Tuesday, October 6, 2015

Why the Forrest trail is so long (Part IV of the case study)

If you just got here, this is one of about a week-long series of blog posts about Singapore Math and number sense, and how Singapore Math techniques can help kids through The Wall, that barricade of "this makes no sense" that most kids run into somewhere between long division and elementary algebra. Much of this material will be appearing later in my forthcoming book, Singapore Math Figured Out for Parents. The book draws on two roots:
  1. I've done a fair bit of science and technology journalism and understand educationese pretty well too; I'm used to explaining more-technical matters to a less-technical audience.
  2. I tutor math to elementary and middle school students for Tutoring Colorado, and I've seen how well these methods can work. 
Another qualification of sorts: I've spent a fair bit of time teaching ADLs, Adult Disadvantaged Learners, people in their 20s-50s who are having to painfully pick up what they never got in school.  That has given me an all-too-clear picture of what the dead end of innumeracy really looks like, why it matters that just as many kids as possible get a decent start in math, and how hard it is to recover from a bad start later. I  really wish I'd known many of the Singapore Math tactics when I was teaching remedial college pre-algebra and beginning college algebra!
The series to date has included
a questionnaire to evaluate your own number sense (if you're going to help your kid get it, it helps to have it or acquire it yourself)
•and three episodes before this one following a case study of the mathematical adventures of a beginning fourth-grader named Forrest. Despite being a composite of several different students with difficulties, Forrest made quite a bit of progress in those episodes, progressing through
1.  a general diagnosis of a memory problem and a conceptual difficulty with perceiving numbers as existing apart from what was being counted, to
3. the breakthrough moment when Forrest caught on to numbers as numbers, which ended with the warning note that breakthroughs are only beginnings, and that it's the practice afterwards that cements the breakthrough and makes it last.
And now, about that practice. If you don't read any other post in this series, this might be the one that gives the clearest idea of what Singapore Math is all about (at least, if I understand it correctly and I'm doing my job, two things of which you must be the judge).
Now that Forrest had a real idea of what numbers were, and how they connected to each other and to the world, he could see why his parents and teachers had been on his case to learn addition facts. He also had a much better understanding of what addition facts might have to do with the rest of math. All of that gave him much more motivation, but that didn't necessarily make the addition facts any easier to learn. If anything, it increased the urgency and made him impatient.
Forrest's mother confirmed he was continuing his practice at home with the addition table board, and was beginning to complain that he was bored and it had become too easy. That meant it was the perfect time to introduce a more complicated trick.
"Let's try you out on this one," I said. "Whenever you have one value, you have all the values around it." I put a tile down at 6+3=9 (that is at the intersection of the 6 row and the 3 column, I put down a 9 tile.)

"Now, instead of a row, you're going to make a spiral. Watch how this works. You put the tile down to the right of the first one -- 6+4=10." I point and he does; so far, of course, this is just like doing a row.
"Then we wrap around." I point to successive squares and ask him to say the sum and place the tile. "7+4=11, 7+3=10, 7+2=9, 6+2=8, 5+2=7, 5+3=8, 5+4=9 ... " 
Here's what it looks like, with red arrows to show the order in which they are placed.

"You see? Now next we wrap around some more, so 5+5= ..."
"Right. Do you see where we go next?" I have to correct and steer him a few times, but soon he's doing the spiral pattern correctly, and gaining speed as he goes.  When it seems to be well-established, with another circuit and a half completed, I say, "So you see how it works. when you go up or to the right, the sum goes up by one. When you go down or to the left, the number goes down by one. And as you lay the tiles out in a spiral, you form that spiral pattern."

"What do I do when my spiral hits the edge of the board?"
"What do you think you should do?"
"Maybe skip down to the nearest blank space?"
"That might work."
"Or I could just start a new spiral on the board somewhere, and grow it till it runs into this one."
"That would work too. Why don't you try it a couple of different ways and tell me what you think?"
Long experience has taught me that boisterous kids like to make spirals run into each other, and then have some complicated rule for managing the collision. Quieter kids, especially ones who just want to get done, tend to try to figure out ways to get things back to running up and down rows and columns. After some debating, Forrest hesitantly made the boisterous choice, and started growing a new spiral around 8+8=16.
"Now there's something else you need to do. Every time you turn a corner, take a long breath, and look at the tiles you've laid down. Just imagine your mind is taking a picture of it. Try that now."
Pretty soon he had a rhythm going, and started building simultaneous spirals, taking turns adding to each one, so that they would collide. That small-child passion for patterns kicked in, embracing saying the addition facts as he did them. For a kid in remedial math, he seemed to be having a pretty good time.
Then a moment of panic: he stumbled at "eight plus nine". He tensed up all over.
"Deep breath," I said, "and look at what you already have.  You've got eight plus eight on one side of it, and seven plus nine below it, and you know what they are, so the square you want has to be -- "

