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Pre-service elementary teachers' understanding of geometric similarity: the effect of dynamic geometry software.

Since the use of technology in mathematics classrooms has increased dramatically during the past few decades, critical issues such as the role of technological advances in the learning and doing of mathematics need to be addressed (Balacheff & Kaput, 1996; Leitzel, 1991; National Council of Teachers of Mathematics [NCTM], 2000). Connell (1998) reported that a technological environment can enhance construction of knowledge and influence learning. Computers are able to aid in visualizing abstract concepts and to create new environments that extend beyond students' physical capabilities. Dynamic software is often employed as a fertile learning environment in which students can be actively engaged in constructing and exploring mathematical ideas (Cuoco & Goldenberg, 1996; Garry, 1997).

Software such as the Geometer's Sketchpad[R] [GSP] (Jackiw, 1995) and Micro Worlds LOGO[TM] (LOGO Computer Systems, Inc., 1996) provide a flexibly structured mathematics laboratory that supports the investigation and exploration of concepts at a representational level, linking the concrete and the abstract. Mathematical ideas can be explored from several different perspectives in an efficient manner, resulting in deeper conceptual understanding (Kaput & Thompson, 1994). Through repetitive experiences of exploring and mathematizing, problem solving skills and one's ability to assimilate ideas are enhanced (Cooper, 1991). In addition, the interactive mode supports active learning--a necessary component for effective construction of mathematical knowledge.

Purpose of the Study

The goal of the study was to determine the extent to which a dynamic geometry learning environment affects proportional reasoning. This study explored the role in the learning of mathematical concepts of the pre-service elementary teachers' classroom environment.

The necessity of requiring pre-service teachers to learn mathematics content and methods of teaching in an environment where technology is an integral component seems clearly evident (Abramovich & Brown, 1996). Students must be given the opportunity to construct and thoroughly develop a deep, interconnected understanding of fundamental mathematics in an environment similar to the mathematics classroom they will oversee (Ma, 1999; Simon & Blume, 1992). Research on rational numbers and multiplicative structures suggests that many K-8 teachers do not have the understanding required in proportional reasoning (Cramer & Lesh, 1988; Harel & Behr, 1995). Hence, based on the assumption that teachers need to be taught in a manner similar to the way they are expected to teach, this study investigated the effect of a dynamic geometry learning interface on performance in the context of similarity tasks. Furthermore, the research sought to better understand whether such an environment assists learning and reasoning, given its ability to efficiently display a variety of mathematical situations.

Theoretical Framework

The theoretical foundation for using a dynamic learning interface emphasizing inquiry and exploration is based on the Piagetian theory of learning as well as the role of repetitive experiences according to Cooper (1991). Current research on teaching from a constructivist perspective follows Piaget's biological metaphor of development and characterizes mathematical learning as a process of conceptual reorganization. The basic tenets of a constructivist epistemology have a direct implication upon pedagogy; that is, acceptance of these premises implies a way of teaching that acknowledges learners as active knowers (Brindley, 2000; Noddings, 1990).

The role of experience is to generate or modify the organization of the relevant cognitive space. As conceived by Cooper (1991), the role of repetitive experience, repeated interaction with the environment, is to create, enhance, and/or reorganize cognitive space. The possibility that repetition induces reorganizations of knowledge, not just of skills, is the rationale that Cooper (1991) gives for use of the term "repetitive experience" rather than "practice." We have known since the days of Brownell (1987) that mathematical "practice" often has little to do with developing habits of reasoning.

When used as intended, the GSP[R] can create a powerful learning environment where geometry lessons can be transformed into active explorations of geometric shapes resulting in stating, testing, and proving one's own conjectures. Furthermore, dynamic software affords the power of manipulation of the geometric objects for efficient exploration of phenomena.

Thus, the dynamic capability can aid in the imagery process to see the invariance of the construction and acquire a richer global view regarding necessary and sufficient conditions for construction. Research indicates that students learn geometry skills with greater efficiency and understand geometry concepts at higher levels as a result of creating and manipulating dynamic constructions on the computer screen (Dye, 2001; Hannafin, Burruss & Little, 2001; King, 1997).

