Month: July 2011

Total 10 Posts

A Train Leaves Chicago Traveling At Who Cares, Ctd.

Some of the responses to my last post focused on narratives to hang around the old math problem – lovers meeting, trains nearly crashing, heroes stuck on train tracks, etc.

Forgive me. I lack faith.

First, I don’t think these efforts engage students (same goes for using student names in boring math problems) but I admit engagement is a fickle, subjective thing.

Second, my broader, more objective criticism of these efforts is that in every one of these cases you will eventually have to create a math problem, to fabricate numbers and dimensions for students to work with and operate on after you’ve (apparently?) piqued their interest in those operations by showing them a clip from that recent Denzel Washington movie. Then, when your students get an answer, you will tell them, “Yes, your math was correct and your answer is correct and the world and your math verify one another.” This isn’t worthless. Your students are practicing operations that require practice. But as a sales pitch for math’s connection to the world outside the math classroom, this is worthless.


Imagine you are going door-to-door selling knives. You get a foot in a door and tell a prospective customer, “These knives are sharp enough to cut through a soup can.”

The customer says, “It so happens I have a soup can right here. Let’s try it out.”

Then you say, “Just take my word for it. These babies are incredibly sharp.”

The customer says, “Are you kidding me?”

Then you say, “Okay, well if you won’t take my word for it, have a look at this pamphlet. Right there. See? ‘Cuts through soup cans.'”

The customer says, “But your company wrote that pamphlet.”

You: “So?”


All the time.

We ask our students all the time to take our word on the presence of math in the world.

Or we ask them to take the textbook’s word on it.

You wouldn’t buy the knives. Why would your students buy the math?

A better sales pitch? Give them a video from which they can draw their own measurements, on which they can operate mathematically, at the end of which they can verify for themselves whether the trains meet at 2:37.

They’ll buy that. But don’t take my word for it. Try it yourself.

A Train Leaves Chicago Traveling At Who Cares?

Here is a basic assumption of my recent work, put as concisely as I can:

Even an application problem as stereotypically and quintessentially lame as the one above can become a powerful testimonial to math’s presence in the world if the problem begins and ends with an accurate representation of itself. Which is to say, in this case, if it starts with a side-by-side video of the two trains leaving their stations that continues unbroken until they meet each other.

If anyone’s been looking for an invitation to take a whack at that assumption, consider yourself invited.

See also: [WCYDWT] Bean Counting

Aside: The problem with paper is that it accurately represents a lot of things, but the world isn’t one of them. Not at the kind of fidelity your students need, anyway.

[3ACTS] Pyramid Of Pennies

The Goods

Download the full archive [33.9 MB].

Act One

Act Two

Act Three


  • I have $1,000,000.00 in pennies, how big of a pyramid can I make?
  • Each stack has 13 pennies which is a strange number to choose. Why do you think Marcelo Bezos chose it? [Hint: not out of an abundance of superstition.]
  • Bezos says he can tell you the number of pennies in a pyramid with this equation:

    where s is the number of pennies in a stack and b is the number of pennies on one side of the square base of the pyramid. Does this work? If so, prove it.


Here’s my burning question: is that enough? Is that skinny outline enough for you to use this in your classroom?

Check for understanding: what happens during the first, second, and third acts of a mathematical story? What are your moves? What questions do you ask your students?

Act one is about visuals, context, and perplexity. Act one hits you in the gut, not the head. Act one eagerly invites questions like, “What is that?” or “Why did he do that?” If your students are anything like the teachers who have worked with this image, you’ll get a fair number of them wondering, “How heavy is that?” and “How much is that worth?” both of which tie into the most popular (by far) question, “How many pennies is that?” Have them write down a guess along with numbers they know are too high and too low. Share guesses. Stir up some competition.

Act two is about tools, information, and resources. “What do you need to know to figure out the answer to your question?” Dimensions of the base? Number of pennies in a stack? The change from one level to the next? Give them what they want.

