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Operations and Symbols

What is the Correct Order of Operations in Math?

 November 6, 2020

By  Mrs. Post

PEMDAS and the Correct Order of Operations in Math

In order to remember the correct order of operations in math, most of us probably memorized some version of PEMDAS or  Please Excuse My Dear Aunt Sally.  If we don't calculate in the correct order, we'll get it wrong. 

P
E
M
D
A
S

Parentheses
Exponents
Multiplication
Division
Addition
Subtraction

This little mnemonic device is supposed to help students remember what to do and when, when calculating. Except, if it worked, there wouldn't be so much confusion about how to apply it.  Like these questions that came from a mom in my Facebook group recently:

What’s the answer to 13 - 6 + 3 + 1? My calculator says 11, but when my daughter uses the rods she gets 3. What am I missing here. Also, when do you do the addition first then subtraction? I learned BODMAS many moons ago?  

T. Risi Homeschool Mom

What IS the Correct Order of Operations in Math?

PEMDAS (BODMAS or any other version of the same) isn't wrong. We just don't know how to use it.  First I'm going to give you the basic run down as most teachers teach it.  And then, I'm going to tell you why I hate it, and provide an alternative to teaching PEMDAS or any other mnemonic.

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So let's say we are working with an equation like this:

2 x ( 6 + 3) – 4²   =

P:  Do What is in the Parentheses First

The first thing we need to do is add the 6 and the 3.

2 x ( 6 + 3) – 4²   =    2 x 9 – 4²

2 x ( 6 + 3) – 4²   =    12 + 3 – 4²

E: Calculate Exponents (and also Roots)

The second thing we do is calculate 4 squared. We do this before we multiply or subtract.

2 x 9 – 4²   =    2 x 9 – 16

2 x 9 – 4²   =    2 x 5²

MD: Multiply and Divide 

Next, we multiply and divide in the order in which they appear in the equation - NOT the order in which they appear in PEMDAS.

2 x 9 – 16   =    18 – 16

2 x 9 – 16   =    2 x (- 7)

AS: Addition and Subtraction

Finally, we add and subtract in the order in which they appear in the equation - NOT the order in which they appear in PEMDAS.

18 – 16   =    2

Why I Hate PEMDAS for Teaching Order of Operations 

The problem isn't  PEMDAS exactly. Rather, it's that PEMDAS is necessary.

For almost all of elementary, students spend the bulk of their time providing the answers to 30 problems on a page - - every day. We spoon feed them mathematical symbols and then expect them to somehow overcome their shallow understanding with something like PEMDAS. 

Because we use memory to teach nearly all of math, most people remember PEMDAS but not how to use it or why it is that we calculate the way we do. They calculate multiplication and subtraction in the order that it appears in PEMDAS.  They aren't sure what to do with roots - and they will add before they subtract even if the subtraction sign comes first -- as in my Facebook group example.

 All of it is scary and gives most people a headache.

The Problem:

We don't give students enough time to learn what the symbols are saying or how to use them. Students are never required to actually use math symbols to take the math ideas in their heads and put them on paper.

The Solution:

Stop giving kids ready-made math symbols. Instead, give them exercises that allow them to take the math ideas in their heads and put them on paper. A productive struggle that requires communicating what's inside a student's head will actually solve nearly every problem that PEMDAS attempts to solve.

Order of Operations: What We Do Differently

In Learning Well at Home Academy Courses, learning the correct order of operations in math begins in the first course as soon as we introduce the idea of a Cuisenaire Rod train.  

We get students reading their own rod creations with their first formal lesson. They don't read a worksheet full of train pictures. They build and read their own trains, which represent addition.

Students begin by reading this train yellow, red, light green, purple, dark green. Within a short time, they will write it:

y + r + g + p + d   

Correct Order of Operations and Transformations

The symbols tell us what happens in time. They document what the writer of the symbols did first, second, third and so on.  As students build their trains, they visually connect what it is that they did first, second, and third to the math symbols they are using. 

This is why transformation are taught very early on as well. Transformations are about making small changes to our work and discovering what happens when we do.  

Every change that's made shows up in the reading of the rods. 

It also shows up in the writing of the rods. 

The above is a red, light green, dark green, yellow, purple train. They will write that:

 r + g + d + y + p 

Our focus from the very beginning is the symbols and what happens when we manipulate them.

How does it change the outcome (answer)? 

We explore the kinds of changes that only affect  the expression and changes that affect both the expression and outcome.  

We don't memorize PEMDAS - we actually study math. 

The Order of Operations Makes Sense

An interesting thing happens when you study math instead of memorize - - you realize  that the rules aren't arbitrary.  Very rarely would a student ever make this mistake:

2 x 9 – 4  =  10

That mistake isn't made because we know that multiplication tells us how much of something we have. The two goes with the nine not the four.  It wouldn't make sense to subtract the four first. We'd read that as, "The difference between two of the nines and four."

Our first year students know that in order to make the nine go with the four we have to show that we did it first in time. To do that, we use brackets.

2 x (9 – 4)  =  10

We would read the above, "Two of the difference between nine and four."

The student uses math symbols to express things that they have either done by physically manipulating rods or mentally manipulating symbols. Either way, it is an expression of what the student is doing rather than meaningless generic examples supplied by a curriculum provider.

Only after writing lots of math expressions for the manipulations that they have created do we expect students to read the expressions of others.

PEMDAS is a non-issue as students understand how the symbols work and what they mean. The following equation just wouldn't make sense: 

2 x 9 – 2²  =  2 x 49

That would be like saying, "The hat belongs to she." It's just not how we use math symbols.

Two squared is another name for four. If you want to connect it to the nine, you are going to need write it like this:

2 x (9 – 2)²  =  2 x 49

We write it that way because, in time, the first thing that happened was two was taken from nine. When would student's know this? Depending on age, their second or third year of math. 

Studying math symbols this way has the benefit of developing students who are adept at both using and reading math symbols.

 In addition, their expressions are much more complicated than we would ever give them.  

Students who would balk at solving problems under normal conditions take great delight  creating sophisticated math expressions that would make "high-schoolers cry".  

As they work to put their ideas on paper, they master a great many algebraic manipulations that would take weeks and weeks for a teacher to teach. The difference -- it's theirs. They own it. 

It's time to Excuse Dear Aunt Sally.

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