

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: C \cdot \frac{9}{5} + 32 = F 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:


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

\begin{aligned}  C \cdot \frac{9}{5} + 32 &= F \\  C \cdot \frac{9}{5} &= F - 32 \\  C \cdot 9 &= 5 \cdot (F - 32) \\  C &= \frac{5}{9} \cdot (F - 32)  \end{aligned}

— 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.

7 Responses

  1. I agree with you, and I think the same can be said for much of mathematics. For example, which when I Google ‘mathematics’, it returns the definition, “(1) The abstract science of number, quantity, and space. (2)The mathematical aspects of something: ‘the mathematics of general relativity’ “.

    I’m pretty sure that defining math as a science kind defeats the purpose of mathematics. Maybe if the “experiment” is the proof, then yeah. But if the experiment is mere observation where if we see enough examples we can conclude that its true. Plus this definition leaves out what’s probably the most important part of mathematics and that’s that its based on logic and reason. Most students in a set theory class don’t do much counting until they get to set cardinality and operations on sets, but there’s a lot of math going on before that.

  2. Sometimes it seems like the mere usage of symbols not only does not immediately indicate algebra, but is an indication of that a lack of algebraic thinking is about to take place. That formality often seems to create a barrier to the logical thinking that we all are capable of.

    If you google “proof without words” there are many examples of ideas that can be expressed nicely with diagrams. It would be good to see more problems (and solutions) that were less about language and more about math.

  3. Would you mind adding you definition of Algebra to this document? And of course you should also add anything else you can think of:



    • I’ll think about it (I don’t know if it is clear from this/other posts of mine, but I get unusually anxious about this sort of thing) but I’ll put something up when I feel like I have it right.

      • No pressure! All posting is anonymous and it will get edited. Although if debate makes you anxious this might be a good one to sit out and watch- we’ve had some good conversations about slope and solving equations! I’ll add the word to our vocab list just to see what happens.

  4. […] himself seems to have a confused idea of what algebra is like, but he’s not alone. (See: my prior post on what is algebra and why you might have the wrong idea.) Also of note: some countries don’t bother to factor quadratics. (I haven’t made a map […]

  5. […] get too excited yet — the big one (Why Algebra Works, you’re best off reading this post for an idea of how it is being written) isn’t […]

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