Act three is the resolution. When groups of students start finding answers, ask them to check the answers against the bounds they set up earlier. Challenge them with one of the sequel problems while you help other students. Bring students up to explain their different solution strategies to each other. Then pay off their hard work and show them the answer.

Release Notes

  • Teachers in my PD session love this one and, as their facilitator, so do I. They each come up with their own interesting question and yet the math doesn’t change. Whether you’re curious about weight, duration, quantity, or cost, we’re all going to work with area and series. That’s a win for every stakeholder in the room.
  • I’m especially fond of this one because everyone has a place to start. You can seriously start counting the pennies one-by-one if that’s the highest level of abstraction you can handle. We’ll beef up your skills over the course of the problem.
  • How many students will factor the number of pennies per stack, saving themselves a load of work? ie. 13*1 + 13*4 + 13*9 + … + 13*1600 vs. 13(1 + 4 + 9 + … + 1600) It’s going to be fun comparing work around the room.
  • A compelling visual is its own classroom management. If you put up a visual that’s a) simultaneously strange and familiar, b) larger than life, and c) aesthetically clear and interesting, the class is yours. Maybe only for a moment, but that moment is yours to lose. The class has given you permission to take them somewhere interesting. I’m not sure I can say the same for a worksheet. A worksheet brings with it a very different set of bags.
  • This one is courtesy of Dan Anderson. I’m drinking your milkshake here, Dan. Where were you on this story?

2011 July 8: Changed one of the sequels per David Wees’ remarks in the comments.

2011 July 15: Elizabeth Bezos, wife of Marcelos, the guy who made the pyramid, stops by to say hi in the comments.

Making It All Worthwhile

I was at urinal during a break in my Grand Forks session. “I’ll give you this,” the guy said next to me. “You walk your talk.”

Two things:

  1. The etiquette on urinal interaction must be a little more relaxed in North Dakota than in California.
  2. You get the subtext right? “I’m not buying any of this stuff, obviously, but you put on a good show.”

I knew exactly what he meant.

I’ve facilitated enough PD to not feel new at it. I’ve taken enough coursework in PD at Stanford to feel like I get some of the theory behind teaching adults about teaching children. Whenever I’m planning a session or a talk, though, I don’t lean on the theory or my experience half as hard as I do on the fear that I’ll be working with a teacher who’s exactly like me, and he’ll hate me. Which is to say, rather, that I’ll hate me.

My urinal buddy helped me understand that whenever I blog or facilitate PD or give a talk or drive in traffic or cook a meal or talk to my friends, subconsciously, I’m always wondering, “Would I hate me?” It’s a coin flip, really, whether that’s evidence of personal integrity or flagrant self-absorption.

David Labaree’s Three Rules For New Education Researchers

2011 July 8: Labaree just sent our class this same sermonette only now edited for publication. So take your pick. Polished or off-the-cuff.

This is a transcript of the “sermonette” David Labaree (author of two of the best papers I cited last week) delivered last week at the end of Stanford’s spring quarter proseminar. He gave me permission to share it with you. The tl;dr version is this:

  1. Be irrelevant.
  2. Be wrong.
  3. Be lazy.

Hell of an enticement, right? I’m posting it primarily for myself. I’m positive I’ll revisit it and find it intriguing in different ways the more I work in this field.

Three Rules for New Education Researchers

David Labaree
June 1, 2011
Stanford University

Be Irrelevant

Be irrelevant. Even in a field where the problems of the field are so important and so demanding, the argument that Augier and March (2007) make is really smart. They say, on the one hand, the push to be relevant causes two kinds of problems. One has to do with what they call myopia and the other is ambiguity.

