Page 241 - A Mind For Numbers: How to Excel at Math and Science

Terrence Deacon, author of The Symbolic Species, notes the inherent complexity of the
encryption/decryption problem of mathematics:
“Imagine back when you were first encountering a novel kind of mathematical concept, like recursive
subtraction (i.e., division). Most often this abstract concept is taught by simply having children learn a
set of rules for manipulating characters for numbers and operations, then using these rules again and
again with different numbers in hopes that this will help them ‘see’ how this parallels certain physical
relationships. We often describe this as initially learning to do the manipulations ‘by rote’ (which is in
my terms indexical learning) and then when this can be done almost mindlessly, we hope that they will
see how this corresponds to a physical world process. At some point, if all goes well, kids ‘get’ the
general abstract commonality that lies ‘behind’ these many individual symbol-to-symbol and formula-
to-formula operations. They thus reorganize what they already know by rote according to a higher-order
mnemonic that is about these combinatorial possibilities and their abstract correspondence to thing
manipulation. This abstraction step is often quite difficult for many kids. But now consider that this
same transformation at a yet higher level of abstraction is required to understand calculus.
Differentiation is effectively recursive division, and integration is effectively recursive multiplication,
each carried out indefinitely, i.e., to infinitesimal values (which is possible because they depend on
convergent series, which themselves are only known by inference, not direct inspection). This ability to
project what an operation entails when carried out infinitely is what solves Zeno’s paradox, which seems
impossible when stated in words. But in addition to this difficulty, the Leibnizian formalism we now use
collapses this infinite recursion into a single character or the integral sign) because one can’t actually
keep writing operations forever. This makes the character manipulation of calculus even less iconic of
the corresponding physical referent.
“So the reference of an operation expressed in calculus is in effect doubly-encrypted. Yes, we’ve
evolved mental capacities well-suited to the manipulation of physical objects, so of course this is
difficult. But math is a form of ‘encryption,’ not merely representation, and decryption is an intrinsically
difficult process because of the combinatorial challenges it presents. This is why encryption works to
make the referential content of communications difficult to recover. My point is that this is intrinsic to
what math is, irrespective of our evolved capacities. It is difficult for precisely the same reason that
deciphering a coded message is difficult.
“What surprises me is that we all know that mathematical equations are encrypted messages, for
which you need to know the key if you want to crack the code and know what is represented.
Nevertheless, we wonder why higher math is difficult to teach, and often blame the educational system
or bad teachers. I think that it is similarly a bit misplaced to blame evolution.” (Personal communication
with the author, July 11, 2013.) 9 Bilalić et al. 2008.
10 Geary 2011. See also the landmark documentary A Private Universe, available at
http://www.learner.org/resources/series28.html?pop=yes&pid=9, which led to much research into
misconceptions in understanding science.
11 Alan Schoenfeld (1992) notes that in his collection of more than a hundred “videotapes of college and
high school students working unfamiliar problems, roughly sixty percent of the solution attempts are of
the ‘read, make a decision quickly, and pursue that direction come hell or high water’ variety.” You
could characterize this as focused thinking at its worst.
12 Goldacre 2010.
13 Gerardi et al. 2013.
14 Hemispheric differences may sometimes be important, but again, claims in this area should be taken with
caution. Norman Cook says it best when he notes: “Many discussions in the 1970s went well beyond the
facts—as hemisphere differences were invoked to explain, in one fell swoop, all of the puzzles of human
psychology, including the subconscious mind, creativity, and parapsychological phenomena—but the
inevitable backlash was also exaggerated” (Cook 2002, p. 9).
15 Demaree et al. 2005; Gainotti 2012.