Once again, as a researcher who is currently employed in "helping"
students to learn arithmetic, I'd like to raise, at least in
principle, the importance of doing such research. Around the U.S.
and the world, a lot of students spend a lot of time either learning
or avoiding learning arithmetical skills, and it would seem vital
that such a large pool of effort should be well spent!?!
I'll add my personal fear here: having grown up in the "school" that
considers arithmetic important as a basis for higher mathematics,
I fear the widespread tendency towards dropping or bypassing the basics.
It is my considered opinion that learning to use a calculator is
a dangerous substitute for learning to perform mental arithmetic.
It seems to me that if this path is taken without understanding the
costs vs. benefits, we may unwittingly lead the future into undesirable
pathways. Though the cost and difficulty of such research might be
great, it would seem to me the costs of "flying blindly" into a
mountainside are likely to be greater. Even so, at some level,
what used to be "basics" must be relegated to history, or civilization
as a whole will not advance. I believe the evolution of education
should occur with conscious deliberation and the guidance of research.
------------------
In this section, I'll present a couple of summary/excerpts that
illustrate the dichotomy of views.
PRO:
A very-pro-learning-arithmetic-facts view was voiced by
Mark Greenwood <MLGREENWOOD@CSUPOMONA.EDU>: Mark wrote of several
situations that either test or use a person's grasp of arithmetic
to reach life-affecting decisions. Getting a job. Taking various
entrance examinations. Working as a carpenter or clerk. He
goes on to say
>We have a crisis in math. ... The high school ... in my town just
>announced that the incoming students have the worst math scores yet.
...
>Students in developmental math do need to have the ability to
>calculate in their heads. I think that problem setup for them,
>in the real world, is no more important that being able to
>confidently perform simple arithmetic in their heads.
CON:
One recent anti-learning-arithmetic-facts response was especially well
written, and touches on several of the issues raised above and in
other replies. Gerald Kulm <GKulm111@AOL.COM> writes:
>In recent mathematics testing, whenever large amounts of time have
>been spent on drilling on computation such as during the 1980s,
>test scores on problem solving have suffered. Often, children who
>are really gifted in mathematics are overlooked because they are
>bored by the computational emphasis in school. On the other hand,
>learning arithmetic can be interesting and develop deep
>understanding that will lead toward advanced work in mathematics,
>if it is done correctly. If patterns are emphasized, if reasons
>for doing things are explained and if students have a chance to see,
>touch, and experience numbers while they learn their arithmetic.
[This paragraph eloquently voices some of the arguments against
learning arithmetic facts. Note also the reliance on anecdotal
evidence and "lore." It's also interesting that both the
respondents use "decreased math test scores" as evidence in
their arguments! -wcm]
-----------------------
Finally, I'll turn from the big, imponderable question to respond to
a specific question about the math tutor I'm developing, again from
Gerald's message:
>I would be very interested whether your computer tutor can and will
>do the latter. I'd like to talk about it.
This raises an analogous dichotomy in teaching methods. One school
of teaching arithmetic (that I was raised with) says that you
teach arithmetic by teaching the facts. While the student learns
the facts, good things like recognizing patterns, seeing
relationships, and learning concepts happen spontaneously, because
the learner is a thinking being. I've learned things this way, and
it does work, at least for some people. The other school says,
teach the concepts and the facts will be easily learned as a
consequence. This raises my skeptic's hackles, since I believe it
is all too easy to say, "Yeah, I understand the concept" and QUIT
before learning the facts. If a student learns that 3+4 is the same
as 4+3, that's useful, but what if he/she doesn't learn that the
answer to either problem is 7?!?
Currently the adaptive tutor is very strong on learning the facts,
and treats concepts by presenting the facts in groupings that
encourage the student to observe patterns and generalize. That was
the chosen emphasis or Phase I of the project, and it has kept me
plenty busy for the past six months!
But the panel of consultants that advises me is very much in tune
with Gerald's interests, and strongly urges greater direct
conceptual content. Phase II of the project is about to be proposed,
and I will likely place some additional emphasis on conceptual
presentations.
One question that remains in my mind about this approach is: can
today's computer, which is essentially a two-dimensional
reading-drawing- talking device, do this type of instruction as
well as real-world classroom/teacher experience with manipulables?
It may simply not be cost effective to use a $2000 box with high
tech programming to simulate what some $20 plastic parts and cards
can do under teacher supervision already. Where is the right
dividing line between computer and teacher/classroom responsibilities?
How far should the responsibility for conceptual understanding be
delegated to the computer "teacher"? How well is this need met
by currently available CAI programs? Fully met? Good enough to
encourage further, better efforts? A miserable failure?
I'm thinking hard about these issues.
Thanks for your interest, especially if you've made it to the
bottom of this somewhat long "argument". I'll gladly continue
to read and think about any comments sent to me directly, or
to follow whatever postings people consider relevant to the list
[BUT- let's not be too redundant or circular-- I think the issues
are well delinieated, and the list would probably not benefit
from continued rehashing of the same themes...]
William C. Mead
wcm@ansr.com
Visit Adaptive Network Solutions Research, Inc. on the web
at http://www.ansr.com/ansr !