Sunday, August 21, 2011

Is Math Black and White?

Math people like RIGHT answers. We feel comforted in knowing that there's one solution that will be considered RIGHT and all the rest are just wrong.

Tutorial and Practice at
One invention made to insure one right answer is the Order of Operations. In math, the Order of Operations is like the rules of grammar are to writing. You could easily compute or write without following these sets of rules, but to insure you get your message across as accurately as possible, they are good to use.

At a recent math conference, the Order of Operations was put to the test. Take a look at this seemingly simple expression. What is the solution?

20 ÷ 5(2 + 2)

Depending on how you interpret the Order of Operations, there are two possible solutions for this expression. If you consult a scientific calculator, your solution will depend on how the programmers interpreted the Order of Operations. Did you just choose to use the short-cut notation for multiplication? Or are you implying that the 5 is part of the calculation with the parenthetical operation, but left out the brackets to make this implication more obvious? Either way there's some interpreting of intention going on. So it seems, there are two "right" answers.

When the teachers at the math conference were presented with this problem (not aware of this difference in interpretation even among programmers), they wanted to get to the bottom of it. They wanted to know which solution was RIGHT. They couldn't allow for multiple interpretations. In their mind, their HAD to be one right answer -- that is the sole purpose of the Order of Operations. One way was RIGHT, the other way was a misuse of the Order of Operations. And it was doubly important that they know which solution was RIGHT so that they could teach their students the one RIGHT way.

Personally, I find this slight ambiguity as a healthy sign that opens up avenues to communication. Even mathematicians have to be careful with their notations and assumptions. They can't rely on a single rule to insure that their communication isn't misinterpreted. In other words, there has to be some discussion, not just a single missive. People involved in the discussion have to make agreements upon what is meant.

The importance of communication in the math classroom has come up recently in other posts I've made.  (If you'd like to see these posts, check here and here.) I just noticed this recently. If I had been keeping a blog last year, I doubt communication would have come up at all. It's not something that first jumps to my mind when I think of math.

As a student, my experience of math was a solo journey. I worked the problems (completed the worksheets) as if I was solving a puzzle. My problems would be graded based on whether or not I got the solution to that puzzle correct. The only time their might be any communication about this result is if the source of the problems had supplied an incorrect answer. There was always a right or wrong answer, never a gray area.

For me, I'll be on the watch for more gray areas in math knowing that they can lead to very interesting conversations that promote understanding and reasoning.


  1. My favourite math professor at university told me, "It's not true until everyone agrees that it is." This certainly speaks to the notion that math is not as objective as some would claim.

    We could create a definition that would handle this case, but given the ambiguity of language, there will always be some areas which are open to interpretation. Mathematics itself has developed regional dialects, although nothing as severe as the fracturing of our day to day language.

  2. I wouldn't say that 20 ÷ 5(2 + 2) is ambiguous. If you follow the rules literally then you have to read it as (20 ÷ 5) × (2 + 2). There is no commonly accepted rule that gives implicit multiplication higher precedence than explicit multiplication.

    But math is a language meant for human communication. When we read a mathematical expression, we should try to divine the author's intended meaning. In the case of 20 ÷ 5(2 + 2), the intended meaning seems to be at odds with the literal meaning, so it is confusing to the reader.

    My personal feeling is that we should avoid showing students examples of bad mathematical writing. It is better to write clearly, and to teach students to write clearly, than to ask students to interpret unclear mathematical writing.