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.

Well, I said, calculating out loud, sticking to one digit accuracy,

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.

•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.

TWO

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?"

THREE

(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)."

FOUR

(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."

FIVE

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.

SIX

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.")

"Really?

"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.

SEVEN

(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.