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About Calculus&Mathematica

Welcome to Calculus&Mathematica at Math Everywhere, Inc.

Calculus&Mathematica (C&M) is a computer based calculus course consisting of a complete interactive electronic text combined with a rich set of well chosen problems for the students to do. The driving forces behind C&M are our beliefs on the basic questions of how mathematics should be taught, with special emphasis on Calculus. The program is devoted to teaching calculus in a new and effective manner through the use of technology and the socratic teaching style, a learning environment that brings student learning to the foreground. The environment is a partnership between teacher and student. Teachers focus on student understanding, but leave the responsibility of learning on the students' shoulders.

These courses are different, and are taught differently. The first thing we learned when we started down this path 10 years ago is, "I can't teach you." I can help you learn. I can help you find those things that are yourgaps, and help you patch them up. I can direct your progress, but the responsibility for learning is yours. This makes problems and expectations different. It makes what you are presented to work from look different.

Tom West will tell you that there's a new and valuable learning style that likely dominates in our culture. There are lots and lots of very smart people who are most comfortable in working and learning in a visual environment. To you, that means that new ideas will be presented first, and frequently, in a visual mode, usually before any calculations. The easy access to powerful technology makes lots of experimentation easy. You can try things out before you have mastered the ideas and techniques to get even more comfortable with new ideas.

The courses treat many topics in different ways. There is no section in the courses devoted to just manipulating trig functions, their dervatives and integrals, and the useful identities that make many calculations work. Instead, Trig and exponential functions and complex numbers appear in most lessons throughout the course. Euler's formula takes care of trig identity memorization, and is the connecting link between the Electrical Engineer's e^{j w t} and some other folks' cos(w t + p). Students tend to understand it, how it is derived, and how to use it. That's special.

Some people say that the main reason to study calculus is to understand what differential equations do for us in modelling all sorts of processes. Once again, we learned very early on in our voyage that many students come with little or no idea of what it means for something (like a specific value of a variable) to solve a given equation. More than that, it is highly unlikely that a student who has never seen a differential equation before will see immediately that the equation is telling the reader how some quantities are growing. Differential Equations appear for study in each and every course here. Similarly to trig and exponential functions, they are simply part of the meat of the whole course.

Ask any mathematically literate person about the Mean Value Theorem. It's important, but is usually used as a tool for proving things and rarely for building intuition first. In C&Mcourses, it appears in its active voice; The Race Track Principle. This is a form suited to most uses of the principle, and it's impossible to forget. Read it, and try to forget it.

Many techniques taught in math courses have been honed and polished for years and years. Nice algorithms have been developed which, if remembered properly, can make quick work with paper and pencil calculations. There's a hook in that last sentence. Do you see it? Many of the tasks we are asked to do can be reduced intuitively to setting up and solving equations and sets of equations. One great example of intuition versus memorized algorithm comes with something you probably think you should know: completing the square. Check it out. There are tons and tons of other places where working from a knowledge of what the process is to accomplish, knowing how to set up equations, and solving them works reliably. Additionally, you can remember what it is and how to do it. The generic name for this sort of thing is the method of undetermined coefficients.

There's a war on, folks. There shouldn't be, but there is. Math afficionados throughout the world are taking positions and defending them strongly. There's a right wing and a left wing. The right wing people say that the left wing people want to eliminate all ability of students to perform routine calculations correctly with paper and pencil. The left wing people say that the right wing people want to have students do nothing but master those low level skills. Both sides are full of it. Both of those statements are wrong. Both are held as immutable positions by many combatants. How silly is that? We'd probably be called left wingers. Do we believe in acquisition of hand calculational skills? Of course we do. Should those skills come at the expense of a deeper understanding of processes and concepts? I think not.

Here's a list of Frequently Asked Questions about Calculus&Mathematica and other MEI courses.