[Squeakland] the non universals

Fri Aug 24 06:48:47 PDT 2007

Hi Mark --At 05:01 AM 8/24/2007, Guzdial, Mark wrote:

Snip

>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?

I think most of the children after a few months of using "increase 
by" in various ways, do recognize rates in many other contexts.

>   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.

I would doubt that his understanding of these kinds of DEs is total 
or even "supremely comprehensive", but it is "operational" very along 
the lines that any mathematician would characterize as "mathematical 
thinking". Our goal was to make an environment in which more than 90% 
of the children exhibited real fluency in this kind of thinking. 
"Real fluency" implies a degree of understanding above an important threshold.

>- 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.

A much more important question is "will Tyrone understand mathematics 
by the time he gets to college?". If the answer is "yes", then he 
will recognize them as the same thing. If "no" then everything will 
be special cases of rules (which they are to most college students).

>   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.

Again, the question is whether he is actually learning math or not. 
It has nothing to do with 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?

All I can say is that this was very thoroughly studied in the 60s (as 
was deep habit formation). What they were testing were not memories 
of isolated unusual incidents (nor of "movie recognition memory" 
which is also from one trial). What they were doing was testing 
changes of paradigms in outlook, and for most children these took 
immersion in an environment for well over a year to be strongly 
detectable years later.

>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.

To me, these are not as interesting (nor are they parallels) to large 
scale epistemological shifts.

Cheers,

Alan

>With best regards,
>   Mark
>
>
>
>-----Original Message-----
>From: squeakland-bounces at squeakland.org on behalf of Alan Kay
>Sent: Thu 8/23/2007 6:49 PM
>To: bradallenfuller at yahoo.com; squeakland at 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>http://www.am 
> azon.com/gp/blog/post/PLNK13L1MC1Q3613J
> >_______________________________________________
> >Squeakland mailing list
> >Squeakland at squeakland.org
> ><http://squeakland.org/mailman/listinfo/squeakland>http://squeaklan 
> d.org/mailman/listinfo/squeakland
>
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