Reading: Beyond Formulas in Mathematics and Thinking
So the author describes an insanely dry lesson, pure procedure, utterly disconnected from its real world application. But then he contrasts it to the "pump" problem, in which the students are presented with a real-world application with almost no connection to the underlying procedures.
Middle ground, maybe? Alas. The author clearly thinks that goofy GeoSketch thingy (different name, same rose-colored software) has value. I have tutored kids who have been given Geosketch homework--kids of "privilege", as progressives so sensitively put it--and they call it Garbage Sketch or Goofy Sketch. Just last week, I worked with my Algebra Support students (most assuredly not kids of privilege) on the package, and half the class did nothing,while the other half diligently did the packet we'd given them--no discovery, just getting the job done. All explanations were compromised by the annoying need to get the software to work. The two kids who don't really need this class "explored" the system by making like Jackson Pollock, drawing bizarre paintings with brilliant color, and then abandoned Geosketch in favor of exploring the system files and seeing if they could hack in to Admin. Geosketch may have value for 1 out of 10 students, but generally speaking it's a monumental waste of time that my CT and I won't be taking on again.
I have also spent quite a bit of time, as it happens, working with these same students on the slope intercept equation. Discovery? Not, thank goodness. But we stress the relationship between the algebraic and graphic representations. We talk about about the ability to graphically estimate a solution by finding the intersection of two lines, and then show how the algebra works out as well. One of the nice "aha" moments on Geosketch day had, of course, nothing to do with Geosketch. A student had just managed to painstakingly create two lines with the same slope and obediently written down that they went "in the same direction". With a bit of questioning, we established that "going in the same direction" meant "not intersecting" and that this meant they were parallel.
"Remember what we call it when lines intersect?" I asked.
"Um. We call it a solution? Like where they equal the same thing?"
"Right. So how many solutions do parallel lines have?"
Pause, as she works it out. "Oh, I see! They would never have a solution!"
"So do the algebra on these two lines and see what happens."
She "solved" them and said "The x goes away! So there's..." "no solution!" we chorused together.
She didn't need Geosketch for that insight. She just needed two parallel lines with designated slopes so she could figure that out. (And in a quiz a week later, she--and about half the class--successfully recalled that parallel lines meant the same slope.)
So yes, there is a middle ground, but I didn't see evidence of it in the article. There's no one big "aha" moment in Algebra. There's constant reinforcement, constant reminders, and tasks that focus on learning procedures with a reminder that yes, these do apply to real life. Even so-called "traditional" textbooks do that, and have for years.
What tasks work best for student discovery? I know, I've answered this many times before in other "contexts", but what the heck--I like procedural instruction and a fair amount of repetition. In the early days of math instruction, connections to reality are helpful, but I see little value to long discovery processes. The point of discovery is the connection--and the only kids who reliably make the connection and engage in the discovery are the kids who would probably get the procedure without any context at all (something that even I don't think is advisable). There are kids who like discovery who aren't good at math, but I rarely see them link the discovery to the math.
BTW, I believe that the "Lampert" that Chazan referred to is the "Lampert" of "On the Nature of Accomplishment" that we read a couple weeks ago, she of the "credit/nocredit/partial credit is better than right and wrong" belief.