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.

Saturday, October 3, 2015

Forrest Clearing (a tutoring case study, part III) . Also tyrannosaurs on Mars, and why I don't like Rumpelstiltskin

Latecomers and accidental wanderers-in: This is one of a whole series of blog posts about Singapore Math, number sense, and how Singapore Math techniques, when properly used, build and develop number sense and ultimately gets math-blocked kids moving again. The series began with some description of what number sense is and a questionnaireto see how your number sense is; since then, we've been following a case studyof the mathematical adventures of a beginning fourth-grader named Forrest. I made up Forrest from bits and pieces of half a dozen students whom I've tutored at Tutoring Colorado, where we use Singapore Math methods to unblock the frustrated and retrieve the lost. Eventually much of this material will be in my forthcoming book, Singapore Math Figured Out for Parents.
When we last left Forrest, I had finished the diagnostics and begun to assign him exercises. 
Diagnostics had revealed that he had a pretty severe conceptual problem: he didn't understand numbers as abstract entities in their own right, but rather as temporary names for things, so that for him, counting "1, 2, 3 ..." was really not much different from naming "Grumpy, Happy, Sneezy ..."; the count was just the last name he arrived at, and there wasn't necessarily any reason to think that if he counted the same things twice, it would come out to "Dopey" both times. 
As explained in the previous two pieces about him, this had also made it very difficult for him to learn elementary addition and multiplication facts, or to see any reason why the various algorithms/procedures were anything other than completely arbitrary.
He'd been given an addition table board and sent home to practice with it; his mother had been shown what he needed to do and had assured me he'd be doing it. 
And now that you're up to date, we're ready for his next visit.
Forrest's mother had assured me that he'd been practicing regularly, with occasional minor nagging and reminding from her. He was also complaining that the addition table board was getting dull, and had mentioned that I had promised him that eventually we'd be playing some games on it. So far, for Forrest, math tutoring was occasionally different, but it still bore an uncomfortable resemblance to plain old math.
I had filled in about two thirds of the addition facts board, leaving few numbers adjoining, when Forrest came in:

  He found this mildly interesting. I added, "I'm holding the tiles over here in this rack -- it's the same one we'll use later when we're playing games -- so that you can't see them. What you have to do is point to a square and ask me for the tile that goes there -- by number, you can't just say 'May I have this tile.' Ready to try?"
He nods, points to an easy one, 3+4, and says, "I need a 7 for this one."
I give it to him. He quickly realizes that he can do this -- the location of most of the blanks at intersections of sequences makes it very easy -- and picks up speed and confidence. When he has most of them, has made no use of his fingers, and is going very quickly, I throw him a little bit of a curve.
He points to the 5+8 box and asks for a 13. I ask him, "So is it true that 5+8=13?"
He starts to count; I say, "Whoa. You did that problem already, several times, this past week, right? And you just pointed to it."
"But you just asked me again."
"I asked you if it was true."
"Well, I don't know if I counted right."
"It is true, Forrest. It will be true whether you count it or not, it was true before there was anyone to count, it will be true forever, even if we never put the tile on the board or we take it off and put a wrong number in its place."
"Okay." He sounds very doubtful and looks confused. Nonetheless, it's a healthy confusion.
I push him a little more. "If you have five Pokemon cards and I give you eight, how many do you have?"
He starts to look at his fingers. I hand him the tile he asked for, and he puts it down, almost unconsciously.
"Look at the board." I point to the five. "Say after me. 'I have five Pokemon cards, I get eight more, and now I have ...'"
He's been mumbling along, but now, firmly, he says, "Thirteen Pokemon cards."
"Because -- " I point back to the five.
"Five plus ... " I point again.
Suddenly he's pointing, as he's been practicing all week, but with much more enthusiasm. "Five plus eight equals thirteen!"
"Right. Now let's do it with dollars."
We do dollars in a bank account, books on a shelf, and at his suggestion, zombies in the graveyard, and at my suggestion, tyrannosaurs on Mars, even though there aren't any really.  Each time he finishes with "because five plus eight equals thirteen."
It's time to see if he sees the point. "So, five anything plus eight anything makes -- "
"Thirteen anything!"
"So when you are doing a problem fifty years from now, and you're an old guy like me, no matter what you are adding, if it's five and eight -- "
"It's always thirteen."
"And way in the future, when you're counting up something that hasn't even been invented or discovered yet, if there are five of it in one bunch and eight in the other bunch -- "
"Thirteen." He looks a little astonished and even, still, a bit confused.
"What if there was a group of eight dinosaurs and a group of nine tyrannosaurs on Mars?"
His hands start to come up to count, but he stops himself before I can, and silently points for a moment before he says, "There would be seventeen tyrannosaurs on Mars."
"For sure?"
"For sure!" He's looking at the whole table now, as if it were a pirate's treasure map or the secret pathway to Oz or Middle Earth. I suppose in a way it is. I wouldn't be able to explain it to him, but he's just taken that step into abstraction, and found out that numbers are not arbitrary. He may not ever like it, but at least he knows a little more of what he's dealing with.
One thing I have always disliked about case studies in psychology texts and self-help books is what I call the "Rumpelstiltskin cure." If you remember that fairy tale,
Nice king you got there. You want him asking about the baby?
Rumplestilstkin tells the former-millers-daughter-now-queen that he will cease tormenting her if she can learn his name.
Once she learns it, by dint of a well-paid spy, she asks him,
"Are you called Rumpelstiltskin !"
"A witch has told you! a witch has told you !" shrieked the little Man, and stamped his right foot so hard in the ground with rage
The groundskeeper still talks about what a job cleaning up that was.
that he could not draw it out again. Then he took hold of his left leg with both his hands, and pulled away so hard that his right came off in the struggle, and he hopped away howling terribly. And from that day to this the Queen has heard no more of her troublesome visitor.

