Notations for Sequencing and Nesting (Parentheses)
In this series, I’m exploring mathematical notation as it is used in schools, and how these notations help or hinder students’ understanding of mathematics. In the first installment of this series, I spoke positively about the practice in US schools to write negative numbers with a raised minus sign. In the second installment, I explored the use of an asterisk (*) as an alternative to the multiplication sign starting in middle school or slightly before.
In this post, I want to explore the various ways that sequencing of operations is indicated in schools. But to set the stage, let’s first take a look at how we indicate to normal adults what sequence of operations to perform. Here is an example, from one of the first Google entries of the “order form” search:
In such an order form, certain information must be provided, like Item Number and Description, that indicates what kinds of things are being ordered. Then there are the columns labeled Quantity, Price and Amount. You will most likely recognize this as requiring the Amount, on a given row, to be determined from the Quantity and the Price by multiplication. The row labeled Subtotal will then be obtained from the Amounts by adding them all up. To this Subtotal, a Discount will be applied, presumably some fixed percentage of the Subtotal, or some sliding scale. The row labeled Total will be obtained by subtracting the Discount amount from the Subtotal. The row labeled Tax (CO) presumably is a known percentage of the Total; the row labeled Shipping may be a fixed amount or may be found from a table. Last, the AMOUNT DUE is obtained by adding Total, Tax (CO) and Shipping amounts. In this set up, a sequence of actions is implied, leading from Quantities to the AMOUNT DUE through the Prices of the Item Numbers. It makes little sense to compute the TAX (CO) before the first Quantity has been multiplied by the first Price. On the other hand, there is some latitude in what order the multiplications of Quantity and Price are done, as long as all or done before the Subtotal is attempted.
I suggest we’ve all seen variants of this order form, and thus are familiar with the implied sequencing of actions in it. Note that this is communicated to us in a relatively standard way, and yet there is no parentheses nor My Dear Aunt Sally to be found.
In secondary school, we may encounter formulas that look like ((75+10) × 1.09 ) + 4, and to make sense of this, we must understand parentheses and “order of operations”. In elementary grades, it is not very common to see multiple operations combined in a single formula. It wouldn’t be uncommon to see it represented in elementary grades as follows:
75 + 10 = _______ × 1.09 = ______ + 4 = _____ ; and even though many teachers would balk at this (mis)use of the equals sign, students don’t seem to have any problem with the sequencing aspect. They seem to get they are supposed to proceed from left to right: 75 + 10 = 175, then 175 × 1.09 = 190.75 and finally, 190.75 + 4 = 194.75.
Like the order form we showed, there are ways to indicate the sequence of arithmetic operations that don’t look at all like mathematical formulas with parentheses. In an earlier post, I worked with representations like the following:
which can be used equivalently to . Note that each box works on its own input and produces an output in return. The sequence of arithmetic operations is obvious from the way the boxes are connected.
None of these examples is intended to suggest that parentheses are bad, merely that there are many other ways to indicate in what sequence operations should be performed. When you learned the quadratic formula, did you ever notice that it doesn’t have parentheses? If you’ve forgotten: one of the solutions of the equation is . Here, the length of the horizontal line of the division is used to indicate what belongs to the numerator; similarly, the length of the horizontal part of the square root sign is used to indicate what belongs ‘inside’ of the square root.
Now let’s look at the kinds of formulas where traditionally we might use parentheses:
In standard notation, the formula on the top would be written as ; the bottom one would be written as . In the picture, parts of the formula have been encapsulated in bags – like crude ellipses. What the picture on top tries to suggest is that the formula is basically a sum of two parts. One part is , the other part is . Similarly, the picture on the bottom tries to suggest that the formula is basically a product of two parts. One part is , the other part is .
We can think of parentheses as an in-line version of the ellipses used in the picture: a rendering of the left part and the right part of the ellipses while leaving out the ceiling part and the floor part. And, indeed, the formulas could be written as and , respectively.
From my observations with students in middle school, the notion of using parentheses is not that difficult to grasp. More difficult for the students is getting facile with when these parentheses are left out. In the standard notation, parentheses are only written “when needed”, by which is meant that parentheses are only written if leaving them out would alter the meaning of the formula. In the standard system of mathematical notation, there is a fairly good reason for wanting to eliminate as many parentheses as possible from a formula: when you get too many parentheses, especially if they are nested, things become a bit confusing. How easy is it to make sense of the following formula?
x = (-b + sqrt(b*b-4*a*c))/(2*a)
This is another rendition of a solution to the quadratic equation. It has three left-parentheses: “(“, and three right-parenthesis: “)” – and each left-parenthesis is matched with a particular right-parentheses. The matching is relatively easy to do with a pencil if the thing is printed on paper, but not quite so easy to do just in reading it.
The adult’s insistence on dropping all unneeded pairs of parentheses has an unexpected consequence for students: students start to think of parentheses as having something to do with multiplication. They see and are asked to simplify it and they expand it to , but then if you ask them to simplify , they likewise treat it as a multiplication instead of an addition. The sight of the parentheses seems to make them think that multiplication is in order – after all, that’s where they have seen parentheses the most: in factors. Sometimes teachers explicitly encourage students to think of the parentheses as indicating multiplication: they’ll tell their students that has a hidden multiplication by 1 in it, so that the formula really means , which is then subject to the distributive property and expanded to , which then becomes and finally .
My proposal is to introduce the bags early and frequently as ways to annotate a formula and indicate the order in which things happen. Bags can be nested, meaning you can have bags inside of bags. This use of bags, as in the picture below, can start in elementary grades as multiple operations are introduced.
Soon after, ‘real’ parentheses can be introduced as an abbreviation for drawing the bags, as follows:
After a while, kids can imagine the whole bags, just from seeing the parentheses. The teacher can draw bags or parentheses interchangeably, while noting that the textbook only uses parentheses.
In middle school, I propose teachers use bags and parentheses whether they are “needed” or not, whenever their use can clarify the structure of a formula. Teachers should use parentheses both for multiplication and addition, so students don’t automatically associate parentheses with multiplication as they tend to do today.
At the end of middle school, students can be eased into the standard notation, through being encouraged to write parentheses only when needed, like the textbooks already do.