Agreed, mostly. I understand what you said about the Japan model,
but could you go into more detail of the Germany model? I find
this very interesting. I am also interested in how other
countries differentiate. How do they handle the mixed abilities
in the room so the low kids learn and the advanced kids don't
stagnate?

I never said we weren't behind. I said that they are comparing
apples and oranges. I'm sure we are behind. I don't blame NCLB
for that, but I blame it for making it worse. I think advanced
kids ought to be given the rein, let them advance. I think we are
holding them back. But instead they are grouped with average and
low, used as tutors or left to their own devices because they can
already pass the test. (Prepared for attacks on that one.)

And about the repeated addition thing. I used it as an example
of, for lack of a better term, order hierarchy, not as a teaching
technique.

On 6/24/09, Burt wrote:
> I agree with much of what has been written in this tread.
> One thing I don’t agree with is that the United States is
> not behind other industrialized countries in math
> education. We are behind, we have a problem.
>
> Has anyone read The Teaching Gap by Stigler and Hiebert?
> It describes the results of a video study of a carefully
> chosen random sample of classrooms in three countries: the
> United States, Japan, and Germany. It was part of the
> Third International Mathematics and Science Study (TIMSS).
> What struck the team of researchers was that each country
> had a theme, or cultural approach to teaching. The image
> of teaching in Germany was “developing advanced
> procedures;” in Japan it was “structured problem solving;”
> in the United States it was “learning terms and practicing
> procedures.” One math educator on the team said that in
> Japan, the students engage in mathematics and the teacher
> mediates; in Germany, the teacher owns the mathematics and
> parcels it out to students, giving facts and explanations
> at the right time; in the United States, there is
> interaction between teacher and students, but he couldn’t
> find the mathematics. He didn’t see any real math in
> memorizing terms and procedures.
>
> We have a cultural mindset that thinks of math in terms of
> procedures and calculating an answer and not in
> understanding mathematical concepts and relationships. As
> Stigler and Hiebert point out, we aren’t even aware of
> this mindset. It was in studying classroom teaching in
> different countries that they realized this cultural
> tendency existed.
>
> The TIMSS video study also showed how difficult it is to
> break the pattern. About 70% of the teachers said they
> were implementing reforms such as those published by the
> NCTM, and they pointed to places in the video where they
> were doing so. When the researchers looked at the video,
> they found only surface changes; the lessons were still
> consistent with the image of memorizing definitions and
> practicing procedures. In fact, in many cases the
> teachers’ actions were worse than what they might have
> done otherwise.
>
> It is such a part of our way of thinking that we don’t see
> it. Here is an example: multiplication is repeated
> addition. Is that the concept or the calculation
> procedure? Do we differentiate between the concept and the
> calculation procedure? Mathematician Keith Devlin has a
> blog for the Mathematical Association of America called
> Devlin’s Angle at http://www.maa.org/devlin/devangle.html.
> His June 2008 post was titled, “It ain’t no repeated
> addition.” He got a lot of pushback from teachers. His
> next two posts were on the same subject, which was very
> unusual for him.
>
> Even the NCTM sample lessons teach multiplication as
> repeated addition. Yet their own research companion book
> warns against doing so. Devlin quotes that book:
> Thompson and Saldanha's article Fractions and
> Multiplicative Reasoning, in Kilpatrick, Martin, and
> Schifter (Eds.), A Research Companion to Principles and
> Standards for School Mathematics, pp. 95-113, published by
> the National Council of Teachers of Mathematics, 2003.
> They say (page 103):
>
> [...] multiplication is not the same as repeated addition.
> [...] One may engage in repeated addition to evaluate the
> result of multiplying, but envisioning adding some amount
> repeatedly cannot support conceptualizations of
> multiplication. [...] Generally, most students do not see
> proportionality in multiplication.
>
> The authors go on to acknowledge (lament?) that a lot of
> instructors continue to perpetuate the problem:
>
> In fact, a large amount of curriculum and instruction has
> the explicit aim that students understand multiplication
> as a process of adding the same number repeatedly. But an
> extensive research literature documents how "repeated
> addition" conceptions become limiting and problematic for
> students having them (de Corte, Verschaffel, & Van
> Coillie, 1988; Fischbein et al., 1985; Greer, 1988b;
> Harel, Behr, Post, & Lesh, 1994; Luke, 1988).
>
> So it is a problem. We need to make a special effort to
> differentiate concepts from procedures, and not to think
> only in terms of procedures
>

Share This Post | Report This Post

Next Post >>

Posts on this thread, including this one