Silences in Teaching

Les voix du silence dans l’académie


Silence! reviving an oral tradition in mathematics education
Eva Knoll

Silence, in my experience, is what can link doing and creating with learning, certainly in mathematics. Mathematics is not only about knowing how to do something with numbers or space, it is also about knowing how to figure out how to do something with numbers and space.

There are four types of knowledge in mathematics. First, there are the names of objects, triangles, right angles, even numbers, centimetres: or conversely, what names and symbols mean, like 22 stands for two-tensand- two, not two-and-two.

There are the mathematical facts: 2 + 2 = 2 x 2 = 22, the diagonals of a square are of equal length, if you multiply two even numbers you get an even number. These first two types make up “knowing how.”

Then there are ideas that help a learner (and aren’t we all always learning more?) understand what she or he observes, and make conclusions: for example, when adding fractions, the denominators have to be the same. This is “knowing why.”

Finally, there are the concepts that underlie these ideas about the mathematics of what is observed. This last type on the list is the most elusive. I call it “knowing when” because it helps one to see when a situation is similar to another that the learner already understands.

Mathematics, they say, is the science of patterns. To see the patterns, and to see the usefulness of a pattern to the understanding of a phenomenon that embodies this pattern, requires “knowing when.” The difficulty in learning at that level is that being told doesn’t work. What is needed is silence. Silence is an absence. Silence has no direction. It does not say “Go this way,” or “Don’t go that way.” It says, “Go any way you like, see where it takes you.”

One of the most rewarding – and most productive – educational moments is when a learner experiences the “Aha!” If it is connected to the underlying idea of mathematics, this moment is even more precious. Learners then experience the creative mathematician in themselves. The problem, of course, lies in how to make these moments happen. They cannot be forced; there are ways, however, to solicit them.

They happen when we don’t tell the answer, when we resist the temptation to take over and tell the story ourselves. When we stay silent.

But how can we trust that they will find the right answer? The problem with the “right answer” is that it pulls relentlessly in a specific direction. It does not let the learner go in any direction she or he pleases. Self-reliant creativity is lost.

To illustrate, Andrew Wiles, well known for his solution of Fermat’s last theorem, described his experience of doing mathematics in terms of a journey through a dark, unexplored mansion:

You enter the first room of the mansion and it’s completely dark. You stumble around bumping into the furniture, but gradually you learn where each piece of furniture is. Finally, after six months or so, you find the light switch, you turn it on, and suddenly it’s all illuminated. You can see exactly where you were.

Groping in the dark is an integral part of doing mathematics. It is then that the light bulb moment is the most powerful. It is then that the “knowing when” develops. Takes hold. This “knowing when” enables the learner to progressively understand more and more complex mathematical situations.

“Knowing when” is essential to a trustworthy learning process; it can only emerge in learners through silence.

 

Work Cited:

Solving Fermat: Andrew Wiles (Interview). Nova. PBS. [http://www.pbs.org/wgbh/nova/proof/wiles.html].