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.


Wednesday, September 30, 2015

From tunneling through the sun with dimes, to Every Time two different times, to the meaning of number sense: a new Singapore Math blog series.

REVISION ON 10/1. My spouse would like to link to this blog to promote our tutoring business, and of course I'm all in favor of that. But, she said, the only image on this was one of me, and she has always hated that picture. "You look drunk in it," she says. As the owner of Tutoring Colorado, she doesn't like that in her main subcontractor for math, apparently. 
So this is exactly the same text as before, except that at the request of my wife, partner, trusted advisor, and employer, I have put in a few illustrations which are NOT me looking drunk.  The things we do for love.

So, first, seven short pieces, seven little things to have in mind. Then, some thoughts about Singapore Math and number sense, kicking off my "almost a week of math-y posts." Those of you who don't find this stuff interesting, who read the blog for some other reason, well, there will probably be something for you, too, sooner or later. But this week we're back to Singapore Math.

The seven things:
ONE (the long one)
"How many dimes would it take to make a bridge from here to Mars?" Years ago, when it was trendy, an HR person at a major corporation that I don't need to name tried it out on me during a job interview.

The general idea only continuous, made out of dimes, and going all the way to Mars. If this image makes you curious, it's part of something really cool at this piece by Ross Andersen in Aeon.

Well, I said, calculating out loud, sticking to one digit accuracy,
•a dime is about three quarters of an inch across, so there are about 16 dimes to the foot.
•the sun is around 90 million miles away,
•and Mars is about 50% further from the sun than Earth is.
That bridge had better be enclosed if people or vehicles are going to pass along it, so figure it's a tube twelve feet in diameter. That should be big enough for someone with a motorocycle, bicycle, or small car, and roomy for a skateboarder or runner. (Never mind where the air is coming from).
So the bridge would be made of rings of dimes laid edge to edge. That's leaky, we'll have to wrap it all in SaranWrap when we're done.
Anyway, each ring would each be 3/4 of an inch thick. If they're twelve feet in diameter, that's thirty-six feet around (for one digit accuracy, pi is three). An inch is 4/3 of a dime, so a foot is 16 dimes, 36 times 16 is 360+216, 576, so call it 600 dimes per ring, rounding up again.

Well, then, still going one digit all the way, when Mars is in conjunction (in a line with the sun and Earth, with the sun between) it's 2.5 times as far away as the sun, which would be 180+45, 225 million miles.
So if a mile is about 5 thousand feet and you need 16 rings to make a foot of bridge, that's about 80 thousand rings per mile, times 600 dimes in a ring, 48 thousand thousand dimes per mile, close enough to 50 million dimes per mile.
225 million miles is its maximum length and every mile is 50 million dimes. 225X50=(200+20+5) * 5 *10, (a thousand plus a hundred plus twenty-five) times ten, 11,250 times a million million. A million million is a trillion, a thousand trillion is a quadrillion, so, right around eleven quadrillion, give or take.
I rattled all that off in about a minute and a half; a life of writing hard sci fi will do that to you.  As afterthoughts, I added that you'd need to figure out things like what to do when the bridge cut through the sun, whether the bridge needed to be sealed, and how to have the dimes slide over each other so the bridge could be telescoped down to shorter lengths.
Now, that particular company supplied interview feedback afterward (something I understand everyone has given up now because of lawsuit anxiety) and in my feedback on that question, the HR person noted:
  1. Did math in head, so figures were not exact. Needed to ask to borrow a calculator.
  2. It sounded like an enclosed tube, which is a tunnel, not a bridge.
  3. Did not consider cost of so many dimes
  4. Very uncreative answer, one of the most uncreative we have ever had. He just calculated. Creative answers we have had for this included
  • "I would pay everyone on earth a dime every time they thought about a bridge to Mars, and when there were enough positive thoughts, someone would build the bridge."
  • "I would just need one dime, that we would spin out into a silver wire and stretch between Earth and Mars. Then I would ride there on my unicycle."
  • "I would ask NASA how much it would cost, multiply by ten, and ask the March of Dimes how many dimes they get per year, and divide."