In half or more of the Hollywood movies about mental illness you've seen, that's the ending; the clever therapist (or the clever patient, or someone clever) figures out the one thing causing all the patient's problems (Rumpelstiltskin, abuse, some traumatic event), and as soon as it is named, the patient's problems vanish, leaving the patient all better.
Freud seems to have started the whole genre of "Rumpelstiltskin cures" with his paper about the Wolf Man, whose problems supposedly originated from having walked in on his parents at That Awkward Moment.
Even in the much less upsetting realm of math difficulty, Rumpelstiltskin is not how it works. Just naming the problem is handy, but it's not even close to the solution.
Conceptual breakthroughs are often very important, but they are the beginning, not the end, of the process. Forrest still had to learn all the math he hadn't learned before, and re-think all the math he thought he knew, and practice until the correct concepts became the center of how he knew that math. Going back, seeing the first wrong turn in the road, and correcting that turn, still leaves you with a lot of driving to do.
And driving, in this case, was a metaphor for "practice." The next and final part of the story is less dramatic (no mystery to it) but it's where Forrest did the real work of Singapore Math and finally caught up with his classmates. 
The genius of Singapore Math is that it teaches the student to think about the right concept at every moment of practice; it's never just a procedure, it's a procedure and the idea behind it. Forrest had seen what the right idea was, after years of living with the wrong one.  But for the right ideas to fully displace the wrong, so that he was forever on the right track, he'd have to practice, practice, and practice, and it would have to be the right kind of practice, by which I mean the Singapore Math kind.
That's the real finish of the story, when the most important parts happen, and I'll tell you about that tomorrow.

Friday, October 2, 2015

Another search through Forrest: the second diagnostic meeting (a tutoring case study, part II)

For those of you who just came in, you might want to drop back a day and read Part I,  which is not terribly long (at least not by my ultraverbose standards). It explained that this is a case study, and like most case studies, the characters in it and their difficulties are composites, examples of the common and the typical pulled from a number of real kids and real math situations that I've encountered since I began handling the math tutoring duties for Tutoring Colorado.
In our last episode, my invented composite tutee, Forrest, had gone through the NumPA interview, and by observing him I'd decided to focus in on two issues: memory problems, which I wasn't at all sure he had, and conceptual difficulties with thinking of numbers as abstractions, which I was fairly sure he had.  Both problems were probably interacting: he had a hard time memorizing because he thought of numbers as a set of temporary, arbitrary names, and he had a hard time moving beyond very rudimentary counting because he couldn't retain enough math facts to see any pattern in them.
So my basic strategy was to tackle the conceptual issue first while I gathered more information about the extent of the memory problems (if indeed there were any). And now on with the story ...
At the second meeting, I showed Forrest an addition-table board and tiles. The addition table board looks like this

and the tiles, which are played on the blank spaces, are the sums, laid out like this:

