In place of discovery and exploration, we have rules and regulations. We never hear a student
saying, “I wanted to see if it could make any sense to raise a number to a negative power, and I
found that you get a really neat pattern if you choose it to mean the reciprocal.” Instead we have
teachers and textbooks presenting the “negative exponent rule” as a fait d’accompli with no
mention of the aesthetics behind this choice, or even that it is a choice.
In place of meaningful problems, which might lead to a synthesis of diverse ideas, to
uncharted territories of discussion and debate, and to a feeling of thematic unity and harmony in
mathematics, we have instead joyless and redundant exercises, specific to the technique under
discussion, and so disconnected from each other and from mathematics as a whole that neither
the students nor their teacher have the foggiest idea how or why such a thing might have come
up in the first place.
In place of a natural problem context in which students can make decisions about what they
want their words to mean, and what notions they wish to codify, they are instead subjected to an
endless sequence of unmotivated and a priori “definitions.” The curriculum is obsessed with
jargon and nomenclature, seemingly for no other purpose than to provide teachers with
something to test the students on. No mathematician in the world would bother making these
senseless distinctions: 2 1/2 is a “mixed number,” while 5/2 is an “improper fraction.” They’re
equal for crying out loud. They are the same exact numbers, and have the same exact properties.
Who uses such words outside of fourth grade?
Of course it is far easier to test someone’s knowledge of a pointless definition than to inspire
them to create something beautiful and to find their own meaning. Even if we agree that a basic
common vocabulary for mathematics is valuable, this isn’t it. How sad that fifth-graders are
taught to say “quadrilateral” instead of “four-sided shape,” but are never given a reason to use
words like “conjecture,” and “counterexample.” High school students must learn to use the
secant function, ‘sec x,’ as an abbreviation for the reciprocal of the cosine function, ‘1 / cos x,’
(a definition with as much intellectual weight as the decision to use ‘&’ in place of “and.” ) That
this particular shorthand, a holdover from fifteenth century nautical tables, is still with us
(whereas others, such as the “versine” have died out) is mere historical accident, and is of utterly
no value in an era when rapid and precise shipboard computation is no longer an issue. Thus we
clutter our math classes with pointless nomenclature for its own sake.
In practice, the curriculum is not even so much a sequence of topics, or ideas, as it is a
sequence of notations. Apparently mathematics consists of a secret list of mystical symbols and
rules for their manipulation. Young children are given ‘+’ and ‘÷.’ Only later can they be
entrusted with ‘√¯,’ and then ‘x’ and ‘y’ and the alchemy of parentheses. Finally, they are
indoctrinated in the use of ‘sin,’ ‘log,’ ‘f(x),’ and if they are deemed worthy, ‘d’ and ‘∫.’ All
without having had a single meaningful mathematical experience.
This program is so firmly fixed in place that teachers and textbook authors can reliably
predict, years in advance, exactly what students will be doing, down to the very page of
exercises. It is not at all uncommon to find second-year algebra students being asked to calculate
[ f(x + h) – f(x) ] / h for various functions f, so that they will have “seen” this when they take
calculus a few years later. Naturally no motivation is given (nor expected) for why such a
seemingly random combination of operations would be of interest, although I’m sure there are
many teachers who try to explain what such a thing might mean, and think they are doing their
students a favor, when in fact to them it is just one more boring math problem to be gotten over
with. “What do they want me to do? Oh, just plug it in? OK.”