I see this sort of thing all over social media, and it bothers me. What’s going on in today’s classrooms is common sense. It actually makes much more sense than the way I was taught, which was called “new math” back then, in the 1970s, but whose final goal was to get me to use the standard procedure shown on the left in the picture above.
15 years ago, I didn’t know this. I would’ve been on the “what the heck?” bandwagon most people my age are when shown one of these “new math” examples. But then I became an editor who worked on math books (books that taught teachers how to teach math this way, because, yes, even teachers, most of whom had been taught the “old” way, needed to learn how to do it), and I found out what the “new math” example represents. I learned how much more I could have gotten out of my math lessons when I was a kid if I’d been taught the way math is being taught in many classrooms today. I visited classrooms where students were excited about doing math (how common was that when you were a kid?), where they were able to do complex arithmetic in their heads, where they eagerly attacked what my friends and I all dreaded: “the word problem.”
Not only am I bothered by these “old math”, “new math” examples that pop up all over the place, but I’m also bothered by some of the articles written by parents, who often end their rants about trying to help with math homework by waving the white flag saying, “And then I tell my kid, don’t worry. I was never any good at math, either. You won’t need to know this to succeed in life.” I want to shake that parent and say, “YES your child WILL need to know this!” There isn’t a single career in life, from homemaker to astrophysicist, that doesn’t require problem solving skills and an understanding of numbers. Most work places are run by one thing: the bottom line. Guess what. The bottom line is a number, a very important number, and your child better understand what it means when that number goes up and down, what it means when an employer expects 5% growth this year, what sales versus profit means. Also, if you don’t understand numbers, you may not understand you can’t really afford that outrageously-priced house the bank is happily giving you a mortgage to buy, until 2008 comes along, and you lose it and have to file for bankruptcy.
It is not acceptable in today’s global economy, where so much of the rest of the world’s children outperform ours in math and science, to think it’s funny to say, “Don’t ask me to add 2 and 2 together. I was never any good at math.” Would you dream of saying, “Don’t ask me to read The Cat in the Hat. I was never any good at reading.”? We need to create a society in which people are as embarrassed to admit the former as they are to admit the latter. Otherwise, don’t come complaining to me when major corporations are hiring people from Singapore and Japan to come over here and fill some very prestigious, high-paying positions. Those corporations need people who understand numbers, who can do the math, and they know where to find them. They certainly won’t find them in a place where people want to keep teaching math in a way that created generations of people who can’t do the math.
First of all, let’s define a few things. Mathematics, in a very simple nutshell, is the abstract study of numbers, quantity, and space. There is no such thing as “old math” and “new math” any more than there is such a thing as “old history” and “new history”. New discoveries, new theories, new problems, etc. arise in the field of mathematics, just as they do in any field, but math itself is old. There’s nothing new about it, and the mathematical theories most kids study in school? Very old.
I’ve also seen this “new math” defined as “common core” math. It isn’t. If you’ve read this far and get nothing else out of this blog post, please don’t make yourself look stupid by calling this “common core math”. Just because more and more schools began teaching math with understanding, using unfamiliar techniques, around the same time the common core curriculum standards were being adopted, doesn’t mean there is something known as “common core math”. The common core is not about technique. It is a curriculum standard. It states that, say, your kindergartner should be able to count to 100. It says nothing about how that child should be taught to count to 100. Oh, and by the way, before the common core came along, your state had curriculum standards. Every state did. The common core is just an attempt to unify these standards across states, so if, for instance, you move from Pennsylvania to Virginia, your child won’t be bored to tears learning how to add 53 and 37, because in Pennsylvania he was taught to do that two years ago.
What you have in this “common sense” example pictured above is arithmetic. Arithmetic is a branch of mathematics, the branch involved with the use and counting of numbers. What you also have are two algorithms or sets of procedures for calculating an answer to a problem. We don’t really know what the problem was here. All we know is that it involved adding the two numbers 53 and 37 together, which means the problem very well could have been something like this, “Last year, my boss said she wanted me to make 53 widgets. I stopped counting once I hit 53, but she tells me I made 37 more than she asked me to make, which is why I’m getting promoted. In this new position, I have to supervise people and make sure each person makes at least as many widgets as I made last year. How many widgets does each person have to make?” You need to add 37 to 53 to get the answer.
