Problems in Geometry at Putnam County High School
In response to calls for reform, in both the teaching and the learning
of school mathematics, the National Council of Teachers of Mathematics has
taken a major stand on the content and emphasis of the mathematics curriculum.
In producing its Curriculum and Evaluation Standards for School Mathematics
(NCTM, 1989) and other documents, professional educators have assumed leadership
roles in two critical areas: (1) the creation of a vision of mathematics
in an increasingly technological society with a diverse variety of needs
and requirements and (2) the design of a set of standards to guide curriculum
revision within this vision. These goals as well as the broader goals of
getting students to value math, gain confidence in their own ability, and
become comfortable as mathematical thinkers place the teacher in the role
as a coach and catalyst for knowledge acquisition.
The state of Georgia has taken a close look at the NCTM Standards in producing
its Quality Core Curriculum (QCC). Indeed, the Geometry or Informal Geometry
Geometry provides students with a way to link their
This small sample of statements from the introduction seeks to link a Georgia
high school course in geometry directly to Standards 1, 2, 3, 7, 8, and
14, and indirectly to the remaining Standards. Despite these affirmations,
the Putnam County High School geometry courses for college-bound students
and the geometry topics for general or vocational students have continued
to be taught predominantly through textbooks and strongly teacher-centered
instruction, especially for lower-achieving students. Manipulatives have
been limited to 3-dimensional models of geometric solids to be held up in
the front of the classroom. Tools have been limited to compass and straight
edge and then used exclusively to complete paper-and-pencil constructions
for college-bound students. Instructors outside the field have rarely been
sought and interdisciplinary lessons have been severely limited by the knowledge
base of individual teachers.
perceptions of the real world with the mathematics that
allows them to solve a variety of problems they will
encounter not only in other disciplines but also in their
lives....Geometry should provide students with visual and
concrete representations that help them gain insight into
important areas of mathematics and their applications to
the real world...High school geometry must extend beyond
the traditional treatment of geometry as a deductive
system and provide students with a broad view of
geometry and its applications...(p. 1).
The role of the teacher as a facilitator rather than as a dictator is perhaps
the greatest distinction between the "traditional" mathematics
classrooms at PCHS and those envisioned by the writers of the standards.
Too many secondary teachers are uncomfortable granting students permission
to move around the room, not to mention allowing interaction with their
classmates! Students are complacently resigned to assume the passive role
of being "talked at" by the teacher. Many adults still view mathematics
classrooms as being the same as they were during personal educational experiences.
Students must now accept a far greater and active role in their own education
to receive far greater rewards. Through group and individual projects, assignments,
and the like, the student is able to explore and see the integration of
mathematical topics that emphasize the body of mathematics as a whole "greater
than the sum of its parts."
The current role of the teacher in many classrooms at PCHS is that of an
information-giver, despite personal desires to create active rather than
passive learners. Teachers need to accept new roles as leaders and inventors
and be facilitated in these roles, but time is lacking to share with and
support other teachers. The mathematics and science classrooms at PCHS are
located across the hall from each other and opportunities to collaborate
have been initiated during the past school year and should continue to be
actively sought. Interdisciplinary teams are nonexistent in most high schools
although the practice of common planning is now part of the middle school
concept. Many primary teachers enhance learning of specific content with
instructional units, integrating a variety of subjects, as opposed to the
isolated content areas in the high school curriculum. Students should be
accustomed to the integration of topics based on experiences at the primary
and middle school levels.
Although publishers have sought to meet the varied mathematical needs of
teachers and students, textbooks can not continue as the main source of
geometry instruction. It is not in the spirit of the Standards nor the interest
of the students to approach applications as side bars or "extra"
exercises. Emphasis on pencil and paper exercises reinforced by correct
answers will not facilitate full appreciation of a subject so visual in
content and scope, yet this has been the nature of the resources used by
mathematics teachers at Putnam County High School. Despite the adoption
of new textbooks which include a variety of supplemental materials, the
focus remains on the textbook as a primary source of instruction. The complaint
raised continues to be lack of time to explore a variety of presentation
methods and materials prior to classroom implementation. Students must be
shown that there is more to geometry than those topics which appear in a
textbook but this engagement can occur only when teachers accept the responsibility
to facilitate rather than dictate learning.
