Archives for the month of: July, 2010

[Edited on 8/29 for the SBG Gala 2]

Shawn is the kick-ass physics teacher I want to be if I ever get to teach physics. In his SBG-routine for physics, along with his content standards he includes throughout his course a number of  inquiry standards, viz.,

  • Student can formulate a testable question (__/40)
  • Student can design a valid experiment (__/40)
  • Student can form reasonable hypothesis from theory (__40)
  • Student can statistically (averages, stdev, & chi) analyze data (__/40)
  • Student can draw reasonable conclusions (__/40)

I am trying to do the same for mathematics; this year that means Algebra II and Calculus. The NCTM offers these process standards:

  • Communication
  • Representation
  • Connection
  • Problem Solving
  • Reasoning and Proof

    I have no quarrel with these as important parts of doing math, but I don’t love these as course objectives because I don’t know how to assess them in this form. I’m trying to develop a more observeable set of general math practices, and came up with:

    • questioning (coming up with questions, conjectures, &c)
    • visualizing (translating words or equations into diagrams, this spans communication and representation)
    • abstracting (or maybe “mathematizing.” Translating a constraint or question into a mathematical representation we can use tools on. Maybe this is NCTM’s “representation,” but I’m focused here on going from the vernacular into mathematical representations more than on moving from one math rep to another)
    • strategizing (or, as Sam says, “take what you don’t know and turn it in to what you do know.” Spans connecting and problem solving)
    • generalizing (which spans problem solving and reasoning and proof)
    • explaining (which spans proof and communication)

    I was also thinking about something like “critiquing” or “debunking” or “skepticism” or something – spotting the errors in a line of argument. But maybe that’s just another way to demonstrate the ability to explain?

    After I came up with these I got some training from College Preparatory Math (thanks for the steer, Riley). For their Algebra 2 course they explicitly call attention to five “Ways of Thinking” which are (from the opening page of the text),

    1. Justifying (explaining and verifying your ideas)
    2. Generalizing (predicting results for any situation)
    3. Choosing a Strategy (deciding which solution methods make sense)
    4. Investigating (gathering information and drawing conclusions) and
    5. Reversing (solving problems backwards and forward).

    I think they have different ways of thinking emphasized in different courses. CPM doesn’t seem to call out any specific “ways of thinking” for Calculus, but I think these same 5 would be a good first cut.

    I also recently checked out the Common Core, which builds on NCTM these with:

    1. Make sense of problems and persevere in solving them.
    2. Reason abstractly and quantitatively.
    3. Construct viable arguments and critique the reasoning of others.
    4. Model with mathematics.
    5. Use appropriate tools strategically.
    6. Attend to precision.
    7. Look for and make use of structure.
    8. Look for and express regularity in repeated reasoning.

    The CPM Ways of Thinking seem to work. They don’t cover the common core standards completely (modeling, precision and structure aren’t stressed), but I can defend grading on them from the common core standards (or from the NCTM standards), and at least in the Algebra book, they are discussed (always in bold) throughout the text so students have support and examples for what I mean by them. So I am tentatively adopting these into my standards list for both courses, and we’ll see how it goes.

    On the other hand, I notice that Shawn doesn’t seem to use anything equivalent to inquiry standards in his own math class. Maybe he knows something I don’t?

    This post is part of a series on soft skills in math teaching incubated and curated by Riley. This conference pushed me and several others over the edge into blogging. My first post went up a week ago, and a couple days ago I got my very first comment (thank you, Ben!), which not only baptised me into Net 2.0 but also provided the grist for today’s post. Ben wrote:

    Again, communicate (body language, tone of voice, and out loud if you have to) that class isn’t moving on without a good answer here. This is the key to everything. They have to know that if they don’t turn their brain back on, everybody is going to sit there awkwardly till they do. (This can be tricky if there are management issues, but these need to be addressed on their own terms. If the kids can’t maintain decorum in the face of a thought-provoking question that they don’t know how to answer at first, the cognitive demand on them is not where the problem lies.)

    There are a number of different ways students may not maintain decorum in the face of a question they don’t know how to answer at first, but the one I dread most is the Soft Mutiny. Perhaps you did not encounter the soft mutiny during your student teaching year, or maybe you’ve forgotten what it feels like. Driving into quicksand? Stepping off the drop-off in the pool at age 6? Outward signs are (a) no open disrespect (that would be a Hard Mutiny), (b) complete refusal by most of the class to answer questions, pick up a pencil, or do anything but look at you and shrug, and (c) a complete closure of all lines of communication between you and the class.

    In the 200 or so class periods I taught during my student year I encountered perhaps 6-10 soft mutinies. Some combination of circumstances or moves on my part caused the room to transform without warning from a reasonably happy, functioning class into a place where students not only were completely lost but were also unable to say to me “We’re completely lost.” Or perhaps it’s “we’re writing you off, and we’re not going to bother to tell you that.” Possible causes of soft mutinies: I’ve inadvertently said something disrespectful, or I’ve given the impression that they “should” know something they don’t, or the question/topic I’ve posed strikes them not just as irrelevant, but appalinglly irrelevant. Or, or, or. The non-communicative aspect of the soft mutiny makes it hard to know just what’s going on.

    At first I tried to just keep going. That didn’t work. (The driving-into-quicksand analogy is helpful here.) Next I went on the theory that they just needed to be shown again what to do. That didn’t work either. (Think backing out of the quicksand and driving right back into it.)

    On subsequent occasions I tried to open the lines of communication: “Have I lost you? What’s going on?” I was surprised to learn that that didn’t work either. I realized over time that if students were lost and feeling discouraged by that, the last thing they wanted to do was talk about it, and if they were upset (with me, with something else), they expressed their mild hostility as “don’t expect us to help you fix this.” Asking them to write instead didn’t work: students had turned uncooperative, and to write when I asked them to would be cooperating.

    Actually, over the course of the year, as relationships all around got stronger, the “what’s going on?” approach did start to work. One or two students would break the ice and give me a hint about how they were seeing things. And along those lines, the better we all knew each other, the less often soft mutinies seemed to happen at all, which is the key to prevention.

    So what is the best response to a soft mutiny? I haven’t mastered it, but I believe there are some key components:

    1. Don’t take it personally. If I got upset, it got worse (think stepping on the gas while in quicksand.)
    2. Ask the students “What would be most helpful for you now?” This gives students input and control without forcing them to voice their own sense of being lost, or, if they’re mad at me or feel I’m doing poorly, without forcing them to say things they think might upset me or hurt me. This question got useful answers that moved the class forward about 50% of the times I asked it.
    3. If that question gets no response, then make a transition to another mode, activity, or task. Acknowledge that “This isn’t working. Let’s shift to a different approach altogether.” This gives everyone a way to leave behind the “stuck” feeling. I don’t always have Plan B in my back pocket, and one doesn’t always occur to me when I need it, but it seems to be key to getting out from under soft mutinies.

    One happy observation is that, although I’ve spent the night following every soft mutiny in despair for my relationship with my students and for my career, the next day students have arrived in a completely different state, as if the soft mutiny never happened.

    As mentioned, these got less frequent as we got to know each other. My aim next year is to prevent these entirely. What are some ways to help kids learn to express, in appropriate ways, when they are feeling shut down or turned off?

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