Now, you can waste the company time as well as resources by getting out a piece of paper and pencil and doing this the “old” way, using the “old math” algorithm on the left, which is basically what you have to do if you were taught arithmetic the way I was, and no one has shown you a better way, which means you’re no good at adding two-digit numbers in your head. Or you can fumble around in your pocket or purse for your phone, scroll through all your apps to find your calculator, punch in the buttons, and get your answer. Or you can quickly add that number in your head and come up with 90, which is basically what that “new math” algorithm is representing. It looks very awkward on paper (and it is), but the point of teaching it this way is that the child will learn to do these calculations in his head. Do you remember that kid you thought was so brilliant because you could turn to her and say, “What’s 345 plus 172?” and within seconds, no pencil or paper needed, that kid would reply with “517.” She probably was brilliant, because without having been taught this procedure, she understood the numbers well enough that she’d figured it out on her own. Most of the rest of us have to be shown how to do it, but that’s what’s happening in classrooms all over the country. Kids are being shown how to do it. They’re learning how to do such calculations in their head.
I can explain to you what’s going on in the “new math” example. The student has been taught how to break numbers down into easier units. He knows that numbers ending in 1, 2, 5, and 0 are very easy to work with, and he can easily count by 1s, 2s, 5s, and 10s. He has also learned, through hands-on representation of those numbers (think five cute little plastic teddy bears, or seven red sticks, or three jelly beans), how to add all single-digit numbers from 1-9 and has them memorized (these are some of the “math facts” you may hear teachers and students talk about). He knows that 37 represents 3 10s and 7 ones. 7 represents 5 ones and 2 ones. Add 5 to 53, and you get 58. Add 2, and you get 60, and now you just add those 3 10s to get 90. That’s just one way to do it. Me? Since I know 7 and 3 is 10, I would’ve chosen to add 50 and 30 to get 80, 7 and 3 to get 10, and added 80 and 10 to get 90, because I find that easiest. The point is to do what you need to do in order to make the numbers easier to work with.
Now, let’s look at the “old math”. If no one had ever shown you how to do that, would it really make any (let alone common) sense to you at all? What on earth is going on there? Why are the numbers stacked like that? What is that odd floating “1” on top of the five up there? And would you have any idea that the person solving this problem worked from right to left, unless someone had shown you that’s how it’s done? In this country, we read from left to right. Just when a child is mastering reading from left to right, we tell her, “Now, I know you’re used to going from left to right, but we’re going to go from right to left when working with big numbers.” The only reason it makes any sense to you at all is that someone taught you how to do it that way.
In one sense, it was a great way to teach arithmetic, because you could fill a page with such problems on a standardized test and have kids prove (or not) they’d memorized a procedure to get an answer. And if you were like me, good at following and memorizing rules, you were pretty good at it. You picked up on that procedure quickly and could get all the problems on a page correct (if you didn’t rush and make careless mistakes), but you didn’t always understand exactly what you were doing (which is why you didn’t catch your careless mistakes), or have any real understanding of how much bigger 90 was than 53 when you were done, which is why, when it came to higher level math, you struggled. In order to do higher level math — the sort of math being required more and more in this technological age — you needed to have a deep understanding of those numbers. Imagine teaching kids individual words without teaching them reading comprehension. Or teaching them grammar without teaching them how to write sentences and paragraphs that make sense.
Yes, the procedure on the right looks awkward on paper (you can probably blame that on standardized testing, too, because such tests require demonstration on paper), but what it represents is very smart, makes a lot of sense. Once a kid gets it, he or she can also be taught the “old” algorithm, which will make more sense to a child who actually understands the numbers than it did to all those people who grew up hating math. Don’t be surprised, though, if he chooses not to use it. It just isn’t as efficient. That kid might also use other techniques for adding numbers, because the fact is, there are almost always multiple ways to get an answer in math (bet no one ever taught you that, either).
Speaking of efficiency, kids are also being taught the best, most efficient way to do a problem. Sometimes — gasp! I know! — this means taking out a calculator to find out what 5,678 divided by 1,897 is. Believe me, by the time the child is taught that the calculator is the way to go with this one, she will know what division is and how to divide (at least, if she’s got a teacher who’s been taught how to teach math this way) and will be ready to move onto higher math.
Still don’t believe me? Take a look at this brilliant example of “old math” versus “new math” from someone who actually teaches math.
Here, I hope you can clearly see that the algorithm, on the right, representing 3000 – 1, the way we were all taught, is an absurd way to do this problem.
My final word? If you’re having trouble helping your child with his math homework, because you never got math, hated math, and find nothing in his work book resembling the algorithms you memorized, don’t despair. You can learn to do math the “new way”. You might even discover you enjoy it. Go online. Talk to teachers. Buy some books. Just, please, don’t tell your child it’s “okay” to be bad at math.