# Distance Education: A Powerful Medium for Developing Teachers' Geometric Thinking.

The purpose of this article is to describe a mathematical
professional development experience for a group (n=l1) of practicing K-7
teachers. Based on research about characteristics for successful
professional development, the workshop/ course was designed to provide
quality experiences for the participants. The content focus of geometry was selected due to requests from half of the group of teachers. The
medium for delivery of the professional development was two-way
audio-video distance education, which enabled several of the teachers to
engage in the otherwise unrealizable experience, due to the distance
necessary for their travel. Comments from teachers' written
reports, lesson plans, and reflections about the level of success of
those lessons provide some evidence of a change in teachers'
geometric knowledge and an advancement in their knowledge about the van
Hiele (1986) theory regarding teaching children to learn geometry.
During the 11 week workshop, all 11 of the teachers demonstrated growth
in knowle dge of geometry and illustrated a working knowledge of the van
Hiele learning theory for geometric knowledge. Professional development
in geometry can be successfully delivered through distance education. It
can also influence teachers' planning of lessons and subsequent
analyses of children's work with geometry as well as impact
teachers' knowledge of geometry.

Over the past two decades, mathematics educators have increasingly turned to constructivism as a reasonable theory to explain student learning and plan subsequent instruction. One way for teachers to use teaching strategies that are consistent with constructivist ideas is to carefully attend to a student's thinking and ask appropriate follow-up questions. Teachers who are able to ask effective questions must draw from strong content knowledge to suitably oversee their students' thinking and learning. However, teachers will teach as they were taught (Lortie, 1975; Russell, 1997; Schifter, 1997; Scholz, 1995). So, not only is it imperative that teachers know content exceptionally well, they must have learned that content through some sort of learning experience that modeled best teaching practice and which was founded on constructivism. There are mathematics teachers for whom these necessary types of learning experiences did not occur. In those cases, professional development is essential. This article describ es a professional development experience that modeled some of those effective practices and discusses the plausibility of using distance education technology as a medium for such an experience.

BACKGROUND

Historically, traditional mathematics classrooms have been characterized by learning concepts and skills in isolation from meaningful contexts and by little connections being made among mathematical ideas or between mathematics and other disciplines (National Council of Teachers of Mathematics (NCTM), 1989 &1991; National Research Council (NRC), 1991). Because many teachers learned mathematics in such an environment, the distressing cycle of teaching (poorly) as one was taught (poorly), continues. To counter their experiences and break the cycle, mathematics teachers must learn to provide students with open-ended problem-solving experiences and other mathematics encounters that generate opportunities for teachers to ask meaningful questions. Unfortunately, such an approach to teaching often stumbles blindly over unseen obstacles when these interactive approaches are attempted in classrooms of teachers who earned their teaching credentials while completing minimal mathematics requirements (Hyde, 1989). Constr uctivist-based mathematics teaching requires that teachers have considerable knowledge of mathematics as well as of how students best learn mathematics (Chapin, 1997).

One curricular area that tends to present more of a stumbling block than other areas is that of geometry (Mathematical Association of America (MAA), 1991; NCTM, 1991). As a discipline, geometry includes theoretical reasoning behind one-, two-, and three-dimensional measurements as well as a strategy for determining the validity of the reasoning for making claims about those measurements. However, many elementary teachers lack a conceptual awareness of the axiomatic nature of the discipline and the applicability of it as well as knowledge about the teaching and learning of geometry. However, meaningful professional development can directly address teachers' knowledge about the teaching and learning of geometry as well as foundational content knowledge of geometry.

Characteristics of Meaningful Professional Development

There are several characteristics of meaningful professional development for mathematics teachers (Aichele & Coxford, 1994; MAA, 1988, 1991; NRC, 1989, 1991). Three primary areas of concern emerge across most references. First, teachers need to be able to analyze the teaching and learning that occurs in their classrooms. Second, they must develop or strengthen their conceptual understanding of worthwhile mathematics. Third, teachers will most often select professional development activities that are conveniently arranged. Productive sessions should be conducted to "minimize participants' travel" (MAA, 1988, p.7) and be at a time so that groups of teachers from a single school can be involved together.

Analyze thinking and learning. In order to understand how a learner comes to know geometry, van Hiele (1986) described a theory about the underlying structures of geometric knowledge. This theory articulates a valid strategy for understanding the development of student thinking about geometric concepts (Burger & Shaughnessy 1986; Fuys, Geddes & Tischler, 1988; Mayberry, 1983). Described in Table 1, the theory consists of five clearly defined stages of understanding of geometric ideas. Students must serially progress through the levels for every geometric concept they acquire. Consistent with constructivism, this theory claims that all learning is built upon or rearranges existing knowledge

Implementing the van Hiele theory requires teachers to connect their understanding of the theory to actual classroom teaching practice (Teppo, 1991). In particular, identifying the exact location of a student who is in the continuum between levels 1 and 2, seems to be difficult (Burger & Shaughnessy, 1986). A teacher has to work carefully to move a child through that crowded place on the continuum. Although the theory does not directly relate stages to chronological ages (van de Walle, 1998), the step from level 1 to 2 is the critical entry level (Senk, 1989) for study of high school geometry, which typically is associated with a learner who is in the 9th or 10th grade. Therefore, it falls upon the shoulders of K-8 teachers to enable students to reach level 2 before entering that high school proof course on geometry. Successful professional development should also enable teachers to expand their knowledge about teaching and learning geometry to include the metacognitive practice of thinking about how they th emselves came to know the content. This approach embeds mathematics content in self-reflection about the nature of teaching and learning and directly addresses the issue of teaching like they were taught.

Understanding worthwhile mathematics. Effective professional development must expand teachers' knowledge about geometry. During teacher preparation, many future elementary teachers take very little mathematics and what they do take often offers few connections to the mathematics they will one day teach. The single math course is usually not powerful enough to counteract any preconceived notions of mathematics (Cipra & Flanders, 1992). So, practicing teachers should be provided with "a qualitatively different and significantly richer understanding of mathematics than most teachers currently posses" (Schifter, 1997, p. 1).

Deepening their knowledge of mathematics will allow practicing teachers to be more flexible when they discuss mathematical ideas with their students (Ball, 1996). Effective classroom discourse is deeply connected to the teachers' content knowledge. The teacher who demonstrates mathematical curiosity while listening to a child models an immensely important piece of successful mathematics learning (Schifter, 1996). Hence, the teaching is successful. For teachers to achieve this comfort with effective interactions with students, teachers must understand how it feels to investigate their own knowledge in an environment that is safe and supportive (Peterson & Barnes, 1996). Recall that teachers will teach as they have been taught, (Russell, 1997; Schifter, 1997; Scholz, 1995). So, there is much need to provide quality mathematics learning experiences for teachers of mathematics (Peterson & Barnes, 1996; Schifter, 1996).

It is also easy to swing too far in the direction of content and away from integrating pedagogy with content. When professional development programs focus only on mathematics content in isolation from pedagogy, teachers are unlikely to actually implement different instructional approaches (Scholz, 1995). When professional development programs merge geometry concepts with pedagogy, teachers change their practices about teaching geometry (Swafford, Jones, & Thornton, 1997).

Utilize a convenient presentation format. To maximize potential for development, the logistics of a professional development experience must be uncomplicated (Acquarelli & Mumme, 1996; Clarke, 1994). The use of distance education can provide a technological solution to the inability of some teachers to participate in professional development. With distance education, the flip of a switch virtually connects teachers to the experience. The teachers' minds are not stressed or exhausted from a commute and they can focus on problem solving, thinking, reading, and discussion. These activities (problem solving, thinking, etc.) are of foremost concern in a successful professional development experience (Clarke 1996). The medium of delivery (face-to-face or distance education) must not diminish teacher reflection. However, teachers often opt into or out of professional development experiences precisely because of the style of delivery, particularly as it relates to convenience.

