I’m not a math teacher. I’m not a mathematician. My father, the risk analyst, is a mathematician. We used to play math ‘games’ and I thought that was normal. I was also a very odd child. My mother was convinced something was ‘wrong’ with me because I did “Number Roll” all day at school for the better part of a year. To anyone not from a Montessori school, Number Roll is a bewildering concept, where you just write numbers, incrementing by one, over and over and over again, getting the numbers as high as you could go.

Now, I was always (am still) a hands on learner. Being forced to learn *anything* by rote memorization is painful. But math is a little different. You can’t ‘understand’ math until you’ve mastered counting. You can’t grasp all the relationships between quantities and numbers without knowing the numbers first. It’s like you can’t learn spelling until you memorized the ABCs.

Number roll is crazy basic. On long sheets of paper, I wrote each number in order, beginning with “1”. I should stress, my mom worried about my intelligence, that I spent days and weeks and months doing this. But what I was really doing was following patterns without knowing it. I mean, I can do my nine-times tables because I know the ‘pattern’ is Plus Minus. Watch:

09 18 27 36

That didn’t make any sense to you? Start with 09. Add 1 to the left and subtract one from the right. Now it’s 18. You do this over and over and over again and it works all the way down. This repetition taught me pattern recognition in a different way and gave me insight into both counting *and* the meaning behind it. The nines work like that because 10 – 1 = 9, so then logically I could apply this to everything! This is where number roll was suddenly magical, as the Montessori concept is that **before** children can gain a meaningful understanding of quantities, numbers, and the relationships between them, they need to learn basic counting, but you should understand what counting means.

In other words, math will make more sense if you can see how the numbers fit together.

This is probably why, when I was a kid, I did my multiplication ‘backwards.’ That is, if you asked me to do 123 x 24, I did it left to right. Let me explain. This is how you probably do it:

123 x 24 ---- 492 246 ---- 2952

Right? You start with the bottom right, so you go “4 times 3 is 12, carry the 1, 4 times 2 is 8 plus one is 9, 4 times 1 is 4.” Most people I know would call this ‘traditional’ math. My math goes left to right, so I get this:

123 x 24 ---- 24 48 72 ---- 2952

1 x 24, then 2 x 24, and finally 3 x 24. I can do this fast because I’ve memorized my times tables, but at one point a friend asked me how this was really left to right, because when you look at 3 x 24, you’re back to the old “3 times 4 is 12, carry the 1. 3 times 2 is 6 plus 1 is 7.” Well, when I do *ALL* all the work, it looks like this:

123 x 24 ---- 2000 (100 x 20) 400 (100 x 4) 400 (20 x 20) 80 (20 x 4) 60 (3 x 20) 12 (3 x 4) ---- 2952

The difference really is I’m breaking apart multiplication into smaller addition steps. And now it makes sense to a lot more people. “100 times 20 is 2000″ and so on. Once it’s spread out, it’s easier for someone new to pick up how I did it, and in a sense, why. It’s true left to right, all the way down. I don’t generally do long-form math this way any more, though, because like everyone else I had to learn the ‘real’ way of doing it, but also I started to memorize the patterns. I know without really thinking that 12 times 2 is 48. It’s a common enough equation that I memorized the answer.

That means I can do all this in that even faster way you saw above. I just know that since 1 times anything is itself, the 4 from 24 goes under the one. Sometimes I have to remember to mark my place, if I’m doing less frequently combined numbers (I don’t seem to use 7s times 9s a lot). When that happens, I usually add on the zeros:

123 x 24 ---- 2400 480 72 ---- 2952

When I don’t, to make sure I keep my place, I go far left top to far right bottom, since those two have to line up. That means I know the “1 times 4″ answer (4) has to be under the 1, and the “1 times 2″ answer (2) is one to the left. But that’s the advantage of understanding how all the numbers work together, and sets. I know how certain numbers combine, I’ve memorized their patterns, and I can apply them backwards and forwards not because I know the equations, but because I see the pattern.

