I'm Bon Crowder and the photos above are both of me - in 1989 and today. I'm a Generation X mom of Generation Z kids.

I began peer tutoring in high school in 1984. MathFour.com is the 2015 version of me helping peers be comfortable in math.

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Tag Archives: order of operations

The Absolute Value of a Number is Always…

This is Day 9 of 31 Days of Math Learning Success. Follow all the days here and check out others that are writing for 31 days here.

Absolute Values of numbers might be simple, but when you start using letters with them, watch out!

“Anytime somebody is absolutely certain about something, they are almost always absolutely wrong.”
~ Jeff Lindsay

Absolute values are weird.

The popular way of handling them is, “Get rid of all the signs.”

And if you’re taking the absolute value of a real number, that works for the most part.

But if you’re working with variables, it might not.

Working with Numbers

Suppose I ask you:

  1. Pick your favorite number.
  2. Subtract 100 from it.
  3. Take off the sign (so if it’s negative, make it positive).

You now have a positive number. But at step 2, you probably didn’t. (The top 10 favorite numbers are all pretty small.)

Using the world’s most popular favorite number (7), we would do this:

  1. Pick your favorite number: 7
  2. Subtract 100 from it: 7-100 = -93
  3. Take off the sign: 93

Working with Letters

Now let’s look at an expression: |x-100|

The absolute value bars indicate to “take off the sign” – our popular definition.

Many people make |x-100| into x+100.

Compare this to our example above:

  1. Pick your favorite number. (We’ll call it x…)
  2. Subtract 100 from it. (…so that would be x-100…)
  3. Take off the sign. (…or do |x-100|.)

For our number 7, we got 93.

But if we claim that |x-100| is the same as x+100, then 93 is the same a 7+100.

Everything You Know is Wrong

Okay, no it’s not. (But that’s a great Weird Al song!)

It all comes back to the squirrelly order of operations (you know – PEMDAS/crazy Aunt Sally). Absolute value bars are a method of isolation. Which means the stuff inside gets ranked as parenthesis.

So you have to “do the inside first,” as they say.

Which means you can’t just get rid of them that easily.

The Bright Side

People believe that absolute value bars turn all negatives into positives.

This is a pretty good way to think, at least in life.

So the next time you’re feeling yucky, grab a couple of sticks or pool noodles and hold them to either side of you.

Now smile – you’re positive!


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6 Responses to Another Reason to Ban PEMDAS

  1. This may be as much a response to your previous PEMDAS post as to this one.

    Firstly, you suggest that PEMA is a fundamental property of the operations. It’s not, at all– for example, there’s no property of multiplication or addition which forces you to do multiplication first. At first glance, distributivity seems to imply we do multiplication first, but it doesn’t: we could just as well write the distributive law as (x*y)+(x*z)=x*y+z, using “PASDME”. (Yes, parenthesis do need to come first)

    At the end of the day, whatever order of operations we use is a linguistic convention, and like any other linguistic convention (e.g., grammar) there’s a roughly agreed-upon set of rules which help us to transform syntactic notions (e.g., the expression 3+4*7÷12) into something with semantic meaning. Getting rid of PEMDAS won’t help people understand this. Although, to be fair, the experiment you suggested in the previous post might: it (could be) an interesting lesson in notational convention and the difference between what we write and what we mean.

    Now, as for all of the other operations you mention in this post–blaming this on PEMDAS is wrong. The misunderstanding of (e.g.) absolute value comes from a failure to understand that absolute value is a unary operator which has semantic, not syntactic meaning–it acts on the thing represented by the expression below it, not on the expression itself. Gaining an intuition for this (even without being able to put it in fancy words) is a huge leap in abstraction; abstraction, not rules, is the thing that makes math hard.

    Moreover, we really do need a notational convention before we get to fancy things like radicals and absolute value (at least, in the path school math is currently taught), so whatever convention you come up with, you’ll have P and then the letters ADEMS and some directional convention (e.g., “from the left”), but operators which “group” will by nature be left out of this, since they show up at a later point in the student’s education. Perhaps, teaching kids to explore notational conventions will help them separate the symbols from their meaning… But the difference between syntax and semantics is something I see graduate math students stumble over from time to time, so I don’t think you’re going to see these problems go away at lower levels, especially with such a simple change as getting rid of PEMDAS.

    • Thanks for your thoughtful comments, Cory. I have speculated that you can indeed switch the operations up as you’d like – provided that we all agree. Alas, I’ve not dug into this enough to agree with it (nor to disagree).

      Regardless, our tradition is to keep the multiplication over addition – and exponents over multiplication.

      And for sure, the neglect of “roots” is a deal breaker for me.

  2. My annoyance at PEMDAS comes from the memorization part. They KNOW that multiplication comes before division, right? That addition comes before subtraction?

