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THE EFFECT OF A DYNAMIC GEOMETRY LEARNING ENVIRONMENT ON PRESERVICE ELEMENTARY TEACHERS' PERFORMANCE ON SIMILARITY TASKS By HELEN GERRETSON A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 1998 Copyright 1998 by Helen Gerretson TABLE OF CONTENTS ACKNOWLEDGEMENTS . ABSTRACT . CHAPTERS I INTRODUCTION . . . . . Rationale . . . . . . Statement of the Problem . . Purpose of the Study . . . Research Questions . . . . Justification for the Study . Definitions of Terms .. ...... Organization of the Dissertation II THEORETICAL FRAMEWORK AND RELEVANT RESEARCH Constructivism . . . . . . . . Repetitive Experiences . . . . . . Phases of Learning . . . . . . . Erlanger Programm . . . . . . . Similarity and Proportional Reasoning . . Computerbased Geometric Construction Environment ... Dynamic Geometric Construction Environment Summary . . . . . . . . . III RESEARCH DESIGN AND METHODOLOGY . . . 13 * 13 * 14 * 16 S. 18 S. 20 * 23 S. 25 * 31 S. 33 Research Objectives . . . Population and Sample .... Instructional Materials . . Pilot Studies . . . . . Instrumentation . . . . Procedure and Data Collection Statistical Design of the Study Summary . . . . . . . . . . . . . . . i i  V . . . . 1 pare, . . . . . 4 3 Research Instrument . . Research Objectives . . Goals of the Study . . Descriptive Statistics Research Hypotheses . . Results of Data Analyses A Reflections Component . Summary . . . . . V SUMMARY AND CONCLUSIONS . Overview of the Study . . Significance of the Study . Research questions ... Results .............. Discussion of Results . . Other Observations . . Limitations Implications and Recommendations Summary . . . . . . . .. . . . . 59 . . . . 58 . . . . 61 . . . . 62 . . . . 62 . . . . 65 . . . . 68 . . . . 69 . . . . 70 . . . . 73 APPENDICES A THE RESEARCH INSTRUMENT . . . B CONTENT OF SESSIONS . . . . C ACTIVITIES WITH RESPONSES . . . D LETTERS OF INFORMED CONCENT ... E: DATA SETS . . . . . . . REFERENCES . . . BIOGRAPHICAL SKETCH . S.74 . .86 S.95 * 105 S110 . . . . 113 . . . . . 124 .43 .44 .44 .45 .46 .49 .53 .58 IV RESULTS . . e O Q Q D CHAPTER I INTRODUCTION As aspects of the current mathematics reform movement become reality in the classroom, researchers question the impact of the restructuring and recontextualizing of the mathematics classroom. Interest must focus upon reform related pedagogy with greater intellectual involvement via appropriatelychosen experiences in a context of problem solving and inquiry. In particular, the effects of implementing technology on the teaching and learning of mathematics and its impact on mathematics as reasoning must be addressed. The question is no longer whether technology belongs in the classroom, but how best to use technology to enhance learning. Rationale Use of technology in mathematics classrooms has increased dramatically during the past two decades and has been encouraged by leaders in the field of mathematics education (Blackwell & Henkin, 1989; English & Halford, 1995; Leitzel, 1991; National Advisory Committee on Mathematical Education [NACOME], 1975; National Council of Teachers of Mathematics [NCTM], 1980, 1989; National Institute of Education [NIE], 1975; National Research Council [NRC], 1989; Suydam, 1976). The changing needs of our society demand that critical issues such as the role of technological advances in the learning and doing of mathematics be addressed. Computers are able to aid in visualizing abstract concepts and to create new environments which extend beyond the physical. In addition to development of problem solving and visualization skills, technology offers enormous opportunity for curriculum reform in mathematics (Abramovich & Brown, 1996). Dynamic software (refer to Definition of Terms) such as Geometer's Sketchpad (Jackiw, 1995),Cabri Geometre II (Laborde, 1994), and Mathematica (Wolfram, 1988) provide a flexibly structured mathematics laboratory: an electronic microworld supporting the investigation and exploration of mathematical 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). Mathematical thinking is now often learned through problem solving activities that bury the mechanical aspects of mathematics behind interesting ideas (Vockell & Schwartz, 1992). Dynamic software allows the computer to be a powerful mathematical tool due to its ability to quickly and accurately plot, construct, measure, and perform computations. Through repetitive experiences of exploring and mathematizing, problem solving skills and one's ability to construct mathematical ideas are enhanced (Cooper, 1991). In addition, the interactive mode supports active learninga necessary component for effective construction of knowledge (Lewin, 1991). Furthermore, the dynamic software environment is motivational for the student because it "promotes mathematical explorations ... and helps the building of conceptual understanding" (Fraser, 1988, p. 216). The Third International Mathematics and Science Study [TIMSS] gives mathematics educators a benchmark toward improvement of the teaching and learning of mathematics. The TIMSS data suggest that when teachers use more challenging methods to teach more complex mathematical ideas, as called for in the Standards trilogy: the Curriculum and Evaluation Standards for School Mathematics (NCTM, 1989), the Professional Standards for Teaching Mathematics (NCTM, 1991), and the Assessment Standards for School Mathematics (NCTM, 1995), students' performance is higher (National Center for Educational Statistics, 1998). Statement of the Problem There is reason to change what and how preservice elementary teachers learn mathematics. Most teachers are aware of the NCTM Standards, but evidence from the TIMSS video component suggests that changes in teaching practice are difficult. In the United States, lessons typically focus on acquiring mathematical skills rather than conceptual understanding, with little mathematical reasoning expected. In contrast, Japanese methodology, requiring students to think and reason, resemble the recommendations advocated in the Standards (National Center for Educational Statistics, 1998). Analyses of data from the TIMSS show, among the content areas addressed, that United States eighth and twelfth graders scored significantly below the international average in geometry and measurement. Moreover, grade twelve students' performance was relatively lowest in geometry; no country scored similar to or below the United States (National Center for Educational Statistics, 1998). This researcher posits that the lack of exposure to transformational geometry, a topic often found in other countries' curricula, may be a leading factor of poor performance. Teachers cannot teach what they themselves do not know (Fennema & Franke, 1992). Teachers must be given the opportunity to construct and thoroughly develop their knowledge of elementary mathematics in an environment similar to the mathematics classroom they oversee (Simon & Blume, 1992). Many mathematics teacher educators advocate that teachers must be taught in ways that are consistent with how we would like them to teach (Committee on the Mathematical Education of Teachers, 1991; Fennema & Franke, 1992; NCTM, 1991). Implementing new technologies in a reformed curriculum is not easy and the education of teachers plays a critical role. The necessity of requiring preservice teachers to learn mathematics content and methods of teaching in an environment where computer use is an integral component is clearly evident. The Mathematical Association of America stresses this point in their report A call for change: Recommendations for the mathematical preparation of teachers of mathematics 5 (Leitzel, 1991). The document states: [Teachers] need experience in using appropriate technology effectively in solving problems so that they can learn and adapt strategies and representations that arise while using that technology. Because they can reflect on their own learning and understanding of mathematical ideas when using technology, teachers will be better prepared to lead their own students in effective mathematical learning (p. 7). The issue is not simply the incorporation of another gadget in the learning milieu. Rather it encompasses the careful consideration of the inclusion of technology commensurate with the current vision for mathematics education reform. Dynamic software is generally accepted as a fertile learning environment in which students can be actively engaged in constructing and exploring mathematical ideas (Garry, 1997). This researcher postulates that the dynamic and interactive medium makes gaining an intuitive understanding more accessible to the [geometry] student (Pea, 1987). Under the psychomotor control of the user, this "dynamic" capability provides an infinite number of cases to be explored. For example, once constructed, geometric figures can be transformed by dragging a point, a vertex, or an edge. As the figure is transformed, patterns and/or relationships become visually explicit for further examination. The theoretical foundation for using a dynamic learning environment emphasizing inquiry and exploration is based on the constructivist theory of learning as well as the role of repetitive experiences according to Cooper (1991). We have known since the days of Brownell that mathematical "practice" often has little to do with developing habits of reasoning (Thompson, 1994). Long term problem solving is primarily a function of recognizing well known embedded principles in new situations. Cooper focuses our attention on the fact that what we want repeated is a composition of situations in ways that motivate generalization and support reflection. The constructivist theory of learning refers to the psychological level of analysis where emphasis is placed on the construction of personal knowledge and how a person acquires this knowledge (Phye, 1997). The philosophical level of analysis, although a legitimate line of inquiry, will not be addressed in this thesis. Purpose of the Study The purpose of the proposed study was to investigate the effect of a dynamic geometry learning environment on preservice elementary teachers' performance in the context of similarity tasks (refer to Definition of Terms). That is, this study was concerned with the mathematical evolution of students majoring in elementary education as a result of instruction with the dynamic geometry software Geometer's Sketchpad (Jackiw, 1995). The basic goal of the study was to detect the extent to which the dynamic environment, given its ability to efficiently display a plethora of mathematical situations, assists learning. Attention was also given to the role of prior knowledge. Research Questions Hence, based on the assumptions that teachers need to learn high quality mathematics and to be taught in a manner similar to the way they are expected to teach, the specific questions under investigation were: 1. After controlling for initial performance on similarity tasks, do preservice elementary teachers who work on similarity tasks in a dynamic geometry learning environment exhibit significantly different posttest performance than those who study the concept of similarity in a paperandpencil learning environment that employs traditional tools such as ruler and protractor? 2. After controlling for prior knowledge as measured by a standardized achievement test, do preservice elementary teachers who work on similarity tasks in a dynamic geometry learning environment exhibit significantly different posttest performance than those who study the concept of similarity in a paperandpencil learning environment that employs traditional tools such as ruler and protractor? Justification of the Study From several perspectives, the problem is important. As indicated earlier, technology is profoundly changing the teaching and learning of mathematics. Models of computer uses in education are needed to show how to incorporate this technology into daily practice (Garciade Galindo, 1994). To define new roles for teachers in a technologically rich classroom environment, it is necessary to explore how computers will modify current ways of teaching and learning. Carleer and Doornekamp (1990) conclude that "to realize a valuable integration of computers in education, it will be necessary to focus the discussion no longer on computers, but on how to think and make decisions with the improvement of education as the goal"(p.5). The acquisition of the concept of similarity is important for development of proportional reasoning and for geometrical understanding of one's immediate environment. One encounters phenomena that require familiarity with projection, scale factor, and other similarityrelated concepts in everyday situations (Friedlander, 1984; Lappan, Fitzgerald, Winter, & Phillips, 1986). Similarityrelated concepts are included in many parts of the mathematics curriculum. Some models for rational number concepts are based on similarity; an integral part of algebraic thinking involves ratio and proportion reasoning skills (Lappan & Even, 1988). Similar geometric shapes would seem to provide helpful imagery for analysis of analogous situations. Dynamic software is relatively new phenomena; in particular, dynamic geometry software was not widely available a decade ago. Computerized dynamic geometry is highly visual and efficient for exploring and discovering properties of similar figures. However, realization of its potential is stifled by the lack of research using dynamic geometry software for middle school mathematics. At this time, there is no information regarding the effect of a dynamic geometry learning environment on K8 9 teachers' learning of the similarity concept. This particular study adds substance to the growing body of literature addressing the significance of dynamic geometry software in the classroom and, in particular, gives evidence whether a dynamic geometry learning environment enhances efficient construction of the concept of similarity. Moreover, this research will help substantiate the need to incorporate innovative technological tools in teacher education. One can be taught techniques and tricks, but expertise results from