New Math Learning Site on Needs You Sunday, Mar 2 2014 

Hey y’all.










A Critical Language for Problem Design Saturday, Jan 18 2014 

I am at the Joint Mathematics Meetings this week. I had a conversation yesterday, with Cody L. Patterson, Yvonne Lai, and Aaron Hill, that was very exciting to me. Cody was proposing the development of what he called a “critical language of task design.”

This is an awesome idea.

But first, what does he mean?

He means giving (frankly, catchy) names to important attributes, types, and design principles, of mathematical tasks. I can best elucidate by example. Here are two words that Cody has coined in this connection, along with his definitions and illustrative examples.

Jamming – transitive verb. Posing a mathematical task in which the underlying concepts are essential, but the procedure cannot be used (e.g., due to insufficient information).

Example: you are teaching calculus. Your students have gotten good at differentiating polynomials using the power rule, but you have a sinking suspicion they have forgotten what the derivative is even really about. You give them a table like this

x f(x)
4 16
4.01 16.240901
4.1 18.491

and then ask for a reasonable estimate of f'(4). You are jamming the power rule because you’re giving them a problem that aims at the concept underlying the derivative and that cannot be solved with the power rule.

Thwarting – transitive verb. Posing a mathematical task in which mindless execution of the procedure is possible but likely to lead to a wrong answer.

Example: you are teaching area of simple plane figures. Your students have gotten good at area of parallelogram = base * height but you feel like they’re just going through the motions. You give them this parallelogram:
Of course they all try to find the area by 9\times 41. You are thwarting the thoughtless use of base * height because it gets the wrong answer in this case.

Why am I so into this? These are just two words, naming things that all teachers have probably done in some form or another without their ever having been named. They describe only a very tiny fraction of good tasks. What’s the big deal?

It’s that these words are a tiny beginning. We’re talking about a whole language of task design. I’m imagining having a conversation with a fellow educator, and having access to hundreds of different pedagogically powerful ideas like these, neatly packaged in catchy usable words. “I see you’re thwarting the quadratic formula pretty hard here, so I’m wondering if you want to balance it out with some splitting / smooshing / etc.” (I have no idea what those would mean but you get the idea.)

I have no doubt that a thoughtful, extensive and shared vocabulary of this kind would elevate our profession. It would be a concrete vehicle for the transmission and development of our shared expertise in designing mathematical experiences.

This notion has some antecedents.[1] First, there are the passes at articulating what makes a problem pedagogically valuable. On the math blogosphere, see discussions by Avery Pickford, Breedeen Murray, and Michael Pershan. (Edit 1/21: I knew Dan had one of these too.) I also would like to believe that there is a well-developed discussion on this topic in academic print journals, although I am unaware of it. (A google search turned up this methodologically odd but interesting-seeming article about biomed students. Is it the tip of the iceberg? Is anyone reading this acquainted with the relevant literature?)

Also, I know a few other actual words that fit into the category “specialized vocabulary to discuss math tasks and problems.” I forget where I first ran into the word problematic in this context – possibly in the work of Cathy Twomey-Fosnot and Math in the City – but that’s a great word. It means that the problem feels authentic and vital; the opposite of contrived. I also forget where I first heard the word grabby (synonymous with Pershan’s hooky, and not far from how Dan uses perplexing) to describe a math problem – maybe from the lips of Justin Lanier? But, once you know it it’s pretty indispensible. Jo Boaler, by way of Dan Meyer, has given us the equally indispensable pseudocontext. So, the ball is already rolling.

When Cody shared his ideas, Yvonne and I speculated that the folks responsible for the PCMI problem setsBowen Kerins and Darryl Yong, and their friends at the EDC – have some sort of internal shared vocabulary of problem design, since they are masters. They were giving a talk today, so I went, and asked this question. It wasn’t really the setting to get into it, but superficially it sounded like yes. For starters, the PCMI’s problem sets (if you are not familiar with them, click through the link above – you will not be sorry) all contain problems labeled important, neat and tough. “Important” means accessible, and also at the center of connections to many other problems. Darryl talked about the importance of making sure the “important” problems have a “low threshold, high ceiling” (a phrase I know I’ve heard before – anyone know where that comes from?). He said that Bowen talks about “arcs,” roughly meaning, mathematical themes that run through the problem sets, but I wanted to hear much more about that. Bowen, are you reading this? What else can you tell us?

