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 tominimizethe 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. (In 2014, Dan Meyer did a number on a related problem and made two terrific dynamic geometry widgets, Act One and Act Two.)

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.