Saturday, November 2, 2013

The Six Acts of a Mathematical Story

Act One: Introducing The Central Conflict

"What's the most ridiculous looking book I can walk out of here with?" I asked. The librarians were incredibly helpful. And I lugged this enormous dictionary back to my classroom and hid it in a filing cabinet.

I told the 4th Graders that I had a surprise for them. In the time it took for me to pull this monstrosity out of my filing cabinet and drag it to the middle of the room, the kids already started asking how many pages long it was. I didn't say a word. I just went to the board and started recording guesses.

Act Two: Confronting The Central Conflict

The first round of guesses were all in the 10,000s, so I took a folder and stuck it after the 100th page. I told them where I'd placed the folder, and we took another round of guesses. The kids seemed split on whether this data made their initial guesses too high or too low. (You can see the lines connecting the initial guess and the second guess.)

That was interesting, so I moved the folder to the 500th page and took a final round of guesses. You can see those in the third column.

Act Three: Resolution of the Central Conflict

[Insert picture of last page here. I'll get it when I get back to school.]

The dictionary ends on page 3,210, and we marched that page around the classroom. (Wikipedia says that it has 3,350, though I don't exactly know how they're getting that number.)

Act Four: Those Clips That Sometimes Show Up After The Credits?

"What would be some interesting math questions that we could ask about this book?"

"How thick is each page?"
"How many words are defined in the dictionary?"
"How many words are in the dictionary, including everything?"
"How many letters are there in the dictionary?"
"How many pounds of ink went into that dictionary?"
"How many atoms are in the dictionary?"
"How many of those Arabian Nights books on the desk would fit into the Big Book?"
"How many words are on a typical page of the Big Book?"
"How many words are there that are not defined in the dictionary?"
"How many pixels is this on a computer screen?"
"How much room is there on each page?"
"How long would it take to read the entire dictionary out loud?" [My question.]
"How many words long is a typical definition?"

The bell rang, and the kids went home, or to some other class, or whatever it is 4th Graders do when they're not in my classroom.

Act Five: Falling Action? Rising Action???

Everybody got a dictionary page. I photo-copied them and handed them out.

How many words are defined in the entire dictionary? What information are we going to need to answer that question? If we could figure out how many words are defined on a typical page, then we could do some crazy multiplication to settle the bigger question.

There's our line-plot. Most kids figured that, based on the data, either 84, 85 or 86 was the typical number of words defined on a page. 

(One girl -- the duck, actually -- insisted that because she had counted 143 bold words on her page, that this was the typical number. I think that she just didn't like the idea that her counting was somehow in vain. Either that or she was burned out from Halloween.)

Act Six: The Sixth Act

There are around 276,606 words defined in the dictionary, though this is tough multiplication for my kids to do.

I'm having trouble finding out exactly how many entries there are in the 1934 Second International Edition, but Wikipedia reports that the Third International Edition "contained more than 450,000 entries, including over 100,000 new entries and as many new senses for entries carried over from previous editions." So that should mean no more than 350,000 for the Second Edition, right? Not bad, kids.

With the rest of class we tried to knock off a few more of their questions. Did you know that linguists estimate that there are roughly 1,000,000 words in the English language? We figured out (roughly) how many words are not defined in this dictionary.

We also had a fun conversation about how to figure out how thick each page of the dictionary is.

[Six Acts]

There you have it. This was a ton of fun, and I certainly recommend checking your local library for a book as silly as this one. I'd love to help those without access to a physical copy to play with this. I uploaded some of my pictures to 101qs, but please let me know if you think of certain pictures that would make this problem more useful to you.


  1. Great initial question: "What's the most ridiculous looking book I can walk out of here with?" Love it!

    In terms of the 6-act structure, I'm curious what made you think the second day's activity was falling action. (Or perhaps rising action). It sounds like the kind of came up with a bunch of "sequels" for the first story and then you facilitated them through one of them. (I'm also curious about the selection process for that--did they choose which one to pursue, or did you choose based on what seemed feasible?) So I guess the overarching question is: what makes this/a 6-act story different from two 3-act stories?

  2. "What makes this/a 6-act story different from two 3-act stories?"

    Dan writes a bunch about how the 3 act structure isn't about pursuing the questions that students ask in a sort of open way. Asking for questions is more a way to get buy-in to an engaging question that the teacher had already picked.

    That's valuable.

    But is there a way to value student's questions in a more committed way? Is there a way to both have control over the initial problem, while also allowing class to be guided by the interesting questions that kids ask? After all, problem posing is an important mathematical skill all on its own.

    One more note: problem posing is hard for kids. Lately, I'm noticing that the best context for problem posing is problem solving.

    Dan's pointed to a certain kind of pedagogical paradigm with his 3 Acts, and I'm trying to offer an extension of his structure. Spend the first day on a 3 Act problem, end by gathering sequels (pitches?), then go home and figure out which of their ideas will work in class for the next day.

    Oh, and don't pay too much attention to the "rising" or "falling" or whatever. I was just trying to be silly.

  3. Trying to parse this out. I'm super-familiar with Dan's 3 act stuff--just gave a talk on using narrative structure in lesson planning at CMC South, in fact--so no need to tease that out further. It seems to me that what you're saying the difference between a "3 act" and a "6 act" is the student agency in choosing the follow-up question, which seems like a reasonable and relevant point of distinction. I also like the metaphor of "pitches" for a sequel to the original problem that the teacher posed. Seems like a good way to value student inquiry in a manageable way.

    I'm not 100% positive that's what you were trying to say...

    1. Yeah, I agree with everything that you said. In addition to sharing a fun lesson, I'm also sharing a manageable way to pursue more student questions that emerge out of a really intriguing context.

      But, again, "6 act" is just me messing around. I know how thoughtful you are about mathematical story-telling, and story-telling more generally, but I wasn't trying to say that there's some sort of 6 Act Narrative Structure that appears in stories and movies or anything.

    2. This is a very fun post, Michael. I see it as a way to encourage any teacher to discover/pursue more from a 3 Act lesson and make it an "n Act" lesson where the number of acts isn't limited, but more catered toward the needs and desires of both the students and teacher. Thanks for sharing.

    3. Hey, thanks! You have me thinking now about what engagement looks like as n --> big. I know Chris Robinson experimented with week-long themes, but I haven't had much success beyond 6 acts.