What is Algebra and What Should be Taught in Algebra

Algebra is: an art, a detached language, a way of thinking, a foundation of the sciences. With very little effort you can find other “definitions” of algebra. According to Keith Devlin “The important thing to realize is that doing algebra is a way of thinking and that it is a way of thinking that is different from arithmetical thinking.”

— http://profkeithdevlin.org/2011/11/20/what-is-algebra/

I and many other mathematicians and teachers of mathematics have long contended that “getting the answer” is not the primary goal in algebra classes. A very good exposition of this point of view is an excellent 17 min. video by Phil Daro. http://vimeo.com/79916037. Everyone with an opinion about what should be happening in our algebra classrooms should watch this video.

In the past I have written that “The purpose of early college level algebra courses is therefore to introduce the student to the use of abstraction, generalization, and deductive reasoning while exploring the patterns and relationships of a variety of algebraic entities including, but not limited to, equations, inequalities, algebraic fractions, polynomials, and functions.”

— http://www.college-algebra.com/essays/what_are_we_studying.htm

I apologize for not regularly including considerations of structure in previous writings. An important goal when teaching algebra (indeed all math) is to help students to see the elegant structure of mathematics and to recognize how such a structural view is important in many situations. Consider the operator of a pump station on a large cross-country pipeline. The operator must view his facility as a part of a much larger structure and must understand how his facility interacts with the rest of the pipeline structure. With a comprehensive structural view the operator is an asset to the company, without that view he may in fact be a hazard.

Should the student learn a bunch of isolated formulas and receive training for solving a hundred or so silly problems or would it be better to become a critical thinker capable of using abstraction, generalization, and deductive reasoning with the ability to examine an entire structure when needed in problem solving situations?

I Will Die a Happy Man

It is only a slight exaggeration to claim that for 20 years I have wanted to produce decent looking mathematics on Web pages and that I wanted to build interactive animated instructional material on Web pages. Finally I have been able to do both and have produced a small demo program http://www.drdelmath.com/sketch_parabola/index.html

This demonstration program is little more than “proof of concept”, but it does not take much imagination to realize that with the six or seven languages/tools I used it is now possible to make mathematics come alive. Instruction can now move away from static boring textbooks and can truly involve the student in the study of mathematics as an engaging subject filled with motion.

As I look at this simple demo program I am proud of my work and somewhat surprised that I was finally able to pull it off. However, I also visualize several, maybe many, additional desirable features which should be added in an attempt to improve its educational value. I hope to add some of these features as time permits.

Caveat: This demo is best viewed with a browser other that Internet Explorer. I do provide a reasonable “fallback” action if you use IE10 or IE11, but your experience will be less than satisfying. I use a type of SVG animation which Microsoft in essence says is not necessary and will not support it in their browsers. As far as I know all other browsers support that aminamtion technique.

I would certainly like to hear any comments you have. Tell me about any errors, suggestions for improvement, possible additions, other kinds of action, etc.

Proper Use of Examples Part I

Examples are an important instructional device used by every mathematics teacher. However, I fear that many of us (myself included) present the wrong examples and use them to illustrate the wrong things. Furthermore I am absolutely convinced that many students do not know the purpose of examples and consequently use them improperly.

For this essay I will concentrate on those examples which are illustrations of a mathematics creature as defined in a formal definition. Such examples are generally a part of the initial discussion and explanation of the definition.

Definitions in mathematics are stipulative definitions as opposed to lexical definitions.

A synonym for lexical definition is extracted definition because it is extracted from common actual usage. Extracted definitions have a truth value—they can be true or false. Beginning algebra students are generally familiar with extracted definitions only. These are the definitions found in a common dictionary (lexicon).

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! Early algebra 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, argument, or discussion.

Because mathematics definitions are stipulative we must depend on them much more explicitly than we do with lexical definitions of words used in casual conversation. For that reason examples are not to be used as a replacement for the definitions in mathematics. This is a lesson that must be learned by many authors, teachers and all others involved in development of instructional material. This is a hard lesson we should teach our students. We should help the student to understand that examination of a bunch of examples will not reveal the stipulative definition. Rather the stipulative definition must be used to determine if the example is indeed the creature defined in the definition.

Suppose the stipulative definition of a jeok is presented and is followed with a bunch of examples. The natural inclination of students is to look at the examples and to use that experience to determine if a newly encountered creature is indeed a jeok. After all, that technique works well enough for identifying dogs (never mind that a hyena is not a dog). This technique lacks the precision required in mathematics and does not acknowledge the stipulative nature of mathematics definitions. In the study of mathematics, and mathematics communication, there must be a dependence on the stipulative definitions.

Examples do indeed help the student understand complexities and nuances of a definition. Students frequently overlook those illustrations of complexities and nuances. Better constructed examples and more discussion could alleviate the problem.

More importantly, examples should be instructional devices designed help the student determine if he/she understands (or does not understand) the definition. Students must be taught to use examples in this manner; it is not natural.

If an example is to be used by the student to evaluate his/her understanding, then we should construct examples to further that goal. In a discussion of the examples the student should be guided through activities which encourage (or force) the student to learn how to use a stipulative definition.

When a student learns to study and use examples in this manner that student will be assured that he/she understands the definition. Furthermore the student will be gaining experience in the use of stipulative definitions. That skill is useful during the construction of an argument whether it is a mathematical argument or is part of some other discipline.

If we teach students to use examples in this way we have taught them something about mathematics, but we have also provided them with a thinking skill which can be used in any part of life. I consider that to be a desirable outcome.