Effective Practices for Distance Education

The primary advantage of two-way audio-video distance education is the convenience of a nearby classroom. The heaviest disadvantage of distance education is solely dependent on the instructor's skill at maintaining interest and activity at the remote sites. The potential for distance education students to develop a feeling of isolation or separateness is high and must be eliminated. Bialac and Morse (1995) found students at remote sites would sometimes sit back and watch as students at the origination site answered questions. Effective and successful teaching strategies in regular classroom situations are particularly tine in distance education classrooms. A substantial variety of teaching strategies must be used in every single class period (Beers & Orzech, 1996).

By spending some of the opening class period teaching students to run equipment (Bialac & Morse, 1995), all students become more comfortable with the microphones, camera, and the other technology (LeBaron & Bragg, 1994). However, with such attention paid to students' comfort with the equipment, a distance education course can unintentionally become a course on technology. So, the instructor must be highly familiar with the hardware and use the technology smoothly and seamlessly. Participants must come to the point where the camera becomes a natural part of their classroom. When such comfort occurs, the technology becomes invisible and all participants can focus on the real objectives for the course, in this case, the teaching and learning of geometry.

This article describes one effort to integrate these three ideas into a professional development opportunity for a group of K-7 teachers. They wanted to know more about the teaching and learning of geometry. But, they taught at a geographically separated set of schools. Distance education provided a solution to that issue.

METHOD

Context

As a result of a preservice and inservice program, a K-5 school collaborated with a local university to become a professional development school with that university. This mathematics-science magnet elementary school included two buildings and served as the field site for the field experiences of a group of the university's undergraduate students. As a result of early efforts to introduce all teachers to strategies for implementing constructivist ideas, a group of teachers from the school requested an opportunity to study more mathematics and pedagogy. Responding to teachers' requests in a manner consistent with their expectations (Clarke, 1994) and involving groups of teachers from a single school (Acquarelli & Mumme, 1996) are key elements of a successful professional development experience. These teachers requested a college course designed to treat two issues: (a) geometry content and (b) theories about teaching geometry.

Participants. The teachers described in this article include that group of six teachers who recognized their need to know more mathematics. Three other teachers from the same district and two other teachers from neighboring districts also enrolled in the course. The participants (n = 11) were practicing teachers in K-7 classrooms from both suburban and urban school districts.

The years of teaching experience varied dramatically among the teachers. Seven teachers had more than 10 years of teaching experience, one teacher had taught for three years, and three teachers were first-year teachers. All participants were female. At the time of this study, four of the teachers were "resource" teachers working with Title I students in schools. Most (10) of the teachers taking the course used the course either to satisfy requirements for an advanced degree or to fulfill requirements for adding a mathematics specialization to their state teaching license. The seventh grade teacher chose to take the course due to personal interest. In addition, one of the first grade teachers recently initiated her doctoral program. Five teachers reported having previously taken teaching pedagogy workshops or courses designed to enhance their teaching of mathematics, but only one participant reported having pursued study of pure mathematics content since graduation from college. All teachers' names in this ar ticle are pseudonyms.

Procedures

The MAA (1988) suggests holding professional development experiences at individual school sites, but two of the participants' schools were 60 miles apart. Due to such disparity in location, the course was offered through distance education. This system provided two-way, fully-interactive video and audio, allowing teachers to participate in the course from a room within a short 10 minute drive from each of their schools. If all participants had to drive to the university campus (or one school site), at least one teacher would have had a one-hour drive each way, substantially increasing the amount of commitment time required by her to participate in the course. In fact, one of these teachers later confided that she would not have taken the course if she had been required to drive to the university campus for every class meeting.

Effective distance education instructional strategies. The instructional strategies practiced during the course were performed in accordance with findings from distance education research. In this way, the course delivery system did not interfere with the course objectives. Although the system can handle several remote sites, the course described in this article limited the number of remote sites to one. The point from which the instructor broadcasts and maintains control of the system is referred to as the origination site and any sites to which students travel, but are not in control of the delivery are referred to as remote sites. Most experienced distance educators have had success teaching the course by traveling to each of the remote sites at least once (e.g., Beers, 1996; Bialac & Morse 1995). The instructor for this course alternated the origination site between the university site and the off-campus site. In addition, each site had the requisite "de facto" technician (Beers, 1996) assuring the teache rs that technological help would be at hand no matter if the instructor was physically present or not.

During a typical distance-education class session, teachers interacted with a variety of hands-on geometry experiments and explorations. Each week, the instructor dispatched packets that contained the necessary materials for upcoming in-class activities. Materials ranged from scissors and construction paper to commercial manipulatives, such as tangrams.

During each class session, all teachers were required to use the system to communicate. One strategy for supporting this requirement was to take attendance by asking an opening prompt, such as "tell me something geometric you have in your classroom," to which each teacher must respond. Another strategy was to pose a mathematics problem, which the teachers must solve in pairs at their respective sites, and then present their solutions, using the technology.

Distance education research verifies the enhancement of the course when students meet face-to-face for either a rather informal class period or a social get-together of some type (Beers, 1996). This three-semester-credit course met every Monday via distance education for two hours for 11 weeks and face-to-face for four hours on three different Saturdays, selected by the teachers. See Table 2 for a course calendar.

Meaningful curriculum. The Swafford, et al. (1997) findings guided the curriculum, assuming the merge of content with pedagogy to be a necessity. The van Hiele theory (1986) was used as a basis for these discussions. The subsequent goal was to enable teachers to determine for themselves on what level of van Hiele a person (including self) operated. Activities ranged from theoretical development of geometry knowledge, such as sorting two-dimensional and three-dimensional figures and defining them, to application of measurement ideas, such as creating "new" tangram sets in which meaningful relationships existed between shapes in the set. Discussion also followed teachers' efforts to understand and apply the van Hiele theory of learning. Teachers openly considered the van Hiele level of any comments made during the course meeting as well as revelations about comments made by children in their own classrooms.

During the Saturday on-campus meetings, teachers learned to construct figures. First, they learned to use a compass and a straight edge to construct several figures, including equilateral triangles, regular hexagons, perpendicular bisectors, and parallel lines. Then, the teachers used the software, Geometer's Sketchpad (Key Curriculum Press, 1995), to explore a variety of conjectures about geometric shapes and transformations of those shapes. Exclusive use of the university computer laboratory offered the instructor the opportunity to closely guide the participants as they applied paper and pencil constructions into the technological realm. During those sessions the teachers also explored such theorems as midsegments of a triangle or Side-Angle-Side (SAS) congruency of triangles. In fact, four teachers purchased the student version of the software for use at home.

The course readings included both regular geometry content from a high school geometry textbook (Serra, 1997) as well as advanced readings from higher level, non-Euclidean, geometry textbooks (Adler, 1958; Greenberg, 1972; Wolfe, 1945). Additional readings included a variety of educational articles, including both research and practitioner papers aimed at K-8 teaching (e.g., Burger & Shaughnessy, 1986; Mitchell & Burton, 1984; Rowan, 1990; Wilson, 1990;). Assignments included communication of geometry content as well as of geometry pedagogy. After completion of the readings and the computer explorations, teachers applied theorems, such as SAS to prove congruency of two triangles and verify the validity of selected geometric arguments. The teachers also planned and delivered geometry lessons with their K-7 students and analyzed their children's responses to those lessons.