Most of these words share with Cody’s coinages the quality of being catchy / natural-language-feeling. They are not jargony. In other words, they are inclusive rather than exclusive.[2] It is possible for me to imagine that they could become a shared vocabulary of our whole profession.

So now what I really want to ultimately happen is for a whole bunch of people (Cody, Yvonne, Bowen, you, me…) to put in some serious work and to write a book called A Critical Language for Mathematical Problem Design, that catalogues, organizes and elucidates a large and supple vocabulary to describe the design of mathematical problems and tasks. To get this out of the completely-idle-fantasy stage, can we do a little brainstorming in the comments? Let’s get a proof of concept going. What other concepts for thinking about task design can you describe and (jargonlessly) name?

I’m casting the net wide here. Cody’s “jamming” and “thwarting” are verbs describing ways that problems can interrupt the rote application of methods. “Problematic” and “grabby” are ways of describing desirable features of problems, while “pseudocontext” is a way to describe negative features. Bowen and Darryl’s “important/neat/tough” are ways to conceptualize a problem’s role in a whole problem set / course of instruction. I’m looking for any word that you could use, in any way, when discussing the design of math tasks. Got anything for me?

[1]In fairness, for all I know, somebody has written a book entitled A Critical Language for Mathematical Task Design. I doubt it, but just in case, feel free to get me a copy for my birthday.

[2]I am taking a perhaps-undeserved dig here at a number of in-many-ways-wonderful curriculum and instructional design initiatives that have a lot of rich and deep thought about pedagogy behind them but have really jargony names, such as Understanding by Design and Cognitively Guided Instruction. (To prove that an instructional design paradigm does not have to be jargony, consider Three-Acts.) I feel a bit ungenerous with this criticism, but I can’t completely shake the feeling that jargony names are a kind of exclusion: if you really wanted everybody to use your ideas, you would have given them a name you could imagine everybody saying.

Kids Summarizing Sunday, Sep 8 2013 


It is an effing game-changer.



* Students were perceptibly more engaged with each others’ ideas.
* The ideas felt more like community products.


A snippet of remembered classroom dialogue to illustrate:



[J raises his hand.]


Me: Raise your hand to summarize J’s thought.





J: Yes.







Me [to class]: Who else wants to know?

[At least half the class raises hands.]



[J’s hand shoots up]


[T calls on J to speak]


T: Oh, yeah, he’s right.

Me: Can you summarize his whole thought?

[T explains the whole thing start to finish.]

Me: Do you buy it?

T: Yes.




I Don’t Get It vs. I Don’t Buy It Wednesday, Jul 24 2013 

I was having a conversation a few weeks ago with a computer programmer and math enthusiast whom I’ll call Dorian. He was arguing very passionately that talking about a square root of -1 was the wrong way to introduce complex numbers. He recounted this moment in his own schooling: 16 year old Dorian, told by his teacher “we introduce a new number i whose square is -1…,” asking, “but I can prove that the square of any number is positive, what about that?!” His teacher wasn’t able to satisfy his objection and made him feel that it wasn’t valid. He left the experience feeling angry and frustrated and that his question had been treated as a failure to understand.

Dorian later learned that complex numbers can be visualized as a plane containing the real line; that addition of points in this plane is just vector addition; and that multiplication is done by multiplying the distances from the origin and adding the angles from the positive real axis (see here for a brief explanation if desired). Here was a concrete model for the complex numbers, with concrete geometrical interpretations of the operations + and \times. And it was clear to him that in this model, there is a point, in fact two points, whose squares correspond to the point -1 on the real axis. But philosophically, this fact is a consequence of the concrete geometrical description of the operations in the plane, rather than an ontologically dubious starting point for the whole project.

Dorian concluded that actually this model, via the geometry of addition and multiplication in the complex plane, is a pedagogically superior introduction to the complex numbers. His argument is that it presents no ontological quandary. Nobody will object to a plane. Nobody will object, at least on philosophical grounds, to these new definitions of + and \times, as long as you can prove they have nice properties and coincide with the old definitions on the real line. You’re not saying anything so wildly speculative as “postulate a square root of -1…”

I am not writing this post to get into the question of whether Dorian is right about this. I see lots to say on both sides. What I am writing this to say is that there is a lesson in Dorian’s story much deeper than the question of how to introduce the complex numbers. That is not the real question here as far as I am concerned.

The real question is this: when you’ve picked your approach and gone with it, how will you deal with the students it doesn’t work for?

