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ENGLISH LANGUAGE PROFICIENCY AND SPATIAL VISUALIZATION IN MIDDLE SCHOOL STUDENTS' CONSTRUCTION OF THE CONCEPTS OF REFLECTION AND ROTATION USING THE GEOMETER'S SKETCHPAD B/ JULI KIM DIXON 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 1995 Copyright 1995 by Juli Kim Dixon This dissertation is dedicated to three generations of life and love. In loving memory of my grandfather, Albert P. Harap, whose belief in me was invaluable, to my parents, Harvey and Joy Inventasch, who instilled in me a respect for education, and to my husband, Marc Todd Dixon, who respected my dream and made it his own. ACKNOWLEDGEMENTS My gratitude is extended to the many people who have provided academic and emotional support throughout the dissertation process. To Professor Mary Grace Kantowski, my chair and mentor, who provided educational challenges and support, I owe my future contributions as a mathematics educator. I am thankful to my committee members, each of whom provided guidance throughout the dissertation process. Professor David Miller shared his statistical expertise. Professor Clemens Hallman led me to a deeper understanding of the needs of students with limited English proficiency and provided financial support through the Bilingual Education and Minority Languages Affairs Fellowship. Professor Donald Bernard contributed to my growth through his experiences as an educator. Professor Charles Nelson enhanced my understanding of mathematics from the viewpoint of a mathematics educator. I am grateful to Susan Lee and the mathematics faculty and students of the central Florida middle school for making the research for my dissertation possible. I am especially indebted to Nick Jackiw of Key Curriculum Press for providing me with the experimental version of The Geometer's Sketchpad used in the study and providing participating teachers with The Sketchpad for the continued use with their students. Nick also provided ongoing and greatly appreciated technical support. I am grateful for the financial assistance provided by Dr. Charlie T. Council and the Florida Educational Research Council in support of this study. I am forever grateful to my family, especially to my parents, Harvey and Joy Inventasch, my husband, Marc Todd Dixon, and my friend, Thomasenia Lott Adams whom I consider family. These very special people have provided me with the love and support that only family can provide. I would never have become the person and mathematics educator I am today without each and every one of them. TABLE OF CONTENTS page ACKNOWLEDGEMENTS .................................iv ABSTRACT ..................................... .viii CHAPTERS I DESCRIPTION OF THE STUDY Introduction ...... ... .. ... .. .. .. ..... .... .... 1 Statement of the Problem. ........................ 5 Significance of the Study ......................... 8 Organization of the Study ......................... 9 II LITERATURE REVIEW Overview .. ....................... ........ 11 Theorietical Background ......................... 11 Studies on Transformations ...................... 17 Visualization ................................ 33 Limited English Proficient Students. ................. 46 Sum m ary ..................................49 III METHODOLOGY Research Objective ............................ 52 M measures .......................... .........53 Pilot Study ....................... ..........56 Sampling Procedures ................... ........ 61 Experimental Treatment and Procedures. .............. 62 Research Design ..............................67 IV RESULTS Descriptive Statistics ................. Limitations of the Study ............... V QUALITATIVE RESULTS M ethodology ........................ Computer Projects ................... Observations ....................... Data from Reflection/Rotation Instruments . Icon Questionnaire. .................... .... 70 . .. .... .. 78 . . 81 .. .... ... 83 . .... .... 90 ......... 105 ......... 108 VI CONCLUSION Sum m ary ......... ................ Discussion ........................ Implications........................ Recommendations ................... Suggestions for Future Research ......... . . 110 . . 116 ... ... .. .. 123 . .. .... 124 ... ....... 127 APPENDICES A PAPER AND PENCIL REFLECTION/ROTATION INSTRUMENT. .. .129 B PAPER AND PENCIL REFLECTION/ROTATION INSTRUMENT OVERLAYS ..................................140 C TRAINING LESSONS ............... .............147 D ANGLE AND DISTANCE MEASURE ACTIVITIES. ........... .150 E EXPERIMENTAL TREATMENT LESSONS ................. 158 F REFLECTION AND ROTATION KEYS .................. .165 G REFLECTIONGAME SCREEN ........................ 170 REFERENCES ................. ...... ............. 171 BIOGRAPHICAL SKETCH ............................. .180 Abstract of Dissertation Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy ENGLISH LANGUAGE PROFICIENCY AND SPATIAL VISUALIZATION IN MIDDLE SCHOOL STUDENTS' CONSTRUCTION OF THE CONCEPTS OF REFLECTION AND ROTATION USING THE GEOMETER'S SKETCHPAD By JULI KIM DIXON May 1995 Chairman: Eleanore L. Kantowski Major Department: Instruction and Curriculum This study was designed to investigate the effects of a dynamic instructional environment, English proficiency, and visualization level, independently and interactively, on middle school students' construction of the concepts of reflection and rotation. Also examined were the effects of a dynamic instructional environment on students' two and threedimensional visualization. The dynamic instructional environment involved exclusive use of The Geometer's Sketchpad with lessons created for the purpose of allowing for student discovery of properties of reflection and rotation. Students were also provided extensive opportunities to perform the transformations. After controlling for initial differences, it was concluded that students experiencing the dynamic instructional environment significantly outperformed students experiencing a traditional instructional environment on content measures of the concepts of reflection and rotation as well as on measures of twodimensional visualization. The students' instructional environment did not significantly affect their threedimensional visualization. There was no statistically significant differences on any of the dependent variables between the performances of limited English proficient students and those of their English proficient peers when the same instructional environment was experienced. Observations provided examples of how students communicate about mathematics, visualize movement, and construct concepts of reflection and rotation while experiencing a dynamic instructional environment. Concrete information was gained regarding the depth of students' knowledge about concepts involved with reflecting and rotating. Insight was obtained into the class attitudes about and understanding of transformations. CHAPTER I DESCRIPTION OF THE STUDY Introduction The Curriculum and Evaluation Standards for School Mathematics (National Council of Teachers of Mathematics (NCTM), 1989) include the following objectives in the standard for middle level geometry: "In grades 5 8, the mathematics curriculum should include the study of the geometry of one, two, and three dimensions in a variety of situations so that students can visualize and represent geometric figures with special attention to developing spatial sense; [and] explore transformations of geometric figures" (p. 112). This standard is not being implemented in many middle schools in the United States. The Second International Mathematics Study: Report for the United States (SIMS) (Crosswhite, 1985) identified topics taught in other countries which were not covered by the majority of eighthgrade classes in the United States. "For example, topics in transformational geometry, taught in some countries, were reported taught by only 12% of United States eighth grade teachers" (p. 20). Furthermore, the results of the study indicated that teachers primarily taught according to their textbook with very little use of manipulatives or other materials not included with the text. This finding was corroborated by data collected in a comparative study with seventh and eighthgrade students in the United States and Japan (Iben, 1988). Reasons for the lack of emphasis on transformation geometry may include the need for an effective and accessible instructional strategy. For students to develop spatial sense, they need many and varying experiences with drawing, measuring, transforming, visualizing, comparing, and classifying geometric shapes (NCTM, 1989). "Spatial sense is an intuitive feel for one's surroundings and the objects in them. To develop spatial sense, children must have many experiences that focus on geometric relationships; the direction, orientation, and perspectives of objects in space; the relative shapes and sizes of figures and objects; and how a change in shape relates to a change in size" (NCTM, 1989, p. 49). The term "spatial sense" is related to what has also been labeled spatial visualization, spatial perception, visual imagery, spatial ability, visual skill, spatial reasoning, mental rotations, and visual processes (Bishop, 1983; Davey & Holliday, 1992; Stanic & Owens, 1990; Wheatley, 1990). The curriculum in geometry and instruction practices must include appropriate experiences for students in the area of visualization as it reflects this important aspect of mathematics (National Council of Supervisors of Mathematics [NCSM], 1989; National Research Council [NRC], 1989). The recent availability of computers in classrooms has provided the tool for such instruction. The dynamic graphic capabilities of the microcomputer allows for geometry to be introduced to students through transformations (Kantowski, 1987). NCSM (1989) supports the use of computers throughout the mathematics curriculum as well as instruction in visualization and transformations. Effective instructional strategies are also needed for use with language minority students. According to Mathematics for Language Minority Students, the NCTM's position statement, "cultural background or difficulties with the English language must not exclude any student from full participation in the school's mathematics program" (NCTM, 1987, n. p. ). The Curriculum and Evaluation Standards for School Mathematics states, "students whose primary language is not standard English may require support to facilitate their learning of mathematics" (NCTM, 1989, p. 80). These students tend to spend most of their time on the prerequisite basic skills and rarely have exposure to higher order mathematics skills (Schwartz, 1991; Secada & Carey, 1990; Stoloff, 1989). The position of NCTM (1989) in general and this researcher in particular is that this current trend is unacceptable. The Curriculum and Evaluation Standards for School Mathematics (NCTM, 1989) and the Professional Standards for Teaching Mathematics (NCTM, 1991), both commonly referred to as the Standards, stress communication as an important part of teaching and learning mathematics. Mathematics can be taught and learned visually; communication does not necessarily refer exclusively to verbal exchanges. Visual communication in mathematics education is especially important to students having limited proficiency in English (Cummins, 1984; Dawe, 1983; Presmeg, 1989). The Standards also stress the need to empower all students through mathematics. Since the assumption persists that mathematics depends only marginally on expertise in English, mathematics typically serves as the first academic course beyond English as a Second Language (ESL) into which counselors place students from foreign countries. However, counselors and bewildered students soon discover that limited English proficiency is a discouraging obstacle to learning mathematics. (Kimball, 1990, p. 604) Limited English proficient (LEP) students can make progress in learning mathematics when the teacher recognizes their special needs and uses communication, manipulatives, handson experiences, the first language of the students when possible, and when the teacher systematically addresses the necessary vocabulary (Kober, 1991). Mathematics taught with an emphasis on visual clues could make the mathematics traditionally taught in a highly verbal instructional mode more attainable to students having limited proficiency in English. Use of The Geometer's Sketchpad (Jackiw, 1991) for the dynamic presentation of rotations and reflections is an appropriate vehicle to facilitate full participation of limited English proficient students while encouraging the development of all students' visualization abilities in the middle school mathematics program. The Geometer's Sketchpad is a dynamic program for the Macintosh and IBM computers. The Sketchpad ". has great potential for teachers and students to use in investigations of ideas in geometry" (Wilson, 1992, p. 157). This tool allows mathematics to be taught visually to the class as a whole, to small groups, or to individuals. Furthermore, the experimental version of the Sketchpad used in this study incorporated icons, or pictorial representations, as clues to appropriate menu selections displayed in English. These menu selections were coupled with symbols designed to provide context clues for the associated vocabulary. Statement of the Problem The study of the rigid motion transformations and experience with visualizing in two and threedimensions are integral parts of the Standards for eighthgrade students in mathematics. These topics are not currently addressed adequately in the middle school curriculum in the United States. Achievement in mathematics in a given language seems