What is Algebra? (and why you might have the wrong idea and why it is important)

I’ve been frustrated lately reading definitions of algebra along the lines of this:

Look: the mere usage of variables or symbols does not immediately indicate algebra. Compare two ways of writing the Celsius to Farenheit formula: vs. “Multiply by 9, then divide by 5, then add 32.” Mere calculation is going on. This is arithmetic.

Keith Devlin gets the essence of the problem right, succinctly, with:

In arithmetic you reason (calculate) with numbers; in algebra you reason (logically) about numbers.

Taking the Celsius to Farenheit formula, and using reasoning to transform it into a Farenheit to Celsius formula —

— now that is algebra. However, the symbols are not required.

To get from Celsius to Farenheit, you multiply by 9, then divide by 5, then add 32. To get from Celsius to Farenheit, you need to do the inverse operations in reverse order. Hence, you subtract 32, multiply by 5, then divide 9.

As Keith Devlin points out, people were using algebra for 3,000 years before symbolic notation.* The two are not equivalent.

Symbolic notation is a massive convenience and once learned it should be used. However, there are good reasons that students in the process of learning should use the real definition of algebra, not the artificial one defined by symbols.

1. You can reason using algebra with words.

The Celsius / Farenheit conversion already given is an example. Most students naturally understand the logic where reversing “add 5” requires “subtract 5” and reversing “add 5 then multiply 6” requires “divide 6 then subtract 5”. Moreover, in this fashion students tend to understand the logic of inverses, not just the mechanics behind a raw procedure.

The students afraid of mathematics tend to like words. It is a comfortable segue for them.

2. You can do algebra without variables.

As practice, it is extremely helpful to perform algebra — that is, reason about arithmetic, not just do arithmetic — with no variables at all. I see many textbooks that introduce the distributive property like this:

Here are two ways to find the value of 6(29 + 24).
Method 1: 6(29 + 24) = 6(53) = 318
Method 2: 6(29) + 6(24) = 174 + 144 = 318
Thus, 6(29+24) = 6(29) + 6(24). This illustrates the Distributive Property of Multiplication over Addition.

The exercises that immediately follow, however, dive straight into variables:

Write each product as a sum or difference.
22. 4(7-4r)
23. (3c + 9)15
24. 3y(7y – 8)

[From Jacobs’s Introductory Algebra 1, Fourth Edition.]

Students can linger on pure numbers for a while, thinking intuitively and using geometric models. The rush to variables seems to occur because of the feeling that without variables it isn’t algebra yet. Google is wrong. Variables are not algebra.

3. You can do algebra with alternate representations.

Elementary teachers are familiar with the question mark substitution

5 + ? = 8

which gives the start of sensing (as John Derbyshire puts it) “a simple turn of thought from the declarative to the interrogative”. However, the question mark is still a symbolic representation.

Rather bolder steps can be made with algebra-as-geometry (for example, tape diagrams, which are now fairly standard in elementary school but usually forgotten by the time high school algebra rolls around):

or even algebra-as-graph-theory-puzzle (solid lines mean multiply, dotted lines mean add):

It is bizarre that something as simple as a definition can restrict thinking, but after reading many textbooks I’m starting to be convinced it is the main obstacle to opening new frontiers in the explanation of algebra.

* He seems to be excluding Diophantus’ Arithmetica from the 3rd century. However, the symbolic notation therein wasn’t really picked up until the 16th century, so his claim still holds.

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