# A Critical Language for Problem Design

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.

## 24 thoughts on “A Critical Language for Problem Design”

1. Ben – I first saw the ‘low threshold – high ceiling’ language in discussion on Dan Meyer’s blog.
Great post here, gets at the heart of some of what I struggle with as a Dept Chair. How can I describe what I want when we don’t have a shared language about what types of tasks we want to use in the classroom?
I’m tweeting out a link to this post.

2. Chris Okasaki says:

The phrase “low floor, high ceiling” goes back at least as far as Seymour Papert’s 1980 book “Mindstorms: Children, Computers, and Powerful Ideas”, where he was applying those terms to a programming language designed for children, sepecifically Logo. Children could get started in Logo very quickly and easily (low floor) but the language was powerful enough that they would not outgrow it for a very long time (high ceiling). However, I do not know if he originated the terms.

Also, as you may be aware, this attempt at creating a powerful set of shared terms is closely related to the “software design patterns” movement from the ’90s (see “Design Patterns: Elements of Reusable Object-Oriented Software), which itself was modeled after Christopher Alexander’s 1970s work in (non-software) architecture (see “A Pattern Language).

3. Just popping by to let you know my last name is Murray, not Murphy. 🙂

1. Yes–just like that! Tessering is an old family secret. 🙂

4. This is a great post, and I love the terms “jamming” and “thwarting.”

Two more:
1. “hiding information” – literally creating contexts in which convenient information is hidden. I’ll be blogging about this over the next few days. I’d filethis under “hiding information” as well.
2. “case, case, theorem” – that thing that you do where you set up a theorem with a few well-chosen cases. PCMI/CME/EDC does this all day long

5. suevanhattum says:

I first heard low floor, high ceiling at the math circle institute, I believe.

1. suevanhattum says:

No. It wasn’t low floor, high ceiling. It was “raising the floor versus raising the ceiling.” Do we do community style math circles to help bring the level up for all (floor), or competitive math circles to improve our top students (ceiling)?

6. I agree — this is an awesome idea! Doug Lemov does something similar with *Teach Like a Champion*, where he gives snappy titles to classroom techniques. The book was required reading in my credential classes, and having that common language really helped up think critically and discuss features of an effective classroom.

In fact, I also participated in part of the Gates MET project; our task was to watch classroom clips and tag them with different labels describing what we saw happening. While not very exciting, that turned out to be a great benefit for me, because it helped me form a language for talking about what was going on — there is definitely something to be said about being about to move further with ideas once those ideas have names (related to Sapir–Whorf, perhaps?)

1. I totally agree that part of the power of Lemov’s book was that it did this. Controversial anthropological theories of the relationship between language and thought aside, it’s totally clear to me how good names can be so useful in practice!

7. I think this would lend itself to a Wiki. Think of TVtropes, but for math ed – something like PSTropes! If several of you here are interested, I am happy to set up an instance of a wiki for this. We can have a live mini-marathon for a couple of hours to start populating it, using Skype or G+.

While we are naming names, Alan Schoenfeld’s works on problem-solving will be a good source. So, I expect, will be works of Ma and the Vygotskians on the Chinese and Russian practices.

Here are a couple of term suggestions:

Extremizing: Considering extreme cases that throw off assumptions, or test procedures and definitions. A lot of number theory definitions break at 0 or 1, for example.

Nonexampling: Asking people to find a non-example, as opposed to counter-example.

8. I’ll take a shot: how about “deconstructing”. Here you would take a problem (like the ski-lift problem Dan uses as an example in his “Math class needs a make-over” talk), and strip it down to a basic element. Have the kids come up with what they believe to be the question, adding information if necessary. Then reveal a little more of the problem, and repeat. Finally reveal the problem in its entirety. They can work on that, along with all the other questions that they have generated. I’ve experimented with this, and will make it the subject of a blog post later this week.
exit10a.blogspot.com

9. I really like this idea, Ben. I’ve doing a fair bit of work on task design myself, as well as looking at a lot of different tasks that have been created (and subsequently mimicked) all over the math education community. I don’t know if a common language exists, but it certainly seems useful.

How are problems of the month different from projects? How are they the same? How can we describe the open-ness of a math task in a shared way? How can we think about the utility of a task to give us information on what students think?

10. Oh, FYI, you can delete this comment, but the font size is so small on the comments on this blog that I had to zoom to 150% to be able to effectively read them.