Dear Aunt Sally … Please be Excused (from math)

In developmental mathematics classes, as in school mathematics, Aunt Sally seems to be everybody’s friend.  As in “Please Excuse My Dear Aunt Sally” as a memory aid for order of operations (aka “PEMDAS”).  I would, indeed, like to excuse Aunt Sally from ever being in my math class. 

In another post, I talked about the “Sum of all Shortcuts”; in this post, the issue is mnemonic aids.  You can improve your students’ “learning” if you minimize the use of these ‘easy to remember’ tricks. 

This may sound counter-intuitive … isn’t it a good thing if students can remember something?  Well, it CAN be a good thing; the issue is what exactly do they remember?  In the case of PEMDAS, they remember ‘do inside parentheses first’ or ‘do parentheses first’.  Fine, to a point — students can evaluate  “8 + 10 ÷ 2” and “(8 + 10) ÷ 2”.

The student then sees these ideas, which do not follow PEMDAS:

• 8x + 2x   (can add before multiply)
• 8x + 2y    (can not add)
• 8(x + 2y)   (can multiply ‘first’)
• (2x²y)³     (can ‘power’ first)
• f(x)=8x + 2    (what does that x mean?)
• f(-3) for that function    (what do we do with the -3 on the left side)

The “P” in PEMDAS is especially worrisome.  Parentheses have multiple purposes in mathematics, and only some of them relate to the order of operations.  We also use other symbols of grouping, some of which are another operation (radicals, fractions, absolute value, etc).

Now, we actually make students do too much with expressions of extra complexity just to see if they can follow the order of operations.  We create our own need for an easy-to-remember tool (PEMDAS) which then results in students having to unlearn later when we do other work in ‘simplifying’.  This is a bit like designing a tool to require disposable parts, in order to keep a business active; I would suggest that our artificial level of difficulty with numeric expressions serves no purpose, not even our own.

It’s important, however, for our students to be literate and comfortable with the basic meanings of expressions and forms.  As I talk with my students, I am impressed by how many of them remember ‘PEMDAS’ years later and by continuing difficulty in doing work that does not involve applying ‘PEMDAS’.  We are not doing our students any favor by giving them an easy thing to remember which does not transfer to future work.

Some readers are likely upset by my suggestion; yes, I know PEMDAS has helped millions of students in their math classes; yes, I am aware of research showing mnemonics help learning disabled students in particular.  However, the benefits for most students do not seem that great to me; the long-term result may be more negative than positive.  [Many of my most struggling students have learning problems, and survived by using tools like PEMDAS; they have difficulty in the situations listed above that do not follow the PEMDAS priorities.]

We know that PEMDAS does not cover most expressions involving variables.  I am suggesting that PEMDAS directly interferes with the algebraic literacy of our students; quite a few students suffer needless discouragement when their algebraic difficulties increase as they painfully discover the real limits of PEMDAS.

Let’s send Dear Aunt Sally on a much needed vacation; she has been used for many years, and perhaps is ready to retire.  Instead, let us focus on basic literacy dealing with reasonable objects of valuable mathematics.

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1 Comment

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  By Holly, September 30, 2011 @ 2:26 pm

  Are there any analyses of the influence of your first and third examples on success in factoring and distributing? As I think about this, I wonder how much the troubles I saw in the department with scores on the factoring tests can be attributed to those examples, misunderstanding of the third in particular.

  As for the fifth and sixth–I think that is the long-term problem I’ve had with the way I was taught order of operations. I did not encounter f(x) at the time, but I run into it all the time now, and there’s often a brief delay while I ‘translate’. This is especially a problem when I’m looking at an integral or derivative with multiple functions of more than one variable, as I’m ‘trained’ to regard parentheses on the right hand side as an order of operations item, not a ‘function of’ or ‘call function where t=3) item.

  A bit of a ramble–for which I apologize, but you ‘pros’ may find my ramble useful at some point, and if I remain in academia I will be teaching coursework with a quantitative component.

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