    2*3-4+5 is very often solved as 6 – 9 = -3 because the multiplication is obviously first but “addition comes next.”

  3. I often tell my students that there are a lot of invisible things in math. A variable standing alone has an invisible
    ‘one’ as a coefficient. Numerators and denominators have invisible parentheses. All variables have an exponent, but sometimes it’s an invisible ‘one’. It helps, I think.

    Funny idea: xkcd.com had one comic that gave a different mnemonic for order of operations: Please Email My Dad A Shark. hee

    • Chakolate – good point! Alas, because we ask them to memorize (as if they’re computers that can etch the algorithms in their carbon based hard-drives) we can’t expect them to think about the “invisible” parts.

      In fact, if we treat them like algorithm following computers, we can’t ask them to think at all.

      Hopefully some of the students are seeing that PEMDAS is a guideline, not a rule. But I see too much of the opposite. #sigh

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2 Responses to Exponents of Negative Numbers

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4 Responses to Math Rules & Their Destruction of Education

  1. Well said, Bon! Rules without understanding are useless. As I read this, I was reminded of a recent experience subbing in an advanced 7th grade math class. We were working with equivalent fractions, and one student told me I needed to multiply the numerator and denominator by the same number. I asked why. He said, “Because you need to do the same thing to mama as to papa.” Huh? Again, I threw out the why question. Nobody knew. I explained it the same way I learned it – http://www.suedowning.blogspot.com/2011/01/equivalent-fractions.html. The response was very positive, and they asked why nobody ever told them that before. That was a question I couldn’t answer.

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19 Responses to PEMDAS and a Stupid Arbitrary Rule

  1. This is ridiculous. Sure, let’s teach children to do it in away that they get a completely different answer from what they will need to get when out in the world. That way, they will be completely confused on what the “agreed upon” way is. I’m not buying your argument.

    • Fair enough, JG.

      And that’s a normal response from people who live in a world where we start crushing novel thinking and curiosity out of students in the first grade.

      I propose that we bring back novel thinking – maybe not in the 7th grade. But we must stop teaching 1st graders that there is a RIGHT answer rather than various ways to get AN answer.

      • So, it’s more important to preserve erroneous “novel thinking” than educating children how to do things the way the world actually works? Is that what you’re saying? That it’s more important they can make up their own rules to get ANY answer, than follow the established ones to get the RIGHT answer?! That’s absurd!

        • Yes, Kaitie-Lynn, if you want to over-simplify it. The fact is that getting the right answer isn’t what employers want. It’s being able to think.

          And “the way the world actually works” is that there isn’t many times when there is a right answer.

          Making up your own rules (and figuring out which of yours will work with everyone else’s, and which won’t) IS the way of the world.

  2. I’m with JR on this one. I think the risk that students will get confused and not remember which is the agreed order of operations is too high.
    On the other hand, having a discussion about the fact that the rule is really arbitrary is quite healthy, I think. That is the sort of thing I wish my teachers had taught, rather than making feel like I must be stupid if I didn’t understand why the rules are the way they are.

    • That “making me feel like I must be stupid” tactic is exactly how grownups work to eliminate the novel thinking in young kids.

      So damaging.

      Thanks for stopping by, Peter.


  3. As a freshmen math teacher, I have students who have been exposed to PEMDAS before but always seem to forget the rules. So to begin the lesson I have pairs of students work on different problems, stating each step. Once completed they share with the class their steps and thought process. Other groups cannot comment until all groups have presented. Then we discuss why would anyone put parenthesis first (because it is grouped and a good starting place). We discuss the rules as created thousands of years ago so that we all can compute the same answers no matter where we live. Students then practice the correct way and once understood create their own rules, create a problem and have a friend solve it. No one is confused and they realize we all need to agree on one set of rules to function world wide:)

    • Brilliant!

      As a high school teacher you’re having to manage years of “math baggage” of incoming students. If it was negative, or some topics were not firmly learned in a positive way, you’re faced with a huge challenge.

      It looks like you’ve figured how to make it work. That’s so awesome to read – thank you for sharing this method!

  4. I struggled for years in math because not once did any of my teachers understand my frustration at how “made up” these kinds of maths were. They’d repeat the lines from the book, give us the tests, and wait for summer. Having really shitty teachers does a disservice to mathematics.

    Now, as a fully grown man with a doctorate in mathematics, I can say with pride that my confusion wasn’t because I was too stupid to understand it (in fact what was responsible for my problem learning it was a keen and rare philosophical insight which I would later go on to write about in my doctoral dissertation!) but rather it was due to having crappy teachers not give a crap about their children’s /understanding/ of the material. In a school system where grades matter more than understanding, the child will fail every time; he’ll fail even if he graduates.

