Problems in Geometry at Putnam County High School


Debra Newsome

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 guide states:

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.

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 experience.

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 mind.


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, 1978.

Jackiw, N. The Geometer's Sketchpad. Berkeley, CA: Key Curriculum Press, 1995.

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, 1973.

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