Hot take: I love Common Core. I think it’s great for our students as they engage with mathematics.
Here’s the thing. It’s not entirely fair for me to make that kind of statement without qualifying that I’m a professional mathematician, and I feel confident and prepared to help my own kids, and anyone else’s kids, with their math homework. When I see something new, I still wonder, “why are they asking you to do it this way.” Because of my background, I’m not faced with the worries of most parents – that I don’t know what to do, that it’s not the way I learned it, and that I won’t be able to help. This is a privilege that is part of my identity. I own that.
Parents’ fears are real, and we need to acknowledge that many more people than ever are facing them as we grapple with remote instruction and trying to help facilitate the education of the young people we have at home. In many cases, it’s true. Our children are being asked to learn mathematics in a way that is not the same as the way we learned it ten, twenty, or thirty years ago.
This is, and I know this can be tough to swallow, a very good thing.
This post about the 4th grade problem that stumped the internet came across my Twitter feed a few weeks ago, and it’s been on my mind ever since. It’s not the only one of its kind, but if we look at it, what we gather from the response is that the way we the parents learned order of operations, Please Excuse My Dear Aunt Sally, steered us towards an incorrect simplification of the expression.
When we start with 6÷2(1+2) we know to start with the parentheses, because that’s what the P stands for, so we have 6÷2(3). The M comes next in PEMDAS, so we multiply the 2 and the 3 and we get 6÷6, which equals 1.
And 1 is not the answer.
The problem is that we understood PEMDAS as a linear set of instructions that we were never taught to question and that gave us an order that doesn’t align with the actual conventions of mathematics. PEMDAS isn’t wrong, it’s just misleading, because it could just as easily be PEDMSA (Please Excuse Dear Mama, She’s Anxious?). Multiplication and Division, when not in any punctuation (parentheses) are tackled left to right. Similarly, addition and subtraction have the same weight, and are completed left to right.
So…
6÷2(1+2)=6÷2(3)=6÷2*3 (to emphasize left-to-rightness)=3*3=9.
The real power in the expectations of Common Core Math Standards, though, won’t make PEMDAS any less misleading. Neither will PEDMSA come into fashion. What will happen, though, is that students whose teachers are using Common Core aligned textbooks will be accustomed to justifying their results and communicating well.
I understand the resistance. When my oldest was in first grade and was asked to justify why 7+5 was 12, he wrote, “That is a math fact.” When his teacher asked him to explain his answer, he begrudgingly drew seven boxes and five boxes, circled the lot, and counted to twelve while pointing to them.
By being asked to justify, my son was given an opportunity to add his voice to the math lesson. He was asked about his thought process, and demonstrated his understanding of a deeper math concept, one-to-one correspondence. He chose to draw, but he could have chosen to verbally explain, or to grab some objects to make his point, and all of those methods of justification would have been correct. His choice, and the different choices made by his classmates, were honored and celebrated. Furthermore, his teacher reinforced the notion that we need to be able to put reason behind our solutions, rather than just calling them “facts” and expecting others to agree. We want students to learn multiple methods for solving problems, so that they can also learn the skill of discerning which works best for them.
We need to think more about why we are asking our students to learn mathematics. Memorizing “math facts” might be where they start, but where do we want them to go, and how do the Common Core Math Standards set them up for success?
We want our kids to maintain their curiosity about the way the world works, and the role that mathematics plays in explaining scientific phenomena. Likewise, our children should be exposed to the arts and humanities, physical and natural sciences, social sciences, physical education, music, foreign language, all of which have conventions and mnemonic devices (like PEMDAS) that we want our children to question and explain, not just repeat.
Stephanie Salomone, Ph.D.
Portland Metro STEM Partnership and the University of Portland
Additional Resources:
Teachers, looking for a way to “create time” by looking for places where Common Core Math, Common Core ELA, and Next Generation Science Standards overlap? This Venn Diagram can help, and teaching content that addresses standards in the intersections will free up some time.
The Math Practices Standards describe habits of mind that we want all children to develop, starting very early on. This site has an explanation of these standards. If you are a parent, it might help to look at these, and think about how exploring these habits might help your child, regardless of their passions and interests. For example, we really do want all people to learn to construct viable arguments, and to be able to appreciate and critique the reasoning of others, regardless of what the argument is about.
If you are looking for an overview of the difference between a Common Core-aligned curriculum and one that isn’t, this is a good resource.
Looking for alternatives to PEMDAS? Here is one that uses a condensed mnemonic, and here is one that uses a visual (The Operation Tower), rather than a string of letters.
An interesting study about math anxiety, and how you can mitigate it. "By developing a student's ability to reflect on past successes -- before maths anxiety sets in -- we can break through some of the negative and emotional beliefs about maths and, hopefully, pave the way for students to accept and engage with maths in the future."