"Seventeen!" He was pretty excited; things were still making sense, after all.
"Say the whole thing, and point. Every tile you put down, say the whole problem. If you don't know the answer automatically, use the layout of the board to see what it has to be, but once you do see it, be sure you say it."
He looked a little stubborn, probably realizing how quickly he could lay out the table if he ignored all that addition stuff and just filled in the sequences.
I asked him, "So what are we doing this for?"
He shrugged. "It's not as boring as flash cards. It's not as hard."
"All excellent reasons, but here's another one. You're training your memory to find its way to the answer. There's four things that build memories, and if you can use all of them at the same time, they make very strong memories that last for a long time.  The first big memory builder is concentrating on what you're doing. Do you see that if you started just laying down the tiles in order, you wouldn't be thinking about the numbers anymore?"
"I guess not."
"You have to think about them and pay attention to them to build the memory. Pointing and saying makes you think about them a little more. It also makes you do the second thing that helps you learn: repeating a thing over and over. So ... get on with it, Forrest. You've probably almost got the whole table already, just from all the repeating and concentrating you've been doing in practice."
He finished a couple more spirals, and now the board held just a scattering of spaces to fill in.
"Now, this is where you can see the other two things that build memory. One is relationship." I pointed to the blank space at 9+6.  "What does that one have to be?"
"Exactly! Now, how many ways did you know?"
He looks puzzled, which is normal at this stage, so I begin with examples. "You knew 10+6=16, you already had that on the board, right? So the 6 stays the same, the 10 goes down one, one down from 16 ... that would be one way to know. Or you knew 9+5=14, nine stays the same ..."
Slowly, he says, "six is one up from five so it's one up from 14, and that's 15."
"That's right. That's another way to know." I tapped my finger over the 8 spaces surrounding 9+6. "You see? Each of these is a clue. So they're all related. This number in the middle has to be the one that all these clues fit.
"That's using the third way, which is relationships, to remember. The more you relate, the better you remember. Going up, down, left, and right, it changes by 1. Going diagonal, it changes by two this way -- see, 13, then 14,15 -- and stays the same this other way. So if you get lost, not only do you have the rows and columns, you've got every square around every square."
I sent him home to practice spirals, and told his mother to let me know if he seemed to be getting bored or resistant.
Sure enough, by the next session, Forrest was good and bored, though he was pretty thrilled that in the special education math class he attended, he had showed a huge improvement with addition facts in a quiz that week.  "Well," I said, "there are lots of other things we can use the board for, and we will, but maybe you'd like to try something else?"
"Yeah!" By now, "something else" probably sounded wonderful. Attentive repetition is highly effective, but even when generously mixed with relation, it's still not much fun.
"Okay, let's see how fast you can set the board up. You can do it in any order and you don't have to say them. I'll time you."
He did it in less than five minutes, noticeably checking his math facts to make sure he was right. His quick confidence was very encouraging.
I drew his attention to the Left-Right-Down diagonals, the ones of identical numbers. Not only did each LR diagonal contain all the same number; the only place that number occurred was on that diagonal. "All the ways of making ten are on that one diagonal," I point out. "And the only place where you find any way of making ten is on that diagonal; the diagonal is the ways to make ten and the ways to make ten are the diagonal.* Why do you suppose that is?"
An advanced fourth-grader might figure out an answer, but a struggling student like Forrest first had to understand the question. (Again, no worry about that: figuring out a hard question begins with understanding it, and this was all valuable practice). His first answer was "Because it goes across like this," making a slashing motion in the air. He meant that it was a diagonal because it looked diagonal.
I said he was right, apologized for my unclearness, and asked him to try again, dropping more hints each time, until it clicked and he said, "Something makes that happen."
"Excellent! Now, here's what makes it happen."
I had him line up ten poker chips on the table and split them into two a group of four and a group of six, and made sure he knew that the number of chips stayed the same.**
"Now point to the first group and say how much it is -- "
"And say 'plus,' and point to the second group -- "
"Four plus six equals ten."

"Exactly right. Just like you do when you're doing the board. Now move one chip from one group to the other, and do it again."
And 4 plus 6 is 10, shooby-doo wa,

  "And again."
And 5 plus 5 is 10, shooby doo wa

"Now, what comes next?"
And 6+4 is 10, bop a a loo bop a bop boom boom bang shooby-doo-wa may be adjusted for cultural and generational reasons

