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.

If you're a Gen-X parent, you're in the right place!

# Tag Archives: subtraction

### The New Way of Subtracting – Crushing a Facebook Meme

“Oh – I see what they’re doing,” I thought to myself after spending 20 seconds looking at the latest bash-the-common-core meme:

The “new way” is really the old fashioned way of giving change, before we had cash registers with computers in them.

### Getting change, the old fashioned way.

I buy something and it costs \$12. So I give you \$32. (Because I’m a freak OR I anticipate not having so many \$1’s in my pocket after this is over.)

Your fancy-schmancy cash register computer is out (or doesn’t exist), so you need to give me change out of your head.

Now theoretically (or in reality) you’d recognize what I was doing and say, “Gee, you want a \$20 bill back,” and you’d hand it to me.

But let’s suppose you didn’t recognize that.

Here’s how you’d do it:

You say, “Your total was 12 and there’s…”

While giving me three \$1 bills, you say, “…13, 14, 15…”

Then you give me a five dollar bill and say, “…20…”

Then you give me a ten dollar bill and say, “…30…”

Then you finish off with handing me two ones and saying, “… 31 and 32.”

Granted, I’d have lots of ones and be annoyed with you. But you would have certainly given me all \$20 that I was due.

### So why all the fuss?

Many people, including me, have written about how the common core isn’t a curriculum. It’s a set of guidelines to help teachers help students think about math.

Nobody says (or should say) that you must do subtraction this way. Instead it’s suggested that there’s more than one way to look at subtraction.

What do you think about the way people are freaking out about the common core state standards? How are you handling it?

Share your thoughts in the comments or tweet it out. And don’t forget to post on Pinterest too!

### Math Picture Book: The Grapes of Math

One of the best math picture books I’ve seen for counting and subitizing. Continue Reading

### PEMDAS and a Stupid Arbitrary Rule

The order of operations is rife with rules created merely for convenience. When teaching these to kids, it’s clear how they see it as a stupid arbitrary rule. Because often it is! Continue Reading

### 19 Responses to PEMDAS and a Stupid Arbitrary Rule

1. JG says:

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.

• Bon says:

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.

• Kaitie-Lynn Cohen says:

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!

• Bon says:

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.

• Bon says:

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. Math Teacher says:

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:)

• Bon says:

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!

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,

• Bon says:

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. John says:

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.

• Bon says:

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. Derek says:

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.

• Bon says:

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. Pete says:

Hi Bon.
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.

Pete

• Derek Schmidt says:

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.

• Bon says:

Thanks for stopping in, Derek!

In fact, the O is for “orders” – like powers or exponents.

• Derek Schmidt says:

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.

• Derek Schmidt says:

Wow. I must correct myself.

2*5+2+6*7 = 10+2+42 = 54. 54 oranges.

### 3 1/2 Ways to Do Math with Jesus

Seems we have more nativity sets than a family should – so why not do a little math with Jesus and the nativity crew? Continue Reading

### Why Engineers Make Bad Math Tutors

While playing Simply Fun’s Math’d Potatoes game with my parents, I was reminded how “math people” can really damper math learning. Continue Reading

### Order of Operations Explained: Addition, Subtraction and Conclusions

The Order of Operations series comes to a conclusion with a discussion of addition, subtraction and how textbooks often disagree! Continue Reading

### 14 Responses to Order of Operations Explained: Addition, Subtraction and Conclusions

1. Jam says:

what is the difference berween 3 + 4 – 2 and (3 + 4) – 2 in terms of writing the number sentence?

• Bon says:

Thanks for writing, Jam.

There is no difference in the value of those two. However, addition and subtraction are actually binary operations, meaning they work on two things (not three), so technically 3+4-2 isn’t “doable.”

Of course, we know it is. But only because we’re humans and we don’t need such formalities. At least not all the time.

Thanks for stopping by!

2. Dessy says:

So… I learned that though addition is before subtraction in “pemdas”, addition can still has to come after subtraction when the problem is: 5-3+2. The only way addition would come before subtraction, is if there were parenthesis around the “3+2” like: 5-(3+2). The correct answer to 5-3+2 is and can only be 4, since there aren’t any parenthesis… Enough thinking bye! I will never return, bye!!

• Bon says:

Thanks for stopping by, Dessy. Sorry to hear you won’t come back – but you won’t ever stop thinking. And yay for that!

3. corbin says:

what is undoing in reverse order?

