>on 6/24/09 wrote:
>could you go into more detail of the Germany model? I find this
>very interesting.

What I understand about the way the German lessons tend to be is
that the teachers explain everything in detail and the students
mostly listen and ask questions if they do not understand. For
example, in our country teachers tend to demonstrate a procedure
and then have the students practice applying that procedure on the
same type of problems. In Germany, the teacher would not just
demonstrate but derive that procedure, explaining why it works and
when it is used. The math is more advanced than in our country.
That is why the description is “developing advanced procedures.”
So the students have a richer conceptual understanding, with more
connections among concepts and understanding of when to use the
procedure and what numbers to plug in where. In this country,
where we only know how, not why, we too often misuse the
procedures, or forget them because they never really made sense.

>How do they handle the mixed abilities in the room so the low
>kids learn and the advanced kids don't stagnate?

I don’t have much information about this. I remember a couple of
things related to this. In Japan, I am pretty sure the classes are
integrated. I remember reading something about more open-ended
tasks where students can work on the tasks at different levels. I
don’t have details about that. In another book that Stigler co-
authored, The Learning Gap, I remember a passage where a group of
researchers were in a Japanese classroom, and one student was
struggling at the board in front of the class. The teacher was
working on something else on another white board. The American
observers were appalled at how the student was allowed to publicly
struggle for so long, but no one in the class seemed phased by it,
not even the struggling student. I think that the student may have
been encouraged by some classmates, not sure it suggestions were
given also. At the end of the class the student finally got it,
and the whole class applauded. Everybody was happy. One point that
was made about this episode was that there was no stigma attached
to being slow. Everyone is expected to struggle and make mistakes.
It is part of the learning process. Another point was that there
is a great respect for student’s efforts. They ascribe achievement
to effort more than ability.

>about the repeated addition thing.

I am certainly not picking on anyone regarding my comment about
repeated addition. It is pervasive. Do a web search on
multiplication and that is what you will find, that multiplication
is repeated addition. That is how we were taught. My point is that
we think in terms of procedures so much, that we miss the concept,
but don’t realize it. Again, that is how we were taught—to think
that math is procedures, not concepts.

There is a strong analogy between using counting to calculate a
sum in addition, and using repeated addition (or skip counting) to
calculate a product in multiplication. They are both good
computational procedures. But we don’t say that addition is
counting. We define addition as combining or joining quantities.
So why do we define multiplication as repeated addition? Devlin
says multiplication is scaling. It is using an intermediate unit
size, a change is scale. I’ll give an example: 3 six-packs of
juice is 3 x 6; 6 is the intermediate unit size and 3 is the count
of how many sixes there are. The product is the conversion of the
units back to the standard unit 1. Multiplication always involves
transforming the units, in this case from 3 six-packs to 18 cans.
Addition is joining like units. The units must be the same size.
Scaling is closely related to proportionality. Rational numbers
require paying close attention to the unit size, what is
considered the whole or 100%. When we treat multiplication as an
additive operation and don’t develop the concept of multiplication
as a multiplicative operation, we set up our students for
difficulty as they progress in math topics.

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