One great idea I took away from NCTM was giving tiered practice work, aka Challenge by Choice. We implemented the idea in our geometry classes' last unit of the year, on coordinate geometry and vectors, and for our final exam review. The way it worked was this: each time, we created two or three sets of in-class problems for students to work on in groups, sorted by difficulty. The lowest-level "A" problems were about as difficult as test problems, and--this is important--were labeled as such. The second level, "B" problems, were a little harder than test questions, and were labeled as such. Sometimes we provided a highest level, "C", which were even harder. For example, our second time out, the problems included:
A1: The midpoint of segment RS is (2, -3, 7). If R = (5, 15, -4) find the coordinates of S.
A2: Two vertices of an equilateral triangle are (-2, 3) and (7, 3). Find coordinates for the third vertex.
B: Regular hexagon HEXAGO has H = (4,0) and A = (10,0). Find the coordinates of point G.
C: Consider the points O = (0,0), A = (x1,y1), C = (x2,y2), and B = (x1 + x2, y1 + y2). What kind of quadrilateral does OABC have to be? Justify your answer.
Often, B & C level problems previewed material we would eventually encounter; later on in the unit, we proved that the points O, A, A + B, and B form a parallelogram. After announcing the activity, students were free to pick their own A, B, or C sheets, and made informal groups with other kids working the same problems.
There are two essential differences between this approach and the traditional homework sets with A,B, and C problems:
Most important, students choose their own level of challenge. As teachers, we worked hard to make the "A" level problems an honorable choice, and we coached some students up who we felt weren't taking on hard enough problems. But because students experience their challenges as choices they themselves have made, they work harder: we saw students struggle for ten or fifteen minutes with problems that they would have given up on as part of regular assignments.
Second, the baseline level is "test difficulty." So the students practicing "A" problems aren't going to be surprised by what's on the test; rather, they know what they're up against in a very explicit way. Students seeking more challenge work harder and get stronger, rather than coasting. The results on my tests were striking: I had more scores above 100 (our tests include 20 points of extra credit, and challenging problems) than in any previous test, and somewhat fewer failures.
Ordinarily we teach a problem-centered curriculum, where students arrive at ideas by solving and discussing one problem at a time. But we've found it useful to spend 45-50 minutes every few weeks reviewing, consolidating, and extending ideas; these tiered practices are far better than anything else we've tried. Everyone was engaged, and everyone got stronger. We can't wait to try them again in the Fall.
What about you? What did you do differently this last month, this semester, or this year?