Asking The Right Question

John Dewey . . . asked a class, “What would you find if you dug a hole in the earth?” Getting no response, he repeated the question; again he obtained nothing but silence. The teacher chided Dr. Dewey, “You’re asking the wrong question.” Turning to the class, she asked, “What is the state of the center of the earth?” The class replied in unison, “Igneous fusion.” (Bloom, Engelhart, Furst, Hill, & Krathwohl, 1956, p. 29)

Did the class know about the center of the earth? The point of this little parable, of course, is not really. Being able to correctly recite the words ‘igneous fusion’ on cue does not make evident that these words had meaning to the students.

Meaning. An ultimate goal of learning, to enrich our lives with meaning.

From this perspective, Dewey’s question was the right one. It connected school learning to meaning in the world.

How do we do that in math class, the place with the most deep-seated rituals of recitation and mindless calculation? How do we move from what English mathematician and philosopher Alfred North Whitehead called ‘inert ideas’ — those which have been received and not utilized or tested or ‘thrown into fresh combinations’?

I am about to go teach a class of pre-service teachers to engage this very set of issues. We are looking in math class at the nature of the tasks being used through the lens of Margaret Smith and Mary Kay Stein’s work on this topic.

I already anticipate some of the arguments that will come up about the richness of particular tasks. Over the years of using this framework with both new and experienced teachers, I can boil the issues down to these questions:

1. What kind of thinking will students do as they engage in this activity?

2. Is there sufficient ambiguity that they can approach it in different ways or re-think familiar ideas in new ways?

I would love to hear from others about your ways of judging good mathematical tasks, as well as your favorite resources for finding them.

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9 thoughts on “Asking The Right Question

  1. What a great anecdote!

    In terms of “ambiguity”: I’m inclined to reserve this word for instances where there is wiggle room for interpretation about what is being *asked*, or asked for. This is one—but not the only, in my mind—entry point to a rich task. For instance, Swan’s measure tasks are an example of ambiguity in the posing of a problem, as is the question (which I like and have explored with middle schoolers): “how many triangles can you make with six lines?”

    Other kinds of problems, though, can have statements that are relatively unambiguous, but where many possible avenues of exploration and analysis are possible. Perhaps, for instance: “what whole numbers can be written as the sum of two perfect square whole numbers?” What is sought is clear enough; finding it is a puzzle entire.

    Do you find this distinction helpful? (I’m tinkering with vocabulary and think I gather and agree with the thrust of your framework.)

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    • Yes! This is good. We often use the metaphor of “low floor/high ceiling” to describe rich problems, meaning that lots of people can have access but there is a lot of room to explore ideas deeply. I find the Stein & Smith task framework to be a good starting place, but one class’s ‘procedural’ problem can be another’s ‘doing math,’ depending on what has come before and the nature of the prompt and set up. I have tried to come up with a way of helping teachers ‘see’ task potential that allows for context but isn’t too mushy. Thanks for the contribution!

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  2. I’d just like to double down on Justin’s example, because I think it’s important to create a distinction between ambiguity and [insert other thing we’re trying to talk about]. To my mind, ambiguous questions are mathematically important, but within a very particular scope: namely, ambiguity (1) creates a need to make ill defined problems precise, and (2) motivates convention. Measure tasks are a great example, and Vaudrey’s mullet task is a great example of a great example. What does it mean for someone to be mullet-y? Is Person A more or less mullet-y than Person B? Mullet-y-ness is obviously not well defined for most kids coming fresh to that problem, so it’s immediately clear that an interesting problem can be ambiguous, and that it’s roughly intractable (which is especially sucky since it’s so interesting) until we are able to state the problem satisfactorily. That’s one good reason we turn to mathematics. And since (by the approximate definition of ambiguity), there’s more than one way to resolve that fuzziness, we have to come to some sort of community consensus on the best/most useful definition of mulletude if we hope to talk about it meaningfully. Which, of course, we do.