Data Generators

This course was designed to demonstrate respect for the teachers' abilities to approach geometry teaching and learning through a metacognitive approach. It was not designed to magnify the (expected) low levels of their geometry knowledge. Direct empirical data about geometry knowledge was not gathered because the instructor believed that such first-day reporting would have set the course on a judgmental note. Prior to the first day of class, several teachers had expressed great fear of the course because of their intense anxiety about geometry. So, the instructor opted to keep a record of anecdotal comments and used perceptions of learning as data. In retrospect, this unobtrusive data collection had the additional impact of forcing these teachers to analyze their own learning. This self-reflection became a daily part of the course (at their request).

Self-proclaimed understanding of geometric terms. On the first day of class, participants were issued a list of geometric terms and asked to note the ones for which they could easily imagine a picture (level 0), the ones for which they could articulate a list of properties (level 1), and the ideas they could deduce from a given statement (level 2). They were also asked to note the terms for which they believed they could engage in a conversation involving formal deductive reasoning (level 3). During the course, participants studied the various van Hiele levels of thinking and were encouraged to recognize the level on which they were thinking as they solved problems from the course assignments. At the conclusion of the course, participants completed a survey noting their changes in knowledge about each of the geometry terms from the first day. Actually defining terms on the first day and then redefining them on the last day might have yielded more specific information about knowledge acquisition. But, the inst ructor evoked her ethics of teaching, standing firm in her foremost desire to create a community of learners. She maintained that the teachers would not volunteer to display their (low) knowledge of geometry on the first day of class.

Geometry lesson plans. As part of other requirements for the course, the teachers planned and taught four geometry lessons to their K-7 students. They fit these lessons into their existing curricula to avoid a novelty effect on the children. The teachers then analyzed the children's responses to those lessons while taking van Hiele's levels of understanding into consideration. The teachers presented a lesson to their children, selected three or four students' responses to that lesson, decided on the van Hiele level of the child for that topic, and developed three follow-up questions designed to move that child to the next van Hiele level. Finally, the teachers furthered their understanding of van Hiele theory by defending their use of the theory to plan a follow-up lesson (or lessons) based on their analyses of students' work.

Geometry papers. The teachers also completed a short paper in which they communicated the biggest idea or most interesting idea they learned from participation in the course. Participants focused the paper on both geometry content and appropriate methods for teaching geometry.

RESULTS & DISCUSSION

Changes in Geometry Knowledge

Definitions. All 11 teachers in this study reported having perceived some personal growth in their abilities to work with geometric figures. For at least 10 geometric ideas, all 11 teachers reported some sort of movement from van Hiele level 0 thinking (ability to visually recognize a figure, holistically) to level 2 (ability to use informal deduction). Some (5) teachers reported level 3 understanding of triangles (ability to prove theorems). This result is not surprising, given the amount of class time dedicated to discussion of triangles, construction of triangles, and development of the four triangle congruency theorems (SAS, Angle-Angle-Side (AAS), Angle-Side-Angle (ASA), and Side-Side-Side (SSS)). In fact, during a class period near the end of the course, participants drew triangles on ping pong balls and discovered that the sum of the angles of triangles on the sphere have more than 180[degrees]. This prompted Rosalie to redefine triangle. At the beginning of the course, she had defined triangle as "a p olygon with an angle sum of 180[degrees]". She recognized this definition to be incorrect for triangles in Spherical Geometry. By her own admission, she "wanted the definition to stand in both geometries," so she changed her definition of triangle to "a polygon with exactly 3 sides." As a class, they had previously included "straight" lines and "closed" figure in their definition of polygon. As a class, they also accepted the notion that a "straight" line on the sphere is a great circle. Clearly, Rosalie is comparing different geometries (level 4), although, at this time she is visualizing (level 0) and analyzing (level 1) triangles in Spherical Geometry.

The nature of geometry. Other changes in geometry knowledge emerged from both the written papers and the reflections on the lessons the teachers delivered. The two basic categories of change were: (a) what geometry is and (b) how to do geometry. Seven teachers (Loretta, Sydney, Amy, Tina, Stephanie, Christy, and Jyll) changed their views of what geometry was. Most (6) of these changes in belief included a just-created grasp of the importance of spatial thinking as a geometric idea. All seven teachers reported not having considered spatial thinking as a valued component of the school curriculum, but rather a "fun," Friday activity. One teacher indicated she had never thought spatial thinking was a part of the mathematics curriculum at all. Now, these seven teachers not only include spatial thinking in their plans, but two of the first grade teachers intend to begin the next academic year with this area of study.

One (Loretta) of the seven teachers now includes deductive reasoning in her definition of geometry. Rather than maintaining beliefs that children should passively receive definitions, four (Christy, Stephanie, Tina, and Jyll) of the seven participants also include enabling children to generate definitions and discuss them as part of doing geometry. When Stephanie asked her first-grade students to build a rhombus with their bodies, they didn't believe it was a quadrilateral. "It was interesting to see how, as teachers, we teach by shape instead of definition. I bet if students learned the definition of polygon before the definition of triangle, what I asked would have been easier." Stephanie's understanding of polygon is demonstrated as she applies this information into her classroom, pondering whether to first introduce sets, subsets, or elements of the sets. A richer knowledge of geometry is emerging for her.

Importance of language. Six of the teachers voiced new beliefs about the use of appropriate and mathematically accurate language. One teacher (Eloise) admitted having called a rhombus a "slanty square" in past lessons. She vowed never to repeat such a description. The fifth grade teacher (Tina) found having her students write their own definitions and use mathematically accurate words seemed to help them grasp the idea of "square" more completely, as well as to be able to write more elegant definitions themselves.

Personal growth in content. All but one teacher recognized some personal growth in their abilities to do geometry. Reporting in their final papers, these 10 teachers included feeling more comfortable in their attempts to solve geometry problems. The only teacher (Zelma) who did not indicate such confidence also did not demonstrate growth in her knowledge of the geometry terms nor in her analyses of student lessons. Also, she described mathematics and the learning of mathematics under the old guise of external authority. She did not demonstrate comfort with allowing children to struggle through a problem, and she herself did not seem to appreciate the importance of advancing through the levels of the van Hiele model. She often asked for definitions, well before she could list properties. There is no evidence to suggest she believed she has any authority to shape her own acquisition of geometric ideas.

The remaining 10 teachers commented on remarkable growth in their understandings.

What a wonderful opportunity to have now to be able to enjoy and grow in concepts in an area I disliked intensely ... I am challenged by the new perspective and reevaluating and renaming 'given' ideas in geometry. (Loretta).

I felt I had a fairly good grasp on defining shapes, until our discussions that night. I had never heard of an "elegant" definition - giving a lot of attributes/description about a shape was more my style (Christy).

I was too ignorant of geometry in general and didn't realize the importance of meaningful experiences at that time (Tina).

These comments indicate evidence that the students were able to comprehend and identify instances of changes in their own geometric knowledge.

The role of geometry in K-7 curricula. One other change in geometric knowledge revolved around teachers' abilities to cite connections between geometry and other areas as well as within geometry, such as their newly observed inclusion of spatial thinking into geometry. These connections mcluded specific statements of observations of geometric phenomenon in the "real-world" (Amy), in symmetry of letters in the alphabet (Beth), as a basis for number knowledge acquisition (Stephanie), and as related to acquisition of motor skills (Jyll), such as eye-hand coordination.

Several of the teachers also voiced a new appreciation for geometry simply for its own sake. "I think we underestimate the power of geometry" (Loretta). This teacher went on to comment on her concerns that geometry continues to be overlooked for many children. "The study of geometry frequently would be left until the end of the year when everything else had been introduced or simply eliminated due to shortage of time... Still today, geometry continues to take a back seat to number concepts in elementary schools" (Loretta). Given the negative attitude typical elementary teachers have about mathematics and the intense fear with which this group of teachers began the course, this result is profound. As this teacher learned more geometry, she came to re-evaluate its place in the curriculum.

