This week’s PreAlg PoW, which told the story of a truck driver in Albequerque who drives to many cities in the west and midwest, inspired many different interpretations.

Students made different assumptions which led to different solutions. Most of the assumptions were ones I was expecting (like whether the driver returns to Albequerque after she visits each other city). But some interpretations of the problem made me scratch my head!

I’ve been thinking a lot about what math teachers call mistakes or misconceptions. I believe there are mistakes in math class, and that not all mistakes are useful. For example, I routinely do things like add 2 and 3 and get 6. Other than reminding me I need to get even better at checking my arithmetic, I don’t learn much from that mistake.

On the other hand, I see a lot of math classes in which teachers have said, “mistakes are great, we can all learn from them!” and when students have what the teacher views as a misconception, the teacher might thank the student for airing it. But then the teacher says, “but *this* is how we do it.” That kind of response, in my mind, shuts the student down.

The fact is, research and experience and common sense all show that when you have an alternate hypothesis in your mind, you won’t let it go until you have come to believe *for yourself* that it isn’t true. We’ve all seen how a student’s misconception can persist in the face of evidence until they really internalize a better theory. Being told “this is how we do this” doesn’t help, because you usually forget what you’re told if it doesn’t agree with your ideas.

What if we called some kinds of misconceptions “working hypotheses” instead? “Working hypothesis” implies to me:

- it should be respected.
- it should be the community’s job to understand it, describe the implications, and test it out.
- it will change, but it will change based on evidence and process, not on being told.

So what does that have to do with this week’s PoW? I wonder if it’s possible for me to look at some of these off-the-wall solutions, describe the working hypothesis, name the implications, and provide the student some evidence to confirm or repute the hypothesis. It will be a challenge and I hope you’ll play along!

First I did 31×5,000.The answer is 30000.

My first thought is that he noticed the 5,000 from the problem, and I wonder if the 31 comes from the number of days in August. Maybe this sentence in the problem, “For the month of August, Lani’s logbook shows that she traveled about 5,000 miles.” gave him the hypothesis, “Lani traveled about 5,000 miles per day, each day in August.” That makes sense if you think of a logbook as something you write in each day.

The implications of that are that Lani spent a *lot* of time driving and she made many, many trips to each city. For the student, I’d want to help him make his units explicit (is 30,000 miles per month? Or some other period of time? What are the units of the 31 and 5,000?) so he knows what his working hypothesis is. Then I’d like him to address the other questions based on his working hypothesis. How many trips to each city might she make each day? Each month? How many hours might she spend behind the wheel. I think he may encounter a contradiction when he thinks about how long it takes to drive 5,000 miles. At that point, he may need to question his hypothesis about the units of the 5,000 miles in the problem (it might be miles per week or per month. What’s reasonable?) Comparing his work with another student’s might also be fruitful, if both students had good skills for listening and comparing thoughtfully.

The answer to the question is she spent 200 hours behind the wheel in the month of August. She went to Chicago 5 times,she went to Denver 16 times, she went to El Paso 3 times and she went to Los Angeles 1 time.

I solved the problem by making a chart for the list of where she went and how many times. On the chart I put how many miles it took her on the way there and on the way back. Then I divided how many miles it took for the whole trip then I divided that by 65 because that was how many miles per hour she was going then i divided that by 8. I divided that by 8 because that was how many hours she worked and I got 5 remainder 1 but rounded that up to 5. 5 is the total amount of times she went to Denver I did that with all the other cities too.

I think this student might have had a valid working hypothesis, and then overgeneralized it. Here’s one possible interpretation of the work. They figured out how many days out of the month the trucker would spend on a trip to Chicago, by dividing the round trip distance to Chicago by 65 mph, and then the number of hours by 8. The 5 that the student refers to would mean that each round trip to Chicago would take 5 working days. That’s a useful calculation to do and could be used to make a reasonable list of trips. However, the fact that we don’t know what percentage of her time the trucker spends driving, nor how many days she works per month, might make any estimate based on hours less accurate.

Then, there’s the issue that the student took a quantity that should have had units of “days spent per trip” and instead used it as “number of trips.” To me, that’s a plain old mistake that stems from not organizing carefully what the numbers in the calculation really mean (Annie wrote about this issue in a blog post last week). I’d start by having the student talk through the calculation, saying the units (what are you counting? What will the result of that calculation tell you?). I’d help him or her to realize the genuine mistake.

Once the student was clear about the units in the calculation and knew 5 measures “days a trip to Chicago takes” I would ask them to think about what they know about the trucker’s work in August. I’d be curious to hear their reasonable estimate for how many days she spent driving, as well as any thoughts they had about possible sources of error. I’d also be curious to hear what other facts and figures they noticed about the trucker that could be used to check their estimates.

790-446=

X=344 miles between LA and Denver

She was behind the wheel for 25 hours

Last but certainly not least is one that’s really got me wondering. I agree that 790 – 446 = 344. I don’t know how they estimated the 25 hours behind the wheel (344/65 = 5ish). I’m also not sure how they chose to focus on the LA to Denver portion, or how they thought LA, Albuquerque, and Denver were situated on the map to be able to subtract to find the miles between LA and Denver. It’s awfully tempting to say that this student doesn’t have any working hypothesis. On their other hand, they clearly did some math and even explained a bit about what they did. Can you name a working hypothesis this student might have, or what experience you’d want them to have next?

**Some “Getting Your Kicks On Route 66″ links** in case you are interested:

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- Information about becoming a Math Forum Problems of the Week Member. Compare prices – consider starting with a $25 membership giving you access to all of this year’s Current PoWs — and now you can create 36 student logins as well!