#### (Archive question of the week)

One day back in November, as I entered the community college where I teach, a man came in behind me with his son, probably about 3 years old. It was the first snowy day of the year, and the father stood on the mat wiping his feet, while the little boy continued on into the tiled hallway. “Better wipe your feet,” the father said – and the boy, right where he was on the tile, starting scuffling his feet, just as his father was doing. “No, come here on the mat and do it.”

It occurred to me as I stood there waiting for the elevator, that this is how we learn – life, math, and anything else. The boy initially copied his father in the way he moved his feet, surely thinking he was doing just what he had been told. But with no understanding of how foot-wiping works, he didn’t realize that doing it where he was would not have the desired effect. He probably didn’t even know why it should be done at all. He *copied the surface attributes* of the action that he was being taught (the visible motion), but missed the *deeper aspects* of it (its purpose, and the conditions under which that will be accomplished). This is perfectly understandable, as this was surely his first exposure to the concept.

Many students see only the *surface* of a math problem and of the methods employed to solve it. They think they are doing what they were taught, and are then confused when it doesn’t work: “But I did just what you told me to do. I followed your example!” They have not yet learned how to hear the instructions, and to see the example, with real understanding, because they are not yet mathematically mature.

Here is an example from the Ask Dr. Math archives that illustrates this point.

Order of Operations Disorder, and Reverse Order

A mother wrote to ask why, when she solved the equation \( 2 \left | 2x – 7 \right | + 11 = 25 \) by a method she had learned, without consciously thinking about rules, she got the “correct” answer, but when her son “followed the rules he’d been taught”, he got a different answer.

First she showed her own solution:

$$2 \left | 2x – 7 \right | + 11 = 25$$

$$2 \left | 2x – 7 \right | = 14$$

$$\left | 2x – 7 \right | = 7$$

$$2x – 7 = 7 \text { or } 2x – 7 = -7$$

$$2x = 14 \text { or } 2x = 0$$

$$x = 7 \text { or } x = 0$$

Then she showed her son’s way:

... My son, applying proper order of operations, took the original equation and multiplied both sides by 1/2 to get rid of the 2 before the absolute value. This gave us |2x - 7| + 11 = 25/2 Then he subtracted the 11 on both sides to get the absolute value expression by itself, leaving us with |2x - 7| = 25/2 - 11 = 3/2 Adding 7 to both sides to get 2x by itself, 2x = 17/2 Multiplying by 1/2 on both sides to get x by itself, x = 17/4 Setting aside the second case, I already see that this first solution does not match either my answers of 7 or 0. By his logic, he is following the order of operations. Doesn't that instruct us to multiply both sides by 1/2 first? I subtracted the 11 first, but I don't know why I did it -- and I cannot explain to him why he is wrong -- so I don't know what to tell him. How do we know which way to do it? And why was I correct and he incorrect?

Her work had been exactly right; but without understanding *what made it right*, she had no way to either correct him or trust her own work. She wanted, and needed, to know *why*, in order to help him. And her son needed to know *what made his work wrong*, in order to understand her correction. Just telling him to do it her way would not be enough.

Two of us (my twin brother and I, as it happens – we are often attracted to the same questions) provided two complementary answers.

First, as Doctor Rick pointed out, the son, in “following the order of operations”, was doing the right thing in the wrong situation: to *evaluate an expression*, you multiply first and then add; but in *solving an equation*, you must reverse the process. This is not uncommon: Students too often learn something like PEMDAS by rote, sometimes even reciting the magic words while they work, without knowing when it applies and when it does not. (They are wiping their feet on the tile!)

The order of operations is the order we follow in EVALUATING an expression. ... ... But in SOLVING an equation, we use the order of operations IN REVERSE. That's because we want to UNDO the operations that we'd do in evaluating each expression, to get back to a "bare" variable -- the x alone on one side of the equation. And to undo an expression, we undo the LAST operation first. It's like your shoes and socks. In the morning, you first put on your socks, then you put on your shoes. But at bedtime, you first take off the LAST thing you put on -- your shoes. Then you take off your socks, once your shoes are out of the way.

To put it another way, the order of operations tells you how to *interpret* an expression you are given; it is not about the order in which you should *do things to* an equation.

But there’s more! As I pointed out in the second response, he was not really doing what he said he was doing in the first place, because when he “multiplied both sides by ½”, he really only multiplied *one term* of the left side – the one his attention was focused on. In fact, if he had really done what he said he was doing, he would have found the correct answer!

You CAN solve by dividing by 2 first, just as you can take off your socks first ... IF you don't mind stretching them out a bit, and pinching your fingers badly. But you still have to take legal steps. When you divide both sides by 2, that means dividing each ENTIRE SIDE, not just the terms you care about.

I went on to demonstrate what really happens if you multiply both sides by ½ first; it’s a little less pleasant, involving fractions, but it all worked out nicely in the end – because I actually *did* what he said he was going to do.

So not only his understanding of PEMDAS, but also his idea of the steps of solving an equation, were superficial; he was again copying the surface behavior of his teacher, but not really doing what he was told to do at all, because he didn’t know what it means to “do the same thing to both sides”, or why that works. He was going through the motions of wiping his feet, but in the wrong way.

When you understand why what you do works, you can break all sorts of "rules" that are really just recommended procedures, finding shortcuts or variations; but if you break the actual "laws" governing the changes that keep an equation's meaning unchanged, you will get a wrong answer.

This is part of our purpose at *Ask Dr. Math*, and now at *The Math Doctors*: to promote genuine understanding, which not only helps us to do the right thing, but also frees us from slavery to mere “rules”, opening our eyes to see alternatives and know that they are legal, even though they are not usual. (If all the coefficients had been even, dividing by 2 first would actually have been a very good choice!) I am always pleased when my students point out such alternatives, rather than feeling they have to do just what I have taught them. That is the hallmark of real understanding.

As the mother said in response,

Thank you so much! You have really helped us to acquire actual UNDERSTANDING -- not just writing down answers. Awesome!

By the way, the mother’s question was very well written, with all the detail we could have asked for; you might want to use it as an example when you write to us. She, too, was awesome!

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