Changes in Understanding of van Hiele

With regard to their understanding of the van Hiele levels of geometric knowledge, several of the participants regularly used the information in classroom discussions as well as lesson analyses. One goal of the course included enabling teachers to recognize their own levels of understanding of geometry, according to the van Hiele model.

The study of the van Hiele levels of knowledge led all but one teacher (Zelma) to voice a change in their perceptions of how children learn geometry. Part of the requirements for the course included teaching four geometry lessons to their elementary students and analyzing the children's responses according to the van Hiele model. After analyzing the children's responses, six of the participants (Christy, Loretta, Eloise, Jyll, Beth, and Tina) instituted a follow-up lesson with their children to gain further insights into the children's thinking. Each of these participants planned their follow--up lesson with careful attention to the van Hiele model. After analyzing the children's responses, the other four teachers planned actions they would take to correct any concerns and described how they would do the lesson differently next year. Three of these teachers who did not take the initiative to institute actions were first-year teachers, the other participant was the seventh grade teacher (Leona) who often char acterized her pace in the familiar vernacular of needing to "cover" the material in an allotted amount of time.

One participant regularly reflected and shared her perception of her level of thinking about a given concept. For example, while using Sketchpad to investigate the midsegments of a triangle, she (Loretta) claimed that two of the segments in Figure 1 were parallel.

When the instructor asked her to explain how she knew the lines were parallel, Loretta said, "Because they look parallel..." After a long pause, Loretta smiled and said, "Well that's level 0 thinking." After several more thinking minutes passed, she directed the program to calculate the slopes of the two line segments and proudly displayed her reasoning to the instructor. "The lines are parallel because the slopes are equal. Now that is level 2 thinking." This participant also shared her personal finding that she is at different levels of van Hiele thinking depending on the concept, a finding supported by literature (e.g. Fuys, Geddes, & Tischler, 1988; Mayberry 1983).

CONCLUSIONS

All of the teachers in this study experienced some change in their understanding of geometry. All but one teacher recognized these changes and described manners in which their teaching would subsequently change. Three factors, intentionally designed into this professional development course, may have contributed to those changes.x

Teaching and Learning of Geometry

All of the teachers were immersed in the study of the van Hiele model of geometric knowledge. They took a metacognitive approach and continually reflected on their personal levels for a given concept. However, as suggested by Swafford et. al. (1997), the teachers also taught geometry lessons to their students and analyzed them according to the van Hiele model of geometric knowledge. In these two manners, participants were introduced to the van Hiele levels of thinking through a hands-on approach in their own classrooms (Clarke, 1994), rather than, for example, being asked to memorize the various levels.

The temptation exists to consider the fact that teachers planned, delivered, and analyzed geometry lessons according to van Hiele theory as evidence that a change in teaching pedagogy occurred. Changing four lessons is minimal at best. However, it was new to all of the teachers. So, in that regard, a change did occur. But, caution should be exercised in attributing cause to this result or in predicting similar future lesson developments and modifications.

Worthwhile Geometry Knowledge

All participants strengthened their knowledge of geometry content. The approach followed Chapin's (1997) suggestion that teachers should be presented with many opportunities to solve problems. In so doing, teachers constructed figures, completed short proofs, discovered and applied triangle congruency theorems, studied spherical geometry, and wrote definitions for several terms.

The fact that two of the first-year, first-grade teachers intend to begin their second year with spatial thinking activities, rather than "number", is a convincing result. These teachers have taken ownership of the mathematics curriculum and can make decisions in an informed, grounded fashion. Moreover, the fact that all teachers came to view spatial thinking as a key component of geometry is encouraging. In addition, the four teachers, who stated their intentions to integrate geometry into other areas of the curriculum, seem to have blossomed in their personal understanding of geometry as a component of the school curriculum. Other advances in geometric knowledge were evidenced as teachers discussed whether or not a triangle on the sphere had an angle sum of 180[degree]. Such rich, meaningful commentaries provided evidence of an enhanced comfort level with the topic of geometry.

As intended, the medium for the professional development course (distance education) was nearly invisible, due to carefully performed instruction. No teachers expressed disgruntlement with the format. In fact, the convenience factor was viewed as important and desirable. For all but two teachers, the distance education classrooms were within five minutes of their schools. The remaining two teachers traveled only 20 minutes to their respective locations. This communicated a respect for the need for professional development to be convenient (Clarke, 1994). The instructor limited the total number of sites to two and alternated her attendance between the sites, never allowing one site to be the "main" site. The teachers learned how to use this technology because they wanted to learn the geometry. Without the technology, several teachers would not have had the opportunity to learn the geometry.

The distance education technology also enabled four of the participants to demonstrate their lessons to the instructor. They transported their classroom of children to their local distance education sites and delivered a geometry lesson over the distance education technology system. Tina taught her fifth graders to use a protractor. Rosalie and Eloise taught fifth graders to use triangle and quadrilateral shapes to illustrate several geometry terms including diagonal, obtuse angle, and midpoint. Leona taught her seventh graders to prove that axial symmetry existed (or did not exist) in a shape, by using congruence. All four teachers simultaneously taught their students to use the distance education technology as well as to do the geometry content. In addition these teachers modeled appropriate use of the distance education technology, never allowing themselves to lecture to their students. Along with the instructor, undergraduate students at the university observed the lessons as part of the requirements in their "mathematics for elementary teachers" course.

The positive results of this experience for each teacher were most likely some combination of the previous factors. The course content was appropriately designed according to suggestions from many authorities in the area of professional development of teachers (e.g., Chapin, 1997; Clarke, 1994; NCTM, 1989 & 1991; MAA, 1988 & 1991; NRC, 1989 & 1991; Scholz, 1995; Swafford et. al., 1997) and the course delivery was consistent with findings from distance education research (e.g., Beers, 1996; LeBaron, 1994).

The successful use of this delivery medium holds power for teacher-leaders to effectively meet the needs of geographically separated teachers. The importance of interacting with others in a teacher's professional development can result from appropriate use of distance education technology. The teachers in this study became more knowledgeable about the teaching and learning of geometry. This result supports the contention that there is much need to provide opportunities for future and practicing teachers to experience quality and appropriate mathematical learning (Chapin, 1997; NCTM 1991; Schifter, 1997). It also supports the contention that distance education can provide a powerful tool to make such tremendous development a reality, particularly when teachers are from quite disparate locations.

Future Directions

Future research will include continued observations of these teachers' classrooms as geometry lessons are taught, to determine the specific nature of the changes in geometry knowledge as well as of changes in pedagogy. Future experiences for these teachers might also include analyzing colleagues' lessons. Such data would provide additional evidence about the abilities of the teachers to use the van Hiele model of thinking.

Acknowledgements

The author gratefully acknowledges the service of Susan Seidenfeld, a teacher from the study who taught all participants, including the author, how to use the Geometer's Sketchpad.

Partially funded by: Exxon Education Foundation.

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Over the past two decades, mathematics educators have increasingly turned to constructivism as a reasonable theory to explain student learning and plan subsequent instruction. One way for teachers to use teaching strategies that are consistent with constructivist ideas is to carefully attend to a student's thinking and ask appropriate follow-up questions. Teachers who are able to ask effective questions must draw from strong content knowledge to suitably oversee their students' thinking and learning. However, teachers will teach as they were taught (Lortie, 1975; Russell, 1997; Schifter, 1997; Scholz, 1995). So, not only is it imperative that teachers know content exceptionally well, they must have learned that content through some sort of learning experience that modeled best teaching practice and which was founded on constructivism. There are mathematics teachers for whom these necessary types of learning experiences did not occur. In those cases, professional development is essential. This article describ es a professional development experience that modeled some of those effective practices and discusses the plausibility of using distance education technology as a medium for such an experience.

