Today, Dana from one of my middle schools let me take over her first period and tackled Robert Kaplinsky’s In-N-Out burger task (with some modifications).
Together, she and I re-created the Problem Solving Framework into a Google Doc. After some deliberation, we decided to print it rather than use Chromebooks.
I don’t know Dana’s class, and it’s possible that one of her students could get bored, Google the lesson, find Robert’s website, and blurt out the cost, ruining the reveal for the whole class.
We decided the tech integration wasn’t worth the risk, and went old-fashioned paper-and-pencil.
(It’s worth noting here that the director of my department is more interested in educational improvement than tech integration, even though we’re the Educational Technology Department.)
I taught first period, she taught second, then we team-taught third. All three went roughly like this:
Teacher: What do you see?
Students: That’s an In-N-Out burger
S: That is deliciousness.
S: I wanna eat that.
S: That is… life.
T: How do you know that’s an In-N-Out burger?
S: At the Habit Burger, like, the lettuce is smaller.
S: McDonald’s burgers have like, no lettuce.
S: Toasted buns
S: I can see the wrapper
S: Special sauce
S: When Mr. Vaudrey clicked PRESENT, I could see the tab was called In-N-Out.
T: Specifically, what kind of In-N-Out burger is it?
T: What parts do you see?
S: Bun, patty, cheese, lettuce, tomato, onion, special sauce
T: What’s this?
S: Oh, baby!
S: That’s a double cheeseburger!
S: No! It’s called a double-double! What, are you new?
S: You should get us that for lunch!
T: What’s the difference between this one and the last one? What parts are different?
Their responses were what you expect; middle-schoolers calling out ideas intermixed with ideals about how fantastic food is.
Them middle-schoolers love food.
We agreed that most double-doubles have two meat, two cheese, but the rest is the same. And that In-N-Out doesn’t charge extra for onions and stuff.
After teaching two classes, I found this post from Hedge, where she used the phrase, “A double-double is a cheeseburger with one extra meat and cheese layer. A 3x3 is a cheeseburger with two extra meat and cheese layers.”
That would certainly be a more straightforward path to generalizing a formula. But I was in Dana’s class to model teacher questioning, so I’m glad that we gave the students very little help.
T: How is this one different from the last one?
S: There’s a lot more meat and cheese.
T: How much more? Tell your neighbor what you think.
<pause for student chatter>
T: This is a “twenty by twenty”–
S: I told you!
T: –that somebody actually ordered, paid for, and finished. This guy.
T: Talk to your neighbor, what are some questions you have about this scenario?
When I taught first period, we used a song to direct student behavior. Dana decided to omit Music Cues, choosing to focus on one new instructional strategy at a time, which was wise.
Our next slide had a link to a Questions doc, with links within that doc so classes couldn’t see each others’ questions.
Dana opted to write their questions on a poster, which was perfect, because I could record them as they happened here:
S: How much does it cost and where can I get it?
S: It’s on the secret menu.
S: I don’t think secret menus are a real thing.
S: Who decided it’d be a good idea to squish all them panties… I mean patties! <whole class burst out laughing>
S: How much does it weight?
T: What else? Talk to your neighbor.
S: Do Hispanics eat this burger?
S: How much cheese?
S: How long to eat?
S: How many calories?
T: I’m curious about your first question, too. I want to know how much it cost. Let’s focus on that question today. Write that down.
S: I lost my pencil.
S: Where do I put it?
S: Is there a sharpener?
S: I literally wanna get a job at In-N-Out.
T: Make a guess, how much do you think it costs?
What do you need to know? What information can I give you to help you figure this out?
S: The unit price.
S: The price of a double-double.
S: The price of a single cheeseburger.
T: It so happens, I have the menu from the guy’s blog.
Note: That’s as big as it gets. I actually had to extract that image from the code for badmouth.net, that’s how bad I wanted the 2005 menu from In-N-Out.
Third period multiplied the double-double price by 10 and felt like they were done. Dana and I struggled each period to explain that ten double-cheeseburgers have a bunch of buns and produce that aren’t present in the picture of a 20x20.
It wasn’t until later I realized that Tim McAffrey built images that described perfectly how the students’ first attempts fell short:
We added those images into the slides for 2nd period and immediately, students declared, “No! If you stacked all those up, you’d have a bunch of extra buns and toppings!”
First period had the highest variation in problem-solving strategies.
This is my favorite No, the student figured that bun, cheese, and “other” all cost the same, so she divided the cheeseburger cost by three.
In a zillion years, I never would’ve tried that. And from me, she got the same, “Thank you. Two claps for Jordan, one, two.”
Four different answers at the top and the word “discharge” for the lettuce. The other teachers whispered to me, “It’s Sex Ed week, so the word discharge is probably not an accident.”
I thanked Madison for having a variety of “possibilities.”
Dana and I were both surprised at how many students just made up numbers that they thought made sense to be charged for stuff. The idea that I can choose any number and run with it will produce some weird problem-solving strategies, which I suppose is good. Even it the idea was birthed from the magician-like reputation of math teachers.