When I set up the room before Forrest arrived, I put the tiles in piles in numeric order just to one side of the board, so that it would be easy for Forrest to find the tile he wanted.
I told Forrest that it was a sort of game or puzzle. The object was to arrange the tiles on the board to make an addition table. I showed him how each cell of the table is the sum of the row and column numbers.
He could put all the tiles into their proper places by any means he liked, but I needed to hear him explain how he was doing it.
 He started with 0+0=0, tentatively, at the lower left, and then picked up speed as he realized how easy the zero and one rows are. As he worked through the 2 row and the 2 column, he slowed down. Confirming previous observations, he seemed unaware of commutative pairs: he had to count out 2+6 and 6+2 separately. 
In fact, he really didn't seem to see any patterns. He made no use I could detect of the left-right-down diagonals, or the sequence of numbers along any row or column. For Forrest, all the addition facts existed as isolated statements. It's a sort of memory that I sometimes call "phone book knowledge": knowing one phone number doesn't really help you to know any other phone number, because there is no relation between them.
Paradoxically, eventually addition and multiplication facts do become phone book knowledge, in older students and adults; eventually you know 7X9 instantly, without reference to any other products of 7 or 9, and that allows you to do arithmetic very quickly. But in the learning stage, 100-169 facts (depending on whether your school district wants students to learn the tables as 1-10, 0-12, or something in between) is simply overwhelming for many children.
Or returning to the metaphor, back before phones had memories, a good salesman or administrator might know that many phone numbers, but that came from months and years of practice. Sitting down and learning that many phone numbers in a few days by just chanting them might be daunting even for adults. You needed a system (area codes for cities, exchanges for large companies, etc.) to find your way through the list, in order to get enough practice to learn, eventually, to do rapid random recall.
Somewhere northeast of 4+4=8, Forrest began to surreptitiously count on his fingers; I told him it was all right to do that, and showed him how to count up. After all, counting up is a baby step in the direction of abstraction, and Forrest needed all the steps in that direction he could take.  He immediately caught on to counting up, but he'd have attacks of doubt every few problems and have to check it by counting total.*
Around 6+6=12, Forrest was bogging down again -- the unfamiliarity of counting up was probably tiring him a little. Not expecting him to get it this time, but preparing a bridge to the next session, I showed him that he could count up on the board even more easily than on his fingers: "see, to add 8+6, start at 0+0=0. Now count up to row 8 ... and count 6 columns to the right ... that's exactly the same thing as the way you count on your fingers. But you could just point to the 8 in the zero column, and count over 6 ... that's exactly the same thing as the way I just showed you."
He did it but I could tell he didn't trust it at all; he kept comparing tiles to fingers, and it was something of a surprise that they kept coming out the same. That surprise was what I planned to build on.
As he worked, as if just making conversation, I asked, offhandedly, other diagnostic questions:
•If you did the same board tomorrow, would it come out the same way? (He answered, "If I did all the numbers the same way." Follow-up questions confirmed that by "the same way" he meant "in the same order" and that he still thought he'd have to count them again.)
•Was 4+7 equal to 11 before you counted it out? ("It can't be a number if you haven't counted it.")
•Could 4+7 ever be a different number? ("I don't know.")
•When your class does a school program, do you have trouble learning the words to songs? ("No, that's really easy.")
•How's your spelling? ("Last year I won the spelling bee twice!")
•"Can you name the starting offensive line for the Broncos?" I had noticed the T-shirts and hats on both Forrest and his father. (He rattled off a quick summary, including the variations. I have no idea if he was right. I just wanted to know if things stayed in his memory and were quickly accessible when he wanted them to be. The answer was obviously yes).
We finished up that second session with some talk about training for effective recall.
"If you want to do this, you have to decide to train your memory. All I can do is show you how, but you will be doing all the training.
"Now, let me show you something you might already have noticed. In the addition table, all the numbers along a row or a column are in sequence,that means in number order. In fact, you could say they look like they're counting up from the row number or the column number. So if you have part of a row or a column, you can always fill in the whole row or column. And you always do have part of it, because you always know the zero row and the zero column, right?"
I showed him how it worked; we filled in part of the sevens row and the eights column, because Forrest felt that he had the most trouble with those two numbers.
 "So now you know how you could do the whole table in three minutes or so, right?"
He's nodding, temporarily happy because it looks like there's a trick to this that will make it easy.
"The trick is to use it to train your memory, to make it stronger and better. The pattern will give you the answer, but getting the answer is just the first step, not the goal. You're trying to train yourself to remember the answer. So as you lay down a row, use the sequence to know what comes next. Then as you add each tile, say the row, say 'plus,' say the column, say 'equals', and then say the number you are putting down. Point to them as you do it. For example, seven plus five equals twelve. Point and say the whole thing."

