I’m writing a paper for a book, and just finished a section whose draft is worth posting. For what it’s worth, I claim here that the book publisher (Springer) will own the copyright and I’m posting this here as fair use and besides, it will get edited.
Here we go:
Modeling activities exist along a continuum of abstraction. This is important because we can choose a level of abstraction appropriate to the students we’re targeting; presumably, a sequence of activities can bring students along that continuum towards abstraction if that is our goal.
As an example, consider this problem:
What are the dimensions of the Queen’s two pet pens?
The Queen wants you to use a total of 100 meters of fence to build a Circular pen for her pet Capybara and a Square pen for her pet Sloth. Because she prizes her pets, she wants the pet pens paved in platinum. Because she is a prudent queen, she wants you to minimize the total area.
Let’s look at approaches to this problem at several stops along this continuum:
a. Each pair of students gets 100 centimeters of string. They cut the string in an arbitrary place, form one piece into a circle and the other into a square, measure the dimensions of the figures, and calculate the areas. Glue or tape these to pieces of paper. The class makes a display of these shapes and their areas, organizes them—perhaps by the sizes of the squares, and draws a conclusion about the approximate dimensions of the minimum-area enclosures.
b. Same as above, but we plot them on a graph. A sketch of the curve through the points helps us figure out the dimensions and the minimum area.
c. This time we enter the data into dynamic data software, guess that the points fit a parabola, and enter a quadratic in vertex form, adjusting its parameters to fit the data. We see that two of these parameters are the side of the square and the minimum area.
d. Instead of making the shapes with string, we draw them on paper. Any of the three previous schemes apply here; and an individual or a small group can more easily make several different sets of enclosures. Here, however, the students need to ensure that the total perimeter is constant—the string no longer enforces the constraint. Note that we are still using specific dimensions.
e. We use dynamic geometry software to enforce the constraint; we drag a point along a segment to indicate where to divide the fence. We instruct the software to draw the enclosures and calculate the area.
f. We make a diagram, but use a variable for the length of a side. Using that, we write expressions for the areas of the figures and plot their sum as a function of the side length. We read the minimum off the graph.
g. As above, but we use algebraic techniques (including completing the square) to convert the expression to vertex form, from which we read the exact solutions. In this version, we might not even have plotted the function.
h. As above, but we avoid some messy algebra by using calculus.
Now let’s comment on these different versions.
Activity (a) is the most concrete, and has the fewest barriers to participation. But is it modeling? Certainly, even though it does not use functions. The experienced teacher can also make the constraint clear (Why can’t you make a little circle and a little square? Because you have to use all the string.) and can even use these concrete materials to clarify one of the most confusing things about the problem—that there is in fact a minimum area in the middle. Most area problems using a fixed amount of fence, after all, maximize area. It only takes a moment to make a single huge square (What if we used all the fence for the sloth?) to see that its area is much greater than any of the “intermediate” examples.
Version (b) uses a graph and a function for the first time. It’s an informal function, but a function nevertheless. Version (c) introduces symbolic mathematics, and here we need technology to help us make this practical in the classroom. In fact, you could do (a), followed by (b) and (c) in rapid succession, to help students see the connections between the string diagrams, the measurements, the graph of data points, and the graph of the function.
Version (d) takes a new step towards abstraction by using a diagram. Not only do students need to make sure themselves that the sum of the perimeters os constant; they probably also calculate the diameter form the circumference and calculate the areas from known lengths rather than measuring.
Using geometry software in (e) is enchanting but complicated; most students will not be able to figure out how to make this work without explicit instruction. But seeing the shapes change dynamically can have a profound effect on your understanding of the situation, especially on the role of the problem constraints and the behavior at the limits.
In (f), we no longer plot data points. We’re doing all the problem setup we would do for calculus, but finishing it using graphing software and approximation. Thus this version includes the important abstraction that many students find so difficult. Note also that this involves recognizing and generalizing the steps from versions (a) through (d).
Versions (g) and (h) are, alas, what we get most often in typical classes. Interestingly, completing the square—which students traditionally learn before calculus—is much harder than taking a derivative.
Looking back at all of these, consider again the question of why it is that this problem involves a minimum rather than a maximum. A successful calculus student might say, “because the coefficient of the first term is positive.” But a student who used the string might say, “because the shape you can make with the whole fence is much bigger than two shapes, each made with half the fence.” One can make a case that the “string” answer shows more insight.