to be related to the degree of proficiency in that language (Secada, 1992a). Contrary to what many people believe, the study of mathematics does not transcend language barriers. The changing demographics of the United States dictates that appropriate mathematical content, teaching strategies, learning tools, and classroom environments must be incorporated in all schools to close the gap of achievement in mathematics between English speaking and LEP students. Current programs designed with the pretense of closing this gap are ". evaluated based on their ability to curtail student drop out and to improve student scores on standardized tests. As a consequence, compensatory programs mimic the tests by which they are evaluated and which are focused on lower level and computational skills" (Secada, 1989, p. 38). This focus on lower level skills is not consistent with NCTM's Standards and is not an acceptable solution to the problem of the achievement gap. The purposes of this study included: (1) to explore the effects of a dynamic instructional environment and students' visualization ability on the identification of and ability to perform the rigid motion transformations of reflection and rotation; and (2) to explore the effects of the relationship between the students' level of visualization and the instructional environment on the identification of and ability to perform these rigid motion transformations. Another purpose of this study was to investigate the effects of students' English proficiency on the identification of and ability to perform reflections and rotations and the effects of the relationship between students' English proficiency and a dynamic instructional environment on the identification of and ability to perform the rigid motion transformations of reflection and rotation. Finally, the study was designed to ascertain the effects of a dynamic instructional environment and students' English proficiency on students' visualization. The treatment in the dynamic instructional environment included exploring the rigid motion transformations of reflection and rotation through a dynamic presentation and subsequent activities using The Geometer's Sketchpad (Jackiw, 1991). The rigid motion transformations of reflection and rotation were addressed based on the view of the researcher that the motion involved with reflection and rotation are very similar with the exception that reflection involves three dimensions and rotation involves two dimensions. The rigid motion transformation of translation was not addressed in this study due to the view of the researcher that the movement involved with translation is not closely related to that of reflection and rotation. The theoretical bases for using a dynamic instructional environment emphasizing student collaboration and exploration as the instructional strategy was based on the constructivist theory of teaching and learning mathematics as well as Vygotsky's (1978, 1986) zone of proximal development. Vygotsky believed that students are provided more opportunities to learn when they are able to collaborate with more capable peers and adults. The constructivist theory promotes collaboration among peers during the construction of knowledge. The choice of the computer program and design of associated student activities was based on Cummins' (1984) theory of context embedded versus context reduced instruction. The Geometer's Sketchpad provides context clues through the dynamic, visual presentation of geometric properties. Together with the activities, the software was used to aid in students' acquisition of the concepts and vocabulary involved with rigid motion transformations in a context embedded rather than a context reduced instructional environment. The objectives of the study were 1. to explore whether students' level of visualization will interact with their instructional environment to affect their ability to identify and perform reflections and rotations in a (a) dynamic testing environment or a (b) static testing environment. 2. to explore whether students' English proficiency will interact with their instructional environment to affect their ability to identify and perform reflections and rotations in a (a) dynamic testing environment or a (b) static testing environment. 3. to explore whether students' (a) instructional environment, (b) visualization level, and (c) English proficiency will affect their ability to identify and perform reflections and rotations in a (a) dynamic testing environment or a (b) static testing environment. 4. to explore whether students' (a) instructional environment and (b) English proficiency will affect their (a) two dimensional and/or (b) threedimensional visualization. Significance of the Study If the transformations are to be studied in their own right, it would seem pointless to do this in a didactic, expository manner. The fact that the transformations can be defined in terms of actions (folding and turning) and their results represented in a very direct manner by drawings means that the topic is ideally suited to a practical and investigative approach. (Hart, 1981) The study of transformations provides an excellent opportunity for a dynamic instructional environment for all students regardless of their proficiency with the English language. Throughout the recent history of mathematics education, individuals and influential reform groups have devoted chapters and entire textbooks to the study of transformations. Transformations often are used to represent physical motions such as slides [translations], flips [reflections], turns [rotations], and stretches dilationss]. Students should use computer software based on this dynamic view of transformations to explore properties of translations, line reflections, rotations, and dilations, as well as compositions of these transformations. These graphic experiences not only help students develop an understanding of the effects of various transformations but also contribute to the development of their skills in visualizing congruent and similar figures. (NCTM, 1989, p. 162) The Geometer's Sketchpad was first published in 1991 by Key Curriculum Press. The Sketchpad is a highly visual and dynamic tool for exploring and discovering geometric properties, but because of the lack of research using the Sketchpad in middle school mathematics its potential is not being realized. This study established the effectiveness of the Sketchpad as a tool for the instruction of all students, including LEP students, to successfully recognize and perform rotations and reflections. The Sketchpad was also useful for improving students' visualization ability. Organization of the Study This chapter included the statement of the problem and its significance in the field of mathematics education. The review of the relevant literature presented in chapter II includes the 10 theoretical bases in which this study has been framed, studies involving instruction with transformation geometry, studies of visualization, and studies involving mathematics and science instruction for limited English proficient students. The design and methodology of the study is reported in Chapter III. In Chapter IV, results of the quantitative analysis and limitations of the study are provided. Qualitative results of the study are reported in Chapter V. A summary of the results, implications, and recommendations for future research are presented in Chapter VI. CHAPTER II LITERATURE REVIEW Overview A review of the relevant literature is presented in this chapter. The areas under discussion are the theoretical bases in which this study has been framed, studies involving instruction with transformation geometry, studies of visualization, and studies involving mathematics and science instruction for limited English proficient students. Theoretical Backaround This study was designed to explore whether an environment in which English proficient and limited English proficient students work together using computers will result in their construction of knowledge of reflections and rotations while simultaneously improving visualization skills. Research must be supported by educational theory. The constructivist theory of teaching and learning explains how teachers can create environments in which students can construct knowledge. Vygotsky's (1978, 1986) zone of proximal development addresses how students working together, with adult guidance as needed, and using proper tools could construct this knowledge socially. Cummins' theory of context embedded language versus context reduced language justifies how English proficient and limited English proficient students could work together in the same instructional setting. Although the constructivist theory of teaching and learning exists in a variety of expressions, there is general consensus on the following principles (Confrey, 1990; Davis, Maher, & Noddings, 1990; Goldin, 1990; Kamii and Lewis, 1990; Noddings, 1990; von Glasersfeld, 1990; Wheatley, 1991): 1. Students build their own knowledge; they do not receive knowledge prepackaged from others. 2. Knowledge is not built passively but through physical and mental action. 3. Truths are not found; rather, interpretations are built to explain experiences. 4. Learning takes place through social interaction. Inherent in these principles is the need for an environment conducive to student exploration and interaction. "Constructivists 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" (Noddings, 1990, p. 10). The constructivist view of learning mathematics is consistent with the belief that students come to the classroom with differing understandings and ways of conceptualizing mathematics (Kamii & Lewis, 1990; Steffe, 1990; Wheatley, 1991). Different students construct concepts in different ways depending on many variables, such as the students' native and learned abilities, aptitudes, dispositions, learning styles, native language, and past experiences. The constructivist view of teaching allows for these differences by providing experiences for children to make sense of mathematics through varying instructional strategies. These strategies include meaningful mathematical exploration and experiences, use of multiple representations, multiple modes of instruction, and, as in the case of this study, a dynamic medium for learning ( Kamii & Lewis, 1990; Noddings, 1990; Wheatley, 1991). "The teacher's main function is to establish a mathematical environment" (Noddings, 1990, p. 13). Viewing mathematical knowledge as a learner activity rather than an independent body of "knowns" leads to quite different educational considerations. Rather than identifying the set of skills to be gotten in children's heads, attention shifts to establishing learning environments conducive to children constructing their mathematics in social settings. Since there is a social dimension of knowledge, this learning environment necessarily includes children talking mathematics with each other. Such learning environments provide opportunities for children to share ideas with peers, both in small groups and within the society of the classroom. (Wheatley, 1991, p. 12) Vygotsky (1978) believed that the pedagogical and social needs of the student could be met through the use of appropriate tools in an environment that promoted collaborative inquiry about concepts. In so doing, the students' zone of proximal development would be formed and learning legitimized. According to Vygotsky (1978, 1986), the zone of proximal development is the distance in developmental level between problems the student can solve independently and problems the student can solve in collaboration with more capable peers or under guidance from an adult. Students' potential to learn is then judged by the problems that they cannot yet solve alone but can accomplish in collaboration with others rather than the problems that they can already perform alone. Vygotsky believed that learning within this zone is thus the only worthwhile learning. He proposed that an essential feature of learning is that it creates the zone of proximal development; that is, learning awakens a variety of internal developmental processes that are able to operate only when the child is interacting with people in his environment and in cooperation with his peers. Once these processes are internalized, they become part of the child's independent development achievement. (Vygotsky, 1978, p. 90) Vygotsky believed that the student was working within his zone of proximal development when he was able to solve given problems only with assistance from more capable peers or adult guidance and was not able to solve the problems independently. This researcher believes that students working in collaboration are able to help each other within their zones of proximal development because, according to the constructivist view of learning, each student brings different experiences to the learning situation and hence may be more capable on different aspects of the same problem. In this situation, the "more capable peer" may alternate from one student to the other while working on the same problem. And, in the event that the students cannot provide the proper catalyst to solve the problem, the teacher, acting as facilitator, can supply the necessary guidance. This role of the teacher is in accordance with both the constructivist view of teaching and learning mathematics and Vygotsky's zone of proximal development (Vygotsky, 1978, 1986; Wright, 1990). For LEP students in particular and all students in general, collaborative inquiry should be placed in a context embedded situation. A contextembedded situation is one that provides comprehensible input so that the students do not need to