The need for further examination of the impact of a dynamic learning environment on geometry instruction effectiveness is clear. Moreover, the lack of research with elementary pre-service teachers warrants investigation in the areas of learning environment and technology use. The specific question under investigation is: after controlling for initial performance on similarity tasks, do pre-service elementary teachers who work on similarity tasks using GSP[R] exhibit significantly different post-test performance than those who study the concept of similarity in a paper and pencil learning environment that employs traditional tools such as compass, ruler, and protractor.

Methodology and Data Source

Fifty-two prospective elementary teachers completed the study. The research participants were university students without prior classroom teaching experience, self-enrolled in three sections of an undergraduate mathematics education course at a major university located in the Southeastern United States. This introductory course, addressing content, methods, and materials for teaching and assessing elementary school mathematics, was a requirement of all students in the elementary teacher preparation program.

The design of the study was a pre-test/post-test control group experiment with randomized blocks; the randomization process, random assignment of individuals to experimental treatments, occurred within each course section. The experimental group worked in the dynamic geometry learning environment; the control group worked in the traditional paper and pencil manner. The instrument was developed by the researcher and based on items used by the Middle Grades Mathematics Project (Lappen, et al., 1986). All participants signed consent forms to participate in the study and agreed not to discuss their work with others until completion of the study. Debriefing and class discussion occurred during the session following the post-test.

The pre-test was administered at the start of the academic semester, eight weeks prior to the learning sessions. The procedure, over the course of a week, consisted of four hour-long learning sessions and the post-test. The sessions for the experimental group involved use of the GSP[R] in a campus computer lab; sessions for the control group were held in a mathematics model K-8 classroom and involved use of manipulatives, protractor, and ruler. A mathematics model K-8 classroom is defined as a classroom equipped with teaching resources and materials commonly found in elementary and middle schools. Written reflections were collected from all participants at the end of each session in order to gain further insight regarding the depth and breadth of the learning that had occurred. During the post-test, administered immediately upon completion of the intervention, participants in the experimental group used the GSP[R] as the tool to construct figures and measure, while control group participants made use of manipulatives, compass, protractor, and ruler.

Each class section met for two hours, five times a week, throughout the semester. During the week of the study, the control group worked with the researcher for the first hour while the other group attended class with the course instructor. The instructor and researcher would then switch groups. Both groups addressed the same content with the researcher using the same instructional method. The initial session presented an intuitive introduction to perspective and use of the GSP[R] or geometry tools. During the remaining sessions, participants explored properties of similar shapes with respect to perspective point, to scale factor, to corresponding sides, to corresponding angles, and to corresponding distances from the point of projection.

The researcher conducted all lessons relating to the study to eliminate differences in results due to instructor variability, but was aware of the possibility that an experimenter effect could threaten the ecological validity of the study (Bracht & Glass, 1968). This was minimized by the use of specific lesson plans implementing printed activities for the participants of both groups (refer to Appendix for an example activity). Participants were not allowed access to the GSP[R] or manipulatives between sessions. Since the study's mathematical concepts were not part of the course content, the participants did not concern themselves with outside-study. The control group members were not exposed to any dynamic geometry software prior to the end of the study. They were informed at the beginning of the study that they would have an opportunity to use the GSP[R] after the post-test.

Performance on the concept of similarity was determined by results from 21 multiple choice items with up to five options for each item. The tasks required students to recall properties of similar figures and proportionality; sample items are presented in Figure 1. The questions exemplify items constructed by the researcher to measure knowledge application and not mere mathematical similarity facts. Items were scored dichotomously with no correction made for guessing. To ensure that the instrument reflected the concept of similarity, a panel of five experts analyzed the twenty-one items to ascertain face validity. The instrument was then field-tested on sixty-six preservice elementary teachers and was administered early in the semester; the retest was given approximately nine weeks later. Both the Cronbach reliability coefficient and the coefficient of stability, calculated for test-retest reliability, were 0.70.