I probably wouldn't have been very happy in that job anyway, I guess.
I was grading papers at the counter in a Starbucks. That's the closest thing there is to an experience of invisibility, I think.  I overheard the following dialogue:
Counterworker (whispering): "Hey, this guy's bill is for $6.87 but he gave me $10.12. What do I do?"
Manager: "Ring it up from $10.12."
Counterworker did, and gave the guy back his $3.25 and his order. After he left:
Counterworker: "Why did he do that?"
Manager: "Probably just didn't want any small coins in his change."
Counterworker: "Yeah, but why did he give me the extra money?"
Manager: "His change would have had thirteen cents in it, so he added twelve to make it a quarter."
Counterworker: "He couldn't know how that was going to happen. I hadn't rung it up yet."
Manager (a bit impatiently): "If there's a twelve cents in the amount he hands over, and an eighty-seven cents in his bill, the change will have twenty-five cents in it."
Counterworker (mix of incredulity and sarcasm; she was clearly not buying into any of this managerial bullshit): "EVERY TIME?"

(short because it's just a reference):
In a previous Singapore Math project blog post, I mentioned my experience with Willard (name changed because he's got a tough enough life to get through as it is), my ADL (Adult Disadvantaged Learner, meaning "grownup who doesn't know math and has trouble learning it) student who had been faking his way through practical shop math while working for a general contractor.
"From there we made a final assault on the concept that elementary algebra depends on: that an unknown number will behave exactly like a known number. (Willard, at first, did not see how we could know that 2x+3x=5x if we didn't know what x was, and could also clearly see that we couldn't possibly perform the experiment of trying all the infinite possible values of x to make sure; nor did he see that we wouldn't have to do that for every possible equation)."

(short because it's intrinsically short)
By my count, since I started counting, I've just encountered the fifth tutee who needed a very simple but very important thing explained: that the equals sign in an equation always means "what is on the left is the same number as what is on the right." It does not mean "write your answer here."
 This kid, however, was a bit more of a math kid than his four predecessors, and as the light dawned, he said, "That's how an equation can be an answer to a question in math. I thought an answer always had to be a number, but it can be an equation, too."  Apparently he had been leaving any problem with "write the equation" or "what is the equation" blank, or just writing down a number (typically by adding all the numbers in the problem), because he didn't see why anyone could be asking him to respond to a question with "write your answer here."

Working with ADLs, one of the safest predictors I've found for "this one is not going to pass " is certain phrases. I do explain, repeatedly, what is wrong with these phrases, but a few students simply keep saying them.
If they're still saying any of these by halfway through the class they are probably not going to get through algebra this time. Indeed, these are practically the Incantation Against Mathematics, or maybe the Common Format Don't Teach Me Any Math Signals.  Here's my top ten list:
  1. All I need is just to review basic math.
  2. Give me a rule that's not so complicated.
  3.  I can do any problem if it's money (or cups and pints, or wood, or cars).
  4. If I always know the right answer anyway, why do I have to do the math?
  5. Just tell me what to do and I'll do it. Don't tell me nothing else.
  6. "Seventeen!" (or any other number) (Often shouted out before anyone else can begin working on the problem, especially when the answer should be an equation, expression, or interpretation.)
  7.  So how come it's a percentage if you just said it's a fraction?
  8. We're never going to use this.
  9. We've been here for t weeks and you still haven't told us what x is. (where t=number of weeks they've been here. Often off by 1 or 2).
  10. You can't add letters.
Nobody looks drunk, either. But at least they didn't waste any time on that algebra stuff. You can't add letters anyway.

A while ago, another tutee was looking at the drawing she'd just done, using the bar model method that Singapore Math teaches. I introduce bar modeling to a lot of kids who are not doing Singapore Math in school because, correctly applied, it will crack nearly all word problems that kids are likely to encounter before algebra. She suddenly pointed to the question-marked bar that represented the answer, and said, "That has to be 9!" (let's say, I don't remember the exact problem and answer).
 "Right," I said, since it was. "Now can you write out the calculation?"
"I still don't see how."
"Just represent each step you did making the drawing ... "
We waded, with only minor struggles, back along how she had drawn her way to the answer, reviewing the big four:
  • when you put two bars end to end, you add;
  • when you put a bunch of identical bars end to end, you multiply;
  •  when you back up along one bar by the length of another bar, you subtract; and
  • when you slice a bar lengthwise, you divide or write a fraction (because fractions and division are the same thing).

Eventually she had the computation in order, and to her amazement, "It's still 9!"
"Yep. And if you actually did exactly this with real things in the real world, there would be nine of them at the end."
She stared at her drawing and calculation, and said, "That's really kind of cool."
"That's the point of word problems." (If she'd been older I might have said "the point of applied mathematics.")
"Yep. Describe the world right, do the correct things to your description, and your answer matches the real world."
"Every time?" Unlike that Starbucks counter worker, she was not being sarcastic.
"Yes, every time, every time from now to the end of the universe, every time back when there were dinosaurs, every time here and in Australia and on Pluto and  in orbit around a star so far away that its light still has not gotten here, every time."
Unlike that Starbucks worker, I think she believed me. At least she improved rapidly in math after that and I occasionally hear from her parents that she is still doing well.  I suppose she still may work at Starbucks some day, but she probably will not stay there.

