Teaching for Skills vs. Understanding
Debate about the relation of skills to understanding dominated
an October, 1995 Roundtable at the University of Wisconsin, Madison. Excerpts
that follow are by John Janty, Mathematics Coordinator, Waunakee High School,
Waunakee, Wisconsin; Sarah Krinke, senior, Waunakee High School, Waunakee,
Wisconsin; Tom Kurtz, Department of Mathematics, University of Wisconsin;
Robert Meyer, Harris Graduate School of Public Policy, University of Chicago;
Margaret Ellibee, Wisconsin Center on Education and Work; Hal Schlais, University
of Wisconsin Centers; and Don Chambers, Wisconsin Center for Educatonal
John Janty: So what are the skills that I need to impart to students?
I hear a lot of talk around this table about the need for higher order thinking
skills, but believe me, we get a lot of messages that students had better
know how to add fractions too. Should we spend our time teaching algebraic
gymnastics or developing conceptual understanding? I'm getting very mixed
We have very limited time for mathematics. How can we use it most productively--on
conceptual understanding so that students can later understand how computer
software functions, or on basic skills? What is it to be--problem solving
Students, I might add, are not buying into drill these days. Sarah is as
likely to say "This is all very nice, Mr. Janty, but when am I going
to use this in my life? When will I ever need to differentiate all these
Sarah Krinke: That's right. I want to be a doctor, and they told
me to take Calc I and II, so I'm taking Calc I and II. When it's over, I'll
be done with math. But I still ask: "When will I ever use it?"
Tom Kurtz: There is no doubt, as John says, that the culture for
mastering skills has changed. When I was in school, if I was told to factor
ten polynomials, I did. But Sarah isn't going to factor ten polynomials--and
she is a good student. So the question is not just whether these skills
are needed, but is there any mechanism to engage students in learning these
essential skills? Factoring polynomials is not so important by itself, but
it is part of the language that one uses when discussing things that are
Robert Meyer: The message universities send is that if you master
certain skills then you can come to the university and pass some entrance
exams. But is there anything there? Do those skills mean anything? The typical
skills on mathematics placement tests are limited to such a narrow repertoire
that in a slightly different context these skills are virtually useless.
So I'd say, even if students pass the test, that they don't really "know
Tom Kurtz: How important are skills, anyway? On one level I agree:
they are not very important. But on another level, I know full well that
when I am explaining something to a classroom of fifty engineering students,
half of them won't have a clue as to what is going on because they are still
trying to figure out what I did when I cancelled two factors two boards
before. A failure to master those kinds of skills is a failure to understand
a language. It is like being locked in a room where the course is taught
in Spanish when you don't know any Spanish.
Margaret Ellibee: The NCTM Standards begin with process standards--math
as communication, math as reasoning--before moving on to more math-specific
areas such as algebra, geometry, and statistics. The [SCANS report] ("What
Work Requires of Schools") looks at skills desired by employers--both
basic skills like reading, writing and mathematics as well as higher order
skills such as reasoning. In that way you have a nice match, in my estimation,
between the NCTM Standards and some of these work force skills standards.
More important is that these skills--whether SCANS or NCTM--incorporate
a student-centered approach that integrates work force skill and mathematics
standards in a context that is relevant to students. School-to-work programs
make learning relevant to the workplace in terms of students' career ambitions.
It all has to be contextual--content, teaching, assessment--or it won't
make any sense.
Hal Schlais: I often teach older students, who have a different need
for skills. They have been away from school for a long time, and need to
take courses to get ready for what they really want to study. Many of them
succeed because they are motivated now to learn things that they never mastered--or
maybe never studied-- when they were younger.
Don Chambers: I believe school mathematics is a cultural artifact
that has been inherited from earlier generations. As such, its features
are deeply embedded in the fabric of our society and are passed on, through
school experiences, to each new generation. Many mathematicians, mathematics
teachers, parents, and members of the public have developed expectations
of school mathematics on the basis of their own school experience. Others
are dissatisfied with this inherited school mathematics and are proposing
changes. One vision of change is expressed in the NCTM Standards.
One important issue on which views differ has to do with the
development of understanding. Some believe that conceptual understanding
develops from a platform of procedural knowledge. Others believe that
conceptual understanding and procedural knowledge develop simultaneously
and support each other. There is some research support for this latter
view, which is built into the NCTM Standards. More research, and more
knowledge of outcomes of recent research, would be helpful in advancing
conversations on school mathematics reform.
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Last Update: 12/18/95