Order Of Operations is referring to Pre-algebra stuff because algebra is now a series of operations, rather than just a more simpler process to follow in early math and arithmetic. When you get to a certain point like fractions you then start to quickly go into pre-algebra stuff. It's not JUST arithmetic stuff anymore and sort of just throws you right in there to the beginning of pre-algebra which can't be helped because of how it just works involving prime factoring, powers, basic divisibility rules, reciprocals... and a few other things... 

KA put PEMDAS in after the basic fractions stuff and before the odd and even numbers and multiplication stuff because of the Cumulative Property. Associative Property, and Distributive Property then Multiplying By Tens (Associative Property) stuff, and with that you have to understand what parenthesis is and why it's used. The way schools use multiplication they tend to throw you into pre-algebra operations using parenthesis (which frankly is confusing doing it in that context and just simply calling it multiplication when it's not i didn't understand a clue what it was back when i was in school lol when really it was part of Cumulative and Associative Property it has a name :P lol).

I guess KA are trying to prepare you for using parenthesis and also powers with fractions when you start learning to prime factor. PEMDAS is also connected to Algebra stuff because it starts to have certain operations you can only do and may use PEMDAS when you get to a certain algebra level. The way i took it is you don't have to bother learning the PEMDAS operations or anything just focus on the parenthesis and exponents stuff while you're still technically in Arithmetic stuff which i think is how they wanted me to take it since we don't need it all yet.

It makes the best sense to just keep calling Arithmetic a process to follow up until you get more into fractions and then start introducing the concept that they're starting to now follow a series of operations in fractions like what would happen in algebra because you are needing to learn several processes in each concept to work out the fractions it's no longer A to B it's more like A to B to C to D in the series of steps you need to take to complete it which now makes it a series of operations. It's just knowing when and where what to call stuff.

Thank you for your interest in discussing this.  It’s not clear to me whether you are advocating PEMDAS, and if so, how PEMDAS better prepares students for the math topics you bring up.
Please take one or two of them (cumulative, associative and distributive properties; fractions; prime factoring, powers)….and explain how PEMDAS is superior to learning by the concepts in my suggestion.

Many math educators think PEMDAS is confusing to elementary math students.  It teaches by acronym rather than concept.  It results in a lot of Q&A in which the discussion is about which letter comes first and whether there is an “or” between the letters.  Wasted time that could have been used to reinforce the fundamentals of math:

addition and subtraction; multiplication and division are…….PAIRS

they are pairs because they are equal but opposite in effect; they reverse/undo each other

If students memorize the line in bold early on - it will help them in several other areas of math: fact families, order of operations, balancing equations, working with fractions,,,
Please let me know if you would like specific examples.

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The method proposed above assumes that we have named the pairs.  Do you think this step is of value?  If so, if you were to name this second group (x/÷) with one word - what you call it?  It needs to be descriptive and easily understood by the intended audience.  

How about “Repeaters?”   

Which would make +/-   ....“Singles?”

This step removes 2 of the 6 letters in PEMDAS….and the two “or” conjunctions.  (Again, it is not an extra step associated with this method - it is part of understanding/simplifying several basic math topics.)

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Exponents are easy to add to Order of Operations after students learn what they are; how they cause EXPONENTIAL change.  A Bigger Change than addition or multiplication.  This way, students are able to understand why expos precede the basic ops using concept/logic…rather than an acronym. 

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As mentioned above, parens (“parentheses”) are not operations but a means to indicate Special Treatment.   Students have seen parens in English, and they will see them in chemistry, biology, physics, computer programming....if they study those disciplines.  Each discipline assigns its own meaning to parens.

In math, that Special Treatment is: Do First

Here’s a graphic that helps students visualize/remember: 

https://commons.wikimedia.org/wiki/File:Hands_Symbolize_Parens.jpg 

Better yet, the teacher can place their own hands over the parens on a chalkboard problem.
The hands are holding the numbers (and operation) to emphasize their importance/urgency.

Hands come in pairs..as do parens…and parents.  

Several ways to visualize/remember this important concept.  

Well i'm saying seeing PEMDAS earlier helps prepare you for those concepts if you see it earlier on before that, in fact i did here on KA when it used to be with the 3rd Grade stuff in the "More on Multiplication and Division" unit, now it's been moved to 7th Grade, however the way they did it is they just showed us PEMDAS and didn't ask us to learn the operations or how it works, i think it was more emphasis on the Powers and Parenthesis stuff if i remember rightly... Just being shown it though and reading through the operations online then learning at least how to make Powers and Parenthesis and what it does... it certainly prepared me for what was to come after it then stuff in Prime Factoring i knew how to make powers since i learned it a while before.

