MathXpert and Mathematical Education
The famous mathematician Karl Gauss called mathematics the Queen of Sciences. Physical science rests on a mathematical basis; engineering rests on physical science; technology rests on engineering. Without mathematics, your car and television wouldn't exist, and your house would be dark and cold. The refrigerator works on principles of thermodynamics that could never have been understood without mathematics, and the same goes for Diesel engines, so without mathematics, you would have to eat what could be grown nearby, and you would have to eat it before it spoiled. Life would be "nasty, brutish, and short."
Without an understanding of mathematics, a person cannot understand the principles on which refrigerators, cars, airplanes, television sets, cell phones, and computers have been designed. Mathematically uneducated persons may simply take these products of human intelligence for granted, treating them as part of the environment. This should not be allowed to happen: every person should have the opportunity to learn and understand the accumulated technical knowledge of the past.
Mathematical literacy is much rarer in our society than verbal literacy. Two main reasons for this are
You must have encountered both of these problems in your attempts to learn mathematics!
I once analyzed the errors my students made on exams in freshman calculus. Eighty percent were not calculus errors at all, but algebra or trigonometry errors. That is, these were errors in things the students were supposed to know before taking calculus at all. Once such an error had been made, the students could not complete the calculus problem correctly. Notice how the two difficulties are connected: for lack of mastery of earlier stages, the students are not able to proceed carefully.
Mastering a subject before proceeding to the next subject is definitely not encouraged by the present educational system. We find students at all levels lacking fundamental skills they should have mastered years ago. Being very careful with your work is not encouraged by our society, either. Examples are easy to find at the individual, corporate, and governmental levels. It is hardly surprising, then, that a subject that requires care and mastery is widely regarded as difficult and frightening. But the roots of "math anxiety" are to be found in the two difficulties mentioned above, not in any inherent difficulty of mathematics itself.
In 1981, I thought that the computer could be harnessed to make a fundamental attack on these two problems in mathematical education. Using a computer, a student can proceed at his or her own pace, so that mastery can be achieved. In a classroom, if a student is a little too slow, he or she can easily get lost, and understand nothing at all of the rest of the lecture. When using a computer projector instead of a blackboard, I learned that about one-third of my class typically lags at least one full blackboard behind the lecture. These students complained that the solutions would scroll off the projector screen before they were finished copying them. In years of lecturing at the blackboard, I had never realized that many students had their attention on the previous board instead of on the developing lecture.
Using a computer privately, these students will be able to look at each screen as long as they desire, and even go back over the problems again and again. Moreover, the computer can offer solid help with the necessity of being careful. If the program is properly written, I thought, it should be impossible to make a mistake. You could, of course, take a less than ideal step, but not a mathematically incorrect step.
If a program is going to help students to achieve mastery, it must be able to present each subject at the appropriate level. Beginning algebra students may need a five-step solution to a simple common-denominator problem such as 1/2 + 1/3. Calculus students must see common denominators as a one-step operation, perhaps even to be performed as a small part of the operation simplify.
There were, even at that time, some mathematical programs in existence, but they had several basic flaws, and I decided in 1985 to write a new program from scratch. This program became MathXpert. The name is a contraction of "Mathematical Expert", because the program is able to show you how to solve most of the problems you would encounter in algebra, trigonometry, and two semesters of calculus. (The term precalculus is used only in the US, and includes trigonometry, logarithms, exponentials, and an introduction to complex numbers, all of which lie within MathXpert's expertise.)
MathXpert has been designed from the beginning to satisfy these principles:
MathXpert is to solving a mathematics problem somewhat as a word processing program is to writing an essay. You still direct the course of the solution, but you use the computer to get the steps on the screen. You tell the computer add these fractions and put them over a common denominator, for example, and it carries out that operation, copying the rest of the line over onto the next line. MathXpert provides an easy method for you to accomplish the task of telling the computer what to do, without having to remember (let alone type) complicated commands or look them up. A lot of work has gone into the design of the "term selection" method that lets you accomplish this easily.
Where MathXpert differs from a word processing program is in the amount of help it can give you in solving your problems. A word processing program can't write your essay; but MathXpert can solve your mathematics problem (most of the time, I hope). This power is made available to you through the Hint, AutoStep™, and AutoFinish™ buttons, whose functions are explained in this manual.
I wrote MathXpert as a tool for you to use in learning mathematics. You will be able to use it to get past the obstacles to doing careful work in mathematics, and mastering mathematics level by level. However, no program, no matter how sophisticated, can do the learning for you. In studying mathematics, you should take it as your ideal to master the subject, and to do absolutely careful work. In the past, this may have been too difficult for you, due to the classroom situation and the lack of immediate feedback when you made mistakes in your homework. Those days are over: now, using MathXpert, you can see every detail at your own pace, and never make another mathematical mistake.
Not only the students who have difficulty with mathematics will benefit from MathXpert, but also the bright ones, who normally are bored in their mathematics classes, and are forced to sit through day after day of trivial repetitions of things they have heard already. These students should fly like birds in the new world that MathXpert will open up to them. They too can proceed at their own pace, and solve problems that will challenge them instead of bore them. I expect to hear from those students about a lot of difficult problems that they were able to solve with MathXpert, which MathXpert's internal algorithms could not solve, or could not solve so beautifully. MathXpert's methods are good, but they are general, and students will find they are sometimes able to improve on MathXpert's auto-generated solutions.
MathXpert is not only for students enrolled in classes. It can also be used for home study. Many mathematics teachers and engineers will be brushing up their mathematics in the evenings using MathXpert. A lot of people who say that math was their worst subject in school secretly wish that hadn't been the case; and some of them will be interested enough to give it another try, especially when no human being will be looking over their shoulder to see their mistakes, and especially when they are guaranteed not to be able to make any mistake.
There is another reason why Gauss called mathematics the Queen of Sciences: she has an austere but compelling beauty. She reveals herself only gradually, after long and patient attention. If you are careful, and seek to master the subject, you too will fall under her spell, and find that you have developed a genuine love of mathematics.
Dr. Michael Beeson