experience and insight. Definitions of Terms The following are some terms that are used throughout the manuscript; they are defined and collected here for reference: A preserviceelementaryteacher is a student, without prior classroom teaching experience. All participants were enrolled in a required introductory mathematics methods course addressing content, methods, and materials for teaching elementary school mathematics. The Geometer's Sketchpad (Jackiw, 1995), developed by Nicholas Jackiw and the Visual Geometry Project at Swarthmore College and now published by Key Curriculum Press, is an example of dynamic geometrv software and was used in this study. Such software transforms the computer interface into a geometric construction microworld capable of quickly and accurately plotting, constructing, measuring, and performing computations. Once constructed, geometric figures can be transformed by dragging a point, vertex, or edge. As the figure is transformed, corresponding measurements and calculations change accordingly. A dynamic geometry learning environment is defined as a learning milieu in which expository teaching is integrated with laboratory activities. The laboratory activities consist of explorations and experimentation using dynamic geometry software that allows for direct construction, manipulation, and measurement of geometric figures on the plane. In contrast, the traditional learning environment is similar to the dynamic learning environment in that expository teaching is integrated with laboratory activities. In this case, however, laboratory activities consist of explorations using manipulatives, ruler and protractor. A variable tension proportional divider (VTPD) is a mathematics laboratory device for drawing similar figures. The VTPD enlarges drawings and pictures in addition to geometric figures. To build a VTPD, knot together identical rubber bands. One end is held down by the thumb of one hand, acting as the projection point, and a pencil is inserted in the other end. The rubber band is stretched so that a knot traces the original figure while the pencil is moved on paper. It is of utmost importance that eyes follow the knot while the pencil draws. Although the final figures are not quite accurate, the image clearly conveys the intuitive "same shape" notion of similarity as one actively creates similar figures. A transformation (onetoone correspondence) on the Euclidean plane is called a similarity if there exists a positive number k such that the distance between the images of any two points is k times the distance between the points. A dilation about the protection point (center) P of factor k is a transformation D in which each point A is mapped to its image D(A) in such a way that the directed distance between P and D(A) is k times the directed distance between P and A (Hall, 1973). In particular, the distance between the images of any two points is k times the distance between the points. A dilation is also referred to as a dilatation, a central similarity, or a homothety (King, 1997). Organization of the Dissertation In this chapter, the rationale for a study designed to examine the effect of a dynamic geometry learning environment on preservice elementary teachers strategies employed and performance on similarity tasks was described and background briefly discussed. Chapter II contains theoretical foundations and a review of related research supporting the study. Chapter III presents a description of the subjects, the composition of the instrument, the methodology and design of the study, and the results of the pilot studies. Analysis, interpretation of the data, and qualitative aspects for the present study will be given in Chapter IV. Chapter V will include a summary of the investigation, a discussion of 12 results, limitations of the study, implications for instruction, and suggestions for further research. CHAPTER II THEORETICAL FRAMEWORK AND RELEVANT RESEARCH In the previous chapter, the study was introduced and background was given. In this chapter, the theoretical basis for the study is presented in detail. This includes a discussion of constructivist epistemology, the role of repetitive experience on learning, phases of learning, and an elucidation of the Erlanger Programm. Also included is a brief exposition on similarity and proportional reasoning in addition to relevant research in the area. The theoretical premise that supports this study is that a dynamic geometry learning environment, supported by use of software such as the Geometer's Sketchpad (Jackiw, 1995), can enhance construction of knowledge and influence learning. Therefore, an examination of research focusing on software for geometric constructions follows; this section develops from static software to dynamic software as research already conducted on the use of dynamic geometry learning software is identified. Constructivism Using von Glasersfeld's radical theory of constructivism (1979, 1990, 1995) as a basis, Kilpatrick (1987) describes the basic tenets of constructivism as an epistemology such that (1) all knowledge is actively constructed, not passively received, and (2) coming to know is an adaptive process that organizes one's experiential world; it is not the discovery of an independent, preexisting world outside the mind of the knower (p. 7). Accepting von Glasersfeld's radical point of view involves rejecting ontological realism and embracing an epistemology that makes all knowing active and all knowledge subjective (Kilpatrick, 1987). This epistemological stance is in complete harmony with Piaget's theory of learning (1964), which demonstrates that human beings acquire knowledge by building it from the inside instead of internalizing it directly from the environment (von Glasersfeld, 1979; Kamii, 1990). Thus, 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 (Cobb, 1995). The basic tenets of a constructivist epistemology have a direct implication upon pedagogy. Nel Noddings (1990) posits that constructivistss in mathematics education contend that cognitive Constructivism implies pedagogical Constructivism; that is, acceptance of constructivist premises about knowledge and knowers implies a way of teaching that acknowledges learners as active knowers" (p. 10). This is particularly meaningful in the case of radical constructivism; it focuses on the individual as a self organizing system. This is not always the case in the many variations of constructivism; some constructivists embrace Vygotsky's (1962) epistemological stance that emphasizes the cultural and social dimensions of development (Cobb, 1995; Confrey, 1995). In contrast, von Glasersfeld (1979, 1990, 1995) presents constructivism as a model of how an individual's experience forms the basis for knowing and communication. Within this model, social interaction constrains (and thus guides) the processes involved in the construction (von Glasersfeld, 1990). Repetitive Experience 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. Because the constructed cognitive space constitutes acquired knowledge, repetitive experience is of utmost importance. As in practicing typing, repetitive experience is used to mean repeating a set of similar and interrelated activities, not doing exactly the same thing over and over again. In addition, the construction of such a cognitive space can serve as the foundation for reflective abstraction from which new knowledge results. 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" (Thompson, 1994). Acquisition of knowledge is experiential in that it must be actively constructed by the learner. Therefore, if cognitive development is generated by successive interactions with one's environment, then repetitive activity is necessary to provide information that facilitates problem solving. Consider the following task: Two equalnumerosity arrays in onetoone correspondence are presented. One is then screened, and n objects are transferred from the visible to the screened array. The objective is to predict how many objects must be added to the visible array to make it equal to the screened array. Children learn with repeated examples that a transfer of n produces a difference of 2n (Piaget, 1974/1980; Piaget, Grize, Szeminska, & VinhBang, 1977; Cooper, Campbell, & Bevins, 1983). Performance on this task highlights the role of repetitive experience because even adults frequently err until they have at least one experience with the task (Cooper, 1991). Robert Cooper's (1991) theory of repetitive experiences provides a strong foundation for reflective abstraction, and hence is a critical element for the development of insightful thought. Phases of Learning According to Karplus (1977), conceptual learning proceeds from an exploration stage to a concept identification stage to an application stage where new ideas are used and extended (Simon, 1992). The application stage triggers a new level of exploration and the cycle recommences. Karplus' learning cycle, derived from Piaget's (1964) mental functioning model, provides a pedagogical framework that complements a constructivist view of learning (Fleener, 1995). Consequently, it provides a sound foundation for the middle stage of each learning session with respect to this research study. Jurascheck (1983) suggests that learning cycle pedagogy be applied to mathematics education. Simon (1992) presents a framework utilized in two research projects for mathematics teacher learning based on what is understood about students' mathematics learning. The framework builds on Karplus' Learning Cycle and identifies a learning cycle that progresses through the following stages: exploration of a mathematical situation, discussion leading to concept identification, and application and extension of new ideas. Also building on Piaget's work, Dienes (Dienes & Golding, 1971) advocates a learning cycle in which learners progress through a series of cyclic patterns, each comprising activities ranging from concrete to symbolic format. The earliest learning phase in each cycle begins with free play to uncover the key features of the structured materials. Following this, experiences are systematically structured to facilitate discovery of inherent relations in the material and in abstracting the concept being represented. Dienes' learning cycle can be compared to the van Hiele's phases of learning. Although much has been communicated regarding the work of the van Hieles in terms of 18 a stratification of human thought (e.g. Senk, 1983), thought levels comprise only one of the three main components of the van Hiele model (Hoffer, 1983); the other components are insight and phases of learning. The van Hieles propose a sequence of five phases of learning: information, guided orientation, explication, free orientation, and integration (Clements & Battista, 1992). Polya (1965), best known for articulating aspects of problem solving, posits that any good teaching device must be correlated somehow with the nature of the learning process and, therefore, outlines some obvious features in the form of three principles of learning: (1) the principle of active learning, (2) the principle of best motivation, and (3) the principle of consecutive phases. Specific to the third principle, Three phases are distinguished: exploration, formalization, and assimilation. Polya (1965) summarizes that "For efficient learning, an exploratory phase should precede the phase of verbalization and concept formation and, eventually, the material learned should be merged in, and contribute to, the integral mental attitude of the learner" (p. 104). The Erlanqer Propramm In 1872, Felix Klein (1893) established a structure for the analysis of geometric concepts, now referred to as the Erlanger Programm, and presented his definition of a geometry: "A geometry is the study of the properties that are invariant when the subsets of a set S are mapped by the transformations of some group of transformations" (Hall, 1973, p.95). Using Klein's definition of geometry, it is possible to categorize and name various geometries according to both the invariant properties under that group and the geometries involved. The transformations of topology are called homomorphisms; a onetoone transformation f is called a homomorphism if it is continuous and reversibly continuous. Shapes can be altered by stretching, compressing, bending, and twisting, but not by tearing or joining (Mansfield, 1985). Projective geometry can be characterized as the study of properties invariant under the group of collineations; collineations are special homomorphisms which transform collinear points into collinear points and, therefore, lines into lines. Affine geometry is obtained as a subgeometry of projective geometry by restricting the group of projective transformations in such a way as to introduce parallelism and distance. In particular, affine transformations map equal distances into equal distances on the same or parallel lines, and midpoints into midpoints. Whereas an affine transformation multiplies distances in the same direction by a constant, a similarity transformation multiplies all distances by the same positive number k, the ratio of the similarity transformation. Thus, the shape of a figure is preserved but not its size. The Euclidean transformation group is a subgroup of the similitude transformation group as a Euclidean transformation preserves distance mensurationn). Similarity and Pronortional Reasoning Fuson (1978) observed that most of Piaget's (1948/1956, 1960) analyses focus on topological concepts, or on Euclidean concepts, leaving the projective, affine, and similitude transformational groups relatively neglected. Nevertheless, Piaget and Inhelder (1948/1956) describe four experiments related to the concept of similarity: (i) drawing similar triangles, (ii) sorting similar cardboard triangles, (iii) choosing and drawing similar rectangles, and (iv) drawing a similar configuration of line segments. These experiments illustrate the gradual procurement of angle and parallelism concepts. Piaget's categorization indicates that children begin to use proportions at the age of eleven; perceptual judgment of similarity is possible two years