Of course, Mark didn't look carefully enough at either the Squeakers DVD or the Kim Rose and BJ Conn book "Powerful Ideas in the Classroom" and other materials which show what we actually do with the kids (actually in 5th grade for this example).
We don't teach any abstractions, but work our way out from various kinds of animated movement in Etoys (constant velocity, random velocities, steadily increasing velocity, etc.). From a number of such examples the children gradually associate both a relationship "increase by" and a history of the movements (shown by leaving dots behind on the screen). Later (about 3 and one half months later, in the case of the first time we tried this) we got them to think about and investigate falling bodies. One example on the Squeakers DVD showed 11 year old Tyrone explaining just how he worked out and derived the actual differential equations of motion (in intellectually honest and mathematical version that computers make very practical). He did this by recognizing accelerated motion in the pattern of pictures of the dropping ball, measured the differences to find out what kind of acceleration (constant) and made the script for vertical motion partly using the memory of how he had done the horizontal motion in Etoys 3 months before. He explained how he did this very well on the video. Also, by luck, I happened to be in the classroom on the day he actually made his discoveries and derivations. Most the children were able to do this.
The important things about this experience was that Tyrone and the other children had learned a model of acceleration and velocity that was quite meaningful to them. Months later they were able to remember these ideas and adapt them to observations of the real-world. According to Lillian McDermott at the U of Wash, 70% of all college students (including science majors) are unable to understand the Galilean model of gravity (which uses a very different pedagogy in college).
The most important piece of knowledge from cog psych is a study done in the late 60s or early 70s that showed exposure to any enriched environment for less than 2 years was not retained. But two or more years of exposure tended to be retained. This also correlates to habit formation and habit unlearning.
So, I would argue that Mark's three examples are very different and don't really deserve to go together. And, in any case, all we know about the 5th graders is that using this pedagogy and Etoys they are generally able to be more successful in both the math and the science of accelerated change than most college students. This particular way of looking at differential equations has become more and more standard as computers have become more and more the workhorses of science (partly because they are in a form well set up for creating a simulation -- and for the kids, because they are much easier to understand than the previous standards for DEs).
Cheers,
Alan
At 03:23 PM 8/23/2007, Brad Fuller wrote:
though I'd pass this along for another viewpoint. Mark Guzdial's latest perspective on powerful ideas, abstractions and design patterns:
http://www.amazon.com/gp/blog/post/PLNK13L1MC1Q3613J _______________________________________________ Squeakland mailing list Squeakland@squeakland.org http://squeakland.org/mailman/listinfo/squeakland