(this was almost all there was)
When I was talking over this series of blog posts with my spouse, I thought I had found a perfect example for how the world looks to a person who has and uses number sense.  I'd just been to Target to buy eggs.
"So you know, Target seemd to never post unit prices. So they had either 30 eggs for 6.99 or 18 eggs for $4.29, and I was wondering which would be the better deal. It was obvious that 18 and 4.29 are both divisible by 3, so dividing both by 3, I got that that was the same price as 6 eggs for 1.43 -- "
"How did you do that in your head?"
"I factored it. 6X7=42, 3X14=42, divide the 9, it was 1.43. Or I could have gone the long way, if it was 4.50 one third of it would be $1.50, the difference between $4.50 and $4.29 is $0.21, and 21 divided by 3 is 7, so 7 cents less than 1.50, which is 1.43. How would you do it?"
"With a calculator. So then how did you know which was the better deal?"
"The package of 18 is 3 sets of 6 eggs, and each set of 6 eggs costs $1.43. The package of 30 is 5 sets of 6 eggs, right? At 1.43 they'd be $1.40 times 5, which is $7.00, plus 3 cents times 5, 15 cents, so 30 eggs would be $7.15. Buying the package of 30 saves 16 cents."
"Why would you do all that math to save 16 cents? And even if you did, why not at least use a calculator and save time?"
"Because I'd rather not give Target more money than I have to, and I did all that in my head in less time than it takes to pull out my phone and select the calculator app. Takes a lot longer to describe than it does to do. In fact," I added triumphantly, "because I did it all in about two or three seconds, say three seconds, that would mean I profited by 16 cents per three seconds, $3.20 per minute, which is $192 an hour. It was probably the most profitable thing I did all day."
I don't know why she gets that facial expression. You'd think I'd been eating bugs in front of her or something. "Well, don't use that as an example."
I generally get in trouble when I ignore Diane's advice -- not from her, but from the universe, which apparently collaborates with her -- but I'm taking a chance this time. Maybe the other six examples will outweigh it or contextualize it or beatify it, or something.

There's a common theme through all seven of those stories: the difference between having number sense and not having it.
Number sense in people who do math is something like musicality in people who dance, situational awareness in martial arts, or eye in fashion; it's very important, hard to define exactly, but if you know what you're doing, you know it when you see it. A definition I currently like a lot is one I wrote by synthesizing some excellent ideas from an article by Dr. Jo Boaler:

People with number sense are people who can use numbers flexi­bly, guided by a conceptual understanding of mathematical ideas.
Any flaws in this definition are more likely to be my misunderstanding than her error.
The key points in that definition are that:
  • the student is flexible in using numbers, meaning that the student knows what a number, a relationship, an operation, or an algorithm is (not just how to manipulate the symbol for it in one narrowly specified context). 
  • the essence of that flexibility is that a student knows what to do with a number, a relationship, an operation, or an algorithm in an unfamiliar context
  • flexibility is achieved by knowing ideas conceptually, meaning that above and beyond memories of the use of the number, relationship, operation, or algorithm in similar problems before, the student grasps the underlying ideas clearly enough to see them in unfamiliar situations and handle them accordingly.
Knowing ideas conceptually, and being able to recognize them in an unfamiliar context, implies that number sense faces in two directions with a single vision, a nice trick if you aren't used to visualizing more advanced topologies.
On the one hand, number sense is about understanding how numbers relate to other numbers -- intuition about ideas like odd/even, factor, prime/composite, natural/whole/integer/rational/real, and what it is that is the same/different about a fraction and a derivative, differential, slope, gradient, tangent, percentage, rate, quantile, or decimal. If you can see that there must be exactly the same infinite number of even and odd numbers, or that zero and one are the only numbers that can be their own square roots, you're using a little bit of number sense.
On the other, number sense is about understanding how numbers relate to the real world. You have at least rudimentary number sense if you know you use addition to total a bill, subtraction to balance a checkbook, multiplication to figure out how much floor tile to buy, and division to figure out how many cookies each kid gets.
In the seven little stories above, I do hope it's fairly obvious who has the number sense (or is acquiring it) and who doesn't, and why there are advantages to having number sense. If numbers are just noises that people use to harass you, or if you see no reason for a number to behave in any way other than whatever the teacher's whim makes it today, or if you just don't see why the world should ever behave in accord with a calculation, you don't have much number sense. That makes it very likely that you're going to go through life without a set of tools that other people have for making sense of things and managing their own lives.
But long before you are regularly cheated, or forced to guess at things that you could have known exactly, or unable to understand what's happening on the job or in the news -- well before all the painful penalties of innumeracy clobber you -- you will for sure have an awful time in math class.  
In the current draft of Singapore Math Figured Out for Parents, my chapter 2 begins with a truism:

A kid with good number sense will learn math, even from indifferent teachers using poor materials; a kid with poor number sense will never really understand math, even in a well-taught excellent curriculum.
This brings up The Wall: for most students who have math trouble, things are fine, math is even easy and fun, up to a fairly abrupt point where it suddenly becomes hard. I like the term The Wall for that point.
Trying to understand The Wall, and the difficulties of otherwise highly capable, bright students in learning math, led to the discovery of number sense.  In 1903, in Lectures on the Logic of Arithmetic, Mary Boole, the widow of that guy who invented Boolean logic, and a very formidable mathematician herself, 
Mary Boole. Looking very not drunk.

set out the basic idea. She was pushing beyond the familiar basic question of what it was about memorizing algorithms as purely symbolic operations that led so many students straight into The Wall?  In doing that , she posed what turned out to be a more productive question: what did the students who avoided The Wall, or encountered it without being defeated by it, do differently?
Rather than study how so many kids failed, she asked how the relatively small group succeeded. Mary Boole observed that students with a strong intuitive feel for what numbers meant naturally and automatically rejected mistakes. When she interviewed those same students, she found that they thought more about the meanings of the digits than about crossing them out or writing them above or below lines. Those few students could quickly recreate any parts of the algorithm they missed, correcting and reconstructing their memories as they went. They grasped the relationships between numbers not as a list of rules to apply, but as connections that were intrinsic to the numbers themselves.
Overall, if the student knew how the numbers related to each other and to the surrounding world, and did not confuse the number with the symbols that stood for it or the procedures for how to construct it, the student was much less easily confused and more easily straightened out. These resilient, capable, Wall-piercing students only needed to know part of each algorithm (and not necessarily the first part) to begin. Students who could confidently rediscover or reinvent were tougher than The Wall. 
In further work with exceptionally successful math students, Boole also found that children who never hit the wall were children who routinely played with numbers, making up patterns that worked in the numbers themselves, generally all on their own. In playing with the numbers and operations, they were practicing both the relations between the numbers and the relations between the numbers and the real world.
Those are still the characteristics of the "naturals" and the "talented" students today. To them, the numbers and their meaning are the fun and interesting point of it all; the algorithms that, to most adults, are the math itself, are just ways of doing the scorekeeping.
What Mary Boole had found in those unusual kids was what we now call number sense. She usually called it "arithmetical faculty," and less often "mathematical faculty."
As "arithmetical faculty" gradually became "number sense" (after being renamed into at least a dozen other terms along the way) in the education literature, its connotations shifted. To Mary Boole, it was the magic that Wall-resistant kids have, and that many more children could have if they weren't frightened, bored, or bullied out of it by procedure-based instruction. Not too surprisingly for a late Victorian reformer, she thought traditional proceduralism made students phobic and the phobia prevented their reasoning.
Decades later, when Dr. Kho Tek Hong was creating Singapore Math, it had been clearly shown many times that although traditional proceduralism often was associated with severe stress, which of course is highly undesirable, the real damage to students' math abilities was more often caused by some strongly misleading premises. Traditional proceduralism hooks students on easy approaches and methods that fail for more complicated problems; the stress and eventual phobia originate in that experience of suddenly not being able to trust what they know. Anyone might have a lifelong fear of something after a bad experience with it, but it's the experience, not the thing, that makes the fear. The particularly terrifying thing about The Wall is usually that students were doing fine and assured they were "good at this" right up till they hit it.
Nowadays, at math teacher conferences, number sense is a reliable subject for a well-attended panel. Still, though we now have a much better idea of where to point, number sense is still "The thing we point at when we say number sense."
So, like every other definition I've seen, "the ability to use numbers flexi­bly, guided by a conceptual understanding of mathematical ideas," is imperfect and sometimes maddeningly vague, but it will have to do till something better comes along.
So here's the deep insight that Dr. Kho brought to math education. The reason most kids don't have much number sense when they need it is because they have been actively trained not to use it when they're doing math.
The kid who hits The Wall and never gets up again learned that you do this, then this, then this, and what you write is your answer. That kid, unless s/he is exceptionally talented, will break down, never to be good at it again, whenever the number of "this" becomes too large or too complicated to memorize.
