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That's why I let my students, as a class, evolve methods
of writing papers that allowed them to communicate more effectively with
each other. After all, any professional writer will tell you that there
really aren't any rules for writers that aren't meant to be broken. With a
lot of practice, you evolve your own rules. Benjamin Franklin did it by
copying the style of Joseph Addison. When I was in high school I used to
copy the style of Dickens. I found that doing that worked wonders for my
writing as well as the grades I got in English.
To this day I find it difficult to come up with a paragraph by a really
good writer that follows the rules for writing paragraphs that are usually
taught in school. I'm sure I never consciously use them myself, and when
my books are edited for publication, my editors never remind me of them.
In my classes I found that once the atmosphere of critiquing each other's
papers was established, the students were all learning from their peers,
with no implication that one person was better than another because that
person had more of the truth or knew better precisely how to put a
paragraph together. Instead, they copied each other's successes and then
improved on them. The rate of improvement in their writing was truly
amazing. In the process, though, it was never clear what the truth was.
What gradually emerged was not the truth, but an evolving understanding of
how to think about and respond to literature. There were no right or wrong
answers; there were only experiences of arousing reader reactions in one's
peers, the challenge I have to contend with right now.
Having said all this, I want to point out that, hypothetically, math could
be taught in a way that resembles what I've described above. It could be
taught in an atmosphere in which students make up problems of interest to
them, perhaps inspired by the chapter they're studying in a math book. As
they worked through a particular student's problem and the solution of it,
they would take turns proposing solutions that might often be erroneous.
Their errors would then be explored by the class to see what was causing
them. Through this the teacher would stand aside, encouraging the students
to work all of it out for themselves. In the process they would figure out
and instruct each other in methods of problem solving.
By this means, students, working together, would make mistakes, understand
why they had made them, evolve new patterns and new understandings, and
have a wonderful time working together to find answers they could all
accept.
Looked at this way, algebra becomes very practical as well as lots of fun.
It is experienced as an exercise in observing the developing of patterns,
and testing your ability to anticipate them. Then you can find those
inconsistencies in your own thinking that interfere with your perception
of the integrity of the pattern.
The more we develop this ability to see through our own mistakes, correct
them, and build what we have thereby learned into a developing set of
personal rules for confronting and dealing with the complexities of our
world, the better able we will be to adjust to the changing demands our
environment makes on us.
In English class we teach students both to read and to write. In other
words, we teach them to process existing essays and also to write them, to
process existing poems, but also to write them, and to process existing
stories and novels, but also to write those as well, in the few cases
where we take the time.
Meanwhile, we're not only teaching math students how to process existing
problems but also to write them. They learn how to process existing
theorems and also to create them. They also learn how to process existing
notational systems and to create those as well. In traditional math
classes it's no wonder so few students practically achieve any permanent
learning during all those years.
We need to exchange our dictatorial methods of presenting material that
has to be learned a certain way. Most students don't learn it at all. We
need to transform instruction to methods of group process that allow
students to learn from each other at an initially slow, but eventually
much faster rate than they otherwise could. When we use effective teaching
that incorporates group process, we soon find that we have included all
the students in a way that allows them to acquire equal status in the
class.
When the usual competitive atmosphere has been replaced with a cooperative
one, we find students working together to derive from their individual
experiences and judgments a collective experience that can enhance all of
their lives.
Article 6: Timely Links with Educational
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