guess about the teacher's intentions but may gather an understanding of the intentions based on linguistic as well as situational cues. This contextembedded language may include pictures, interpersonal interactions, and nonverbal information (Fradd, 1987). According to Cummins (1984) and Fradd (1987), this environment is more compatible to LEP students than is a context reduced situation (the traditional classroom lecture style of environment) where students must rely solely on linguistic cues in a language with which the LEP student is not yet proficient. The degree of cognitive involvement in activities where communication is used can be explained by viewing these activities in a continuum. At one end there exist ". .communicative tasks and activities in which the linguistic tools have become largely automatized (mastered) and thus require little active involvement for appropriate performance" (Cummins, 1984, p. 139); at the opposite ". end of the continuum are tasks and activities in which the communicative tools have not become automatized and thus require active cognitive involvement" (Cummins, 1984, p. 139). Contextual support is most important when the degree of cognitive involvement is the greatest (Cummins, 1981, 1984; Fradd, 1987). When exploring new mathematical concepts with unfamiliar mathematical vocabulary, students, especially LEP students, will not have automatized the associated communicative tools. They may be experiencing just this kind of situation when working within their zones of proximal development. It may seem that the limited English proficient students need help with the mathematics when in actuality the problem is with the language. And, according to Fradd (1987), "comprehensible input is the foundation of effective instruction" (p. 143). Comprehensible input shares both a point of commonality and a point of difference with the zone of proximal development. The two constructs are similar in that advancement is possible to the extent that language input is made comprehensible to the learning by an adult or more capable peer. Moreover, both constructs underscore the importance of being responsive to the learner's perceived needs. The difference lies in the focus each construct places on the role of language learning. Whereas comprehensible input stresses the importance of adjusting speech to the learner's linguistic level of competence, the zone of proximal development focuses on adjusting speech to the learner's interactional level of competence. (DeVillar & Faltis, 1991, pp. 2324) Studies on Transformations Several studies have addressed geometric transformations. Some have involved extensive instruction in performing transformations (Edwards, 1990, 1991; Ernest, 1986; Johnson Gentile, 1990; Pleet, 1990; Williford, 1972); most have included only a brief introduction to transformations, if any at all, before testing or interviewing students on their ability to perform transformations (Edwards & Zazkis, 1993; Hart, 1981; Kidder, 1976; Law, 1991; Moyer, 1978; Schultz & Austin, 1983; Soon, 1989). A few studies have incorporated the use of computers in the presentation of transformations (Edwards, 1991; Edwards & Zazkis, 1993; Ernest, 1986; JohnsonGentile, 1990; Pleet, 1990). An investigation involving students' actual construction of transformation concepts involved the use of the computer software Logo with small groups of students (Edwards, 1990, 1991). Edwards investigated "the learning of a small group of children who interacted over a short period of time with a computer microworld dealing with transformation geometry" (Edwards, 1991, pp. 122 123). The microworld, referred to as the TGEO computer microworld, was based on Logo, which all of the students had previously learned. Prior to the study, the students did not have formal instruction in transformation geometry. Twelve students from grades six through eight were chosen from a small, private school based on their interest in participating in the study and their ability to meet after school. The students were introduced to transformations in their regular computer classes during two 50 minute sessions. After this introduction, the students met in pairs after school once a week for five weeks. The pairs of students worked together on tasks at the computer or on worksheets. The investigator introduced the activities then answered questions while videotaping the session. A pretest was administered after the first small group session and administered again as a posttest at the conclusion of the study. Twelve items of the 24 items on the posttest were identical to those used in a large scale British study (Hart, 1981). As this was a qualitative dissertation, most of the data were gathered via videotape. The results of the study suggest that the microworld and the associated activities were effective in assisting the students to construct a working knowledge of the transformations. The average performance of the students on the written final examination was 70% correct. The subjects in [Edward's] study performed better than the British students on 10 of the 12 items common to both tests. Furthermore, by the time of the final session, all the students were able to carry out any transformation on the computer without error or hesitation. (Edwards, 1991, p. 129) Edward's (1991) study, which was conducted at a small private school, included the components for the necessary environment. Her microworld necessitated facility with Logo, a computer programming language. It is unrealistic to assume that the already overburdened and underfunded public schools will develop curriculum that will be altered to include a computer programming course in the middle school before students can learn to perform transformations. Transformations have been performed on microcomputers using programs other than Logo such as student and teacher generated programs in Basic using Apple II Graphics (Orton, 1990; Shilgalis, 1982). Orton (1990) described a short program in Basic that uses students' knowledge about algebra, transformations, and computer programming to perform reflection and compositions of reflections over user generated mirror lines. This program was presented by Orton in response to a conference of NCTM at which short programs were called for in order to teach and learn mathematics. Shilgalis (1982) also described a program for Apple II graphics that performs reflections as well as rotations, translations, glide reflections, dilations, and compositions of the transformations. Both programs rely on the coordinate axes as references for points, lines, and vectors. The user enters the appropriate information as a response to program generated prompts. Hence, to use the programs, students must have some knowledge of coordinate axes and vectors as they apply to transformations. The large scale British study was part of Concepts in Secondary Mathematics and Science (CSMS). This research program (Hart, 1981) conducted at Chelsea College, University of London, included widescale testing of British 13 to 15yearold students' ability to perform rotations and reflections. The results of the test were reported in Children's Understanding of Mathematics: 1116 (Hart, 1981). Two hundred ninetythree 13yearold students, 449 14 yearold students, and 284 15yearold students (1026 subjects total) were administered a 52item paper and pencil test lasting one hour. The test consisted of three parts: single reflections, single rotations, and combinations of reflections and rotations. The students practiced and received answers to examples of reflections and rotations immediately prior to the test. No other formal instruction was provided by the investigators because reflections and rotations were included in the British curriculum for ages 13 to 15. Nearly all students experienced some success performing single reflections and rotations; however, most students had a difficult time performing combinations of transformations. Ernest (1986), who also used items from Hart's (1981) test, studied the effects of computer gaming on students' performance in transformational geometry. Eighteen 15yearold students of below average mathematics ability received six hours of instruction on geometric transformations. The students were divided into two comparable groups of nine subjects each. Prior to the first computer session, each group was given a pretest involving specific and general transformations. The general transformations involved items from Hart's "Reflection and Rotation Test." Both groups worked in pairs or trios on computer games for two halfhour sessions. The experimental group played transformation games using the computer program Triangles. The control group played computer games without transformations. A posttest of an equivalent form to the pretest was administered immediately after the final computer session. The experimental group performed significantly better on the transformations specifically related to the game. There was no significant difference on the performance on the test of general transformations. Unlike Edwards' (1990) study, the computer games used in Ernest's (1986) study did not require knowledge or experience with programming. While the students had access to computers, it could be concluded that they were not allowed sufficient time to explore transformations through the use of this technology. The students were exposed to the software for two 30minute sessions. Perhaps this brief duration of time explains the successful performance of the experimental group only on transformations directly related to those experienced in the computer game. The results might have been different if students were allowed more time at the computer and/or if there was a longer delay between the computer session and the final. Pleet (1990) compared the use of the computer program Motions to the use of the manipulative Mira on eighthgrade students' ability to perform transformations and mental rotations. Eight teachers at eight different schools in Los Angeles taught one class using Motions and one class using Mira for a three week period. Seven additional teachers at seven additional schools in Los Angeles administered pre and posttests with no transformation instruction to control for test effects. Complete data were collected on 560 subjects. The Card Rotation Test from the Kit of FactorReferenced Tests (Ekstrom, French, Harman, & Derman, 1976) and a researchermade transformation geometry test were given as pre and posttests at the beginning and end of the threeweek period. Each lesson during the threeweek period was structured with a 20minute reviewexplanationdiscussion period followed by a 30minute activity session. The Motions groups transferred to an Apple computer lab for the activity period while the Mira groups stayed in their classrooms. There was no significant difference between the Motions computer program group and the Mira manipulative group on either acquisition of transformation geometry concepts or mental rotation ability. Pleet recommended that teachers who would be teaching transformations using either a computer graphics program or handson manipulatives receive sufficient training in both use of and teaching strategies for the appropriate program or manipulative. JohnsonGentile (1990) also investigated the effects of computer and noncomputer environments on students' achievement with transformation geometry. The researcher examined the effects of a Logo version and a nonLogo version of a "motions" unit developed by Professors Clements and Battista based on the van Hiele levels. Two hundred twentythree fifth and sixth grade students from two different schools participated in the study. One school was a city school with 50% of its students labeled as gifted and talented; the other was an upper middle class suburban school. The students were divided by class into two treatment groups and one control group. The two treatment groups consisted of a Logo based motions curriculum and a nonLogo based motions curriculum. The nonLogo based curriculum was identical to that of the Logo group's in every way except when the students worked on the computer. The nonLogo groups worked with paper and pencil, transparencies, and the Mira while the Logo groups worked with Logo on the computer. The control group did not participate in any motions activities. The Logo groups were part of a larger study involving use of the Clements and Battista curriculum. A pretest of achievement in geometry was given to all groups in the Fall of 1988; the study was conducted during the Spring of 1989. The motions unit lasted two weeks. A posttest consisting of objectives related to the rigid motion transformations was administered to the two treatment groups and the control group immediately following the unit. A subsample of 36 students evenly distributed according to sex and ability was chosen by stratified random sampling for interviews. Each interview was 30 minutes long, the process began within four days of the posttest. The interviews were designed to collect information pertaining to precision of language with respect to transformations and levels of thinking based on the van Hiele levels. Finally, a retention test (identical to the posttest) was administered one month after the posttest. According to the pretest, the groups were equivalent with respect to geometric ability at the time it was given. Both treatment groups scored significantly higher than the control group on the posttest and the retention test. There was no significant difference between the Logo and nonLogo groups on the posttest but there was a significant difference between the groups on the retention test. The Logo group scored