 After a minute or so he got the hang of pointing to the row, then to the column, then to the tile. "One more time on this one," I said.
"5 plus 7 equals 12." He pointed at each number with full confidence; his index finger was the only finger he had up.
"What happens next is, I'm sending you home with a board and set of tiles, yours to keep. Try to keep them clean and organized because eventually you will be playing a lot of different games on them. Twice a day, for not more than twenty minutes at a time, what I want you to do is set up the addition table -- you don't have to finish each time, you can just get as far as you get, then start from that point at your next session. Every time you finish, just sort the tiles into groups over to the side, and start again from the beginning. It's not about getting it done, it's about doing the set-up right each time.
"Now, as you set it up, every single time you put down a tile, I want you to say the  addition fact out loud, pointing to the row and the column and the tile, just the way we just did.  It's very important to really look at every number at every step. Maybe say it two or three times extra if you catch your mind wandering."
We did a few more sevens and eights, rows and columns, together, and at the end he could say them in sequence. He was dubious about it; the addition facts just didn't feel as true to him when he didn't count. But he agreed to fill in as much of the board as he could in two 20-minute sessions every day, and to get the answers from the tiles, not from his fingers, speak each addition fact aloud, and try to concentrate on them as he did it. When his mother came to pick him up, we all went over the assignment together, to make sure she understood it and could remind and guide him.
"And he'll know his addition facts from this?" she asked.
"Probably.If not, I've got a lot more tricks.  And once he's had the practice, I can show him much more about how to use his memory more effectively. But the most important thing is that it gets him ready for the conceptual breakthrough we're trying for. Usually that doesn't take long, if he sticks to the practice.
"Remember, twice a day, but not more than twenty minutes, and it's okay to skip if he's not feeling well. But he's so good at it, he'll probably want to do it when you remind him." That's sort of a self-fulfilling lie; most of the kids I see are so discouraged that to have me tell their parents they are good at a math exercise, no matter how simple the exercise is, tends to make them want to do it; that experience of being good at it is something they enjoy so much and haven't had in so long.
And that's part two of this case study; part three, in which we see that conceptual breakthrough, tomorrow, I think. Always allowing for fate to jump me again, of course.
*Luckily, and I think to his surprise, they did always give the same answer.

