Entry #21 - Feb. 8, 1999
"Parents and others, including friends who marked the tests,
told me of numerous very bright, competent young mathematicians who become
frustrated and disgusted with being asked to 'explain' everything."
There was a half-day workshop this week for some our of our teachers on
evaluating students' work in mathematics. I didn't get to go myself (drat!),
but I pounced on returning colleagues to peruse the materials handed out.
An increased emphasis in our system on problem-solving strategies, written
explanation of procedures, and using math journals has generated (to put
it diplomatically) some controversy among parents, teachers and the public
at large.
Systemwide testing at the third, sixth and ninth-grade levels has recently
emphasized having students explain their thought processes in some detail.
The caricature of all this, of course, is a joke about the student who gets
the wrong answer, but an "A" for a good explanation, while a student
who calculates the problem accurately fails because of an inadequate "journal."
I was anxious to see what sorts of exemplars were being developed of student
work showing achievement at various levels, and what criteria were given
to evaluate these efforts. Being a visual learner myself (as I believe many
teachers are), a picture to me is worth much more than a thousand words!
I'm of two minds about this development myself. On the one hand, the emphasis
on reasoning and explanation is a good one. Who doesn't remember students
who could calculate with speed and precision, possibly even at an advanced
level, but could not apply these skills to actual mathematical situations?
There has always been a sort of dialectical tension between "knowing
the math facts" and "solving problems." An increased emphasis,
in recent years, on teaching thinking processes and specific problem-solving
strategies has helped bridge this chasm, but I believe the polarity will
always be with us.
Surely kids DO need to "know the number facts," to the level of
automaticity if possible (for some students with learning disabilities this
is not a reasonable goal, however), but they need not only to be able to
apply their math skills, but also to reflect and analyze on these applications.
This is a "real-life" skill, not just a "school" skill.
In many life situations (and surely, I would think, in the business world),
one needs not only to be able to propose a solution, but also explain how
it would work and be able to predict the effect if one or more input variables
were changed. To the query, "How do you know this will work?"
the answer, "I just know!" would not be good enough! In the past,
we accepted the "I just know" answer in most instances without
pushing students to develop their analytic and expressive language skills
further. So as far as that goes, we're moving in the right direction.
But are we overdoing it? In our systemwide math tests for third-grade mathematics,
students were asked to "explain their reasoning" at nearly every
step. Parents and others, including friends who marked the tests, told me
of numerous very bright, competent young mathematicians who become frustrated
and disgusted with being asked to "explain" everything. They had
to show their work for each problem, and in many cases thought that should
have been sufficient to demonstrate their understanding. It's only anecdotal
evidence, of course, but I heard of numerous instances where high-performing
students quit in the middle, or wrote annoyed comments like "Why do
I have to keep explaining this. It's stupid!" Some just refused to
write all the "explanations." They solved the problems and left
it at that. Of course, that is considered "unsatisfactory" for
evaluation purposes.
The overall system results were disappointing. One can only guess to what
degree the exasperation factor may have influenced outcomes, but it is worth
bearing in mind, especially as we extend these tests system-wide to sixth
grade this year. As we know, young adolescents are somewhat less compliant
than eight-year-olds, and may be less willing to elaborate endlessly on
the obvious.
It's not only the gifted students who are turned off by these requirements
-- those with language disabilities, second language, and the like, may
understand the problem well, may be able to solve it accurately, and may
even be able (in an interactive situation) to demonstrate their understanding
and elaborate on their solution. Yet, their less-developed expressive language
skills hinder their performance on the measures as written. And what of
the truly intuitive learner? There are students whose brilliance lies in
a kind of holistic insight, a"gestalt" if you will, whose minds
make leaps and connections to arrive at new understanding. These students
can NEVER "explain their thinking" in such detail, at least not
on short notice. Should we be trying to force them into some kind of mold?
One thing that emerged clearly from the workshop materials was the inherent
complexity of the assessment process. There are so many factors to keep
in mind in evaluating individual students, their needs and accomplishments,
and this is one reason why teacher evaluation of individuals can never be
replaced by standardized tests. On the other hand, the expectations of the
tests have a profound (and not always positive) impact on classroom expectations
and practices, and we are always walking a tightrope between conforming
to system expectations and trying to meet the actual (and unstandardized!)
needs of our real-life students.
We are beginning to think (in between report cards, groan!) of changes to
the program for next year to adapt to some of these new directions. Several
of my colleagues have long been doing a thankless, but essential, job of
reinforcing basic math fact learning in our students, many of whom have
not mastered the additions and subtraction facts, let alone the times tables.
This emphasis is still needed, as students with almost no basic skills cannot
even use the calculator effectively in higher level math, nor can they estimate
or evaluate the reasonableness of an answer.
The new math books being touted by the major publishers are very glossy
but appear, to our jaundiced eyes, weak on emphasis on learning and applying
the basic operations. Our challenge will be to keep the "back to basics"
emphasis that we have never left, while incorporating more of the problem-solving
and explanations activities in ways that enhance our students' overall language
and cognitive development.
In the reading debates, some are calling for an end to the "reading
wars" by having BOTH a strong phonics base and a literature emphasis:
"Balanced Literacy." Perhaps WE can pioneer an end to the "math
wars" with "Balanced Mathematics?" Why not?
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