higher on the retention test compared to the posttest and the nonLogo group scored lower. There was no significant difference between the Logo and nonLogo groups on interview measures but both treatment groups scored significantly higher than the control group on the same measures. It should be noted that the Logo group had been using a different curriculum (Clements/Battista curriculum) previous to the beginning of the study and after the administration of the pretest. The pretest was given in the Fall of 1988 and the study began in the Spring of 1989. The groups may not have been equivalent with respect to experience with geometric concepts at the beginning of the study. The following studies involved little or no instructional intervention. Soon (1989) examined whether the hierarchical levels a student follows while learning concepts in transformation geometry are consistent with the van Hiele theory for geometrical understanding. Textbook material related to transformation geometry in Singapore were also examined to determine whether they were consistent with van Hiele based levels. Twenty female students in their final year of secondary school were randomly selected from one school in Singapore. The students all used the same textbook but were taught by four different teachers. Two interview sessions of 2 1/2 hours were conducted with each student. The first session was videotaped and the second session was audiotaped. Subjects were instructed to think aloud as they solved problems using paper, pencil, and concrete objects. The subjects were given 31 problems. The problems were separated into levels based on the van Hiele levels of learning geometry as they relate to concepts in transformation geometry. The hierarchy of understanding of concepts in transformation geometry the students followed were consistent with the van Hiele theory. Fortytwo and onehalf percent of the students responded according to criteria for the Basic Level which "required students to visually identify and discriminate the following transformations: reflection, rotation, translation, and enlargement" (Soon, 1989, p. 78). Thirtysix and onequarter percent of the students were at Level 1. "At this level questions focused on a knowledge of the properties of each transformation, and what happens to each figure after each transformation" (Soon, 1989, p. 79). Six and onequarter percent were at Level 2 in which "students were required to interrelate the properties of the different transformations. They were expected to deal with composition of transformation and the use of matrices in transformations" (Soon, 1989, p. 93). Finally, 12.5% were at Level 3 which required the use of transformations in proofs. Soon also found that the textbooks did not provide students with opportunities to explore and conjecture about transformations. The levels of the textbooks were consistent with the levels of a majority of the students. Law (1991) also looked at the hierarchy of the acquisition of the concept of transformation in an attempt to determine how preservice elementary school teachers construct the concepts of translation, reflection, and rotation. Eighteen preservice elementary school teachers in a college geometry class volunteered to participate. This class was the second college level mathematics class in the preservice program of study. There were 15 students from the Spring 1989 semester and 3 from the Fall 1990 semester. The three transformations were covered in class through lectures. Each student was interviewed following the coverage of the material. "Based on his understanding of the concept, the researcher conjectured that students learn the concept in the order of learning definition of transformations first, and then singlepoint movement, then figure movement, and finally identification of transformations" (pp. 7273). Furthermore, "the researcher implicitly assumed that a student cannot understand any concept if he or she did not understand the previous concept" (p. 73) in the given order. The examples that Law used during his interview were limiting in number (eight questions) and the fact that each subject received one interview does not seem sufficient to develop a hierarchy. Preservice teachers' understandings of transformations were also examined by Edwards and Zazkis (1993) in a study of student teachers' initial understandings of the concepts of reflection and rotation. Edwards and Zazkis hypothesized that the student teachers, having no previous instruction in transformation geometry, would use primitive definitions of reflection and rotation even if such examples were not readily available. Results based on initial interviews with the 14 student teachers involved in the study as well as written assignments completed by the student teachers after computer experience with transformations supported the previously stated hypothesis. The interviews included situations where the subject was asked to rotate or reflect a figure on a table such as a number nine or a rectangle. The students were not given any parameters. For the most part, the students responded by carrying out the request without asking for more direction. In all responses, the subjects rotated the object around a center that was located on the object or reflected the object over a mirror line that ran through or adjacent to the object. After the interviews, the subjects spent 2 1/2 hours using the TGEO computer microworld described by Edwards (1990). The subjects worked individually during this time playing a game that involved the rotation and/or reflection of a figure so that the image of the figure completely covered a congruent figure. The written assignment that followed the computer experience involved the subject in a similar game using paper and pencil. The written assignment also included problems in which the student was asked to perform reflections and rotations given the mirror line or the center and angle of rotation. Edwards and Zazkis (1993) found that student solutions to the games both on the computer and with paper and pencil showed strategies based on using sequences of transformations to reach a target rather than a single reflection or rotation with the mirror line or center of rotation not located on the object. The solutions to 28% of the reflection and rotation problems were based on incorrect concepts of transformation. The misconceptions seemed to be based on the belief that rotations and reflections affect the object alone rather that the plane containing the object. For example, if the center of rotation was not located on the object, the student would translate the object so that it was located on the center before executing the rotation. Similarly, if the mirror line was not adjacent to the object, the subject would translate the object so that it was adjacent to the mirror line before reflecting the object about the mirror line. Edwards and Zazkis concluded that these prevalent misconceptions should not be ignored but should be addressed so that students may build upon their prior knowledge to understand new situations. Kidder (1976) investigated fourth, sixth, and eighthgrade students' ability to perform single transformations, compositions of transformation, and inverse transformations. Fifteen students identified by their teacher as belonging to the upper threefourths of the class in general ability, were administered a pretest dealing with spatial analogies. Following the pretest, the investigator spent 10 to 15 minutes with each student instructing them in an "operational" definition of each of the three transformations through demonstrations of the transformations and discussion of student attempts at performing the transformations. Rigid figures were transformed according to operations indicated by wire models (lines and or arrows) for each transformation by both investigator and student. The transformation test immediately followed the session. Students were asked to use triangles made of toothpicks to illustrate the indicated individual, composite, and inverse transformations. "The data did not support the experimental hypotheses that adolescents could perform Euclidean transformations at the representational level" (Kidder, 1976, p. 49). It seems reasonable to conjecture that 15 minutes was not ample time to learn a new concept and be able to perform applications of this concept on command, especially for the age group used in Kidder's study. This lack of support might also be due to the choice of model (for both object and operation). Schultz and Austin (1983) studied the effect of transformation direction on the difficulty of performing the transformations. Slides, flips, and turns were investigated in horizontal, vertical, and diagonal directions (for the purpose of consistency, slides, flips, and turns will be referred to as translations, reflections, and rotations respectively). Fifteen tasks involving each of these transformations and directions were performed by 105 first, third, and fifthgrade students. The students had no formal experience with transformations. Prior to collection of the data, the researchers insured that each student exhibited proficiency with copying the task object (a sailboat) in several different orientations. Two transparent platforms, one placed on top of the other, were positioned in front of each student. The bottom platform contained a flat model of a sailboat. The top platform was empty. After watching the investigator perform a transformation of the top platform, the students were to place a duplicate sailboat in the appropriate location and orientation on the top (image) platform. The students' responses to the tasks were scored on a scale from zero to four. The scorer reliability coefficient was 1.0. The 15 tasks consisted of five translations, five reflections, and five rotations. "[Translations] seem to be the easiest transformations for students this age to visualize. However, the direction of the transformation influences the relative difficulty of [rotations] or [reflections]" (Schultz & Austin, 1983, p. 101). Moyer (1978) investigated the connection between the mathematical organization of transformations and the cognitive structure of the learner. The following questions were addressed: Is dynamic presentation more appropriate than static presentation? Are reflections easier to perform than rotations and translations? Under what structure do different age students perform topological like or Euclideanlike transformations? Nine tasks involving translation, reflection, and rotation (referred to as slides, flips, and turns respectively) of marked circles were administered to the students. The tasks were divided into three categories: with color clues and with motion, without color clues and with motion, and with color clues and without motion. The students' responses on the tasks were scored on a scale from zero to three. There were 24 randomly selected students from each of the following grade levels: preschool, kindergarten, first, second, and third (120 subjects total). Analysis of the data indicated that there was a statistically significant difference between rotations with motion and without motion, but motion was not a significant factor for the translation and reflection tasks. The students performed best on the translation task followed by the reflection task. The rotation task was the most difficult for the students. Analysis of the data indicated a direct correlation between grade level and Euclideanlike structures. "The results of this study confirm that mathematical and cognitive structures are not always in total accord" (Moyer, 1978, p. 90). Williford (1972) also focused on primary grade students in his investigation involving transformation geometry. The purpose of Williford's study was to ascertain information regarding second and thirdgrade students' ability to perform transformations after being taught about transformations through a specific teaching strategy as well as the effects this instruction had on the students' spatial ability. Sixtythree second and thirdgrade students of average or above average general classroom performance ability (according to their teachers) were chosen for the study. The pre and posttest consisted of a spatial ability test and an achievement test on congruence and transformations. After administration of the pretest, the students were randomly divided into experimental and control groups. The experimental groups were removed from class for 12 sessions each lasting 25 to 30 minutes over the duration of four to five weeks. During these sessions, the control groups remained in class working on subjects unrelated to transformation geometry. The experimental groups were involved in demonstrations and activities related to congruence and transformations. The control groups received a lesson including an overview of the experimental groups' lessons. The lesson consisted of appropriate vocabulary and examples dealing with each of the three rigid motion transformations, reflection, rotation, and translation. These lessons were designed to compensate for the lack of experience with the terminology of the test. The decision to include the control group treatment was based on information from a pilot study. The researcher hypothesized that control group subjects may gain knowledge of transformations through everyday experiences but may not have the vocabulary to demonstrate this knowledge on the posttest. Following the completion of the lessons, the posttest was administered. The experimental groups performed significantly better then the control groups on the achievement test; however, there was no statistically significant difference between group performance on the