(Written in the 1990s when I
was substitute teaching in the math dept. in high school.)
It is generally recognized
that teaching and learning mathematics has special problems. Many children seem to learn math only
with difficulty. Many teachers of
math seem unable to teach the subject beyond the sort of rote process by which
they acquired their knowledge of it.
And yet there are some people who seem to have a great affinity for
mathematical thinking, who seem to almost know it before they are taught and
only need to be shown the meaning of symbols in order to express their
understanding.
In almost no other human
activity is there such a range of difference than between the mathematically
commonplace and the mathematically sophisticated. It is perfectly possible for there to be two students with
essentially equivalent intelligence, one with a limited capacity for
mathematical comprehension and so needing to learn specific skills by rote, and
another who learns with fluidity and comprehension.
I propose that we compare and
contrast language learning with number learning as a beginning exercise in
developing a theory for both explanation and method for math learning. Language has large areas of the nervous
system dedicated to it. The
anatomy of the respiratory tract is evolved for language vocalization. All this suggests the natural
conclusion that humans have been doing language for a very long time and that
it is part of our physical design.
We know that there are readiness states in infants associated with
language learning, further tying it to our biology.
Language is involved in the
communication of detailed environmental information, the establishment of
social relationships and order and supports the creative process of
thought. We teach language related
skills -- writing, reading, verbal comprehension and aesthetic expression -- as
a communication tool based in the actual transfer of information. Even in so simple a sentence as
"See Jane run," the nascent reader can, in their mind’s eye, ‘see
Jane run.’
Contrast this with number
learning. Initial number learning
-- counting -- is a part of local language process and historically most people
never move significantly beyond this level. Number learning and mathematical thinking doesn't have the
powerful evolutionary history of language, but is subsumed within it as
counting. Further math instruction
has little, if any, communication content and is most often presented as only
‘that which must be learned by rote’ for reasons discussed later. Yet, It is clear from the math ability
of some people that mathematical thinking can be a powerful form of
comprehending and communicating about the world.
Just as there are relatively
few people with the language sensitivity of the great poets, there are few
people with the specialized sensitivities and comprehension of
mathematicians. But this is not
the end of it -- or even the beginning.
If you wish to discuss the writing of Yeats or Roethke, you are still
within the familiar world of language for which there is a dedication of brain
and biology in everyone. Almost
everyone can develop a little tangle, in some a cold sweat, from a sensuous
passage by Ted Hughes even if they could never in a lifetime of word tossing
make such a line for themselves.
Mathematics is yet again different.
On the extreme other end of
this hazy continuum is the so-called idiot savant with special calculating
powers. Here is a person with
often reduced or unusual language expression, but who, with training in symbology and operation, can show remarkable abilities especially with
various forms of very complex counting.
If I say, "57349 times 4274!" and if the answer is given
immediately as "254,109,626," then I have to conclude that some mathematical
processing and judgment occurred somewhere and that this processing preceded by
acceptable mathematical rules. The
only reasonable place to find such processing is within the nervous system of
the savant. Ergo, something allows
the perception to identify the numbers and to very rapidly associate them
through a mathematical process that actually calculates the result by some
physical process. The savant often
simply says that they know or see the number in some form or other -- it just
comes to them.
If I see ‘8 times 8’ and say
‘64’ without any recognized intervening conscious process, it is a skill I have
acquired by rote. The
multiplication of two numbers exceeding a couple of places by ‘recognition’ of
the answer requires that the mathematical operation take place somewhere and
this is almost certainly not my rote process.
But we perform mathematical
calculations regularly at a very high level. There is no mechanical analog design between the wad of
paper in my hand and the small trashcan 20 feet away, yet I can throw 10
different wads of paper of different weights and compaction reliably into that
can. Each one must be given a
different final velocity and arch, the trashcan can be moved and I can throw
from different places all with high levels of success. The force and trajectory calculations
are taking place somewhere! Clearly, I must be measuring the distance with my
eyes, muscularly weighing the paper wads, visually estimating their density and
calculating the final patterns of muscular activity. Somehow in the normal-order disturbance of the savant, these
calculating systems are made more immediately addressable. Mathematical
proficiency must be somewhere between the mentally disrupted design of the
savant and the exceptional sensitivity of the poet -- think of a
three-dimensional space, not a one-dimensional line.
So if this description, in a
very rough, almost poetic, way describes the way math ability works in humans,
what does it mean for teaching this increasingly important skill? First, it should be clear that, as a
whole, math instruction has been failing our children. Large numbers of children deeply
believe that they are ‘math stupid,’ i.e., the most effective thing being
taught to them is that they can’t learn math. Then there are the small number of children who, given even
the slightest chance, will learn math easily, rapidly and often beyond the
ability of their rote-learning teachers to teach them. Another group will dig in, learning by
rote exactly what is being taught, be able to use it for exactly those
applications shown -- if any -- and think that they have learned math because
that is what they will be told.
