Algebraic Notation on the Keyboard and Elsewhere

There are two types of algebraic notation. There is the traditional notation we use when we write with pen or pencil. And there is keyboard notation that we use when we key in math without a good math word processor. As long as we keep things simple, there is little difference between the two. And we should keep things simple when we introduce students to algebraic notation.

The best way to learn algebraic notation is by working with formulas. We could start with simple formulas like P = 4S as a shorthand way to describe the rule that the perimeter of a square is 4 times the length of a side. We can then proceed to slightly more complicated formulas like P = 2L + 2W as shorthand for the perimeter of a rectangle is twice the length plus twice the width. We could also write P = 2(L + W) as shorthand for an alternate way to find the perimeter of a rectangle--twice the sum of the length and the width. In these formulas it was convenient to write 4 times S as 4S, and to write 2 times L as 2L etc. If we generalize this idea and use it to express the multiplication of two variables in the same way we get A = LW as the formula for the area of a square. We might get A = (1/2)BH as the area of a triangle or A = BH/2 as an alternate formula.

Working with formulas like this to solve practical problems can help students become familiar with a simple version of algebraic notation. Students are able to work in a setting where they are actually using the notation to solve interesting problems. At this stage, students are only required to apply the formulas by substituting in the appropriate input variables such as S or L or W or B or H.

Mathematics is a deductive science, It is based upon rules that we follow when we do math. But humans learn inductively. We learn from examples, generalizing what we see in specific cases. The inductive nature of human learning can be seen from the way we learn language. We easily pick up words inductively from the conversations we hear and we find ourselves using them correctly in our own conversation, while dictionary definitions help very little if we want to actually use the words that we learn.

In mathematics we learn best from examples where we can see things used in context, rather than from rigorous definitions. Ultimately we will need to work with rigorous definitions, but these will make little sense to us until we have the experience that will make these definitions meaningful to us.

At a slightly more sophisticated level, algebraic notation includes the use of exponents to represent repeated multiplication. These are usually written as superscripts--but without some level of word processing we may not be able to keyboard in superscripts. Instead we use what I call the hat symbol ^ to write a to the n th power as a^n.

Unfortunately, the hat symbol ^ gives us a less readable way of writing exponents than the traditional superscript notation. This pretty much forces us to do something we really should do anyway. Instead of writing the really ugly A = S^2 as the formula for the area of a square, we will simply write it as A = SS. We can write the volume of a cube as V = SSS.

When we get to multiplying algebraic expressions (x + 5)x = xx + 5x has a simple clarity that would be lacking in (x + 5)x = x^2 + 5x even if we were
able to write our exponent as a superscript. It is important to use exponents to represent a large number of repeated multiplications. It is essential that we use exponents if we wish to represent repeated multiplication where the number of factors in the product could vary. But there is a lot of basic algebra where the use of exponents is optional, and we can achieve greater clarity for beginners by not using exponents.