If you just got here, this is one
of about a week-long series of blog posts about Singapore Math and number
sense, and how Singapore Math techniques can help kids through The Wall, that barricade of "this makes no sense" that most kids run into somewhere between long division and elementary algebra. Much of this material will be appearing
later in my forthcoming book, Singapore Math Figured Out for Parents. The book
draws on two roots:
- I've done a fair bit of science and technology journalism and understand educationese pretty well too; I'm used to explaining more-technical matters to a less-technical audience.
- I tutor math to elementary and middle school students for Tutoring Colorado, and I've seen how well these methods can work.
Another qualification of sorts: I've spent a fair bit of time teaching ADLs, Adult Disadvantaged Learners, people in their 20s-50s who are having to painfully pick up what they never got in school. That has given me an all-too-clear picture of what the dead end of innumeracy really looks like, why it matters that just as many kids as possible get a decent start in math, and how hard it is to recover from a bad start later. I really wish
I'd known many of the Singapore Math tactics when I was teaching remedial college pre-algebra and beginning
college algebra!
The series to date has included
•a questionnaire to evaluate your own number sense (if you're going to help your kid get it, it helps to have it
or acquire it yourself)
•and three episodes before this
one following a case study of the mathematical adventures of a beginning
fourth-grader named Forrest. Despite being a composite of several different
students with difficulties, Forrest made quite a bit of progress in those
episodes, progressing through
1. a general diagnosis of a memory problem and a conceptual difficulty with perceiving numbers as existing apart from what was
being counted, to
2. a specific diagnosis that his conceptual problem was what was causing his memory problems, his very slow calculation, and his inability to progress, leading finally to
3. the breakthrough moment when Forrest caught on to numbers as numbers, which ended with the warning note that
breakthroughs are only beginnings, and that it's the practice afterwards that
cements the breakthrough and makes it last.
And now, about that practice. If
you don't read any other post in this series, this might be the one that gives
the clearest idea of what Singapore Math is all about (at least, if I
understand it correctly and I'm doing my job, two things of which you must be
the judge).
§
Now that Forrest had a real idea
of what numbers were, and how they connected to each other and to the world, he
could see why his parents and teachers had been on his case to learn addition
facts. He also had a much better understanding of what addition facts might
have to do with the rest of math. All of that gave him much more motivation,
but that didn't necessarily make the addition facts any easier to learn. If
anything, it increased the urgency and made him impatient.
Forrest's mother confirmed he was
continuing his practice at home with the addition table board, and was
beginning to complain that he was bored and it had become too easy. That meant
it was the perfect time to introduce a more complicated trick.
"Let's try you out on this
one," I said. "Whenever you have one
value, you have all the values around
it." I put a tile down at 6+3=9 (that is at the intersection of the 6 row
and the 3 column, I put down a 9 tile.)
"Now, instead of a row,
you're going to make a spiral. Watch how this works. You put the tile down to
the right of the first one -- 6+4=10." I point and he does; so far, of
course, this is just like doing a row.
"Then we wrap around." I
point to successive squares and ask him to say the sum and place the tile.
"7+4=11, 7+3=10, 7+2=9, 6+2=8, 5+2=7, 5+3=8, 5+4=9 ... "
Here's what
it looks like, with red arrows to show the order in which they are placed.
"You see? Now next we wrap
around some more, so 5+5= ..."
"Ten."
"Right. Do you see where we
go next?" I have to correct and steer him a few times, but soon he's doing
the spiral pattern correctly, and gaining speed as he goes. When it seems to be well-established, with
another circuit and a half completed, I say, "So you see how it works. when
you go up or to the right, the sum goes up by one. When you go down or to the
left, the number goes down by one. And as you lay the tiles out in a spiral,
you form that spiral pattern."
"What do I do when my spiral
hits the edge of the board?"
"What do you think you should
do?"
"Maybe skip down to the
nearest blank space?"
"That might work."
"Or I could just start a new
spiral on the board somewhere, and grow it till it runs into this one."
"That would work too. Why
don't you try it a couple of different ways and tell me what you think?"
Long experience has taught me that
boisterous kids like to make spirals run into each other, and then have some
complicated rule for managing the collision. Quieter kids, especially ones who
just want to get done, tend to try to figure out ways to get things back to
running up and down rows and columns. After some debating, Forrest hesitantly
made the boisterous choice, and started growing a new spiral around 8+8=16.
"Now there's something else
you need to do. Every time you turn a corner, take a long breath, and look at
the tiles you've laid down. Just imagine your mind is taking a picture of it.
Try that now."
Pretty soon he had a rhythm going,
and started building simultaneous spirals, taking turns adding to each one, so
that they would collide. That small-child passion for patterns kicked in, embracing
saying the addition facts as he did them. For a kid in remedial math, he seemed
to be having a pretty good time.
Then a moment of panic: he
stumbled at "eight plus nine". He tensed up all over.
"Deep breath," I said,
"and look at what you already have.