After a few of those numbers ex machina, we had class conversations about it.
T: All those methods sound fair. Do you think In-N-Out changes their pricing based on the order?
S: <silent thought about it>
S: No? I mean, probably not.
T: I saw a few papers with “$24-25” on them. If you went to In-N-Out and they asked for “Seven or eight bucks,” would you just pay it?
S: For In-N-Out? Absolutely.
S: That’s not fair, though.
T: What do you mean, fair?
S: Like, it’s gotta be the same for everybody.
T: So if they charged this guy for 10 double-doubles, he might be upset that he’s getting charged for a bunch of buns and toppings that he isn’t getting?
S: Yeah, exactly.
T: I can tell you this, all those methods sound fair, and In-N-Out uses a different way of pricing.
S: Oh! I think I know what it is!
T: Come show us.
S: Okay, ignore this top part. I times’d 90 cents by 19 because that’s how many extra layers there are, then I added 1.60 and subtracted 30 cents.
We gave Jeremy two claps and fine-tuned his idea, even drawing pictures on the board to show the cheeseburger with 19 extra layers in it.
Then, we showed this.
T: This is from his blog. What do you think?
S: He got ripped off!
S: No! What about tax?
S: Maybe he got a combo.
T: That’s what we thought, too. We found these pictures on his blog.
S: Look! He got a drink!
S: Maybe two drinks?
S: And ketchup, so he had fries.
S: Ugh. He ate all that and fries?
Once again, I didn’t leave enough time for the closure. I was hoping to read aloud to them a portion of the epilogue here.
And address all the questions they had. Bummer.
When the class ends at 8:52, I should’ve started wrapping up at 8:42.
Dana found herself slipping into her her old ways, leaning on what was comfortable and familiar, validating correct answers quickly. Sometimes, she’d catch herself, though.
“He’s on the perfectly perfect path… but, umm…. does anybody gave a different approach to the … um… problem?”
“I really like one of your approaches.” Which one? “Um… Joshua, can you show us yours?”
“The 8th graders are doing this, too. We’ll see if you did it faster than they did.”
As soon as she said, “That’s the correct way,” the rest of the class shut down. No one volunteered more answers and nobody wanted to show their work.
During first period, we had two other math teachers joining us in the back. One of them dropped this gem in my ear after students were wrestling with the topic for about 20 minutes.
It is so hard to not tell them what to do! We’re so used to just giving them the answer or telling them the “correct” way to tackle the problem. This takes way longer, but all the students are working and engaged.
I regularly make the mistake of assuming that I have all the answers or that the teacher needs my help to make her students into real mathematicians. Some of Dana’s students dropped bombs like this:
“My first answer was too big. It was more than the cost of 20 cheeseburgers, so I knew that it wasn’t fair to be charged that much.”
Dana is doing some good stuff in her class, even if I may cringe when she handles students differently than me.
The awesome parts of Dana’s class were there long before I arrived.
Students also had some comments that showed what type of class they’re accustomed to.
“I did it totally wrong and I’m gonna go sit down now.”
“This was so fun.”
“Can we do another lesson like this?”
“Is this for a grade?”
“This is confusing.”
After the student showed her division of the cheeseburger price by three (an approach that was creative and weird), I was acutely aware of the teachers in the back corner.
The three teachers who were watching me tensed up after that happened. He didn’t tell her she was wrong. What if other students try her method? Does Vaudrey not know if that’s correct? Oh, God; what happens now?
What they didn’t see was me tagging four specific students with four very different approaches to the problem. Those four students were the ones who shared; I didn’t take volunteers. I wanted a variety of weird approaches.
I don’t care about the answer. I care about the process.
When Robert did this lesson for a room full of math teachers, we had seven functions to describe In-N-Out’s pricing.
A gaggle of adults who use abstract math every day had seven ways to approach the problem. Yet in many classrooms, the book shows one method.
How arrogant of us to assume that our “teacher answer” is better than the variety of student answers.
My favorite part of this task is the answer. “About $22.”
It’s definitely not the answer that the students got when they worked out In-N-Out’s pricing guide and applied it. The usual relief students expect when their answer matches the back of the book was absent here.
Students gave a lot of hesitant and uncertain looks.
Imagine if that relief came rarely, if at all.
What kind of students would that produce?
~Matt “Two Cheeseburgers” Vaudrey
UPDATE 7 MARCH 2016: This morning, I met with all three teachers to debrief the process and discuss what they saw. Dana had this to say:
Dana: I had to prep the later classes, even if just for myself. I told them, “Guys, I’m not gonna tell you if it’s right or not.” And they were so hungry for the right answer! They were like, “Am I right? Am I close? Which one of us was the closest?” And it was so hard for me to keep a straight face and just say, “I’m not gonna tell you.” Like, I’m used to affirming them with stuff like, “Thank you, good job, that’s right.”
Vaudrey: So were you able to find ways to validate their process without giving hints at the answer?
Dana: Oh, yeah. I said, “Thank you” a lot. I even put a Post-it note on my podium to remind myself to just say ‘Thank You’ and that’s it.