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
how Would This Work?
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
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.
, 25, 0,
, 25, 0,
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:
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.
Does it Really Work?
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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