You've got eight plus eight on one side of it, and seven plus nine below
it, and you know what they are, so the square you want has to be -- "
"Seventeen!" He was
pretty excited; things were still making sense, after all.
"Say the whole thing, and
point. Every tile you put down, say the whole problem. If you don't know the
answer automatically, use the layout of the board to see what it has to be, but
once you do see it, be sure you say it."
He looked a little stubborn,
probably realizing how quickly he could lay out the table if he ignored all that
addition stuff and just filled in the sequences.
I asked him, "So what are we
doing this for?"
He shrugged. "It's not as
boring as flash cards. It's not as hard."
"All excellent reasons, but
here's another one. You're training your memory to find its way to the answer. There's
four things that build memories, and if you can use all of them at the same
time, they make very strong memories that last for a long time. The first big memory builder is concentrating
on what you're doing. Do you see that if you started just laying down the tiles
in order, you wouldn't be thinking about the numbers anymore?"
"I guess not."
"You have to think about them
and pay attention to them to build the memory. Pointing and saying makes you
think about them a little more. It also makes you do the second thing that
helps you learn: repeating a thing over and over. So ... get on with it,
Forrest. You've probably almost got the whole table already, just from all the
repeating and concentrating you've been doing in practice."
He finished a couple more spirals,
and now the board held just a scattering of spaces to fill in.
"Now, this is where you can
see the other two things that build memory. One is relationship." I pointed
to the blank space at 9+6. "What
does that one have to be?"
"Fifteen?"
"Exactly! Now, how many ways
did you know?"
He looks puzzled, which is normal
at this stage, so I begin with examples. "You knew 10+6=16, you already
had that on the board, right? So the 6 stays the same, the 10 goes down one,
one down from 16 ... that would be one way to know. Or you knew 9+5=14, nine
stays the same ..."
Slowly, he says, "six is one
up from five so it's one up from 14, and that's 15."
"That's right. That's another
way to know." I tapped my finger over the 8 spaces surrounding 9+6.
"You see? Each of these is a clue. So they're all related. This number in
the middle has to be the one that all these clues fit.
"That's using the third way,
which is relationships, to remember. The more you relate, the better you
remember. Going up, down, left, and right, it changes by 1. Going diagonal, it
changes by two this way -- see, 13, then 14,15 -- and stays the same this other
way. So if you get lost, not only do you have the rows and columns, you've got
every square around every square."
I sent him home to practice
spirals, and told his mother to let me know if he seemed to be getting bored or
resistant.
§
Sure enough, by the next session,
Forrest was good and bored, though he was pretty thrilled that in the special
education math class he attended, he had showed a huge improvement with
addition facts in a quiz that week. "Well,"
I said, "there are lots of other things we can use the board for, and we
will, but maybe you'd like to try something else?"
"Yeah!" By now,
"something else" probably sounded wonderful. Attentive repetition is
highly effective, but even when generously mixed with relation, it's still not
much fun.
"Okay, let's see how fast you
can set the board up. You can do it in any order and you don't have to say
them. I'll time you."
He did it in less than five
minutes, noticeably checking his math facts to make sure he was right. His
quick confidence was very encouraging.
I drew his attention to the
Left-Right-Down diagonals, the ones of identical numbers. Not only did each LR
diagonal contain all the same number; the only place that number occurred
was on that diagonal. "All the ways of making ten are on that one
diagonal," I point out. "And the only place where you find any way of
making ten is on that diagonal; the diagonal is the ways to make ten and the
ways to make ten are the diagonal.* Why do you suppose that is?"
An advanced fourth-grader might
figure out an answer, but a struggling student like Forrest first had to
understand the question. (Again, no worry about that: figuring out a hard
question begins with understanding it, and this was all valuable practice). His
first answer was "Because it goes across like this," making a slashing
motion in the air. He meant that it was a diagonal because it looked diagonal.
I said he was right, apologized
for my unclearness, and asked him to try again, dropping more hints each time, until
it clicked and he said, "Something makes that happen."
"Excellent! Now, here's what
makes it happen."
I had him line up ten poker chips on
the table and split them into two a group of four and a group of six, and made
sure he knew that the number of chips stayed the same.**
"Now point to the first group
and say how much it is -- "
"Four."
"And say 'plus,' and point to
the second group -- "
"Four plus six equals
ten."
"Exactly right. Just like you
do when you're doing the board. Now move one chip from one group to the other,
and do it again."
"And again."
 |
| And 4 plus 6 is 10, shooby-doo wa, |
"And again."
 |
| And 5 plus 5 is 10, shooby doo wa |
"Now, what comes next?"
 | ||
| And 6+4 is 10, bop a a loo bop a bop boom boom bang | shooby-doo-wa may be adjusted for cultural and generational reasons |
He hesitated when he ran out of
one group. I pointed to the empty space where it had been and said, "So
how many chips are there here?"
"None."
"What's math talk for
none?"
"Zero. Oh! Zero plus ten
equals ten!"
"Good, now start back the
other way."
He quickly developed a rhythm,
moving the counters and saying what they meant at the same time. Since he was a
little bit of a ham and liked to sing, I encouraged him to sing the
combinations according to a melody that he gradually made up.