BACKGROUND

Historically, traditional mathematics classrooms have been characterized by learning concepts and skills in isolation from meaningful contexts and by little connections being made among mathematical ideas or between mathematics and other disciplines (National Council of Teachers of Mathematics (NCTM), 1989 &1991; National Research Council (NRC), 1991). Because many teachers learned mathematics in such an environment, the distressing cycle of teaching (poorly) as one was taught (poorly), continues. To counter their experiences and break the cycle, mathematics teachers must learn to provide students with open-ended problem-solving experiences and other mathematics encounters that generate opportunities for teachers to ask meaningful questions. Unfortunately, such an approach to teaching often stumbles blindly over unseen obstacles when these interactive approaches are attempted in classrooms of teachers who earned their teaching credentials while completing minimal mathematics requirements (Hyde, 1989). Constr uctivist-based mathematics teaching requires that teachers have considerable knowledge of mathematics as well as of how students best learn mathematics (Chapin, 1997).

One curricular area that tends to present more of a stumbling block than other areas is that of geometry (Mathematical Association of America (MAA), 1991; NCTM, 1991). As a discipline, geometry includes theoretical reasoning behind one-, two-, and three-dimensional measurements as well as a strategy for determining the validity of the reasoning for making claims about those measurements. However, many elementary teachers lack a conceptual awareness of the axiomatic nature of the discipline and the applicability of it as well as knowledge about the teaching and learning of geometry. However, meaningful professional development can directly address teachers' knowledge about the teaching and learning of geometry as well as foundational content knowledge of geometry.

Characteristics of Meaningful Professional Development

There are several characteristics of meaningful professional development for mathematics teachers (Aichele & Coxford, 1994; MAA, 1988, 1991; NRC, 1989, 1991). Three primary areas of concern emerge across most references. First, teachers need to be able to analyze the teaching and learning that occurs in their classrooms. Second, they must develop or strengthen their conceptual understanding of worthwhile mathematics. Third, teachers will most often select professional development activities that are conveniently arranged. Productive sessions should be conducted to "minimize participants' travel" (MAA, 1988, p.7) and be at a time so that groups of teachers from a single school can be involved together.

Analyze thinking and learning. In order to understand how a learner comes to know geometry, van Hiele (1986) described a theory about the underlying structures of geometric knowledge. This theory articulates a valid strategy for understanding the development of student thinking about geometric concepts (Burger & Shaughnessy 1986; Fuys, Geddes & Tischler, 1988; Mayberry, 1983). Described in Table 1, the theory consists of five clearly defined stages of understanding of geometric ideas. Students must serially progress through the levels for every geometric concept they acquire. Consistent with constructivism, this theory claims that all learning is built upon or rearranges existing knowledge

Implementing the van Hiele theory requires teachers to connect their understanding of the theory to actual classroom teaching practice (Teppo, 1991). In particular, identifying the exact location of a student who is in the continuum between levels 1 and 2, seems to be difficult (Burger & Shaughnessy, 1986). A teacher has to work carefully to move a child through that crowded place on the continuum. Although the theory does not directly relate stages to chronological ages (van de Walle, 1998), the step from level 1 to 2 is the critical entry level (Senk, 1989) for study of high school geometry, which typically is associated with a learner who is in the 9th or 10th grade. Therefore, it falls upon the shoulders of K-8 teachers to enable students to reach level 2 before entering that high school proof course on geometry. Successful professional development should also enable teachers to expand their knowledge about teaching and learning geometry to include the metacognitive practice of thinking about how they th emselves came to know the content. This approach embeds mathematics content in self-reflection about the nature of teaching and learning and directly addresses the issue of teaching like they were taught.

Understanding worthwhile mathematics. Effective professional development must expand teachers' knowledge about geometry. During teacher preparation, many future elementary teachers take very little mathematics and what they do take often offers few connections to the mathematics they will one day teach. The single math course is usually not powerful enough to counteract any preconceived notions of mathematics (Cipra & Flanders, 1992). So, practicing teachers should be provided with "a qualitatively different and significantly richer understanding of mathematics than most teachers currently posses" (Schifter, 1997, p. 1).

Deepening their knowledge of mathematics will allow practicing teachers to be more flexible when they discuss mathematical ideas with their students (Ball, 1996). Effective classroom discourse is deeply connected to the teachers' content knowledge. The teacher who demonstrates mathematical curiosity while listening to a child models an immensely important piece of successful mathematics learning (Schifter, 1996). Hence, the teaching is successful. For teachers to achieve this comfort with effective interactions with students, teachers must understand how it feels to investigate their own knowledge in an environment that is safe and supportive (Peterson & Barnes, 1996). Recall that teachers will teach as they have been taught, (Russell, 1997; Schifter, 1997; Scholz, 1995). So, there is much need to provide quality mathematics learning experiences for teachers of mathematics (Peterson & Barnes, 1996; Schifter, 1996).

It is also easy to swing too far in the direction of content and away from integrating pedagogy with content. When professional development programs focus only on mathematics content in isolation from pedagogy, teachers are unlikely to actually implement different instructional approaches (Scholz, 1995). When professional development programs merge geometry concepts with pedagogy, teachers change their practices about teaching geometry (Swafford, Jones, & Thornton, 1997).

Utilize a convenient presentation format. To maximize potential for development, the logistics of a professional development experience must be uncomplicated (Acquarelli & Mumme, 1996; Clarke, 1994). The use of distance education can provide a technological solution to the inability of some teachers to participate in professional development. With distance education, the flip of a switch virtually connects teachers to the experience. The teachers' minds are not stressed or exhausted from a commute and they can focus on problem solving, thinking, reading, and discussion. These activities (problem solving, thinking, etc.) are of foremost concern in a successful professional development experience (Clarke 1996). The medium of delivery (face-to-face or distance education) must not diminish teacher reflection. However, teachers often opt into or out of professional development experiences precisely because of the style of delivery, particularly as it relates to convenience.

Effective Practices for Distance Education

The primary advantage of two-way audio-video distance education is the convenience of a nearby classroom. The heaviest disadvantage of distance education is solely dependent on the instructor's skill at maintaining interest and activity at the remote sites. The potential for distance education students to develop a feeling of isolation or separateness is high and must be eliminated. Bialac and Morse (1995) found students at remote sites would sometimes sit back and watch as students at the origination site answered questions. Effective and successful teaching strategies in regular classroom situations are particularly tine in distance education classrooms. A substantial variety of teaching strategies must be used in every single class period (Beers & Orzech, 1996).

By spending some of the opening class period teaching students to run equipment (Bialac & Morse, 1995), all students become more comfortable with the microphones, camera, and the other technology (LeBaron & Bragg, 1994). However, with such attention paid to students' comfort with the equipment, a distance education course can unintentionally become a course on technology. So, the instructor must be highly familiar with the hardware and use the technology smoothly and seamlessly. Participants must come to the point where the camera becomes a natural part of their classroom. When such comfort occurs, the technology becomes invisible and all participants can focus on the real objectives for the course, in this case, the teaching and learning of geometry.

This article describes one effort to integrate these three ideas into a professional development opportunity for a group of K-7 teachers. They wanted to know more about the teaching and learning of geometry. But, they taught at a geographically separated set of schools. Distance education provided a solution to that issue.

