A "big question" is supposed to do two things in a math class. First, it’s supposed to help students situate knowledge. Second, it’s supposed to make the content more meaningful to students. How does a question have this effect?
My (totally made up) analysis is that we’re trying to bootstrap our math content onto a question that students can quickly recognize as meaningful, and an approach to answering the question that students can quickly recognize as natural. We’re hoping that they care about the question (giving our, say, Alg2 content value) and that they’ll remember the natural approach to answering the question (so that they can associate our, say, Alg2 content with the approach).
A question such as “What simple functions are there?” is no help to students because (a) they’re not interested in the answer and (b) they don’t have any idea how to go about answering it. As a consequence, the question (a) is unable to make Alg2 more meaningful to students and (b) unable to provide students with a framework for their knowledge.
Even excellent metaphors or analogies won't necessarily make great "big questions." Take the idea that functions are analogous to relationships; just as we could catalog human relationships, we could catalog the numerical ones. In question form, that looks like, “What kinds of relationships can numbers have?”
But what makes for a great metaphor or analogy, in this case, doesn't lead to a great question. It fails at both the tasks that a "big question" is supposed to excel at: (a) I don’t think my kids will think that it’s worth answering and (b) I can’t think of a natural way to go about answering the question.
The next step for me is to pick anything--anything at all!--from the three math courses that I teach and try to get some practice finding big questions. Then I'll try to take on NY's Algebra 2.
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Tuesday, June 28, 2011
Sunday, June 26, 2011
The value of big questions
What makes big, course-spanning questions so great is not that they motivate students with a tantalizing question. No question that I ask is going to be able to motivate students 5 months after I ask it. Any moment of curiosity will have passed. For motivation and engagement, we need daily questions, curiosities and (for +10 Meyer points!) perplexities.
I think that the big unit/course spanning questions are wonderful because they provide meaning to the curriculum. It's harder to ask the question "Why are we learning this?" when what we're learning is clearly situated in a larger, obviously meaningful framework. For instance, the question "Is there life on other planets?" naturally leads to the questions "What are the conditions for life?", "How hot are other stars?", "How far away are planets from stars?" and "What's in the atmospheres of alien planets?" Bam, there's your calculus-based intro Astronomy course. And while students in a different class might wonder, "What good are absorption lines?" my bet is that students in this class will (a) be more likely to situate them correctly as helpful in determining temperatures of distant stars or atmospheric content of exoplanets and (b) won't think that astronomy is useless and boring. So that's what we're going for here, I think.
And now, a problem. Let's partition the world of course-spanning questions into the purely mathematical and applied mathematical questions. Let's take an applied mathematical question such as "Can we predict the motion of a basketball?" or "How do electronics work?" or "Can we beat the stock market?" If we really and honestly pursue these questions, we're going to have to go beyond our mathematics, since we're going to need to use the tools of physics, or economics, or engineering. In other words, doggedly pursuing non-mathematical questions quickly leads us out of the mathematical domain.
On the other hand, rich mathematical questions don't typically do the work of being obviously meaningful to students. The best that I can think of is "What's a number?" which I imagine as a narrative arc spanning the first bits of a first year of Algebra.
This is a long-winded way of saying that I think we're either looking for mathematical questions that are big and basic enough to motivate this month-long investigation, or applied mathematical questions that are closed under honest inquiry.
I think that the big unit/course spanning questions are wonderful because they provide meaning to the curriculum. It's harder to ask the question "Why are we learning this?" when what we're learning is clearly situated in a larger, obviously meaningful framework. For instance, the question "Is there life on other planets?" naturally leads to the questions "What are the conditions for life?", "How hot are other stars?", "How far away are planets from stars?" and "What's in the atmospheres of alien planets?" Bam, there's your calculus-based intro Astronomy course. And while students in a different class might wonder, "What good are absorption lines?" my bet is that students in this class will (a) be more likely to situate them correctly as helpful in determining temperatures of distant stars or atmospheric content of exoplanets and (b) won't think that astronomy is useless and boring. So that's what we're going for here, I think.
And now, a problem. Let's partition the world of course-spanning questions into the purely mathematical and applied mathematical questions. Let's take an applied mathematical question such as "Can we predict the motion of a basketball?" or "How do electronics work?" or "Can we beat the stock market?" If we really and honestly pursue these questions, we're going to have to go beyond our mathematics, since we're going to need to use the tools of physics, or economics, or engineering. In other words, doggedly pursuing non-mathematical questions quickly leads us out of the mathematical domain.
On the other hand, rich mathematical questions don't typically do the work of being obviously meaningful to students. The best that I can think of is "What's a number?" which I imagine as a narrative arc spanning the first bits of a first year of Algebra.
This is a long-winded way of saying that I think we're either looking for mathematical questions that are big and basic enough to motivate this month-long investigation, or applied mathematical questions that are closed under honest inquiry.
Sunday, June 5, 2011
Virtual Filing Cabinet: The Blog
I have a lot of summer projects, as this is the end of my first year of teaching. I'm teaching (gulp) Algebra 1, Geometry and Algebra 2 again next year, in addition to a computer programming course. This is way more than I can handle with excellence, but I teach at a small school so I just have to suck it up. I didn't have much of a choice about this.
Anyway, I need to think through all my curricular assumptions, now that I've actually gone through this stuff once, and I need to work it through on paper. I'm going to do as much as that as I can on this blog. My goal is to do some thinking on a ton of things, and to collect as much from the internets as I can. In short, I'm hoping to make this blog a VFC for curricular stuff, with thoughts about the advantages and disadvantages of various resources.
This has the added benefit of potentially helping others, and making a helpful contribution to the online math world. Since my last idea on online sharing didn't really catch on (I still like it, though!) maybe this will prove helpful to some.
Anyway, I need to think through all my curricular assumptions, now that I've actually gone through this stuff once, and I need to work it through on paper. I'm going to do as much as that as I can on this blog. My goal is to do some thinking on a ton of things, and to collect as much from the internets as I can. In short, I'm hoping to make this blog a VFC for curricular stuff, with thoughts about the advantages and disadvantages of various resources.
This has the added benefit of potentially helping others, and making a helpful contribution to the online math world. Since my last idea on online sharing didn't really catch on (I still like it, though!) maybe this will prove helpful to some.
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