So, Dr. Kho said: Don't do that to them. Yes, they love patterns, learn them easily, prefer them to all that hard thinking stuff. But giving them what they like for math is no better an idea than giving them what they like for food (the all-chocolate diet) or for experiences (unless you want your second grader to start driving).
Of course it's easy to teach simple repetitive patterns to primary-grades kids, who love simple repetitive patterns (have you ever been driven mad by a child who sings one verse of a song over and over and over and over ...?). The problem is, as they quickly gobble up the patterns and demonstrate proficiency at pattern without meaning, they will also pick up (from teachers, parents, or both) that that's what this math stuff is all about, like learning all the gestures for "I'm a Little Teapot", or the rules to hopscotch, or the Cup Song. They'll like it and be good at it -- but it is setting them up for a fall.
Rather, said Dr. Kho, keep the idea of what the numbers are really doing in front of them, even when that makes it harder to get to that false grail, the Right Answer.  Teach them the traditional algorithms, yes, of course -- they're highly efficient ways to get to an answer, and the kid will need them eventually. But don't let them learn how  without knowing why. (I borrowed that phrase, gratefully, from Jana Hazekamp's excellent book,  and if you need Singapore Math help before my (also to be excellent) book is done, that's the single best one I know).
Dr. Kho went even farther: make sure they can understand the meaning, and that they apply the meaning to the process of learning the algorithm.(There's a very strong reason in memory theory for doing this, and I've found it works wonders, as you'll see in the upcoming case study). That is, don't just teach them procedures and hope they'll have number sense.  Don't even just teach them number sense along with the procedures.  Rather, teach them via number sense -- that is, make the student exercise number sense constantly while learning and applying the algorithm. 
And that way, when the pre-puberty fade of memory that most people experience hits, and as more advanced procedures require many steps with many decision points, and even when they move into areas where there are no longer any standard procedures to memorize -- that number sense will be there to guide them.  The number sense is the spirit, the algorithms are the law, and we all know which bears the better and more lasting fruit.
All of that would, of course, be an interesting theory, if Kho's methods did not also happen to have a 30 year history of producing the most math-proficient students in the world.  The six top math-score nations in international comparisons are in fact exactly the ones that have been using Singapore Math long enough so that their current high school students started with it at the beginning of school and did it all the way through. The majority of Singapore's math teachers are now products of Singapore math themselves.  When kids and cultures thoroughly absorb Singapore Math, they move into a whole new higher realm of capability.
Unfortunately, in what I would have to say is an absolutely typical move for American education policy, we have set the goal of catching up, and having that better performance, something that took Singapore 30 years, for right now and why not yesterday, dammit!?  The Common Core standards have defined satisfactory math performance as the level that kids who have had Singapore Math training all along achieve, as if we were going to take a track team of people who hardly ever ran and just order them to run like Olympic athletes. 
The truth is, if we do it at all, the road will be long and hard. And despite the touching faith of people like Theodore Roosevelt, John F. Kennedy, and Ronald Reagan in our nation, Americans do not much like long and hard. I really fear that we'll give up on Singapore Math at the first rough spot in the road.
Nevertheless, if we're going to demand that our kids perform like the best in the world, we'd better start training them the way the best in the world do.  And that means Singapore Math, because Singapore Math trains and develops number sense, and number sense is what makes you the best.
All right, that's number sense. For the rest of this week, I'll be laying out some more things about it.  My plan at this point is that tomorrow -- well, later today, it's already Wednesday --  I'll show you a "how good is your number sense" test, because when most people hear about an idea, the first thing they want to do is apply it to themselves. Thursday, I'll describe a typical case study, how I was able to help a tutee move himself from three grades behind to grade level in less than half a year, and how Kho Tek Hong's methods were the key. Friday, some thoughts about why it's going to be so hard to implement Singapore Math in the United States, and Saturday, perhaps, a little broader view of why all this matters so much.
Unless, of course, other crises interrupt.  The weekend and Monday were packed with hassles that ate up all the time to get this ready; I should probably have given up and let it slide another week, but I just hated the idea.  So, most of you will see this sometime Wednesday, though I'd hoped to have it up for Monday morning. Life's that way ... see you soon, I hope, barring more swarms of hassles.