Latecomers and accidental
wanderers-in: This is one of a whole series of blog posts about Singapore Math,
number sense, and how Singapore Math techniques, when properly used, build and
develop number sense and ultimately gets math-blocked kids moving again. The
series began with some description of what number sense is and a questionnaireto see how your number sense is; since then, we've been following a case studyof the mathematical adventures of a beginning fourth-grader named Forrest. I
made up Forrest from bits and pieces of half a dozen students whom I've tutored
at Tutoring Colorado, where we use Singapore Math methods to unblock the frustrated
and retrieve the lost. Eventually much of this material will be in my
forthcoming book, Singapore Math Figured Out for Parents.
When we last left Forrest, I had
finished the diagnostics and begun to assign him exercises.
Diagnostics had
revealed that he had a pretty severe conceptual problem: he didn't understand
numbers as abstract entities in their own right, but rather as temporary names
for things, so that for him, counting "1, 2, 3 ..." was really not
much different from naming "Grumpy, Happy, Sneezy ..."; the count was
just the last name he arrived at, and there wasn't necessarily any reason to think that if he counted the same things twice, it would come out to "Dopey" both times.
As explained in the previous two pieces about
him, this had also made it very difficult for him to learn elementary addition
and multiplication facts, or to see any reason why the various
algorithms/procedures were anything other than completely arbitrary.
He'd been given an addition table
board and sent home to practice with it; his mother had been shown what he
needed to do and had assured me he'd be doing it.
And now that you're up to date,
we're ready for his next visit.
§
Forrest's mother had assured me
that he'd been practicing regularly, with occasional minor nagging and
reminding from her. He was also complaining that the addition table board was
getting dull, and had mentioned that I had promised him that eventually we'd be
playing some games on it. So far, for Forrest, math tutoring was occasionally
different, but it still bore an uncomfortable resemblance to plain old math.
§
I had filled in about two thirds
of the addition facts board, leaving few numbers adjoining, when Forrest came
in:
He found this mildly interesting. I added, "I'm holding the tiles
over here in this rack -- it's the same one we'll use later when we're playing
games -- so that you can't see them. What you have to do is point to a square
and ask me for the tile that goes there -- by number, you can't just say 'May I
have this tile.' Ready to try?"
He nods, points to an easy one,
3+4, and says, "I need a 7 for this one."
I give it to him. He quickly
realizes that he can do this -- the location of most of the blanks at
intersections of sequences makes it very easy -- and picks up speed and
confidence. When he has most of them, has made no use of his fingers, and is
going very quickly, I throw him a little bit of a curve.
He points to the 5+8 box and asks
for a 13. I ask him, "So is it true that 5+8=13?"
He starts to count; I say,
"Whoa. You did that problem already, several times, this past week, right?
And you just pointed to it."
"But you just asked me
again."
"I asked you if it was
true."
"Well, I don't know if I
counted right."
"It is true, Forrest. It will
be true whether you count it or not, it was true before there was anyone to
count, it will be true forever, even if we never put the tile on the board or
we take it off and put a wrong number in its place."
"Okay." He sounds very
doubtful and looks confused. Nonetheless, it's a healthy confusion.
I push him a little more. "If
you have five Pokemon cards and I give you eight, how many do you have?"
He starts to look at his fingers.
I hand him the tile he asked for, and he puts it down, almost unconsciously.
"Look at the board." I
point to the five. "Say after me. 'I have five Pokemon cards, I get eight
more, and now I have ...'"
He's been mumbling along, but now,
firmly, he says, "Thirteen Pokemon cards."
"Because -- " I point
back to the five.
"Five?"
"Five plus ... " I point
again.
Suddenly he's pointing, as he's
been practicing all week, but with much more enthusiasm. "Five plus eight
equals thirteen!"
"Right. Now let's do it with
dollars."
We do dollars in a bank account,
books on a shelf, and at his suggestion, zombies in the graveyard, and at my
suggestion, tyrannosaurs on Mars, even though there aren't any really. Each time he finishes with "because five
plus eight equals thirteen."
It's time to see if he sees the
point. "So, five anything plus eight anything makes -- "
"Thirteen anything!"
"Always."
"Always!"
"So when you are doing a
problem fifty years from now, and you're an old guy like me, no matter what you
are adding, if it's five and eight -- "
"It's always thirteen."
"And way in the future, when
you're counting up something that hasn't even been invented or discovered yet,
if there are five of it in one bunch and eight in the other bunch -- "
"Thirteen." He looks a
little astonished and even, still, a bit confused.
"What if there was a group of
eight dinosaurs and a group of nine tyrannosaurs on Mars?"
His hands start to come up to
count, but he stops himself before I can, and silently points for a moment
before he says, "There would be seventeen tyrannosaurs on Mars."
"For sure?"
"For sure!" He's looking
at the whole table now, as if it were a pirate's treasure map or the secret
pathway to Oz or Middle Earth. I suppose in a way it is. I wouldn't be able to
explain it to him, but he's just taken that step into abstraction, and found
out that numbers are not arbitrary. He may not ever like it, but at least he
knows a little more of what he's dealing with.
§
One thing I have always disliked
about case studies in psychology texts and self-help books is what I call the
"Rumpelstiltskin cure." If you remember that fairy tale,
 |
| Nice king you got there. You want him asking about the baby? |
Once she learns it, by dint of a well-paid spy, she asks him,"Are you called Rumpelstiltskin !"
"A witch has told you! a witch has told you !" shrieked the little Man, and stamped his right foot so hard in the ground with rage
The groundskeeper still talks about what a job cleaning up that was.
that he could
not draw it out again. Then he took hold of his left leg with both his hands,
and pulled away so hard that his right came off in the struggle, and he hopped
away howling terribly. And from that day to this the Queen has heard no more of
her troublesome visitor.
In half or more of the Hollywood
movies about mental illness you've seen, that's the ending; the clever
therapist (or the clever patient, or someone clever) figures out the one thing
causing all the patient's problems (Rumpelstiltskin, abuse, some traumatic
event), and as soon as it is named, the patient's problems vanish, leaving the
patient all better.
Freud seems to have started the
whole genre of "Rumpelstiltskin cures" with his paper about the Wolf
Man, whose problems supposedly originated from having walked in on his parents
at That Awkward Moment.
Even in the much less upsetting
realm of math difficulty, Rumpelstiltskin is not how it works. Just naming the
problem is handy, but it's not even close to the solution.
Conceptual breakthroughs are often
very important, but they are the beginning, not the end, of the process.
Forrest still had to learn all the math he hadn't learned before, and re-think
all the math he thought he knew, and practice until the correct concepts became
the center of how he knew that math. Going back, seeing the first wrong turn in
the road, and correcting that turn, still leaves you with a lot of driving to
do.
And driving, in this case, was a
metaphor for "practice." The next and final part of the story is less
dramatic (no mystery to it) but it's where Forrest did the real work of
Singapore Math and finally caught up with his classmates.
The genius of Singapore Math is
that it teaches the student to think about the right concept at every moment of
practice; it's never just a procedure, it's a procedure and the idea behind it.
Forrest had seen what the right idea was, after years of living with the wrong
one. But for the right ideas to fully
displace the wrong, so that he was forever on the right track, he'd have to
practice, practice, and practice, and it would have to be the right kind of
practice, by which I mean the Singapore Math kind.
That's the real finish of the
story, when the most important parts happen, and I'll tell you about that
tomorrow.
