No connectedness without discrete skills

A mathematician once said that math was "a seamless whole" inside her head.

I don't know if this ties in with the idea of a seamless whole, but it has occurred to me that discrete skills are needed first before one can appreciate the connectedness of math. Without these discrete skills, math is more like a seamless black hole.

This became apparent to me again while teaching a group of seventh and eighth graders brought up on EM and currently using CMP who are a tabula rasa when it comes to the simplest bits of math knowledge. They can't do any operations with fractions (e.g. change mixed numbers to improper fractions let alone addition and division), can't divide decimals, don't have knowledge of even rudimentary geometry... One wonders what they have been doing for seven and eight years.

The seventh graders are currently in the CMP stretching and shrinking stage. Their homework consisted of finding the scale factor of two rectangles the width of which goes from 1.5 cm to 3 cm. So the idea was to divide 3 by 1.5 (they can't do it because they can't divide decimals). When I tried to show an alternative way of division using fractions to demonstrate the connectedness of math (seamless whole), I ran into trouble, too. They don't have the discrete skills of seeing 1.5 as 1 1/2, then changing this mixed number to 3/2 and dividing 3 by 3/2 (they absolutely can't divide fractions and moreover don't see 3 as 3/1. It would have been spectacular to make them experience with understanding that the more complicated decimal division problem 3/1.5 virtually solves itself when you divide the respective fractions (3 divided by 3/2). Invert and multiply but they have never heard of reciprocals and how they work. The 3 cancels and 2 is left standing without much ado!

So the upshot is: they use Connected Mathematics but can't see the connectedness of math because they don't have discrete skills (skills they could have learned through drill and kill but haven't). So to them, math is a seamless black hole from which not even light can escape.

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?

Lift all boats

Education Week reports on a new study that shows that smaller classes don't help close the achievement gap.

Class-Size Reduction of Limited Value on Achievement Gap, Study Finds

Reducing class sizes—a popular policy among parents, teachers, and lawmakers—has long been viewed as a way to increase student achievement.

But while shrinking the number of students in a class can lead to higher test scores overall, it might not necessarily reduce the achievement gaps that exist between students in a given classroom, a new study suggests.

Reviewing data from Project STAR—a longitudinal research study on class-size reduction in Tennessee and the most famous experiment on the topic—Spyros Konstantopoulos, an assistant professor of education and social policy at Northwestern University, in Evanston, Ill., said that it’s a “tempting” idea to think that having fewer students assigned to a teacher will reduce the achievement gaps between students.
Instead, he found, “manipulating class size” doesn’t appear to narrow those gaps. In fact, the range from the lowest achievers to the highest achievers—what he calls “variability”—was greater in the smaller classes of 13 to 17 children than it was in larger classes of 22 to 26 students. He came to that conclusion after looking at the performance of all students in the STAR study, as measured by the Stanford Achievement Test.

“In the present study, I found that high achievers benefit more from being in small classes than low achievers,” Mr. Konstantopoulos said in an e-mail. “This indicates that the achievement gap is larger in small classes than in regular-size classes.”

He suggests in the article, which is being published in the March issue of Elementary School Journal, that the higher achievers, perhaps, are better at taking “advantage of the opportunities or teacher practices that take place in small classes.”

It's probably true that high achievers would benefit more from such a setting. But why fret over gaps if both high and low achievers benefit from smaller classes? Just not at the same rate.

Student body reform

A grand experiment spearheaded and financed by Bill Gates aimed at reforming high schools in Chicago has suffered a setback. In addition to the ongoing high school transformation, Gates' other big idea is to smash high schools into smaller pieces, part of the so-called small-schools movement.

A high school in a tough neighborhood was turned into small schools six years ago. Performance continued to be dismal. Now, the idea is to put the pieces back together again. Oh, and don't forget to fire all the teachers in another merry-go-round.

See this Chicago Tribune article:

Chicago Public Schools to fire hundreds at 8 under-performing schools -- 200 teachers, 7 principals face ax after years of poor performance

Beginning in 2002, Orr was broken up into three small schools -- Vines Preparatory Academy, the Applied Arts Science and Technology Academy, and EXCEL-Orr Academy -- which were supposed to improve performance both through their scale and through curriculum specialties.

[snip]

Under the plan, Orr would be re-created as a single school, a teacher-training academy. It would be operated, along with its feeder elementary schools, by the Academy for Urban School Leadership, the non-profit teacher preparation and school management organization that runs Sherman.

Panacea after panacea has been tried with this school. What hasn't been tried is student body reform.

Biofuels not "green"

The New York Times reports on scientific studies that show that biofuels are not the vaunted "green" fuel many believed it to be. The findings makes sense. Insatiable demand for fuel will necessarily lead to the destruction of natural habitats when the fuel comes from plants.

UPDATE: concernedctparent called my attention to this great article in National Geographic on the subject.