I saw this on Facebook today, and it really set me off: (I actually really dislike the accompanying article’s explanation, as I think it still perpetuates the problem instead of ACTUALLY developing the depth of understanding necessary):

Problems like this are why I appreciate the Boss Triangle & the NCTM’s article, “The Problem with PEMDAS.” It drives me nuts when people try to talk about math as being arbitrary based on examples like this. What I REALLY think these “viral math problems” illustrate is a lack of numeracy and a lack of understanding of the relationship between the mathematical operations AND an over reliance on mnemonic devices/acronyms to remember procedures… but, those mnemonic devices actually mask the inherent relationships necessary to figure it out (because, as Pam likes to say, math is figure-out-able).

If you understand division to be “multiplying by the reciprocal,” and you convert the divison by 2 to be multiplying by 1/2 (and thus .5), then there is literally no question that the value is 9.

You get 6 * .5 (1 + 2) as your new expression, and you now have flexibility in how you evaluate it: whether you start by taking half of 6 (which is 3) & then multiplying it by the sum in the parentheses (also 3), the product of which is 9, or you distribute the half to the two terms in the parentheses and get 6 (.5 + 1), which is 6 (1.5): still 9. There are no longer possible paths to get 0, 1, 3, or 6, which are the common “wrong answers” when trying to evaluate the original seemingly ambiguous expression. (And, if a mathematician WANTED you to divide by the PRODUCT of 2(2+1), they would have used a vinculum, often known as the “fraction bar” and put that whole expression in the denominator, with only the 6 in the numerator – grouping symbols also help mathematicians to be less ambiguous I. their intentions).

The REAL problem here is that people do not develop a deep enough understanding of the relationship between the operations to understand the ORDER of the operations. And instead of taking the time to fully develop those relationships, we fast forward students through relationships and to procedures.

The order of operations is NOT arbitrary, it is based on two key relationships:

Addition & subtraction are inverse ops & Multiplication & division are inverse operations (and radicals & exponents are inverse operations), and thus you can rewrite any radical, division, or subtraction as its inverse operation. For subtraction, use the additive inverse with addition or for division, use the multiplicative inverse (often called the reciprocal) with multiplication. You can even write radicals with fractional exponents: in this case, you indicate the inverse operation by using the multiplicative inverse of the exponent (i.e. root 2 is with an exponent of 1/2, which is the multiplicative inverse of 2, which is the power you’re trying to “undo” by taking the square root).

Thus, in considering the order of operations, you actually need only consider exponents, multiplication & addition. This is step 1 for deepening your understanding, because it explains why “PEMDAS” should actually have the MD and AS on distinct levels of the hierarchy: they are equivalent to their inverses in the order because you can eliminate division and subtraction by replacing them with M & A.

Secondly, the operations are prioritized in order of power: more powerful operations are done first (hence exponents are done before multiplication and multiplication before addition). I define powerful here to mean the operations that would change a number more if the specific numbers in question were kept constant: for example, ab will change a more than a+b, and a^b will change a even further (certainly for the integers, and even the rationals, but I think this is true for all real numbers). (I use “change” in a large part here to mean the difference between your sum or product and the original a).

I like explaining this relationship as being about power, because then it holds true even with the rationals, but students with less experience with the operations may find it helpful to also consider the fact that exponential notation (with whole numbers only) can be rewritten as repeated multiplication, and multiplication (of whole numbers) can be rewritten as repeated addition (of whole numbers). I hesitate to ONLY use that (even with kids who have less well developed relationships between the operations), because this type of understanding of the operations limits a more in-depth understanding of rational number arithmetic (i.e. addition, subtraction, multiplication, division, and exponentiation of numbers with fractional/decimal/negative parts). But that’s a whole separate issue.

Grouping symbols, like parentheses, often get thrown in to the order of operations as “coming first,” but actually, that is not at all a correct or complete understanding of the meaning or use of grouping symbols!

I also think it is confusing that we use parentheses (and brackets and braces) to group expressions together (within) and imply multiplication between quantities (one inside and one outside the parentheses), as people have to first ask themselves whether the parentheses mean multiplication or “group this expression together and treat it as a single quantity” – or worse, both meanings simultaneously!

Now, the truth is, grouping symbols are NOT required to be dealt with “first,” rather, it is that the terms/expression contained within a grouping symbol must be treated as a single quantity rather than each part of that expression treated as separate quantities that are “free” to be operated on by other parts of the overall expression.

That’s why you need to either simplify the expression inside the parentheses and then multiply OR use the distributive property to distribute the “divided by 2” (as a factor of 1/2) before you can do anything to the 1 and 2 inside the parentheses (in the original expression that inspired this post today). This could bring me to a rabbit whole about how we don’t deal with quantities that are not single numbers well, but that’s a whole different post! For now, I will simply say that experience with growing patterns and treating “chunks” of the visual as a being represented by a single can support the development of this type of understanding.

Which brings me to the last point about the order of operations: you can use the properties of numbers to make equivalent expressions: associative, commutative, distributive properties… all of which apply to MULTIPLICATION & ADDITION (but not division & subtraction), which is the other reason why if you convert the division and subtraction to multiplication and addition, the expression becomes easier to evaluate (and not as ambiguous). Both subtraction & division specify a left to right order inherent in how we record them (we do something differently to the subtrahend and the minuend or the divisor and the dividend), but because addition and multiplication are both commutative and associative, we simply call ALL of the chunks of them by the same name: factors in multiplication and addenda in addition.

These “viral math problems” are only ambiguous to evaluate if you DON’T understand the RELATIONSHIPS between operations and instead rely on acronyms and mnemonic devices to remember “what to do.”

I think if we spent MORE time on developing an understanding of these relationships between numbers and relationships between operations, we wouldn’t need to “teach” the order of operations at all: it would be a logical conclusion to these questions. It would be “figure-out-able.” In fact, if I were to sum up what I’ve learned from my work with Pam Harris, I think it directly connects:

If we develop a deep understanding of RELATIONSHIPS & QUANTITIES, then math is figure-out-able.

(And, typically, the development of that understanding comes from repetition of a relationship or a quantity & analyzing the patterns seen in the reptition OR looking for structure within quantities or relationships>> now I’ve connected the three avenues of thinking from Amy & Grace to Pam Harris’s work).

Oh, I also just felt some sparks connecting my ideas from this to the ideas that numbers represent either quantity or a relationship & the ideas of “math as a second language” & the ideas of adjective-noun that were recently shared with me on twitter… but that connection will have to wait for another post.