## Thursday, February 21, 2008

### Explicit and interactive math teaching

I came across comments made by Lee Stiff, past president of NCTM on how math used to be taught.

STIFF: Parents are upset because, when they visit classrooms, they see activities that they're not used to. When they were students in school, they probably sat in rows neatly lined up, and the teacher just talked and talked and they used paper and pencil, and that's how they learned their mathematics.

When they see students engaged and talking with one another, when they see teachers allowing students to question and think thoroughly about the mathematics and the relationships, they wonder if the basics are going to be achieved. But the test results show that they are, their students are learning the basics.
It seems to me that keeping this caricature of traditional math teaching alive plays a vital role in perpetuating fuzzy math. By setting up a false dichotomy, the caricature provides the rationale without which the fuzzy project would collapse.

The scenario described by Stiff sounds more like an idee fixe, a hallucination or just a plain lie. What teacher would teach math without encouraging student participation through questions and having students work problems or come to the board?

Recently, I achieved amazing success with a small group of usually refractory and definitely lagging 7th and 8th graders through a combination of direct instruction and the Socratic method. The problem I put on the board was a circle inscribed in a square, one of my favorite mini think problems (an alternative is two circles in a rectangle). The task was to calculate the area not covered by the circle. Only the measure for a side of the square was given. The creative jump was to see that subtracting the circle area from the square area would get to the answer and that the known length of the side of the square would reveal the radius of the circle.

It was amazing to see that with a little bit of prodding and filling knowledge gaps the students actually got the answer with FULL UNDERSTANDING.

SAT brain teaser:

To make an orange dye, 3 parts of red dye are mixed with 2 parts of yellow dye. To make a green dye, 2 parts of blue dye are mixed with 1 part of yellow dye. If equal amounts of green and orange are mixed, what is the proportion of yellow dye in the new mixture?

fifteenweeks said...

So, you say you're an instructivist, but I say you're a REconstructivist, same as I am. Instead of either/or, our best bet is both/and: we teach facts and concepts, then arrange some sort of concrete experience to hammer them home. We don't attempt to teach the big picture without teaching its component building blocks, but your example seems to show that you don't teach the building blocks without at least indicating what big picture they eventually serve.

I'd be interested in developing a sound hypothesis on why ed schools have gone off the deep end on this constructivism thing. Folks like Kozloff seem content to call them names, some of which they no doubt deserve; but that helps not at all in the attempt to change minds.

Instructivist said...

"So, you say you're an instructivist, but I say you're a REconstructivist, same as I am."

I am trying to figure out how to interpret the prefix re. Help!

I am in favor of explicit, interactive instruction more or less in the form of V.'s ZPD. Interactivity includes asking probing questions and student effort. I also believe in lots of practice. I am against calculator overuse. Calculators make sense for higher levels once the basics are mastered. Most of that is anathema to constructivists.

"I'd be interested in developing a sound hypothesis on why ed schools have gone off the deep end on this constructivism thing."

That question interests me, too. I think the trouble started when academic knowledge became detached from pedagogy. Ed schools then lost touch with reality. That made them susceptible to quackery. The trouble goes back over a hundred years (even farther if you include the influence of Rousseau, Pestalozzi, Froebel, Spencer...) . The historical Progressives were an anti-intellectual bunch who despised the academic curriculum.

rocky said...

orange -> 3 red : 2 yellow
green -> 1 yellow : 2 blue

Orange has 5 volumes, and green has 3. To make equal parts we need 3 orange plus 5 green.

orange -> 9 red : 6 yellow
green -> 5 yellow : 10 blue

So that would make 11 yellow for every 30 volumes of total.