spatial ability test. The experimental groups did not show success on the application questions on the achievement test of transformations. This may explain the lack of significance between groups on the spatial ability test, assuming that spatial ability is actually an application of transformations. This inability may also be explained in part by the age of the subjects (primary) or by the lack of exploration involved in the use of technology. Studies dealing with the rigid motion transformations that have included substantial instruction on transformations have used Logo or Motions (Edwards, 1990; JohnsonGentile, 1990; Pleet, 1990). Logo involves a language for the computer that must be taught to children in order for them to perform the transformations on the computer. Children do not typically find the program to be complicated; however, Logo takes time to learn. Motions is a transformation program based primarily on Logo. The majority of studies dealing with transformations have included little or no instruction on transformations (Hart, 1981; Kidder, 1976; Law, 1991; Moyer, 1978; Schultz & Austin, 1983; Soon, 1989). After reviewing the literature relevant to instruction with rigid motion transformations, it became evident that there is a lack of consensus between conclusions based on studies involving this topic. The studies previously reviewed have provided a basis for the discussion of the requirements for an appropriate instructional environment for successful construction of the concepts of reflection and rotation. According to the constructivist view of teaching and learning mathematics, a thorough investigation of students' ability to perform transformations must allow for an environment which includes student opportunities for communication with each other and the teacher as well as time to explore the mathematical concepts so knowledge may be built. The appropriate technology and a well organized instructional strategy will help to create this environment. Visualization Visualization studies tend to be involved with either the interaction between students' spatial ability and performance in specified areas of mathematics (Battista, Wheatley, & Talsma, 1982; Connor & Serbin, 1985; FerriniMundy, 1987; Kiser, 1990; Moses, 1977; Perunko, 1982; Tillotson, 1984) or students' trainability with visualization (Baker, 1990; Battista, Wheatley, & Talsma, 1982; BenChaim, Lappan, & Houang, 1988, 1989; Brinkmann, 1966; Connor, Schackman, & Serbin,1978; Connor & Serbin, 1985; FerriniMundy, 1987; Lord, 1985; Mendicino, 1958; Miller & Miller, 1977; Moses, 1977; Pallrand & Seeber, 1984; Smith & Schroeder, 1979; Tillotson, 1984). The current research involved both issues: hence, a review of related literature in both areas is presented below. The relationship between aptitude with spatial visualization and success in a computer enhanced instructional environment was examined by Kiser (1990). Fiftytwo students in two intact college algebra classes taught by the same instructor at an aeronautical university made up the subjects in the study. The Paper Folding Test and Form Board Test (Ekstrom, French, Harman, & Derman, 1976) were used to measure students spatial visualization ability. Both classes in Kiser's study experienced the same course objectives and criterion measures. The class instruction differed in mode of presentation for one week during which both classes were to learn the procedures for solving linear absolutevalue inequalities. The treatment group was exposed to a highlyvisual, computerenhanced instruction (CEI) version of the lesson based on a discovery approach to learning. The students were able to make conjectures about solutions and test them using the graphic visual feedback of the computer software while the teacher verified the algebraic solutions on the chalkboard. The control group experienced a more traditional presentation of the topic with the instructor working out algebraic solutions on the chalkboard while using the overhead projector to display graphic solutions. Kiser concluded that there was a significant interaction between spatial ability and the treatment. Furthermore, he found that "spatial ability is a significant predictor of achievement of the CEI group but not the traditional group" (Kiser, 1990, p. 95). The relationship between spatial ability and mathematical problem solving performance as well as the effect of instruction in perceptual tasks on spatial ability were explored by Moses (1977). The Card Rotation Test and Punched Holes Test were administered in conjunction with the Figure Rotation Test, Form Board Test, and Cube Comparison Test (Ekstrom, French, Harman, & Derman, 1976) to acquire a spatial ability score. Moses created a problem solving instrument to test mathematical problem solving performance. The subjects were fifthgrade students in four classes at one elementary school, two classes in the control group and two in the treatment group. All classes were administered all tests as protests. The control group received no training. The treatment group received training with perceptual tasks during one class period per week for nine weeks. The perceptual training consisted of three instructional phases. The first phase involved hands on experience with threedimensional objects. The second phase involved drawing of two and threedimensional objects. The third and final phase involved the translation of mathematical word problems into two and threedimensional representations. The researcher taught all lessons to each class in the treatment group and administered all tests to both treatment and control groups. The tests were given again directly following the training to all classes as posttests. Moses found that the Card Rotation Test and the Punched Holes Test were good measures of visualization ability for fifthgrade students. Moses found a positive correlation between spatial ability and mathematical problem solving performance. Training in perceptual tasks positively effected spatial ability but not mathematical problem solving performance. The experimental group experienced gains in the ability to solve spatial mathematical problems whereas the control group did not experience such gains. Among several relationships investigated by Perunko (1982) was the relationship between spatial ability and mathematical problem solving. Ninetythree community college students were given tests including measures of students' ability to rotate visual material (a measure of spatial ability) and students' ability to solve algebra and geometry problems. The students had not experienced success in high school mathematics courses and were enrolled in developmental algebra II and geometry classes at the community college. Perunko found a positive correlation between spatial ability and performance on visual mathematics problems but not verbal mathematics problems. It was also noted that students approached mathematics problems that combined visual and verbal components with visual techniques. Baker (1990) also investigated issues of spatial visualization with college students. Results from two related studies with undergraduate students revealed that students preparing for spatial related fields outperformed students preparing for less spatial related fields on tests of spatial reasoning. These same spatially oriented students reported more academic and nonacademic spatially enriching experiences then their less spatially oriented peers. Baker concluded that spatial reasoning could be improved through training. However, this conclusion was based on instruction in four scheduled mathematics classes; according to Baker, there was no control group and there was high experimental mortality. Tillotson (1984) and Connor and Serbin (1985) were interested in both spatial visualization ability as a predictor of success in mathematics and as a trainable skill. Tillotson worked with sixth grade students and Connor and Serbin worked with seventh and tenthgrade students in a series of three studies. All used the Card Rotation Test and Punched Holes Test among other instruments to measure spatial ability. Tillotson added the Cube Comparison Test to determine the subjects' spatial index. The specific purposes of Tillotson's research were to determine if spatial visualization ability was a predictor of problem solving performance and if training in spatial visualization would improve spatial visualization ability as well as problem solving performance. Students from two elementary schools all in the sixth grade made up the subjects in the study. Students at one school acted as the treatment group while students at the second school made up the control group (102 subjects total). The treatment lasted eight weeks and consisted of one 45minute session per week taught by the researcher. The pre and posttest were administered the week preceding and following the treatment respectively. The visualization training for the treatment group consisted of manipulating and imagining the manipulation of three dimensional models, performing transformations of two dimensional drawings, and solving problems. The control group received no visualization training. Based on the collected data, Tillotson concluded that spatial visualization is a trainable skill and a predictor of problem solving performance. However, instruction in visualization skills did not affect high visualization students differently from low visualization students. After testing seventh and tenthgrade students with a wide variety of both spatial orientation and visualization instruments and measures of mathematical achievement in two separate studies, Connor and Serbin (1985) concluded that spatial ability is a predictor of mathematical achievement for both boys and girls. A third study with eighthgrade students involved in a series of half hour spatial ability training sessions resulted in significant increases in spatial ability. BenChaim, Lappan, and Houang (1988, 1989) conducted a study with sixth through eighthgrade students that was motivated by the researchers' observation that middle school students tended to have difficulty communicating information about three dimensional solids. Building description tasks were created "to determine whether students' preference for representation mode and rate of success on the task would be affected by instruction in spatial visualization activities" (BenChaim, Lappan, & Houang, 1989, p. 123). The tasks required the students to describe a three dimensional solid constructed out of cubes. The students were not instructed how to communicate the information and the mode of representation was not implied in the directions for the tasks. Seven classes in each of the three grade levels participated in the study. All subjects were administered a building description task as the pre and posttest with three weeks of spatial visualization activities in the interim. A wide variety of representations for building descriptions were present in both pre and posttests, however, out of the three modes of representation (verbal, graphic, and mixed), the verbal mode was nearly eliminated after the visualization activity intervention. The rate of success was also dramatically improved following the visualization activities. It is clear that BenChaim, Lappan, and Houang's (1988) study supports the claim that training can improve spatial visualization skills. "It should be noted that a significant positive training effect was also evident in students' performance on the Middle Grades Mathematics Project (MGMP) Spatial Visualization Test. In addition, a persistence over time of the effects was demonstrated" (Ben Chaim, Lappan, & Houang, 1989, p. 142). As with the previous study, Brinkmann (1966) improved eighth grade students' spatial visualization through training. The training in Brinkmann's study was in the form of programmed instruction in elementary geometry including topics about points, lines, line segments, rays, angles, planes, plane figures, and solids. The program stressed problem solving rather than mathematical proof. Sixty students from two eighthgrade classes were assigned to treatment and control groups so that the groups were approximately matched according to visualization pretest score and sex. The treatment group followed the described programmed instruction during mathematics class for three weeks. The control group received no training. The treatment group performed significantly better than the control group on a geometry inventory based on the instruction as well as on spatial visualization. Success in visualization training was also experienced in studies as shown by Miller and Miller (1977) and Connor, Schackman, and Serbin (1978) with firstgrade students. Miller and Miller illustrated this success in three similar experiments involving visual perspective. Each study compared the students' ability to coordinate projective space after receiving different levels of training in perspectives from teacher directed visualization activities to no visualization activities. In all but the first study, the students experiencing the visualization training performed significantly better on a test of perspective ability then students not exposed to such training. The first study showed significant improvement for boys receiving the training but not for girls. Connor, Schackman, and Serbin (1978) found testretest improvement with the control group as well as effects from training in spatial visualization with the treatment group. The test was similar to the training; when given a different test of spatial ability, the same gains in performance was not realized. Smith and Schroeder (1979) investigated but did not find differences between