I’m thinking a lot about how my 6th graders responded to a pre-lesson task in “Interpreting Multiplication and Division” — a lesson from Mathematics Assessment Project (MAP) .

MAP lessons begin with a set structure:

Before the lesson, students work individually on a task designed to reveal their current levels of understanding. You review their scripts and write questions to help them improve their work.

I gave the students this pre-lesson task for homework, and 57 students completed the task.

I’m sharing students’ responses to question 2 (of 4) only because there’s already a lot here to process. I’m grouping the kids’ calculations and answers based on their diagrams.

**Each pizza is cut into fifths.**

About 44% (25/57) of the kids split the pizzas into fifths. I think I would have done the same, and my hand-drawn fifths would only be slightly less sloppy than theirs. The answer of 2 must mean 2 slices, and that makes sense when we see 10 slices total. The answer 10 might be a reflection of the completed example in the first row. “P divided by 5 x 2 or 5 divided by P x 2” suggests that division is commutative, and *P* here must mean pizza.

**Each pizza is cut into eighths.**

Next to cutting a circle into fourths, cutting into eighths is pretty easy and straightforward.

**Each pizza is cut into tenths.**

I’m a little bit surprised to see tenths because it’s tedious to sketch them in, but then ten is a nice round number. The answer 20, like before, might be a mimic of the completed example in the first row.

**Each pizza is cut into fourths.**

I’m thinking the student sketched the diagram to illustrate that the pizzas get cut into some number of pieces — the fourths are out of convenience. The larger number 5 divided by the smaller number 2 is not surprising.

**Each pizza is cut into sixths.**

It’s easier to divide a circle by hand into even sections, even though the calculations do not show 6 or 12.

**Each pizza is cut into fifths, vertically.**

Oy. I need to introduce these 3 students to rectangular pizzas. :)

**Five people? Here, five slices.**

Mom and Dad are bigger people, so they should get the larger slices. This seems fair. We just need to examine the commutative property more closely.

** Circles drawn, but uncut.**

I’m wondering about the calculation of 5 divided by 2.

**Only one pizza drawn, cut into fifths.**

Twenty percent fits with the diagram, if each person is getting one of the five slices. The 100 in the calculation might be due to the student thinking about percentage.

**Only one pizza drawn, cut into tenths, but like this.**

I wonder if the student has forgotten what the question is asking for because his/her focus has now shifted to the diagram.

** Each rectangular pizza is cut into fifths.**

Three kids after my own heart.

**Five portions set out, each with pizza sticks.**

I wonder where the 10 comes from in his calculation.

**Five plates set out, each plate with pizza slices.**

Kids don’t always know what we mean by “draw a picture” or “sketch a diagram.” This student has already portioned out the slices.

What diagrams and calculations would you expect to see for question 3?

There’s important work ahead for us. The kids have been working on matching calculation, diagram, and problem cards. They’re thinking and talking to one another. I have a lot of questions to ask them, and hopefully they’ll come up with questions of their own as they try to make sense of it all. If I were just looking for the answer of 2/5 or 0.4, then only 12 of the 57 papers had this answer. But I saw more “correct” answers that may not necessarily match the key. We starve ourselves of kids’ thinking and reasoning if we only give multiple-choice tests or seek only for the answer.

That’s why Max Ray wants to remind us of why 2 > 4.

## 7 Comments

Your blog system just timed me out and I lost my comment.

I am going to do it in notepad and try again.

GRRRR!!!

So sorry, Howard. Not sure how to fix this as you’ve mentioned this before. I’m all ears if anyone knows what I can check into to fix this pesky feature.

This is fascinating.

some thoughts on no. 3

1: Square cake for this one

2. How would they cope with the cake cut into 12 pieces, and Max eats two of them

Some possible follow-up questions for the pizzas:

1: Can you explain how you got this calculation?

2: Do you think it is right/can you check your answer?

Some thoughts inspired by George Polya (I am a Polya fan)

1: Is there a simpler related problem that I can solve?

2: None of them thought of putting one pizza on top of the other one.

I was also thinking about how they would proceed if they were not asked to produce a calculation.

The last person in your first table (1/5 + 1/5) clearly ignored the instructions, and probably saw (visualized) the process almost straight away.

Too much specification of “how to do it” can limit the brain activity, and not only that, it goes against the Common Core philosophy. I have just posted a criticism of one of the PARCC sample high school geometry test items on these grounds.

I haven’t taught a MAP lesson yet but I gave the preassessment for the Repeating Decimals 8th grade lesson and can’t wait to teach it. It’s a lot of prep but I think teachers need to realize that the lessons that require the most prep tend to be the richest.

It seems to me that the kids did remarkably badly on this question, relative to the standard “math-teacher” answer: 2/5 of a pizza.

My hypothesis is that the students were struggling mathsemantically to identify the terms in which their answer should be expressed and were inhibited by their real world experience of sharing pizzas. In vernacular, we count multiple whole pies using “pizza” as the unit and fractions of a pie as “slices.” In a math classroom context, “slice” is not a sensible unit because the whole point is to distinguish differently sized pizza fractions and “slice” is not a uniformly defined amount of pizza.

As to real-world pizza sharing, the common restaurant experience is to receive pre-cut pizza, typically sliced into 8 pieces (though 4 and 16 are also standard for large or small pies) and divvy up the slices. Remainders are seized by those with the highest product of gluttony times reflex speed.

Finally, I can’t resist harping on the overwhelming problem of using pizza as a unit for even whole pies. In the problem why should we believe that the two pizzas have the same size and, even if they do, they probably don’t have the same toppings. Can we really trust that 2/5 of a pizza has any useful comparability (for fair sharing purposes) to 2/5 of a different pizza? My children would loudly argue: no!

So, I suggest being very careful using pizzas for fraction questions, even though it is so tempting. Fractions of a single pie: ok. Fractions across multiples pies: avoid.

If any of this is correct, then the students should do a lot better with the next cake question.

Have you shared calculation, diagrams and problem cards? I love that idea!

Download .pdf: SEWELL, Martin, 2011. History of the efficient market hypothesis . Research Note RN/11/04, University College London, London.