Monday, October 21, 2013

Are Parentheses Multiplication?

Are parentheses multiplication? My remedial algebra students will pretty universally answer "yes" to this question; I guess they must be taught that explicitly in other courses. I'm pretty damned sure that the answer is "no", and I try to pound it out of them on the first day of the class.

Even professional researchers exploring common mistakes in algebra education are prone to saying "yes" to this question, for example:
Misconceptions: Bracket Usage -- Beginning algebra students tend to be unaware that brackets can be used to symbolize the grouping of two terms (in an additive situation) and as a multiplicative operator [Welder, "Using Common Student Misconceptions in Algebra to Improve Algebra Preparation", slide 7; references Linchevski, 1995; link]
But are parentheses a multiplicative operator? It seems clear that the answer is "no". Now clearly all of the following are multiplications of a and b: ab, (a)b, a(b), (a)(b), etc. But notice that the parentheses make no difference at all in this piece of writing. These are multiplications because of the usage of juxtaposition; any two symbols next to each other, barring some other operator, are connected by multiplication. Obviously, if there were some other written operator like + - / ^,  between the a and b it would be something different; but granted that multiplying is probably the most common operation, we read the absence of a written operator to indicate multiplication.

The chief problem with telling students that parentheses indicate multiplying is that they then routinely get the order-of-operations incorrect. Assuming a standard ordering ([1] inside parentheses, [2] exponents & radicals, [3] multiply & divide, [4] add & subtract), students want to perform multiplying with any factors in parentheses in the first step, before exponents. One of the first and frequently repeated side-questions I ask in my class is, in an exercise like "Evaluate 2+3(5)^2" -- "Yes or no, is there any work to do inside parentheses?" On the first day of algebra, almost the entire class will answer "yes" to this (and want to do a multiply), at which point I explain that the answer is actually "no". If there is no simplifying inside the parentheses, then the first piece of actual work will be to apply the exponent operation. And that's all that parentheses mean. (There is of course a multiplication here -- not because of the parentheses, but because of the juxtaposed 3, and it must take place after the exponent operator.) A majority of the class will pick up on this afterward, but not all -- some proportion of a class will continue to say "yes" and be confused by this particular question throughout the semester. (As another example, some students are prone to evaluate something like "(5)-2 = -10" for this and other reasons.)

Whether a factor is juxtaposed next to something in parentheses or not is irrelevant to the multiplication; parentheses are a separate and distinct issue. What say you? Have you ever said that the parentheses symbols actually mean multiplying?


  1. I've never seen the misconception you raise, but there is a common confusion that is inherent in standard math notation: is f(g) the function f applied to value g, or f*g? That is, are parentheses just a grouping symbol or are they a marker for a function call?

    There is good reason why most computer languages do not interpret juxtaposition as multiplication.

    1. Indeed, whenever the same symbol gets used for multiple meanings you get problems. Same could be said for a -1 superscript (reciprocal or inverse trig function), an overbar/vinculum (fraction, grouping, or repeating decimal), etc. But I suppose the real reason for no juxtaposition in programming is that it would be ambiguous with multi-character variable names?

    2. FYI, I totally saw this misconception again today in the first class of a college algebra course. On the board: 6÷2(1+2). I ask: Is there any operation inside the parentheses? Yes. I ask: Exactly what operation is it? And while most of the class says "addition", one student audibly says "multiplying", and then looks around surprised at what the rest of the class said.

    3. is the answer to this problem 1?

    4. No, it's 9. In detail: 6÷2(1+2) = 6÷2(3) = 3(3) = 9. Recall that any operations of the same precedence are read left-to-right across the expression (as for the divide-then-multiply in this case). See also Google

  2. "Evaluate: 2+3(5)^2". If solving this problem myself, I would start by rewriting it (at least in my head if not on paper) as: "Evaluate: 2+3(5^2)", and perhaps further as: "Evaluate: 2+(3*(5^2))". This makes it clear that the operation inside the (innermost) parenthesis is exponentiation.

    Similarly for "Evaluate: (5)-2=?", I would recognize this as "Evaluate: 5+(-2)=?", and rewrite it as simply: "Evaluate: (5-2)=?" (with or without the parenthesis).