earlier (p.374). The failure to perform well in similarity tasks at an earlier age is attributed to functioning in preoperational thought and, later, the inability to use mental constructs (i.e. proportions) involving secondorder relations (Lunzer, 1968). Freidlander's (1984) study of 675 suburban, middle class, midwestern, predominantly white students addressed four similarityrelated topics: (1) basic properties of similar shapes, (2) proportional reasoning, (3) area relationships of similar shapes, and (4) applications. Their teachers had volunteered to teach the Similarity Unitan instructional unit developed by the Middle Grades Mathematics Project (Lappan, Fitzgerald, Winter, & Phillips, 1986) for grades six, seven and eight. Pre and postinstructional performance, at a significance level of 0.05, show achievement increased as a function of grade level. In a study involving 119 high school geometry students, Chazan (1988) reported the results of an investigation into high school students' understanding of similarity. A unit was constructed for use with the Geometric Supposers (Schwartz & Yerushalmy, 1985). Students were observed as they learned similarity with this unit and were given protests and posttests on fractions, ratio and proportion, and similarity. Data from the study show that even those students who show an understanding of multiplicative ratio may exhibit that understanding with certain geometric configurations and not with others. It is important to note that this study did not compare students who used the Geometric Supposers (Schwartz & Yerushalmy, 1985) with those who did not; instead this study focused on students' understanding of aspects of similarity. Fleener,et al (1993) investigated the following questions: (1) Does knowing a standard algorithm for solving proportion problems interfere with the development of proportional reasoning? (2) Are stronger mathematics students more flexible or intuitive than weaker students in applying proportional reasoning strategies to solve problems? and (3) What is the relationship between a student's general level of reasoning ability (concrete, transitional, or formal) and the strategies used for solving proportional reasoning tasks? Sixteen ninth grade students engaged in proportional reasoning tasks and computational proportional problem solving in their science classes made a general positive gain in Lawson's Classroom Test of Scientific Reasoning scores, although gains for concrete learners were mixed. Explicit teaching of the concept of ratios and student engagement in exploratory studies of the relationships between and among ratios provoked development of proportional reasoning for average students. The researchers concluded that students can benefit from experiences from which the crossmultiplyand divide algorithm can be derived. The understandings involved in proportional reasoning as embedded in similarity tasks are complex. According to Piaget (1975), the essential feature of proportional reasoning is the involvement of a complex quaternary relation, a relationship between two relationships, rather than a simple relationship between two directly perceivable quantities (English, 1995). Tournaire and Pulos (1985) posit one should expect this type of reasoning to develop slowly over several years. Research rational numbers and multiplicative structures suggests that many teachers do not have this understanding (Cramer & Lesh, 1988; Harel & Behr, 1995; Lacampagne, Post, Harel, & Behr, 1988). The aforementioned studies provide a basis for the discussion of the requirements for an appropriate instructional environment to maximize the learning of the concept of similarity and, in general, the development of proportional reasoning. In view of the eclectic epistemological framework presented by this researcher, a thorough investigation of performance on similarity tasks necessitates an instructional environment where exploration and inquiry dominate the climate of the classroom. Computerbased Geometric Construction Environment Several studies (McCoy, 1991; Yerushalmy, 1991; Yerushalmy & Chazan, 1990; Yerushalmy & Chazan, 1993) show that, when used as intended, the Geometric Supposers (Schwartz & Yerushalmy, 1985) create a powerful learning environment: geometry lessons are transformed into active explorations of geometric shapes resulting in stating, testing, and proving one's own conjectures. Thus, the conception of mathematics shifts from something the learners encounter and observe to something they do and invent. McCoy (1991) compared the geometry achievement of a tenth grade class (n=29) which used the Geometric Supposers (Schwartz & Yerushalmy, 1985) periodically during one school year and a similar class (n=29) which did not use the software. Analysis of Covariance was used to determine the effect of the treatment on post geometry treatment scores. To control for individual differences, the covariate was mathematics scores from the pretest, the SRA Achievement Test. The final examination provided by the publisher of the adopted textbook was used as the posttest. The items were examined and classified according to Bloom's Taxonomy (Bloom, 1956). Results revealed the experimental group had higher scores on upper hierarchical (analysis, synthesis, and evaluation) level problems and there was no statistically significant difference on lower hierarchical (knowledge and comprehension) level questions. Fortyeight eighth graders who worked with the Geometric Supposers (Schwartz & Yerushalmy, 1985) outperformed a comparison group of ninety students on a test that measured knowledge of basic geometry concepts even though the comparison group was taught the same concepts and topics during the same amount of time. Yerushalmy (1991) found that the main difference between the groups was that the experimental group did not exhibit some of the frequently observed, persistent misconceptions such as having a stereotyped image of certain geometric concepts and shapes. Moreover, Yerushalmy & Chazan (1993) demonstrated that high school students working with the Geometric Supposers (Schwartz & Yerushalmy, 1985) seemed to have greater flexibility in interpreting figures and diagrams; in fact, the comparison group showed greater difficulty in overcoming visual obstacles such as the inability to perceive a diagram in different ways. To explore which teaching strategies may adapt to a computer environment that facilitates discovery learning and encourages cooperative learning, Garciade Galindo (1994) compared the teaching strategies and evaluation methods of preservice high school geometry teachers observed in traditional classes and classes implementing the Geometric Supposers (Schwartz & Yerushalmy, 1985) and/or the Geometer's Sketchpad (Jackiw, 1995). Comparative analysis was used to categorize the teaching strategies, teaching styles, evaluation methods, classroom interactions, and computer uses of four selected teachers. The main results were that the preferred teaching strategies and evaluation methods of the IntuitiveFeeling, IntuitiveThinking, and SensingFeeling teachers may adapt to computer environments that encourage discovery and cooperative learning. The SensingThinking teacher may prefer the model of teachercentered instruction when teaching high school geometry with computers. The four teachers reported plans to use computers in different ways. The IntuitiveFeeling teacher related plans to do calculations; the IntuitiveThinking teacher, to generate conjectures; the SensingThinking teacher, to teach the curriculum; and the SensingFeeling teacher, to review basic skills. Developed for students to conjecture about Euclidean geometry, geometric construction software provides an effective means to collect a wealth of visual and numerical data to analyze. By observing the relationships revealed by the data, insightful thinking is exercised. This in turn generates need for further exploration to collect necessary data to test the conjectures and the learning cycle begins anew. Dynamic Geometric Construction Environment In comparison to the geometric environment described above, dynamic construction software affords the added power of dynamic manipulation of the geometric objects for efficient exploration of geometric phenomena. The construction of, for example, a quadrilateral, thus 26 represents the mathematical object in its fullest form due to its ability to be transformed into, in this case, another quadrilateral. Thus, the dynamic capability aids in the imagery process to see the invariance and acquire a richer global view regarding necessary and sufficient conditions for construction and analysis of the constructed objects. Elchuck (1992) explored the effects of the dynamic capability of the geometric tool software Geometer's Sketchpad (Jackiw, 1995) on conjecture making ability. One hundred fifty seven grade nine students were randomly assigned to either of two treatment groups, based on capabilities of the tool software. The dynamic group had access to the full power of the Geometer's Sketchpad (Jackiw, 1995); the static group had access to all but the drag capabilities of this software. All participants were administered tests to determine their mathematics achievement level, spatial visualization skill, locus of control, and van Hiele level of geometric thought prior to engaging in an instructional unit. Both treatment groups underwent identical instruction in the same geometric content. The study demonstrated that mathematics achievement is a statistically significant predictor of conjecture making ability. In a posthoc regression analysis, the type of software (dynamic versus static) was found to predict conjecture making ability when the factor school was also included in the regression model. The purpose of Foletta's (1992) case study was to describe the nature of four high school geometry students' inquiry by observing how the students used Geometer's Sketchpad (Jackiw, 1995) and by characterizing their small group interactions. Factors contributing to the students' inquiry included the role of the Geometer's Sketchpad (Jackiw, 1995), the design of the investigations, and the nature of peer interactions. Foletta (1992) posits that the students adapt the Geometer's Sketchpad (Jackiw, 1995) as an extension of the paperandpencil medium. A posttestonly controlgroup quasiexperimental study was conducted by Lester (1996) to address the problem of improving achievement of geometric knowledge through instructional use of the software program Geometer's Sketchpad (Jackiw, 1995). An inductive reasoning approach was the pedagogy of instruction. Fortyseven female high school geometry students participated in the study. The Geometer's Sketchpad (Jackiw, 1995) was the tool used by participants in the experimental group; the control group used traditional geometry tools: ruler, pencil, protractor, and compass. The three dependent variables measured on a posttest were: geometric knowledge, construction, and geometric conjecturing. Descriptive findings for the dependent variables geometric knowledge and construction were not statistically significant at an alpha level of 0.05; descriptive findings for the dependent variable geometric conjectures were statistically significant at an alpha level of 0.05. Results from the study indicated that students learned geometry skills with greater efficiency and understood geometry concepts at higher levels as a result of creating and manipulating dynamic constructions of geometric objects on the computer screen. The relationships among 158 high school geometry students' spatial visualization ability, mathematical ability, and problem solving strategies with and without the availability of the Geometer's Sketchpad (Jackiw, 1995) was explored by Robinson (1994). Following instruction using the Geometer's Sketchpad (Jackiw, 1995), participants were randomly assigned to one of two groups: with or without software access. The availability of the computer was not a significant factor for performance on problems related to mathematical locus. Teaching specific skills resulted in similar strategies used by participants with and without access to the Geometer's Sketchpad (Jackiw, 1995). Robinson (1994) posited that strategies learned with the technology transfer to paper and pencil situations. Dixon (1995) investigated the effects of a dynamic instructional environment, English proficiency, and visualization level on the construction of concepts of rigid motion transformations. Two hundred fortyone middle school students were trained on use of the Geometer's Sketchpad (Jackiw, 1995). The control group was taught using the traditional textbook approach while the treatment group worked in a Macintosh computer laboratory. After controlling for initial differences, Dixon (1995) concluded that students experiencing the dynamic instructional environment significantly outperformed students experiencing a traditional instructional environment on content measures of rigid motion transformations, as well as on certain measures of visualization at an alpha level of 0.01. The purpose of Choi's (1996) study was to investigate secondary school students' development of geometric thought during instruction based on a van Hiele model and using dynamic computer software as a tool. In particular, the students' learning process was traced in relation to van Hiele levels of geometric thought with geometric topics using an interactive computer environment. The clinical interview procedure was used. It was evidenced that the instruction based on the van Hiele's five instructional phases was well integrated with the use of the dynamic nature of the software since all students showed extensive development of geometric thought. Also, the use of the dynamic capabilities of the geometric construction software, Geometer's Sketchpad (Jackiw, 1995), was found to provide an advantage to students because it facilitated the movement from symbol to signal and then to implicatory character. Research has also been conducted to investigate how dynamic geometry learning environments qualitatively impact psychological aspects of geometry learning. Yousef (1997) studied the effect of using Geometer's Sketchpad (Jackiw, 1994) on high school students' attitude toward geometry. The sample consisted of two groups with two classes in each group. All classes involved in the study participated in exploration activities. The exploration activities for the students in the experimental group involved the use of the Geometer's Sketchpad (Jackiw, 1995) and for the control group involved paperandpencil work only. Measurement of attitude toward geometry was conducted before and after the implementation of the experiment for both groups. Results 30 indicated that the scores of the pretest and the posttest of the students in the experimental group were significantly different and there was a significant difference between the gain in the scores of the control and experimental group. Qualitative data were collected from two sources: observation and interviews. The results supported the quantitative data. Melzcarek (1996) focused on the effects of problem solving activities using dynamic geometry computer software on readiness for selfdirected learning. Attitude towards the learning of mathematics, the use of computers, and their mediating effects were also explored. Six high school classes received the experimental treatment and visited the mathematics computer laboratory once a week for a period of six weeks to work on problemsolving activities that were designed to be used with the Geometer's Sketchpad (Jackiw, 1995). One class served as the control group. The results of this study indicated a positive relationship between the use of dynamic geometry computer software and readiness for self directed learning through the mediating effects of attitude towards dynamic geometry computer software. A taskbased clinical interview procedure was used by Manouchehri (1994) to study the cognitive actions of two preservice elementary teachers and their interactions with the Geometer's Sketchpad (Jackiw, 1995) as they worked on computerbased geometry explorations and problem solving activities. A final interview session was conducted to solicit the participants perspectives on their experiences with the computerbased activities. In the process of active engagement in exploring with the interactive computer 31 software, the participants moved from a naive intuitive mode to an analytical mode of thinking. Some difficulties resulted from subjects' lack of knowledge about how to explore, their lack of reflection on their operations and findings, and their inadequate content knowledge base. The collective impact of these difficulties led the subjects to drawing false conclusions and overgeneralizations. Both subjects appreciated being in control of their own learning. They articulated, however, that such experience was confusing and frustrating at times. Summary The review of the literature presented in this chapter commenced with a theoretical framework grounded in constructivist epistemology. According to von Glasersfeld (1995), new knowledge is constructed and reconstructed from prior knowledge within an environment of active participation as one strives to organize one's experiential world. A discussion of what repetitive experience means to research is included in addition to theories of instruction focusing on conceptual development of mathematical ideas. A review of geometric construction software, including the Geometric Supposers (Schwartz & Yerushalmy, 1985) and/or the Geometer's Sketchpad (Jackiw, 1995), revealed that it has been considerably discussed, tested, and used. The incorporation of a dynamic aspect to geometric construction software warranted further discussion. 32 The research provided a strong foundation for investigation into geometry learning within a dynamic environment. The research also uncovered the need for further examination of the impact of a dynamic geometry learning environment on geometry instruction effectiveness. Furthermore, the lack of research with preservice elementary teachers warrants investigation. It is the synthesis of the information gained from the review that provided the foundation and justification for this study. CHAPTER III RESEARCH DESIGN AND METHODOLOGY The study methodology is detailed in this chapter which commences with the research objectives. The population addressed by this study and the participants of this particular research study are described next. This is followed by a discussion of the materials and the measurement instrument to be used. Then, a summary of the pilot sessions is described. Finally, the design of the study and the procedures to be undertaken are detailed. Research Objectives This research study was designed to investigate the effect of the dynamic geometry learning environment Geometer's Sketchpad (Jackiw, 1995) on preservice K8 teachers' performance on similarity tasks. In light of the given eclectic theoretical framework and the literature review that reports the results of research on aspects of geometry learning, task performance was identified as a core category to analyze how preservice teachers respond to learning in a dynamic environment. The hypotheses for the study were generated from the questions posed earlier and an inspection of pilot study data. In order to facilitate quantitative statistical analyses, following each question is the list of hypotheses pertaining to that question used for analyses of data: 1. After controlling for initial performance on similarity tasks, do preservice K8 teachers who work on similarity tasks in a dynamic geometry learning environment exhibit significantly different posttest performance than those who study the concept of similarity in a paperandpencil learning environment that employs traditional tools such as ruler and protractor? la. Hypothesis: There is no significant effect on posttest scores due to the interaction of treatment, course section, and pretest scores, using pretest scores as a covariate. lb. Hypothesis: There is no significant effect on posttest scores due to the interaction of course section and pretest scores, using pretest scores as a covariate. ic. Hypoothesis: There is no significant effect on posttest scores due to the interaction of treatment and pretest scores, using pretest scores as a covariate. id. Hypothesis: There is no significant effect on posttest scores due to the interaction of treatment and course section, using pretest scores as a covariate. le. Hypothesis: There is no significant effect on posttest scores for students grouped by course section, using pretest scores as a covariate. if. Hypothesis: There is no significant difference on posttest scores for students grouped by treatment, using pretest scores as a covariate. 2. After controlling for prior knowledge as measured by a standardized achievement test, do preservice K8 teachers who work on similarity tasks in a dynamic geometry learning environment exhibit significantly different posttest performance than those who study the concept of similarity in a paperandpencil learning environment that employs traditional tools such as ruler and protractor? 2a. Hypothesis: There is no significant effect on posttest scores due to the interaction of treatment, course section, and SAT scores, using SAT scores as a covariate. 2b. Hypothesis: There is no significant effect on posttest scores due to the interaction of course section and SAT scores, using SAT scores as a covariate. 2c. Hypothesis: There is no significant effect on posttest scores due to the interaction of treatment and SAT scores, using SAT scores as a covariate. 2d. Hypothesis: There is no significant effect on posttest scores due to the interaction of treatment and course section, using SAT scores as a covariate. 2e. Hypothesis: There is no significant effect on posttest scores for students grouped by course section, using SAT scores as a covariate. 2f. Hypothesis: There is no significant effect on posttest scores for students grouped by treatment, using SAT scores as a covariate. Population and Sample The research population consisted of preservice elementary education teachers attending an American university. The 52 participants of this study were College of Education students enrolled in three sections of a mathematics methods course during a spring semester. This course, addressing methods for teaching elementary school mathematics, is a requirement of the teacher preparation program in elementary education at a major landgrant university located in the Southeastern United States. The participants were randomly assigned to two groups within each section. One group worked in the dynamic geometry learning environment; another group worked in the traditional paper and pencil manner (refer to Definition of Terms). Instructional Materials The experientialactivity orientation of the Middle Grades Mathematics Project [MGMP] was used with both experimental and control groups. These materials follow recommendations of the Piagetian research on geometrical development and the instructional implications of the van Hiele studies (Friedlander, 1984). Each activity utilizes an instructional model consisting of three stages: (1) launch introduces new concepts, clarifies definitions, reviews old concepts, and issues some challenge; (2) exploration entails gathering data, looking for patterns, making conjectures, or developing other types of problem solving strategies; (3) summary demonstrates ways to organize data, discusses the used strategies, and refines these strategies into efficient problem solving techniques. The launch and summary stages are conducted in a wholeclass mode to purposefully generate takenasshared knowledge (Cobb, et al., 1992); while at the exploration phase, individuals investigate the problem situation and follow their own path of cognition in analysis of the given task. The MGMP material presents the concept of similarity at the van Hiele Level I, properties of shapes, and Level II, relationships among properties of shapes, of geometric development (Friedlander, 1984). Consideration of the Van Hiele levels in the teaching of geometry in the United States facilitates progress in the learning and understanding of geometrical concepts (Burger & Shaughnessy, 1986; Carpenter et al., 1981; Clements & Battista, 1992; Crowley, 1987; Fuys, Geddes, & Tischler, 1988; Jaime & Gutierrez,1995). Pilot Studies In May, 1997, a pilot study was conducted using undergraduate students as defined by the population sample. The students participated in four ninetyminute sessions utilizing the Geometer's Sketchpad (Jackiw, 1995), a dynamic geometry software package. The first session was simply an introduction to resources made available to students enrolled in the class and did not constitute part of the treatment. Table 3.1 in Appendix B summarizes the academic content of the learning sessions conducted by the researcher. As a result of the initial pilot, adjustments were made regarding the structure of the study. The length of each session was limited to a fiftyminute session and the number of learning sessions decreased. Participants considered the sessions too lengthy and the researcher noted some students were off task after approximately one hour. In addition, the "launch" phase for each learning session was viewed by participants as too indepth and in need of revision for use with preservice teachers. Although appropriate for middle school students, adjustments were made to reflect the curricular needs of college students. A second pilot study was conducted during the Fall of 1997. Undergraduate students enrolled in the mandatory introductory mathematics methods course for the preservice teacher education program participated in three fiftyminute learning sessions according to random placement in one of two groups as described above. Again, adjustments were made regarding the structure of the study. Content was further parametrized to linear measurement, therefore eliminating introduction of the tangential topic of area growth. As a consequence of the change in curriculum, the research instrument was appropriately altered, with items addressing area growth omitted. Additionally, items hypothesized to address learning attributed to the dynamic component of the geometric construction software were developed and placed into the final version of the research instrument. Table 3.2 in Appendix B summarizes the academic content of the learning sessions conducted by this researcher. Instrumentation Performance on the concept of similarity was measured by an instrument developed by this researcher and based, with permission, on items used by the Middle Grades Mathematics Project [MGMP] (1986). The instrument, given in Appendix A, consisted of 21 multiple choice items with up to five options for each item. Items were scored by assigning a 1 for a correct response and a 0 otherwise; no correction is made for guessing. The total score is considered as a general indicator of the level of task performance. Content validity and reliability were considered. To insure that the instrument reflected the content domain of transformational geometry regarding the concept of similarity, a panel of five experts analyzed the twentyone items to ascertain validity. The experts, three mathematics educators and two research mathematicians, reviewed each problem with consideration to mathematical content. The instrument was fieldtested in Spring, 1998, on sixtysix preservice elementary teachers enrolled in two sections the required elementary mathematics methods course at an American university located in the southeast sector. Measuring internal consistency of the test, the Cronbach reliability coefficient calculated from the data is 0.70. The coefficient of stability, calculated for testretest reliability, is 0.70. Regarding the covariates used in this study, multiple partial correlations were computed to assess to the strength of association between the dependent variable and the covariate while controlling for the other independent variables treatment and section. The research instrument, administered as the pretest, served as the covariate to statistically analyze the hypotheses related to the first question. For the second question, prior