gebra through the use of rich media. By embracing these media, we can engage students while synergistically meeting the needs of math teachers. Indeed, we have already seen our curricular approach, described below, help students raise their algebra scores.
Let’s make this vision concrete. Algebra textbooks contain exercises that ask students to determine the next entry in a table, such as Table 1, or to create a general “variable expression” that computes any arbitrary entry of the table. In Table 1, students are expected to say that 5 comes with 25 and x comes with x · x. We might even hope to teach the student the notation f(x) = x · x, but why would they care? This function means nothing to them outside their homework.
We can, however, show these students that modern arithmetic and algebra do not have to be about numbers alone. They can just as well involve
Table 1.
images, strings, symbols, Cartesian points, and other forms of “objects.” For example, Figure 1 is an arithmetic expression involving images in addition to numbers. The operator placeImage takes four arguments: an image (the rocket), two coordinates, and a background scene (the empty square). The value of such an expression is just another image, as shown in Figure 2. That is, algebraic expressions can both consume and compute pictorial values, enabling students to manipulate images using algebra.
Imagine asking students to determine a rising rocket’s altitude after a given period of time. We could start with a table and the simplifying assumption that rockets lift off at constant speeds, as shown in Table 2. Because students understand that functions can produce images, not only numbers, we could even express this exercise as a problem involving a series of images and asking students to determine the next entry in Table 3.
1
1
2
4
3
9
4
16
5
?
…
…
x
?
figure 1.
placeImage (
, 25, 0,
)
figure 2.
placeImage (
, 25, 0,
) =
Table 2.
0
0
1
10
2
20
3
30
…
…
t
height(t) =
By asking the student to define the function rocket, we are asking for a “variable expression” that computes any arbitrary entry of the table—just as we asked in the case of numbers. We would hope to get an answer like the one shown in Figure 3. A teacher may even point out here the possibility of reusing the results of one mathematical exercise in another, as shown in Figure 4. Students thus see the composition of functions and expressions, all while using mathematics as a programming language. In addition, students are motivated to learn more about mathematics and physics to improve these little programs.
With one more step, students can visualize this mathematical series of images and get the idea that constructing such mathematical series can be an aesthetically pleasing activity:
showImages(rocket, 28)
This expression demands that rocket be applied to 0, 1, 2, 3, 4, 5, etc., and that the result be displayed at a rate of 28 images per second. (Note how show-Images furtively introduces the idea of functions consuming functions, because its first parameter—rocket—is itself a function.) Now we can tell students that making animated movies is all about using the “arithmetic of images” and its algebra.
Readers shouldn’t be surprised to find out that what we’ve described and illustrated here isn’t just imagination or a simple software application for scripting scenes. A form of mathematics can be used as a full-fledged programming language, just like Turing Machines. In such a language, even the design and implementation of interactive, event-driven video games doesn’t take much more than algebra and geometry. As students develop such programs they “discover” many concepts on their own simply because they want to add luster to their games—and, to formulate their improvements, they learn new mathematics and physics.
We have field-tested the beginnings of such a curriculum in the context of our TeachScheme! project for the past five years with a family of teaching languages that support images as first-class values. These languages
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