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?