mathematical knowledge, as measured by the mathematics section of 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 for this study and, concerning the covariate pretest, 0.5925 was calculated. Thus, statistical analyses supported confidence for the utilization of the specified variables. Procedure and Data Collection The procedure consisted of a pretest, three learning sessions, and a posttest. The pretest was administered eight weeks prior to the learning sessions. The sessions for the experimental group were held in a computer laboratory on campus and involved use of the dynamic geometry software Geometer's Sketchpad (Jackiw, 1995); sessions for the control group were held in the traditional mathematics laboratory and involved use of manipulatives, protractor, and ruler. Written reflections were collected from all participants at the end of each learning session in order to gain further insight regarding the depth and breadth of the learning that had occurred. During the posttest, administered immediately upon completion of the research study learning sessions, participants were given access to whichever tools they had used during the course of the research study. All components of the study were administered by this researcher during scheduled meeting times. The researcher conducted all lessons to eliminate differences in results due to instructor variability, but was aware of the possibility that a threat to validity may be introduced by the researcher influencing results. This was minimized by the use of specific lesson plans including printed activities for the participants of both groups. The contents of the learning 41 sessions are given in Table 3.3 of Appendix B with exemplary printed activities, showing student work, presented in Appendix C. Prior to the study the Institutional Review Board of the given University granted permission for the investigation to take place. Students were informed; both researcher as principal investigator and those who agreed to participate in the study signed the consent form. An addendum was proposed and permission granted to access participants SAT scores. In a similar manner, students were informed and the parties involved gave written approval. The letters of informed consent are given in Appendix D. Statistical Design of the Study The effect of environmental intervention was examined by use of a pretestposttest controlgroup design with random assignment. The Borg and Gall (1989) configuration is shown as R 0 X 0 RO 0 0 where X represents the experimental treatment, 0 represents pretest or posttest measurement of the dependent variable, and R indicates that the experimental and control groups were formed randomly. The randomization procedure occurred within each course section. A literature review of studies focusing on aspects of geometry learning suggested use of several statistical procedures (Dixon, 1996; Fleener, et al, 1993; Friedlander, 1984; Lester, 1996; McCoy, 1991; Melzcarek, 1996; Robinson, 1994; Yousef, 1997). To analyze the data collected during the study, descriptive statistics were obtained for all variables under consideration and analysis of covariance [ANCOVA] was conducted. The ANCOVA allows one to test for interactions; in particular, the interaction of the treatment and course sections is of particular interest to this study. Furthermore, ANCOVA does adjust for random differences in the two groups. In addition to quantitative statistical analyses, reflections collected in written form were examined to further elucidate the geometry learning experience. Summary This chapter commenced with the research objectives. Information regarding the population, research sample, and materials followed. Next, the instrument used in this particular study was described and the pilots were ,presented. This chapter ended with the design of the study which included a description of the procedures utilized in the study. CHAPTER IV RESULTS This chapter contains the results of the analyses described in the previous chapter. Descriptive statistics for the variables under investigation and ANCOVAs are presented along with the results of the qualitative component detailing the geometry learning experience. The chapter begins with descriptions of the research instrument, research objectives, and the goals of the study. Next, descriptive statistics are presented and the research hypotheses that reflect the research questions are articulated. Then, the analyses of the data are given in light of the hypotheses. Finally, an exposition of written reflections is presented to elucidate the geometry learning experience. The Research Instrument The research instrument measured performance on similarity tasks in a multiple choice format with up to five options for each of the twentyone items. Appendix A contains the research instrument. Items were scored by assigning a 1 for a correct response and a 0 otherwise; no correction was made for guessing. The total score was considered as a general indicator of the level of task performance. Research Objectives To test for significant difference in task performance between participants grouped by learning environment, questions were developed to explore the effect of a dynamic geometry learning environment on preservice elementary teachers' performance on similarity tasks. Hypotheses were formulated to identify significant interactions between various independent variables and to adjust for initial performance on similarity tasks as measured by the pretest or by prior mathematics knowledge as measured by a standardized achievement test. The comparison of scores on the research instrument that assessed performance on similarity tasks, administered prior to and at the end of the study, served as the basis for the analysis of the first question. The second question focused upon the effect of a dynamic geometry learning environment on preservice elementary teachers' performance on similarity tasks with prior mathematics performance adjusted for by a standardized achievement test. Hypotheses pertaining to each question allowed for statistical testing. The results of the analyses are reported at an alpha level of 0.05 in all cases. Goals of the Study Based upon the assumptions that teachers need to learn high quality mathematics and to be taught in a manner similar to the way they are expected to teach, two questions of interest were generated for this particular study, focusing upon the effect of the dynamic geometry learning environment on preservice elementary teachers' performance on similarity tasks as measured by the research instrument: 1. After controlling for initial performance on similarity tasks, do preservice elementary teachers who work on similarity tasks in a dynamic geometry learning environment exhibit significantly different posttest performance than those who study the concept of similarity in a paperandpencil learning environment that employs traditional tools such as ruler and protractor? 2. After controlling for prior knowledge as measured by a standardized achievement test, do preservice elementary teachers who work on similarity tasks in a dynamic geometry learning environment exhibit significantly different posttest performance than those who study the concept of similarity in a paper andpencil learning environment that employs traditional tools such as ruler and protractor? Thus, the goals of the study included answering the research questions in light of the given assumptions. Descriptive Statistics For the sake of completeness, the data set for the sample population is given in Appendix E, sorted first by treatment and then by gain score. This inclusion is to afford the reader the option to compare the relative performance of the participants on the posttest. Descriptive statistics for responses to the pretest, SAT mathematics component, and posttest for the entire sample, by group, and by group and course section, are presented in Tables 1, 2, and 3, respectively. The sample size, mean, and standard deviation are listed for each response category. The posttest mean 46 scores, adjusted for the covariate, were computed as 14.3699 for the control group and 16.2568 for the experimental group. Table 1 Descriptive Statistics / Combined Number Mean Standard Deviation Pretest 52 12.2885 3.4376 SAT 48 568.3333 80.2213 Posttest 52 15.5769 3.0120 Table 2 Descriptive Statistics / by Group Group Number Mean Standard Deviation Pretest control 26 12.2308 3.2287 experimental 26 12.3462 3.6980 SAT control 24 580.0000 73.0098 experimental 24 556.6667 86.8115 Posttest control 26 14.6538 2.6542 experimental 26 16.5000 3.1145 Research Hypotheses The following hypotheses were generated to test for significant effect in mathematics achievement regarding preservice teachers' performance on similarity tasks as measured by the research instrument. For the first research Table 3 Descriptive Statistics / by Group and Section Standard Section Group Number Mean Deviation Pretest SAT Posttest a control experimental b control experimental m control experimental a control experimental b control experimental m control experimental a control experimental b control experimental m control experimental 10.0000 10.8000 13.8182 13.1667 12.0000 12.1111 585.0000 556.0000 597.0000 563.6363 555.0000 547.5000 13.7143 14.0000 15.0909 17.2500 14.8750 16.4444 question, initial differences in mathematics achievement were controlled for through the use of the research instrument administered as a pretest. The latter set of hypotheses addressed the second question and controlled for initial differences in mathematics achievement via the mathematics component of the standardized achievement test SAT. la. Hypothesis: There is no significant effect on posttest scores due to the interaction of treatment, course section, and pretest scores, using pretest scores as the covariate. 3.7417 4.3818 2.5226 3.7376 2.6726 3.3706 55.0454 99.9000 75.1369 87.4383 83.3238 89.2428 1.4960 3.5637 2.4680 2.7010 3.6425 3.3582 lb. Hvypothesis: There is no significant effect on posttest scores due to the interaction of course section and pretest scores, using pretest scores as the covariate. ic. Hypothesis: There is no significant effect on posttest scores due to the interaction of treatment and pretest scores, using pretest scores as the covariate. id. Hypothesis: There is no significant effect on posttest scores due to the interaction of treatment and course section, using pretest scores as the covariate. le. Hypothesis: There is no significant effect on posttest scores for students grouped by course section, using pretest scores as the covariate. if. Hypothesis: There is no significant effect on posttest scores for students grouped by treatment, using pretest scores as the covariate. 2a. Hypothesis: There is no significant effect on posttest scores due to interaction effect of treatment, course section, and SAT scores, using SAT scores as the covariate. 2b. Hvypothesis: There is no significant effect on posttest scores due to the interaction of course section and SAT scores, using SAT scores as the covariate. 2c. Hypothesis: There is no significant effect on posttest scores due to the interaction of treatment and SAT scores, using SAT scores as the covariate. 2d. Hypothesis: There is no significant effect on posttest scores due to the interaction of treatment and course section, using SAT scores as the covariate. 2e. Hypothesis: There is no significant effect on posttest scores for students grouped by course section, using SAT scores as the covariate. 2f. Hypothesis: There is no significant effect on posttest scores for students grouped by treatment, using SAT scores as the covariate. Results of Data Analyses The initial analysis of covariance, controlling for pretest performance, yielded a pvalue greater than the set alpha level of p=0.05 and the null hypothesis was accepted (Refer to Table 4). That is, no significant effect on posttest scores due to the interaction of treatment (trt), course section (scn) and pretest scores (pre), after controlling for pretest performance, was found (p=0.1017). Therefore, the following null hypothesis could not be rejected: Using pretest scores as the covariate, there is no significant effect on posttest scores due to the interaction of treatment, course section and pretest scores. Finding no significant threeway interaction, the three way interaction term was removed from the model. After removal of the threeway interaction term, hypotheses lbld were then tested. There were no statistically significant twoway interactions based on the results of the ANCOVAs. The pvalues for hypotheses Ibld were 0.2939, 0.5337, and 0.7871, respectively. Therefore, the following null hypotheses could not be rejected: Using pretest scores as the covariate, there is no significant effect on posttest scores due to the interaction of course section and pretest scores. Using pretest scores as the covariate, there is no significant effect on posttest scores due to the interaction of treatment and pretest scores. Using pretest scores as the covariate, there is no significant effect on posttest scores due to the interaction of treatment and course section. Table 4 Initial Analysis Of Covariance with Pretest as Covariate Dependent Variable: Posttest Source DF SS MS F Value Pr > F trt 1 0.3639 0.3639 0.07 0.7935 pre 1 134.4845 134.4845 25.66 0.0001 scn 2 19.1402 9.5701 1.83 0.1742 pre*trt 1 0.7196 0.7196 0.14 0.7129 trt*scn 2 24.0913 12.0457 2.30 0.1135 pre*scn 2 23.4062 11.7031 2.23 0.1204 pre*trt*scn 2 25.3796 12.6898 2.42 0.1017 Upon finding no significant twoway interactions, the interaction terms trt*pre and pre*scn were removed from the model. The term trt*scn was left in the model to guarantee control of the blocking variable course section. The ANCOVA information is presented in Table 5. After removal of the interaction terms, course section showed no statistical significance (p=0.8332) but treatment was statistically significant at the alpha level of 0.05 (p=0.0275). Thus, the following null hypothesis to the initial research question could not be rejected: Using pretest scores as the covariate, there is no significant effect on posttest scores for