You’ve got me interested in the opinion that you aren’t explicitly stating.

And **Dan Meyer** says he was:

Mostly confused. Those two [Larry and Stu] are strange.

(via Twitter)

So forthwith I offer a deconstructed, annotated version of the conversation. All links from the original post have been deactivated in order to avoid a second round of massive pingbacks. My comments include a few new links; see the original post for the original references.

Larry and Stu have been teaching high school math for 10 years. They take a load off in the teachers’ lounge at lunchtime.

I am always thinking about what my own students (whether the context is middle-school, college or professional development) take away from the experiences we have in the classroom. How does what I *say* match what they *hear*? And vice versa.

In a particularly vivid example, I recently worked with a group of teachers on a *Connected Mathematics* implementation. Several of these teachers had been to Michigan State and had met the authors of the curriculum. One of these teachers claimed definitively that Betty Phillips had told her that the units in the curriculum could be taught in absolutely any order. Problem is, Betty is our loudest voice on the *importance* of the order of mathematical ideas and the sequence of units in the curriculum. Surely what Betty said is not what this teacher heard.

So I’m always curious about the relationship between what is said and what is heard. I wonder about *the takeaway message*, in common business parlance.

Larry and Stu are two typical high school math teachers with enough experience to have settled in to a standard American mode of mathematics teaching, but also not so deep into their careers that they aren’t interested in what else might be out there. I imagine them being good at what they do, caring about students and proud of their work.

**Larry: **Y’know Stu, I’ve been doing a lot of reading on the Internet. There’s some great stuff out there. And I’m convinced that Math Class Needs a Makeover.

Surely someone has emailed Larry and Stu a link to Dan Meyer’s TED talk. I didn’t see it until February of this year but most of the people I talked to afterwards had already. In May, a friend posted it to Facebook.

**Stu: **What do you mean, Larry?

**L:** We gotta make math real-world.

**S: **Huh?

This is not new language. Many people have burned many calories thinking about what it means to make math “real-world”. The Dutch *Realistic Mathematics Education* program has done the best job of defining the “real” part of “real-world”. Something is “real” if it can be brought to life in the imagination of the student. If students can operate on it in their minds, it matters little whether the thing actually exists.

Dan recently was on the cover of an Education Week supplement with the phrase, “Real-world learning” in large type next to his photograph. He expressed discomfort with this phrase (calling it “a phrase I try to avoid”). Meanwhile, Karim over at Mathalicious unambiguously defends the real world as “an underused launching point in most math classrooms”.

I understand Dan’s hesitance to adopt the “real-world” label as it has been used and abused to justify all sorts of instructional decisions. It is perhaps only a bit more meaningful and a bit less bullying than “good for kids,” as in “I just want to do what’s good for kids.”

**L:** OK, see there are these Russian Dolls. And, oh yeah, these escalators. And there’s a whole mess of these graphing stories. Kids watch these videos and they make a graph showing the relationship between time and height, or time and distance, or heck-there’s even one on time and time!

I have indicated on this blog and in the comments on Dan’s that these are all brilliant contexts and great contributions to the field. After I met Dan back in February, my mind was buzzing for weeks. I spent a lot of time sorting out what exactly I could learn from Dan’s work. Ask my colleagues; I wouldn’t shut up about it.

**S:** Sounds interesting, Larry. Tell me more.

**L:** Right, so you show this video and kids ask the question. You don’t have to give them some phoney-baloney task, the video leads them right to it. (By the way, do you know what exogenous means?)

I am all about purging the mathematics curriculum at all levels of phoney-baloney tasks. But here is an interesting tension I have come across in following Dan’s work.

Dan is working very hard to help math teachers to understand what it means to pose a problem in a compelling way to students who are not predisposed to seeing the world mathematically. I’m 100% on board with that agenda, and I have learned a lot from it.

I am more critical than he is of the accepted, enacted curriculum in the U.S. I love his visual reformulation of the house painting problem as a counting beans problem, for instance. But I’m not willing to give the math curriculum a pass on the value it assigns to the work equation. There’s the tension for me.

My office mate and I have debated many times the value of this very equation. I argue that the work equation unnaturally emphasizes time per job, rather than the more natural job per unit of time. So if I’m painting a house, I am more likely to be operating on the “fraction of a house I can paint in an hour” than “the fractions of an hour it takes to paint a house”. And if I think about houses per hour, I can’t use the work equation.

I am more interested in eliminating the work equation from the standard curriculum than I am in retooling the problem we use to get at the work equation. Lounge chairs, deck of the *Titanic*, etc.

And the “exogenous” thing is a good-natured jab at my man, the erudite Karim who wrote the lovely sentence on Dan’s blog, “I think the prohibition on exogenous questions may be a bit too strict.”

**S: **No idea. But, OK, kids ask the question. Cool. Then what?

**L:** Well, then you give ’em whatever information they need to solve the problem. And you let them work, or maybe some days you lecture. But the point is, they’re motivated to do the work because they asked the question and they identify with the context.

