Monitor Projector


Intermediate Algebra
Elayn Martin-Gay

"One cannot apply what one knows in a practical manner if one does not know anything to apply."
-- Robert Sternberg


Section 0.0: General Guidlines

Getting the answer is not your primary goal.

Your goal is:Your goal is to learn concepts, learn how to use those concepts in new situations, and to communicate correctly and effectively about mathematics. You are to learn deductive reasoning, critical thinking, how to generalize, and how to create and work with abstractions. Those are the skills that will work to your benefit after college. You are expected to learn the techniques, procedures, and concepts presented in the textbook, on the website, and in the lectures (with the caveat that in a few instances I will advise to not use a technique presented in the textbook).

Getting the answer is not your primary goal: For example: If I ask you to find the product of two numbers written in scientific-like notation you are expected to show me that you have learned how to take advantage of the notation to perform the operation. Converting a number like 3.298X10-8 to 0.00000003298 and the performing whatever operation will earn you ZERO points because you are telling me that not only didn't you learn how to operate with numbers in scientific notation but you are not interested in learning.

Writing Mathematics: You are expected to learn correct writing styles for presenting mathematics. Your author, the website, and my lectures all present appropriate styles for writing mathematics. Any serious student will try to imitate those writing styles. Your test responses should convince me that you are learning those styles. There must always be some mathematically correct structure, phrase, or sentence that links the pieces of your work in such a way that I can begin reading at the top-left and continue reading in a "to the right-and-down" manner (like all writing in the US) until I have reached the end of your argument. That end should make the conclusion clear and should also convince me that you know and understand the conclusion as well as each individual step in the process. There is no need (nor will it be tolerated) to "circle your answer".

This advice goes far beyond the classroom: When you intend to use a formula, state the formula, substitute values into the formula, then calculate. These pieces MUST generally be connected with an = symbol. To throw an = symbol in at the last step is completely wrong and leads me to believe that maybe you think the = symbol is used to mean "the answer is". If your understanding of the = symbol is "the answer is", your previous teachers should be beat with a stick. If two expression are equal, that fact should be indicated with the use of the = symbol. If two expression are not equal, do not insert an = symbol between them.

Do not get hung-up on techniques you learned in high-school: As you proceed through college courses you will always extend what you have previously learned. That statement is true for life-long learning as well. You are expected to grow mathematically as you travel through Intermediate Algebra and College Algebra. Your work should convice me that you are growing.

Section 0.1: Definitions in Mathematics

The Need

The need for mathematical definitions for communication about mathematics is clearly established in my essay Mathematics without Definitions.

Scholars frequently organize definitions into five categories: Persuasive, Precising, Theoretical, Lexical, and Stipulative. The last two are of interest to the mathematics teacher and student. A common name for a lexical definition is an extracted definition because it is extracted from common actual usage. Extracted definitions have a truth value -- they can be true or false. Definitions familiar to beginning students are generally extracted definitions.

The Difference

Definitions used in mathematics are very different. Definitions in mathematics are always stipulative definitions. They are stipulative in the sense that they specify usage rather than report usage. Stipulative definitions do not have a truth value. They are neither true nor false -- they just are! Beginning students are generally completely unfamiliar with stipulative definitions. A definition in mathematics does not announce what has been meant by the word in the past or what it commonly means now. Rather it announces (stipulates) what will be meant by the word (or term) in the present work. Unlike extracted definitions, stipulative definitions cannot be reliably learned by repeated exposure to instances of the definition. They must be memorized.

Definitions used in mathematics differ from ordinary lexical definitions in other very important ways. Definitions used in mathematics are very precise in the sense that they contain no extra words nor have any words been omitted. There is no ambiguity associated with the definitions used in mathematics. Words used in mathematics do not have multiple meanings as do many words used in ordinary discourse.

Ref: B. S. Edwards Oregon State Univ., M. B. Ward, Western Oregon Univ. "New Practices in Teaching Mathematical Definitions."
Ref: B. S. Edwards Oregon State Univ., M. B. Ward, Western Oregon Univ. "The Role of Mathematical Definitions in Mathematics and in Undergraduate Mathematics Courses."
Ref: B. S. Edwards Oregon State Univ., M. B. Ward, Western Oregon Univ. "The Role of Mathematical Definitions in Mathematics and in Undergraduate Mathematics Courses."

The Value

Verbal communication is a critical component of teaching, learning, and studying mathematics. Communication always involves two parties: the presenter and the reciever. In order for verbal communication to be effective, it is necessary that the presenter and reciever agree on the meaning of words used. In normal everyday communication the reciever can use context to gain some understanding of an unfamiliar word. This is not the case with mathematical communication because the only definitions used in mathematics are stipulative (as opposed to lexical). Moreover, the definitions in mathematics are extremely precise.

No longer is there any doubt that "All new learning requires a foundation of prior knowledge."
Memorization of stipulative mathematics definitions provides a form of foundational knowledge upon which to build a complete concept.
Simple mathematics concepts are constructed with a definition as a central core tightly wrapped in several layers of illuminating information.

Therefore in order for any one of teaching, learning, or studying to be effective it is necessary the both the instructor and the learner know precisely the stipulative definitions of all the words being used.

The Problem

There are at least four very significant problems associated with a lack of knowledge of definitions.

The Remedy

Realize that memorization of a mathematics definition and understanding that definiton are not oppposing ideas.
Realize that the definitions are the foundation of every problem solving technique to be learned.
Then begin by memorizing each definition.
Try to understand every aspect of the definition.
Study examples and non-examples to determine why they do or do not satisfy the conditions of the definition. Create your own examples and non-examples.

The Debate

A recommendation to memorize mathematics definitions immediately triggers the reaction that understanding is more important than memorizition. Make no mistake about this: Memorization of a mathematics definition and understanding that definiton are not oppposing ideas. Intelligent rote memorization is a good starting point toward gaining understanding. In most instances after gaining understanding (based on the precise memorized wording) that previously memorized definition becomes the only option and the memory synapse retrieval path becomes very strong. Ideally that path becomes so strong that recall of the definition seems natural and has little to do with the initial memorization task.

Intelligent rote memorization has several important components.

The following infographic presents the virtues of flashcards as a learning device.