Method 4

Create a diagram [7].

Class of Method

A Primary Problem-Solving Strategy.

Use When

When your problem involves information that can be visualized and your problem is not trivial.  When a diagram would help you to more clearly understand what you "want" in its various aspects, and/or would likely provide you with some intuition on how to relate the given information to get what you want.

How

1.  Depict the entities in your problem pictorially or geometrically.  Assume nothing in your diagram, and avoid depicting specifics that your problem does not provide [7].  For example, do not diagram a triangle as an equilateral triangle if such was not specified in the problem.  Try drawing a more oblique triangle to give you a more general intuition of your problem.

-Assume what needs to be done is already done in your figure.  Be sure to use colors to emphasize important parts of your diagram.

2.  Give variables and information in your diagram names or notation.  Do this by following method 5 (Create suitable notation).

3.  Check to make sure that your drawing is legitimate.

4.  If you still need a clearer understanding, then restart this method to continue experimenting with other diagrams in your problem that specialize in viewing other aspects or cases of your problem.  Diagram extreme cases to analyze information that may be contained in such cases that may relate to your problem.

5.  Taking into account your success or failure of applying this method, choose the next heuristic (method) you will use based upon good defeasible reasoning or by using the general method given on page 100.

Notes

Creating a diagram also assists your ability to use your defeasible reasoning.

Method 5

Create suitable notation [7].

Class of Method

A Primary Problem-Solving Strategy.

Use When

Whenever you need to create notation.  When notation you currently use becomes cumbersome or time consuming to manipulate, consider creating better notation.

How

Create notation that satisfies the following principles:

Give the unknowns, and the information in your problem names with proper letters or notation.  However be careful how you define your notation because improper use of notation can significantly complicate the problem-solving process and become burdensome.

-Give the mathematical entities you are working with a name by the use of notation.

-If possible, avoid creating or using notation with similar symbols in the same problem unless necessary.  Doing this is important in order to avoid confusing mathematical entities.  For example, using a1 and a2 could cause you to make errors.  Using α and lowercase a in the same problem could also create confusion.  Using π as a variable could cause confusion with 3.14...  Never use the same symbol for two different mathematical entities in the same problem either.

-Try to use notation that directly reminds you of what the notation represents in order to avoid distracting your mind or your memory when trying to juggle information during the problem-solving process.  This allows you to better concentrate on the problem instead of being burdened and overwhelmed by the notation.  For example, you may use v for velocity, d for distance, and t for time.

-Use similar notation used in similar problems because this helps to also relate, recall, and imitate the methods used in the similar problem.

-Let your mind get into notational “habits” meaning that you tend to represent notation similarly when mathematical circumstances are similar.  Use the same notational methods for similar mathematical entities.  This also helps to allow you to better concentrate on the problem and not get distracted by the notation.

-Try to keep the notation as simple and uncluttered as possible.  Sometimes standard notation can become cluttered for some problems and you may need to invent a simplified notation to represent the same ideas in order to think and manipulate mathematics more clearly.

-Try to recognize ways to simplify notation and definitions in order to simplify the statements of theorems.  For example, when defining exponents using xn to represent n x’s all multiplied together, the theorem xn/xm = xn-m is easily found.  But the theorem xn xm = xn+m could easily include n and m as negative numbers thus including the previous theorem by defining x-n :=1/ xn . This type of notation creation is important for simplifying mathematics and theorems.

-Categorize variables by notation.  For example letting a, b, and c, be the given information, and letting x, y, and z be the unknowns.  Or using capital letters for one type of entity, and lowercase letters for another type.  Often angles in geometry are represented by Greek letters whereas sides of figures are represented by capital Roman letters.

-Do not define unnecessary notation that clutters the problem-solving process.

However, sometimes it is useful to use more than one symbol for the same mathematical entity, but only do this if there is a use for it.

Notes

Choice of notation can make or break your ability to solve your problem.

Some of these notational suggestions are originally from G. Polya in [7] under “Notation” in his dictionary of heuristics.

Example

The summation symbol significantly simplified notation for adding entities.