Here’s another tension for me, and it has been a common topic of discussion over on Dan’s blog. Let’s say we craft this perfect mathematical problem, kids ask the question and really, badly want to know the answer. Then what?

This question is outside the domain of Dan’s WCYDWT work. He’s working on setting up the task. That’s good. And yet it is a very real problem of practice to identify how students are to learn some mathematics from the task. It requires an entirely different set of skills to move students forward mathematically than it does to pose the problem in the first place.

I think Larry and Stu could very well show the escalators video, get everyone to the point of wondering how long it takes Dan to go up the down escalator, and then not really know where to go from there. If they are going to implement these ideas in ways that support student learning, they’re going to have to improve other aspects of their practice as well.

**S:** Good. Go on….

**L:** OK. Then after they answer the question, you don’t just look in the back of the book for the answer, you show the answer in the video. They see the results of their calculations. If they’re right, they get validated. If they’re wrong they know they’re wrong. They get that math solves real problems.

This is a lovely feature of WCYDWT, I think. If the problem has been posed with multimedia, the solution to the problem can be found there too.

By the way, I am frustrated that I cannot get my escalators answer to agree with reality within 3 seconds. Even making the most generous assumptions about things we can’t quite see in the video I’m off by at least 3 seconds. I want to be **right**!

**S: **Nice. Do we do videos every day?

**L: **No. That would get old. Once every couple of weeks or so.

**S: **What do we do in between?

**L:** Well, what we’ve always done.

**S:** Oh, so not all of Math Class Needs a Makeover? Just 10% of it?

OK, so here’s where I tip my hand and get Dan’s goat (see comment on original post). Of course I know that he doesn’t think only 10% of Math Class Needs a Makeover. Of course I know that his laser focus on one aspect of teaching doesn’t imply endorsement of everything else.

But the use of “Math Class” is ambiguous and I have played with that ambiguity in my mind as I’ve thought about Dan’s work. Does “Math Class” refer to a single class session? Or does it refer to the mathematics curriculum in a more general sense? Larry and Stu are sorting through this ambiguity in their conversation.

**L:** Maybe. Or maybe we flip the classroom; I’ve read about that, too.

There are others on the Internet working on changes in the ways we teach math. Perhaps you have run across Khan Academy?

Most of the discussions of “flipping the classroom” that I have come across have been founded on a set of faulty assumptions about the kind of makeover math class needs. Dan’s scope is limited to one class period, one problem at a time. I agree with his assumptions about how people learn-by being interested in the topic at hand, by being puzzled by something and wanting to figure it out, by connecting new knowledge to prior knowledge, etc.

But KA and much of the larger “flip the classroom” rhetoric is founded on assumptions that students learn best when they are told clearly and repeatedly and by practicing what they have been told. So Khan tells clearly in his videos and he can tell repeatedly because students can watch the video as many times as they like. No need for the teacher here. Then the teacher becomes the tutor who helps students with their practice in real time. If you buy that this is what learning mathematics is about, then flipping the classroom is a substantive improvement. But if you question the curriculum, or if you question the effectiveness of telling, then KA and flipping don’t have much to offer you, I’m afraid.

For me, there’s not much interesting in these ideas. I’m not opposed to them, I just disagree with the assumptions they represent and I don’t see that they offer much for me to learn. So I’m not gonna put in the kind of work I did for Dan.

*fin*

The main thing to understand from all of this is that I respect Dan’s ideas and work enough to spend time critically questioning them. Not dissenting so much as asking hard questions.

I thought the initial post/dialogue was both critical and respectful, and really enjoyed it. My takeaway wasn’t that Dan is wrong to focus on his 10% (or whatever), but that there’s a larger spectrum, each section of which demands its own approach. I view Dan’s work as being at one end of this spectrum; it’s open-ended and intentionally deconstructed, because that’s what the section demands. Meanwhile, the folks at Connected Math live more on the nuts-and-bolts end of the spectrum, which requires its own set of rules.

In the end, it’s not a question of which part of the spectrum is better (which seems silly and misunderstands continuity), but rather whether people who spend their time in each do so intentionally & thoughtfully. And having been fortunate to get to know both Dan & Chris, I’m heartened–indeed, inspired–to witness how seriously they take the work that they’re doing, as different as it may *seem*.

Thanks Chris – that was a great read. I guess I never thought about whether the bean problem actually implies that work equation problems are an acceptable and important part of the standard curriculum. For me (who has never formally taught any work, mixture, etc. problems) I was intrigued by it as an answer to those who do put emphasis on work equation problems – but more so just as an interesting problem. One of those problems where I’m interested in what the kids would do with it.

In contrast to what I just said, I love Dan’s work, I think about it a lot, but I have never used any of it. It would take me a lot of time to start with a WCYDWT or an #anyqs and craft it into something that I see going exactly where I want it to go. I think much much much more time needs to be spent here so that what we come out on the other side with is a coherent, well structured course of study.