He hesitated when he ran out of one group. I pointed to the empty space where it had been and said, "So how many chips are there here?"
"What's math talk for none?"
"Zero. Oh! Zero plus ten equals ten!"
"Good, now start back the other way."
He quickly developed a rhythm, moving the counters and saying what they meant at the same time. Since he was a little bit of a ham and liked to sing, I encouraged him to sing the combinations according to a melody that he gradually made up.
Once he had it well worked out, I said, "So, do you recognize the words?"
He looked puzzled.
"Try doing that song and pointing to numbers going down that ten diagonal on the board."
He started, stopped, and looked up in confusion.  "It's the same as it is with the chips. I'm singing the exact same words."
"So why do the tens all fall on a diagonal?" At that point, I shut up and waited. This is one of those things where if a kid can say it for himself, you've won.
"'Cause a diagonal goes one right and one down, and that's like moving a chip from one to the other, kind of."
At that moment, however primitively, Forrest was doing real mathematics.
This is one of the foundational teaching tricks in Singapore Math: students are guided to come at things more than one way, then learn to integrate the ways. It's another way of building memory/retention through the relationship pathway, and also through the fourth avenue (anticipation, often known as "guessing ahead" or "self-testing.")
Parents often ask about this. Many really don't see why a student has to know more than one way to do anything, and why that way can't just be the memorized traditional algorithm. I usually offer them this analogy: "If you are going somewhere completely unfamiliar in a town strange to you, you follow the directions exactly to get there, and the moment you get off the directions you back up, or try to figure out or find new directions. But if you are going between two familiar spots in your hometown, you have a real understanding of where they both are, and you just take what you know will be the best route between. The objective is to move your kid from that lost-in-a-strange-place, must-stay-on-the-directions state to inhabiting mathematics like it's his/her hometown."
To put it a little more abstractly, once a kid learns to see the patterns as manifestations of underlying causes -- to realize, for example, that the first group can be assigned to a row and the second group to a column, and that a move that goes Row-1,Column+1 is a diagonal move on the board, and sums to zero -- that kid actually understands the math, rather than just playing the pattern.  Which is to say, the kid has learned to use number sense.
Or putting the issue another way (you see how you can use this method for anything?): to learn one algorithm, all you need to do is to memorize. To learn more than one algorithm, you just need more memory at first. But to understand why two or more different algorithms are actually doing the same thing requires number sense. And if a kid does those "why are two methods really the same, just written differently" exercises enough, s/he starts to learn to reach for the number sense to understand any algorithm. That means, for example, that when the kid hits fractions, the question will probably be "what does it mean when the numerator is bigger than the denominator?" instead of "which number do I write on top?" (The second question leads to much more understanding than the first.)
Not long after he started singing the groups-of-chips songs, I pointed out to Forrest that he could just picture the chips in his mind, or even imagine the diagonal on the board, and sing it just as well. I had him demonstrate it by singing the sevens diagonal while blindfolded. As soon as he finished, he insisted that his mother watch him do it.*** We agreed that he'd try to sing all the diagonals from the table a few times a day, but didn't have to use the board or the chips unless he wanted to.
The next week, I handed him a randomized list of all the addition facts. In less than fifteen minutes, he had gotten them all right.
When Thomas Vowler Short figured out and systematized his much better way of teaching fractions somewhere in the 1830s, he was astonished at how students went from slowly, carefully plodding to soaring. It still startles me.
Breakthroughs take time and patience. Exploiting the breakthrough fully, making it part of how the student sees math and the world, takes attentive practice, so it is often a much slower process, and subject to setbacks. Keeping a kid focused on the idea while practicing is hard and requires a lot of inventiveness and close attention.
Once they have learned a few fundamental ideas through the whole process, from insight to practice to complete familiarity, they really know what math is about. And after that, the kids who were "never any good at math" move with amazing speed, often moving up a full grade level in a couple of months.
That blissful state doesn't last forever, of course, though it's great while it does. Sooner or later the kid faces another conceptual barrier, but the next time, it's with the experience of getting through or over a barrier, and of knowing that s/he has seen a block like this before, and made it through.
The student knows to look for an idea, not a rule about where to write things, and how to practice the idea via concentration, repetition, relationship, and anticipation until it is really second nature.
After two to four times working through conceptual blocks in this way, most kids are true "math kids" regardless of whatever talent they started with. They know how to push into the difficulty, how to work their way through the conceptual problems, and ultimately how to have their own breakthroughs.
All that moves "Aha!" out of the realm of intuition and miracle, and into something that can be deliberately worked for and achieved. And with that power, students can go about as far as they need or want to go, without nearly as much fear or anxiety as in traditional methods. Math has become their own common sense of how the world works, rather than an arcane ritual adults use to prove you're dumb.
*I don't know for sure that this will give him a head start on graphing functions in a few years, but I am inclined to think it might.
**This is not usually a problem with a nine or ten year old, even one with severe math problems, but it's worth checking because now and then a child who is delayed on the Piaget scales may think that rearranging a group of objects can change how many there are. These children may grow up to become investment bankers and should be watched carefully.
***Luckily, she thought it was cute.