Jay Mathews, in "New teacher jolts KIPP", writes about Lisa Suben, a new teacher in the KIPP schools, who had her math students jump from the 16th to the 77th percentile in a single year. That's an unbelievably huge jump! How'd she do it? Theoretically, she says:

"My primary goal as a teacher is to help my students understand the reasoning behind math rules and procedures. I have several core beliefs about this: (1) Understanding is constructed by the learner, not passively received from the teacher. (2) Understanding is built by making connections between as many strands of knowledge as possible. (3) Understanding is galvanized through communication. (4) Understanding is only valuable when you reflect on it and question it."

Items (2) and (3) are related. That is, communication can (but need not) present more strands of knowledge to enter the picture that allows more connections to be made. It's not the connections per se that build understanding but rather the contradictions among them. Contradictions are the driving force of learning. On item (4), reflecting and questioning can improve one's understanding, of course, but most understanding is unconscious. That doesn't make it unvaluable.

Suben translated her theory into the following practice:

The core of her method is the workbook she produced last year on the fly. It "lets students build their own notes and create their own examples. It is incredibly active learning," she said. They were encouraged to write down the meaning of important terms and strategies they used that worked with certain kinds of problems.

Suben, I imagine, is differentiating between a traditional lecture form of teaching and Deweyan "learning by doing". It's not clear that one type of learning is more active than another. All learning is active. Of course, I can also imagine that students focus more on something they are "doing" as opposed to "receiving," and thus they spend more "effective time on task," the crucial element in learning. Thus, Suben's having her students create their own notes, examples, and meaning is an excellent way to (1) focus them more effectively on the tasks at hand and (2) bring them into contradictions between their declarative and procedural knowledge (see ACT-R Theory) and so improve their understanding.

Related posts on the five-paragraph essay:
Forget IQ. Just Work Hard!
The Expert Mind
Learning: A State of Disatisfaction
Learning with Examples