Expressions and Formulas: Standard Notation
In this series, I’ve been exploring issues of mathematical notation and their impact on student learning in school. In the last installment I veered a bit from the straight path and showed a way to denote negative numbers that is very familiar in computer science and yet completely missing from school math.
In this post, I’ll get back to the center of mathematical notation in middle school: expressions and formulas. Before embarking on various alternative notations in subsequent posts, let me use this post to review the standard notation and all that it implies.
First, a clarification on terminology. In an equation like , we call an expression. And as far as I can tell, formula is just a synonym for expression; other people claim that a formula is the kind of thing like F = C*9/5 + 32 where you convert from degrees Centigrade to degrees Fahrenheit by applying a recipe written in mathematical language – I’ll stay neutral on the issue here. Beyond middle school, the whole equation might be seen as an expression, in this case one with a value of true or false. The use of “=” as an operator, with possible values true or false, is one I deliberately skipped in my earlier posts about the overloading of the equals sign.
In middle school, we know what expressions consist of, since we’ve all learned the Please Excuse My Dear Aunt Sally thing about the order of operations. By implication, we’re looking at the composition of exponentiation, multiplication, division, addition and subtraction, where composition is done by use of parentheses. And this pretty much matches what students see in middle school, with only the occasional absolute value |x|, square root √x or trigonometry functions sin(x), cos(x) and tan(x) adding a wrinkle.
Since in the standard order of operations the last action we do is add or subtract, we can coin a name for those things that we add or subtract. Actually, there is an already existing name that will do fine, it is called a term. So an expression is either a term or it is the sum or difference of terms. According to the order of operations, what comes before additions and subtractions is multiplication or division. This means that these terms, these things we are adding or subtracting, are themselves products or divisions. The things we are multiplying or dividing, the are commonly called factors. So a term, in turn, is either a factor, or a product or division of factors. A factor, in turn, is either a primary, or a primary raised to an expression. Finally, a primary is either a number, or a variable, or a parenthesized expression.
What I described in words in the previous paragraph, is usually expressed with what is called a generative grammar:
expression : term
expression: expression + term
expression: expression – term
term: term × factor
term: term / factor
factor: primary expression
primary: ( expression )
The little grammar above assumes that it is clear what a number looks like, and what a variable looks like. This grammar can be extended to deal with roots, absolute value and trig functions by adding rules for primary, e.g.
primary: | expression |
primary: trigfunction ( expression )
These rules together make for a language of expressions that can be uniquely parsed; under broad circumstances, the ×symbol can be omitted and expressions can still be uniquely parsed. This language is compact and terse; it is fairly easy to appreciate why this language won out among professionals and spread throughout the world. Yet just because it is compact and can be parsed uniquely doesn’t mean that it is easy to interpret by student learners. In earlier posts I’ve alluded to the confusion students have between parentheses and multiplication: students see parentheses, and are primed to expect that there is a multiplication involved. Sometimes there is, as in , and sometimes there is not, as in , and yet I’ve routinely seen even ‘good’ students treat as some kind of multiplication. The standard method of writing expressions is linear and monochrome. We can do better in a learning environment, and subsequent posts will explore various alternatives.