Shape

42 Introduction: Tell Me All About It
with the same alphabet (vocabulary). Letters and inches were alike—only so many fit
on a line. Naturally, counting was next. Both girls used manipulatives in the lower
grades to see what was happening. They started with unit cubes (points) that were
small enough to pick up with their hands and count. The cubes went together in
groups of ten to produce a ten-bar (a line). Then ten ten-bars were placed side by side
in a hundred-square (a plane), and finally ten hundred-squares were stacked in a
thousand-cube (a solid). These materials weren’t very attractive
but the method was clever. There was counting and place value—even geometry
and Cartesian coordinates—all at once. But it was odd, too. Units weren’t independent
in ten-bars, hundred-squares, or thousand-cubes. Lines were etched in their faces in
a grid. You couldn’t take units out. They fused in unusual sums when their number
and arrangement were right. Number and arrangement—that’s why Peirce said his figures
were the same. Yet fusing things together was different. Somehow, shapes were
involved. What was there to see? Did ten-bars, etc., always look the same? And it
went on in this way—at ten thousand, there were alternatives. There might be a tesseract
and cubes of higher dimension, or bigger bars and squares up to a million-cube.
The recursion was neat either way. Only which one was it; how did you decide? Counting
wasn’t ambiguous—it was just the way it looked.
Of course, fathers can be silly when it comes to their daughters. Counting is
something everyone needs to learn. And today, there are standards and tests to make
sure you do. ‘‘A student who’s tested is a student who’s taught.’’ Do your eyes make a
difference? Do children see what I see, or use shapes in the same way? Listen to Alexandra
when she was seven years old. What she shows is more than I could possibly
make up. ‘‘This’’ (draws a loop)
‘‘is a cursive l. You can use it in an i’’ (draws a small circle)