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

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