METHOD

Context

As a result of a preservice and inservice program, a K-5 school collaborated with a local university to become a professional development school with that university. This mathematics-science magnet elementary school included two buildings and served as the field site for the field experiences of a group of the university's undergraduate students. As a result of early efforts to introduce all teachers to strategies for implementing constructivist ideas, a group of teachers from the school requested an opportunity to study more mathematics and pedagogy. Responding to teachers' requests in a manner consistent with their expectations (Clarke, 1994) and involving groups of teachers from a single school (Acquarelli & Mumme, 1996) are key elements of a successful professional development experience. These teachers requested a college course designed to treat two issues: (a) geometry content and (b) theories about teaching geometry.

Participants. The teachers described in this article include that group of six teachers who recognized their need to know more mathematics. Three other teachers from the same district and two other teachers from neighboring districts also enrolled in the course. The participants (n = 11) were practicing teachers in K-7 classrooms from both suburban and urban school districts.

The years of teaching experience varied dramatically among the teachers. Seven teachers had more than 10 years of teaching experience, one teacher had taught for three years, and three teachers were first-year teachers. All participants were female. At the time of this study, four of the teachers were "resource" teachers working with Title I students in schools. Most (10) of the teachers taking the course used the course either to satisfy requirements for an advanced degree or to fulfill requirements for adding a mathematics specialization to their state teaching license. The seventh grade teacher chose to take the course due to personal interest. In addition, one of the first grade teachers recently initiated her doctoral program. Five teachers reported having previously taken teaching pedagogy workshops or courses designed to enhance their teaching of mathematics, but only one participant reported having pursued study of pure mathematics content since graduation from college. All teachers' names in this ar ticle are pseudonyms.

Procedures

The MAA (1988) suggests holding professional development experiences at individual school sites, but two of the participants' schools were 60 miles apart. Due to such disparity in location, the course was offered through distance education. This system provided two-way, fully-interactive video and audio, allowing teachers to participate in the course from a room within a short 10 minute drive from each of their schools. If all participants had to drive to the university campus (or one school site), at least one teacher would have had a one-hour drive each way, substantially increasing the amount of commitment time required by her to participate in the course. In fact, one of these teachers later confided that she would not have taken the course if she had been required to drive to the university campus for every class meeting.

Effective distance education instructional strategies. The instructional strategies practiced during the course were performed in accordance with findings from distance education research. In this way, the course delivery system did not interfere with the course objectives. Although the system can handle several remote sites, the course described in this article limited the number of remote sites to one. The point from which the instructor broadcasts and maintains control of the system is referred to as the origination site and any sites to which students travel, but are not in control of the delivery are referred to as remote sites. Most experienced distance educators have had success teaching the course by traveling to each of the remote sites at least once (e.g., Beers, 1996; Bialac & Morse 1995). The instructor for this course alternated the origination site between the university site and the off-campus site. In addition, each site had the requisite "de facto" technician (Beers, 1996) assuring the teache rs that technological help would be at hand no matter if the instructor was physically present or not.

During a typical distance-education class session, teachers interacted with a variety of hands-on geometry experiments and explorations. Each week, the instructor dispatched packets that contained the necessary materials for upcoming in-class activities. Materials ranged from scissors and construction paper to commercial manipulatives, such as tangrams.

During each class session, all teachers were required to use the system to communicate. One strategy for supporting this requirement was to take attendance by asking an opening prompt, such as "tell me something geometric you have in your classroom," to which each teacher must respond. Another strategy was to pose a mathematics problem, which the teachers must solve in pairs at their respective sites, and then present their solutions, using the technology.

Distance education research verifies the enhancement of the course when students meet face-to-face for either a rather informal class period or a social get-together of some type (Beers, 1996). This three-semester-credit course met every Monday via distance education for two hours for 11 weeks and face-to-face for four hours on three different Saturdays, selected by the teachers. See Table 2 for a course calendar.

Meaningful curriculum. The Swafford, et al. (1997) findings guided the curriculum, assuming the merge of content with pedagogy to be a necessity. The van Hiele theory (1986) was used as a basis for these discussions. The subsequent goal was to enable teachers to determine for themselves on what level of van Hiele a person (including self) operated. Activities ranged from theoretical development of geometry knowledge, such as sorting two-dimensional and three-dimensional figures and defining them, to application of measurement ideas, such as creating "new" tangram sets in which meaningful relationships existed between shapes in the set. Discussion also followed teachers' efforts to understand and apply the van Hiele theory of learning. Teachers openly considered the van Hiele level of any comments made during the course meeting as well as revelations about comments made by children in their own classrooms.

During the Saturday on-campus meetings, teachers learned to construct figures. First, they learned to use a compass and a straight edge to construct several figures, including equilateral triangles, regular hexagons, perpendicular bisectors, and parallel lines. Then, the teachers used the software, Geometer's Sketchpad (Key Curriculum Press, 1995), to explore a variety of conjectures about geometric shapes and transformations of those shapes. Exclusive use of the university computer laboratory offered the instructor the opportunity to closely guide the participants as they applied paper and pencil constructions into the technological realm. During those sessions the teachers also explored such theorems as midsegments of a triangle or Side-Angle-Side (SAS) congruency of triangles. In fact, four teachers purchased the student version of the software for use at home.

The course readings included both regular geometry content from a high school geometry textbook (Serra, 1997) as well as advanced readings from higher level, non-Euclidean, geometry textbooks (Adler, 1958; Greenberg, 1972; Wolfe, 1945). Additional readings included a variety of educational articles, including both research and practitioner papers aimed at K-8 teaching (e.g., Burger & Shaughnessy, 1986; Mitchell & Burton, 1984; Rowan, 1990; Wilson, 1990;). Assignments included communication of geometry content as well as of geometry pedagogy. After completion of the readings and the computer explorations, teachers applied theorems, such as SAS to prove congruency of two triangles and verify the validity of selected geometric arguments. The teachers also planned and delivered geometry lessons with their K-7 students and analyzed their children's responses to those lessons.

Data Generators

This course was designed to demonstrate respect for the teachers' abilities to approach geometry teaching and learning through a metacognitive approach. It was not designed to magnify the (expected) low levels of their geometry knowledge. Direct empirical data about geometry knowledge was not gathered because the instructor believed that such first-day reporting would have set the course on a judgmental note. Prior to the first day of class, several teachers had expressed great fear of the course because of their intense anxiety about geometry. So, the instructor opted to keep a record of anecdotal comments and used perceptions of learning as data. In retrospect, this unobtrusive data collection had the additional impact of forcing these teachers to analyze their own learning. This self-reflection became a daily part of the course (at their request).

Self-proclaimed understanding of geometric terms. On the first day of class, participants were issued a list of geometric terms and asked to note the ones for which they could easily imagine a picture (level 0), the ones for which they could articulate a list of properties (level 1), and the ideas they could deduce from a given statement (level 2). They were also asked to note the terms for which they believed they could engage in a conversation involving formal deductive reasoning (level 3). During the course, participants studied the various van Hiele levels of thinking and were encouraged to recognize the level on which they were thinking as they solved problems from the course assignments. At the conclusion of the course, participants completed a survey noting their changes in knowledge about each of the geometry terms from the first day. Actually defining terms on the first day and then redefining them on the last day might have yielded more specific information about knowledge acquisition. But, the inst ructor evoked her ethics of teaching, standing firm in her foremost desire to create a community of learners. She maintained that the teachers would not volunteer to display their (low) knowledge of geometry on the first day of class.

Geometry lesson plans. As part of other requirements for the course, the teachers planned and taught four geometry lessons to their K-7 students. They fit these lessons into their existing curricula to avoid a novelty effect on the children. The teachers then analyzed the children's responses to those lessons while taking van Hiele's levels of understanding into consideration. The teachers presented a lesson to their children, selected three or four students' responses to that lesson, decided on the van Hiele level of the child for that topic, and developed three follow-up questions designed to move that child to the next van Hiele level. Finally, the teachers furthered their understanding of van Hiele theory by defending their use of the theory to plan a follow-up lesson (or lessons) based on their analyses of students' work.