The myopia problem is that you’re looking at something that’s presented to you. Fix this. And so you burrow into this. But this is typically located in time and space. And the circumstances are such that you pull it in close. (That’s why he calls it myopia. It’s a nice image. You pull something up close if you have myopia. You’re nearsighted.) And so in the process of looking at it up close, the whole context disappears. And you start treating the problem as though it were not connected to that context. And then you often end up engineering a change that may or may not work in that setting but is not transferable because it’s actually involving things that are contingent on a particular context – a particular time and a particular place – and efforts to then generalize from that show that the insight was actually pretty irrelevant in terms of workability.

Also in terms of time. If you’re burrowing in too much on fixing the particular problem, by the time the work gets out, that problem has already evolved into something else. That was then, this is now. And even in that setting it may not work. It has already been outgrown because you are missing the evolutionary component of what’s going on.

So there’s a certain sense in which relevant research may actually have a very short shelf life. It may start to smell bad after awhile. It may have to have a buy-by date on it that says “After this, it may not be good anymore. It may not apply in any other location.” So in some ways, not trying to be so relevant may actually come up with insights that are more transportable, more useful, and are actually more applicable, even though that wasn’t your intention.

The other issue they raise that makes me say “be irrelevant” is that relevance is kind of a rhetorical plane and one of the things you have to say is “relevant for whom?” Is it relevant in the schools? Is it relevant for teachers? For students? For administrators? For superintendents? For policymakers? For politicians? For parents? It depends. Maybe what’s relevant for one is not good for the other. High-stakes testing is highly relevant for policymakers in order to make the claim of accountability in schools. It may be very harmful for teachers and students. So it’s relevant. Research supports it but it may be a relevance that depends very much on a relevance “for whom?” That’s a claim that’s not generic. It has to be established but it’s not generalizable.

There’s also the relevance “for what?” For what end? What are we trying to accomplish in schooling? Are we trying to make better citizens or more productive workers or help people get ahead or reduce social inequality or what are we trying to do? Well, the relevance of the research depends on the relevance to which of these claims it’s focusing on. One argument is that in some ways it’s a fool’s game to try to be too relevant in a field like this and, counter-intuitively, the most useful research may be the stuff that doesn’t seem to have an immediate application when you’re actually doing it. And an effort to be slavishly useful may give your work a limited purview and a very short shelf life.

Be Wrong

All right. One piece of advice that nobody’s going to follow: “Be irrelevant.” Another one nobody’s going to follow, and that’s “be wrong.”

I think one of the dangers in programs like ours is that we encourage people to find answers, to be right, and that makes you risk averse. And my argument is that it’s much more useful to be interesting, and to provoke thought with your ideas, even if you’re wrong, than it is to be right in a manner that’s not very interesting, not very provocative, and not very likely to spur anyone else to do anything.

You never establish claims for all time. Truth is an ideal you pursue but you never reach it. And if you ever waited to nail down everything before you published something, you would never publish anything. Whatever you do, you have to recognize it’s going to be a partial statement. It’s going to be at best a partial truth. It’s something that’s true under certain circumstances and under certain conditions and with certain limitations about it and that’s enough, actually.

You’re not in the position where you need to make everybody’s counterargument to your argument. You just need to make your argument effectively and say to yourself, “Is this something that’s not in the conversation that should be in the conversation? If so, I should get it out there. And I can picture what some people will say in response but I don’t have to make that. Let them make that. I want to make a strong case in this direction. I don’t want it to be easily dismissed or laughable. I want it to have solidity, validity, and rhetorical effectiveness but it’s not my position to find out what the absolute final truth is because that’s not findable in my lifetime.

“So I’m going to be part of a conversation and the conversation is what matters and I’m going to learn from the conversation and in the process I’m going to revise what I did and I’m going to admit that some parts of that were wrong and I’ll move ahead and I’ll publish something else that is also a contribution to the literature. It helps with the conversation, but it also has plenty of possible responses to it. I’m going to learn from those responses too and I’m going to continue working on this in a somewhat new form after having acknowledged that certain parts of what I did before were not that good.”

