Marveling At The Historical

Math Oldies But Goodies

  • About This Blog

    This blog is mostly about math procedures in textbooks dated from about 1825-1900. I’m writing about them because some of the procedures are exquisite and much more powerful, and simpler, than some of the procedures in current text books. Really!

    I update this blog as frequently as possible ... every 2-3 days. And, if you are a lover of old texts and unique procedures, you might want to talk to me about them, at markdotmath@gmail.com. I’m not an antiquarian; the books I have are dusty, musty, brown-paged scribbled-in texts written by authors with insights into how math works. Unfortunately, most of their procedures have vanished. They’ve been overcome by more traditional perspectives, but you have to realize that at that time, they were teaching the traditional methods.

Don’t use PEMDAS, Just Underline Terms

Posted by mark schwartz on April 25, 2016

Introduction

Introducing order of operations to remedial/developmental community college students tends to be messier than would be expected. There are several considerations: first, most of these students have had exposure to order of operation but didn’t master it well. This shows up in the first day assessment they take. Second, since they’ve had the exposure, they tend to believe they know it and therefore don’t have to listen to its being presented again. Given these, I realized that teaching order of operations the same old way would give the same old results. Obviously, I wanted the students to avoid this. So, the following is an approach that students found more manageable.

The Story

Most likely students learned order of operations as a system that has PEMDAS as an acronym, or perhaps a word variation on PEMDAS. “P” stands for parentheses, meaning do the work inside the parentheses first; “E” means exponents (and roots); “MD” means do multiplication and division, from left to right and last, “AS”, do all addition and subtraction, left to right.

This works, but it can also cause issues, most notably the left to right thing and also typically in problems like 2 + 3(5  ̶  1). There is a tendency to add the 2 and 3 before doing the multiplication indicated by the parentheses. In problems like 24÷6⦁2, the left to right rule “doesn’t apply” (as stated by some students!) because the “M” in PEMDAS comes before the “D”. Also, students tend to stumble on notation, which is intended to clarify what to do when. Look how order of operations interacts with notation in problems such as 2 + 2⦁32 … (2 + 2)⦁32 … 2 + (2⦁3)2 … and 2 + 2⦁(3)2. Again, in a first day assessment, these 4 problems in order give students the heebee jeebes.

The point is that if PEMDAS were as simple and straight forward as it seems, students wouldn’t be so stymied by it. Somehow, the interaction with the notation seems to disrupt the straightforward application of PEMDAS.

There is another way to work on expressions without using PEMDAS. It comes from Algebra, when students will be introduced to recognizing terms and doing operations with like and unlike terms. It also works when you solve equations. So, way out there in their math future is something that can help them with order of operations now. So, by the time they get to equations and other Algebraic operations, they’ll already know how to do some of it.

Generally, when you are given an arithmetical or Algebraic expression, you will be given directions to “simplify the following”. Sometimes the directions say “do all the indicated operations.” I prefer the latter because once you’ve done all the indicated operations … well, you’re done.

How does it work? Take the following example:   2 + 3(5 −1) − 8 + 16 ÷ 4. It has parentheses, addition, subtraction, multiplication and division.

Here’s how to do this problem using what I call the “underline method”.

First, start underlining from the left. Put your pencil under the “2” and underline until you come to the “+” sign. Don’t underline the “+”. Lift your pencil and start underlining again under the “3” and continue until you come to the “−“outside the parentheses. Lift your pencil again and start underlining again under the “8”. In this case, only the “8” gets underlined before you again come to a “+”. So underline the “8”, lift your pencil past the “+” and start underlining again under the “16” and continue until you come to another “+” or “-“. In this case, you don’t so “16 ÷ 4” gets underline.

What this does is identify the terms in the expression. And, this comes from the Algebraic idea of identifying the number of terms in an expression, and doing this so that like and unlike terms can be identified and handled according to the indicated operations. Here, we have only like terms … numbers.

So, after underlining, we have 2 + 3( 5 −1) 8 + 16 ÷ 4. There are 4 terms in this expression. The point now is to take each term, one at a time, and do all the indicated operations within each term. For 2, there are no operations to do, so we have ‘2’. For 3( 5 −1), the notation indicates to multiply 3 times whatever is in the parentheses. The only way to know what’s inside the parentheses is to do the subtraction inside the parentheses, so this term becomes 3(4) and then ‘12’. Basically, what’s inside an arithmetic grouping symbol is one number, after all the indicated operations are done.  The 8 has no operations to do, so we have ‘8’. And, 16 ÷ 4 gives ‘4’.

So, after having done all the indicated operations within each term (or it might be expressed as ‘simplify each term”) we have 2 + 12 – 8 + 4. And now there are alternative strategies for finishing this and we’ve covered these already.

This underlining method pushes thinking about what to do, rather than rummaging for the rules. Once each term is identified by simply underlining, the operations within any term are to be done based on thinking about what is being asked. In the term 3(5 −1) it’s not a rule that needs to be employed, rather just think about what the problem states – multiply 3 times what’s inside the parentheses and you don’t know what it is until you do the ‘5 −1’.

With numbers, the order of operations emerges from identifying each term in an expression and then simplifying each term completely and then finally doing the remaining addition and subtraction. And, this parallels exactly what PEMDAS does.

Here’s another example, underlining inside and outside the parentheses.

Do all the indicated operations:      26 + 6 ÷ 2 – 2(4 + 8 ÷ 2) – 7

First, underline outside the parentheses, which gives 26 + 6 ÷ 22( 4 + 8 ÷ 2)7. There are 4 terms in this expression. Now underline inside the parentheses, you get 4 + 8 ÷ 2. This shows that there are 2 terms and clarifies what to do inside the parentheses.

Then, looking at each term, the ‘26’ and the ‘7’ have no operations to do. The ‘6 ÷ 2’ gives ‘3’ and the term ‘2(4 + 8 ÷ 2)’ again indicates to multiply ‘2’ by what’s inside the parentheses, and we’ll know what’s inside once the operations are done. Since there are two terms inside the parentheses, look at them one at a time. The ‘4’ term has no operations to do; the ‘8 ÷ 2’ term gives 4, so inside the parentheses is now ‘4 + 4’ or 8. The term ‘2(4 + 8 ÷ 2)’ is now 2(8) or 16. The expression is now 26 + 3 – 16 ─ 7. Doing the remaining operations gives 6.

In summary,

  • Identify each term in the expression by underlining each term. Terms are separated (of if you like, connected) by “+” and “−“ signs outside any parentheses. You can also identify terms inside the parentheses by this same underlining procedure, if there are a bunch of operations inside a parentheses.
  • Simplify each term completely by doing all the indicated operations.
  • After all terms have been simplified completely, do the remaining operations. One way to do this (but not the only way) is to do the operations from left to right. Another way is to do all the additions and then do all the subtractions from the sum of all the additions. For example, 2 + 12 – 8 + 4 gives18 (which comes from 2 + 12 + 4), then subtract 8. The associative and commutative properties had been demonstrated and discussed previously, and they can always be applied which gives options for doing the final additions and subtractions.
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