Produced by Charles Wells.     Home     Website TOC      Website Index  

Back to Useful Behaviors head

Last edited 12/21/2006 3:59:00 PM

MORE USEFUL BEHAVIORS

Contents

Think of examples. 1

Guessing. 1

Trial and error 2

Appendix. 2

Think of examples

One characteristic of people that are good at math is that when they are faced with a new concept, they immediately start thinking of examples. 

(See [Selden & Selden].) 

Example

Suppose you just learned the definition of prime number.  (A positive integer n is prime if its only positive divisors are 1 and n.)  You might have thoughts like this go through your head:

2 is prime, 3 is prime, 4 is not prime because 2 is a divisor, 5 is prime, 6 is not prime because 2 is a divisor, 7 is prime, 8 is not prime because 2 is a divisor…

Hold on!  If n is prime it has to be odd, because otherwise 2 is a divisor!  Wait: 2 is a prime anyway even if it is even.  So the only even prime is 2. 

So if n is odd it must be prime, because it is not divisible by 2.  Hold on!  It might be divisible by something else.  That’s right, 9 is odd but it is not a prime! 

And so on…

This is a baby example of what you might go through when you first learn about primes.  It is vital to generate example of any new concept (if you can!) because that is the fastest road to understanding the concept.

 More examples

Guessing

Guessing at the answer to a problem and then using a theorem to prove it is correct is legitimate.  Some students don't believe this!

Example

You need to find .  You remember that you just did that integral yesterday!  Wasn’t the answer ?  Let’s see, the derivative of  would be , which is .  So the Fundamental Theorem of Calculus says that the answer  is correct!

This is a perfectly respectable way to do math.  You need not use a specific method (for example substitution or integration by parts) to get an integral.  Any way of coming up with the answer is OK as long as you can check it out using differentiation. 

Of course, if you do  have a method, you may be better off than you are if you can only guess.  The integral  is easy to do by substitution:  Let  and .  Then

Put links to integration, differentiation, Fundamental  theorem.  Give another example not involving differentiation.

Trial and error

Example

Suppose you need to know the largest integer n for which .  One way to do it is to calculate:  4! = 24, 5! = 120, so the answer is n = 4.   When I gave a problem that came down to this calculation in my discrete math class, most students solved it correctly, but several wrote apologies on their paper for doing it by trial and error.

Of course, trial and error is a method.  In the case of this problem it is probably the easiest way to do it.   Even if you need to know the largest integer for which n! < 4,000,000 it makes sense to do it using a calculator or a program such as Mathematica or Maple. 

More Examples

Appendix

Answers to questions

Why did you say having a method “may” be better?

Answer:  Because in some cases guessing may be easy and the method may be hard to apply.  Example.

Return