Thursday, October 1, 2015

Rescuing Forrest from the trees: a tutoring case study, part I

Charming book, by the way.
Like case studies in self help, psychoanalysis, business management, and so forth, this is a composite tale. In using Singapore Math methods to coach kids with math problems at Tutoring Colorado, I've seen several kids who shared a problem or two with Forrest, and no kids who were exactly like him.  When I realized that this particular tutoring tale was getting very long for a blog post, I decided to break it up, so, here's part one.  Part two, probably, tomorrow.
Forrest, who was just starting fourth grade, was a little shy when he came in with his parents for the assessment. His mother described how upset and frustrated he became while trying to do his math homework. His father added, "His school says he's about two years behind, but I think it's worse; there's a lot of first grade math he can't do." His mother quickly listed the commercial tutoring services they had tried.
It did not take a child psychologist to see that Forrest was not looking forward to going into a room to do math with two strangers. He sat down, squirming a little, and stared down at the table.
We use the NumPA (Numeracy ProjectAssessment) developed by the government of New Zealand,  because it's a very well done conditional-levels assessment: structured rather like a choose-your-own-adventure, if students get a problem right, the next problem is harder; if wrong, the next problem is easier. The highest level the kid gets consistently right is the score in each category. "Getting it right" often includes explaining a correct method for getting their answers, so luck and guessing play a minimal role.*
In the first session of the assessment, my spouse/partner administers the questions; I sit off to the side and watch what the students actually do while they try to arrive at an answer. Forrest was cooperative but numb; to him, this was just one more exhibition of his ineptitude.
The score showed that Forrest's father was basically right. Forrest was an entering fourth-grader, but the highest level where he was consistently right was below middle-of-first-grade. Sometimes, but not consistently, he got a short streak of right answers well above his usual level: for example, he did fine at "which fraction is bigger" as long as the fractions were one over some low integer.
The directions explicitly tell students that they can count on their fingers, because we're trying to assess where they are starting, so if that is what they actually do, we want to see them do it. But Forrest was counting on his fingers under the table; he wasn't going to let any authority see that. He often lost his place or became confused when he couldn't sneak a good look at his fingers. It was interesting, too, that he didn't know what fingers he was holding up without looking.
He had to start counting on the little finger of his left hand every time. When he started on any other finger, he went back to start the problem over. About half the time, he would then forget the problem, and although he was told he could ask to have it repeated, he usually just said, "I don't know."
He never "counted up" while adding; he only "counted total." That is, to solve 2+5, he didn't start from a closed fist as "two", holding up one finger each for "3-4-5-6-7." Instead, he had to put up two fingers (and they had to be the little and ring fingers on his left hand), then put up five fingers (slowly counting them under his breath: 1-2-3,middle-index-thumb on the left, 4-5,thumb-index on the right), and finally counting all seven fingers to say "seven."
When he wasn't finger counting, he often dug a knuckle into his thigh, twisting it back and forth, before restating the problem, often followed by firmly stating a wrong answer.
That drilling-into-his-thigh maneuver suggested memory problems. His tendency to forget the original problem within a few seconds, especially if he panicked, started on the wrong finger, and had to go back, seemed to confirm this.
But other clues suggested that Forrest's memory problems were probably more deficient skill plus abundant anxiety, not his memory per se. Clearly he remembered the size rule for 1-over-an-integer fractions, for example; he was visibly relieved as soon as those problems appeared. He assured us three times during the hour that he knew his "times tables" for 0, 1, 2, 5, 10, and 11, offering to recite them for us. It was suggestive too that those are the sequences that fall into an easily discerned regular pattern.
Possibly, to Forrest, those number facts were magic spells for getting rid of unwanted adult attention, and recall was just something he did to please adults rather than for any purpose of his own. It is quite common for even students with good memory not to have learned how to retain/recall information for intermediate steps, or for any purpose other than pulling up "fact nuggets" to please adults.
 Forrest's way of finger counting suggested a common major conceptual problem: though he had learned the natural (counting) numbers as names to be matched with things, as most three-to-five year olds do, he hadn't taken that step into abstraction where numbers become as entities in their own right. Where you or I or most second-graders would see seven puppies (and might quickly count once to confirm the number), Forrest would see a group of puppies and name them after the fingers of his left hand, plus the thumb and index finger of his right: 1, 2, 3, 4, 5, 6, and 7.  "Number of puppies" was not a property of the group of puppies; it was the finger where he last named one of them. Thus, if he recounted them starting with a different puppy, there was no particular reason to expect the last one to be named "seven" -- there wouldn't be "seven puppies" until he named the last puppy after his right index finger.
Possibly Forrest hadn't taken the basic step into abstraction of understanding that the "seven" in "seven dwarves," "seven sheep," "seven feet," and "seven dollars" are all the same "seven." That ability to move from the concrete world of objects or pictures into the abstract world of numbers is a foundation of number sense**, and Forrest did not seem to have it.

These seemingly small misconceptions can make big trouble. Some of the most common misconceptions allow the student to misunderstand what s/he's doing for quite literally years, all the while appearing proficient and collecting praise, exactly until that foundational error makes a difference; then suddenly nothing will make any sense, which is, of course, The Wall . By the time they realize something is wrong (if they ever do), the wrong idea has spread incorrect versions of many other topics through their whole understanding of mathematics, creating permanent misunderstandings, unclear things that should be easy, and whole other areas where they gave up and either memorized a simple procedure or just decided they would never understand it.
Forrest's particular misconception, that numbers are just a sequence of arbitrary temporary names, often leads to difficulty in learning math facts. Without any sense that number facts refer to anything permanently true, there's no reason to store or save any of them for later. After all, if for some reason sometime in the future, we need to know what 7+6 is, we'll just count it then. Even if it is 13 today, who knows what it will be by the time we need it? And math facts are not intrinsically fun facts to know, like the names of dinosaurs, nor are they useful for making Grandma exclaim how smart you are, at least not once she's seen you do them once.
So for the second part of the assessment, I would be concentrating on two questions:
1)                       Did Forrest have a bad memory, just lack the skill to use his memory effectively?
2)                       How much of an abstract concept of number did Forrest have?
*Thanks to better and better software and more available computers, eventually we will be able to test all math this way, which will be a solid blessing to every good teacher and student. Human performance should be rated on the level mastered, not on how many times the student succeeds at repetitive tasks. For example, we score high jumpers on the highest bar they clear, not on how many times out of 50 they can clear a one-meter bar, and pianists on whether they can handle a Chopin etude, not on how many times they play a scale acceptably in one hour.
** Do you? Check out this questionnaire.  

