Thanks! That is very informative and interesting. I believe I will be
getting that book.

On 6/24/09, Burt wrote:
>>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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