Shape

125 A Second Look at Calculating
on the verge of doing superstars. Once you start out, you just want to add to the series
until you get them all—it’s the little matter of distinguishing one, two, and three. But
the feeling of completion (exhaustion) that comes with an enumeration of constituents
and their configurations is debilitating. The eye is sated, so that seeing more is
superfluous. Still, there’s always more to see. That’s why visual calculating is so important.
Seeing—using my kind of rules—trumps counting with its units, names, lists, and
definitions. Shapes aren’t everyday arrangements—their constituents can alter when I
look at them. And taxonomies (topology or syntax) aren’t stable—they change dynamically
as calculating goes on. John Dewey applies this relationship that puts seeing
ahead of counting in How We Think. He maps the hazards and drawbacks of ‘‘logical
analysis’’ and ‘‘anatomical and morphological method’’ in education. Evans tries the
former, and I use the latter in my catalogue of triangles in table 1. (In fact, Miss H——’s
exercise—connecting dots in lines to make planes—also shows what Dewey finds.
Sooner or later, counting is meaningless—and then it’s time to see.)
Even when it is definitely stated that intellectual and physical analyses are different sorts of operations,
intellectual analysis is often treated after the analogy of physical; as if it were the breaking
up of a whole into all its constituent parts in the mind instead of in space. As nobody can possibly
tell what breaking a whole into parts in the mind means, this conception leads to the further
notion that logical analysis is a mere enumeration and listing of all conceivable qualities and relations.
The influence upon education of this conception has been very great. Every subject in the
curriculum has passed through—or still remains in—what may be called the phase of anatomical
or morphological method: the stage in which understanding the subject is thought to consist of
multiplying distinctions of quality, form, relation, and so on, and attaching some name to each
distinguished element. In normal growth, specific properties are emphasized and so individualized
only when they serve to clear up a present difficulty. Only as they are involved in judging some
specific situation is there any motive or use for analyses, i.e. for emphasis upon some element or
relation as peculiarly significant.
I’m not so sure there’s a huge difference between intellectual (rational) and physical
analyses. It’s hard to imagine a final test for either. But in ‘‘normal growth,’’ there’s
the logic of inquiry, and this is how parts are defined as rules are applied. In fact, had
Dewey gone on to say that properties aren’t fixed but change even erratically from moment
to moment—that embedding isn’t only identity—he would have said a lot more.
But then, he wasn’t thinking about rules and calculating with shapes. The damage
done when logical analysis is taken too seriously is all too evident in the host of school
subjects Dewey surveys, from reading, writing, and arithmetic to drawing and geography.
The same kinds of problems are plain today in computer graphics (drawing, modeling,
visualization, etc.) and geographical information systems. It’s Evans’s syntactic
approach again, and the scientific approach to urban morphology, too, with atomic
units of description, cellular automata, and fractal geometry. Drawing and geography
are slow to change when there’s only counting. But why is counting naturally assumed
to be reliable and trustworthy—to get it all—when it leaves out so much? Repeatability
isn’t the goal when there’s always creative work to do now. Counting misses everything
that’s new—there’s nothing to see.