Multiple partial correlations were computed to assess the strength of association between the dependent variable and the covariate while controlling for treatment and section. Prior mathematical knowledge, as measured by the SAT, served to adjust for initial random differences in the two treatment groups. The multiple partial correlation with respect to the covariate SAT was 0.4281 and for the pre-test as covariate, 0.5925 was calculated. Thus, statistical analyses supported confidence for the utilization of the specified variables SAT and pre-test. To analyze the data collected during the study, descriptive statistics were obtained for all variables under consideration and analysis of covariances [ANCOVAs] were conducted.

Results of the Study

Table 1 contains the descriptive statistics for responses to the pre-test, SAT mathematics component and post-test by group. The post-test mean scores were adjusted by the covariate. Further analysis of the data was undertaken to better understand the large standard deviations. Stem and leaf plots of the data revealed that the control group data plot was essentially bimodal and the experimental group data contained outliers creating a negatively skewed plot.

The comparison of scores on the research instrument that assessed performance on similarity tasks served as the basis for the analysis. The results of the analyses were significant at the 0.05 level in all cases. ANCOVAs were conducted with the following covariates: pre-intervention achievement on the research instrument and SAT scores.

Initially, ANCOVAs were applied to assure that interaction effects were not present. These analyses, in both cases, indicated that there was no statistically significant effect on post-test scores due to either three-way or two-way interaction of the independent variables. The term "treatment with course section" was left in the model to guarantee control of the blocking variable "course section."

ANCOVAs were conducted using the final models. Based on the analysis, there exists supportive evidence that the difference in learning environment between the two treatment groups had a significant effect on pre-service elementary teachers' performance on similarity tasks as measured by the research instrument. Fundamentally, software users outperformed non-software users even when prior knowledge variability was taken into consideration. The ANCOVAs are presented in Table 2.


This study focused on the effect of dynamic geometry learning software use on pre-service elementary teachers' performance on similarity tasks; one can be reasonably confident that this effect is attributable to the dynamic technological learning environment for three reasons: participants were randomly assigned to the experimental and control groups, extraneous variables were effectively controlled, and the observed differences between groups on post-test performance are statistically significant. Moreover, given the fact that a common Piagetian constructivist pedagogy was assumed and parallel sets of curricula materials were used, the question was specifically focused on the effect of a dynamic software interface on student learning, as measured by achievement on similarity tasks.

In addition to statistical analyses, consideration was also given to qualitative aspects intrinsically rich in interpretive value. The activity worksheets (refer to Appendix for an example) not only gave structure to the sessions, participants' responses throughout the guided discovery process provided insight into their construction and reconstruction of knowledge. During the first session, participants in both groups were observed struggling with the novel approach to understanding concepts that many believed they had learned in middle school. Their procedural understanding of ratio and proportion was neither sufficient nor helpful for understanding properties of similar figures.

By the end of the first session, the experimental group had measured and analyzed many different polygonal figures. Observation of the control group indicated that, although they had rediscovered the principles of similarity, control group participants were often confused about similarity properties whenever the polygonal figures were unfamiliar shapes (e.g. concave). Results obtained from the statistical analyses suggest that the experimental group had acquired a greater knowledge base to access, network, and apply. If learning is generated by successive interactions with one's environment (Cobb, 1995), then repetitive activity is necessary to provide information that facilitates learning (Cooper, 1991).


Classroom teaching is undergoing great change when innovative dynamic software is used. A major challenge facing mathematics education research is development of an in-depth understanding of the role technology provides, not only to revise the mathematics content that we teach, but also to elucidate our understanding of the teaching and learning processes. Models of computer uses in education are needed to show how to incorporate this technology into daily practice.

To ensure proper use, teachers must be trained appropriately in order to bring to the classroom the experience, confidence, and enthusiasm necessary to effectively facilitate student learning. Dynamic software is a relatively new phenomenon; in particular, the GSP[R] is highly visual and efficient for exploring and discovering properties of similar figures.

To date, there is very little research regarding the effect of novel classroom environments on elementary teachers' learning of the similarity concept. This particular study adds to the growing body of literature addressing the effect of dynamic geometry software in the classroom. Specifically, it gives evidence that the GSP[R] can enhance efficient construction of the concept of similarity. The research helps substantiate the need to incorporate innovative technological tools in teacher education. This study indicated that a dynamic learning environment can provide an effective and efficient environment for the development of proportional reasoning. Instructors can use the GSP[R] to provide their students with a powerful tool for investigation of mathematical concepts such as similarity.


