## Devlin's Angle |

Much of my rationale for believing that the way forward in math ed is to be found in the DMZ of the war comes from my recognition that on both sides you find significant numbers of smart, well-educated, well-meaning people, who genuinely care about mathematics education. Unless you are of the simplistic George W. mindset, that the world divides cleanly into righteous individuals on the one side and "evil doers" on the other, it follows, surely, that both sides have something valuable to say. ("Evil doers" always stuck me as a strange phrase to hear uttered in public by a grown-up, by the way. It sounds more like the box-copy description of a band of orcs in a fantasy video game.) The challenge, then, is to reconcile the two views.

In brief, the gist of my Mercury News
opinion piece was this. While it sounds
reasonable to suggest that understanding
mathematical concepts should precede (or go
along hand-in-hand with) the learning of
procedural skills (such as adding fractions
or solving equations), this may be (in
practical terms, given the time available)
impossible. The human brain evolved into its
present state long before mathematics came
onto the scene, and did so primarily to
negotiate and survive in the physical world.
As a consequence, our brains do not find it
easy to understand mathematical concepts,
which are completely abstract. (This is part
of the theme I pursued in my book *The
Math Gene,* published in 2000.)

However, although we are not "natural born mathematicians," we do have three remarkable abilities that, taken together, provide the key to learning math. One is our language ability - our capacity to use symbols to represent things and to manipulate those symbols independently of what they represent. The second is our ability to ascribe meaning to our experiences - to make sense of the world, if you like. And the third is our capacity to learn new skills.

When we learn a new skill, initially we simply follow the rules in a mechanical fashion. Then, with practice, we gradually become better, and as our performance improves, our understanding grows. Think, for example, of the progression involved in learning to play chess, to play tennis, to ski, to drive a car, to play a musical instrument, to play a video game, etc. We start by following rules in a fairly mechanical fashion. Then, after a while, we are able to follow the rules proficiently. Then, some time later, we are able to apply the rules automatically and fluently. And eventually we achieve mastery and understanding. The same progression works for mathematics, only in this case, as mathematics is constructed and carried out using our language capacity, the initial rule-following stuff is primarily cognitive-linguistic.

Of course, there is plenty of evidence to show that mastery of skills without understanding is shallow, brittle, and subject to rapid decay. Understanding mathematical concepts is crucially important to mastering math. The question is: What does it take to achieve the necessary conceptual understanding, and when can it be acquired? Certainly my own experience is that conceptual understanding in mathematics comes only after a considerable amount of procedural practice (much of which therefore is of necessity carried out without understanding). How many of us professional mathematicians aced our high school or college calculus exams but only understood what a derivative is after we had our Ph.D.s and found ourselves teaching the stuff?

In fact, I can't imagine how one could possibly understand what calculus is and how and why it works without first using its rules and methods to solve a lot of problems. Likewise for most other areas of mathematics. In fact, the only parts of mathematics that I find sufficiently close to the physical and social world our brains developed to handle that there are innate meanings we could tap into, are positive integer addition and subtraction for fairly small numbers, and perhaps also some fairly simple cases of division for small positive integers.

Interestingly, those were the *only*
examples cited by the readers of my
Mercury News article who argued against my
suggestion that understanding comes only
after a lot of procedural practice. Now it
may be that in those particular areas,
understanding can precede, or accompany the
acquisition of, procedural mastery.
Personally I doubt it, but I have yet to see
convincing evidence either way. But, leaving
those special (albeit important) cases
aside, what about the rest of mathematics?
Here I see no uncertainty. Understanding can
come *only* after procedural
mastery.

For example, physics and engineering faculty
at universities continually stress that what
they want their incoming students to have
above all is procedural mastery of
mathematics *as a language* - it is,
after all, the language of science, as
Galileo observed - and the ability to use
various mathematical *tools* and
*methods* to solve problems that arise
in physics and engineering. Since even
first-year physics and engineering involve
use of tools such as partial differential
equations, there is no hope that incoming
students can have conceptual understanding
of those tools and methods. But by a
remarkable feature of the human brain, we
can achieve procedural mastery without
understanding. All it takes is practice. One
of the great achievements of mathematics
over the past few centuries has been the
reduction of conceptually difficult issues
to collections of rule-based symbolic
procedures (such as calculus).

Thus, one of the things that high school
mathematics education should definitely
produce is the ability to learn and be able
to apply rule-based symbolic processes
*without understanding them.* Without
that ability, progress into the sciences and
engineering is at the very least severely
hampered, and for many people may be cut
off. (This, by the way, is the only
rationale I can think of for teaching
calculus in high school. Calculus is a
supreme example of a set of rule-based
procedures that can be mastered and applied
without any hope of anything but the most
superficial understanding until relatively
late in the game. Basic probability theory
and statistics are clearly far more relevant
to everyday life in terms of content.)

Is mastery of rule-based symbolic procedures
the only goal of school mathematics
education? *Of course* not. The reason
I am not focusing on conceptual issues is
that much has been written on that issue -
of particular note the National Research
Council's excellent volume *Adding it Up:
Helping Children Learn Mathematics,*
published by the National Academy Press in
2001 (a book I have read from cover to cover
on three separate occasions). My intention
here is to shine as bright a light as
possible on a mathematical skill that I
think has, in recent years, been overlooked
- and on occasion actively derided - by some
in the math ed community. Life in today's
society requires that we acquire many skills
without associated understanding - driving a
car, operating a computer, using a VCR, etc.
Becoming a *better* driver, computer
user, etc. often requires understanding the
technology (and perhaps also the science
behind it). But from society's perspective
(and in many cases the perspective of the
individual), the most important thing is the
initial mastery of use. If something has
been so well designed or developed that
proficient use can be acquired without
conceptual understanding, then the rapid
acquisition of that skillful use is often
the most efficient - and sometimes the only
- way for an individual to move ahead. I
think this is definitely the case with
mathematics.
I believe we owe it to our
students to prepare them well for life in
the highly technological world they will
live in. In the case of mathematics, that
means that one ability we should equip them
with (not the only one by any means -
*Adding It Up* lists several others,
for instance) is being able to learn and
apply rule-based symbolic processes
*without understanding them.* That does
not mean we should not provide explanations.
Indeed, as a matter of intellectual
courtesy, I think we should. But we need to
acknowledge, both to ourselves and our
students, that understanding can come only
later, as an emergent consequence of use.
(No shame in that. It took 300 years from
Newton's invention of calculus to a properly
worked out conceptual basis for its rules
and methods.)

Devlin's Angle is updated at the beginning of each month.