# From Arithmetic To Differential Calculus (A2DC)

A short plea for such a sequence appeared in the March 2013 issue of the Notices of the American Mathematical Society. The offprint can be downloaded here.

It all started after the publication by the Office of Instutional Research at my school of a longitudinal study that had followed 1732 students starting with Arithmetic all the way up to and including Differential Calculus. The results were rather dismaying: Less than one quarter of one percent of the students entering Arithmetic had passed Differential Calculus.

It immediately struck me that the mere length of the squence had to have had an impact and, based on my experience teaching Arithmetic and Basic Algebra, and since the only prerequisite for Lagrange Differential Calculus had been polynomial algebra, I had no doubt that it should be possible to design a prequel to Lagrange Differential Calculus that "developmental students" could reasonably cope with in one four-hour semester and which would then result in a three four-hour semester sequence as opposed to the "traditional sequence": five three-hour semesters followed by one four-hour semester.

And so the first thing I did was a "what if" spreadsheet to see what the likely effect of a reduced length might be. It turned out that, in order to get a 0.25% passing rate in a three semester sequence, one really has to work at it and assume extremely low passing and continuing rates that are completely inconsistent with the numbers that the same Office of Instutional Research had found for the two-semester Lagrange Differential Calculus sequence.

So, I started to work on a detailed table of contents for the "prequel"with the idea that the second and third semester would just be a rewrite of Lagrange Differential Calculus. However, it became quickly painfully obvious that the rewrite would have to be a total one if the conceptual flow was to be smooth and efficient.

Given the size of the undertaking, though, I thought that I would just write a "proto-text" , that is only a proof of concept, and leave the actual implementation to whomever might be interested. Barbara Rives, the then Editor of the AMATYC Review, "serialized" the first few chapters.

However, I soon had to bow to the wisdom of the Hestenes dictum and started to work on a Text for A2DC.

But then, I started using parts of the prequel in my classes and thus needed Homeworks, Reviews and Exams. So, it occurred to me that standalones could be extracted from A2DC and that, in fact, it would probably be better to begin with them but within an overall framework in which A2DC should still be relatively simple to produce once the standalones were done. (This, by the way, is the reason for the discontinous numbering of the contents files in the source of RBA.)

The prequel was specified through a kind of reverse engineering, namely as what was strictly necessary for Lagrange Differential Calculus. Here are a few examples:
• In order to define, say, the exponential function as solution of the Initial Value Problem f '(x) = f(x), f(0) = 1, it is necessary to see an equation as an a priori specification of something that may or may not exist—as opposed to being a "question" to be answered—and the idea needs to be introduced a long time before this point so as to give students the time to get completely used to it. So the idea is developed in the middle part of RBA as the investigation of Basic Problems, Affine Problems, etc.
• In order to feel comfortable with the idea of (locally) approximating a function, it is necessary to see approximation as a natural thing to do—as opposed to finding "the" (exact) answer to the given question. The idea is thus introduced in RDA where, for instance, 1/3 = 0.3 + (...) or 1/3 = 0.33 + (...) or 1/3 = 0.334 + (...), etc with (...) standing for "a little bit of no significance in the present situation".
• In order to realize that the + in 3x5 + 4x2 is not the symbol for addition but for a (linear) combination—and thus we really ought to use &, it helps to have already seen that this is the exact same situation as in 3 Apples + 4 Bananas, 3 Hundreds + 4 Tens, 3 Halves + 4 Thirds, 3f + 4g, etc—And of course as in 3e1 + 4e2 = (3, 4) in linear algebra.
Hence the three parts of A2DC:

Part I. Decimal-Metric Arithmetic, Arithmetic Functions, Comparisons and Operations, Equations/Inequations Problems, Laurent Polynomial Algebra.
Part II. Algebraic Functions: Power Functions, Polynomial Functions, Rational Functions.
Part III. Transcendental Functions as solution of Initial Value Problems: Exponential Functions, Logarithmic Functions, Circular Functions, Hyperbolic Functions.

and the corresponding standalones:
• Reasonable Decimal Arithmetic. (RDA) Essentially the arithmetic part of Part I of A2DC.
• Decimal-Metric Number-phrases
• Approximation
• Arithmetic Functions
• Comparisons and Operations
• Reasonable Arithmetic and Algebra. (RAA) An assembly of RDA and RBA that corresponds to a reduction of Part I of A2DC.
• Decimal-Metric Number-phrases
• Comparisons and Operations
• Equations/Inequations
• Polynomial Algebra
• Reasonable Algebraic Functions. (RAF) A reduction of Part II of A2DC that de-emphasizes derivatives.
• Power Functions
• Polynomial Functions
• Rational Functions