Tuesday, September 4, 2012

Parenthetically Speaking

I imagine that you, like me, have taught, or retaught, or referred to parentheses in the traditional manner:
Parentheses are used in mathematical expressions to denote modifications to normal order of operations (precedence rules). In an expression like (3+5)×7, the part of the expression within the parentheses, (3+5)=8, is evaluated first, and then this result is used in the rest of the expression. Nested parentheses work similarly, since parts of expressions within parentheses are also considered expressions. Parentheses are also used in this manner to clarify order of operations in confusing or abnormally large expressions. (from Wolfram Math World)
Wolfram goes on to define seven other mathematical uses for parentheses, including interval notation[0,5), ordered pairs (0, 5), binomial coefficients (n; k), set definitions (a,b,c), function notation f(x), etc. With so many uses, it's perhaps no minor miracle when students are able to emerge with any working facility of parentheses at all!

Honestly, I feel for my students. Even to me, mathematical definitions can sometimes seem inconsistent and confusing. Like the difference between terms (things that are added) and factors (things that are multiplied).
 I can hardly keep my own head on straight to describe the number of terms (2) and factors (0) in the following expression:

(To be fair, the first term consists of two factors, each containing two terms each, and the second term has four factors.)

And then recently, I reviewed a prealgebra curriculum that described parentheses as symbols that tell us to "treat part of the expression as one quantity." (from onRamp to Algebra) The book goes on to further implore the teacher to forgo the order of operations description in lieu of the 'one quantity' idea.

I knew that... 

So WHY have I NEVER thought to describe it that way??? 

Parentheses are grouping symbols that tell us to treat the group as a single entity. Period. No confusion. 

A function input is a single argument.
An ordered pair is a single location.
An interval is a single, uninterrupted region.
A matrix is a single array.
A set is a single collection.
A binomial coefficient is a single combination.
An expression in parentheses is a single quantity.

For some reason, this simple statement of an idea that seems so obvious is completely enlightening to me. The Common Core lists "Look for and make use of structure," as one of its Standards for Mathematical Practice. To me, this 'single entity' idea is paramount to the mastery of this standard: the key to seeing structure in long complicated algebraic expressions.

I find a subtle beauty in tiny moments of enlightenment, even if it is only my own. It's got to rub off on someone!


  1. Love this! What a great way to explain parenthesis. I will be sharing this with my students.

  2. Interesting! I had never though of it that way, but I see the appeal. I'll have to keep that structure in mind. I especially how you connected the idea to the Standards for Mathematical Practice. The descriptions of them are long-winded, but despite that everyone's left looking for practical examples to help understand them better. Thanks for sharing!

  3. My light bulb just went on! Thanks for sharing this way of looking at parenthesis. I've been explaining that parenthesis are grouping symbols along with others like square-root, division (---), brackets, etc. In fact, we use GEMDAS instead of PEMDAS. But I like the idea that when something is a group, it is a single quantity or entity. They will relate to this (family, soccer team, etc).