Hello, all.

I, too, agree with much of what has been said in this thread.

And to DD, I have The Teaching Gap on my bookshelf right now. It
was one of the first books I read when I first became concerned
about the state/quality of mathematics teaching in this country,
including my own.

Let me preface the next sentence by saying that I have no problem
with procedures in mathematics. That said, one of the most
INTERESTING points you made below, DD, is that EVEN WHEN AWARE,
it is very difficult to enhance/augment this procedures-only or
procedures-mostly math teaching style (and therefore math
learning conditions).

Costly financial example of this: Here in my state, they dropped
a concepts-based curriculum into the schools. It cost MILLIONS.
(It was also heavy in constructivist-compatible teaching and
treated like a textbook instead of one resource of many, but
that's another post!)

When the teachers began implementing it, one of the MAIN problems
was that students weren't used to analyzing a mathematical object
(i.e. a procedure, concept, aspect, fact, feature) and refine
their thinking about it so that deep, connected, conceptual
understanding was grasped. I have concluded (and I might be
wrong) that students were mainly used to having information
poured into their heads, so it was an absolute shock to the
system for a math teacher, for example, to have students think
about the underlying concept that a pattern or mathematical
observation implied. And it seemed that teachers weren't
accustomed to eliciting student thinking, observing it, and
guiding it. Everybody (including parents) were accustomed to math
class as being equivalent to "calculate and label" class. This is
just my own opinion, and I might be WAY OFF.

Point is: Within a year, the expensive adoption was abandoned.
And . . . teachers lost face in the whole process! :(

Anyway, this is not about constructivist-compatible teaching. I
am a HUGE fan of using direct instruction whenever it's best,
which I believe is often. This is about concept-based teaching
that is teaching for UNDERSTANDING. A huge issue is that LEARNING
(as opposed to just DOING) is mediated by thinking. I know that I
hardly ever asked my students to profoundly KNOW and to truly
UNDERSTAND content for themselves based on the actual
characteristics of a mathematical object and not on my "expert"
status. I really just unconsciously directed them to BELIEVE that
what I was saying was true, based on my position as a math
teacher. I didn't do this consciously. In other words, I didn't
ever FEEL like I was saying to the students, "Believe what I'm
telling you about this math topic . . . well . . . because since
it's coming from my mouth as a math teacher it's true and . . .
um . . . here are some other superficial/random reasons that also
suggest/support that it's true." But that's pretty much what I
did for all intents and purposes now that I think about it. I
even think that sometimes you might need to do this (ask students
to believe you for a moment), and then go back and place the
conceptual understanding there at a later time. Problem is, in
the past I didn't do go back. "Do this procedure like this" was
the extent of what I had to offer.

When I took a hiatus from the classroom and tutored math using a
method created by a 30-year vet, I was FLOORED at the HUMONGOUS
difference it makes to concisely teach (traditional) PROCEDURES
conceptually and concepts . . . well . . . conceptually for deep
understanding. (I contrast the word conceptually with
constructivist-like because they aren't exactly the same.) What I
took away from that stint: When math is taught and learned
conceptually, math and learning math becomes so much simpler for
most (not all) students, and all the little math tidbits start to
seem for them very unified and related and continuous and
similar. And therefore sensible, manageable, cohesive, fun, neat,
etc. It is MUCH more manageable for most students, and math is
much less scary, overwhelming, and "busy" for most students this way.

It's kind of scary, but I had no CLUE about this until I
accidentally signed up to work with this math tutoring franchise.
That chance experience is the main reason why I can sit in a room
with advocates of constructivist-compatible teaching and
advocates of efficient procedures and fact memorization and see
the huge space for overlap in their goals: teaching procedures
with conceptual understanding and creating the conditions for
students to see the big, underlying themes that resonate through
mathematics. And then checking to see that students are seeing
these connections and reinforcing the process.

I think about how various educators respond to the idea that
"Teaching is telling." I think a PART of teaching IS telling, but
I also think that seeing is understanding. That's what a lot of
teachers cite as what they love to hear--as a main part of why
they came into teaching in the first place: That "Oh, IIIIII
SEE!!!!" This is different from, "I see that you see, so I'll
take your word for it." I think it takes both: a
well-orchestrated combo of telling and also of presenting
mathematical objects (concepts/facts/procedures) for students to
really see. And, I think really seeing is confidence boosting.
And, I think that really seeing is CRITICAL if American students
are to move out of the fixed mindset of learning (see Dweck) that
says that learning is fixed and innate versus mediated by effort
(the growth mindset).

In the past, I never really asked my students to do too much
seeing and thinking (save for when I randomly happened to have a
method for doing so). All I asked for, basically, was just doing,
whether consciously (with understanding) or not. Doing with
understanding wasn't a staple--a feature--of my classroom. I
mainly asked my students to calculate and label. I'm sure there
were some students who would have fallen in love with math if I'd
shown more of the beauty, continuity, and connectedness of
mathematics. I'm sure fewer students would have felt that math
rules were so arbitrary and so overwhelmingly numerous to memorize.

I've typed for entirely too long! I apologize for that, and for
any typos!

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
>

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