Maybe this is a small footnote to the discussion, but the thing I was
struck by was the used of color-coded math sheets corresponding to
different levels of difficulty:
> I noticed T was sitting at the same desk as before so I walked up to
> her to see if she was working on math again. Sure enough she had the
> blue (medium difficulty) math sheet. I exclaimed, "Blue again, common,
> where's the green!" T looked up and smiled and then pulled out the
> green (most difficult) sheet as I pulled up a chair.
The point, for me, is that the problems have been differentiated into
levels of difficulty not by the student, but by the teacher. So such an
approach is unlikely to develop the child's reflective understanding
of the boundary of his/her knowledge.
In case it's of interest: In our math classes we use a technique I
observed in Russia, where several different problems are put up on the
board and the children are asked to vote on: (1) which problem is the
most difficult, (2) which problem is the most interesting, and (3)
which problem do I want to work on today.
It is often difficult for children to rank the problems in level of
difficulty because they will focus on empirical details of the problems
rather on the mathematical relations they contain.
After solving the problem of their choice independently and checking
with a partner, a child or a pair of children present their solutions
at the board, and the process they use to solve the problem is assessed
and evaluated by the class as a whole.
The remarkable thing is that at the end of the class the children vote
once again on (1), (2), and (3), and what the children believe to be
"difficult," or "interesting," or "do-able" will have changed during
the course of the class. That is, the kids will have refined their
reflective understanding of the boundary between what they can do and
what they cannot do. For example, by the end of the class many kids
will have concluded that they can attack problems that use algebraic
instead of concrete numbers. And others will have concludes that a
superficially "difficult" problem is actually one that they can solve
easily.
But all of this derives specifically from the levels of difficulty of
problems NOT having been defined beforehand by the teacher.
But perhaps such a technique is easier to implement in the collective
work of a class than in a child's interaction with a single adult.
Peter
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