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| Charming book, by the way. |
§
Forrest, who was just starting
fourth grade, was a little shy when he came in with his parents for the
assessment. His mother described how upset and frustrated he became while
trying to do his math homework. His father added, "His school says he's
about two years behind, but I think it's worse; there's a lot of first grade
math he can't do." His mother quickly listed the commercial tutoring
services they had tried.
It did not take a child
psychologist to see that Forrest was not looking forward to going into a room
to do math with two strangers. He sat down, squirming a little, and stared down
at the table.
We use the NumPA (Numeracy ProjectAssessment) developed by the government of New Zealand,
because it's a very well done conditional-levels assessment: structured rather
like a choose-your-own-adventure, if students get a problem right, the next
problem is harder; if wrong, the next problem is easier. The highest level the
kid gets consistently right is the score in each category. "Getting it
right" often includes explaining a correct method for getting their
answers, so luck and guessing play a minimal role.*
In the first session of the
assessment, my spouse/partner administers the questions; I sit off to the side
and watch what the students actually do while they try to arrive at an answer. Forrest
was cooperative but numb; to him, this was just one more exhibition of his
ineptitude.
The score showed that Forrest's
father was basically right. Forrest was an entering fourth-grader, but the
highest level where he was consistently right was below middle-of-first-grade.
Sometimes, but not consistently, he got a short streak of right answers well
above his usual level: for example, he did fine at "which fraction is
bigger" as long as the fractions were one over some low integer.
The directions explicitly tell
students that they can count on their fingers, because we're trying to assess
where they are starting, so if that is what they actually do, we want to see
them do it. But Forrest was counting on his fingers under the table; he wasn't
going to let any authority see that. He often lost his place or became confused
when he couldn't sneak a good look at his fingers. It was interesting, too,
that he didn't know what fingers he was holding up without looking.
He had to start counting on the
little finger of his left hand every time. When he started on any other finger,
he went back to start the problem over. About half the time, he would then forget
the problem, and although he was told he could ask to have it repeated, he
usually just said, "I don't know."
He never "counted up"
while adding; he only "counted total." That is, to solve 2+5, he
didn't start from a closed fist as "two", holding up one finger each
for "3-4-5-6-7." Instead, he had to put up two fingers (and they had
to be the little and ring fingers on his left hand), then put up five fingers
(slowly counting them under his breath: 1-2-3,middle-index-thumb on the left,
4-5,thumb-index on the right), and finally counting all seven fingers to say
"seven."
When he wasn't finger counting, he
often dug a knuckle into his thigh, twisting it back and forth, before
restating the problem, often followed by firmly stating a wrong answer.
§
That drilling-into-his-thigh
maneuver suggested memory problems. His tendency to forget the original problem
within a few seconds, especially if he panicked, started on the wrong finger,
and had to go back, seemed to confirm this.
But other clues suggested that
Forrest's memory problems were probably more deficient skill plus abundant
anxiety, not his memory per se. Clearly he remembered the size rule for
1-over-an-integer fractions, for example; he was visibly relieved as soon as
those problems appeared. He assured us three times during the hour that he knew
his "times tables" for 0, 1, 2, 5, 10, and 11, offering to recite
them for us. It was suggestive too that those are the sequences that fall into
an easily discerned regular pattern.
Possibly, to Forrest, those number
facts were magic spells for getting rid of unwanted adult attention, and recall
was just something he did to please adults rather than for any purpose of his
own. It is quite common for even students with good memory not to have learned
how to retain/recall information for intermediate steps, or for any purpose
other than pulling up "fact nuggets" to please adults.
Forrest's way of finger counting suggested a
common major conceptual problem: though he had learned the natural (counting)
numbers as names to be matched with things, as most three-to-five year olds do,
he hadn't taken that step into abstraction where numbers become as entities in
their own right. Where you or I or most second-graders would see seven puppies
(and might quickly count once to confirm the number), Forrest would see a group
of puppies and name them after the fingers of his left hand, plus the thumb and
index finger of his right: 1, 2, 3, 4, 5, 6, and 7. "Number of puppies" was not a
property of the group of puppies; it was the finger where he last named one of
them. Thus, if he recounted them starting with a different puppy, there was no
particular reason to expect the last one to be named "seven" -- there
wouldn't be "seven puppies" until he named the last puppy after his
right index finger.
Possibly Forrest hadn't taken the
basic step into abstraction of understanding that the "seven" in
"seven dwarves," "seven sheep," "seven feet," and
"seven dollars" are all the same "seven." That ability to
move from the concrete world of objects or pictures into the abstract world of
numbers is a foundation of number sense**, and Forrest did not seem to have it.
§
These seemingly small
misconceptions can make big trouble. Some of the most common misconceptions
allow the student to misunderstand what s/he's doing for quite literally years,
all the while appearing proficient and collecting praise, exactly until that foundational
error makes a difference; then suddenly nothing will make any sense, which is,
of course, The Wall
. By the time they realize something is wrong (if they ever do), the wrong idea
has spread incorrect versions of many other topics through their whole
understanding of mathematics, creating permanent misunderstandings, unclear
things that should be easy, and whole other areas where they gave up and either
memorized a simple procedure or just decided they would never understand it.
Forrest's particular
misconception, that numbers are just a sequence of arbitrary temporary names,
often leads to difficulty in learning math facts. Without any sense that number
facts refer to anything permanently true, there's no reason to store or save
any of them for later. After all, if for some reason sometime in the future, we
need to know what 7+6 is, we'll just count it then. Even if it is 13 today, who
knows what it will be by the time we need it? And math facts are not
intrinsically fun facts to know, like the names of dinosaurs, nor are they
useful for making Grandma exclaim how smart you are, at least not once she's
seen you do them once.
So for the second part of the
assessment, I would be concentrating on two questions:
1)
Did
Forrest have a bad memory, just lack the skill to use his memory effectively?
2)
How
much of an abstract concept of number did Forrest have?
§
*Thanks to better and better software and more available
computers, eventually we will be able to test all math this way, which will be
a solid blessing to every good teacher and student. Human performance should be
rated on the level mastered, not on how many times the student succeeds at
repetitive tasks. For example, we score high jumpers on the highest bar they
clear, not on how many times out of 50 they can clear a one-meter bar, and
pianists on whether they can handle a Chopin etude, not on how many times they
play a scale acceptably in one hour.
** Do you? Check out this questionnaire.
