Dan Meyer read yesterday’s post!! And commented!! Thoughtful as always, it deserves a post of its own—not just a reply—in reply.
The first question in a lot of the activities is “Predict what the relationship will look like.” What advantages and disadvantages does that question have over a question like, “Predict how tall a stack of 500 cups will be,” or another question that requires the relationship but which involves a more concrete objective.
Dan makes a good point. That sort of task promotes using the function. I think it’s especially powerful when it’s late in a sequence of questions. A prototype might be the handshake problem, where early questions help the kids understand what we’re asking with small numbers. The small-number, intuitive, draw-able, act-out-able cases help students get the notion of a sequence and start to organize their data. Then, as he points out, the question about a large number makes using symbols practical and desirable. Furthermore, the concrete example presumably helps bridge that abstraction gap.
In this case, a good question of that sort might be, How many triangles do you have to make before the “spokes” are one meter long? That would be a good addition, but I’m not sure that the beginning is the right place to add it.
Let me explain: In this book manuscript, that opening question,
How will [the sequence number and the side lengths] be related?
Predict: What do you think the relationships will look like?
asks the kids to make a prediction before they measure anything. I think it makes a difference because it’s about data, and because the relationships are not intuitive.
This prediction business is tricky. I don’t pretend to know exactly what I’m talking about, but it’s informed by what I’ve learned using Sonatas for Data and Brain, and from some wonderful work in physics education by Eugenia Etkina and her colleagues.
The nub of what the Etkina folks say is that, in physics, it’s inhumane (or at best pointless) to ask students to predict a specific number when they don’t yet have a model; the point of prediction is to test whether your model is any good.
In the case of the handshake problem, early prediction may have a purpose in setting up surprise; if you think 100 people require only 50 handshakes, learning the real answer may wake you up. But it may also send the message that you’re bad at predicting.
In contrast, the task in the manuscript really is to predict what the relationship will look like, for example, by drawing a graph that shows the relationship, with as much detail as you can manage. I should check the excerpt I posted to see if this is clear. This prediction could include a function, written in symbols, but usually doesn’t.
This type of prediction serves several purposes. For students who (like many) are not comfortable with data and measurement, it helps students focus and pre-think what they’re going to do:
- It helps students imagine what the data will look like.
- It helps them figure out what the variables are.
- It helps them think about which ones are dependent.
- It gives them an idea about what their goal might look like.
We look at the prediction graphs to see if the axes are labeled and whether they’re scaled; whether we see data points and whether there is a model.
If students are more comfortable or experienced,
- The prediction graph prompts students to imagine the shape of the relationship (Is it linear? Does it curve?)
- It gives students a chance to analyze the situation and figure out what the function should be.
Initially we generally get crummy graphs. As bad as sketchy axes and a “curve” that’s a version of y = x. But through the feedback of finishing these mini-investigations, and seeing actual graphs—with data and models and (eventually) residuals—and comparing the early predictions to these graphs, the predictions gradually improve. (One important step in evaluating the predictions in the “recapitulation” phase is to ask, what about your prediction is right? Just recognizing that the spokes get longer is something…)
So: I love Dan’s question later in the process, but not at the beginning. I should ask more questions that demand the function—questions that are impractical without it.
I think there are also other kinds of questions that highlight the formal relationship, however, and don’t take the occasionally-dangerous step of extrapolating far beyond the data. In this case, an example is, How do you know the relationship isn’t linear?
4 thoughts on “Questions to Encourage Actually Using Functions”
Thanks for the Etkina reference, Tim. Something I should check out, I’m sure. I suppose I’m curious what kind of model (and at what level of abstraction) a student needs in her head before she can make a guess that falls within the ballpark. I mean, if I’m an eighth grader, I may not be fully aware of these models they call “quadratic” (in any of their abstract forms) but I can make a pretty confidence prediction whether or not a basketball sailing towards the hoop is going to go in.
Dan’s point is well-taken; one can legitimately encourage using something engaging (as Dan is famous for doing), especially from the real-world, but often in video, that promotes prediction, making things quantitative, finding the need to climb the “ladder of abstraction,” and so forth. And by those measures the spiral of triangles and the associated questions fall short: there is no real need, no burning question, no perplexity involved in finding the function of “spoke lengths.” No one cares how big the hundredth triangle is in the same way as they care whether the basketball goes in. And most kids certainly don’t care that the relationship is exponential as opposed to something else.
But to answer the question about what kinds of models students need, I should say that for questions like the ones in EGADs, I’m expecting some exposure to all of the underlying tools.
To get everything in the spiral activity, students need to understand similarity, have seen exponential functions, and have a decent understanding of soh-cah-toa trig. That is, I wouldn’t use it before grade 11. The point is that the data you get from measuring shows relationships that you can model with elementary functions. Students can understand why those particular functions are the right ones, that is, how the geometry connects with the symbolic expressions, and how the data follow the functions appropriately. Often, pretty-good-fit parameters from the modeling have meaning; in this case, the base of the exponent is related to the central angle.
Yet you can get something out of it even if you can’t make all the connections. You can predict that the relationship is not linear. Or you might discover that the relationship is not linear. You might see it curve upwards and decide it’s a parabola, and then figure out from residuals that it’s not. Or try an exponential and succeed in fitting the data—without understanding why it’s exponential.
So it’s rich and has multiple points of entry, multiple ways of dealing with the problem. But it’s not as real and engaging as the question about the basketball. I’m feeling that Dan is chastening me—in a good way—for getting the fundamentals wrong. So I have to ask: can we do everything with great three-act, video-based problems? Or is there a place for this kind of task, where students work with an admittedly concocted exponential situation that is nevertheless more physically real (in that you measure it) than an amortization problem (yawn) and more realistic than the ones where you start with ten bacteria in a Petri dish and they double every 2.5 hours?
Hi Tim, thanks for the follow-up. I should probably clarify that my question stands apart from any “real world” or three-act math concerns. (The answer to this question is a resounding “No” from me: “So I have to ask: can we do everything with great three-act, video-based problems?”)
I’m asking — even in pure math — what the difference is between saying “Model y as a function of x where y = i^x.” and “What would i^2013 equal?” I think I’d ask the second question first. It’s more concise and lighter on formal mathematical language and it also requires the student to grapple, at least informally, with the first question. I’d probably explicitly have to ask the first question later.
I think the difference between those two questions takes on additional resonance with applied math where we can often use media to provoke students to wonder questions that require our models. But I wouldn’t say my question applies only to applied math, or that we should use applied math in every case for that matter.
I think, seriously, that at some point students should have some level of mathematical curiosity. I did a lesson, one in which students discovered and created trig tables by careful measurement. (If I could three-act that lesson….). The kids were into it! Not, let’s look like we’re enjoying it, but really enjoying it.
Math that looks cool has its place, no offense Dan… I’m sincerely an admirer. But math that has the potential to show how cool and interesting math can be, without the glitz, needs to be done.
We never would have become math people if we had not at some point seen the wonder of real mathematics. I think predicting is a great way of getting kids thinking of how things change. I had a 9th grader ask me, kind of in a tentative way, at tutoring one afternoon (not one of my students, btw) if lunar cycles were an absolute value kind of function. I’m looking forward to having him in algebra 2 next year.
Lots of respect to both of you…