**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. |

•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:

Did math in head, so figures were not exact. Needed to ask to borrow a calculator.It sounded like an enclosed tube, which is a tunnel, not a bridge.Did not consider cost of so many dimesVery 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

**mean "write your answer here."***not*
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:

- All I need is just to review basic math.
- Give me a rule that's not so complicated.
- I can do any problem if it's money (or cups and pints, or wood, or cars).
- If I always know the right answer anyway, why do I have to do the math?
- Just tell me what to do and I'll do it. Don't tell me nothing else.
- "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.)
- So how come it's a percentage if you just said it's a fraction?
- We're never going to use this.
- 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).
- 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

**-- 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.***how numbers relate to other numbers*
On the other, number sense is
about understanding

*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.***how numbers relate to the real world**.
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

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?

*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.