Archive for November, 2014

What is Algebra and What Should be Taught in Algebra

November 20, 2014

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.”

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. 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.”

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

November 4, 2014

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

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.

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