The typical secondary math education includes:
This list is not very descriptive of what is being learned. For instance, why do we call it "Algebra"? The word carries no descriptive meaning in English, The word calculus (pebbles) is also misleading. Using the word "pre" is also non-descript. It makes math educators seem unknowledgeable and uncreative. Yes, each subject comes with a variety of topics, but we should be able to characterize what is generally covered. So below I try to rename the above list to be more descriptive of their contents.
Pre-algebra mainly covers the dreaded "Fractions". The word shares the same root as "fracture", as in a broken bone. So fractions are broken numbers of the form a/b, for example 3/4ths. So we could simply call the first secondary school course "Fractions". That includes ratios, proportions (equivalent ratios) and percents.
The Hindu-Arabic number system with the succinct schemes or algorithms of addition, subtraction, multiplication, division forms the basis of working with numbers in grade school. The word "algorithm" is the transliteration of “Al-Khwarizmi”, the name of the man responsible for developing the power of the Hindu-Arabic number system. He wrote a book called, "Hisab al-jabr w'almuqubalah", which literally means, "Science of transposition and cancellation". The book made "al-jabr" synonymous with the science of equations. The word "al-jabr" comes from a medical term connecting and disconnecting bones. Ancient medical doctors would "al-jabr" a shoulder into and out of its socket to learn its secrets. Now algebra means connecting and disconnecting numbers and variables. (I like to use the word "al-jabr” instead of algebra and I use it as a verb when I tell my students to "Al-jabr it"!) Algebra was once known as the “Cossic” art because the Latin word for the thing is “coss”. With this understanding we could rename “Algebra” to “The Cossic Art of Al-jabring”. This would be a much more interesting, enlightening, useful and creative title. In Al-jabring 1 students learn to apply the Cossic Art to solve mathematical problems.
As we pointed out Geometry means "Earth-measurements". It is the study of shapes that are used to model things on the earth assuming the earth is flat. The measurements include lengths, area and angles. We could call it "Shapes"—which might increase high school enrollment. Linear and quadratic expressions are the building blocks of polynomials. We build polynomials using the operations of addition, subtraction and multiplication.
In Al-jabring 2 we must now deal with division. Division has several difficulties as expressed in two mathematical commandments:
1. Thou shall not divide by zero, and
2. Thou shall not take roots of negative numbers.
In Al-jabring 2 we discuss these difficulties which are encountered in rational and radical equations that involve division or ratios of some sort. We may say Al-jabring 2 is the study of “Ratios”, including ratios in the exponents, ratios of polynomials, ratios of similar triangles or trigonometric ratios, ratios of determinants, negative bases raised to fractional exponents which leads to imaginary numbers, etc.
The worst descriptor of all is Pre-calculus. It is as if we are saying, "Before you study Calculus (pebbles) we want to tell you something else", but what? There happens to be a specific reason for Pre-calculus. To learn what that is, we have to look at all that has come up to that point and what is coming next. Previously students learned how to find numeric solutions using the Cossic Art. But Calculus involves derivatives and integrals, both of which are functions derived from a function. It is no longer just looking at numbers, but functions or mappings from a set, domain for x, to a set, range for y, that assigns a unique number y for each x. So Pre-calculus is needed to get students to think in terms of functions rather than just numbers. Each function is graphed in the coordinate plane so that they can be visualized; see as it were the face of each function. Pre-calculus is the study of “Functions in the Coordinate Plane”.
The word Calculus means "pebbles" and makes reference to mathematics because pebbles were used for counting. In higher math Calculus is referred to as Analysis. Analysis is a new way of thinking that avoids the problem of dividing by zero by asking what happens when something gets close to (but does not equal) something. So it would be preferable to refer to Calculus as “Analysis: A New Way of Thinking”.
Renaming the secondary school math curriculum would look like this:
Algebra 1……………………The Cossic Art of Al-jabring”.
Pre-Calculus……………The Faces of Functions
Calculus……………………Analysis: A New Way of Thinking
Now you're excited to get started learning the Cossic Art, studying shapes, picturing functions and thinking in a whole new way!