GEOMETRY COURSE PHILOSOPHY--ST. STEPHEN'S
Ken Rutkowski
The "Content" Imperative
"What do we teach?" and "How do we teach it?" are
questions of central concern in mathematics education today. For classroom
teachers such concern often translates to a working reality based on intimate
knowledge of the sequential nature of the mathematics courses they teach.
Math teachers are acutely aware that certain facts and procedures must be
covered before students can expect to progress to the next math course. The
traditions built around this awareness--as codified in textbooks and curriculum
guides--are usually what determine course content.
Assuring that math courses cover all relevant facts so that students can progress
up the ladder, however, should not be the only rationale driving curriculum.
Students, certainly, want to know why they even have to climb the ladder and
what awaits them at the top. For their part, teachers want to make sure that
their students have become creative and analytical thinkers and problem-solvers
as they progress through the math curriculum. Ideally, a math course should
strike a unique balance between content and cogitation, a balance based on
the recognition that either one in and of itself does not guarantee the other.
From this perspective, Geometry affords a truly unique opportunity in the
math curriculum. A very informal survey of the HRW text's table of contents
shows that only half to two thirds of the sections can be considered crucial
to other math courses or to standardized tests (see addendum). This does not
mean that the remaining material is not important. It does, however, mean
that geometry teachers have the opportunity and the obligation to make the
development of mathematical modes of thought a central emphasis of their curriculum.
General Themes and Components
Geometry's influence on other math courses is often subtle but potentially
powerful. It has become somewhat of a cliché to say that students should
be taught to be thinkers and problem-solvers. The cliché takes on meaning
in a geometry course when students are involved in activities which foster
their intellectual development in the following ways:
By teaching students to conceptualize and analyze any mathematical problem
from a geometric perspective.
By challenging students to function as working mathematicians, observing
patterns, formulating conjectures, and validating statements as parts of a
larger system of knowledge.
By necessitating that students articulate connections between
concepts and specific problems so that they are able to bring previous knowledge
to bear on novel situations.
Above and beyond specific content issues, these are the global components
that should stand central to a course in geometry.
Activity-Based Classrooms
The teaching reality of many math classrooms today is essentially the
same as it has been for many years. This lecture based system consists of
teacher presentation of new material, practice/example problems and discussion,
and assignment of homework. This system works well--especially with motivated
students--as an effective and efficient way of delivering information in a
timely manner. More often than not, math teachers have been forced to rely
on it in order to get all pertinent material covered.
Geometry classrooms have been some of the first math classrooms to become
activity based. Classes begin with an exploration--done either in small groups
or as a teacher-led "follow along. " This is followed by whole class
discussion where students articulate their conjectures and write down notes,
guided practice examples, and assignment of homework.
Activity-based classrooms give students an initial opportunity to discover
course content independently or in collaboration with their peers rather than
having it given to them by the teacher. It is important to note, however,
that this teaching model is basically the traditional lecture system with
an additional over-lay. As such, students get multiple opportunities to assimilate
material and to have it reinforced. Also, teachers need not feel like they
must choose between two opposing systems. They can adapt as the exigencies
of time and their own comfort levels permit.
Activity-based learning is particularly well-suited to the geometry classroom
since the "content imperative" is not as much of an issue as it
is in other math courses. Since every part of the process is important, it
obviously takes more time to cover material in an activity-based setting.
Teachers are forced to make critical choices about the relative value of course
content in order to foster a learning environment that maximizes student involvement.
Knowledge transfer--uncharted territory
Math teachers are often amazed at the inability of even bright students
to relate previously learned concepts to new situations. Math texts rarely
require students to articulate the specific connection between a concept taught
in a lesson and a question posed in a homework problem. There is an assumption
that the student must make the connection in order to do the problem. Many
students, however, are at a loss to explain how what they learned enabled
them to solve a problem. They are simply satisfied that they can do it.
Students are being asked to collect their work in portfolios and to keep journals
where they communicate their thought processes. Homework, quiz and test questions
can also be rewritten to incorporate more than answers and student work. Still,
designing questions and activities that teach students how to rise above what
seems to be their natural intellectual inclination to compartmentalize rather
than to relate and connect remains the great unknown in mathematics education
today.