Ethical use and expectations

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Ethical use and expectations

        IÕve been watching the recent lively discussion on this thread (seems
to be a case of feast or famine here) all week, without having time to join in.
Much was said again that has been said before, though much of that (in my
opinion) is worth saying again a time or two. Maybe I can add just a bit of
different perspective, or at least give it a try.

        It seems to me that a lot of what is being discussed has to do with
expectations -- the expectations we have about what our students should be able
to do by the time they arrive in our courses, expectations others will have of
them after they leave our classes, and expectations students should develop
about their own capabilities (a matter that is as important a focus of
attention in our teaching as is development of actual skills and
understandings).

        Should our students, and our colleagues (say, everyone on this list),
be expected to be able to mentally square two-digit integers? I can do this --
itÕs not at all hard if you learn a few basic tricks and practice a little. But
no, I donÕt really expect my students (and all of you) to pick up this skill. I
do make the effort to let my students know each quarter that this is something
that any reasonably intelligent person _can_ learn to do, and to give them an
idea of how useful this capability can be if you are constantly working with
numeric calculations (electronically assisted or not).

        Should our students be able to add single digit integers by inspection
(another way to say ÒmentallyÓ).? Next question! Somewhere between these two
extremes lies the ground that people have been trampling all week in the
messages flying back and forth. I think what is important is to realize that
velocity in this respect is as significant as position -- different individuals
will always exhibit different degrees of capability with mental computation,
but everyone needs to monitor their own progress, strive for personal
improvement, and take pride in ground gained. ItÕs a lifelong goal for an
educated person.

        There is another whole set of expectations to be considered when we
move to talking about pencil-and-paper computation. Should students be expected
to be able to find the difference of two double-digit integers given these
tools (or, if they can perform this task by inspection, something better than
this)? I would say yes, myself -- it is reasonable to expect that having pencil
and paper handy should, in a pinch, extend oneÕs mathematical reach. Floating
point computations are another matter. Even in the old days, those who had to
multiply and divide regularly generally found ways to shove it off -- to a
slipstick, or tables, or some such (IÕve just run across a fascinating account
of how the ancient Babylonians used to do it 4000 years ago). And we can note
that even extended digital computations (e.g. finding the sum of a couple of
dozen entries) are better suited to other modes, an abacus or a digital
calculator -- mechanical or electronic both work just fine.

        Of course, calculation with pencil and paper has been carried, by
exceptionally talented and dedicated individuals, to fantastic levels over the
centuries. The highest general level of expectation in this regard was probably
the extraction of square roots by digital methods, which used to be a
universally-taught skill. Today, this is a curious anachronism, and the ÒrightÓ
way to find a square root, if one is needed, is to use a two-buck pocket
calculator.

        An interesting point comes out here -- reasonable expectations with
regard to mental calculation have not really changed much, but there has been a
significant shift with regard to p&p calculation. Looking back at high school
and even college textbooks of 75 years ago or so one is struck by the fact that
a good portion of the assigned work would nowadays be seen as a waste of time.
ThatÕs a good thing, one which we donÕt realize often enough when we decry
so-called lowered expectations in modern teaching -- we are really much freer
to focus attention on process, quality assurance, communication and all of the
other important matters that go into the proper solution of a mathematical
problem.

        But proper advantage is usually not taken in this regard. The final
point to make about expectations is that often our students are not expected to
do things they _should_ be able to do with the tools that are in their hands,
either by their instructors or (if it happens that the students _were_ Òbrought
alongÓ by a savvy instructor) those who work with them following their
experience in a modern mathematics class. No, we shouldnÕt expect the average
student to be able to extract a square root with pencil and paper, though we
should still expect the student to be able to give us a ball-park estimate of a
called-for square root, by inspection. If a calculator (any calculator) is
available, we ought to expect the student to do a lot more than just press the
square root key and read off the result. He or she should, at the very least,
be able to analyze the error introduced by rounding off the result to, say, two
places.

        Part of the reason why those setting the expectations do not set them
nearly high enough is lack of familiarity on their part with the powers of the
calculator. Perhaps they were themselves brought up with the old technology. Or
else they were taught in turn by someone who did not push properly. It is truly
the case that computational technology is progressing faster than we can make
proper use of it. Perhaps the ÒgoodÓ will slowly drive out the ÒbadÓ, and the
general level of expectations will rise as regular calculator use slowly
penetrates. It is certainly frustrating, though, to find oneÕs students held to
old expectations and not being given credit for the new things they _can_ do
(which, I think, was where this strand all began about a week ago).

RWW Taylor
National Technical Institute for the Deaf
Rochester Institute of Technology
Rochester NY 14623

>>>> The plural of mongoose begins with p. <<<<

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