Monthly Archives: May 2010

Contest Math

Many people first become interested in mathematics through contest math in middle school or high school. Others, however, progress to higher level mathematics without participating in the contests at all. What, then, is the role of math competitions in attracting people to mathematics? Is contest math beneficial for doing math beyond competitions?

This, of course, depends on what we mean by “contest math”. Of course, a competition to produce research results would certainly be different from a mental arithmetic contest. For now, I define contest math to be in between these two extremes; the problems are not straightforward, but they are not at the difficulty of research questions. In particular, I classify Mathcounts, ARML, AMC, USAMO, and Putnam in the category of contest math.

One positive effect of contest math is that it tricks students into liking and doing math at an early age. The mathematics education system in America is not conducive to positive impressions of mathematics, and contest math provides an alternative world where problem solving ability is emphasized above rote computation. For example, the Mathcounts contest hooks many students in middle school, providing an introduction to math and leading them to a continued interest in mathematics.

Contest math hooks students partially through its competitive nature. Unlike the typical mathematics classroom, contest math provides a challenge. There are always harder problems and puzzles to consider, and students never run out of room to explore. The feeling of solving a difficult problem is always exhilarating, and pushes the student to do more problems. Furthermore, the contest math community is full of students who are always competing to be the best. The rankings at each math contest push students to work harder than they would if left alone, and the competitive spirit injects energy and excitement into mathematics.

Along with competition, contest math brings together a group of people who are interested in the same thing: Solving math problems. This gives contest math a social aspect as well, challenging the conventional notion that doing math and being social are mutually exclusive activities. A good example of this phenomenon is the ARML contest. In many regions of the country, ARML brings together the best students and gives them an opportunity to get to know each other through practices and an annual trip to the national competition. This social aspect of contest math even extends to the Art of Problem Solving forum, for example.

After contest math hooks its victims, it also indoctrinates them in good mathematical principles. In particular, since contest math values ingenuity over knowledge, it ends up very a good teacher of problem solving. This is important because the ability to think about and solve problems is the key to success in higher level mathematics. The best mathematicians are the ones who can make connections and devise clever arguments. In this way, through the bait of math competitions, contest math pulls students toward real mathematics.

Despite the positive effect of producing more mathematically-minded humans, contest math has many deficiencies. In particular, contest math is not representative of how mathematics is actually done. In reality, mathematical research requires much more knowledge than Euclidean geometry, and pure ingenuity often isn’t sufficient. Also, math competitions teach students to expect solutions to their problems within minutes or hours. Though this is much better than the seconds-long attention spans of many high school math students, it is far from the days, months, or years that are needed to solve difficult research problems in mathematics. Thus, in some sense, contest math forms a different (but related) subject from mathematics, and by doing contest math exclusively, students sometimes lose sight of what mathematics is actually about.

Thus, contest math taken to excess is almost never a good thing. By overtraining themselves for math competitions, students do nothing to improve their mathematical ability. Some students even become resistant to the idea of doing any math beyond the contests that they’ve focused on for years, rendering their mathematical talent useless. This is especially true for more computationally oriented contests such as Mathcounts. Though they provide good introductions to math, they are too far from actual mathematics to be meaningful for any more than a year or two. In this way, contest math can often be harmful for the future mathematician.

In the end, we see that contest mathematics is good for providing an introduction to the world of mathematics to those who haven’t seen it before. However, it cannot replace mathematics, and eventually, math competitions must lead into higher level mathematics. Even among math contests, some are more meaningful than others; not all math contests are equal. But that’s a topic for a future post.


Problem Solving Heuristics

We confront problems all the time in our lives, and success in life is largely determined by how we confront and resolve these problems. Thus, it’s important to know how to approach problem solving.That’s a big issue that I don’t feel capable of tackling, so let’s consider a more specialized question: How does someone approach solving a mathematical problem? From answers to this question, we can hopefully extrapolate about the real world.

The most important skill in mathematical problem solving isn’t knowledge of theorems and facts; instead, it’s more important to have the right mindset for approaching a problem. Here’s some general advice:

Before beginning to approach a mathematical problem, the best problem solvers are psychologically prepared. They are confident in their own ability to solve problems, and they are not worried about failure. In the meantime, they are willing to try new and creative attacks. They are persistent and never give up, and they are also courageous — willing to try any technique that might seem to succeed.

  • Be confident. Believe in yourself.
  • Be creative.
  • Be persistent.
  • Be courageous: “Fearless courage is the foundation to all success.”
  • If you worry then you will fail… so don’t worry!
  • Be brave and try it.

When confronted with a problem, first try to understand it. There are a number of typical strategies for doing this:

  • What are we trying to find?
  • What information are we given?
  • What conditions do we have on the problem?
  • Get your hands dirty: plug in numbers, draw a diagram, do some simple calculations.

Then, try to find a solution.

  • Try some special cases: plug in simple numbers, extremal values, guess and check.
  • Estimate. Order of magnitude estimate, draw a careful diagram.
  • Choose convenient notation.
  • Pursue symmetry, in geometry, in algebraic expressions, and in notation.
  • Find a pattern, make a hypothesis. Can you prove it?
  • Work backwards, go back to the definitions, wishful thinking.
  • Have you used all of the data and conditions? Can you do a more general problem?
  • Change the data and the conditions. How much can they vary?
  • Have you seen the problem before? Do you know of a related problem?
  • Look at the problem in a different way: convert between different topics in math.
  • Divide and conquer: Separate into easier subproblems and attack each separately.
  • “If you can’t solve a problem, then there is an easier problem you can solve: find it.”

Once you’ve solved the problem, check your answer.

  • Check your work. Are you sure that every step is correct? Is every step valid?
  • Estimate. Is the answer reasonable?
  • Do you see another way to solve the problem? Does it give the same result?

These problem solving heuristics do not consider any particular formula, theorem, or method; they apply to every problem. By being independent of specific techniques, they transcend such techniques and become universal; some of these heuristics (“Be brave and try it”, for example) could also apply to the real world.

For more information on problem solving heuristics, see George Pölya’s classic book How to Solve It. I leave you with a quotation from his book:

A great discovery solves a great problem but there is a grain of discovery in the solution of any problem. Your problem may be modest; but if it challenges your curiosity and brings into play your inventive faculties, and if you solve it by your own means, you may experience the tension and enjoy the triumph of discovery.
— George Pölya, How to Solve It

More quotations

It’s been three weeks since my last quotation collection, and there aren’t very many new quotations. People don’t say enough quotable things.

20 April 2010:

  • “This is far too technologically advanced.” — Moor (discussing pushing a straw through the top of a milkshake cup)
  • “I’m thoroughly disappointed in you.” — Melissa (speaking to a poster that had fallen down)

29 April 2010:

  • “Why should we care about living people?” — Moor

The argument: After the living people die, they won’t know or care any more, so it doesn’t really matter what happens to them. I’ll expand on this idea in a future blog post.

9 May 2010:

  • “Usually 1+1=2, but sometimes it can be 4 or 0 as well.” — A physics TA

This was sent to me by Anand, supposedly from a discussion of interfering waves.

10 May 2010:

  • “Just set pi equal to zero.” — Moor

Referring to a math problem that can be solved by the heuristic of replacing distracting numbers by more convenient values.