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发帖数: 19398 | 1 Lessons in math don't have to be so boring. Physicist Frank Wilczek on logic
puzzles, games of chance and other ways to entice students
By Frank Wilczek
[Dr. Wilczek, winner of the 2004 Nobel Prize in physics, is a professor at
the Massachusetts Institute of Technology and the author of "A Beautiful
Question: Finding Nature's Deep Design."]
You might not expect to find challenging mathematics on supermarket magazine
stands, but it is there in abundance. New collections of cerebral puzzles
are always coming out (including a weekly feature in this section). Many,
such as Sudoku, involve pattern recognition. Others, such as Kakuro and
Kenken, bring in simple arithmetic. Still others smuggle in graph theory and
topology. My favorites are logic puzzles.
Though never labeled as such, all of these puzzles involve the same kinds of
thinking as formal mathematics. Yet many people think of "math" as
something scary and of puzzles as something fun. The reason for this paradox
is that they've been misled about what mathematics is. Their main exposure
to something with that name, in school, is often an off-putting ritual
featuring memorization and mindless replication of useless abstractions.
Can we do better? An idea from economics, "revealed preferences," may be
helpful. To understand what people enjoy, look at what they choose. Those
racks at the supermarket are telling us something important.
Whole magazines are devoted to logic puzzles, which come graded in
difficulty from one star (suitable for beginners) to five stars (fiendishly
difficult). The situations that they describe vary widely, from the everyday
to the surreal. You might, for example, have lists of characters, presents
and holidays, and the problem will be to figure out, from a bunch of clues,
who gave what to whom, when.
If you consider the clues as axioms and the solution as a theorem, you'll
recognize that these puzzles embody the same logical structure as Euclid's
geometry. But they are easily digestible miniatures, self-contained and
attuned to the human taste for narrative. In a wonderful variant called
logic art, you deduce instructions for filling in a grid that ultimately
produces a picture.
The branch of mathematics that governs logic puzzles is called propositional
calculus. It is fundamental not only for mathematics but also for computer
science. It's a fascinating and open-ended problem to program a computer to
solve them. (When my daughter Mira was in high school, we played around at
this. Our programs got to the point that they could solve the four-star
problems in a few minutes on a 2000 vintage laptop, but we didn't do as well
with the five stars.) Recreational logic problems can be a gateway, leading
to a serious commitment to thinking and programming.
Another revealed mathematical preference, this time in geometry, comes to us
from the Italian Renaissance. Around 1413, Filippo Brunelleschi discovered
perspective-the art and science of capturing, in a drawing, the proportions
of how things actually look. Contemporary artists including Masaccio,
Donatello and da Vinci took up Brunelleschi's constructions enthusiastically
. Within a few decades, they created masterpieces that people have enjoyed
and admired ever since.
Perspective introduced a new kind of geometry, called projective geometry,
into mathematics. The concepts of projective geometry permeate the most
vibrant, advanced parts of contemporary mathematics and computer graphics.
Yet they can be introduced, following Brunelleschi, in rules for drawing
that allow students to create splendid, convincing town squares and
buildings within minutes.
I myself only learned these techniques recently, and it's been a magical
experience to play with them. To me, it is a no-brainer that this experience
should be a very early part of the geometry curriculum. It's another way
into serious thinking and programming-and, of course, into art.
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We know that people like games of chance and gambling. These lead naturally
into adventures in probability and statistics, which can be tested in
entertaining experiments. And these adventures scale up. Just a few steps
take us into hot developments in big data.
I should admit that some very important branches of mathematics don't have
immediate entertainment value, at least for most people. Linear algebra, for
example, is the language of quantum physics. Learning it is an essential
part of understanding how the physical world works. Yet the early parts of
linear algebra are quite dull and abstract. One must have patience to
persevere until the more advanced, and spectacularly beautiful, parts of the
subject open to view.
-Dr. Wilczek, winner of the 2004 Nobel Prize in physics, is a professor at
the Massachusetts Institute of Technology and the author of "A Beautiful
Question: Finding Nature's Deep Design." [See http://www.amazon.com/Beautiful-Question-Finding-Natures-Design/dp/1594205264 ] |
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