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: division

2 Ways to Do Math Problems

This is Day 17 of 31 Days of Math Learning Success. Follow all the days here and check out others that are writing for 31 days here.
Many math problems boil down to being a choice between two things. Why not try them both out?
There are two kinds of people in the world — those who divide everything in the world into two kinds of things and those who don’t.
~Kenneth Boulding

When I go to calculate my gas mileage, I do it twice. One way is always wrong, and one way is always right.

I know that I have to do something with the number of miles and the number of gallons. In particular, I know I have to divide them. But I can never remember which number divides by which. Or which is the dividend and which is the divisor.

But it doesn’t really matter. There’s only two numbers. And there’s only two ways to do it.

I drive 245 miles on a tank of gas that’s about 20 gallons. I know it’s either 20÷245 or 245÷20. The first one gives 0.0816, And the second one gives 12.25.

Neither of these is that desirable, I’ll admit. I drive a 1995 Chevy pickup. And I love it.

But there’s no way my gas guzzler gets 0.0816 miles per gallon. If it did, driving to the neighbor’s house would use a whole tank of gas.

So I know that the second answer, 12-ish, is the correct answer.

Experiment with numbers.

The “translate statements to expressions” is another common “two-way” problem.

“7 less than x” screams subtraction. And as soon as you have a little experience with this, you know that it could be 7-x or x-7. But which one?

Well, suppose x is 539 (pick your favorite crazy number here). Then 7 less than 539 is 532.

So that was 539-7 or x-7.

Try it both ways.

It’s like trying on a wedding dress. You don’t have to know what dress you want. You can try them on and see which works better.

And unlike wedding dresses – there are only a few (or two) ways to do many problems.

Picking a wedding dress, well, you have bazillions.


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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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One Response to Long Division with Sumology Number Tiles

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One Response to The Order of Operations Explained: Parenthesis

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7 Responses to The Order of Operations Explained: Intro and Mnemonics

  1. I don’t use a mnemonic. I point out that addition and subtraction are low level (basic) operations taught first in school. Multiplication and division are built on the concept of repeated addition, so they are higher level operations and are taught after + and -. Exponentiation, and its inverse taking roots, are built on repeated addition. They are the highest level operations.

    The rule is to do higher level operations first. Use parentheses whenever your intention is to deviate from this rule.

    • For some things I don;t use mnemonics, you explain the logic very well. Soh-cah-toa though I was also taught (it’s sos-cas-toa in Dutch though…).

      Mnemonics might be useful in the beginning, but teaching the logic behind them would pay off much more. There is an increasing trend (in general, not just OoO) of the system putting value on memorization as opposed to understanding.

      These are the kinds of kids who are taught a “trick” and will do fine on most tests. If you however change the context or question they will be lost and have no idea what to do…

      • So true, Mentor! The tricks only take you so far before you run into something new and novel. If you understand the logic, you can adjust the tricks to fit. Or just figure out what to do next.

        Thanks for stopping by!

    • That’s a great way to remember it – and teach it, David. I wonder if they’re totally comfortable reversing it when they do algebra. I would think practicing it this way should help it stick and become internal for them. But I would love to hear if that was true.

      Thanks for your comment!

  2. I am firmly planted in the conventions field. I used an analogy to getting dressed, using underwear as the 1st thing we put on out of the bath, and so forth. I began in fourth grade and they, and later fifth graders, found mnemonics helpful. Now in middle school, I try and teach more depth. Soon they come to other grouping symbols (GEMDAS) and find old favorites need to be adjusted. when using devices I have found it useful for many to create their own personalized versions. It is important (I think) to keep in mind these are really only aids to get us started.

    • I completely agree, Tom. I’ve never tried the underwear analogy. I use pants and shoes. I think perhaps the underwear analogy will be a little funnier. I’ll give it a shot.

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