    Good math teachers connect with and create understanding in their students. Kudos to you, Bon, for breaking the poor mold imposed on teachers. By doing this you’re giving your kids a fair chance at exploring the beauty of the world of mathematics. No regrets in that.

    Good on you,

    Adam S.

    • Thanks so much, Adam, for sharing your story and for the kind words. I’m certainly looking forward to perusing your dissertation – you have me curious!

  5. I realize you wrote this blog entry 2 years ago, so my comment is hardly timely. However, the PEMDAS algorithm is not at all arbitrary.

    I’ll grant you PEMDAS is difficult for students to remember and that it causes quite a bit of frustration. However, allowing them to do it any other way isn’t about allowing student decision making or ‘not crushing novel thinking’. Doing so will only damage their ability to understand math.

    Like it or not, math is a logical structure with firm rules. Progress in understand comes from mastering the rules and understanding their interactions.

    The priority of the order of operations is not arbitrary. It exists because some operations are shortcuts or condensations of others. Multiplication has a higher priority than addition because multiplication is condensed addition.

    3 + 2 x 5 = 13, not 25, because 2 x 5 is actually 2 + 2 + 2 + 2 + 2. So 3 + 2 x 5 = 3 + 2 + 2 + 2 + 2 + 2. By expanding the problem, the students can see that multiplication is merely a shortcut for addition.

    If you let the students do 3 + 2 x 5 as 3 + 2 = 5 and 5 x 5 = 25, there is no way you can show them how to expand the multiplication as addition.

    Worse yet, allowing ‘choice’ in order of operations will kill the students ability to see patterns because terms will not have consistent values. If 3 + 2 x 5 = 13, then 4 + 2 x 5 = 14 because we are merely adding 1 more, which can be shown as 1 + 3 + 2 x 5. But the other way gives 4 + 2 = 6 and 6 x 5 = 30, which is not 1 more than the wrong answer of 25 we got from doing doing the first problem the wrong way.

    Exponents come before multiplication for the same reason. Exponents are condensed multiplication so in an expression like 3 x 2^5, doing the multiplication before the exponents means you cannot expand the the expression. 3 x 2^5 = 3 x 2 x 2 x 2 x 2 x 2 = 96. If 3 x 2^5 = 3 x 2 = 6 and then 6^5, how can you show the expansion? 6 x 6 x 6 x 6 x 6? Where did all the 6’s come from?

    I understand your desire to create student engagement and reduce frustration, but in this case you are just wrong. Teaching the students that PEMDAS is a social justice matter in which they have been deprived of their choice does nothing but mathematically handicap them.

    • Well, John, I’m pretty sure PEMDAS (regardless of how you feel about it) is not going out of fashion any time soon – so your timing is still good!

      There are many good reasons for the order of operations. But there are some serious problems with forcing the PEMDAS mnemonic. Even if we decide that it’s not arbitrary, it still leaves out roots, logs and absolute values.

      And I think Keith Devlin might take up the argument of multiplication being repetitive addition: http://www.maa.org/external_archive/devlin/devlin_01_11.html

  6. This here is why I insist that the only reason math exists is because at some point, some very practical individual needed some way to solve a real-world problem. From there, it did delve into the abstract and invented problems yielded higher math which, again, has practical applications. The trick is to know what you’re given, acknowledge what you’re assuming, and from there, estimate what is expected of you.

    So now, how do you solve 3 + 2 * 5?

    The real question is what does 3 + 2 * 5 even mean? It’s quite obviously shorthand for something. Is it shorthand for 3 oranges + 2 oranges/pair * 5 pairs of oranges? Is it 3 oranges/batch + 2 oranges/batch * 5 pairs? Which makes more sense for it to be abbreviating? Unless these numbers represent something, there’s no point in solving it and there’s no way of understanding what the solution even means.

    This is why I insist that word problems are the ones that should be taught. The rest is convention. Yes, as much as you want to you can say convention is artificial, but it is an important thing when dealing with communication, including traffic signals (why is RED the color to signal a stop, and GREEN the color to signal that it is appropriate to proceed?) and which side of the road we drive on and what “flammable” and “inflammable” mean and why we use a new word “nonflammable”.

    I would love it if parents and teachers comprehended that units are of the utmost importance in learning and applying mathematics at any level.

    • Nicely put, Derek!

      I taught at Halliburton for a while (wrote the Oilfield Math textbook too) and that was the BIG thing. If you don’t have the units, it’s all useless.

      Thanks for stopping in and sharing!