How's Your Number Sense? Not quite a quiz

Yesterday I talked about number sense in quite a bit of detail, and sure enough, several of my Twitter buddies began talking about it with me. Even more naturally, they all wondered how good their number sense was and where they stood compared to everyone else.
This is not a surprise. Every living thing that has eyes seems to love a mirror.
I've never met readers or students who learn a new idea without wondering if it applies to them. This is why so many medical students suffer from hypochondria, law students become fascinated with petty grievances to themselves and their families, and I'm told by a friend who went through a full set of factory training as an auto mechanic that it was at least a year before he could drive without hearing every little stray sound from the engine. And whenever I've found myself explaining number sense to the parents of my tutees at Tutoring Colorado, sooner or later the parents have wondered about their own number sense.
So I am guessing that you might be wondering how good your own number sense is. My quick answer is, "probably pretty decent, since you're reading this, and most people with really bad number sense won't read about math at all." Then again, some people will endure almost anything, even fractions, if they think it will benefit their kids.  Possibly you are even wondering if the whole problem is that your own number sense is deficient, so you never really learned real math, and now you can't help your kids. It's a bit like asking, before you start reaching for the victims and pulling on their arms, whether you yourself are standing on quicksand, and it's a very good question.
In that case, please take some comfort in this: helping your kids with Singapore Math will boost your own number sense tremendously. I often send my tutees home with Singapore Math-based projects to work on with their parents, and I've lost count of the number of times I've heard, "So while I was trying to help him I suddenly got it myself. I never got that before!", and sometimes the even more enjoyable, "She did fine. She got it before I did, and she was so proud of herself for being able to explain it to me." As you come to understand what should be happening in/with/for their number sense, you're going to rapidly improve or reawaken your own. You may also become a great role model for how to handle intellectual difficulty, and help the kid see that though knowing matters, learning matters more. 
So don't worry about how much number sense you have now. It's not a quiz. It's not a competition. There is no generally accepted scale for measuring raw number sense anyway; a good score might only mean you are a fast snail or a big mouse, a bad score might mean you're a slightly less beautiful eagle or a smallish whale. Most likely of all, it might mean I'm a poor questionnaire writer.
Nonetheless, let's see if this gives you a picture of where you are, and maybe some idea of where you want yourself or your kids to be.
This questionnaire is based on material I've used with older kids in the tutoring business. It aims to show how much you already use (or don't use) number sense in your approach to math, and I hope therefore clarifies what this number sense thing is all about. Please accept one hug, pat on the back, or small medal for voluntarily taking a math test in the hopes of helping your child. If that's not parental love, I don't know what else could be.
It will help to have some way of recording the results as you go along, so you might want to open a note window or grab pencil and scratch paper. There's a full explanation of the answers at the end, but I strongly suggest you do all the questions before reading through the answers. On the other hand, even if you get them all right, you will still want to look at the explanations to see if you were actually using number sense to get the answers.
Read each question carefully. Figuring before thinking is probably the leading warning sign of poor number sense.
Do not time yourself, or do anything else to give yourself an incentive to be fast with an answer rather than clear about why it is right.
For each problem, record two pieces of information: You might want to draw up a little table with 2 columns, "answer" and "NS level", and 20 numbered rows, if you're one of those people who likes to keep neat records. "Answer" needs to be the widest column. Here's one to copy to paste to your note window if you like:
Problem number
NS level




















