Views on Effective Pedagogy
Comments on pedagogy and on the possible need for "multiple
mathematics" from an October 1995 roundtable discussion at the University
of Wisconsin at Madison by John Janty, Mathematics Coordinator, Waunakee
High School, Waunakee, Wisconsin; Robert Meyer, Harris Graduate School of
Public Policy, University of Chicago; Thomas Romberg, Wisconsin Center for
Educational Research; Jim Miller of the Wisconsin Policy Research Institute;
Mazie Jenkins, Abraham Lincoln Elementary School, Madison, Wisconsin; Hal
Schlais, University of Wisconsin Centers; and Judy Ann Jones, Madison Area
Technical College, Madison, Wisconsin.)
John Janty: If students acquire higher order skills--the ability
to apply what they learn to new situations--then we do not need multiple
mathematics. Higher order goals--to think mathematically, to communicate,
to make connections--are the essence of the NCTM Standards. When
we start talking about skill sets, then confusion sets in. The business
community wants one set of skills, university mathematicians want a different
set, and scientists want still other skills. It is all this talk about different
skill sets that leads to speculation about multiple mathematics.
If we could get students to think mathematically, to be able to formulate
their own mathematics in any situation--whether in business or in the university--then
we would not need to worry about multiple skills or multiple tracks or multiple
mathematics. But what kind of curriculum do we put students through so that
they well be able to make all those connections and do all those transfers?
That's what I'd like to know.
Rob Meyer: The 1982 data set from the "High School and Beyond"
study shows that a tremendous amount of mathematics is learned outside the
mathematics curriculum--especially in science and vocational courses. This
suggests two things about the movement to reform mathematics education.
Most important, it shows that NCTM's emphasis on learning in context seems
reasonable. Students learn most mathematics through its applications. But
the study also suggests that limiting mathematics reform to mathematics
teachers is quite inadequate. To meet the needs of different students, we
need to tackle the organizationally difficult task of using all our resources--in
different subjects--for the improvement of mathematics education.
Tom Romberg: I like to think of mathematics as a language like English,
a language that people use at different levels. To friends we speak one
way; in writing we use language more formally. Some people study language
and become linguists. Yet despite these differences there is just one English
language. I like Bill Thurston's image of mathematics as a banyan tree--with
many different roots that grow together. This interconnectedness yields
just one language, and just one mathematics.
Jim Miller: I really believe that we need to get teachers out of
the classroom and into businesses to see what kinds of problems businesses
are trying to solve. Education starts when teachers enter their classrooms
and close the door. We need to be sure that teachers have the knowledge
and skills required for their work when they close that classroom door.
Mazie Jenkins: Someone asked what turns kids on. When kids construct
their own knowledge, that's what turns them on. It doesn't have to be fancy;
it doesn't have to be flashy. But if they accomplish it for themselves,
that turns them on. We've taught algebra and geometry by memorizing: Do
this, do that... No wonder students don't remember anything. If they construct,
they understand; then they can recreate what they need later, when they
Hal Schlais: Many students at the UW Centers are over 30, and I can
assure you that the idea of constructivism applies there as well as it does
for school-age children. This concept of constructing your own knowledge
is an extremely powerful idea. Returning adults--often older women--are
just not going to do 95 simplification problems. On the other hand, they
are more than happy to take on a major term paper using all the technology
(calculators and computers) at their disposal, including a lots of reading
and library work. This way they can deal with complex optimization problems
with very few algebraic skills. However, our curriculum in the Centers is
modeled after the university mathematics major, so it doesn't provide enough
of these kinds of opportunities.
Judy Ann Jones: I teach technical mathematics, as well as courses
in intermediate and advanced algebra. I am always struck by the realization
that a lot of my college algebra students couldn't handle the kinds of mathematics
that I teach to my technical students. It is very applied, yet they need
this as a basis for their future careers. Students need to understand that
they cannot get by with general math.
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Last Update: 11/19/95