Reading Mathematics Tip 4

Tip 4 from Dr. Del

Learn the vocabulary.
Why are definitions important?

If a reader does not know requisite definitions, mathematical statements become meaningless, textbook presentations are confusing at best, and lecture explanations have absolutely no value to the learner.  In addition, if the learner does not know requisite definitions his/her own written or spoken statements are incorrect and in many cases completely nonsensical.

Every mathematical term in the following sentence, taken from an elementary Algebra textbook, has been replaced by a randomly chosen word from a foreign language.

To natus one desenvolvemos by another, natus each consectetur of the first sagte by each geschiedensboek of the second liever and nesciunt dignissimos hálito.

Clearly, no one can learn mathematics from statements such as this.  However, that is precisely what a reader attempts when definitions are not learned.  After a few attempts it becomes clear that “the book doesn’t help at all” and the student stops reading the text and simply tries working problems. That strategy always leads to failure.

There are other reasons why definitions are important in the study of mathematics.

“Preteaching vocabulary in the mathematics classroom removes cognitive barriers that prevent children from grasping new content.”[Chard]

Here is what professor Stephen Maurer of Swarthmore college writes about the role of definitions in mathematics.

“Most disciplines don’t need to make definitions explicit nearly so often as mathematics does – they don’t need to be so precise nor do they deal so regularly with situations outside common experience.”[Maurer]

Definitions play a much more important role in mathematics than they do in any other area of study.  A word may (in fact probably will) have a different meaning in mathematics than in normal discourse.  One of the dictionary definitions for the word function is: “the action for which a person or thing is specially fitted or used or for which a thing exists”.  The definition of function in mathematics is very different.  That mathematical definition will be the subject of much of the content of any College Algebra course.

Definitions in mathematics form a solid and completely adequate foundation upon which we base all our mathematical reasoning.

A mathematical definition of a concept gives necessary and sufficient conditions for a creature to be an instance of that concept.

For example, the definition of a prime number is:

A number is a prime number if and only if it is a natural number greater than 1 with exactly two divisors.

From this definition it is possible to conclude that 7.32 is not a prime number because it is not a natural number and therefore violates the necessary condition that a prime number be a natural number.  From this definition we can also conclude that 6 is not a prime number because it has four divisors and therefore violates the necessary condition that a prime number have only two divisors.  From this definition we can also conclude that 7 is a prime number because it is a natural number greater than 1 and it has only two divisors and therefore satisfies the necessary conditions stated in the definition.

Non-mathematical concepts on the other hand are frequently defined in a hazy and flexible manner.  Find any definition of prime real estate and note that it does not give necessary and sufficient conditions.  Rather it will be hazy, deliberately quite flexible, and open to personal interpretation.

In an article about teacher preparation, Professor H. Wu from the Mathematics Department at Berkeley claims that precise definitions form the basis of any mathematical explanation.  He correctly states:

“logical explanations – the essence of mathematics no matter how mathematics is defined – cannot be given without precise definitions.”[Wu].

Finally, I should point out that many/most of the exercises in College Algebra are simple one or two step logical consequences of a definition.  For example, a firm understanding of the term “graph” is the key to answering many questions. Just one example of this is the fact that an understanding of the word “graph” provides the basis for finding the points of intersection of two graphs.

Definitions are the first tool used when attempting to answer a mathematics question.

The importance of, and the role of, definitions in mathematics is a complex topic. I am currently working on an in-depth paper which examines the type of definition used in mathematics, how those definitions are learned, and their significance in mathematics.

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