  7. Hi Bon.
    Interesting read. Thanks.
    I recently became interested in the subject of arithmetic order of operation after taking a test on Facebook, which I failed. I failed because I worked from left to right. I didn’t take into account this thing called BODMAS. I’m an electronics engineer and I’ve been solving long, complicated equations on a regular basis for many years. It had me a little concerned that I’d never heard of BODMAS or arithmetic order of operations before. How can this be? I’ve done a lot of research and reached the conclusion that I’ve gotten away with it so the same reason that 80% of people failed the Facebook test – no-one really ever uses it. (At least, most of it).
    What I have discovered from my research and many conversations is that people are on one side of the fence or the other. It’s been difficult for me to persuade either side to see the other. It’s incredible how loyal people are to their beliefs.
    So, let’s clear a few things up… one of the arguments i have heard a few tines is “you need to have rules”. I think it’s safe to say that we all agree that we need rules. We all need to be following the same processes to avoid a mighty. I’m also confident that we can all agree that this is a rule that was decided upon a long time ago, as the method that we will all use. That’s all it is – the one way that was chosen – one way of many possible ways. Now… MY argument is that I do not believe that it is the easiest / most logical way.
    Next, I’d like to point out an obvious flaw in the BODMAS rule: BRACKETS first – no problem… Of – huh?… DIVISION – ok… MULTIPLICATION – the acronym implies that division comes first, whereas it has equal power to multiplication, you solve them from left to right. The same is true for ADDITION / SUBTRACTION.
    This is not a straightforward rule. We need to remember an order, and that some of the operators have equal power. If you come across 2 together, there is an exemption that we will now work left to right.
    Another common excuse I’m given for using this rule is that it means you don’t have to put brackets everywhere… even with this rule, brackets are still frequently required. This rule hasn’t added much value yet.
    I’d also like to counter the fans of this rule out there that talk about the wrong way and the right way. Just bare in mind that 80% of people do it the “wrong” way. Remember why we have this rule? So that we can all follow the same process.
    There is one final thought I’d like to discuss from my research. I’ve heard the exact same statement from 2 different people. “You can’t just go making up rules just cos it suits you!” Why not? How did we get BODMAS in the first place? If the complaint is actually meant to be ” you can’t go changing rules…” then again… why not? How come women vote? How come gays get married? Because we realised that the old rules didn’t work anymore.

    Lastly, regarding John’s post:
    This is a fine example of being firmly planted one side of the fence. John raised some points which don’t take much imagination to answer. Such as: “there is no way to show them how to expand the multiplication as addition.” Sure there is. You just did it. Simply apply the expansion technique that youve demonstrated to the 5’s. 5x5x5x5x5. Simple. You also said “where did all the 6,s come from?” Another simple answer: the same place all your 2’s came from earlier.

    I move to replace the order of operations with a simple left to right rule. If an equation solved in thus order does not achieve the answer you wanted, I put it to you that you must have written it in the wrong order.

    That said… I’m keeping an open mind. There might be someone who can actually demonstrate to me that the current rule is better that left to right.


    • You refer to BODMAS (which I do not see mentioned anywhere in this article or any comments). I believe you might be thinking of BEDMAS instead, which is brackets, exponents, division, multiplication, addition, and subtraction. (That explains your look when assuming that O stands for “of”.)

      Processing division before multiplication has no effect on processing multiplication AND division as it comes across from left to right.

      a*b*c/d*e*f, when processed left to right (as a spreadsheet does) gives you the same result as a*b*c*e*f/d. Processing division first gets you a*b*(c/d)*e*f which gives you the same result. However, processing multiplication first (which one might argue that PEMDAS implies) gives you (a*b*c)/(d*e*f) which yields a different result. For this reason, I prefer BEDMAS for stuff like spreadsheets.

      However, reading 1/2x with BEDMAS in mind, you should get x/2. This reading would be rarely expected in such a small equation.

    • Regarding your comment “I move to replace the order of operations with a simple left to right rule. If an equation solved in thus order does not achieve the answer you wanted, I put it to you that you must have written it in the wrong order.”

      Consider the problem where we have 2 groups of 5 oranges, one pair of two oranges, and 6 groups of 7 oranges. Using PEMDAS or BEDMAS, we can compute this as 2*5+2+6*7. In what other order can we calculate the total number of oranges? Reading it with BEDMAS or PEMDAS, we get 10+2+13=25.

      Reading it left to right, we get 10 from the 2*5, add 2 for a subtotal of 12 so far, add 6 for a subtotal of 18, and multiply that by 7, we get 126, which makes no sense at all.

      You claim to be an electronics engineer. I’m afraid I have to call you out on that.

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2 Responses to How to Teach a Weird Math Rule

  1. Maybe my reasoning is wrong but for some reason I usually think of negative numbers in terms of money owed. So a negative times a positive would come out negative. ex. If I owe three children five dollars each for an allowance -5 X 3 would be -15 or 15 dollars owed.

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2 Responses to The Order of Operations Explained: Exponents

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