Creating Good Mathematical Definitions
In order to clearly communicate yourself to others, you need to use words whose definitions are understood by everybody in the same way. Picture a young man saying to his girlfriend, “I love you.” And his girlfriend replying, “I love you, too.” Are they using the word love in the same way? Do they understand precisely what the other one means when they use the word love? Probably not. Love, as a word and as a concept, can carry an enormous variety of meanings and implications and generate very different ideas in the minds of different users of the word. It may well be that it is a good thing for a word like love to be so vague; we probably need powerful-but-vague concepts in our world.
However, at other times, particularly when precise, shared understanding is critical, we need to be able to use words that carry precise, specific meanings that all users understand in the same way. Mathematical practice is one such example.
All mathematical definitions depend on a shared understanding of some ideas built up from descriptions that we collectively choose to agree upon. In geometry, those key building block ideas are point, line, and plane. Here are the descriptions (not definitions):
1. A point is the basic building block of geometry. It has no size. It has only location. You represent a point with a dot, and you name it with a capital letter, for example, P.
2. A line is a straight, continuous arrangement of infinitely many points. It has infinite length but no thickness. It extends forever in two directions. You name a line by giving the letter names of any two points on the line and by placing the line symbol above the letters, for example, AB or BA.
3. A plane has length and width but no thickness. It is like a flat surface that extended infinitely along its length and width. You represent a plane with a four-sided figure, like a tilted piece of paper, drawn in perspective. You name a plane with a script capital letter, such as R.
TASK 1: Models aren’t the actual object, but represent the object as a useful approximation. Draw or write examples of both mathematical models and physical models of these building block concepts.
Building Block / Mathematical Models / Physical modelsPoint
Line
Plane
TASK 2: Create a definition of “Geobug” that will allow you to successfully distinguish between “true” Geobugs and figures that are not truly Geobugs.
A. Use the Geobug Definition-Builder form to identify essential and non-essential (i.e. possible distracting) characteristics and then craft a definition.
B. Check your definition with the others in your group to see if all agree that your definition will clearly distinguish between Geobugs and non-Geobugs.
C. Use your definition to determine which of the following are Geobugs.
Developing good mathematical definitions for a word or concept is a three-step process:
1. Classify the concept: What is it? What to what larger category of objects or concepts does it belong?
2. Differentiate the new word. How does it differ from other closely related objects or concepts? What makes it unique?
3. Test your definition by trying to find counterexamples.
TASK 3: Looking for counterexamples.
A. Complete the following sentence:
A square is a ______that ______.
B. Freddy answer for A above was “A square is a figure with four equal sides.” Sketch 3 different counterexamples to Freddy’s definition to help him (and you) to improve it. A counterexample is an object (or concept) that satisfies the definition as it is stated, but which isn’t actually a true example.
C. Write a definition for a square that is precise enough that it doesn’t have any counterexamples, but broad enough to include all true “squares.”
TASK 3: In your groups, use the three-step process to define the following terms:
A. Line Segment
B. Angle
C. Parallel lines
D. Perpendicular lines
E. Circle