PEMDAS (boo) & Bathroom Signs (yea)

14 years into my teaching career I transferred from a high school to a K-8 order-of-bathroomschool in my district.  It was quite the culture shock.  I was in awe of the K-5 teachers and thought ‘I never want to teach little kids’.  I soon learned that most of the K-5 teachers in my school avoided teaching Kindergarten like the plague.  When I asked them why they said “I don’t want to teach students how to tie shoes….etc”.

The K-8 school had a phenomenal Kindergarten teacher named Kate.  I loved watching her in action.  My favorite thing I saw her do most days was stand outside of the boys bathroom making sure her little darlings were learning to negotiate bathroom etiquette.  I would hear say things like “Austin, don’t come out here until you’ve washed your hands” or “Samuel, don’t put your hand in the toilet” or even “Zip up your pants, we don’t touch each others private parts”.  I thought ‘WOW, I am so glad I do not have to talk about these things as a middle school teacher.

Today, the teacher I co-plan with daily reminded me of Kate when she used the sign above (& below) at the start of her Advanced Algebra lesson, asking students to notice & wonder.

notice-wonder-math-order-of-op

We are currently teaching students how to solve equations that involve exponents & logarithms and several students were struggling to remember the order of operations. To help, Morgan did a quick review of the Order of Operations with them.  She used this visual to talk about the need for a standard order of doing things. (note: the students had a great conversation ’cause talking bathroom behaviors seems to have magical ways of increasing engagement – even with HS students)

I loved her use of the bathroom sign because I think bathrooms and PEMDAS have a lot in common.  My wish is for you to STOP using PEMDAS and begin doing something better.  Let me continue….

A few days earlier at our team meeting I told Morgan that (like the K-5 teachers avoiding teaching Kindergarten) I now avoid teaching PEMDAS like the plague.  This led to a great conversation with our team as to why this is and what to do instead.  I thought I’d share a bit of or conversation and share what we use instead.  I also encourage you to work to eliminate the tricks in your curriculum and consider how you teach concepts.

nix-the-tricksI started by pointing my team to Tina Cardone’s (FREE download) book ‘Nix the Tricks‘.  In her book Tina gives reasons or avoiding 50+ common ‘tricks’ math teachers have used forever to teach math, including asking us to Nix (quit) using PEMDAS.  What I love about Tina’s book is that not only does she explain why to eliminate a trick but provides a fix to replace it with.  Go now and download her book.  She talks about PEMDAS starting on page 9.

I personally started moving away from using PEMDAS 15+ years ago when I last taught HS.  My students would continually apply the order incorrectly when simplifying expressions like this:

pemdas-fails

Many students would first multiply 2 and 5 resulting in an incorrect solution of 3.  Their mistake was believing multiplication always proceeded division because M is before ‘D’ in the PEMDAS acronym.

As a high school teacher I knew I had to disrupt their use of PEMDAS.  It is kind of like the backwards bike that I am trying to learn to ride.  I can ride a normal bike so unconsciously that any change to these skills require such effort to rewire my brain.  My students first understanding of PEMDAS has several misconceptions but they apply them almost unconsciously.  To change their bad habits I had to change things up and stop using PEMDAS altogether to disrupt the pathways in their brain.

One way that I disrupt PEMDAS is to use this simple warm-up with secondary students at some point at the beginning of the year.

ooo-warm-up

All 4 of these expressions lead to potential misconceptions often with the misuse of PEMDAS.  These expressions have led to great discussions with my students about why we should review the standard order of doing math and  learn a better list of the order of operations that works best for HS.  #1 is the classic doing multiplication before division misconception.  #2 is a super famous Facebook expression that keeps floating around every few months with millions of comments with incorrect solutions.  Students will often do #3 correct without a calculator, but then I tell them to type it into their calculator (often they type in 24+8/4+12 resulting in an incorrect solution) which leads to great conversations about the term ‘grouping’.  #4 includes a great discussion about whether 4(3) induces parentheses or multiplication.

With Advanced Algebra we then gave out a 1/2 pager to glue into their notebooks with a ‘new & improved’ order of operations.

my-order-of-op

Morgan and I made this and we spend time talking about what ‘grouping’ symbols mean.  We also make sure students see radicals and exponents as related.  We intentionally wrote Division before Multiplication (and Subtraction before Addition) on our 1/2 pager in an effort to further disrupt incorrect bad pathways related to OOO in students brains.

We are not 100% in love with this 1/2 page of notes yet, if you have suggestions for improving it, please let us know.  Here is the word doc of our current 1/2 pager:  3-4a-order-of-operations  When I teach middle school or 9th grade I don’t just hand this 1/2 page out.  I do a fuller lesson on order of operations and we develop this list together.  In my Advanced Algebra courses, I speed this up a bit and have the conversation of this concept through a warm-up followed by this 1/2 pager.

A few extras related to Order of Operations for your classroom:

#1  I love using this GIF.  Good stuff!

order of operations game show

#2:  Read this great NCTM article titled “12 rules that expire in middle grades“.  Check out #5 – more great information.

nctm-12-math-rules-that-expire-in-the-middle-grades

#3:  Use Kate Nowak’s great ROW GAME for Order of Operations.  This is great for partners to self check their work motivating them to rely on each other to find their mistakes.

row-game-order-of-operations-1

#4:  My favorite Order of Operations Lesson that I used when I taught Middle School was taken from an episode of Survivor (Spinning Propellerheads) that that got the math wrong.  I first have students work in pairs to simply this expression (Here is the word doc: day-3-survivor-order-of-operations-answer:

survivor-order-of-operations

Students (at least those in middle school) really struggle with this.  After 10-15 minutes I tell students that this expression lost someone $1 million dollars and I have the video to prove it.  ms-van-annoyed

I then give a bit of background (what an immunity challenge is, the prize for winning the show…) on the TV reality show Survivor and proceed to show them this video:

Before showing the video I tell my students that there are at least 3 math errors in this video and their job is to finjt-pemdasd them.  I then make exaggerated sounds of disgust at 0:53 seconds when Jeff Probst (the host) calls operations ‘math symbols’ and continue my exaggerated disgust when I see J.T. write
‘PEMDAS’ (he does not win) on his chalkboard at timestamp 3:46 as the host says ‘JT is doing the math’ (UGH!).  I scream at the video when the winning answer is worked left to right completely ignoring the order of operations all together.  This is wrong.

answer

The students and I then have a great conversation about why we have a standard order for operations in mathematics.  I follow this conversation up with some practice (here is a HW I used with my middle schoolers: u2-order-of-operations-homework or this partner task in class: u2-order-of-operations-partner-task)  for students and a challenge to add ‘grouping symbols’ to the expression above to make it equivalent to one (the incorrect solution preferred by Survivor’s host).  parenthese

Note:  Here is the solution the TV show Survivor should have accepted if the winning solution had followed the order of operations: day-3-survivor-order-of-operations-answer

right-answer

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