Now on to the rather controversial image I posted recently:

This shows you two ways to solve a problem. First is the ‘traditional’ way, or as I’ll call it, the fast way:

32 - 12 ---- 20

At it’s heart, this is a simple equation. Most of you went “Sure, 3 minus 1 is 2, the 2’s are the same, so 20.” Some of you went “1 plus 2 is 3, so it’s a 2…” Both are correct. Then you get the ‘new’ way:

32 - 12 = __ 12 + [ 3] = 15 15 + [ 5] = 20 20 + [10] = 30 30 + [ 2] = 32 -------------- 20

And a bunch of adults just when “LolWHUT!?”

When I saw this math problem, the first thing I did was the same as you “Why 15!? What?” I mean, we’ve all been told “Show your work, don’t pull numbers out of thin air!” Then I thought back to when I was a kid trying to understand this whole math thing. Fives were easy to remember: 5 10 15 20. It’s either a 0 or a 5, and the number in front went up by 1 every 0. We all kind of got that pretty fast. Number Roll (see?) taught me that concept really early on. That was my lightbulb moment.

“OH! We’re adding X to 12 to get to the 5s, then we add Y to get to the tens, then Z to get to the base of 32 (30), and add the leftovers Q. Add up X, Z, Y, and Q, you get 20!”

This is what I would call “the long way” however the thought occurred to me that this was a number roll-less way to try and teach children how numbers came together! Common Core (which is where this comes from) is actually sneak-teaching kids algebra, while at the same time giving them a reference for that rote memorization they had earlier. You remember your 5 times tables? This is how we use that information in a practical application!

Part of the difference comes in if you think about subtraction as ‘Something new’ or ‘backwards addition.’ I tend to think of it as backwards addition, and multiplication is ‘Faster addition’ (division is ‘faster backwards addition’). I was fairly young when I realized that all math was really, at it’s heart, the same, it was just the formula you slapped in to make it messy. Everything comes down to adding for me, always. We’re all just playing fast ways to do things and solve problems, and this is starting with the long way first.

All this comes back to what Richard Feynman wrote in the essay New Textbooks for the “New” mathematics:

If we would like to, we can and do say, ‘The answer is a whole number less than 9 and bigger than 6,’ but we do not have to say, ‘The answer is a member of the set which is the intersection of the set of those numbers which is larger than 6 and the set of numbers which are smaller than 9′ … In the ‘new’ mathematics, then, first there must be freedom of thought; second, we do not want to teach just words; and third, subjects should not be introduced without explaining the purpose or reason, or without giving any way in which the material could be really used to discover something interesting. I don’t think it is worth while teaching such material.

It’s his third point that I believe Common Core is trying to address. How many of you were taught the purpose of your times tables, after all? How many of you understood the reason besides ‘so I can pass the class’ that we learned to think of numbers and how they were put together? A lot of people seem to think that Feynman didn’t like kids to learn the application of math, to understand what it meant, but that’s incorrect. He rallied *against* new math because it lacked word problems and applications of use! Yes, you hated those word problems, but they were meant to teach you application. Instead most people learned how to pick out the important bits and do the math as a simple formula to which they could apply that rote memorization.

There’s a problem with this, though, and Common Core has the same problem that New Math does and that the ‘traditional’ way did back when I was a kid, so this is nothing new. It forces kids to learn in one way, and one way only. I was incredibly lucky in that my father let me do math my own way (he found it interesting), and once I showed my work (see above) he and my teachers saw that I had in fact achieved the absolute goal of number roll: **I internalized the connection between math equations and the numbers**.

Rote memorization has a place. You memorize the tables, you can do math *faster*, and things like calculus will be surprisingly easier to you because all you have to do is put the numbers into the formula. At the same time, some of the other concepts will be a struggle because you don’t get the connections, you only know memorization and implementation.

I will note that once you’ve memorized this stuff, it’s all a lot faster. I tend to count on my fingers when I’m trying to math days of the week (like today is the 5th, so next Wednesday is 12th) because I’m messing with names (Wednesday) and numbers, and then I have to remember how many days are in March, but I can do all this in my head, including calculating tax. And no, I don’t think it’s ‘cheating’ to use a calculator. The point is understanding what the relationships between the numbers are, knowing what formula to apply when and where, and enjoy it.

