Pizza Time!

The Internet is in a tizzy yet again about the evils of mathematics education. At least Common Core isn’t being demonized quite as front-and-center as in the recent past, but still.

This time it’s about pizza. Which means every mathematics educator reading this will know it’s about fractions, because that’s why we ever mention pizza in math class. (Random comment: “tizzy” and “pizza” have the same cryptographic profile. Good to know.)

At any rate, Marty and Luis were eating some pizza. Even though Marty ate 4/6 of his pizza and Luis ate 5/6 of his, Marty had more. How is that possible?

The student gave the most obvious answer: Marty’s pizza was bigger. (Alternative answers include that Marty also ate other people’s pizzas and that “pizza” is a mass noun, so Marty had several pizzas.)

The teacher marked it wrong, saying that the problem itself was describing an impossible scenario. The correct answer to the worksheet’s “How is that possible?” should be “That is not possible.”

… and cue the Internet table-pounding. “Moron!” declares the Internet of the teacher. Of the situation, A Plus declares it “so senseless, so confoundingly stupid, and so frustratingly obvious”, elaborating: “Here’s why the question and the teacher’s answer suck.”

Now, the teacher was incorrect. I do not disagree with that. Whether it was a momentary lapse of reason by an otherwise competent third grade teacher or the act of a “f***ing idiot” who “should not be teaching math at all” (as the fine pedagogical experts of the A Plus comments section feel), I can’t say. I don’t know the teacher. I’ve made similar gaffes. Heck, Terence Tao called 27 a prime number on national TV (at 3:00 in the clip), and I dare anyone to say he’s a “f***ing idiot” who “should not be teaching math at all.” G’won, I dare you. So I have no comment about the general mathematical abilities of this particular teacher, but this was a mistake. The student is correct.

What is not clear is the assumption that the textbook was likewise incorrect or poorly worded. I can’t find the answer key for that specific worksheet, but I did find keys for similar sheets* from the same publisher. Here’s the relevant question-and-answer: HalfASalad

As you can see, Pearson (the publisher) is fully aware that the same type-of-thing (salads and, presumably, pizzas) can come in different sizes. This is a better question than the Marty/Luis question for a few reasons:

  • The word “Reasoning” is a little more accessible to the average third-grader than “Reasonableness” (although it’s still Tier II).
  • It’s easier to see salad as coming in a variety of sizes, as opposed to pizza (especially in math class, where pizza is usually just one size).
  • Students are overtly asked to assess who is correct, rather than being given a situation that may well contradict their expectations and being forced to overcome that confusion.

But regardless, there’s nothing so poorly phrased about Pearson’s question about pizza so as to render its answer even remotely controversial: The student is clearly correct, the teacher is clearly incorrect. Indeed, with the salad question, there would at least be some justification for the teacher’s claim, but the pizza question doesn’t ask for an assessment of the validity of the situation. It plainly says that the situation exists and asks why.

I’m not letting Pearson off by any means. My defense of Pearson stops at saying that the question was clear and that the student answered it correctly. The question was not poorly phrased, but it was very poorly presented.

Have I mentioned this is a worksheet for third graders? I did find it online*, just not with an answer key. On the sheet, questions 1 to 6 are plain procedural questions: Write >, <, or =. The fractions all have the same denominator, so all students have to do is identify whether 3 is less than, greater than, or the same as 2. Message drummed. No units. Question 7 asks students, “Why is \(\frac{6}{8}\) greater than \(\frac{5}{8}\) but less than \(\frac{7}{8}\)?”

At this point, students have been primed. They have been given multiple problems with the same denominator, without units. They are implicitly told to not worry about units.

Then comes question 8: “Reasonableness Marty ate \(\frac{4}{6}\) of his pizza and Luis ate \(\frac{5}{6}\) of his pizza. Marty ate more pizza than Luis. How is that possible?” The units appear to be the same (“his pizza”). So what gives?

Of course, students are supposed to recognize that “Marty’s pizza” is one unit and “Luis’s pizza” is a different unit, and we have no way of knowing from the information given whether those units are the same or not. But how does a student pick up on this? The one major clue that this question is different is a five-syllable Tier II word that the teacher may not have prepared them for. Given that the teacher didn’t have this particular answer top-of-mind themselves, I’m guessing not.

There are two more questions on the sheet. Question 9 is another thing that looks like a story problem (i.e., it’s a bunch of words): “Two fractions have the same denominator. Which is the greater fraction: the fraction with the greater numerator or the lesser numerator?” The answer to that question appears to directly contradict the answer to question 8. Then there’s another procedural question, this time in multiple choice form so the third-graders are properly trained for the ACT/SAT they’re going to be taking for keepsies in eight years.

So, the entire sheet is beating the drum that units don’t matter, that the denominator of a fraction is effectively a unit (something the Common Core itself gets close to doing), and so on. In the middle of this is a question the answer to which relies on realizing that units do matter. Students should realize that units matter. They should be reminded that story problems carry assumptions. But this question is a complete gotcha in this context: All the other questions on the page prime students to disregard units, and pizzas are one of the go-to objects for fractions in math class (so students might no longer think of “math class pizza” and “real world pizza” as being the same sort of thing). The teacher can’t be entirely faulted for being in the “math class” zone, for that matter.

Point being: It’s a useful question in an unfair context. Pearson should reframe it (and perhaps they have, which is why I found a different version of the exercise).

* These links might fail if Pearson finds them and C&Ds them into oblivion. As they should, but since the links work now, there you go. If they’re gone by the time you click on them, sorry.



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