    Speaking of measure tasks, it seems that what you’re really getting at with your second question is some measure of a problem’s openness. Something like, “How strongly does this problem suggest its own solution?” A bit of that might be inherent in the problem, but probably more in how it’s posed. It’s tough to find problems that don’t legitimately have multiple paths to a solution; I suspect a large part of presenting rich tasks is how to suggest a productive path for those students who otherwise wouldn’t know how to begin without lighting it up so brightly that it robs others of the lovely opportunities for meandering. Of course, if you manage to quantify that satisfactorily, you’ll let me know.

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    • I think you raise a number of good issues. I especially think the observation that the “openness” of a problem depends largely on how it is posed cannot be stressed enough! I don’t think I will ever quantify that — it is highly dependent on the kids and what has come before.

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  3. I’m late to the conversation but to come at the question another way, a couple of things I suggest teachers (pre-service or in-service) do to “open up” tasks are: create opportunities to use multiple representations or strategies and especially, to make connections between representations/strategies; require explanation and justification; and present incomplete work or errors for students to analyze. Of course these strategies aren’t foolproof, but what I like about these little steps is that they make it possible to take a task that looks pretty routine and make it dramatically less so, without having to always look for magical new tasks. It has been my experience that teachers often get bogged down in the hunt for tasks–none of which are ever context-proof, to link to one of your more recent posts, Lani–and neglect to attend instead to those critical things that make tasks work in their contexts (e.g., What’s the really key mathematics, and how do I support my particular students to grapple with it? What might they get stuck on, and what strengths do they have that I can get them to leverage when they get stuck? If they aren’t getting stuck, but they’re blazing through it, how do I slow them down? Etc., etc.). Or they get so hung up on “good tasks” that they start to feel like unless they can devote big chunks of class time to big, open-ended, exploratory stuff, they can’t teach the ways we’re asking them to.

    So, I guess that’s just to say, I agree that it is important and helpful to engage teachers in thinking about what makes a good task vs. a poor one, but I’m trying to get them to move on from Good Task-worship and into thinking more about how to use the humble tasks they have in front of them to make interesting things happen in their classrooms.

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    • Yes, thanks for this Nicole. I’ve been thinking about this this morning. One of the things you’re saying, I think, is that even in the small routine activities, there’s a need for a variety of approaches and ownership, for explaining and listening to each other?
      It’s a really good point. I’ve been finding that in doing regular counting circles, the sharing of strategies is a really worthwhile moment, even though it’s not a big part of the lesson.
      Why it got me thinking though, is that I really like looking for new and better ways to get the students exploring a topic. So I as a teacher get more ownership, and hopefully the students get more control too. Maybe I have more “wiggle room” in my curriculum, but whatever the syllabus there’s usually a space for creativity about lessons. I do like a good task, a magical one even. Can’t we have the tasks *and* the attention to the details of what makes the task work for the students??

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  4. I like what you’re saying about ambiguity, Ilana.
    Following on from what Justin says, I think what I look for more is the linked idea of the possibility of divergent paths, within constraints.
    Because I’m a primary teacher, the arts, and writing are a great model of this for me of how a polyvalent stimulus can work as a starting point, and I’m trying to bring over what works so well in those areas. So, for instance this last term,” here are some Paul Klee pictures, write a hundred word story about one of them”:
    http://y4ist.blogspot.fr/search/label/Paul%20Klee
    The ambivalence (more, the surrealism) of the Klee’s paintings helped to push the students off in different directions, even when they started from the same image. The instruction, to echo Justin’s reply, is clear and unambiguous, but the responses very varied.
    The ownership, the “I know that it’s mine” that comes with this divergence is a precious thing. The students have their own intentions, their own images, and they know they can produce what they want to, even if they’re not perfectly satisfied with it.
    So, in maths, instead of giving a pattern to think about, “make your own”:
    http://y4ist.blogspot.fr/search/label/cube%20patterns
    then we’ll look at what’s going on in these patterns.
    I’m still finding new ways to do this, and it’s great being a pioneer, and in such good company, in the great Wild West of Ambi-mathematics!

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    • I am growing to believe that ownership and learning, identification with the objects of learning, is critical and not supported enough in school in general — and math class in particular. I agree that this is often what helps put the light in students’ eyes — that precious kind of engagement you describe. Thanks for sharing your thinking.

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