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OBSERVER
Finding Your Inner Mathematician
By KEITH DEVLIN
Many people assume that it takes a special
kind of brain to be able to do mathematics -- that unless you were
born with some kind of "math gene," you simply are not
going to be able to get math, no matter how hard you try. As someone
who struggled hard with math in school until I was 15, and then got
it all at once, I never believed the math-gene theory. What made the
difference for me was that everything suddenly made sense -- perfect,
simple, elegant sense.
Having taught mathematics for 30 years, I am convinced that everyone
has the capacity to do mathematics, at least through high-school
algebra and geometry. In fact, all you really need to do math is nine
basic mental abilities, which our ancestors developed thousands of
years ago to survive in a hostile world:
1. NUMBER SENSE. This is not the same as
being able to count. It's much more basic than that, and includes the
ability to recognize the difference between one object, a collection
of two objects, and a collection of three objects -- and to recognize
that a collection of three objects has more members than a collection
of two. Number sense is not something we learn. Child psychologists
have demonstrated conclusively during the past 20 years that we are
born with number sense.
2. NUMERICAL ABILITY. This does involve
learning -- both to count and to understand numbers as abstract
entities. Early methods of counting, by making notches in sticks or
bones, go back at least 30,000 years. The Sumerians are the first
people we know of who used abstract numbers; between 8000 and 3000
B.C., they inscribed numerical symbols on clay tablets.
3. SPATIAL-REASONING ABILITY. This
includes the ability to recognize shapes and to judge distances
accurately, both of which have obvious survival value. In addition to
forming the basis for geometry, this ability is important for a lot
of mathematical thinking that is not, on the face of it, visual or geometric.
4 . A SENSE OF CAUSE AND EFFECT. Much of
mathematics depends on "if this, then that" reasoning, an
abstract form of thinking about causes and their effects.
5. THE ABILITY TO CONSTRUCT AND FOLLOW A CAUSAL
CHAIN OF FACTS OR EVENTS. A mathematical proof of a
theorem is a highly abstract version of a causal chain of facts.
6. ALGORITHMIC ABILITY. An algorithm is
a step-by-step procedure for performing a certain mathematical task
-- the mathematician's equivalent of a recipe for baking a cake. In
elementary school, we are taught algorithms for adding, subtracting,
multiplying, and dividing whole numbers and fractions. Secondary-school
algebra requires that we learn algorithms to solve equations.
Algorithmic ability is an abstract version of the fifth ability on
this list.
7. THE ABILITY TO UNDERSTAND ABSTRACTION.
Humans developed the capacity to think about abstract notions, along
with acquiring language, 75,000 to 200,000 years ago.
8. LOGICAL-REASONING ABILITY. The
ability to construct and follow a step-by-step logical argument is
fundamental to mathematics. It is another abstract version of the
fifth ability.
9. RELATIONAL-REASONING ABILITY. This
involves recognizing how things and people are related to each other,
and being able to reason about those relationships. Much of
mathematics deals with relationships among abstract objects.
The human brain acquired those nine abilities at least 75,000 years
ago. They are basic mental attributes crucial to our daily lives. The
question is: What does it take to put those abilities together and do math?
The key is the ability to handle abstraction -- No. 7 on the list. We
can all use our brains to reason about physical objects we are
familiar with, and we can carry out the same kinds of reasoning about
imaginary variants of those objects -- for example, the characters in
a Harry Potter book or a Star Trek television show.
Mathematical thinking involves one more step: reasoning about purely
abstract objects. The trick is to make those abstract objects seem
real -- to fool the brain into thinking that it's dealing with real
objects. Once you have taken that step into the world of the
abstract, the rest is comparatively easy. After all, the mind is then
performing tasks that it finds natural and instinctive.
Although making the abstract seem real sounds hard, we all do much
the same thing whenever we read a novel or watch a movie. So am I
saying that to do mathematics, you have to treat it like reading a
novel or watching a movie?
In fact, I'm going a step further. When you start reading a novel, or
you watch a movie for the first time, you have to familiarize
yourself with the characters and the situation in which they find
themselves. In the case of mathematics, the characters never change,
only their situations. You have to familiarize yourself with the
characters just once, and from then on everything amounts to finding
out new things about them.
What does that remind you of? It reminds me of a television soap
opera, like the long-running As the World Turns. That isn't a
joke. The secret to being able to do mathematics is to think of math
as a soap opera.
I'm not talking about the love lives of mathematicians here -- it's
math itself that constitutes the soap opera. The characters are not
fictitious people but mathematical objects: numbers, geometric
figures, topological spaces, and so on. The facts and relationships
of interest are not births, deaths, marriages, love affairs, and
business deals, but mathematical facts and relationships like: Are
objects A and B equal? What object has property P? What is the
relationship between objects X and Y? Do all objects of type X have
property P? How many objects of type Z are there?
Mathematicians think about mathematical objects and the relationships
among them using the same mental abilities that most people use to
think about physical space or about other people.
Mathematicians don't have a different kind of brain. They have
learned to use a standard-issue brain in a slightly different way.
What distinguishes a great mathematician from a high-school student
struggling in a geometry class is the degree to which each one can
cope with abstraction. The mathematician learns to create and hold an
abstract world in her mind, and then reason about that world as if it
were real.
The importance of abstraction and the brain's difficulty in handling
abstract objects have three clear implications for mathematics
teaching. First, we should start with what is familiar and concrete,
and move gradually into the abstract. Second, we must realize that
the key -- the real challenge -- is for the student to come to view
the abstract objects of mathematics as real. Third, we need to accept
the fact that a period of repetitive training is unavoidable --
because repeated use is the only way to make abstract objects seem
sufficiently real for the brain to process them.
Much of the current debate about mathematics teaching is focused on
whether rote learning of basic math skills is still important in an
age of electronic calculators and computers. That debate misses the
point. The real value of learning basic math skills today is not that
you will need to use those skills per se; chances are you won't.
Rather, the benefit is to make the abstract objects of mathematics
become so familiar -- and seem so real -- that you can reason about
them using the same mental capacities you use to reason about
everyday things. Unless you can get to that stage, you'll never be
able to master the more sophisticated kinds of mathematics that today
are part of the jobs of stockbrokers, architects, scientists,
builders, Olympic coaches, physicians, and many other people.
Of course, not everybody will use those forms of math in their daily
lives. But mastering mathematical abstraction, like learning a
foreign language, is much easier when you are young. Good, effective
instruction in math should be part of everyone's education, so that
no one is shut out of such an important area of modern life.
Keith Devlin is dean of science at Saint Mary's College of
California and the "math guy" on National Public Radio's
Weekend Edition. This essay is based on his most recent book,
The Math Gene: How Mathematical Thinking Evolved and Why Numbers Are
Like Gossip, published in August by Basic Books.
http://chronicle.com
Section: The Chronicle Review
Page: B5
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