[Edu-sig] a non-rhetorical question

[Jay Bloodworth]

I'm joining the discussion late; I'm going to respond on a couple of
points that resonated with me, but forgive me for neglecting a few
attributions.  Also, I'm going to ramble a bit through some of the
things I think about as a math educator.  Hopefully I can make it work
the trip for you to follow along.

I don't teach programming.  I teach math in a public school - algebra
and geometry to 7th and 8th graders.  While certainly there are
differences, I think the core issues in teaching the style of analytical
thinking are the same for programming and formal math.

I've been teaching math for 8 years.  I'm department head, National
Board certified, my test scores are pretty good, the high school
teachers generally say my students know their stuff.  That being said, I
honestly have no idea how to teach algebra.  I'm not positive anyone
does.

I present the algorithms correctly.  I give mnemonics, organizers, and
shortcuts wherever I can.  I show numerous examples, give the students
plenty of practice, accurately diagnose (at least the proximate cause)
and correct mistakes when students make them.  Plenty of students are
very successful.  But some aren't, despite their and my (apparent) best
efforts.

In the more extreme of these cases, I tell the student and parents that
he or she is simply not ready for algebra, that they can repeat the
course next year and with a year of brain growth under their caps they
will probably be very successful.  And they almost always are.

So at one level "not ready for algebra" is an entirely correct
diagnosis.  But then the question becomes: What's going on in the brain
of the "ready" student that is different from the "not ready" student?
Call that question A.  Once we've answered questioned A, question B is
obviously: Is there a different way to teach the "not ready" student so
he/she can be successful earlier?  Important question, but I don't think
we have much chance of answering it until we make some progress on A,
and from where I am in the trenches, I don't even see any work being
done on it.

Bit of an aside: The hip thing in education now is "brain research".
Some consulting firm sells the school 200 copies of their book and
charges us beaucoups of money for a three day inservice to tell us
things like the average human brain can hold 3-10 "chunks" of data at a
time; the amygdila is the part of the brain that controls attention and
that we need to engage the amydilas of students if we wish to teach
them; that students learn better when they're happy than when they are
sad or mad.  I'm quoting selectively only because my brain (amygdila and
all) shuts down when I have to sit through this garbage.  It's
frustrating not because the notion of brain research in the service of
is a bad idea, but because the brain research we need is the kind that
answers question A, not the kind that tells us that students learn
better if the walls are blue.

Now, getting back to Andy's question.  My first response to this was
that as a teacher he should have had a pretty good idea of how students
would do on this assignment because they should have been assessed
informally on assignments like this one before.  But as I thought about
it before, I realized that there is a lot of room to debate what is
meant by "assignments like this one".  Does that mean an assignment with
precisely the same logical structure but just different strings?
Another "loop until" with slightly different conditions?  Loops, input,
and conditions separately?  Or pairwise, but never all three?  I
constantly see example of students who are fluent with two skills
individually but flail wildly when asked to compose them (e.g. they can
do 2x+3x and 2/5+3/4 but not 2/5x+3/4x).  Why some students fail at
these composed tasks even when the interface between the modules is very
clean is a big part of the answer to my "question A", and as I've said I
don't think we're very far along in terms of answering it.  As a
practical matter, Andy, I think my answer to your question is that you
need to give your students a lot more practice that is a lot more
similar to what you plan to assess them on than your gut tells you you
need.

Whew.  Sorry for the brain dump.  My thought were a lot more focused in
my head.  The upshot is I don't think there is an easy answer.
Programming computers is easy.  Programming brains to program (or to
combine like terms, or to solve systems, or whatever) is hard,
particularly since a good portion of the brains at large seem to
implement the algorithms according to different semantics than than the
ones we intend.  Hopefully one day we can reverse engineer these brains
enough that we can program them effectively as well.

Jay