Geometry papers. The teachers also completed a short paper in which they communicated the biggest idea or most interesting idea they learned from participation in the course. Participants focused the paper on both geometry content and appropriate methods for teaching geometry.

RESULTS & DISCUSSION

Changes in Geometry Knowledge

Definitions. All 11 teachers in this study reported having perceived some personal growth in their abilities to work with geometric figures. For at least 10 geometric ideas, all 11 teachers reported some sort of movement from van Hiele level 0 thinking (ability to visually recognize a figure, holistically) to level 2 (ability to use informal deduction). Some (5) teachers reported level 3 understanding of triangles (ability to prove theorems). This result is not surprising, given the amount of class time dedicated to discussion of triangles, construction of triangles, and development of the four triangle congruency theorems (SAS, Angle-Angle-Side (AAS), Angle-Side-Angle (ASA), and Side-Side-Side (SSS)). In fact, during a class period near the end of the course, participants drew triangles on ping pong balls and discovered that the sum of the angles of triangles on the sphere have more than 180[degrees]. This prompted Rosalie to redefine triangle. At the beginning of the course, she had defined triangle as "a p olygon with an angle sum of 180[degrees]". She recognized this definition to be incorrect for triangles in Spherical Geometry. By her own admission, she "wanted the definition to stand in both geometries," so she changed her definition of triangle to "a polygon with exactly 3 sides." As a class, they had previously included "straight" lines and "closed" figure in their definition of polygon. As a class, they also accepted the notion that a "straight" line on the sphere is a great circle. Clearly, Rosalie is comparing different geometries (level 4), although, at this time she is visualizing (level 0) and analyzing (level 1) triangles in Spherical Geometry.

The nature of geometry. Other changes in geometry knowledge emerged from both the written papers and the reflections on the lessons the teachers delivered. The two basic categories of change were: (a) what geometry is and (b) how to do geometry. Seven teachers (Loretta, Sydney, Amy, Tina, Stephanie, Christy, and Jyll) changed their views of what geometry was. Most (6) of these changes in belief included a just-created grasp of the importance of spatial thinking as a geometric idea. All seven teachers reported not having considered spatial thinking as a valued component of the school curriculum, but rather a "fun," Friday activity. One teacher indicated she had never thought spatial thinking was a part of the mathematics curriculum at all. Now, these seven teachers not only include spatial thinking in their plans, but two of the first grade teachers intend to begin the next academic year with this area of study.

One (Loretta) of the seven teachers now includes deductive reasoning in her definition of geometry. Rather than maintaining beliefs that children should passively receive definitions, four (Christy, Stephanie, Tina, and Jyll) of the seven participants also include enabling children to generate definitions and discuss them as part of doing geometry. When Stephanie asked her first-grade students to build a rhombus with their bodies, they didn't believe it was a quadrilateral. "It was interesting to see how, as teachers, we teach by shape instead of definition. I bet if students learned the definition of polygon before the definition of triangle, what I asked would have been easier." Stephanie's understanding of polygon is demonstrated as she applies this information into her classroom, pondering whether to first introduce sets, subsets, or elements of the sets. A richer knowledge of geometry is emerging for her.

Importance of language. Six of the teachers voiced new beliefs about the use of appropriate and mathematically accurate language. One teacher (Eloise) admitted having called a rhombus a "slanty square" in past lessons. She vowed never to repeat such a description. The fifth grade teacher (Tina) found having her students write their own definitions and use mathematically accurate words seemed to help them grasp the idea of "square" more completely, as well as to be able to write more elegant definitions themselves.

Personal growth in content. All but one teacher recognized some personal growth in their abilities to do geometry. Reporting in their final papers, these 10 teachers included feeling more comfortable in their attempts to solve geometry problems. The only teacher (Zelma) who did not indicate such confidence also did not demonstrate growth in her knowledge of the geometry terms nor in her analyses of student lessons. Also, she described mathematics and the learning of mathematics under the old guise of external authority. She did not demonstrate comfort with allowing children to struggle through a problem, and she herself did not seem to appreciate the importance of advancing through the levels of the van Hiele model. She often asked for definitions, well before she could list properties. There is no evidence to suggest she believed she has any authority to shape her own acquisition of geometric ideas.

The remaining 10 teachers commented on remarkable growth in their understandings.

What a wonderful opportunity to have now to be able to enjoy and grow in concepts in an area I disliked intensely ... I am challenged by the new perspective and reevaluating and renaming 'given' ideas in geometry. (Loretta).

I felt I had a fairly good grasp on defining shapes, until our discussions that night. I had never heard of an "elegant" definition - giving a lot of attributes/description about a shape was more my style (Christy).

I was too ignorant of geometry in general and didn't realize the importance of meaningful experiences at that time (Tina).

These comments indicate evidence that the students were able to comprehend and identify instances of changes in their own geometric knowledge.

The role of geometry in K-7 curricula. One other change in geometric knowledge revolved around teachers' abilities to cite connections between geometry and other areas as well as within geometry, such as their newly observed inclusion of spatial thinking into geometry. These connections mcluded specific statements of observations of geometric phenomenon in the "real-world" (Amy), in symmetry of letters in the alphabet (Beth), as a basis for number knowledge acquisition (Stephanie), and as related to acquisition of motor skills (Jyll), such as eye-hand coordination.

Several of the teachers also voiced a new appreciation for geometry simply for its own sake. "I think we underestimate the power of geometry" (Loretta). This teacher went on to comment on her concerns that geometry continues to be overlooked for many children. "The study of geometry frequently would be left until the end of the year when everything else had been introduced or simply eliminated due to shortage of time... Still today, geometry continues to take a back seat to number concepts in elementary schools" (Loretta). Given the negative attitude typical elementary teachers have about mathematics and the intense fear with which this group of teachers began the course, this result is profound. As this teacher learned more geometry, she came to re-evaluate its place in the curriculum.

Changes in Understanding of van Hiele

With regard to their understanding of the van Hiele levels of geometric knowledge, several of the participants regularly used the information in classroom discussions as well as lesson analyses. One goal of the course included enabling teachers to recognize their own levels of understanding of geometry, according to the van Hiele model.

The study of the van Hiele levels of knowledge led all but one teacher (Zelma) to voice a change in their perceptions of how children learn geometry. Part of the requirements for the course included teaching four geometry lessons to their elementary students and analyzing the children's responses according to the van Hiele model. After analyzing the children's responses, six of the participants (Christy, Loretta, Eloise, Jyll, Beth, and Tina) instituted a follow-up lesson with their children to gain further insights into the children's thinking. Each of these participants planned their follow--up lesson with careful attention to the van Hiele model. After analyzing the children's responses, the other four teachers planned actions they would take to correct any concerns and described how they would do the lesson differently next year. Three of these teachers who did not take the initiative to institute actions were first-year teachers, the other participant was the seventh grade teacher (Leona) who often char acterized her pace in the familiar vernacular of needing to "cover" the material in an allotted amount of time.

One participant regularly reflected and shared her perception of her level of thinking about a given concept. For example, while using Sketchpad to investigate the midsegments of a triangle, she (Loretta) claimed that two of the segments in Figure 1 were parallel.

When the instructor asked her to explain how she knew the lines were parallel, Loretta said, "Because they look parallel..." After a long pause, Loretta smiled and said, "Well that's level 0 thinking." After several more thinking minutes passed, she directed the program to calculate the slopes of the two line segments and proudly displayed her reasoning to the instructor. "The lines are parallel because the slopes are equal. Now that is level 2 thinking." This participant also shared her personal finding that she is at different levels of van Hiele thinking depending on the concept, a finding supported by literature (e.g. Fuys, Geddes, & Tischler, 1988; Mayberry 1983).