• Bon says:

Good question, Corbin. It’s just sloppy talk.

Technically it would be going in reverse order, or undoing what you’ve done. Not really both.

Thanks for stopping by!

4. Actually, you made an error. 5-3+2.
If you did the addition first, you would have -3+2, you can not ignore a number’s sign. -3+2 = -1. 5-1 = 4. Sometimes this is easier for people not to mess up if they write out all subtraction as it truly is (an addition of a negative number). 5-3+2 is shorthand for 5 + (-3) + 2. Easy to forget.

• Bon says:

I don’t believe I made an error. My point is that you could (and students often do) look at addition separate from subtraction. If that’s the case, then “doing addition first” would mean that the minus sign doesn’t have the meaning “negative 3.”

It’s a subtle difference that math teachers, or even people well versed with math, can’t see – we’re too unconsciously competent.

But math students who haven’t mastered it don’t see the subtraction as “negative addition.”

See?

5. Fred says:

Just want to share a nice line where kids can practice addition and subtraction. It is free.

http://mathskillbuilder.org

It helps to my students.

• Bon says:

Thanks for sharing, Fred!

6. I disagree with you and Coach G about the order of operations being convention. I’ve noticed this a lot among math educators. They seem to think that the order of operations is just an arbitrary set of rules that people just decided some day, and the world jumped on the bandwagon. THIS IS NOT TRUE! The order of operations is NOT arbitrary. There is a REASON that we do multiplication before addition, and that is because multiplication IS addition: It’s iterated addition.
Example: 4*5 is really 4+4+4+4+4, which is 20. So given a problem 3+4*5, we MUST do 4*5 first, because the problem must be re-written as 3+4+4+4+4+4 (which yields 23). The problem CANNOT be written as 7+7+7+7+7, which is what you would get if you had done the 3+4 first.
Summary: the order of operations is NOT arbitrary. It is NOT convention. We as mathematicians didn’t suddenly “agree” on an order. The order of operations is the way that it is because it’s mathematically accurate!

• Bon says:

I think some would disagree with you on the face that multiplication is iterated addition. (I’m not one of those, though.)

Regardless, if you want to sit and have a glass of wine someday and play, we could certainly work out a number system where we switch around the order of operations and prove a bunch of interesting things.

In fact, for our real world purposes, we do need them the way they are. But not all of math is real world. Thus, the convention and arbitrary-ness of it.

7. John says:

Could you cite a credible source (preferably third party (in lieu to another link on this site)) that would prove that you do addittion before subtraction or you would do multiplication before division?

• Bon says:

There is no proof that you would do addition BEFORE subtraction. If a book (or set of people) decided that you should do that, then you would need to follow that convention.

See?

### The Order of Operations Explained: Exponents, Multiplication and Addition

Next in the series explaining the order of operations is this discussion on the relation of exponents, addition and multiplication and how to remember the order of operations… the natural way! Continue Reading

### The Order of Operations Explained: Exponents

Next in the series explaining the order of operations is this discussion on exponents and everything this includes. And roots and logs are in this one! Continue Reading

### 2 Responses to The Order of Operations Explained: Exponents

• Bon says:

Thanks for the link, Rebecca! Kate’s suggestion is awesome. I’ll start using it. I even tweeted it!

Another thing I sometimes do is teach the International Log Song – totally unrelated to math logs, but fun to sing. Totally loosens the students up! http://www.youtube.com/watch?v=zkNj1WXqDmw&feature=related

### Why Learning to Subitize Is Important

Learning to subitize is more important than learning to count – and knowing how to subitize comes before learning to count! Continue Reading

### 2 Responses to Why Learning to Subitize Is Important

1. I remember subitizing as a little girl because I got mixed up chanting to count, or lost my place. But I could see triangles, quadrilaterals, and groups of doubles, so I’d make shapes, then add them up.

I think it just happened during block play when my Mom needed me out of her hair while she made dinner.

Why I felt I needed to count my blocks I’m not sure though; maybe I was imitating counting books or Seasame street?

• Bon says:

That’s really neat Christine. I wonder if there’s research on shape seeing – there’s most definitely some better, fancier word! I’ve never been good at “getting” spacial stuff but I know others (my dad) who are very good at it. Are you good with spacial stuff – like being able to “see” things in 3D when they are just drawn or you only see one part of them?

And I’m sure Sesame street had a lot to do with all of our desires to count. What a great show!

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