students grouped by course section. However, the following null hypothesis could be rejected: Using pretest scores as the covariate, there is no significant effect on posttest scores for students grouped by treatment. The analyses relevant to the first question indicated a statistically significant difference in performance on similarity tasks between the experimental group and the Table 5 Analysis Of Covariance Final Model with Pretest as Covariate Dependent Variable: Posttest Source DF SS MS F Value Pr > F trt 1 29.0884 29.0884 15.19 0.0275* pre 1 136.3718 136.3718 24.23 0.0001 scn 2 2.0534 1.0262 0.18 0.8332 trt*scn 2 6.7176 3.3588 0.60 0.5534 *statistically significant at alpha level 0.05 control group. LSMeans, with respect to use of the pretest as covariate, was calculated at 14.74 for the control group and 16.30 for the experimental group. The remaining hypotheses, addressing the second question, were then analyzed. Again, no significant difference on posttest scores due to interaction effect of treatment (trt), course section (scn) and SAT scores (sat) was found (p=0.7926) after controlling for prior mathematics performance. That is, the analysis of covariance, controlling for prior mathematics performance, yielded a p value greater than the set alpha level of p<0.05 and the null hypothesis was accepted (Refer to Table 6). Therefore, the following null hypothesis could not be rejected: Using SAT scores as the covariate, there is no significant effect on posttest scores due to the interaction effect of treatment, course section and SAT scores. The threeway interaction term was removed from the model upon finding no significant threeway interaction. After removal of the threeway interaction term, hypotheses 2b2d were then tested. There were no statistically Table 6 Initial Analysis Of Covariance with SAT as Covariate Dependent Variable: Posttest Source DF SS MS F Value Pr > F trt 1 1.5549 1.5549 0.19 0.6681 sat 1 62.8058 62.8058 7.59 0.0091 scn 2 0.9755 0.4877 0.06 0.9428 sat*scn 1 0.2838 0.1419 0.02 0.9830 trt*scn 2 3.2057 1.6028 0.19 0.8247 sat*trt 2 0.2638 0.2638 0.03 0.8592 sat*trt*scn 2 3.8684 1.9342 0.23 0.7926 significant twoway interactions based on the results of the ANCOVAs. The pvalues for hypotheses 2b2d were 0.9550, 0.7317, and 0.8297, respectively. Therefore, the following null hypotheses could not be rejected: Using SAT scores as the covariate, there is no significant effect on posttest scores due to the interaction of course section and SAT scores. Using SAT scores as the covariate, there is no significant effect on posttest scores due to the interaction of treatment and SAT scores. Using SAT scores as the covariate, there is no significant effect on posttest scores due to the interaction effect of treatment and course section. The interaction terms trt*sat and sat*scn were removed from the model upon finding no significant twoway interactions. The term trt*scn was left in the model to guarantee control of the blocking variable course section. After removal of the interaction terms, course section continued to show no statistical significance (p=0.2134). However, after controlling for initial differences on SAT mathematics scores, a statistically significant difference in means existed at the alpha level of 0.05 between treatment groups (p=0.0271). The ANCOVA information is given in Table 7. LSMeans, with respect to use of the mathematics component of the SAT as covariate, was calculated at 14.37 for the control group and 16.26 for the experimental group. Table 7 Analysis Of Covariance Final Model with SAT as Covariate Dependent Variable: Posttest Source DF SS MS F Value Pr > F trt 1 38.8572 38.8572 5.26 0.0 sat 1 79.2114 79.2114 10.71 0.0 scn 2 23.7177 11.8588 1.60 0.2 trt*scn 2 2.9500 1.4750 0.20 0.8 *statistically significant at alpha level 0.05 Thus, the following null hypothesis to the second research question could not be rejected: Using SAT scores as the covariate, there is no significant effect on posttest scores for students grouped by course section. However, the following null hypothesis could be rejected: Using SAT scores as the covariate, there is no significant effect on posttest scores for students grouped by treatment. 271* 022 134 199 h A Reflections Component Written reflections on the learning experience, collected from the participants at the end of each learning session, afforded the researcher valuable insights into the preservice elementary teachers' perception of what they had learned and the value of this knowledge in their teaching. During each session, the 52 participants logged responses on activity worksheets that prompted for description of the thought processes. After each session, participants were verbally asked to write what they had learned mathematically, their reaction to the methodology, and their views as to use in the mathematics classroom. The data gathered from responses on the activity worksheets used in this study were collected to illuminate the cognitive process occurring within each individual during active participation. Examples are presented in Appendix C. The purpose of the collecting of reflections was to gain further insight by providing opportunities for a richer story to emerge and for reflective meaningmaking to take place. Critiquing of the written reflections granted the researcher an opportunity to gain knowledge regarding the preservice elementary teachers' perspective of the experience. The participants shared their opinions and knowledge about the mathematics content, the instruction, and the technology, in addition to the learning. Affect, as well as performance, seemed to have improved due to the correspondence from the preservice teacher to the researcher. The following excerpts, taken from the written reflections collected at the end of each session during the course of the research study, exemplify the effect of the dynamic geometry learning environment upon cognitive development. Both treatment groups participated and are represented. Experimental group preservice elementary teacher #1: Day 1: "Today I learned it is easier to see this idea than if I had to draw inaccurate polygons each time to compare them. I also learned how the buttons work and how to make measurements and calculations. I would like to know how to make calculations such as a2+b2=c2 to prove what I am doing on the computer. Very cool program!" Day 2: "I liked today's activity because the student is guided through a process but the discovery is left to them! It asks the right questions. It really provides the basis for pregeometry study. I don't know if the student will walk away from this exercise with a grasp on proportion geometry, but at least it gets them thinking in the right mode." Control group preservice elementary teacher #2: Day 1: "We learned about projective geometry mostly through the use of an instrument made up of a pencil and rubber bands. The activity was unique to say the least! The most important thing that I learned from this activity was the value of practice. I was able to participate in the activity as a novice. I had no idea what I was doing and I learned everything as I went along. As a future teacher, this is a valuable lesson and will help me better understand my students' learning." Experimental group preservice elementary teacher #3: Day 1: "Today I did not LEARN a whole lot. I did, however, discover quite a bit!! I did not have enough time, nor enough experience to feel as if I learned. Perhaps further practice will allow me to learn next time!" 56 Day 2: "Today I became comfortable with the commands on the computer!! I think because this is such a new concept, I am enjoying 'playing around' with it! Again, I AM fairly comfortable w/ dilation, rotation and transformation as we are learning this in geometry class now!" Day 3: "I said I was not learning because I already know this stuff. But I said I discovered because I could make a conjecture and immediately see the outcome. I'm not very good on computers but I see that this tool will be great for use w/ my kids. Being able to see dimensions change is awesome. Gives one the big picturequickly and accurately." Experimental group preservice elementary teacher #4: Day 3: "I learned a lot because you gave us a different way to look at the concept. I could pull together old stuff and ideas and make connections. The dynamics made a difference, like the old way cartoons were made, it made things possible to see in a short amount of time." Control group preservice elementary teacher #5: Day 3: "I learned I could fail a test and then pass it with a bit of practice. It was neat to see that the first test actually had some background to it. When I first took it I thought it really was a bunch of madeup information. After the couple of days, I felt like I could ace the test." The reflections portray the preservice elementary teachers' perspective regarding the learning experience. It was interesting to note that, in both groups, many 57 participants regarded the active handson approach a form of practice that was enjoyable in addition to being productive. Furthermore, connections were established between previous knowledge and new ideas so that insightful revelations (ah ha!) occurred; this was more often evidenced during learning sessions with the experimental group. For example, teacher #3 quickly mastered the basic skills necessary to explore Euclidean constructions and was quite enthusiastic regarding his explorations. He would "conjecture and immediately see the outcome," evaluate the situation, and repeat the cycle. Regarding affective aspects, teacher #5 was quite skeptical regarding participation in mathematical activity. As a prospective teacher, she believed the course requirement a waste of time since she intended to teach only in the primary grades. However, by the end of the study (near the end of the semester) her participation had increased dramatically; teacher #5 had constructed a firm foundation of elementary mathematics that was deemed personally relevant. In particular, she had developed an informal, intuitive understanding of the concept of mathematical similarity by way of a novel approach. Then, teacher #5 reconstructed her knowledge base regarding the concept to conclude that she had become mathematically empowered! Of the participants in the experimental group, most became comfortable with the software during the launch phase when participants were given time to experiment with the tool. Teacher #1 expressed great interest in the capabilities of the Geometer's Sketchpad (Jackiw, 1995), asking for assistance to develop a geometric proof of the Pythagorean Theorem. Able to transcend physical limitations of manual dexterity, she explored the capabilities of the dynamic geometry software in order to verify the theorem. The experimental group participants, for the most part, were quite impressed by the dynamic capability of the tool. The participants in the control group, exhibiting enthusiasm in a similar manner, reported plans to share manipulatives and techniques with their future students and colleagues. Summary Results of the statistical analyses for the study have been described in detail in this chapter in addition to presentation of a sampling of collected reflections that exemplify certain perceptions and themes held by the research participants. Discussion of the findings, including implications and recommendations for further research, are presented in the concluding chapter. CHAPTER V SUMMARY AND CONCLUSIONS This final chapter discusses the results presented in the previous chapter. An overview of the study is provided first, including a description of the sample population. Next, the significance of the study is articulated, followed by the research questions. Finally, the results and a discussion of the conclusions based upon the research results is presented. Implications from the present study are given next. The chapter concludes with recommendations for possible future research. Overview of the Study The purpose of the study was to explore the role of the learning environment with respect to the mathematical evolution of preservice elementary teachers as a result of active inquiry using the dynamic geometric construction software Geometer's Sketchpad (Jackiw, 1995). That is, this study was designed to investigate the effect of a dynamic geometry learning environment on performance in the context of similarity tasks. The basic goal of the study was to detect the extent to which the dynamic environment, given its ability to efficiently display a plethora of mathematical situations, assists learning. 60 Complete data were available on 52 preservice elementary teachers enrolled in three sections of an undergraduate elementary mathematics methods course in the College of Education at a major landgrant university located in the Southeastern United States. The design of the study was a pretestposttest control group design with random assignment. Detail on the design, procedures, and methodology were presented in Chapter III. The randomization procedure occurred within each course section. The procedure consisted of the pretest, three learning sessions, and the posttest. Sessions for the control group were held in the traditional mathematics laboratory and involved use of manipulatives, ruler, and protractor; the sessions for the experimental group were held in a computer laboratory on campus and involved use of the dynamic geometric construction software Geometer's Sketchpad (Jackiw, 1995). To analyze the data collected during the study, descriptive statistics were obtained for all variables under consideration and analysis of covariance [ANCOVA] was conducted. The research instrument, given to all participants eight weeks prior to the instructional sessions, served as the covariate to statistically analyze the hypotheses related to the first question. For the second question, prior mathematical knowledge, as measured by a standardized achievement test, served to adjust for initial differences in the two treatment groups. In addition to the quantitative statistical component, written reflections were collected from all participants at the end of each session. The purpose of this exercise was to gain further insight regarding the preservice elementary teachers' perspective of the instructional environment and to ascertain what had been learned during the session. Significance of the Study Implementing new technologies into the mathematics curriculum is not easy and the education of teachers plays a critical role. The necessity of requiring preservice teachers to learn mathematics content and methods of teaching in an environment where computer use is an integral component is advocated by many mathematics teacher educators (Committee on the Mathematical Education of Teachers, 1991; Curcio, Perez, & Stewart, 1994; Leitzel, 1991; McNerney, 1994; National Council of Teachers of Mathematics, 1991) It is, therefore, imperative to explore how software will modify current ways of teaching and learning in order to define new roles for teachers in a technologically rich classroom environment. Research Questions Based on the assumptions that teachers need to learn high quality mathematics and to be taught in a manner similar to the way they are expected to teach, the specific questions under investigation were: 1. After controlling for initial performance on similarity tasks, do preservice elementary teachers who work on similarity tasks in a dynamic geometry learning environment exhibit significantly different posttest performance than those who study the concept of similarity in a paperandpencil learning environment that employs traditional tools such as ruler and protractor? 