That was the goal of New Math, you know. To make math something kids wanted to do. You should read Feynman’s “Surely you’re joking, Mr. Feynman!” and follow his account of being on the board to set up these new curriculums, and you’ll see exactly why they continue to fail over and over. It’s a pity, too, since I bet some kids are looking at the Core method and there’s a lightbulb going on over their heads. I hope parents aren’t scaring their own kids off math because the adults don’t understand this new stuff.

Of course, a lot of this is the fault of the school system, in that the parents aren’t taught what the kids are learning or *why*. If you’re learning something at school, and at home your parents go “What? This is bullshit!” you’re going to have a harder time learning and accepting. Don’t believe me? Creationism. You’re welcome. The point being you have to reinforce what a kid learns at school in the home or they have to come to terms with the dichotomy of difference at a stage when they don’t understand enough of the world to get what that meant.

Not that having multiple choices is great for every kid. Some people freak out when there’s more than one right answer, especially in math which in the beginning is remarkably straightforward (like spelling). There’s *one* right answer, but now you’re giving them multiple paths (spelling has this too, by the by: color, colour; grey, gray). It breaks brains. This, perhaps, is a little bit why WordPress is “Decisions, not options.” Maybe we’re giving people the options too soon, but when it comes to learning, we adults should already know there isn’t one ‘right’ way to learn and master skills. And with math, there isn’t going to be one perfect way to get those base concepts into their minds.

I loved this post. Our kids are in a charter school where we home school 3 days a week and send them to classes 2 days a week. As a Feynman fan, and someone who learned math in alternative ways (to keep up with my younger brother’s conversations with my dad at the dinner table), I appreciate how well you articulated things.

@Chris Lema: I admit, the state of the school system makes me think two things about having kids.

1) I would move back to Solana Beach/Del Mar and send my kids to the Montesorri school I went to.

2) I would home school.

It’s not really their fault. They’re trying to mass produce education and get ‘most’ kids at the basic level of reading and writing. Of course, learning like that has a cost (I have a massive rant about how kids today are taught to memorize, not to think). But … what can you do when people are struggling to make ends meet and don’t have the time to sit and teach kids math like my dad and Taffy did for me? You have to dump your kid in school and hope for the best.

Math is one of the easier aspects of our education system flaws to point at, since it’s got static answers. 1+1=2 and that’s what it is. How you get there, that is how you learn, is the part the schools are left with, and even though we KNOW there’s not one right and perfect way for all kids, we don’t give teachers, schools, or parents the ability to do anything more than what we’ve done. *sigh*

I may think too much on this, for someone who doesn’t have kids…

I appreciate your post and perspective. My daughter is 6, in first grade, doing very well in math, but I’d like to understand “new math” better. The picture you tweeted yesterday and then explained yesterday on Twitter helped me understand how they’re learning their addition right now. Tonight I was able to help her with her homework in the same way she’s learning at school because of your explanation. Of course, I still have my “traditional” ways to revert to if I need to, but I can see how these methods are helpful.

Tonight, she was adding x + 10 = 52, and said “20 + 20 + 12″ is the answer” but it then took her a few minutes to get to 40 as the actual answer. In my mind, I thought to myself, “It’s like she’s doing beginning algebra to get to her answer (x + y) + z = (a)” I thought it was interesting then that you said as much in your post.

She is good at math, and I want to encourage her to not be afraid of math. Your post is encouraging to me. Thanks.

@Angie Meeker: Funnily enough, I ended up writing this because a parent friend of mine absolutely refused to even consider this as a ‘valid’ way to learn math. She’s a little stuck on how there’s only one way to learn a thing still :(

Mika is correct and brillant. Long live the Number Roll. Mika, don’t you do the number roll on a big roll of adding machine tape?

Now what’s really exquisite are mathematical proofs. Here is an easy one:

Proof that the squares of odd numbers are always odd

1. 2n+1 is an odd number by definition (2n is always even because it can be divided by 2 evenly and any even number plus 1 is an odd number)

2. (2n + 1)**2 = 4n**2 + 4n + 1

3. 4n**2 + 4n + 1 = 2*(2n**2 + 2n) + 1

4. let 2n**2 + 2n = m

5. therefore (2n + 1)**2 = 2m+1 which is odd by definition

And proofs by mathematical induction are magic.

@woody: I believe so, but I can’t be sure because mom didn’t save all my number rolls!