boys' and girls' spatial ability. In this study, fourthgrade students were randomly assigned to two groups within three intact classes. One group received the spatial visualization test prior to training in tasks similar to those tested, while the other group was tested after the training. The training and test were both based on Tangram activities. Results indicated that fourthgrade boys and girls were equally trainable in spatial visualization and significantly outperformed students who were not similarly instructed. Findings from FerriniMundy (1987) and Mendicino (1958) were not consistent with the success in visualization training illustrated in the studies previously discussed. FerriniMundy investigated the effects of a training program in spatial skills on college students spatial ability and achievement in calculus. A random sample of 334 students was selected from 1054 preregistered Calculus I students and randomly assigned to seven classes making up two treatment groups and one control group. The treatment was made up of six lessons presented via audiotapes and slides with students completing a worksheet with each lesson. One treatment group did not use accompanying manipulatives the other treatment group used the manipulatives along with the lessons. The lessons involved two and threedimensional tasks including rotations and translations of twodimensional representations to threedimensional images. The control group received no spatial training. Based on pre and posttest scores, FerriniMundy found no treatment effect on the calculus or spatial measures. "The case for enhancing college students' spatial scores through the type of training in this study was not supported by the results" (FerriniMundy, 1987, p. 136). Students' scores might have been different had the training involved teacherstudent and/or studentstudent interaction. Mendicino (1958) compared the spatial perception scores of tenth grade students enrolled in a vocational machine shop class to students not enrolled in a vocational class of that nature. Pretests of mechanical reasoning and space perception were given to both groups at the beginning of the year and posttests were given at the end of the year. No difference was observed between the group receiving vocational machine shop instruction and the group not receiving the instruction. Lord (1985) worked with students on a program similar to FerriniMundy's but he experienced success in positively effecting college students' spatial ability through training. He worked with undergraduate biology students, randomly assigning 84 students to treatment and control groups. The treatment group received training with two and threedimensional figures; the control group received no training. The treatment group met thirty minutes per week for twelve weeks in the intervention. During the lessons, the students worked with both manual and mental operations dealing with three dimensional figures. Most of the lessons involved imagining the crosssections of sliced threedimensional objects. The intervention was effective in enhancing the scores of the treatment group on tasks similar to those practiced during the lesson. However, differences in pre and posttest scores on spatial visualization and spatial orientation were also significantly in favor of the treatment group. Another study centered around improving the spatial ability of undergraduate science students illustrated success in increasing spatial ability (Pallrand & Seeber, 1984). Four groups of students from a community college made up the subjects for this study: three groups from an introductory physics course and one group from a liberal arts course whose students had not taken and were not planning to take physics. One physics group received the treatment, one physics group received the placebo treatment, and one physics group received no treatment. The liberal arts group was tested to compare the spatial ability of physics students to nonphysics students. All students were given the same pre and posttests consisting of perception, spatial orientation, and spatial visualization measures. Included in the tests were the Paper Folding Test and the Card Rotation Test. The treatment intervention lasted 65 minutes per week for ten weeks and consisted of perspective drawings, geometry of lines, planes, angles, solid figures, and transformations, and relative position activities. The placebo intervention was also 65 minutes per week for ten weeks but consisted of lessons on the history of science. The control and liberal arts groups received no intervention. All groups experienced gains in spatial ability, however, the liberal arts group received the smallest gain leading Pallrand and Seeber to believe that taking physics may improve spatial ability of the students involved. Additionally, "Visualspatial scores of the liberal arts group were lower than those of the physics sections, suggesting that visual spatial ability influences course selection" (Pallrand & Seeber, 1984, p. 507). The following account of educational research with undergraduate students focused on preservice elementary teachers. Battista, Wheatley, and Talsma (1982) investigated the importance of spatial visualization and cognitive development for learning of geometry. They further investigated whether the geometry learning would enhance spatial visualization of the same preservice elementary teachers. Data were collected on 82 students enrolled in a geometry course. Pre and posttests were given to the students at the beginning and end of the semester respectively. During the semester, the students experienced lessons in symmetry, paper folding, tracing, and Mira math as well as transformation geometry through handson activities. Not only were spatial visualization, as measured by the Purdue Spatial Visualization Test: Rotations, and cognitive development, as measured by a modified version of the Longeot test of cognitive development, directly related to performance in the geometry class, but students spatial visualization significantly increased by the end of the semester. This latter finding suggests that the type of geometry lessons used in the course may actually improve students' spatial visualization ability. Further proof of students' trainability with respect to visualization skills was also exemplified in Iben's (1988) comparative study between seventh and eighthgrade students in Japan and the United States. Iben investigated students' development of spatial relations and found that Japanese students had significantly more developed spatial relations than American students of the same grade. Four explanations were offered for this observed difference. Japanese students have spent approximately 2.4 school years more in school by the time they are in seventh or eighthgrade than U. S. students of the same grade. The Socratic method is used to teach Japanese students whereas students in the U. S. are primarily taught using direct instruction. Students in the U. S. are generally given computational spatial relations activities while Japanese students have handson spatial relations activities. And, spatial relations is a large part of the early Japanese curriculum through paper folding activities such as origami. Students in the United States do not tend to be exposed to spatial relations activities until seventh or eighthgrade. Based on Iben's observations, Japanese students' superior performance on spatial relations activities compared to the performance on the same activities by U. S. students may be due in part to the substantially greater amount of time spent in such training as well as the type of training received. As with the SIMS (1985) study, students in the United States tend to experience much less handson activity in the classroom then students in other countries. Many of the reviewed visualization studies concluded that students' visualization ability could be improved and that visualization ability was a predictor of achievement with mathematics. Study of the rigid motion transformation seems to be an appropriate setting for visualization ability to be improved as transforming geometric figures can involve the visualization of the figures as they are flipped, turned and slid. This improvement of visualization ability is important to increase students' achievement in mathematics. Limited English Proficient Students Bilingualeducationprogram research and evaluation have been driven by concerns for the development of English and of academics among LEP students. These studies have taken for granted the school mathematics curriculum that LEP students are exposed to and, even when problems in instruction are noted, those concerns get cast in terms of language development. (Secada, 1992b, p. 218) This same dilemma does not currently exist in school science (Sutman & Guzman, 1992). Cheche Konnen (Rosebery, Warren, & Conant, 1992; Warren & Rosebery, 1993; and Warren & Rosebery, 1992) is a program conducted by the Teacher Education Research Center in Cambridge. Working with bilingual teachers and LEP students, Rosebery and Warren ". are attempting to elaborate an approach to science teaching and learning that supports the development of scientific sensemaking communities in the classroom" (1992). Data have been collected describing the training of teachers and the resulting experiences of the students. The teachers in the program include bilingual teachers, English as a second language teachers all with no training in teaching science, and a science specialist. In pilot projects, the researchers found that the training of the teachers must include both science content and science pedagogy to make effective facilitators, coinvestigators, and mentors to the students. Teachers needed to experience scientific sensemaking in a setting conducive to collaborative inquiry themselves. The teachers are trained by the researchers in a setting similar to that in which the students are placed. The teachers and the researchers maintain contact throughout the program. The emphasis on collaborative inquiry in the training of the teachers and the experiences of the students build on Vygotsky's (1978) belief, in that "robust knowledge and understandings are socially constructed through talk, activity, and interaction around meaningful problems and tools" (Rosebery, personal conversation, 2/17/94; Warren & Rosebery, 1992, p. 280). The collaborative inquiry approach allows for shared responsibility in learning. This is especially helpful for the LEP students as they are contending with learning the language as well as the science content. Rosebery and Warren might argue that language and science cannot be separated. Data have been collected with students in kindergarten, seventh and eighth grade, and tenth grade. Exploration was based on diverse content areas including weather patterns, water taste tests and bacteria content, and swamp life. The science that the students experience is partly controlled by the inquiries the students generate. Results from data collection gathered from the aforementioned settings suggest that students' knowledge of the topic explored and their ability to reason about hypotheses and experiments in an organized manner increased between September and June. A major goal of Cheche Konnen is to forge links between learning science and doing science, and among science, mathematics, and language. This is in large part what makes it a powerful model for language minority students, in particular, and perhaps for all students. (Rosebery, Warren, & Conant, 1992, p. 3) Sutman and Guzman (1992) reported a thematic based program similar to Cheche Konnen for teaching science to LEP students. The primary emphasis of this program is illustrated through its assumption that instruction in science and the English language can coexist effectively without placing excessive emphasis on the students' first language. The authors believe that use of the students' first language by the teacher may enhance instruction when used judiciously, however, it is not entirely necessary. Thematic lessons are used in this approach to scientific inquiry and discovery to provide students the opportunity to become accustomed to the vocabulary and syntax associated with one concept. The authors described lessons called IALS (Integrated Activity (based) Learning Sequences) that have been developed for both the elementary and secondary curriculum. The lessons include preparation and materials for teachers, objectives, handson activities for students, and questions that students might ask. Sutman and Guzman contend that even though a discovery/inquiry lesson has been outlined, it is up to the teacher to maintain the environment in which students are free to discover and inquire about the concept. Existing research in science education focused on providing meaningful science instruction for LEP students is related to meaningful mathematics instruction for LEP students. The instruction of LEP students in both science and mathematics has as its focus the language development of the students in a content oriented situation or the students' development of conceptual knowledge of the content while the associated language is developed. The research in science education included in this review had the latter as its focus as does this research. Summary Numerous studies have provided evidence that students can learn to identify and perform the rigid motion transformations of reflection and rotation and improve their spatial visualization. Results from research have provided evidence that spatial visualization is linked to achievement in mathematics. Projects have set the framework for environments in which LEP students may work