2. After controlling for prior knowledge as measured by a standardized achievement test, do preservice elementary teachers who work on similarity tasks in a dynamic geometry learning environment exhibit significantly different posttest performance than those who study the concept of similarity in a paperandpencil learning environment that employs traditional tools such as ruler and protractor? Results The hypotheses that were statistically tested, with consideration of the aforementioned two questions, are listed below. Following each set of hypotheses, the results of the analysis are presented for consideration. la. Hypothesis: There is no significant effect on posttest scores due to the interaction of treatment, course section, and pretest scores, using pretest scores as a covariate. lb. Hypothesis: There is no significant effect on posttest scores due to the interaction of course section and pretest scores, using pretest scores as a covariate. ic. Hypothesis: There is no significant effect on posttest scores due to the interaction of treatment and pretest scores, using pretest scores as a covariate. Id. Hypothesis: There is no significant effect on posttest scores due to the interaction of treatment and course section, using pretest scores as a covariate. le. Hypothesis: There is no significant effect on posttest scores for students grouped by course section, using pretest scores as a covariate. if. Hvpothesis: There is no significant effect on posttest scores for students grouped by treatment, using pretest scores as a covariate. To investigate whether the dynamic geometry learning environment affects performance on similarity tasks as measured by the research instrument, ANCOVAs were conducted with achievement on the research instrument as a covariate. Initially, ANCOVAs were applied to assure that interaction effects were not present and to develop the final model with the following terms as independent variables: pretest, treatment, course section, and the interaction of treatment with course section. The term treatment with course section was left in the model to guarantee control of the blocking variable course section. The initial analyses indicated that there was no statistically significant effect on posttest scores due to either threeway or twoway interaction of the independent variables. Therefore, with the exclusion of the interaction term of treatment with course section, the interaction terms were removed. An ANCOVA was conducted using the final model. Based on the analysis, there exists supportive evidence that the difference in learning environment between the two treatment groups had a significant effect upon the preservice elementary teachers' performance on similarity tasks as measured by the research instrument. Fundamentally, the participants in the experimental group outperformed the participants in the control group even though initial variability was taken into consideration. Let us now focus upon the hypotheses generated to address the latter question. 2a. Hypothesis: There is no significant effect on posttest scores due to the interaction of treatment, course section, and SAT scores, using SAT scores as a covariate. 2b. Hypothesis: There is no significant effect on posttest scores due to the interaction of course section and SAT scores, using SAT scores as a covariate. 2c. Hypothesis: There is no significant effect on posttest scores due to the interaction of treatment and SAT scores, using SAT scores as a covariate. 2d. Hypothesis: There is no significant effect on posttest scores due to the interaction of treatment and course section, using SAT scores as a covariate. 2e. Hypothesis: There is no significant effect on posttest scores for students grouped by course section, using SAT scores as a covariate. 2f. Hypothesis: There is no significant effect on posttest scores for students grouped by treatment, using SAT scores as a covariate. In a similar fashion, to further investigate whether the dynamic geometry learning environment affects performance on similarity tasks as measured by the research instrument, ANCOVAs were again conducted. For this set of hypotheses, prior knowledge, as measured by SAT mathematics scores, served as the covariate. Once more, ANCOVAs were applied to assure that interaction effects were not present and to develop the final model. There was no statistically significant effect on posttest scores due to either threeway or twoway interaction of the independent variables. With the following independent variables as terms: SAT, treatment, course section, and the interaction of treatment with course section, a final ANCOVA was conducted. It was concluded that the participants in the experimental group outperformed the participants in the control group when consideration was given to prior knowledge as measured by SAT mathematics scores. This study focused on the effect of a dynamic geometry learning environment on preservice elementary teachers' performance on similarity tasks as measured by the research instrument. One can be fairly confident that this effect is attributable to the dynamic learning environment for three reasons: classes were randomly assigned to the experimental and control groups, the observed differences between groups on posttest performance were statistically significant, and extraneous variables were effectively controlled. Discussion of the Results Constructivism fully agrees with Kuhn's idea of paradigm shifts (Kuhn, 1970), interpreting paradigm shifts as "change in the habitual way of constructing." The more often a construct is repeated and the greater the number of larger structures in which it is involved, the more indispensable it becomes and the more "given" or "objective" it seems (von Glasersfeld, 1979, p. 121). 66 The researcher concluded from the data analyses that the results support the dynamic geometry learning environment as a viable enhancement to the traditional mathematics laboratory approach. This study focused on the impact of the dynamic geometric construction software Geometer's Sketchpad (Jackiw, 1995) upon preservice elementary teachers' performance on similarity tasks. Human beings acquire knowledge by building it from the inside instead of internalizing it directly from the environment (Kamii, 1990). Thus, current research from a constructivist perspective characterizes mathematical learning as a process of conceptual reorganization (Cobb, 1995). If learning is generated by successive interactions with one's environment, then repetitive activity is necessary to provide information that facilitates problem solving (Cooper, 1991). Polya (1954, 1957, 1965) advised, when faced with a problem, to look at examples and special cases, to look at analogous situations for hints and, whenever possible, to use specialization and generalization. Polya (1965) claimed that problem solving is a practical art, like playing the piano. Such arts can only be learned by imitation and practice. This type of software allows for direct construction, manipulation, and measurement of geometric figures. Users are actively involved in the learning process; the software supports efficiency in that it allows for dynamic manipulation of the geometric figures in order to analyze more cases in a minimal amount of time spent. As in Lester's (1996) and Yerushalmy's and Chazan's (1993) investigations of high school students, results from the study indicated participants learned with greater efficiency as a consequence of constructing geometric figures on the computer screen. Graphic support seemed to enhance participants ability to visualize mathematical relationships and connect geometric properties to facilitate inquiry. Calculation of posttest group means presented the researcher with additional statistics worthy of consideration. It had been previously mentioned in Chapter III that the research instrument (refer to Appendix A) was adjusted in lieu of findings from the pilot studies. Items hypothesized to address learning attributed to the dynamic component of the geometric construction software were developed, reviewed by a panel of experts, and placed into the final version of the research instrument. Thus, it was deemed appropriate to investigate individual items. A most interesting phenomena was that posttest means on parallel items, items #1 and #12, hypothesized to address learning attributed to the dynamic component of the geometric construction software, were at least 0.25 points greater for the experimental group. This study suggests that preservice elementary teachers are able to progress quite well with the use of the Geometer's Sketchpad (Jackiw, 1995) through the guess investigateconjectureverify process inherent to the formalization of the "phases of learning" (Karplus, 1977; Simon, 1992) and inquiry that are supported by constructivist epistemology. Polya (1965) summarizes that "for efficient learning, an exploratory phase should proceed the phase of verbalization and concept formation and, eventually, the material learned should be merged in, and contribute to, the integral mental attitude of the learner" (p. 104). Other Observations In addition to the rightful inspection into quantitative aspects of this study, consideration must also be given to qualitative aspects intrinsically rich in interpretive value. The activity worksheets (refer to Appendix C) 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 either group could be observed struggling with the novel approach to understanding a concept many believed they had learned in middle school. Participants in the control group slowly drew similar geometric figures using the Variable Tension Proportional Divider (Friedlander & Lappan, 1987) to develop an intuitive notion of similarity whereas the participants in the experimental group used the dynamic capability of the Geometer's Sketchpad (Jackiw, 1995) to investigate basic principles of similar figures. By the end of the first session, the experimental group had measured and analyzed many different polygonal figures while the control group, although they had discovered the principles of similarity, were unsure of whether nonconvex polygonal figures would observe the principles. The control group did, however, 69 convey an enthusiasm to use the Variable Tension Proportional Divider (Friedlander & Lappan, 1987) with students as often as the experimental group voiced opinions regarding the usefulness of the Geometer's Sketchpad (Jackiw, 1995) as a classroom tool. Much information can be gained through analysis of activity worksheets and clinical reflections. The qualitative component of this study offers a mere sampling of the preservice elementary teachers' cognitive processes in action. Limitations This study had limitations that affect the generalizability of the results. Since the study lasted over a period of several days, variability of attendance was a problem. Absences were taken into account for the testing days, but adjustment could not be made to compensate for instructional time lost. Concern was present that the small sample size of 52 preservice elementary teachers would have an effect upon the outcome of this research study. It is a fact that statistical power increases automatically with sample size. Furthermore, the larger the sample, the more likely its mean and standard deviation is representative of the population mean and standard deviation. Although statistical analysis supported no statistically significant effect, course section variability was possible in the amount of orientation the students had regarding software use prior to the study. Specifically, the researcher planned a similar amount of exposure to the software for all involved in the study, but there was no documentation of utilization time. The researcher conducted all lessons to eliminate differences in results due to instructor variability, but was aware of the possibility that a threat to validity may be introduced by the researcher influencing results. This was minimized by the use of specific presentation plans and formalized activity worksheets. In summary, the results of this study are generalizable only to situations involving similar participants, classroom environment, and classes of variables. Implications and Recommendations Our society has advanced from the industrial age into the information age where numeracy and technological literacy are important aspects of effective citizenship. The mathematics classroom must evolve from a "chalk and talk" environment into an "active" conceptuallyoriented habitat that models the contemporary workplace. Classroom teaching is undergoing great change with innovative