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Saturday, October 27, 2012

Teaching and Learning Math

(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.