# 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.