I didn't learn how to actually do PEMDAS though because there was no need to at that stage and they weren't asking me to either, but i feel being shown it earlier on worked well, i think it's a good idea just showing it and using it as a base to explain why the parenthesis is around the numbers, and then how to write powers before you start on doing Commutative and Associative Property stuff then using Multiplication to show how to use Parenthesis.

Back when i was in school up to 2005 and before it nobody showed us PEMDAS i hadn't seen it once and because of that i didn't know why i was using parenthesis i didn't know a lot of stuff really because they didn't show half the the things there is on KA, if i was shown PEMDAS i bet i would've understood why we do equations inside parenthesis, and like i don't see why schools can't do several things instead of always trying to find the one method that works it would've been nice to see several explanations to compare to.

But learning all of PEMDAS i think should be reserved for when you start to need it because you won't understand what it's doing or really why... and you don't really need the rest of it while in Arithmetic stuff except Powers and Parenthesis. I guess that's why they probably moved it to 7th Grade stuff because it's right there as you truly start on Pre-Algebra stuff, but i still found it of some use being shown it within the 3rd Grade stuff.

With PEMDAS, if i replaced that with just other words like PAIRS, Repeaters, Singles, or DO FIRST it wouldn't tell me or you which order to do them all in and we'd be confused as to what takes priority over each other, and both Multiplication & Division then Addition & Subtraction take priority over each other... and along with it, it then also depends the order in which the equation is written in. So if Multiplication came before Division in the equation you'd do multiplication first and if Division came first in the equation you'd do division first instead, the same thing with Addition & Subtraction. 

Using those words to help identify that Parenthesis & Exponents are SINGLES, and Multiplication & Division then Addition & Subtraction are PAIRS along with PEMDAS would be helpful, but if you wasn't using PEMDAS you wouldn't know or remember what order to do each of them in just calling them SINGLES or PAIRS. I don't know if there would be an easier way to explain that it depends on the order the equation is written in i don't think so though... that's the part that makes things a little less simple. Again though i think it's better only learning all of PEMDAS stuff when you start to get to Pre-algebra stuff.

Using little pictures of hands around the Parenthesis though for kids i think that would be a good idea the Grades are now more for schools now than older self-learners like me so i think that may be a good option to use for kids using KA at school or at home. However recently they cut back on content to help easier maintain stuff so i don't know if they'd do that or not, that's probably one of the reasons why moved PEMDAS over to 7th Grade/

The goal of the proposed method is to teach the reason and logic that underlies Order of Operations (ie, Mathematics).  Using an acronym (PEMDAS, BODMAS, etc.) does not accomplish that. Acronyms result in classroom discussions about which letter comes before which, and is there and "or" between the letters.....when the time could have been used to reiterate some of the key concepts of math.

The operations were set up long ago based on the order of Bigger Change.  Bigger Change is a concept that is easy to illustrate/visualize on a number line.  Take a number and add 5 to it; then multiply it by 5.  Which moves the result further down the number line?  (Yes, there are exceptions, but we don't teach concepts or make 'rules' by exception.)

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Regarding Repeaters and Singles: You wrote, “Parenthesis & Exponents are SINGLES”  and “Multiplication & Division then Addition & Subtraction are PAIRS”

That is not correct.  Here’s a diagram:

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You wrote:  "So if Multiplication came before Division in the equation you'd do multiplication first and if Division came first in the equation you'd do division first instead, the same thing with Addition & Subtraction."

This issue is not complex, and it is even simpler after you name the basic operations in pairs.  You simply mark the Repeaters and calculate them left to right.  This is just like reading a sentence.  Students have read from left to right for years before they encounter Order of ops.  Why not relate to something they already know?

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Students need to solve equations that involve both mult/div and add/sub before they get to expos.  Do you agree?

Here’s an example of how SIMPLE the first level of Order of Ops can be: 

Thus far, the Order of ops can be summarized by just two rules that young students already know (or should already know):

1. Do the most important things first - in life and in math.  Repeaters cause Bigger Change - they are more important.
2. Perform the ops “just like you read a sentence.” 

That’s it!

Does this make sense so far?   

(Please address the above from the point of view of the intended audience: a middle-grade elementary school student.  The nice thing about this method is you build on it as you advance thru math.  The same concept of Bigger Change first all the way thru.  No acronym  required because the student learns by concepts and logic.)