Mathematics and Logic - Skill and Concept Development

Adding Words to Algebra

While some calculations may be described with words - think of the perimeter calculation example, many calculations can be represented or given by an algebraic expression or formula that is best seen and grasped in a silences in a glance. Formulas like the compound interest or growth formula and like the quadratic formulas fall in this category of being seen and grasped in silence. The use and development of formulas for lengths, areas, volumes, speed, distance, time, simple interest, compound growth and even or odd numbers, introduces students to the shorthand role of letters and symbols in mathematics.

A. Naming and Identifying Formulas

Outside of algebra, a picture is worth a thousand words. Inside, formulas to awkward to read aloud term by term may also be worth a thousand words. However, both pictures and formulas may be named or identified with words. Outside of algebra, the phrase Mona Lisa identifies a painting of Leonardo da Vinci. Inside of algebra, formulas may be identified by name

• the compound interest formula
• the quadratic formula

Inside algebra, descriptive or identifying phrases also identify formulas.

1. Triangle area calculation formula
2. Square area calculation formula
3. Circle Perimeter Formula
4. Circle area formula
5. Sphere or ball surface area formula.
6. Sphere volume calculation
7. Cube surface area expression
8. Box volume formula

In these phrases, the words formulas, expression and calculation may be used interchangeable.

In dealing with fractions, we may speak of general and/or efficient methods or formulas for adding, subtracting, multiplying and dividing. At the higher level in the mathematical subject of calculus, the phrases product rules, quotient rule, chain rule, integration by parts identify operations with words.

Words to name or identify formulas and operations is a simple way to expand the role of words in mathematics.

B. Talking about Numbers, Amounts and Quantities

Following the appearance of algebra and logic in and apart from mathematics education, the oral element introduce above may expand. The first skill for algebra which I introduced in fall 1983 emphasized that we can talk about numbers, amounts and quantities without doing any arithmetic and even apart from or parallel to the use of letters and symbols to denote them. So we may talk about and describe numbers, amounts and quantities as being known or not, measure-able or not, private or not, confidential or not, forgotten or not, constant or not, and varying or not. Some of these descriptive terms are not usually part of mathematics, but their presence suggests how we may understand and explain the concepts of a numbers, amount or quantity being known or not, constant or not, or variable or not. I have written an essay What is a Variable that informally introduces and expands the concept before algebra begins. While pure mathematics and logic introduce technical definitions of what is a variable, those definitions are too complex for people just learning algebra. In the first instance, when a letter that denotes a number, amount or quantity that is unknown, constant or variable then the letter too will be called an unknown, constant or variable, respectively. And in the use of formulas, letters often denote numbers or measure whose value is to be given or found. What I am advocating here is a simpler use of language, a use closer to everyday use. While numbers, amounts and quantities may be described or talked about apart from algebra or the use of letters to denote them or their values, doing so while denoting them by letters or symbols expands the role of words in algebra and so in mathematics.

C. Using Formulas etc Forwards and Backwards.

Every formula met in mathematics, accounting, science, technology etc may be used directly and indirectly, that is forwards and backwards.

This message needs to be given explicitly and early in secondary mathematics. Otherwise the underlying skill become part of the hidden, or silent and unspoken, agenda in mathematics courses.

First Site Example

Volume 2, Chapter 10, in discussing Direct use of A =WL assumes W and L are given. Indirect use assumes A and one of W and L is given, and leads to the calculation or formulas W = A/L or L = A/W. The explanation of those formulas is a step towards algebraic reasoning - the direct and indirect or forward and backward use of formulas.

Chapter 10 before the forward and backward use of a formula goes further in showing how to describe a the calculation of a box V = H(WL) and show how to employ substitution (a new concept for students) to go between this formula and V = HA where A = WL. Details are given in the chapter. The details may be easier to grasp if numerical examples are added to this exposition.

Algebraic Exercises:

1. Find a formula for the area of square in terms of its perimeter (easy)
2. Find a formula for the area of circle in terms of its perimeter (easy)
3. Find a formula for the perimeter of square in terms of its areas (harder)
4. Find a formula for the perimeter of circle in terms of its areas (harder)

The exercises will be easier after reading the first sections of Chapter 15 and Chapter 14 in Volume 2, Three Skills for Algebra.

Remark. In arithmetic to calculus and beyond in mathematics and the physical sciences, and in the use of implication rules in or from logic, all rules and formulas will be used forwards and backwards, some time bothways in the same problem, especially those involving proportionality constants. In the latter, finding one represents a backward use of a formula, after which the formula may used forward or backwards. use.
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