dynamic software available. A major challenge facing mathematics and mathematics education research is the development of an indepth understanding of the function of the power technology provides, not only to revise the mathematics that we teach, but also to transform our understanding of the teaching and learning processes. To ensure proper utilization, future teachers must be trained appropriately in order to bring to the classroom the experience, confidence, and enthusiasm necessary to effectively facilitate student learning. This study added to the body of knowledge that already exists on technology and its impact upon curriculum and instruction. However, replication is needed before any definitive conclusions can be drawn about the effect of a dynamic geometric construction environment upon learning. Further investigations can build upon this research and continue to expand the knowledge base for educational practices in a technologically rich environment. Future research should strive for inclusion of other factors relevant to effective learning. The impact of spatial factors, such as visualization or orientation, warrant investigation. This is particularly relevant in light of the mathematical content addressed. Furthermore, affective factors should be given consideration in order to suggest direction for proper reconceptualization of the classroom environment. In particular, differences in attitudes regarding computer or software use are rich avenues that have only recently opened up for investigation. There is also a need to continue the investigation of grade differences and learning style differences with respect to the dynamic environment. Calculation of posttest score means presented the researcher with additional statistics worthy of consideration. It has been previously mentioned in Chapter III that the research instrument (refer to Appendix A) was adjusted in lieu of findings of the pilot studies. Items hypothesized to address learning attributed to the dynamic component of the geometric construction software were developed and placed into the final version of the research instrument. Thus, it was deemed appropriate to investigate individual items. Research should be undertaken to study assessment of learning in a dynamic geometric construction environment in addition to the development of appropriate test items. Moreover, research should give consideration to the evaluation of programs that utilize a dynamic learning environment. In addition to the rightful inspection into quantitative aspects of this study, consideration must also be given to qualitative aspects intrinsically rich in interpretive value. The activity worksheets (refer to Appendix C) not only gave structure to the learning sessions; participants responses throughout the guided discovery process provided insight into their construction and reconstruction of knowledge. During the first session, participants in either group could be observed struggling with the novel approach to understanding a concept many believed they had learned in middle school. Participants in the control group slowly drew similar geometric figures using the VTPD (refer to Definition of Terms) to develop an intuitive notion of similarity whereas the participants in the experimental group used the dynamic capability of the Geometer's Sketchpad (Jackiw, 1995) to investigate basic principles of similar figures. By the 73 end of the first session, the experimental group had measured and analyzed many different polygonal figures while the control group, although they had assimilated the principles of similarity, were unsure of whether nonconvex polygonal figures would observe the principles. In other words, participants of the control group were confident regarding the limited number of examples they were exposed to, but were unsure when analyzing unknown cases. Further research should be undertaken to consider possible misconceptions and, also, levels of geometric thought. Summary This final chapter discussed the results presented in the previous chapter. An overview of the study was provided first, including a description of the sample population. Next, the significance of the study was articulated, followed by the research questions. Then, discussion of the findings based upon the research analysis was presented. Limitations followed; the chapter concludes with implications and recommendations for future research. APPENDIX A RESEARCH INSTRUMENT 1. Point projections are made of the given triangle first from point P and then from point Q. Each projection uses a scale factor of 3. Which of the following is true? A. The triangle image from point P has angles of greater measure. B. The triangle image from point Q has angles of greater measure. C. The triangle image from point P has a larger area. D. The triangle images from point P and point Q have the same area. E. The triangle image from point Q has a larger area. o 0 P Q 2. Which of these transformations was used for the given rectangle and its image: image A) (x,y) > (2x,2y) B) (x,y) > (x,2y) C) (x,y) > (2x,y) D) (x,y) > (2x,4y) E) (x,y) > (4x,2y) 3. Which projection point was used to dilate the shaded rectangle? 0 D 4. What scale sailboat? factor has been used to enlarge the small C) 4 D) 6 A) 2 B) 3 E) 1/4 5. The given rectangles are similar. Find the missing length. A) 10 B) 11 C) 12 D) 13 E) 14 6. A man who is 6 feet tall has a shadow that is 8 feet long At the same time, a nearby tree has a shadow that is 32 feet long. How tall is the tree? A) 30 B) 21 C) 24 D) 42 E) 48 feet feet feet feet feet 7. Given rectangles of dimensions 1x6 and 4x24, the area of the larger triangle is how many times bigger than the area of the smaller rectangle? A) 4 times B) 6 times C) 8 times D) 16 times E) 18 times 1 ~ZIIIIZZ~ 8. If two figures are similar, which of the following might be different? A) number of sides B) lengths of corresponding sides C) shape D) size of angles E) ratio of corresponding sides 9. A 2 meter stick has a shadow of 1/2 meter at the same time that a nearby tree has a shadow of 3 meters. How tall is the tree? A) 6 meters B) 12 meters C) 1.5 meters D) 3 meters E) 15 meters 10. The given rectangles are similar. Find the length of the missing side. A) 7 B) 9 C) 10 D) 11 E) 15 2 6 5 ? 11. A 1x5 rectangle grows into a 4x12 rectangle. The area of the new rectangle is how many times larger than the area of the small rectangle? A) 3 times B) 4 times C) 5 times D) 15 times E) 16 times 12. Point projections are made of the given triangle first from point P and then from point Q. Each projection uses a scale factor of 2. Which of the following is true? A. The triangle image from point P has angles of greater measure. B. The triangle image from point Q has angles of greater measure. C. The triangle image from point P has a larger area. D. The triangle images from point P and point Q have the same area. E. The triangle image from point Q has a larger area. 0 P 13. Given the bolded segment of the shaded triangle, the corresponding segment on the projected image is: A) a B) b C) c Q i\ / I 0== a  14. Given the bolded segment of the shaded polygon, the corresponding segment on the projected image is: A) a B) b C) c D) d 0gd b 81 15. Which projection point was used to dilate the shaded rectangle? 0D D C 0 0 A 16. The point of perspective is: A) point A B) point B C) point C D) point D E) point E 83 17. Choose the scale factor of the similar figures. The bolded line segment is the preimage.The scale factor is: A) 1 B) 2 C) 3 D) 4 E) 5 18. Choose the scale factor of the similar figures. The shaded circle is the preimage. The scale factor is: A) 1 B) 2 C) 3 D) 4 E) 5 19. Which projection point was used to dilate the shaded rectangle? 20. Which projection point was used to dilate the shaded trapezoid? 0 C o0 D 0E 21. Solve the proportionality a/? = w/x. ? = A) a B) b C) c D) d E) e 0 /! /1 / Iy a ,/ , / I / I / 0. "' c d   APPENDIX B LEARNING SESSIONS Table 3.1 First Pilot Study Presession Information regarding UFIRB form. Pretest. Computer lab orientation. Session 1 Launch Terminology review. Use of construction tool and measurement tools. Exploration Properties of similar polygons. Summary Class discussion of conjectures generated during individual exploration time (Corresponding angles congruent. Ratio of 2 sides of a polygon is equal to the ratio of the corresponding 2 sides of any similar polygon and they are equivalent fractions) 88 Session 2 Launch Exploration Summary Review of properties of similar polygons. Introduction to intuitive Projective Geometry. SCALE FACTOR with DILATE tool. Properties with respect to (1) point of perspective and (2)corresponding sides. Class discussion of conjectures generated during individual exploration time. (The ratio of the distance from the perspective point to a vertex of a similar polygon over the distance from the perspective point to a vertex of the given polygon is equal to the scale factor. The ratio of the measure of a side of a similar polygon over the measure of the corresponding side of the given polygon is equal to the scale factor. Properties of similar polygons are invariant as the position of the point of perspective varies.) 89 Session 3 Launch Exploration Summary Connection of concept of similarity to ratio and proportional reasoning. Given preconstructed sketches, indirect measurement tasks. Class discussion of approaches generated during individual exploration time. Session 4 Launch Review of properties with respect to (1) point of perspective and (2)corresponding sides. Use of POLYGON INTERIOR tool. Exploration Relationship of areas of similar polygons. Summary Class discussion of conjectures generated during individual exploration time. (The ratio of the area of a similar polygon over the area of the given polygon is equal to the square of the scale factor.) Table 3.2 Second Pilot Study Session 1 Launch Exploration Summary Terminology review. Use of construction and measurement tools. Introduction to intuitive Projective Geometry. Dynamic geometry exposition with animation or the Variable Tension Proportional Divider. Properties of similar figures. Class discussion of conjectures generated during individual exploration time. (Corresponding angles congruent. Ratio of 2 sides of a polygon is equal to the ratio of the corresponding 2 sides of any similar polygon and they are equivalent fractions) 91 Session 2 Launch Define scale factor. Review properties of similar polygons. Exploration Properties with respect to (1) point of perspective and (2)corresponding sides. Summary Class discussion of conjectures generated during individual exploration time. (The ratio of the distance from the perspective point to a vertex of a similar polygon over the distance from the perspective point to a vertex of the given polygon is equal to the scale factor. The ratio of the measure of a side of a similar polygon over the measure of the corresponding side of the given polygon is equal to the scale factor. Properties of similar polygons are invariant as the position of the point of perspective varies.) 92 Session 3 Launch Discussion of how to find the area of a figure. Exploration Relationship of areas of similar polygons. Summary Class discussion of conjectures generated during individual exploration time. (The ratio of the area of the image over the area of the given polygon is equal to the square of the scale factor.) Launch Review of properties with respect to (1) point of perspective and (2)corresponding sides. Connection of concept of similarity to ratio and proportional reasoning. Exploration Given preconstructed sketches, indirect measurement tasks. Summary Class discussion of approaches generated during individual exploration time. (The measures of corresponding parts of similar triangles can be set onto a proportion to find the unknown measurement.) Table 3.3 The Study Session 1 Launch Exploration Summary Terminology review. Use of construction and measurement tools. Introduction to intuitive Projective Geometry. Dynamic geometry exposition with animation or the Variable Tension Proportional Divider. Metric properties of similar figures. Size and position of the image with respect to the numerosity of the scale factor. Class discussion of conjectures generated during individual exploration time. Session 2 Launch Define scale factor. Review properties of similar polygons. Exploration Numeric properties of similar shapes with respect to (1) point of perspective and (2)corresponding sides. Summary Class discussion of conjectures generated during individual exploration time. 94 Session 3 Launch Review of properties with respect to (1) point of perspective and (2)corresponding sides. Connection of concept of similarity to ratio and proportional reasoning. Exploration Numeric properties of similar shapes with respect to (1) scale factor and (2)corresponding distances. Summary Class discussion of approaches generated during individual exploration time. APPENDIX C ACTIVITIES WITH RESPONSES Sketch step 1: In the Display menu, choose Line Weight "thick" and Shade "100%." step 2: Construct any polygon and change the color of each side to be distinct. step 3: Construct a point outside the polygon and mark it as center in the Transform menu. step 4: Select your entire polygon and dilate by a scale factor of 2/1 (Place 2 into top box and 1 into the bottom box). J Investigation: Corresponding Sides SU ta w j step 5: Measure the ratio of a side on the dilated polygon with the corresponding side on the original polygon. step 6: Repeat step 5 using a different side. What is this ratio? .50 step 7: Drag a vertex to change the length of a side. What changes? What stays the same? 0 ',u< 'V A\^Or ro ^ rwhos sWa~ so\ n ,c]M Investigation: Projection Point step 7: Measure an angle of the dilated polygon and the corresponding angle on the originalpolygon. flQxAL, w(& 'Wc. ^$4T< step 8.tS rag the point of projection to different locations. Regarding the e.aO, figures, what changes (measures, locations,...)? What stays the same? Make sure to include placing the projection point on a vertex of the original polygon! (",, ? 0 ln! projcc / /ii o n o o 4U lorTu4 I t b i ^ . ... ... '*^ V 