It is amazing to me that after 2 1/2 years on this journey of relearning a lifetime of mathematics rules in a conceptual way that I still haven’t asked all the questions yet. Like an onion, I peel back one layer of understanding only to find yet another piece of learning to unearth. It’s both beautiful and mind-boggling at the same time. The most recently discovered layer of this onion is the rule for the Order of Operations known as PEMDAS (otherwise known as Parantheses, Exponents, Multiplication, Division, Addition, and Subtraction). While this Order of Operations rule has long been a part of my mathematical life, I just recently realized that I had yet to ask the question, “Why does this work?” And, as all good questions do, it got me thinking about other questions, like…
- “Why does multiplication and division come before addition and subtraction?”
- “Why do parentheses come first?”
- “Why must multiplication and division be solved from left to right?”
- “Why is the same true for addition and subtraction?”
- “Does it really have to be solved from left to right????”
It was like a mathematical floodgate had opened in my mind. Once I started asking questions, I couldn’t stop. I needed to do some exploring. I needed to make some sense of these questions. I decided to solve an expression that required the Order of Operations, but pushed myself to answer the “whys” of every move.
When I thought about the expression above, I knew the following:
- If I work from left to right, the solution will be 3
- If I solve using Order of Operations, it will be -47.
Big difference, right? How could I prove to myself that I needed to multiply before I subtracted? First, I needed to know what the expression actually meant. When I put it in words, it sounded like this-
Twenty-four groups of three, less twenty-five
This wording was key for me to say for certain that you must work with your “groups of” (multiplication/division) first before adding or subtracting from those groups. Knowing the meaning of the operations was critical to making sense of the meaning of that expression. Alas, I finally had an answer to go with my heaping pile of questions.
Next, I aimed to determine why the parentheses must be solved first. I was struck by Wolfram Alpha’s definition:
Parentheses are used in mathematical expressions to denote modifications to normal order of operations (precedence rules). In an expression like , the part of the expression within the parentheses, , is evaluated first, and then this result is used in the rest of the expression.
In mathematics, some items must simply be agreed upon like the symbol = meaning “the same as”. Piaget calls this type of learning Social Knowledge. One learns this type of knowledge by being told or through demonstration. It is not something that can be discovered. The parentheses are grouping symbols and therefore must be evaluated first. Another question answered!
I began to feel that I was making progress. This particular layer of mathematical learning was beginning to gain momentum. I was left with my last group of questions. Must multiplication and division be solved from left to right? How about addition and subtraction? Could there be exceptions to this rule? I decided that finding a non-example would be all the justification I needed to make my decision. But where to start?
As I did some Googling (I know, I know…I was desperate!), I landed upon an enlightening article entitled 12 Rules That Expire in the Middle Grades written by Karp, Bush, and Dougherty. This article claims that the PEMDAS rule expires in 6th grade when students are given expressions that do not require this rigid sequence. This example is given to support the claim:
For example, in the expression 30 – 4(3 + 8) + 9 ÷ 3, there are options as to where to begin. Students actually have a choice and may first simplify the 3 + 8 in the parentheses, distribute the 4 to the 3 and to the 8, or perform 9 ÷ 3 before doing any other computation—all without affecting an accurate outcome.
There it was. This one non-example was all I needed to prove that solving does not always move from left to right with addition and subtraction or multiplication and division. In addition, the article goes on to state that PEMDAS limits students to thinking only about parentheses, when there are additional grouping tools like brackets and horizontal fraction bars that must also precede the other operations for the same reason.
One big question still remained. I needed to understand why the Order of Operations works. CCSS-M 6.EE.2c calls refers to the Order of Operations as the “conventional order”. To some extent, I believe that like many mathematical practices, we had to socially agree upon a procedure. But as I considered it further, I also now see that there is a heirarchy of groupings within the Order of Operations that is quite logical.
Socially we agree that parentheses (and other grouping symbols) take precedence over any other operation. That makes these grouping symbols top of the grouping food chain. Beyond that, exponents can be interpreted as repeated multiplication, thereby creating groupings of groupings. Logically, this is a larger grouping than making or breaking groups through multiplication and division and should preceed that work. Finally, as addition and subtraction aren’t grouping symbols, it makes sense that we would address them last.
Just like with all the other discoveries I have made on this journey from rules to true understanding, I have found that we MUST use our sense of numbers and operations when engaging in mathematics. These “rules” that we were taught all seem to have an expiration date. However, the need for mathematics never expires. Therefore, my quest for deeper mathematical understanding continues.