Tyrone is eloquent in his explanations--I believe he understands what he's doing.  Here's my concern: Does he really understand differential equations?  Let me break that down into two parts.

- When Tyrone is faced with another problem related to rates (maybe disease propagation, rates of decay, etc.) in eToys, can he use those tools to analyze the new situation?

  Does he recognize the situation as similar and that his same tools would apply?  That would convince me that he has developed an understanding of the powerful idea of differential equations.

- When Tyrone gets to college and studies differential equations, will he recognize them as the same thing?  I doubt that.  They won't look the same.

  His calculus course may not even relate to differential equations to modeling gravity.  He will have too few cues to make that connection.  A reasonable response to this should be that the calculus course might be taught with eToys, too, and that would help make the connection.  I would agree.  It's just unlikely that many (any?) college calculus courses will use eToys.

What I do believe is that the students in BJ's course have developed an understanding of the power of computation (*programmable* computation) in problem-solving and knowledge transformation.  That's tremendous, and likely will transfer to other situations using computers.

I'd like to argue with your claim from cognitive psychology, though.  "Length of exposure" is an ill-defined variable which has since been better refined and tested.  What does "length of exposure" mean?  One hour a day for two years?  One hour a week for two years?  Here's a brief thought experiment to address this point: I'll bet everyone on this list remembers exactly where they were and what they were doing when they first learned of the 9/11 attacks.  That wasn't a very long exposure, yet everyone remembers it.  Why?

The two new variables that are more often studied are:
- Time on task.  The more time you spend on an activity, the more likely that you will remember the experience and lessons of that activity later.
- Amount of reflection.  The more often that you reuse an association, the more often you think about and talk about an experience, the more likely you will retain it.  That's the best explanation I know for the 9/11 effect (or the Challenger effect, or the JFK assassination effect).  You thought about that moment later that day, and the next day, and you've discussed it with your friends.  That leads to longer term learning.

With best regards,
  Mark



-----Original Message-----
From: squeakland-bounces@squeakland.org on behalf of Alan Kay
Sent: Thu 8/23/2007 6:49 PM
To: bradallenfuller@yahoo.com; squeakland@squeakland.org
Subject: Re: [Squeakland] the non universals

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
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>Squeakland mailing list
>Squeakland@squeakland.org
> http://squeakland.org/mailman/listinfo/squeakland

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