The Basis of Basic Algebra: PEMDAS or Order of Operations or ??

My professional work focuses on helping students who have generally completed their K-12 mathematics though they are not able to place in to a college level math course.  Based on doing this for a long time, I share the following conclusions:

• Most students (even those who can place ‘college ready’) have dismal abilities and understanding about arithmetic relationships.  However, this (perhaps surprisingly) has little impact on their success in college.
• The primary issues preventing success in college (in terms of quantitative outcomes) deals with fundamental concepts of basic algebra: expressions & simplifying; equations & solving.  The most fundamental of these issues is order of operations.

So, let’s make this concrete.  We are doing really basic expressions and equations in our Math Lit course; one of the problems for today’s group work was the following:

Solve   15 = -3(y + 2) – 3

Because we are finishing up a unit for a test, we have been doing a lot of distributing in class.  We’ve talked about concepts of order of operations as it relates to expressions like the left-hand side of that equation.  In spite of that, students claim that:

(y + 2) = 3y  (because there is a 1 in front of the y)

Now, it is very easy to tell a student that their work is incorrect; it’s easy to say “you should distribute first” (though we don’t always want to distribute).  I am more interested in diagnosis … WHY is that mistake there?  What understanding needs to change to know what to do with all problems we will see?

It is very disturbing to learn that many “bad things” students do are based on being told in the past to “use PEMDAS”.  In this problem, students honestly think that they have no choice — they MUST combine y and 2; since they know that y=1y, they add 1+2 to get 3y.  Somewhat reasonable … if the requirement to combine were true.

We need to avoid misleading (or incorrect) rules about calculating which lack a sound mathematical basis.  PEMDAS is such a rule; I have written before on this, so I won’t repeat myself (not too much anyway).  See prior posts:  PEMDAS and other lies 🙂 , More on the Evils of PEMDAS! and What does ‘sin(2x)’ mean? Or, “PEMDAS kills intelligence, course 1”.

Our students would be better served if we focused on the relationships between operations and how that helps with ‘order’ questions — even if we don’t present such complicated (and contrived) problems.    Simple problems are sufficient for much of what we need students to learn:

• -5²  and (-5)²
• 4x²   [does the square apply to the 4?]
• 8+2(x+3)   vs 8+2(6+3)

Algebra is about properties and choices.  Students focus on what they have been told is really important, and PEMDAS is often in this category.  This conflicts with the goals of basic algebra — and with most mathematics our students will work with.  I would rather spend an hour in class exploring the 3 different ways to solve the equation 15 = -3(y + 2) – 3 than in redundant examples drilling “one way” to simplify or “one way” to solve.

Correct answers from PEMDAS are worse than worthless.  Success in basic college math and science classes is based on understanding (thoroughly) a few concepts.  Nobody should be ‘teaching’ PEMDAS, because we should never deliberately harm our students.  Understanding is what enables students to reach their dreams; quick fixes — whether in the form of PEMDAS or ‘co-requisite remediation’ — are more about correct answers than they are about student success or mathematics.

Are you so focused on ‘correct answers’ that you either limit your student’s knowledge or unintentionally cause them harm?  As I tell my students:

Correct answers themselves are almost worthless.  The value comes from our understanding.

2 Comments

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  By Eric Neumann, November 18, 2019 @ 1:39 pm

  Yes!!! I couldn’t agree with you more. And yet… Do your students push back on this as hard as mine do? When I stand my ground and insist that PEMDAS is wrong, and insist that, when an approach gets you the right answer sometimes and the wrong answer other times, it’s probably not a good approach, and insist that it’s valuable and in fact necessary to actually know definitions of these math jargon words (what does simplify mean, after all? What is a factor?) … a handful of students trust me and try to leave the safety of their trusty PEMDAS (And “a negative and a negative is…” and simplify means to “break it down”, etc.), but many simply write me off as a self-righteous intellectualist (though not in so many words) and bide their time to take the course the following semester with a more traditional instructor.

  How do you get buy-in from your students on this when we are still in the minority? How do you ease the pain of cognitive dissonance and accepting that the way they’ve always done it is not, in fact, the best (or even a good) way.

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  By Jack Rotman, November 19, 2019 @ 8:28 am

  I don’t actually see resistance on this, partially because I usually frame my initial critique in terms of ‘aunt sally’ (and what did she do that we had to ‘excuse’ her). I am, perhaps, too gentle in my approach — I suspect that the typical cognitive reaction is that “this guy is funny, but I am not letting go of PEMDAS” though little is said. Many students still write ‘pemdas’ when doing tests.
  My strongest reaction against PEMDAS comes from the routine error of saying that “5(2x + 1)” is “5(3x)” … because we ‘must’ do within parentheses first; very resistant to correction.

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