Okay, let’s start at the beginning. For some reason, we have decided to screw kids up by saying that ‘equality means the same as’. If you are a parent or a teacher or an aunt or an uncle, or anyone who is talking to a kid about math – re-think. Equality does not mean the same as. Equality means the same value as. I know that sounds unimportant to those of you who think about other (more important?) things like flowers, or thank you cards, or sports teams, or beer festivals, or what you are going to eat for dinner, but I am going to argue that this is the root of all our problems (in case you haven’t heard, the U.S. is pretty lousy at math).

Remember when you were hating algebra and you had to deal with things like 3x – 36 = 0 ? You immediately dug your nose into your book and said to yourself, ‘okay, I’ll add 36 to both sides because Mr. Brook told me to (and wow, his polyester shirt is really sticking to his back today, I wonder if he can feel that?)’. But you could have just said to yourself, ‘well I know that 3x must be 36 because that’s the only way that side of the equation could equal 0 (and that must itch)’ (Note: remember that you get 0 by taking away all of whatever you have – I know you know this because you have played poker before and lost your entire pile of chips). Again, I know you maybe don’t care now, but if you know a kid who is suffering through this stuff get him or her to think about the fact that the equals sign matters and that all it means is that two things have the same value, not that they are ‘the same thing’. Start young. The example I use in work shops is two trucks being equal in weight to an elephant. I have no idea, actually, what trucks and elephants weigh, but let’s just pretend. If I know that those two things are equal in weight, does that mean they are the same thing? Really? Do you see what I mean? Won’t that play with a third grader’s mind?

So, instead of saying yes! You are so wonderful, 3 + 4 is the same as 7. Say instead, well no, 3+4 is not exactly the same thing as 7, 3 + 4 is decomposed into two parts (trust me they can handle the word decomposed and anything else accurate that you throw at them) and the 7 is in simplified form. But they do have the same *value*, and we call that equal. Quick – would you rather have 3 quarters and 4 dimes for a total of 7 coins or 4 quarters and 3 dimes for a total of 7 coins? Oh, yeah, that’s right, it doesn’t matter, they’re just the same thing…

Also, kids in the United States think an equals sign means ‘put your answer here’. So if they are given the problem 9 + 4 = ____ + 7, they confidently put in the blank spot, ’13’ (if you don’t believe me, try it on three elementary school kids and see what happens – if they get it right, write a thank you note to their teacher, or go to a beer festival). This happens because we only ever show kids equations like this 3 + 4 = 7 and never like this 7 = 3 + 4. Whoops. Turns out they were paying attention to us and we blew it by just doing with them what adults did with us. But, we can’t do that anymore. We have to think more about how we talk about math with kids. And we have to get them to see that when two things are equal it is an important piece of information. But we can write numbers and think about the same values in many different forms. And that’s when math gets interesting–deciding what form of a value you want. Is it the same thing to say that Susie O made 365/489 foul shots as to say that she shot them at 75%. They are considered equal, but the two different forms tell you different things because, well dammit, they are not the same. The first one tells you she got fouled a lot and is strong from the free throw line, the second form just helps you to see better that she makes 75 out of every 100 shots she takes (or 3 out of 4 if you want to think of another form altogether). It doesn’t actually mean she took that many shots, and in fact, someone could shoot 75% and literally only have taken 4 shots and made 3. oh. that’s weird. I thought they were the same thing because they were equal? math is stupid.

I don’ agree with you

3+4 is THE SAME as 7, in the sense that they mean(or denote) the same ABSTRACT mathematical number!!!!

An abstract mathematical number can be presented in many many “forms”, but all of them mean the SAME number!

We just have two “names” for that number.

LEt’s give an example from reality: Barack Obama=the president of United States=the husband of Michelle OBama, etc. ALL these are THE SAME PERSON. It’;s just that we can describe this same person in many different ways, but each way/form identifies the person uniquely. It is Barack Obama in all “Descriptions”.

What you tell with “the value” is more confusing, i think. What’s the “value” of a number?

An arithhmetic expression like 1+2+3-5 is a NUMBER FROM THE MOMENT YOU WRITE IT.

Of course, the number is the same as one, we mean, we have another, much simpler description for this number.

If you don’t agreee with me, and keep your version with “values” what is “a+b”, with a,b unspecified? It really is a number, even if we don’t have(and cannot have at all) a much simpler “form ” for it, if we don’t know exactly who a and b are.

IF you teach math, you may want to “meditate” more at stuff like this, your version is more confusing.

And also, if children give wrong answer to problems like 3+4=?+7, it is not entirely the teacher’s fault, it’s because the simply IGNORE the “+7”.

Call it lack of attention in reading/analyzing the problem.

“3+4=?” is one thing

but

“3+4=?+7” is a completely different.

Children have their own part of fault here, for answering the second one with the answer of the first.

Apologies for the delay in my reply! I just noticed this. Thank you so much for engaging in this conversation. It really helps to see your articulation of this argument.

First, I totally see what you mean. The underlying number is the same. That is absolutely true. And that is what I mean by same value. The underlying number is the same. But the EXPRESSION Of the number is not the same. And different expressions are used for different intents.

To teach children to think like mathematicians they must understand that they have to evaluate the form of an expression. As a writer I might choose to write “Today I watched the President of the United States give a speech” Or I might say “Today I watched Barack Obama give a speech” Or I might say “Today I watched the husband of Michelle Obama give a speech”. Yes they all refer to the same man, but each choice as a writer is a very different choice. They may Denote the same thing, but they Connote different things. Would an English teacher say “oh, it’s the same thing whether you choose to say ‘the husband of Michelle Obama’ or ‘The president’. No. I can tell you for sure that my English teacher* made a big deal over the difference between the connotation of a word and its denotation. As a writer I was asked to consider Both of these things if I was to develop my sophistication. And to notice clearly that these choices are not the same thing.

I ask my students to notice that while 3 + 4 denotes the same thing as 4 + 3, they are NOT the same expression and so are not completely the same thing. The square root of 9 has a different connotation than a simplified 3 does. The square root of 9 implies that we are considering a geometric shape, perhaps. I want my students to see these distinctions. I disagree with you that the distinction I make here makes it more confusing for students. I have seen just the opposite. Students who are utterly confused by the odd statement that 3 + 4 is the same as 4 + 3 light up when you point out that they are NOT exactly the same. They are two different ways to name the same number. If 4 – 3 is the same thing as 4 + -3 then why did you bother to make that change in the way you wrote the number? You did so exactly because 4 – 3 is NOT the same thing as 4 + -3. 4 + -3 gives you access to the commutative property and 4 – 3 does not. Now, the number they represent is the same, and that’s why you can make that change without violating the value of the expression.

In my experience, students benefit greatly from this conversation and this acknowledgment that the two expressions are both visually different AND operatively different. As a mathematician you must gain appreciation, just as you would if you were a writer, for the nuanced difference between expressions. A teacher is not getting at the meat of the issue if he or she glosses over the differences in the expressions and instead said ‘yeah, they are just the same thing’.

I hope you will consider these arguments. But please feel free to counter them in writing here.

* Ms.Joanne Coley