Developing your Sixth (Number) Sense.

confusion

Classes are starting up all over the place this week and many students are returning to math instruction. Occupy Math is going to try to help by supplying some perspective. A recent article found that teaching students philosophy helped them to get better at math. This is very much in the same direction as an earlier Occupy Math post, math is not a form of ritual magic. People who understand how and why math works find it much easier to deal with mathematics than those that do not. One of the most common questions asked in math class is “what will I use this stuff for?” This question assumes the only thing you can do with math is to directly apply it — a belief that helps more people flunk math classes than any other.

This question also assumes math is a form of ritual magic.

Current math instruction often presents math as a series of triads: problem, solution method, answer. Rinse. Repeat. The greatest benefit of mathematics lies in generalization. Generalization permits your understanding to grow and it enables you to solve problems you have never solved before. The triad instruction method never gets to the step where you generalize, integrate your knowledge into a coherent whole, and gain understanding. The triad of mathematics instruction method carefully avoids looking for patterns. Many of our math classes carefully avoid teaching real mathematics in favor of teaching narrow techniques of calculation that will, in all probability, be completely useless to the students during the rest of their lives.

What, then, is the elusive benefit that the study of mathematics can convey? Occupy Math begins by turning to his hero, Charles Darwin.

“I have deeply regretted that I did not proceed far enough at least to understand something of the great leading principles of mathematics; for men thus endowed seem to have an extra sense.”
— Charles Darwin

Modern math educators even have a name for this extra sense perceived by Darwin: the number sense. Wikipedia defines the number sense as “an intuitive understanding of numbers, their magnitude, relationships, and how they are affected by operations.” Occupy Math feels this is a good beginning, but it misses the full scope of the matter. The remarkable Keith Devlin observes that “All the mathematical methods I learned in my university math degree became obsolete in my lifetime.” This is the title of an essay he wrote that includes a good discussion of number sense. He talks about the incredible changes caused by modern computational technology, but a key point is that …

Since machines can perform routine calculations, the valuable skill for people is understanding which calculations to perform.

A common finding in math education research is that students are too dependent on calculators. A person without the number sense finds it difficult to understand what calculations to perform, no matter how powerful their calculator is. A person with number sense is able to perform useful calculations even with modest computational resources. Students are given calculators too early, before they develop their number sense, and the calculator actually prevents the development of the number sense. Students see mathematics as useless, not irrationally, but because they are prevented from learning the useful skills within mathematics. The math they are taught is often useless unless partner skills are added.

It is an aphorism among runners that the first mile of a run is the hardest. This means that a person that does not run far enough will only experience the hardest part of running. In a similar way, a person that uses brute force calculation and solution methods off a tedious, memorized list will do only hard, unpleasant mathematics. The soul of mathematics is to find general understanding while avoiding all but necessary calculations. Almost all the mathematics that K-12 students do is the kind of calculation that a mathematician would hand off to a computer or completely avoid. Let’s look at an example.

Problem: Suppose that people at a party, considered in pairs, are either friends or strangers. Is it possible to have a party at which each person has a different number of friends at the party?

The normal method a student uses when presented with this problem is to try to find an example of a party in which each person has a different number of friends. A good way to diagram this situation is to represent people as small disks and to connect two disks if the people they represent are friends. An example of this type of diagram appears below, with the numbers showing the number of friends each person has.

party

This example is close — only two people have the same number of friends — but it doesn’t answer the question in the affirmative. The answer to this question is arrived at, not by computation of examples, but by logic.

Consider the number of friends a person can have. As in the example above, the smallest possible number of friends for a person is zero. The largest possible number of friends is one less than the number of people at the party: everyone else. This means the available numbers of friends are 0,1,2,…,N-1 where N is the number of people at the party. This is a list of exactly N numbers so, for everyone to have a different number of friends, each of these numbers must appear as the number of friends of some person. This is not possible! If one person is friends with everyone else then it is not possible for another person to have no friends. The answer is thus that it is not possible for everyone to have a different number of friends.

The solution to the friends-at-a-party problem is to prove something is impossible — not a skill that is often taught. The only way that computing examples can help is that, at some point, the pattern that there are always (at least) two people with the same number of friends becomes clear. Observing this pattern can lead to the intuitive leap that the key aspect of the problem is considering possible numbers of friends, leading to discovery of the real answer.

Students trained in what is now the standard fashion find this problem incredibly frustrating.

Notice that the technique used to solve the friends-at-a-party problem is either “logic” and so incredibly general or (this is Occupy Math’s view) wildly special purpose until placed in a broader context. This means the only point of this problem (beyond the fact it is useful in a field called graph theory) is to place it in a larger context. The pigeon hole principle says that if you place pigeons in pigeon holes and there are fewer pigeon holes than pigeons, then one hole must contain (at least) two pigeons. For the friends-at-a-party problem, the pigeon holes are the numbers of possible friends and the pigeons are the people at the party. Many clever solutions in mathematics are versions of the pigeon hole principle. This principle is an example of “real” mathematics.

Let’s conclude with some advice to those of you heading back to class.

  1. Always try and place your problems in the broadest context you can. Do not be shy about asking your instructor what the relevant context is.
  2. Hesitate before flailing away with your calculator or computer. Think about the problems, see if there is a clever shortcut or method of avoiding calculation.
  3. If you think you see a pattern, think it through. Math instructors often use such patterns that make the problems easy for them.
  4. If you’re spending more time trying to improve your grades than learn the material, change your strategy. Some modest concern with your grade is healthy, but good grades follow — mostly — from learning the material.
  5. If you think you’re intrinsically bad at math, you’re probably wrong: being bad at math is a curable condition as long as you don’t lose the will to try.
  6. If you don’t understand, ASK! Hiding from math never, ever works.

Occupy Math hopes this advice helps. It is based on decades of teaching experience from 7th grade to post-graduate and spans courses from remedial math to graduate work on the creation of previously unknown math. Occupy Math supported himself in university by one-on-one tutoring of people that were taking required mathematics and statistics classes. He has watched too many failures and helped many people succeed. Do you have advice that should be added to this list? Comment or tweet!

I hope to see you here again,
Daniel Ashlock,
University of Guelph,
Department of Mathematics and Statistics

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