Think how unlike language learning this is.
The essence of language
learning is to be able to express new ideas, your own feelings and thoughts, in
ways that will, if not excite at least, be somewhat clear to others. And to be able to receive similar
expressions with comprehension and appreciation. The essence of math learning has been -- well -- to do math
as it is prescribed in a book.
What this little analysis
suggests about teaching math is that we first recognized that humans have a
complex relationship to math learning unlike much of the general learning that
we expect of everyone. In general,
math needs to be taught as a form of communication to the vast majority. All formulations and operations need to
be put in a form like language – information, no matter how silly or
simpleminded, needs to be presented.
This is not to say that all problems need to be language problems, many
of these are just as devoid of communication as is ‘a2 + b2=c2’
for most people (or this equation can be seen as an exquisite and exact
communication about the relationships of sides of right triangles – deeply
poetic in fact). When the 60 MPH
train from Chicago meets the 55 MPH train from Detroit may have no
communication value even though framed in words. The math work and learning needs to be about real
communication.
Certainly there is rote
learning required: Sums, times tables, order of operations, the various names
of things, etc., but the math itself needs to routinely communicate
information. Numbers need to come
from somewhere, be a certain kind, any answers need to actually answer some
question. Operations and formulae
need to be driven by the questions. The symbols and operations need to be seen
as a language with special properties.
People who use math in their
daily lives and work use math in this way. It is not some meaningless exercise. There's no reason that it should be
taught as a meaningless exercise that only a rare few find the secret of by
either special aptitude or accident.
I took my seventh-grade
daughters (home schooling) out to some power-lines with tall wooden poles. The question was, "what is the
diameter at the top of the pole?"
We had with us only a 50 foot tape measure. It was late in the afternoon and the shadow of the pole lay
tantalizingly on the ground at their feet. Many possibilities were considered, but ultimately they
collected a series of numbers that were set up as various proportions,
assumptions were stated, an answer produced. Their original, "about this big," demonstrated
with the finely metered distance between outstretched palms, had become:
"if the pole changes diameter consistently over its length, then the top
is 9.5 inches in diameter."
Much multiplying and dividing had been going on as well as various acts
of measuring and counting.
While helping tutor some
algebra I students with their first graphing assignments, a bewildered young
man got it immediately when I asked him to show me the football when it was on
the 20 yard line and 10 yards in from the sideline. He might need some help in the future in understanding that
a point is dimensionless, not an oblong ball, but that goal line can be crossed
when he gets to it.
It would be nice if there
were a series of solid mathematically strong texts that had a communication
base with lots of examples of activities and hints for enriching exercises [1].
But existing books can be used as long as teachers understand that the majority
of their students are going to learn math best as a communicating tool; that a
balance between rote practice and activities that serve a communication
function can be struck. There will
always be a few math-poets to challenge the teacher. The trick is recognizing them, realizing that they
experience the numbers and the operations in a different way. It is not that they are simply smart
and other students are not smart; they are processing this highly specialized
symbolic system in a different way from those who are using what is essentially
a language model to do math. Every
student should be given the chance to become a Fermet or a Bertrand Russell,
but the fact is that just as the vast majority of people get along just fine
without running a 10 second hundred yard dash, most math students will not make
math into a full-fledged, separate, beautifully spoken language. And quite frankly there are far too few
math-poets teaching math to allow for a renaissance in math learning anyway.
These examples are nothing
special -- the point is to look for how to frame communication paradigms from
minimal mathematical designs.
Certainly the richness of Cartesian coordinates comes from an abstract
quality that inheres in concepts of variable. These concepts are not presented in gridiron examples, but
each level of understanding advances toward that goal.
Ultimately the failure of
math instruction has to rest at the feet of a system that places teachers
without requisite understanding, ability and tools in the classroom, but if we
begin to recognize that our very natures as math learners can be remarkably
different, we might begin to make a change for the better. This is not to say that some children
can’t and should not learn math—quite the opposite. All but the most abstract math can be learned and used
effectively by people who are taught math as a communication tool inside the
design of spoken language.
However, this will not happen if confused teachers teach children that
their only available approach to math learning is wrong and that they are
somehow damaged goods if they can’t do it the way some text (that the teacher
really doesn’t understand either) says to.
The many changes that math
education has gone through over the last 50 or so years have been a response to
the recognition that math was not being especially well taught and learned as
education became more and more factory-like in its design. It is not that the “old way” is the
best way, unless ALL of the old way is made available. When intimacy and caring were natural
to education, many ‘curriculum problems’ went unnoticed. Today we have little room but to
be accurate with how we approach these matters.
[1] When I wrote this piece
the Integrated Math Program (IMP) was not generally known. While far from perfect, this 4 year
high school program (I’ve not seen the middle school version) is consciously
directed toward many of the concerns presented here. I took the short-course IMP training and used the program
for a time until the school where I was teaching adopted confusion over consistency and lost the courage to follow through with such deviation from existing approaches.
