Thanks for pushing back in this good way. I’m glad I remembered to go back and reread this this year!

But a set of well-designed situations, some which are intuitive and some which are counterintuitive (like the rocket graph of angle vs. time), could possibly be the trick to getting them to engage/care. I need to scaffold less and let them play around more, and related rates is a perfect place for them to just play. I did something like that last year and this year to lead into optimization which went over well, and they were overall engaged [http://samjshah.com/2012/03/15/optimization-an-introductory-activity-project/].

Thanks for getting me think about my practice. As always.

Sam

Are we to assume that climbing down the ladder of abstraction means students are moving from a task with more cognitive demand to one with less?

Not necessarily. Those two spectrums (“cognitive demand” and “the ladder of abstraction”) don’t correlate exactly. For just one example, if someone spends too much time just working on one level of abstraction, they can show symptoms of what Hayakawa called “dead-level abstracting” which would make a downward move on the ladder very demanding work.

I did want to give credit to where credit is due for the Rocket problem and image… Bowman Dickson [bowmandickson.com] posted it on Geogebra tube [http://www.geogebratube.org/material/show/id/2187]

It’s exactly this sort of dynamic visualization, this idea of getting a concrete sense of what’s going on in a situation, and THEN going forward to mathematize that situation… this is where I think related rates can be less stupid. (Because right now I’m still on the fence about them.)

As I pined on my blog a while ago:

“The more I mull it over, the more I think that geogebra has to be central to my approach next year… teaching students to make sliders to change one parameter, and having them develop something that dynamically illustrates how a number of other things change. And then analyzing how those things change graphically and algebraically.

(A simple example: Have a rectangle where the diagonal changes length… what gets affected? The sides, the angle between the diagonal and the sides of the rectangle, the area, the perimeter, etc. How do each of these things get affected as the diagonal changes?)”

Sam

I’d say no. Looking at the common core that Dan referenced a couple of posts ago, I’d say that

[quote]interpreting the results of the mathematics in terms of the original situation,
validating the conclusions by comparing them with the situation, and then either improving the model or, if it is acceptable,
reporting on the conclusions and the reasoning behind them.[/quote]

are all moving down the ladder of abstraction. They require different skills (i.e. synthesis instead of analysis) than moving up the ladder, but they’re just as cognitively demanding.

The students know when that’s the case. I have seen it in geometry (especially with angle measures). Until the students learn enough about what a degree is as a unit, and how to make some general connection in their minds about the look of an angle and its corresponding degree measure, there is very little value in having them predict… and the students are reluctant to do so.

I think you’ve hit the nail directly on the head. Students don’t want to be wrong – especially out loud and ESPECIALLY in a math class where many of them have become convinced that there is 1 right answer and that’s it. Of course, in many cases there is one right conclusion you;d like to reach but the journey there should be much more interesting than it is for many.

The timing of this post is perfect as I am using Sam’s worksheet that preceded the rocket one in my class today.