TITLE: Teachers Need Relational Understanding
AUTHOR: Eugene Wallingford
DATE: October 10, 2010 10:55 AM
DESC: 
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BODY:
Last week I was pointed in the direction of Richard
Skemp's 1976 article,
<A HREF="http://republicofmath.wordpress.com/2010/01/08/richard-skemps-relational-understanding-and-instrumental-understanding/">
   Relational Understanding and Instrumental Understanding</A>,
on how peoples' different senses of the word
"understanding" can create dissonance in learning.
Relational understanding is what most of us
usually mean by understanding: knowing how to
solve a problem and knowing on a deeper level why
that works and how it relates to the fundamental
ideas of the domain.  Sometimes, though, we settle
for instrumental understanding, simply knowing the
rules that enable us to solve a problem.  Skemp
describes the cognitive mismatch that occurs when
a teacher is thinking in terms of one kind of
understanding and the student in terms of the
other.  Throw in a textbook that aims at one or
the other, and it's no wonder that teachers and
students sometimes have a difficult time working
together.

The paper is a little long for its message, but
Skemp does try to cover a lot of ground in his
own slow realization that teachers need to see
this dissonance as a concrete challenge to their
efforts to help students learn.  He even considers
the sort of situations in which a teacher may have
to settle for giving students an instrumental
understanding of a topic.  But one thing is clear:

<BLOCKQUOTE><EM>
... nothing else but relational understanding can
ever be adequate for a teacher.
</EM></BLOCKQUOTE>

I know that when I am weakest as a teacher, it is
either because I am underprepared for a particular
lesson or because my understanding of a topic is
instrumental at best.

I often hear teachers at all levels talk about
teaching a new course by staying one chapter ahead
of the students in the textbook.  While there may
situations in which this approach is unavoidable,
it is always less than ideal, and any teacher who
does it is almost necessarily shortchanges the
students.  Teaching is so much more than presenting
facts, and if all the teacher knows today is the
facts his or her students will be seeing a week or
so hence, there is no way that student learning can
tie ideas together or push beyond 'how' to 'why'.

When I think about teaching computer science and
especially programming, I think of three levels
of activity that give me different levels of
understanding:
<OL>
<LI> reading about something, even extensively </LI>
<LI> doing something, applying knowledge in practice </LI>
<LI> understanding something at a deeper level </LI>
</OL>

At the third level, I know not only how to solve
problems, but when and how to break the rules, and
when and how to reason from first principles to
create a new method of attack.  When I am at my
best as a teacher, I feel fluid in the classroom,
knowing that my deep understanding of an area has
prepared me for nearly any situation I might
encounter.

I'll close with this quote from Skemp, which alone
was worth reading the paper for:

<BLOCKQUOTE><EM>
... there are two kinds of simplicity: that of
naivety; and that which, by penetrating beyond
superficial differences, brings simplicity by
unifying.
</EM></BLOCKQUOTE>

Many people talk about the virtue of simplicity,
but this sentence captures in fewer than two
dozen words two very different senses of the
word and expresses that the best kind of
simplicity both grasps differences and unifies
over them.

That is what relational understanding is about.
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