I haven’t met students for whom parentheses show up as problematical. Students seem to get the idea of parentheses as indicating a preferential grouping – the idea that whatever is inside of the parentheses, however many pieces it consists of, that that whole thing counts as one thing with regard to the rest. This is a key idea, and I’ve been surprised at how well students get that. Yet what I’ve seen indicates that students mostly get it in a passive way: they can interpret the parentheses that somebody else wrote. This doesn’t translate into facility with active use of parentheses. I’ve often seen students write something like when they meant what you and I would write as . Yet the gap between passive understanding of parentheses and active facility with using parentheses doesn’t need a sophisticated explanation: when you ask kids about their they are likely to report that they knew what they meant. For students, active mathematical notation is mostly for scribbling on a piece of scratch paper in response to a problem from a textbook, and isn’t seen as a way to communicate mathematical ideas among buddies.
Parentheses are also used to indicate very different mathematical ideas: they are used in the function notation f(x), and for coordinates (3,4) and in a number of other places. For this post, I will ignore all these other uses for now, and focus solely on parentheses as used in mathematical expressions.
To more fully appreciate the job parentheses play – and play so well – I think it is useful to note that there are alternatives to the use of parentheses. One of these alternatives is known by various names, including postfix notation, RPN (Reverse Polish Notation) or “Hewlett Packard calculator notation,” the latter after a style of calculator design built and marketed by Hewlett Packard, with a very loyal following. On one of those Hewlett Packard calculators, you would input as follows: . If you care, you can learn how this works and why it works here, suffice it say here that it does work, and hence that parentheses aren’t needed for this job we always thought they were needed for. The strange name “Reverse Polish Notation”, by the way, was coined in honor of a Polish mathematician, Jan Łukasiewicz, who put the operators in front, like this: and who may have been the first to introduce a parentheses-free notation.
A very different approach that avoids parentheses, specifically also including parentheses as used in f(x), was introduced by Haskell B. Curry in the late 1920s and relies on combinators. The Wikipedia article I’m linking you to mentions an earlier mathematician, Moses Schönfinkel, who is said to have preceded Curry in this invention. Curry’s work, together with Alonzo Church‘s work on the Lambda Calculus was a fairly direct influence, much later, on the design of programming languages for electronic computers. I had the pleasure of meeting Curry and Church, both at very advanced age, at a convention on functional programming languages in the 1980s, where they were being honored. Curry was pleased but sort of non-plussed with the all the fuss – computer languages weren’t exactly on their mind when they did their work in the twenties. Church looked more than anything like a giant child, taller than everybody else and with the biggest feet I’ve ever seen on anybody. Curry died soon after, Church lived another fifteen years. And lest you think that somebody paid these folks a lot of money, during the Depression, no less, to come up with high-falutin’ tricks to avoid a few parentheses here or there – the un-necessity of parentheses was a side-effect of what they were working on, and by no means the main course.
Yet both these fundamental ways to handle grouping without parentheses are artifacts of the line, of a linear way of expressing what is going on. In modern terms, we might say it reflects a concern for what can be done on the keyboard. If you free yourself from a linear way of expressing, there have always been ways of indicating grouping without parentheses. In this example:
no parentheses are used. And still, it is clear that x + 1 is done first, the result is divided by 2, and the result of that is the subject of the square root. On a single line, on the keyboard, you can write the same thing as sqrt((x+1)/2). This is an example of what are called “nested parentheses”, which some people (teachers, especially) avoid like the plague.
Yet kids rarely have any trouble picturing parentheses as the “remnants” of the bag, like an ellipse, that holds all the relevant stuff.