CONCLUSIONS

All of the teachers in this study experienced some change in their understanding of geometry. All but one teacher recognized these changes and described manners in which their teaching would subsequently change. Three factors, intentionally designed into this professional development course, may have contributed to those changes.x

Teaching and Learning of Geometry

All of the teachers were immersed in the study of the van Hiele model of geometric knowledge. They took a metacognitive approach and continually reflected on their personal levels for a given concept. However, as suggested by Swafford et. al. (1997), the teachers also taught geometry lessons to their students and analyzed them according to the van Hiele model of geometric knowledge. In these two manners, participants were introduced to the van Hiele levels of thinking through a hands-on approach in their own classrooms (Clarke, 1994), rather than, for example, being asked to memorize the various levels.

The temptation exists to consider the fact that teachers planned, delivered, and analyzed geometry lessons according to van Hiele theory as evidence that a change in teaching pedagogy occurred. Changing four lessons is minimal at best. However, it was new to all of the teachers. So, in that regard, a change did occur. But, caution should be exercised in attributing cause to this result or in predicting similar future lesson developments and modifications.

Worthwhile Geometry Knowledge

All participants strengthened their knowledge of geometry content. The approach followed Chapin's (1997) suggestion that teachers should be presented with many opportunities to solve problems. In so doing, teachers constructed figures, completed short proofs, discovered and applied triangle congruency theorems, studied spherical geometry, and wrote definitions for several terms.

The fact that two of the first-year, first-grade teachers intend to begin their second year with spatial thinking activities, rather than "number", is a convincing result. These teachers have taken ownership of the mathematics curriculum and can make decisions in an informed, grounded fashion. Moreover, the fact that all teachers came to view spatial thinking as a key component of geometry is encouraging. In addition, the four teachers, who stated their intentions to integrate geometry into other areas of the curriculum, seem to have blossomed in their personal understanding of geometry as a component of the school curriculum. Other advances in geometric knowledge were evidenced as teachers discussed whether or not a triangle on the sphere had an angle sum of 180[degree]. Such rich, meaningful commentaries provided evidence of an enhanced comfort level with the topic of geometry.

As intended, the medium for the professional development course (distance education) was nearly invisible, due to carefully performed instruction. No teachers expressed disgruntlement with the format. In fact, the convenience factor was viewed as important and desirable. For all but two teachers, the distance education classrooms were within five minutes of their schools. The remaining two teachers traveled only 20 minutes to their respective locations. This communicated a respect for the need for professional development to be convenient (Clarke, 1994). The instructor limited the total number of sites to two and alternated her attendance between the sites, never allowing one site to be the "main" site. The teachers learned how to use this technology because they wanted to learn the geometry. Without the technology, several teachers would not have had the opportunity to learn the geometry.

The distance education technology also enabled four of the participants to demonstrate their lessons to the instructor. They transported their classroom of children to their local distance education sites and delivered a geometry lesson over the distance education technology system. Tina taught her fifth graders to use a protractor. Rosalie and Eloise taught fifth graders to use triangle and quadrilateral shapes to illustrate several geometry terms including diagonal, obtuse angle, and midpoint. Leona taught her seventh graders to prove that axial symmetry existed (or did not exist) in a shape, by using congruence. All four teachers simultaneously taught their students to use the distance education technology as well as to do the geometry content. In addition these teachers modeled appropriate use of the distance education technology, never allowing themselves to lecture to their students. Along with the instructor, undergraduate students at the university observed the lessons as part of the requirements in their "mathematics for elementary teachers" course.

The positive results of this experience for each teacher were most likely some combination of the previous factors. The course content was appropriately designed according to suggestions from many authorities in the area of professional development of teachers (e.g., Chapin, 1997; Clarke, 1994; NCTM, 1989 & 1991; MAA, 1988 & 1991; NRC, 1989 & 1991; Scholz, 1995; Swafford et. al., 1997) and the course delivery was consistent with findings from distance education research (e.g., Beers, 1996; LeBaron, 1994).

The successful use of this delivery medium holds power for teacher-leaders to effectively meet the needs of geographically separated teachers. The importance of interacting with others in a teacher's professional development can result from appropriate use of distance education technology. The teachers in this study became more knowledgeable about the teaching and learning of geometry. This result supports the contention that there is much need to provide opportunities for future and practicing teachers to experience quality and appropriate mathematical learning (Chapin, 1997; NCTM 1991; Schifter, 1997). It also supports the contention that distance education can provide a powerful tool to make such tremendous development a reality, particularly when teachers are from quite disparate locations.

Future Directions

Future research will include continued observations of these teachers' classrooms as geometry lessons are taught, to determine the specific nature of the changes in geometry knowledge as well as of changes in pedagogy. Future experiences for these teachers might also include analyzing colleagues' lessons. Such data would provide additional evidence about the abilities of the teachers to use the van Hiele model of thinking.

Acknowledgements

The author gratefully acknowledges the service of Susan Seidenfeld, a teacher from the study who taught all participants, including the author, how to use the Geometer's Sketchpad.

Partially funded by: Exxon Education Foundation.

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Table 1 Levels of the van Hiele Model and Their Characteristics Level 0 Visual "Student identifies names, compares, and operates on geometric figures according to their appearance." 1 Analytic "Student analyzes figures in terms of their components and relationships among components and discovers properties/rules of a class of shapes empirically." 2 Informal Decuctive "Student logically interrelates previously discovered properties/rules by giving or following informal arguments." 3 Formal Deductive "Student proves theorems deductively and establishes interrelationships among networks of theorems." 4 Nature of logical laws "Student establishes theorems in different postulational systems and analyzes/compares those systems." (Fuys, Geddes, Tischler, 1988, p.5) Table 2 Brief Description of Course Content 1 Consider list of terms, Sort quadrilaterals, define square, rhombus, rectangle, Discuss elegant definitions (necessary & sufficient). Define triangle. 2 Define circle, cylinder - compare a "how to draw it" definition to a "what it is" definition. Discuss such conversation with students. 3 * Construct simple two-dimensional figures with paper/pencil Use Geometers Sketchpad to construct figures. 4 Discuss the difference between examples, counter examples, & definitions and how to present these ideas to children. 5 * Duplicate (Copy) triangles. SSS SAS emerged, (discuss these using sophisticated language, associated with writing proofs) 6 Use of geoboards to measure areas of polygons 7 Develop ASA and MS congruency with constructions Discuss feelings associated with level 3 van Hiele thinking. 8 * Determine angles of pentablocks without the use of a protractor. Discuss assumption that opposite angles of a parallelogram are congruent. Use Sketchpad 9 Resume parallelogram discussion. Couch entire discussion in van Hiele rhetoric. Create new tangram. Descdbe relationships between the pieces. 10 Continue van Hiele discussion about parallelograms. Use comments to guide thinking toward 5th postulate. Attempt first foray into non-Euclidean geometry: 11 Draw non-Euclidean triangles on spheres. Discuss fifth postulate and ramifications in "new" environment Discuss thinking (levels 3 & 4) 12 Build similar shapes W/ coffee stirrers and pipe cleaners. Calculate and compare volumes, and areas of faces, and lengths of edges. Introduce Golden ratio 13 Wrap up, re-visit terms from the first day. (*)Course meeting held on campus, not through distance education technology

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Author: | SHARP, JANET M. |
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Publication: | Journal of Computers in Mathematics and Science Teaching |

Geographic Code: | 1USA |

Date: | Jun 22, 2001 |

Words: | 7207 |

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