关于解题策略的书。
This is a book about mathematical problem solving for college-level novices. By this I mean bright
people who know some mathematics (ideally, at least some calculus), who enjoy mathematics, who
have at least a vague notion of proof, but who have spent most of their time doing exercises rather than
problems.
An exercise is a question that tests the student’s mastery of a narrowly focused technique, usually
one that was recently “covered.” Exercises may be hard or easy, but they are never puzzling, for it
is always immediately clear how to proceed. Getting the solution may involve hairy technical work,
but the path toward solution is always apparent. In contrast, a problem is a question that cannot be
answered immediately. Problems are often open-ended, paradoxical, and sometimes unsolvable, and
require investigation before one can come close to a solution. Problems and problem solving are at
the heart of mathematics. Research mathematicians do nothing but open-ended problem solving. In
industry, being able to solve a poorly defined problem is much more important to an employer than
being able to, say, invert a matrix. A computer can do the latter, but not the former.
A good problem solver is not just more employable. Someone who learns how to solve mathematical problems enters the mainstream culture of mathematics; he or she develops great confidence and
can inspire others. Best of all, problem solvers have fun; the adept problem solver knows how to play
with mathematics, and understands and appreciates beautiful mathematics.
An analogy: The average (non-problem-solver) math student is like someone who goes to a gym
three times a week to do lots of repetitions with low weights on various exercise machines. In contrast,
the problem solver goes on a long, hard backpacking trip. Both people get stronger. The problem solver
gets hot, cold, wet, tired, and hungry. The problem solver gets lost, and has to find his or her way.
The problem solver gets blisters. The problem solver climbs to the top of mountains, sees hitherto
undreamed of vistas. The problem solver arrives at places of amazing beauty, and experiences ecstasy
that is amplified by the effort expended to get there. When the problem solver returns home, he or she
is energized by the adventure, and cannot stop gushing about the wonderful experience. Meanwhile,
the gym rat has gotten steadily stronger, but has not had much fun, and has little to share with others.
While the majority of American math students are not problem solvers, there does exist an elite
problem solving culture. Its members were raised with math clubs, and often participated in math
contests, and learned the important “folklore” problems and ideas that most mathematicians take for
granted. This culture is prevalent in parts of Eastern Europe and exists in small pockets in the United
States. I grew up in New York City and attended Stuyvesant High School, where I was captain of
the math team, and consequently had a problem solver’s education. I was and am deeply involved
with problem solving contests. In high school, I was a member of the first USA team to participate in
the International Mathematical Olympiad (IMO) and twenty years later, as a college professor, have
coached several of the most recent IMO teams, including one which in 1994 achieved the only perfect
performance in the history of the IMO.
