SUGGESTED PROBLEM SOLVING TECHNIQUES

James J. Cochran

     There are almost as many approaches to solving problems as there are problems to be solved. The following is a short list of suggested approaches to solving mathematics problems. Keep in mind that this list is neither mutually exclusive nor collectively exhaustive.

 

Pattern Identification or Bottom - Up Approach

     When we are solving a problem in mathematics, we frequently have access to similar problems for which we know how to calculate the solution. In this approach, the same steps used to solve the problem with a known solution are applied to the problem for which we do not know the solution. This is the problem solving approach most frequently taken by students in an introductory mathematically oriented course.

     Consider the very common formula for the sample MAD (mean absolute deviation):

as applied to the following four sample values for some random variable x: 6, 9, 8, 5. Suppose also that we know the calculations for the sample MAD for the following five sample values for some random variable y: 10, 9, 14, 15, 12. These calculations are

We can apply a similar approach (substitute the sample mean for random variable x into the MAD formula and complete the necessary calculations), i.e.,

Top - Down Approach

     For more complex problems we can again make use of a similar problem for which we know how to calculate the solution. However, in this approach we start with the final solution and work backward through the steps necessary to solve the problem. Suppose we are trying to calculate the sample mean of the following four sample values for some random variable x: 6, 9, 8, 5. Suppose also that we know 12 is the sample mean for the following five sample values for some random variable y: 10, 9, 14, 15, 12. The formula for the sample mean applied to the sample values for random variable y is

Even if we do not understand how to calculate the numerator of this equation, we can use a little algebra to find

From this point it is fairly easy to deduce that the numerator of the formula for the sample mean is the sum of the observed values in the sample, i.e. 10 + 9 + 14 + 15 + 12 = 60. From this point we can apply our newly acquired understanding to calculate the sample mean of the random variable x.

Rocking

     Often we can not completely figure out how to solve a problem through either the Top-Down Approach or the Bottom-Up Approach. In the Rocking Approach, we alternately utilize the Top-Down and Bottom-Up Approaches. This is done until we have made enough progress in each direction to link the two up and complete the solution process.

 

Trial & Error

     Quite often there are only a few conceivable ways to solve a simple problem. Consider the very common formula for sample variance:

as applied to the following four sample values for some random variable x: 6, 9, 8, 5.

Students who are unfamiliar with statistics and its notation often confuse the order of operations in the numerator of this calculation. They are uncertain whether to

  1.  subtract the sample mean from each observation, sum these differences, and square this sum

or

  1.  subtract the sample mean from each observation, square these differences, and sum these results

Since there are really only two reasonable and simple ways to progress (1 or 2), we can try both and observe the results. Perhaps the correct approach will become obvious upon our inspection of these results. Since the sample mean is 7.0, we have

or

From these results it is obvious that approach 1 can not be correct (only a constant has a sample variance of zero, and our random variable x is not a constant). Approach 2 is the correct method for calculating the sample variance.

Note that this approach is particularly valuable when you are uncertain of the output that an analytic software is providing you. Use the software on a similar problem for which you know the correct answer, then compare the output to the correct answer!

 

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