FRANK QUINN
Abstract. Many common terms have very different meanings in the two
communities,
and sometimes neither is appropriate. Actual solutions will require
us to recognize and transcend termino logy problems .
1. A Search for Meaning
A few years ago a draft K-12 Standards Document arrived at the AMS for
review.
This happens from time to time and while as far as I can tell AMS feedback has
no effect, it is flattering to be asked. However this Document was accompanied
by
a guide for reviewers that included the question:
"Do the standards specify a range of cognitive skills to be expected,
including some
range of the following?
•Remembering: recognizing, recalling
•Understanding: selecting, interpreting, illustrating, classifying, summarizing, inferring, comparing , explaining
•Applying: using, executing, implementing, computing, translating
•Analyzing: differentiating, organizing, attributing, synthesizing
•Evaluating: checking, critiquing, justifying
•Creating: generating, hypothesizing, planning, designing, constructing"
Say what?? Are these ranges of cognitive skills or ranges
of synonyms?
Math educators generally reject use of careful definitions
so one cannot just look
these up. However there is an extensive literature from which we could try to
infer
meanings, and we can see how these things actually play out in students. Two
conclusions emerge: first, as expected, these are for the most part synonyms and
reflect a richness of language rather than of content. A more troubling
conclusion
is that when when these terms do have specific meanings they are quite different
from the meanings used in the mathematical community.
2. Misunderstanding Understanding
Every discipline develops terminology adapted to the
discipline. Specialized
meanings for common terms lead to "talking past each other" communication
failures.
We illustrate this with the term "understand".
The mathematical community has evolved a rather strong
meaning for "under-
stand": roughly "complete mastery" including full facility with working
problems.
Weaker meanings have been found to be dysfunctional in the sense that they do
not provide a foundation for further mathematical learning.
FRANK QUINN
The educational community has a much weaker meaning for
this term. My
guess is that it reflects something about human learning: people learn some
things
(e.g. inferring patterns from examples) quickly and easily. Fixing errors in
this
natural learning is a different process and much harder, so it makes sense to
have
terms for the first step . "Understand" may be one of these. At any rate the
mathed
meaning for "understand" is closer to "show evidence of exposure". Teachers
can say "you can't work the problems but I see that you basically understand, so
I can give you partial credit". And when students get to the college level they
say
"I really do understand it, but just can't work problems. Can't you give me
partial
credit?"
There are similar mismatches with most other terms. Does
"recall the quadratic
formula" mean "know and be able to use the quadratic formula " or "recall having
seen the quadratic formula"? Does "know multiplication facts " mean "know there
is a multiplication table" or "be able to multiply numbers with facility "? Terms
such as "synthesizing", "justifying", "creating", "discovering", etc. refer to
highly-
structured activities that have little in common with the mathematical meanings.
3. Right, Wrong or Different?
To a degree these terminology issues can be seen as
cultural: they have their
meanings, we have ours, and it is neither necessary nor appropriate to declare
one
or the other "wrong". We just have to be mindful of the differences and very
careful
when trying to communicate.
There are, however, cases where one meaning really is
wrong. The slogan "we
should put less emphasis on rote learning and mechanical calculation , and more
emphasis on understanding" has strongly influenced math education in the last
few
decades. It is certainly very attractive. But remember that there is a job to be
done:
students should emerge with a good foundation for further mathematical learning.
We are not at liberty to use any convenient meaning for "understand" but must
use one that actually gets this job done. The math-ed meaning is dysfunctional
in
this regard and so-in the context of the slogan-is actually wrong.
I do not believe that use of a dysfunctional meaning for
"understand" is an evil
plot de signed to cripple higher education in mathematics, even if it is working
out
that way. The K-12 system is rather self-contained and the curriculum adapts
to whatever students can do. The failure to "provide a foundation" only becomes
unavoidable and acute at the college level. K-12 educators are pretty
unresponsive
to complaints from the college level, but in their defense it must be said that
these
complaints are often incoherent.
4. Plea
It is important to realize that the real problems will not
have terminology solutions.
Mathematical understanding is too demanding to be appropriate at the
school level. The mathematically-adapted meaning for "understand" might make
the "understanding, not rote calculation" slogan correct but would also make it
unrealistic. The proper goal may be a mostly-subconscious template for
mathematical
understanding. The math-ed meaning will likely play an important role
in it's development. Making effective sense of something like this would take
deep
insights into both human learning in general and the needs of long-term learning
MATH/MATH-ED TERMINOLOGY PROBLEMS
in mathematics. It is likely to require cooperative e ort
by both the math and
math-ed communities.
Unfortunately, we won't be able to formulate or agree on
the real problems, much
less solve them , until we sort out the terminology issues.