One reason for the existence of a narrow approach to the teaching of high
school geometry is tradition. Teachers tend to adopt teaching styles and
methods most like those to which they themselves were exposed. Many people
hold the childhood views of school as authoritarian institutions in which
somebody smart stands in front of a room and tries to pass information on
to large groups of students. The content of textbooks, too, is often a reproduction
of those used by previous generations despite the current penchant for visual
displays. Emphases in content topics may change, but the typical classroom
activities carry on as they always have. Society must realize that not only
can students (and teachers) learn from each other, but also from manipulatives,
other tools and technologies, a spectrum of qualified personnel, and a wide
variety of situations.
Tradition breeds the idea that schools should operate on the assumptions
of the past and therefore schools are having a difficult time adapting to
the ever-changing needs of the present. Even with the large volume of, the
retrieval speed of, and the variety of sources of information available
today, our school finds it difficult to stay abreast of current trends.
Rural schools, like Putnam County High, are especially at risk for lagging
behind, since they are often further removed by geography from major business
and industry influences. Information as to the current and future needs
of employers and post-secondary institutions may be disseminated by informal
or other means which may be unavailable or inaccessible to rural districts.
Smaller population centers lack the financial and human resources that are
taken for granted in urban and surrounding suburban centers. The low population
density in rural areas limits the attention given to education by the media
and hence even fewer resources may be identified or accessed (Bracey, 1992).
Availability of financial resources continues to be a major player in the
determination of curriculum and its implementation. The diversity in size
and resources impinges negatively on rural communities in the applications
of geometry to which students are exposed. Smaller systems or those less
financially able, have had to be content with less technology, specifically
computers, software, and multimedia applications. Rural schools face strong
competition for resources from suburban schools whose populations include
a larger tax base to provide money for materials, experimental and/or innovative
programs, and instructors of the highest caliber. Furthermore, a wider variety
of business and industry personnel in non rural areas provides resourceful
contacts that can increase motivation for students to learn geometry, especially
if knowledge acquired can be applied in a local job or is directly related
to a field chosen for further study.
The world apart from school depends on the successful conception, implementation,
and completion of projects that involve the cooperation of many individuals.
Students are too often left to forage through the same old curriculum, in
the same old manner, with the same old results, namely poor student achievement,
motivation, and inspiration. An education system suffused with individual
and group projects, particularly apprenticeships and hands-on experiences
can fill the void in genuine student understanding. Assessment, although
often looked upon with disfavor by students, must become part of the learning
process rather than the objective. Alternative assessment advocates maintain
that the proof of a person's capacity is found in their ability to perform
or produce, not in their ability to answer on cue. Students value the opportunity
to discover for themselves what they have mastered, without the need for
teacher approval. Mathematics teachers at PCHS are seeking new ways to assess
students, but these processes require further study since no department
member is experienced in alternate assessment methods.
Geometry offers a wonderful opportunity for students to expand their knowledge,
gain confidence, and explore new interests. Projects completed independently
of the classroom or cooperatively within the classroom have allowed students
at PCHS to explore new areas of geometric applications while pursuing their
own interests. Art, architecture, construction, design, drafting, forestry,
history, map-making, photography, research, teaching, and many other areas
all provide opportunities for students to use geometry in a field of interest.
My students have explored a variety of topics including tessellations, scale
models, art, blueprints, fractals, geometry in the workplace, and bridges.
These projects, when presented to classmates, offered other opportunities
for personal growth and inspiration for peers. As their teacher, it has
been a joy to experience the interest of this aspect of my students' geometric
Teachers, too, can benefit from the experiences of their peers and expand
their knowledge base to provide interesting activities for their students.
Multi-disciplinary committees can be formed to plan for units that would
reinforce concepts in all areas. Math conference participants have shared
a variety of projects and plans for implementing programs designed to provide
innovative opportunities for students to connect and integrate concepts
and activities within the disciplines of mathematics, science, and technology
education. If business personnel can be persuaded to provide input, perhaps
offering real problems occurring on the job, students may experience the
power of geometry through sources outside of the classroom. Study for an
advanced degree has offered opportunities to extend and explore standard
topics within the framework of technological innovation. The door that has
been opened by this knowledge will benefit teachers and students at PCHS
for years to come.