Once he had it well worked out, I
said, "So, do you recognize the words?"
He looked puzzled.
"Try doing that song and
pointing to numbers going down that ten diagonal on the board."
He started, stopped, and looked up
in confusion. "It's the same as it
is with the chips. I'm singing the exact same words."
"So why do the tens all fall
on a diagonal?" At that point, I shut up and waited. This is one of those
things where if a kid can say it for himself, you've won.
"'Cause a diagonal goes one
right and one down, and that's like moving a chip from one to the other, kind
of."
At that moment, however
primitively, Forrest was doing real mathematics.
§
This is one of the foundational
teaching tricks in Singapore Math: students are guided to come at things more
than one way, then learn to integrate the ways. It's another way of building
memory/retention through the relationship pathway, and also through the fourth
avenue (anticipation, often known as "guessing ahead" or
"self-testing.")
Parents often ask about this. Many
really don't see why a student has to know more than one way to do anything,
and why that way can't just be the memorized traditional algorithm. I usually
offer them this analogy: "If you are going somewhere completely unfamiliar
in a town strange to you, you follow the directions exactly to get there, and
the moment you get off the directions you back up, or try to figure out or find
new directions. But if you are going between two familiar spots in your
hometown, you have a real understanding of where they both are, and you just
take what you know will be the best route between. The objective is to move
your kid from that lost-in-a-strange-place, must-stay-on-the-directions state
to inhabiting mathematics like it's his/her hometown."
To put it a little more
abstractly, once a kid learns to see the patterns as manifestations of
underlying causes -- to realize, for example, that the first group can be
assigned to a row and the second group to a column, and that a move that goes
Row-1,Column+1 is a diagonal move on the board, and sums to zero -- that kid
actually understands the math, rather than just playing the pattern. Which is to say, the kid has learned to use
number sense.
Or putting the issue another way
(you see how you can use this method for anything?): to learn one algorithm,
all you need to do is to memorize. To learn more than one algorithm, you just
need more memory at first. But to understand why two or more different
algorithms are actually doing the same thing requires number sense. And if a
kid does those "why are two methods really the same, just written
differently" exercises enough, s/he starts to learn to reach for the
number sense to understand any algorithm. That means, for example, that when the
kid hits fractions, the question will probably be "what does it mean when
the numerator is bigger than the denominator?" instead of "which
number do I write on top?" (The second question leads to much more
understanding than the first.)
Not long after he started singing
the groups-of-chips songs, I pointed out to Forrest that he could just picture
the chips in his mind, or even imagine the diagonal on the board, and sing it
just as well. I had him demonstrate it by singing the sevens diagonal while
blindfolded. As soon as he finished, he insisted that his mother watch him do
it.*** We agreed that he'd try to sing all the diagonals from the table a few
times a day, but didn't have to use the board or the chips unless he wanted to.
The next week, I handed him a
randomized list of all the addition facts. In less than fifteen minutes, he had
gotten them all right.
§
When Thomas Vowler Short figured
out and systematized his much better way of teaching fractions somewhere in the
1830s, he was astonished at how students went from slowly, carefully plodding
to soaring. It still startles me.
Breakthroughs take time and
patience. Exploiting the breakthrough fully, making it part of how the student
sees math and the world, takes attentive practice, so it is often a much slower
process, and subject to setbacks. Keeping a kid focused on the idea while
practicing is hard and requires a lot of inventiveness and close attention.
Once they have learned a few
fundamental ideas through the whole process, from insight to practice to complete
familiarity, they really know what math is about. And after that, the kids who
were "never any good at math" move with amazing speed, often moving
up a full grade level in a couple of months.
That blissful state doesn't last
forever, of course, though it's great while it does. Sooner or later the kid
faces another conceptual barrier, but the next time, it's with the experience
of getting through or over a barrier, and of knowing that s/he has seen a block
like this before, and made it through.
The student knows to look for an
idea, not a rule about where to write things, and how to practice the idea via
concentration, repetition, relationship, and anticipation until it is really
second nature.
After two to four times working
through conceptual blocks in this way, most kids are true "math kids"
regardless of whatever talent they started with. They know how to push into the
difficulty, how to work their way through the conceptual problems, and
ultimately how to have their own breakthroughs.
All that moves "Aha!"
out of the realm of intuition and miracle, and into something that can be
deliberately worked for and achieved. And with that power, students can go
about as far as they need or want to go, without nearly as much fear or anxiety
as in traditional methods. Math has become their own common sense of how the
world works, rather than an arcane ritual adults use to prove you're dumb.
§
*I don't know for sure that this
will give him a head start on graphing functions in a few years, but I am
inclined to think it might.
**This is not usually a problem
with a nine or ten year old, even one with severe math problems, but it's worth checking because now and then a child who is delayed on the Piaget scales may think that
rearranging a group of objects can change how many there are. These children may grow up to
become investment bankers and should be watched carefully.
***Luckily, she thought it was
cute.