within their zones of proximal development constructing meaningful mathematics while simultaneously building their associated English vocabulary. It is the synthesis of the information gained from these valuable studies and the theoretical framework previously described that provides the foundation and justification for the current study. This study extends the research of Edwards (1990, 1991) by placing the students within an entire class during class time rather than after school in groups of two or three, using computer software that is much less restricting and much easier to learn than Logo, and including LEP students in the sample. The duration of this study which included three weeks of instruction using the Sketchpad to construct the concepts of reflection and rotation was influenced by the lack of generalizability of knowledge reported by Ernest (1986). The students in Ernest's study received only six hours of instruction with transformations. Pleet's (1990) recommendation combined with information gained from programs in the science education of LEP students lead the researcher to make the decision to teach the experimental treatment lessons due to experience using the Sketchpad in computer laboratory settings. The instruments used for this study were chosen based on information gained from a review of the related literature. Hart's (1981) test of students' ability to identify and perform reflections and rotations, the basis of many previous studies, influenced the types of problems and grading keys used in this study. The Card Rotation Test and Paper Folding Test used as measures of spatial visualization for this study were used for similar purposes in many of the studies reviewed in this chapter. The choice of instructional environment was predominately influenced by the theoretical bases for the study. The constructivist view of teaching and learning mathematics combined with the zone of proximal development requires that the instructional environment of the learner be such that collaborative inquiry is emphasized and accommodated. Students working in pairs at computers to discover the properties of reflection and rotation using computer software that allows for student conjectures and provides visual feedback is consistent with these requirements. Results from research conducted by Soon (1989), Edwards and Zazkis (1993), and Kiser (1990) reinforced the instructional environment used in this study and Moyer (1978) provided evidence that the motion involved with use of the Sketchpad might aid students' acquisition of the concepts involved with rotation. Many studies provided evidence that spatial visualization is a trainable ability, however they also lead this researcher to believe that the type of lesson used to increase students' spatial visualization effects the extent of the increase in spatial visualization. Instructional environments providing studentstudent and/or studentteacher interaction and handson activities seemed most effective. This study attempted to provide an environment incorporating such interaction and activities. CHAPTER III METHODOLOGY Research Obiective The purposes of this study included: (1) to explore the effects of a dynamic instructional environment and students' visualization ability on the identification of and ability to perform the rigid motion transformations of reflection and rotation; and (2) to explore the effects of the relationship between the students' level of visualization and the instructional environment on the identification of and ability to perform these rigid motion transformations. Another purpose of this study was to investigate the effects of students' English proficiency on identification and performance of reflections and rotations and the effects of the relationship between students' English proficiency and a dynamic instructional environment on the identification of and ability to perform the rigid motion transformations of reflection and rotation. Finally, the study was designed to ascertain the effects of a dynamic instructional environment and students' English proficiency on students' visualization. The treatment in the dynamic instructional environment included exploring the rigid motion transformations of reflection and rotation through a dynamic presentation and subsequent activities using The Geometer's Sketchpad (Jackiw, 1991). The following null hypotheses were tested: 1. Students' level of visualization will not interact with their instructional environment to affect their ability to identify and perform reflections and rotations in a (a) dynamic testing environment or a (b) static testing environment. 2. Students' English proficiency will not interact with their instructional environment to affect their ability to identify and perform reflections and rotations in a (a) dynamic testing environment or a (b) static testing environment. 3. Students' (a) instructional environment, (b) visualization level, and (c) English proficiency will not affect their ability to identify and perform reflections and rotations in a (a) dynamic testing environment or a (b) static testing environment. 4. Students' (a) instructional environment and (b) English proficiency will not affect their (a) twodimensional or (b) threedimensional visualization. Measures The measures for the study consisted of three covariates (one continuous and two discontinuous) and four posttests. The covariates consisted of the Card Rotation Test and Paper Folding Test of the Kit of FactorReferenced Cognitive Tests (Ekstrom et al., 1976) as well as the Language Assessment Battery (LAB). Two posttests were designed by the researcher and were versions of a Rotation/Reflection Instrument. The Card Rotation and Paper Folding Tests were also used as posttests. Paoer Folding and Card Rotation Tests The Card Rotation Test, S1 of the Spatial Orientation Factor and the Paper Folding Test, VZ2 of the Visualization Factor were used to measure students' visualization. The Card Rotation Test was used to control for initial differences between the subjects and was used as a continuous measure of students' twodimensional visualization. The students were divided into thirds according to their scores on the Paper Folding Test. Based on their scores, students were categorized as having high, medium, and low visualization levels. It was also used as a continuous measure when investigating students' change in threedimensional visualization. The Card Rotation Test required that the student differentiate between figures that were equivalent in every way but orientation from cards that had been reflected. The Paper Folding Test requires that the student imagine a piece of paper being folded and having holes punched through all thicknesses according to drawings. The student was then to choose the appropriate result once the imagined paper had been unfolded. Both tests contain two parts and each part has a time limit of three minutes. Each of the tests calls for spatial orientation and visualization skills required to identify and perform rotations and reflections. The Card Rotation Test involves the ability to visualize movement in twodimensional space and the Paper Folding Test involves the ability to visualize movement in threedimensional space. The Card Rotation Test and Paper Folding Test were each used by Edwards (1991) in her investigation of students' construction of the concepts of transformations. Language Assessment Battery The Language Assessment Battery was used as a measure of English proficiency. Scores were provided by the students' middle school and were used to categorize students as either English proficient or limited English proficient according to district mandated levels. Students scoring at or above the sixtieth percentile were assumed to be English proficient, students scoring below the sixtieth percentile were assigned to the limited English proficient category. Reflection/Rotation Instruments A paper and pencil version and a computer version of the Reflection/Rotation Instrument were designed by the researcher, used in the pilot study, and validated for content by four experts. The experts were two mathematics education professors, a mathematician, and an educational technology professor. The instruments were adapted, with permission, from the examination designed for a large scale British study which was part of Concepts in Secondary Mathematics and Science (Hart, 1981). The computer and paper and pencil versions of the Reflection/Rotation Instrument are similar in content and difficulty. However, the computer version was more dynamic in nature than the paper and pencil version as the computer version involved motion. The paper and pencil version was more static in nature compared to the computer version as the students were not exposed to motion while taking the examination. The paper and pencil Reflection/Rotation Instrument can be found in Appendix A. The acetate overlays used to grade the paper and pencil version of the instrument can be found in Appendix B. The objectives of both versions of the Reflection/Rotation Instruments are given in Table 1. A description of the motions for the computer version of the Reflection/Rotation Instrument is given in Table 2. The computer and paper and pencil versions of the instrument were not equivalent forms of the same test, therefore, the students' performance on the two tests could not be statistically compared. Pilot Study A pilot study was conducted during May 1994. The study included three classes taught by the researcher. The researcher trained the classroom teacher in use of The Geometer's Sketchpad during the six weeks preceding the pilot study. The pilot study began with the administration of a personal inventory designed by the researcher to determine students' language background and proficiency with English. All classes were then given the Card Rotation Test and Paper Folding Test. Following the administration of the covariates, the classroom teacher and researcher instructed the students from all three classes in use of The Geometer's Sketchpad. Lessons on angle measure and distance were used to train the students After the students became familiar with the A computer program, the paper and pencil and computer versions of the Reflection/Rotation Instrument were administered to all classes as protests. Following the tests, two classes were given the reflection Table 1 Objectives of Reflection/Rotation Instruments Items Objective: Test the student's ability to . 14 recognize examples and nonexamples of reflections. 58 perform reflections given a figure, a mirror line, and a background grid. 912 perform reflections given a figure and a mirror line. 1316 recognize examples and nonexamples of reflections when the mirror line is hidden and draw mirror lines where appropriate. 1720 recognize examples and nonexamples of rotations. 2124 perform rotations given a figure, a center point, and a reference circle. 2532 perform rotations given a figure and a center point. 3336 draw the center of rotation given a figure and its image. 3740 determine the correct composition of transformations given the figures and their images without mirror lines or center points. Table 2 Motion in Computer Version of Reflection/Rotation Instruments Items Description of motion: The figure moves from the preimage ... 14 to the image following the properties of a reflection when the example portrays a reflection. 512 to the mirror line then disappears. The movement follows the properties of a reflection. 1316 to the image following the properties of a reflection when the example portrays a reflection. 1720 to the image following the properties of a rotation. The center of rotation may or may not be correctly drawn in each example. 2132 approximately 200 in the direction of the given reflection then disappears. The movement follows the properties of a rotation. 3336 to the image following the properties of a rotation. The center point is hidden in each example. 