That’s okay. That’s actually considered a successful career. That’s doing your job as as a social scientist. If you’re trying to nail it down and be right you probably won’t publish anything. You’ll keep waiting until you get it right. It has to be good enough to be provocative. Research is a provocation of thought. If you’re provoking thought with people and if you’re giving them a slice through a situation that’s a little different, that makes them think and reframe their understanding of something in an interesting way, that’s a successful piece of research. Even if it’s wrong, in a lot of ways.

As you know, it’s very easy to take even the best study and trash it. On methodological, theoretical, or other grounds. So that’s not the test of a good study. The test of good studies is, did it have an impact on you? Did it provoke your thinking? If so, it’s worth doing.

Be Lazy

Be irrelevant. Be wrong. Third one. Don’t tell your advisor about this, but “be lazy.”

There’s a real danger in educational research that you just plow ahead. “I’ve got a big pile of data in front of me. I’m just gonna wrestle with it. I’m gonna run this data set using every single test in all of the statistical programs I’ve got and keep plowing ahead until I’ve got every permutation. I’ve got all this qualitative data. I’ve got all these tapes. I’ve got interview tapes. All these other kinds of things. So what I need to do first, of course, is to transcribe everything and then to code everything. And then try to put it all together.”

No – don’t do that. Don’t do that. You want to transcribe very little of it. Most of it’s garbage. Most of it’s noise. You were there listening. You know what was in there. You don’t have to transcribe every bit. Same thing with a big data set. You shouldn’t be wallowing in the data hoping it’s going to speak to you. It won’t. You have to make it speak. You’re looking for the music in the data. Most of what’s in data is noise. So your task is not to somehow encompass the whole thing. It’s to find a strategic route through the data that provides some kind of insight that’s not out there in the literature right now.

And often that means not being diligent. Diligence can be a dangerous trait in a grad student. It means “I’m just plodding ahead day every day. I’m going through another test. I’m transcribing another tape. I’m doing research.” No. You’re not. You’re transcribing tapes. Research means you’re actually trying to figure something out, you’re thinking your way through it.

Shortcuts are very nice. Shortcuts. “Do I have to go through all these data? Maybe only some of it matters. Maybe some of that whole issue is over there.”

I spent two years of my life working with a quantitative data set that I generated, coded, keypunched, analyzed, and had print-outs coming out my ears and it ended up when I published the book, I had a colleague, David Cohen. He looked at the book and said, “All of the data you had in there seem to be a footnote to the claim, ‘Central High School had meritocratic achievement.’” Two years of my life. A footnote. It turns out it was an interesting finding. It was counter-intuitive. But the actual interesting stuff was elsewhere in the data that didn’t take me two years of my life plowing through all of the stuff.

So don’t ignore the low hanging fruit and don’t assume that the only way to get from here to there is the longest possible route through the most amazing morass of data. It’s okay to think your way through and around a problem. That’s a good thing to do. Sometimes you find something and you’re gonna have to plow through it. But you want to have some confidence that you’re doing it for a good cause and you’re not just doing it in a kind of Stephen Colbert way. It’s “researchiness.” Researchiness means “I need to analyze data. It’s what researchers do. Give me some data. I need some more data.”

No, you’re supposed to come up with something interesting to say and it may be that only a little piece of data are actually germane to that and it may be that it’s an entirely different data set way over there that you want to be working on so why waste your time on this.

So as I said file that way. Never follow any of this. Don’t tell your adviser about this. But you might want to keep it in the back of your mind as a kind of cautionary tale about how you don’t want to get caught up in the aphorisms and the common senses of what research is. You have to keep in mind, “What am I doing this for? What am I trying to do here? What am I trying to get out of this? And how can I go about doing that in a way that’s productive and not just busy?”


Augier, Mie & March, James G. (2007). The pursuit of relevance in management education. California Management Review, 49(3) (Spring), 129-146.