have them discuss what variables would go into that spreadsheet and why. Have each of the groups write a letter to the district office explaining what factors they think are relevant and why, and asking for a response explaining how the actual *adults* made that decision.

If it turns out that the adults do, in fact use some sort of math, then you can have the students play with it and try to understand why the adults did what they did. If not, maybe have the students propose something?

10. #### Dan Meyer

January 30, 2014 - 3:44 pm -

@Molly, thanks for the taking the task head-on. Can you give a brief rationale that I can quote for why one task is ineffective and the other one effective?

11. #### Molly

January 30, 2014 - 4:55 pm -

@Dan, as I learned from you, giving students the exact information they need to solve a problem in the exact format in which they need it is ineffective! The idling buses are a real world situation, but turning it into a real-world formulaic question isn’t an actual improvement over what they do in traditional textbooks. The intriguing part of the idling bus depends on your background and passions; do you care because it might waste money? because of environmental harm it might be causing? It’s a great opportunity to explore the concept of skewed information – the data you collect and the formulas you use might be directly related to the point you are trying to prove, pro or con.

12. #### Bruce James

January 30, 2014 - 9:13 pm -

re Molly: “I”m not sure calculating the cost would actually be the point!”
Too often, our reflex in math is “what’s the answer? Am I right?”
Calculating the cost is simply the period at the end of the sentence. The glory lies in the sentence, in the construction, in the overcoming of itchy, scratchy puzzlement that happens up to the period that ends the sentence that brings the problem to closure.
The idling bus problem is thick with potential. You can spend lots of time just estimating and defending and revising. Every step of the problem is begging you to critique the reasoning of your peers.
As an ancillary to this you might have a mini lesson called How cold is cold?”(here come integers)
We keep math questions sequestered in discrete chunks of calculations because those are the ones that bring the fastest closure.
I think someone commented on the Magic Octagon, or some other one with, “great, where’s the lesson?” The lesson is in your ability to see and hear and dream up the long and diversified narrative that can spin out of such problems.
What’s kind of shape was that hand holding?, How did you know? Draw an octagon, draw a regular octagon, how many degrees are in an octagon?, Hey, how do you use a protractor, and on and on unto a full problem.

13. #### Michael Pershan

February 1, 2014 - 7:55 pm -

I’m having a hard time actually doing good work on this particular problem, but I have an observation to toss into this discussion.

How is it possible for kids to become engaged in hard problems?

A problem needs to be doable for kids to get interested by it. But if a problem’s doable, then what makes it hard?

(We often cheat by saying that “It’s not too hard or not too easy.” No wishy-washiness here. Can the kid solve the problem, or can she not?)

Here’s what I think is going on in a lot of these problems. We give kids a question that they do have the tools to answer…poorly. Maybe their tools are inefficient, slow. Maybe their tools are inaccurate, sloppy. Either way, the question is immediately doable with their poor methods, and hence potentially interesting. The key is to find problems that pair lousy quick answers with better, tougher ones.

14. #### Dan Meyer

February 2, 2014 - 9:44 pm -

Michael Pershan:

Here’s what I think is going on in a lot of these problems. We give kids a question that they do have the tools to answer…poorly. Maybe their tools are inefficient, slow. Maybe their tools are inaccurate, sloppy. Either way, the question is immediately doable with their poor methods, and hence potentially interesting.

This harmonizes nicely with Harel’s writing on need. It’s great if students know it’s possible to solve a problem given their weak, small tools but so undesirable they’ll be interested in learning about stronger, larger tools.

It’s a nice way to turn arithmetic into algebra. Sure you could figure out the 2,014th term in the arithmetic sequence 13, 14.5, 16, etc., by adding but it sounds annoying. Anybody got a better tool than arithmetic?

15. #### Evan Weinberg

February 3, 2014 - 6:13 am -

Dan and Michael:

To toss my hat into the ring, I’d suggest that a computational tool (e.g a spreadsheet or Python/JavaScript program) would be a nice intermediate step in moving toward the abstraction of algebraic tools.

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