Everyone
loves a good story. In fact, narrative is so essential to our humanity that God
reveals himself to the world in a story. Art and the humanities are full of
narrative elements. Symphonies and string quartets develop themes and motives
to create sonic stories which include rising tension, a climax, and a
resolution. Even 30-second commercials present stories of despair-turned-to-joy
and promise us the same happily-ever-after if we buy their products.
Mathematics
too is like a story. That may seem a strange idea, yet it seems strange only
because we don’t usually teach mathematics as if it were one of the
humanities––which it is. We are happy to teach literature and history and
theology through discussion and essay assignments, but suddenly change tactics
for mathematics. But what if we taught mathematics in a more human (and perhaps
humane) way? What if we taught math in narrative contexts? First, however, I
should explain what I mean by saying that math is like a story.
One
of the clearest examples of the narrative quality of mathematics can be found
in the greatest geometry text to come out of the ancient world: Euclid’s Elements. The
Elements is divided into 13 books, the
first 6 of which form the basis for what is still taught in high school
geometry.
Euclid
begins The Elements
by presenting definitions, postulates, a few constructions, and the
side-angle-side theorem for the congruence of triangles––and with that
foundation laid, a world is opened for discovery. I can’t help but compare this
beginning to the opening of Genesis. God creates the world out of nothing in
six days, and Euclid creates a world out of concepts in 4 propositions.
After
this grand opening Euclid unfolds several themes: first the triangle, then
parallel lines, and finally parallelograms. Euclid’s handling of these themes
is much like the development in a mystery novel where little observations soon
uncover a complex web of intrigue. The climax of Book I occurs in the second to
last proposition, which we know as the Pythagorean Theorem. Here, theorems on
parallelograms and triangles suddenly unite to prove a beautiful and surprising
truth: that squares built on the two sides of a right triangle are equal to the
square built on the hypotenuse. Taken out of context and put in plain language
as I have just done, the proposition seems hardly surprising or beautiful. But
this too The Elements
shares with the mystery novel. Unless we have followed the story it is not
shocking to find out that a cook murdered a millionaire. Likewise, Euclid’s
propositions are not surprising to anyone who is merely given them; the story
must be read through from the beginning.
But
reading the story through from the beginning is not an opportunity frequently
given to students. As beautiful and precise as The Elements is, it is no longer used to teach geometry. The
reasons for this are twofold. First, it is inefficient to wade through 46
propositions in order to teach the Pythagorean Theorem. It is far easier to
simply tell students that A2 + B2 = C2.
Second, textbooks are usually organized to present material in the most
expedient way, rather than the way in which concepts actually developed and
were discovered.
In
other words, most textbooks take a CliffNotes approach and tell you all you
need to know about the narrative without allowing you to encounter the
narrative itself. This may seem an efficient way to prepare for tests but it
misses the story and gives the student the impression that mathematics, like
Athena, sprang full-grown from the head of Zeus (or from the textbook writers,
in this case). The hidden tragedy is that test answers are much sooner
forgotten than stories.
As
a result, we often teach mathematics as if it were merely a list of skills to
learn and concepts to memorize, and consequently treat students as robots to be
programed rather than as souls to be cultivated. Students usually have no idea
why certain concepts and formulas were developed, and even less idea where they
came from. Consequently, students grow bored and frustrated and decide that
they hate math.
This
is precisely what happened to me in high school. I grew annoyed and confused
and decided that math was less important––and certainly less interesting––than
literature. Literature had a narrative I could understand; mathematics was
nothing but a list of abstractions. Thus, teaching mathematics without properly
introducing the narrative and expecting students to remain interested is like
telling the punchline without telling the joke and expecting to get a laugh.
For
the textbook, it doesn’t matter that Viete’s algebra was too bound to geometry
(he didn’t believe one could add a squared number to a cubed number because it
doesn’t make sense to add an area to a volume). Nor does it matter that a few
years later Descartes simultaneously merged algebra and geometry and, through
greater abstraction, liberated algebra from the constraints of geometry. If
narratives like these do not matter to the textbook writers, we should not find
it surprising when mathematical concepts and rules matter little to our
students. But to a student who understands and appreciates the story, concepts
do matter, because Descartes’s revision of Viete becomes a triumph on the level
with the French discovery of the Rosetta Stone.
Despite
my dislike of textbooks, there are several things they do well. They relay
information efficiently and home schooling would be more difficult without
them. Teaching from primary texts is, after all, messy and inefficient (but
then, so is raising children). If you introduce a primary text into your math
curriculum, you’ll find that it won’t fit neatly anywhere. Book I of The Elements won’t fit into your geometry
textbook in any one place, nor will Viete’s The Analytical Art fit neatly into Algebra 1 nor
Descartes Geometry into Algebra 2. The concepts
from these and other works fall into textbooks in distilled, abridged, and
reorganized forms––like meat in Spam or bologna.
Yet
there is good news! Although textbooks remain the only option for many, growing
numbers of home schoolers and classical Christian schools are finding ways to
put primary texts back into math curricula. A good place to begin is with the
first book of Euclid’s Elements. Although, as I said, it won’t
fit neatly into your curriculum, Book I makes a great supplement to any high
school math curriculum. It presents the basics of high school geometry in a
logically complete system with fewer than 50 propositions, which makes it
relatively easy to use alongside a modern textbook.
You
may find, however, that the narratives, even in Euclid, are subtle and
difficult to discover. But the work is worth the reward. When perception dawns,
you may feel, as I have felt, that for a silent moment you stand upon a mountain
overlooking a sun-splashed valley. And when this happens, you are thinking no
longer as a mere student, but as a mathematician.
Daniel
Maycock is the founder of Polymath Classical Tutorials (www.polymathclassical.com)
where he teaches Classical Mathematics and offers summer workshops in writing
and mathematics. Daniel also works for Memoria Press Online Academy where he
teaches Composition, Literature, and Material Logic.