In the answer column, write the correct answer, if you can see how to get it. I've used the current Colorado fifth-grade math standards (i.e. the last year before middle school in a state that has average math scores and happens to be located immediately around me) to devise the questions, so there is always a way to the answer through elementary school math. You are very welcome to use any higher math you know, however (and often that will be much easier). If you don't see any way to the correct answer, write "guessed", "?", or something else to remind you of your process.
In the NS level column, write a letter from the list below; what level of number sense did you use to solve the problem? You should be trying to work at the highest level of number sense you can, so you should probably read through the levels first:
a.      You just knew the answer, and why it had to be right, right away. For example, most people can correctly answer "Which is bigger, 1448+5 or 1448+6?" right away, without calculating, because they notice that the only actual difference is the one between 5 and 6, so it feels like they "just know." They would record "a" for their level.
b.     You could arrive at the right answer after some thinking about it, but you didn't have to calculate. For example, whether or not you know how much a quadrillion is, you can probably answer "What is half of six quadrillion?" by thinking of an analogy (what is half of six dollars? half of six sheep? half of six gallons?) and referring to a math fact. That would be a "b." If you actually had to do something to compute half of six, either because you don't have that in memory or because you don't see that "half of six" is the same number no matter what the units or the multiplier are, then you would record "c" or below.
c.      You calculated correctly and got the right answer. For example, most people would calculate to answer, "What is 162 divided by 6?", by long division, mental short division, or factoring, and would record "c" for it. If, however, right after doing that you realized either "oh, wait, I know 6x25=150 and 6X2=12 so it had to be 27" that might be more like "b". (Notice this is truly an honor system; the difference between "b" and "c" is more about how much calculating you had to do than about not doing any or having to calculate every tiny step. Are you mostly thinking, or is the pencil or calculator really busy?).
d.     You thought you knew how to calculate but then realized you weren't getting the right answer, or you got confused in the middle of the calculation, or you couldn't decide which of several possible calculations to do.  "d" is probably best described as "I used to know that, I think."
e.      You can see there must be a way to calculate this, but don't know or remember enough to see how to do it yourself. In other words, you're pretty sure the answer is in there (without my having to tell you it is -- though there's one trick question where the answer is there's no answer), but you really don't have any idea how to go in there and drag it out. Most people who know what a cube root is will concede, for example, that there must be some way of finding the cube root of 864 without a calculator or spreadsheet, but they wouldn't know where to begin, so they'd put a question mark for the answer and an "e" for number sense. (There's not actually any problem that hard below, by the way).
f.       You have no idea at all; don't even see how an answer could be calculated. This isn't the same thing as the terms being unfamiliar; those should be marked "unfamiliar" or "didn't know the words" in the answer space. For example, since most people don't know what a hyperbolic cosine is, if I asked you to calculate one (I won't!) you would write "unfamiliar" in the answer space and leave NS level blank. On the other hand, if the problem is that it costs 72 cents each to make the first gallon pitcher of lemonade, and each successive gallon is 13% cheaper, you always sell exactly one gallon at $1.00 per glass on a sunny day with 80 degree temperatures, you sell an extra quart for every degree the temperature goes above 80, and sales double for every 20% price reduction, if the temperature is 91 degrees, how much lemonade should you make and at what price should you sell it for maximum profit?, (about a college sophomore level economics problem -- don't worry, nothing like that below either). Then if you see there's a way to get an answer, even though you couldn't do it yourself, that's an "e." If you don't see any way that anyone could get any answer at all, give it an "f."
Record the highest level of number sense you could have used, whether or not it was your first thought. This second score is about the highest level of number sense you can work at, not about what level you usually work at. (Though if you notice you're always calculating first and then number-sensing afterward, that information might be useful or interesting also.)
Again, record BOTH your answer (if any) and the level of number sense you were able to approach the problem with (whether you got the right answer, or any answer at all). We'll be looking at both the answers and the NS level at the end.
All right, grab your pad and let's begin. (Sorry about the weird formatting of what follows; I haven't mastered all the nuances of getting math notation to work in Blogger's interface. I decided to prefer size and readability to style, as well as to staying up all night figuring it out.  If you have elderly eyes like mine, click on any panel and it will pop up as  a separate, easily enlarged window).

The answers, and how people with number sense might know them without calculating. 

You can count the answers right/wrong in any conventional way you like. The problems were taken mostly from the Grade 5 advanced standards with some additions from the Grade 6 regular, so if you got 14 or more right, you're about on par with what we expect of a brainy, well-trained 11 year old in Colorado.
For number sense, count the frequency of a, b, c, d, e, and f. (If you're ambitious you might even do a histogram).  If you have ten or more a's and b's combined, that looks like pretty good number sense to me; if most of your answers are c's and d's, you probably have fairly good number sense but learned math procedurally, so you may have to work on your own number sense to coach Singapore Math well. e's and f's mean you probably really need to work on your own number sense at the same time you are trying to help your kids. Be sure to admit you're trying to figure it out together -- seeing you struggle and catch on may very well be exactly the model the kid needs.