With graphing calculators and computer software, students can now experiment
with parameter changes that previously had to be time-consumingly drawn
by hand and visualize how the graphs are effected without the tedium of
re-drawing. Many maintain that these visualizations allow students to make
abstract connections, yet students are only now beginning to experience
technology at Putnam County High School. Graphing and computer technologies
which have been available at other schools throughout the state have only
now been made available to students at PCHS, and presently only on a severely
limited basis. As recently as the past school year, students have been graphing
exclusively using paper-and-pencil, which has had obvious limitations to
the depth of mathematical experiences. Technology is quite expensive and
financial considerations have often been given as justification for lack
of spending in Putnam and other rural counties. Extra effort must be made,
however, to supplement direct expenditures on technology with alternate
sources of materials acquisition such as aid-in-kind. Putnam County leaders
must overcome their aversion to seeking financial sources outside of the
local budget. Grant money may be sought, business leaders may donate or
lend materials and/or speakers, or appeals to the community may result in
volunteers who can share their real-world uses of geometry with students.
Despite statements in curriculum guides or mandates from administration
as to the content to be taught in geometry, teachers make the ultimate decisions
as to the depth and scope of these objectives and how they will be carried
out in the classroom. It is evident that student-centered activities must
come from teacher leadership positions. Moves to broaden the approach to
teaching geometry will not occur without a direct effort by teachers in
the classroom to refocus the direction and methods of instruction. Most
teachers in Putnam County have not been trained in the use nor the application
of technology toward teaching methodology in the classroom. This training
process, though slow to be implemented, will surely benefit all parties.
Many sources offer suggestions on how to make mathematics more interesting.
The current high school student seems to be under the influences of "entertainment".
Leisure and fun are pervasive in their lives in the form of fast-paced computer
and video technology and students seem to expect education to come in the
form of entertainment. Many books activities have been written to make mathematics
fun. The "hook" of fun is designed to increase success and encourage
the further study of mathematics. Activities can be designed that are fun
but also require application of concepts learned in the classroom. For example,
rope and chalk can be used outside to build a hopscotch or a basketball
court after a unit in geometry on compass and straight edge constructions.
Origami activities produce objects of beauty and geometric significance.
Tessellation exploration and creation can be as practical as quilting or
as creative as design opportunities.
The use of manipulatives is recommended by professional mathematics teachers
and their associations (NCTM, 1989). Teachers are inundated with catalogs
offering manipulative merchandise for sale, but these accouterments cost
money. Creative teachers and students, however, can generate many items
with available scrap materials, donations, and redesign of projects. For
example, a hypsometer can be used to measure the angle of elevation or depression
but a crude representation can be constructed of tag board or adapted from
a protractor and a paper clip. Low budget activities such as paper folding,
geoboard lessons, tangrams, etc. are often readily shared among teachers
at professional conferences or work sites.
Practice lab experiences, although normally confined to science classrooms,
can provide opportunities for students to brainstorm problem solving techniques
and explore results. For example, Hunt (1978) describes three methods that
are commonly used to determine the height of an object. Working cooperatively,
students are likely to determine these and other methods through discovery
or research. The same problem can be solved in geometry after a unit on
basic trigonometric functions, employing a hypsometer or its equivalent.
Labs can be involved and require additional resources or as basic as paper
folding explorations. Explorations using The Geometer's Sketchpad await
the geometry students at PCHS for this school year. NCTM yearbooks and addenda
offer other resources for exploration. Materials included with adopted textbooks
also offer suggestions for lab activities and these may be fully accessible
to student groups.
Opportunities for changing the traditional narrow approach to high school
geometry exist. Students, teachers, business and community leaders, and
parents can exert a positive influence over the materials and methods used
in the classroom to provide innovative and interesting explorations in geometry.
Every successful innovation has the power to spark the imagination of a
Bracey, G. W. The second Bracey report on the condition of public education.
Phi Delta Kappan, 74, 104-117,1992.
Coxford, A. F. Curriculum and Evaluation Standards for School Mathematics
Addenda Series: Geometry From Multiple Perspectives. Reston,
VA: The Council, 1991.
Froelich, G. W. Curriculum and Evaluation Standards for School Mathematics
Addenda Series: Connecting Mathematics. Reston, VA: The Council, 1991.
Georgia Department of Education. Quality core curriculum: Geometry
or informal geometry, 1990.
Hunt, J. D. How high is a flagpole? Arithmetic Teacher, 25,42-43,
Jackiw, N. The Geometer's Sketchpad. Berkeley, CA: Key Curriculum Press,
National Council of Teachers of Mathematics. Curriculum and Evaluation
Standards for School Mathematics. Reston, VA: The Council, 1989.
National Council of Teachers of Mathematics. Geometry in the Mathematics
Curriculum , Thirty-sixth Yearbook of NCTM Reston, VA: The Council,