PEMDAS - And Other Atrocities

Going Wrong
Intuitive Learning - The Results
The Failure of Intuition
The Monster Study and Intuition
Success, Failure, and the Need to Succeed
Free Will and Randomness
Occam's Razor
PEMDAS - And Other Atrocities
Math Lessons
For Further Reading

Mnemonics and the Failure of Education

Teachers love mnemonics, it seems.
A mnemonic is a device for remembering built around using a key word (or phrase) in which each letter (or first letter) stands for one of a list of things to be remembered.
For example, begining music students learn Every Good Boy Does Fine, which represents E G B D F.
In arithmetic, we have PEMDAS: parentheses, exponents, multiplication, division, addition, subtraction.
If neonatals (new borns) had teachers, I have no doubt that would be required to memorize BIBO:  Breathe In, Breathe Out.
And there you have it.

The idea of learning, we have seen, is to achieve intuitive mastery, so that we can do a particular activity without thinking about it.
Mnemonics is the opposite.  It is a security blanket that is so seemingly effective that almost no child, having learned some mnemonic, will willingly let go of it.
The problem is, what should be a right brain activity triggers a left brain activity decoding the mnemonic and applying it some process.
To truly comprehend this, imagine having to use BIBO.
BIBO = breathe In, breathe out.  So breathe in, then breathe out.  Now hold your breath and think.
BIBO = breathe In, breathe out. So breathe in, then breathe out. Now hold your breath and think.
BIBO = breathe In, breathe out. So breathe in, then breathe out. Now hold your breath and think.
and on and on and on.
Ludicrous?  Of course it is.

But it is worse than ludicrous.
It is crippling.
The child who has learned to dodge memorizing by using a mnemonic becomes addicted to the process, and wants more mnemonic to help with each new memorization challenge.
The thing about memorizing is that it is a fragile process.  It works on trust.  The moment you begin doubting what your right brain serves up is the moment you begin losing memory accuracy.
Memory is like a puff of smoke.  Make the slightest motion of distrust, and it fades away.

So how does a child learn the order of operations?
The order of operations is a visual process.
Parentheses are not shaped the way they are by accident.
(  )  suggests hands or a bag containing something.  The very shape of the pair of parentheses is that of two facing pieces of a circle, a container.
So (  ) are meant to suggest that what is contained within them is done as a unit distinct from what is not inside them.
Notice I did not say that the contents of a pair of parentheses has to be computed first.  In ordinary arithmetic, entities are separated by addition and subtraction signs.  Entities are parts of the problem that can be computed separately from each other, and in any order.
3x4 + 5 - 6 + 9x10 + 3x(4 + 1) can be worked and will produce the same answer if each entity is computed in ANY order.
3x4 + 5 - 6 + 9x10 + 3x(5) = 12 + 5 - 6 + 9x10 +3x(5)
                                       = 12 + 5 - 6 + 90 + 3x(5)
                                       = 12 + 5 - 6 + 90 + 15
                                       = 17 - 6 + 90 + 15
                                       = 11 + 90 + 15
                                       = 101 + 15
                                       = 116
is the strict order of operations.
However, work all entities first, then combine terms using the sign in front, in any userful to you order:
3x4 + 5 - 6 + 9x10 + 3x(4 + 1) = 12 + 5 - 6 + 90 + 3 x 5
                                             = 17 - 6 + 90 + 15
                                             = 11 + 105
                                             = 116
This is learned with the minimum of explanation, giving explanation only to correct an error or in response to a question.  The student's right brain will soon start learning real arithmetic.

Enter supporting content here