For one of my final projects, I wrote the first of three lesson plans for a high school course on plane geometry. When designing learning environments, it’s important to work around four dimensions that affect learning. They are to what extent your classroom is **knowledge-centered**, **learner-centered**, **community-centered**, and, of course, **assessment-centered**. Sadly there is no absolute consensus about what those words actually mean. And even worse, there has been considerable emphasis on learner-centered and assessment-centered environments to the near exclusion of the other two. And even worse still, many politics have tricked the general population into thinking that there is a zero-sum binary between learner- and assessment-centered classrooms. The fact of the matter is, a good instructor will make sure to provide classroom that is well balanced among all four components.

Knowledge-centered is perhaps the easiest of the four concepts to pin down. Make sure there is substance to what you’re doing. Teach something. Knowledge-centered environments require just that: knowledge. My lesson guides to geometry are filled with—you guessed it—geometry. Passing mention of concepts from real analysis and abstract algebra show up. Were I to write a fourth installment, you’d read about symmetry groups, group representations, and addition. A proper discussion about measurement would dive deep into the definition of number itself, equivalence relations, and probably prove Euclid’s so-called Common Notions. (That A=A; if A=B, then B=A; and if A=B and B=C, then A=C. Yes, students should be able to explain why self-evident facts are true, too.)

Student-centeredness takes into account what the learner already knows—or doesn’t know, or misunderstands for that matter. For this reason, my lessons are written for the instructor but led by the students. I use a list of questions that the teacher can use as a model. Taken together they form a cohesive mathematical narrative. But since the point of student-centered environments is that each classroom ought to be tailored to the individual needs of the particular students in the seats, the idea of a student-centered lesson plan that has been blindly written and mass-distributed is somewhat antithetical to its own aim. The Socratic question-and-answer method gets around that. Instructors have both the license and responsibility to dovetail the lessons in a way that best suits the students in the class.

Because of the individual nature of the plans, assessment becomes a problem. How do you figure out if the students have figured out the material if there is not one but several possible right answers? There are over 350 published proofs of the Pythagorean theorem, for example. And all of them are equally correct.

Student-directed learning has assessment built right into it. The teacher can constantly monitor student responses to gage their depth of understanding. The count of prompting questions (given by the teacher) to achieve a particular response can be used an index of mastery over the material. This sort of examination is not obvious to the students and therefore relaxes the pressure associated to more conventional means of testing. Moreover, sustained dialogue between students and the teacher promotes a collaborative, community atmosphere within classroom. Students and instructor exchange roles dynamically, which fosters all sorts of other leadership qualities and instills intrinsic motivation and proactiveness within the students. Having students talk and draw on the board takes care of three of the target components all at the same time.

So, all that you really need my plans for are the knowledge. And the notes are pretty insightful, if I do say so myself. At least have a gander at the very pretty marginal glosses. I employed some artful information mapping techniques. You’ll find that the diagrams are rather palatable. I’d be interested to know what other teachers have to say about them, how I should change them, and if I should write more.

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