1407 Please Stop Making Excuses for My Dear Aunt Sally

Please Excuse My Dear Aunt Sally. You’ve heard math teachers say that phrase many times. Supposedly, Aunt Sally is supposed to help you remember Parenthesis, Exponents, Multiplication/Division, Addition/Subtraction as the correct order to do operations when simplifying math problems.

I say, please stop making excuses for my dear Aunt Sally!

My Dear Aunt Sally. People think they know her, but too often they really don’t. Lots of people have tried to please her. Sometimes they succeed, but just as often they fail. She seems to relish the fact that so many people misunderstand her.

I clearly remember my first year teaching a classful of seventh graders at a new school. I was trying to develop a good relationship with my students and be the best teacher I could be. One of the first lessons I was supposed to teach them was order-of-operations.

I wish I knew about the better mnemonic PEMA back then, but I didn’t. Instead, I brought my dear Aunt Sally to class with me: I introduced her to my students and tried to make it clear that multiplication and division were equals so they must be done in order from left to right whichever one comes first. The same is true of addition and subtraction.

“That’s not what we learned last year!” students responded. Their teacher last year brought Aunt Sally to class, too. but she gave them the impression that all multiplication was supposed to be done before any division, and the same for addition and subtraction. Yeah, Aunt Sally went to class their previous year and didn’t say a word when their teacher gave them misinformation. Now that I was telling them the truth about her, she didn’t speak up and tell them I was right either. Instead, she allowed me to lose credibility with my students that day as I insisted on sticking with the truth. If I had retold the lie, the students would have believed me more. I also discovered that for some problems in the textbook, you would get it right either way.

I seriously couldn’t believe that their teacher from the last year would have given them the wrong information. Surely the students misunderstood what had been taught. However, since that day, I have heard more than one teacher incorrectly tell students to do all the multiplication, division, addition, and subtraction in that order from left to right. Those teachers put the students’ next teachers in a catch-22:

That is why I prefer to keep “my dear Aunt Sally” away from kids. She always shows up at the beginning of the school year when students and teachers are trying to start off on the right foot.  She torments students and immediately causes them to feel bad about themselves or mathematics. She makes them question the teaching of their current teacher or their past teachers. She gets a kick out of making children and even adults feel like there’s no way to understand math:

Why do we allow Aunt Sally to abuse children like this? I want to shout, “please, stop making excuses for my dear, Aunt Sally!”

Let me tell you the story of when I decided not to introduce this abusive aunt to children every again.  It was 2016. I was substituting in a 5th-grade class. I wrote an expression I saw on twitter on the board and told the students it was my favorite order-of-operations problem. Here’s what I wrote:

10 + 9 + 8 × 7 × 6 × 5 – 4 + 321 = 

I, along with my dear Aunt Sally,  encouraged the students to figure it out. The students knew that 8 × 7 was 56. I watched them struggle to multiply 56 by 6 and then by 5. When I mentioned that they could multiply the 6 and the 5 first to get 56 × 30 to make the problem easier, they argued that doing that wasn’t allowed. They said that the order-of-operations demanded that the multiplication be done in ORDER from left to right.

They thought that order-of-operation makes multiplication no longer commutative?!!  How do you counteract that misinformation? After that day, not only do I not invite my dear Aunt Sally to meet my students, but I also avoid the phrase “order-of-operations”!

Order-of-operations is just an ALGORITHM! It doesn’t trump the commutative property, and it doesn’t even have to be used to solve these kinds of problems!

Jo Boaler’s tweet especially applies to this kind of problem and this algorithm.

Besides, are these kinds of problems still necessary since typing on a computer no longer has the same limitations as typing on a typewriter? I hope you think about that! If you insist on using an algorithm, I suggest you use PEMA instead.

Since this is my 1407th post, I’d like to tell you a little bit about that number:

  • 1407 is a composite number.
  • Prime factorization: 1407 = 3 × 7 × 67
  • 1407 has no exponents greater than 1 in its prime factorization, so √1407 cannot be simplified.
  • The exponents in the prime factorization are 1, 1, and 1. Adding one to each exponent and multiplying we get (1 + 1)(1 + 1)(1 + 1) = 2 × 2 × 2 = 8. Therefore 1407 has exactly 8 factors.
  • The factors of 1407 are outlined with their factor pair partners in the graphic below.

1407 looks interesting when it is written in some other bases:
It’s 111333 in BASE 4,
21112 in BASE 5, and
727 in BASE 14.

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