Now you can always obsess about how to introduce a topic, and I believe there is basically always value in thinking and talking about the pedagogical consequences of different ways of looking at things. And I think some models for ideas are legitimately better than others. But no model will speak to every student. This point is so important, and was so lost on me as a young teacher, and is lost on so many (especially young) teachers that I have spoken with, so excited that they are about the way they have thought of to present negative numbers or whatever, as though miraculously everyone in the room will get it this time, that I need to repeat it:

There is no model that is the right model for each and every student, each and every time.

No matter how awesome your idea for how to think about XYZ concept is, there will be somebody in your class who will have no idea what you are talking about. To me, the big question here is, what are you going to do about it?

More specifically, how are you going to treat their thinking?

Now, I like to think that nobody reading this blog would be so callous as to intentionally make a student feel stupid for asking an honest question. But there are far subtler ways to do it. The one I most want to warn you against is the sin I know I’m guilty of: being so wrapped up in the awesomeness of your presentation that the kid who doesn’t get it does not compute to you. You say whatever you say out loud but in your mind you’re like, “wait – you don’t understand? Huh?” Or, you’re like, “oh my goodness can’t you just see it as I do?”

Regardless of what you say out loud, having such a response in the back of your mind invalidates whatever obstacle the student is facing. I want to suggest an alternative:

Take the case that any earnest failure of a student to see your point of view is actually coming from a legitimate mathematical objection.

This is how you treat dissatisfaction with honor.

I don’t care what the kid’s IEP says. Mathematical convention does not require us to check somebody’s Wechsler results before they are allowed to raise an objection. If they don’t buy it, they don’t buy it. Now it’s your turn to understand their objection and answer it.

“I don’t get it.” “I don’t buy it.”

A student I’ll call Manny, whom I had in my 2003-4 AP Calculus class, came to me around March and said something like, “this entire class is based on a paradox.” He objected to my (retrospectively totally hand-wavy) discussion of limits. It never gets there, so how can you talk about what happens if it were to get there?

I tried to answer Manny’s objections; I spent some time with him on it; but he left the conversation unsatisfied. Retrospectively it is clear to me that this is because (a) I didn’t get what the problem was, and (b) to my shame I didn’t consider the possibility that there was really much to it. Then, less than a year later, I read The Calculus Gallery, whereupon I learned that actually Manny’s objection was more or less exactly Bishop Berkeley’s famous objection that in due time forced mathematicians to invent real analysis. For a sense of the importance of this development, let me mention that I have read, though I don’t recall where right now, that the development of real analysis was really the event that led to the birth of modern mathematical rigor.

So, yes, I am on record as having treated as essentially invalid an objection that actually led to the creation of modern rigor. Don’t let that be you.

If they don’t get it, take the case that there’s a legitimate mathematical objection behind that. Treat their “I don’t get it” as “I don’t buy it.” Now getting them to buy it is your job.

Sh*t I F*cking Love (Wherein I Am Moved to Profanity by Enthusiasm) Friday, Apr 5 2013 

Shawn Cornally doing his thing.




Deborah Ball and Lucy West are F*cking Masters Sunday, Mar 31 2013 








Here was the problem:

What fraction of the big rectangle is blue?

What fraction of the big rectangle is blue?


He is, of course, wrong.



(a) clearly acknowledged the sound reasoning behind his answer


(b) clarified that he missed something key.


Ball: Who can summarize what [Kid A] said?



Kid B: Oh.



Kid A: Yeah.



Ball: And what were the two parts?


Ball: And are they equal?

Kid A: Yes.


Many students: Yes.



Kid A: Oh.




Here is the other video he refers to in the video:




I am cross-posting my review of this book on







Good Brawls and Honoring Kids’ Dissatisfaction Friday, Mar 8 2013 


Is 1 a prime number?


Point 1: This is awesome.





Multiplying by 10,



9 = 9x

so x=1







Dividing by 10,



-.9 = .9x

so x=-1










p.s. Obliquely relevant: Teach the Controversy (Dan Meyer)

A Note to My Fellow White People Friday, Jan 18 2013 


A Note to My Fellow White People












Thereby proving his point.


How about now?

Featured comment



Notes from the Learning Lab: How to Dull My Curiosity Friday, Dec 14 2012 





« Previous PageNext Page »


Get every new post delivered to your Inbox.

Join 156 other followers