In this post, I’d like to reflect on story problems and the purpose of mathematics education. Consider this problem:

Every week, Julie invites some friends over for pizza. Last week, she had four friends over and they ate one whole pizza. This week, she had two more friends over. If she orders two pizzas, how much left over can she expect to have?

On its surface, this is a fairly typical mathematics story problem. It even involves pizza, which is the Universal Clue that fractions will be involved. It has the added bonus that, even on the “reasonable expectations” level that these questions are expected to be approached on, it’s a trick question.

### The “Reasonable Expectations” response

The simple approach to this problem is to convert it to an algebra problem and solve that.

In week 1, there were five people eating pizza. If you said there were four people, there’s the “trick question” part: Surely Julie will be eating as well. Those five people ate one pizza, so if \(x\) is the amount of pizza each person eats, \(5x = 1\) and \(x = \frac{1}{5}\).

In week 2, there will be seven people eating pizza. Those seven people are eating from two pizzas. Let \(y\) be the amount left over, \(y = 2 – 7x\) and \(y = \frac{3}{5}\).

If you had omitted Julie from both weeks, you would have gotten \(y = \frac{1}{2}\).

This is a perfectly acceptable approach to the story problem as given. Students usually get the instruction to “solve this”, and that is indeed a solution.

### Assumptions, so many assumptions…

There are a number of assumptions that are made in forming this solution. These include both linguistic and “real-world” assumptions.

**Linguistic assumption:** “They” in sentence two has the antecedent “Julie and her friends”. This is a fair assumption, but not a trivial one. This could be particularly troubling for English Language Learners (ELLs). Structurally speaking, the most logical antecedent for “they” is “four friends”, excluding Julie. Why would Julie be excluded? Perhaps she had her own pizza. Perhaps she didn’t eat at all.

In context, because it would be strange for Julie to host a pizza party and then not eat anything, we tend to assume that “they” includes Julie. Linguistically, though, that involves constructing an antecedent group from the subject and the object of the previous clause.

Compare the second sentence to this one: “Last night, Julie had to lock her cats in separate rooms because they were having a fight.” The most obvious interpretation there is that the fight did not include Julie: “They” applies to the cats. We rely on context to decide the contextual antecedent of “they”, including whether to build it using multiple antecedent groups.

**Linguistic assumption:** In the third sentence, “more” means that she invited these friends in addition to the ones she’d previously invited. Compare that to the assumption you’d most likely have made with: “If she orders two more pizzas…”. You probably would not have assumed that she’d ordered three total pizzas; she already ordered one, and she ordered two more.

With pizzas, we assume she’s replacing the previous (consumed) order with a new order. With friends, we assume she’s adding to the previous count. But perhaps she has reason to change her social commitments week after week. Perhaps the first weeks are for her knitting circle, and the second weeks are for her field hockey friends, and so on.

We could create parameters that at least two people ate pizza in the second week and at most five people ate pizza in the first week; at most seven people ate pizza in the second week and at least four people ate pizza in the first week. We could use those parameters to solve the problem as a range, and take the midpoint of that range as her expectation.

But that brings us to the real-world assumptions….

### The Real World isn’t as polite as all that

If we assume all of our diners consumed to their hearts’ contents, then we can assume that \(\frac{1}{5}\) of a pizza is “typical”. But maybe Scott was particularly hungry last week, when he chowed down on half a pizza by himself, and now he’s on a diet. And maybe Francine was hungry for a lot more but only got a slice before the masses descended.

Several commentators on this question on G+ and FB pointed out that we only know that the friends ate until they were out of pizza. We don’t know what the typical consumption rate is. Indeed, as one person put it, “Our appetite responds to situational cues. Hence, if there is more pizza, we will eat more.” The most likely scenario, indeed, is that if there are seven friends eating two pizzas, there will be two pizzas consumed.

We’re also assuming that Julie is ordering the same size and type of pizzas. If Julie had ordered a ten-slice medium pizza last week, she might be better off ordering a fourteen-slice large pizza this week. Or perhaps she’s ordering two eight-slice small pizzas, in which case an inferred two-slices-per-person rule would result in two slices (\(\frac{1}{4}\)) left over.

One commentator expressed an assumption that all of Julie’s friends are female by suggesting that the results would be different with males. I wouldn’t be surprised if this were a common assumption, just not one generally expressed.

We also assume that pizza is the only consumable variable involved: What about salad? What about bread sticks? Beverage?

### So… what?

As an educator, I continue to have mixed feelings about story problems.

On the one hand, the original problem is the sort of thing a host might think through, in a more general way, when preparing for a party: They have a sense of how much, in the past, people tend to eat on a per-person basis. A good host will provide an appropriate amount of food, so the goal is to guess how much food will be needed, and then offer just a little bit more. What Julie is doing is what people in that position generally do: *In my experience, people tend to eat two pieces of pizza at my parties, so I’ll order a little more than that. *Anyone who’s ever been at the “pizza ordering” portion of a party no doubt has experience of that sort, where the host goes around trying to count the number of pieces based on either survey or experience.

On the other hand, the burden of the assumptions and the typical exactness of the answer work against keeping the problem in the real world. The question is, How much left over can she expect to have? One respondent answered, “Around half a pizza.” This is a more realistic answer, in my opinion, than “three-fifths of a pizza”. “Three-fifths of a pizza” carries the technical weight of accuracy; even “about six slices” (which makes an assumption about the pieces-per-pizza) feels more like what someone would actually say. The practical purpose of such a math problem would be to use estimation and predictions based on past experience to make plans, and “I’ll have around half a pizza left” is a more typical real-world prediction than “I’ll have three-fifths of a pizza left” is.

But the problem does have a deeper purpose in mathematical education. Because of standardized testing, we tend to teach students how to parse questions quickly, extract the numeric relationships, discard the details, and come up with a final value. That leads to “How Old is the Shepherd?“, where most students won’t even pause long enough to find out if the question even makes sense.

What if we also taught students to dig deep into these problems? One commentator suggested pointing out these flaws was being a smart ass, and another wrote, “Any time a question might penalize a student for thinking correctly beyond the intended confines… [it] should be reworded.”

My suggestion, though, is not to offer this problem to students as one that needs a solution, but rather as an exercise for precisely this sort of discussion: What assumptions are we making? What additional information would we need to ask Julie about? Is Julie’s decision to order two pizzas, given her past experience, a reasonable one?

In that context, it would be better to make that goal clear from the framing of the question, such as:

Every week, Julie invites some friends over for pizza. Last week, she had four friends over and they ate one whole pizza. This week, she’s having two more friends over. How much pizza should she order?

Incidentally, the discussion I’ve seen has failed to comment on the other data that’s presumably available to Julie. “Every week” implies that this has been going on for a while, and yet all the comments I’ve seen have assumed that Julie only has one week of data at her disposal. Was last week typical? What happened two weeks ago? Three weeks ago? If this is a new “weekly pizza party” concept that she’s just trying out, she’d be better off experimenting with different amounts of pizza; if this has been going on for a while, she should have a clearer understanding of what to expect in a given week.

This is the “what are we ever going to use this for?” stuff of mathematics. If we’re teaching mathematics, I feel we need to find a way to be encouraging this, rather than creating a feeling that students will be penalized for thinking outside of the box.

Google Plus discussions: Mathematics Education / Mathematics

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