(As submitted by email to Sylvia Riselay on April 24)
Mathematics and Silence
“Imagine a line, straight and without thickness, running along my arms in
both directions, out through the classroom wall into the playground,
farther and farther, on into space – a line without end, infinitely long.”
The mathematics teacher stretched out his arms, and, following the line as it extended
from his index finger, this twelve-year old was caught up in the vision of something
perfect, something against which, for a brief moment all else seemed flawed, indistinct,
and mundane. Nothing happened in that classroom, not in the usual sense, not at that
moment. There was no problem to solve, no algorithm to learn, not even a clever
explanation to savour. It was above all a moment of silence. It was the moment that
made me a geometer.
Mathematics is replete with problems to engage a young mind. I remember my father
challenging us with mathematics problems at the dinner table. Especially good were
puzzles that became simple once you shifted your point of view. One such was the
problem of the bee flying at constant speed between the noses of two bicyclists
approaching each other on a straight road. Given the initial separation of the bicyclists
and their speeds, my siblings and I were challenged to calculate the distance covered by
the bee as it flew back and forth over shorter and shorter distances. We imagined it
crushed between the noses of the two bicyclists and wondered how we could ever add all
those smaller and smaller distances. At some point it is not the effort that unlocks the
problem, but the unexpected shift in focus, the reflection, the silence.
Mathematics is surprising in its power to shed light on so much. The physicist Eugene
Wigner wrote of "The Unreasonable Effectiveness of Mathematics in the Natural
Sciences,"1. Though much of our early mathematical understanding is awakened by
intuitive grappling with the physical world, it is surprising, and delightful, that the
potentialities that seem present in our brains from birth are so well tuned to our
experience. It allows me, as a teacher, to treat a lot of mathematics as carefully examined
common sense.
Plato’s vision centred on ideal objects that were reflected in our experience by imperfect
copies. Only someone who had taken the time to attend to those ideals would recognize
these reflections for what they were. Only he or she would be able to step back and see
the whole picture, to see through to the solution of a problem.
More current views of learning stress the autonomy of the learner, the time needed to
construct understanding, the delight of knowing for oneself, while others stress the role of
community in the education enterprise. Even so, when a difficult concept opens up to our
understanding, perhaps after frustrating effort, we recognize it not as something of our
1
Communications in Pure and Applied Mathematics, vol. 13, No. I (February 1960).
own making, but as something universal, belonging to everyone and to no-one. Seeing it
we are reduced to an inarticulately uttered ‘Ah!’, or to mere silence.
Diana Laurillard2, in her discussion of attempts to discover how students learn, points out
that, when students are observed, and overheard, struggling with a mathematics problem,
the critical insight is always preceded by a moment of silence. That is the time when the
connection is made. At that moment, speech would get in the way. Even our inner
speech seems suspended at the time of an insight. The hard work of checking, justifying,
correcting and refining an idea requires articulation, but its birth is attended by silence.
The geometer Coxeter, whose courses I took at the University of Toronto, understood the
role of silence in mathematics. Though he died only recently, already when I was an
undergraduate did Coxeter appear to represent a different era. His classes seemed quaint,
and shockingly quiet to our eager, restless minds. There was no attempt to motivate
material in terms of applications. A proof was a picture, carefully drawn in coloured
chalk; to understand it required contemplation - not the usual string of “ifs” and
“therefores” ending in a flourished “Q.E.D.” He would stand there, smile, and wait till
we saw the connection.
The power of mathematics is both a blessing and a curse. Since the subject is considered
important, great efforts are made to teach it as quickly and thoroughly as possible, and to
present it as useful rather than beautiful - quite unlike poetry. We feel we must teach the
necessary skills, now; careers depend on it. Such teaching leaves little time to listen to
music, to wait for insight.
Students in my first year calculus class, high achievers all, arrive at university ready to do
all the problems in the textbook. Over-ambitious and over-extended, they have no time
for reflection – no time for silence. I advise them to do just one or two questions; to
persist in solving these; and, when a solution has been found, to meditate on it.
The best insights always come suddenly, as surprises, as gifts. They may come after
hours of frustration, when we think we are ready to give up; but when they come we
should pause to marvel that our small brains can grasp something so intricate and so
simple; so bound up with the still beauty of the world.
Leo Jonker
2
Rethinking University Teaching, page 49