Experimental Group Explorations

Sketch -- open a new sketch

step 1: Construct any quadrilateral and change the color of each side to be distinct.

step 2: Construct a point P outside the quadrilateral and mark it as center in the Transform menu.

step 3: Select your entire quadrilateral and dilate by a scale factor of 2/1 (Place the numerator into the top box and the denominator into the bottom box).

Write a description of what you notice about the quadrilaterals; that is, write about how the size and shape of the original quadrilateral compares to the image (the dilated figure).

Investigation: Corresponding Angles

Measure an angle of the image (the dilated polygon) and the corresponding angle on the pre image (the original polygon). Write a conjecture about corresponding angles of similar polygons. Repeat, using a different angle to test your conjecture. Drag the vertex of the angle (on the pre image) to change the measure of the angle in order to further test your conjecture.

Investigation: Corresponding Sides

Measure the ratio of a side on the dilated polygon with the corresponding side on the original polygon. Repeat, using a different side. Write down what you think this ratio is. Now, write a conjecture about corresponding angles of similar polygons.

Drag a vertex to change the length of a side you have measured. Write a short paragraph to address the following questions:

* What numeric changes occur?

* What stays the same?

* Do the results of your analyses support your conjectures?


Control Group Explorations

Sketch -- Use a pencil and ruler

step 1: Construct any quadrilateral and label the vertices M, A, T, H in clockwise order.

step 2: Construct a point P outside the polygon; draw four rays beginning at point P and intersecting each of the vertices.

step 3: Mark point M' on ray PM so the measure of segment PM' is twice as long as segment PM. Repeat the process to mark points A', T', H'.

step 4: Connect the points M', A', T', H' in clockwise order to form a quadrilateral. Write a description of what you notice about the quadrilaterals; that is, write about how the size and shape of the original quadrilateral compares to the image (the dilated figure).

Investigation: Corresponding Angles

Measure an angle of the image (the dilated polygon) and the corresponding angle on the pre-image (the original polygon). Write a conjecture about corresponding angles of similar polygons. Repeat, using a different angle to test your conjecture. Trade sketches with your neighbor and repeat the measurement process to further test your conjecture, if time permits.

Investigation: Corresponding Sides

Measure the ratio of a side on the dilated polygon with the corresponding side on the original polygon. Repeat, using a different side. Write down what you think this ratio is. Now, write a conjecture about corresponding angles of similar polygons.

Repeat the process on other quadrilaterals. Write a short paragraph to address the following questions:

* What numeric changes occur?

* What stays the same?

* Do the results of your analyses support your conjectures?

Table 1 Descriptive Statistics

 Group Number Mean Standard Deviation

Pre-test control 26 12.23 3.23
 experimental 26 12.35 3.70
SAT control 24 580.00 73.01
 experimental 24 556.67 86.81
Post-test control 26 14.65 2.65
 experimental 26 16.50 3.11

Table 2 Analysis of Covariance Final Model with Pre-test as Covariate

Dependent Variable: Post-test

Source DF SS MS F Value p-value

treatment 1 29.09 29.09 15.19 0.03*
pre-test 1 136.37 136.37 24.23 0.00
section 2 2.05 1.03 0.18 0.83
treatment by section 2 6.72 3.36 0.60 0.55

*statistically significant at alpha level 0.05

Analysis of Covariance Final Model with SAT as Covariate

Dependent Variable: Post-test

Source DF SS MS F Value p-value

treatment 1 38.86 38.86 5.26 0.03*
SAT 1 79.21 79.21 10.71 0.00
section 2 23.72 11.86 1.60 0.21
treatment by section 2 2.95 1.48 0.20 0.82

*statistically significant at alpha level 0.05


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Helen Gerretson

University of Northern Colorado
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Author:Gerretson, Helen
Publication:Focus on Learning Problems in Mathematics
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Date:Jun 22, 2004
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