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
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.
