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The Wikipedia Calculus Article (retrieved 2008-05-27) starts with the following definition: "Calculus (Latin, calculus, a small stone used for counting) is a branch of mathematics that includes the study of limits, derivatives, integrals, and infinite series, and constitutes a major part of modern university education."
This definition is problematic for me. First off, my goal was to redesign the curriculum so that Calculus was taught in high school. Secondly, I de-emphasized limits and infinitesmals.
The nature of what I wished to do required a different definition. I called my new, stipulated definition "The Calculus of Change," which I define in the next section.
Definition: "Calculus is the mathematical description of change."
At heart, derivatives and integrals are equations that describe change. The derivative describes how the slope of a line changes. The integral (also known as the anti-derivative) describes how the area between a curve and the x-axis changes.
In my view, limits and infinitesimals are simply tools used to discover derivatives.
Some textbooks go as far as to define the derivative as a limit. It appears to me that this definition confuses the method used to discover something with the thing itself. For example, scientists invented a microscope. They pointed to microscope at a drop of water and discovered protozoa swimming in the water. Scientist discovered protozoa with a microscope, but the protozoa is not a microscope.
I believe that the derivative transcends the method used to discover it. You can find a derivative with limits, infinitesmals or a lucky guess. This does not mean that the derivative is a limit, infinitesmals or lucky guess.
The main problem defining the derivative as a limit is that this method transfers all of the logical complications associated with limits onto the derivative.
In the case of Calculus, we find that the derivative is a relatively simple concept. The limit is a horrifically complex idea that takes years to master. An infinitesmals is an abstract idea can lead to paradox and absurdities.
If mathematicians were to accept that the derivative transcends the method used to find the derivative, then they would come to the stark realization that primary function of Calculus is to describe change.
The reason that calculus is fundamental in all sciences is that change is fundamental to all sciences.
The Calculus book I envisioned would be written for students who had mastered high school algebra. The book would open with a section on visual perspective and then move on to Calculus proper.
If I were designing an entire curriculum (from K1-12); I would do the following: I would start by teaching basic structure of language and reasoning. The Quadrivium teaches grammar, logic, rhetoric and arithmetic.
In the middle years, I would present a series of classes that I call "The Calculus of Space." This includes geometry, algebra and analytic geometry. The goal of this set of classes is to teach students how to describe objects in a three dimensional space. The Calculus of Space builds on the skills developed in the Quadrivium.
The "Calculus of Space" would culminate with the study of visual perspective. Visual Perspective is interesting in and off itself. The study shows how to project a three dimensional space onto a two dimensional surface.
Having mastered the Calculus of Space, the students would move on to The Calculus of Change. Here the student would learn or derivatives, integrals and other tools for describing change in a multidimensional space.
An even quicker overview:
With this basic structure, students will be able to move on to physics, trignometry and other advanced subjects.
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