What Is This Thing Called Three?
Math teachers in college generally teach two different types of classes:
classes in which relatively high-level material is taught to students who
will use it in their careers, and classes in which relatively low-level mate-
rial is taught to students who, given the choice of taking the class or a root
canal without anesthesia, might well opt for the latter. The second type of
class includes the math courses required by the business school—most
of the students in these classes believe they will someday be CEOs, and in
the unlikely event they ever need a math question answered they will hire
some nerd to do it. It also includes math for liberal arts students, many of
whom believe that the primary use for numbers are labels—such as “I
wear size 8 shoes”—and the world would function better if different la-
bels, such as celebrities or cities, were used instead. After all, it might be
easier to remember that you wear Elvis shoes or Denver shoes than to re-
member that you wear size 8 shoes. Don’t laugh—Honda makes Accords
and Civics, not Honda Model 1 and Honda Model 2.
Fortunately (for at my school all teachers frequently teach lower-level
courses), the second type of math class also includes my favorite group of
students—the prospective elementary school teachers, who will take two
semesters of math for elementary school teachers. I have the utmost re-
spect for these students, who are planning on becoming teachers because
they love children and want to make life better for them. They’re certainly
not in it for the money (there’s not a whole lot of that), or for the freedom
from aggravation (they frequently have to teach in unpleasant surround-
ings with inadequate equipment, indifferent administrators, hostile par-
ents, and all sorts of critics from politicians to the media).
Most of the students in math for elementary school teachers are appre-
hensive on the first day of class—math generally wasn’t their best sub-
ject, and it’s been a while since they’ve looked at it. I believe that students
do better if they are in a comfortable frame of mind, so I usually start off
with Einstein’s famous quote, “Do not worry about your difficulties with
mathematics; I assure you mine are far greater.”^2 I then proceed to tell
them that I’ve been teaching and studying math for half a century, and
they know just about as much about “three” as I do—because I can’t even
tell them what “three” is.
Sure, I can identify a whole bunch of “threes”—three oranges, three
cookies, etc.—and I can perform a bunch of manipulations with “three”
such as two plus three is five. I also tell them that one of the reasons that
mathematics is so useful is because we can use the statement “two plus
three is five” in many different situations, such as knowing we’ll need $5
14 How Math Explains the World