I want it both ways I guess. I want to throw Meyer problems out to my kids just to see what I get back and explore where ever they take us, but I also need the “where ever they take us” part to be where ever we are supposed to be. Obvious, I guess, but also the kind of thing that makes what we do exciting and alive.

I am more interested in eliminating the work equation from the standard curriculum…Amen to that. I often find myself, when confronted with someone’s ideas of how to teach a certain “math topic” better, asking why the importance of that topic is taken for granted, or why it is being presented to students at that particular point in their mathematical education.

Dan Meyer’s ideas are wonderful, the Khan Academy flip-the-classroom approach is intriguing, and your musings are also good food for thought. However, no matter which series of topics are included in a text, curriculum, or video series, there will always be debate about the topics that were left out or left in.

While we all seek to teach a set of concepts and skills, I think we often define them too narrowly. If we step back and advocate teaching “broader” topics, it may be easier to achieve consensus. Suppose our goal was to have students able to “make up your own mind in a quantitative situation”… that could cover many bases, might take a few years to build the skills needed, can be approached in a variety of ways (depending on the teacher and the students), and can also be integrated with the study of other subjects. A “real-world” approach can be taken, or not, or in between – depending on the teacher and the class.

However, such an approach would also require the teacher to be able to productively choose from an array of rich activities (which is/are most likely to engage my class?), then simplify or extend each activity (initial questions must be doable by all, later ones should be a challenge for the most advanced). But, as a teacher, where do I find such activities? Where do I find the time to adapt them? How can I improve my skills in adapting them?

These issues should be more easily addressable in the age of the Internet than they could have been in times past. The activity bank need not be distributed in print, but can be web-accessible and constantly improving. Teachers from around the country with similar classroom compositions and curricular goals could (if they can find one another each year) collaborate to customize and improve activities they decide to use.

And most importantly, each activity needs to be presented in a way that makes it quick and easy for a teacher to:

a) Find it using a couple of keywords

b) pick and choose among customizations that have already been developed by others

I have not yet come across a resource that does the above well.

Here’s a question: if you were completely unaware of the history and context of the house painting problem within math education in the US, would it change your desire to know how long it’s going to take Dan and Chris to fill the container on their own?

Aside: this has been an extremely interesting problem for teachers in my workshops. I ask them to guess how long it’ll take Chris & Dan to fill the cup together and the guesses are

amazing. Utterly amazing. Jaws will drop later. Foreheads will be smacked. Math teachers can be just as impatient with irresolution as anybody. Bad curriculum wrecks studentsandtheir teachers.Easy: you show them how to use math to find the answer. You lecture. I mean, that isn’t my best case scenario where you’re 1:1 with a pupil and you determine exactly what tools she brings to bear on the problem and ask pointed questions to strengthen those tools. But, seriously, assuming they came to the question honestly and they honestly want to know the answer, what’s the downside to showing them how to use math to find it?

Personally, I want to know (and more importantly) figure out how long it takes Dan and Chris to fill the container. And I’m convinced my kids would also. But that seems unrelated to whether or not one likes or dislikes the work equation and house painting problems. Does the bean counting problem have to lead to the house painting problem?

And I get lecturing or KA-ing the math behind it, but is there a point where after we have lectured and done whatever practice we are going to do, that the novelty of the beans wears off and the kids realize they are in the middle of just another math topic? And does that matter?

Right, that’s my point.

I read Christopher to mean the bean counting video has a strike against it on account of its association to a really lame standard in US math ed. I’m saying, our students don’t see that association. They just want to know how long it will take these two clowns to fill the cup together. My general rule of thumb here is that if a) my students wants to know the answer to a question and b) the answer can be most accurately found through mathematics, it’s a problem worth doing, regardless of its cultural baggage.

Dan:This is a good question, Dan. I don’t dispute at all that the problem as you have presented it is a compelling one. I like it without regard to whether the sociocultural history of mathematics curriculum. I felt very satisfied with my exactly correct answer when I worked it shortly after you posted it.

But whether I or others were motivated to solve the problem wasn’t the basis of my critique. No my critique has to do with this question…Would you ever in a million years have dreamed up the counting beans problem were it not for the prevalence of the work equation/house painting problem? That is, you are solving a problem of teaching by reformulating house painting as bean counting, right? The problem of teaching that you solve is one of motivation to study the curriculum as it exists. I am interested in pushing a big picture agenda.

They’re not mutually exclusive agendas, yours and mine. I have learned some profound things about task design from reading your work closely. By pushing back on questions of curriculum, I’m hoping to return the favor.

Predictably, I’m going to push back on these rules of thumb. I think they are excellent places to begin with curriculum design. But stay at it long enough and you’re going to have (1) a surfeit of problems, among which you’re going to need to make choices, (2) the responsibility to make choices about what order to have kids work through these problems, and (3) some gaps where important skills are just not teachable with the same level of engagement. I believe these to be important issues to deal with, and they require different sets of rules for dealing with.