3740 to the image to the image after composition following the properties of rotations and reflections as appropriate. All mirror lines and center points are hidden. and rotation unit using the computer. The third class was given the reflection and rotation unit without the computer. Upon completion of the units, all students were given the paper and computer versions of the Reflection/Rotation Instrument as posttests. The Reflection/Rotation Instruments were scored by the researcher using acetate overlays similar to those used to score the CSMS exams (Hart, 1981). Reliability estimates were calculated for the paper and computer versions of the Reflection/Rotation Instrument based on the scores from the protests using the Kuder Richardson (KR20) formula. The estimates were found to be .87 and .84 respectively. There were no statistically significant differences between the three classes on the posttests after controlling for initial differences between students using the protests and the visualization tests. Several observations by the researcher may explain this lack of treatment effect. All students were exposed to aspects of the treatment in various forms from the pretest and teaching. The computer version of the Reflection/Rotation Instrument was used as a pretest. This version of the Instrument involves motion similar to that experienced by the treatment group in the dynamic instructional environment. Hence, while learning in a traditional instructional environment, the control group may have been envisioning the movement experienced during the administration of the pretest and in this way, experiencing the treatment. The researcher taught all classes. Due to the researcher's extensive experience using The Geometer's Sketchpad to teach transformations, the teaching of the control group was contaminated with the treatment. This observation was dually noted by the classroom teacher who indicated that the static lessons were not taught using traditional means of explanation. Lastly, the dynamic unit experienced by the treatment groups did not include a sufficient number of lessons to highlight the properties of reflections and rotations. This lack was due to the belief held by the researcher at that time that the students would not be capable of using the Sketchpad for lessons of such complexity. The following changes were based on previously described observations made during and directly after the pilot study. The paper and pencil and computer versions of the Reflection/Rotation Instrument were not used as protests, only as posttests. The Card Rotation Test was found to be highly correlated with the pretest and was therefore sufficient for the continuous covariate. The Paper Folding Test was to be used as a discontinuous measure of students' level of visualization to determine whether or not this level affected performance on the dependent measures. Classroom teachers with no previous experience using The Geometer's Sketchpad taught the control classes. Lastly, more advanced lessons designed to address the properties of reflection and rotation using The Geometer's Sketchpad were created to be used with the treatment groups. Sampling Procedures Letters were sent to district mathematics supervisors of all counties in Florida believed to have high percentages of LEP students. The supervisors were asked for assistance in locating middle schools having Macintosh computer laboratories and LEP students enrolled in English speaking, heterogeneously grouped mathematics classes. Based on the responses, a central Florida public school district was chosen for the study. This district most closely met the requirements of the study. The supervisor of that county directed the researcher to a middle school having the necessary attributes for the purposes of the study. The researcher met with the chairperson of the mathematics department of the school to request her participation in the study. The teacher and her principal agreed to participate in the study and a proposal to conduct research was submitted to the school district and subsequently accepted. Both the pilot study and the study were conducted in the same school. The pilot study was conducted during the school year preceding the school year in which the data were collected for the study. The pilot study was conducted with the chairperson present using three heterogeneously grouped mathematics classes usually taught by the chairperson. The remaining two eighthgrade teachers were contacted during the summer between the pilot study and the study and agreed to participate in the study. This middle school had three eighthgrade teams, each eighth grade team had one mathematics teacher who taught four mathematics classes. One mathematics class on each team was made up of "advanced" students chosen at the end of their sixth grade school year based on mathematics grades. The other three classes consisted of the remaining students, heterogeneously grouped. All classes contained LEP students. The researcher began the school year as a longterm substitute teacher replacing the chairperson for the first five weeks of the school year. Three classes normally taught by the chairperson and one class taught by the remaining teachers were used as the Treatment Group and taught by the researcher. The first three classes made up all of the heterogeneously grouped classes taught by the chairperson and the fourth class was chosen based on heterogeneity of grouping and ease of scheduling. The remaining five classes, two taught by one of the remaining teachers and three by the other, made up the remaining heterogeneously grouped eighth grade mathematics classes in the school. These five classes made up the Control Group. Experimental Treatment and Procedures Training of Teachers The teachers of the Control Group did not undergo training in the use of The Geometer's Sketchpad This was necessary to insure that the study would not be contaminated in the manner observed in the pilot study. The teachers were briefed about the study and the need for the teachers to avoid use of the Sketchpad until the completion of the study. The teachers from the Control Group were promised training in use of The Geometer's Sketchpad after completion of the study. Training of Students The students from the Treatment and Control Groups were taught how to use The Geometer's Sketchpad prior to the collection of the data. All training was conducted by the researcher and lasted one class period each. The middle school followed block scheduling, therefore the students spent approximately two consecutive hours in the mathematics class period every other day. The training lesson was designed by the researcher to require students to use all of the tools necessary to successfully complete the computer Reflection/Rotation Instrument. At no time during the training did the students witness or perform reflections or rotations using The Geometer's Sketchpad. A detailed account of the lesson is in Appendix C. Captured computer screens from the activities associated with the lesson were printed and can be found in Appendix D. The teachers for the Control Group administered the Card Rotation and Paper Folding Tests to the Treatment Groups while their classes were trained by the researcher. Data Collection Following the training on use of The Geometer's Sketchpad, all classes were given the Card Rotation Test and Paper Folding Test. After completion of the above measures, the Control Group was taught about the concepts of reflection and rotation using the traditional textbook approach. The control group teachers used Chapter Eight of Transition Mathematics (Usiskin et al., 1990), the adopted textbook for eighthgrade at this school. This chapter addresses the rigid motion transformations. The researcher taught the lessons in the computer lab using The Geometer's Sketchpad. The treatment classes were held in the Macintosh computer lab throughout the unit on reflections and rotations. Since there were 15 computers and from 28 to 32 students in each mathematics class, the students usually worked in pairs at the computers. For the most part, the students chose their own partners. Several pairs included students whose first language was Spanish. These students tended to converse in their first language when discussing properties of the geometric transformations as well as problems posed by the researcher. The researcher assumed the role of facilitator and problem poser for the Treatment Group. An experimental version of The Geometer's Sketchpad was used throughout the study. The transformation commands; translate, rotate, reflect, mark center, and mark mirror were accompanied by descriptive icons (pictorial representations) for each command. The adjusted menu of the experimental version was requested to aid in the students' acquisition of vocabulary related to transformations. The researcher did not call attention to the icons at any time during the study. A subsample of the Treatment Group was given a short questionnaire about the usefulness of the icons approximately two weeks after the administration of the posttests. The reflection and rotation units for the Treatment and Control Groups each lasted approximately eight class periods over three to four weeks. The Card Rotation Test and Paper Folding Test as well as the paper and computer versions of the Reflection/Rotation Instrument were administered to both groups at the end of the units. Treatment Lessons All experimental treatment lessons were preplanned so that any teacher with adequate knowledge of geometry, middle school pedagogy, the computer software, and computer lab pedagogy could teach using the lesson plan. Lesson plans for all experimental treatment lessons have been included in Appendix E. A brief outline follows: Lesson 1 Objective: to have students learn how to execute reflections and rotations using the Sketchpad. Procedure: students worked in pairs on exercises requiring students to conjecture about positions of mirror lines and centers of rotation so that the images of figures after the transformations would rest entirely within the interior of given squares. Students were given keys to help with the steps to perform reflections and rotations (see Appendix F). A typical screen is located in Appendix G Lesson 2 Objective: to have students make conjectures about positions of the images of figures after a reflection or a rotation. Procedure: students worked in pairs conjecturing about the position images after reflections or rotations given a figure and either a mirror line or a center and angle of rotation. The students checked each other's conjectures by performing the given transformation using the Sketchpad. The class then played a game involving more practice with similar conjectures. Lesson 3 Objective: to allow for more practice in conjecturing with rotations and to introduce the computer project. Procedure: The students practiced conjecturing using the same format used during the previous lesson. The instructor then introduced the computer project. The students spent the remaining class time working on the computer projects. The computer project is described in detail in the description of the lessons in Appendix E. Lesson 4 Objective: to have students work on computer project. Procedure: students worked in pairs on computer project introduced during lesson 3. Lesson 5 Objective: to have students complete and present computer projects. Procedure: Students spent first part of class time completing projects and remaining class time presenting projects to classmates. Lesson 6 Objective: to have students present remaining projects and to review properties of reflections and rotations. Procedure: students spent first part of class time presenting projects to classmates and remaining class time reviewing the properties of reflections and rotations with the instructor. Research Design Quantitative Methodology A three factor, nonequivalent controlgroup design was used for the study. This quasiexperimental research design involved a 2X2X3 matrix to examine three factors. The three factors were the level of computer use, the level of English proficiency, and the level of visualization of the students. There were two groups, one Treatment Group and one Control Group, with four classes in the Treatment Group and five classes in the Control Group. The Treatment Group used the computer unit on reflections and rotations. The Control Group followed the lessons included in the text used by the middle school. An analysis of covariance (ANCOVA) was used to control for initial differences between groups. Two tests from the Kit of FactorReferenced Cognitive Tests (Ekstrom, et. al., 1976) and the Language Assessment Battery (LAB) served as covariates. The Paper Folding Test was the discontinuous measure used to assign students to visualization levels. The Card Rotation Test was the continuous covariate used to control for initial differences between groups. The objective of the design was to determine the effects of the independent variables (computer use, English proficiency, and visualization level), individually and interactively, on the dependent variables (posttest scores). Of interest were the interaction between computer use and English proficiency and the interaction between computer use and visualization level. The performance on two dependent variables was investigated, the computer version and the paper version of the Reflection/Rotation Instrument. Two separate ANCOVAs were run to determine the effects of the independent variables on each of the dependent variables. The first ANCOVA described the effect of the independent variables individually and interactively on the performance of the sample on the computer version of the posttest. The second ANCOVA described the effect of the independent variables individually and interactively on the performance of the sample on the paper version of the posttest. 69 Additional hypotheses were tested using a 2X2 matrix to examine the effects of a dynamic instructional environment and English proficiency on students' visualization. The last two ANCOVAs were run with the purpose of describing the effects of the two independent variables on the performance of the sample on the Card Rotation Test and Paper Folding Test. CHAPTER IV QUANTITATIVE RESULTS Descriptive Statistics Sample The sample consisted of all students enrolled in heterogeneously grouped, English language, eighthgrade mathematics classes in a central Florida middle school. Data were collected from 241 students, 109 in the Treatment Group and 132 in the Control Group. Table 3 describes the sample according to race and Table 4 describes the sample according to sex. Table 3 Frequency and Percentage of Race by Group Race Treatment Group Control Group Sample African American 8 (7.3%) 9 (6.8%) 17 (7.1%) American Indian 0 (0%) 1 (0.8%) 1 (0.4%) Asian American 3 (2.8%) 1 (0.8%) 4 (1.7%) Caucasian 63 (57.8%) 79 (59.8%) 142 (58.9%) Hispanic American 35 (32.1%) 42 (31.8%) 77 (32%) Table 4 Frequency and Percentage of Sex by Group Sex Treatment Group Control Group Sample Female 50 (45.9%) 67 (50.8%) 117(48.5%) Male 59 (54.1%) 65 (49.2%) 124 (51.5%) Instruments Descriptive statistics for the Paper and Pencil Reflection/Rotation Posttest, Computer Reflection/Rotation Posttest, Card Rotation Test, and Paper Folding Test for the entire sample, Experimental Group, and Control Group are given in Tables 5, 6, and 7 respectively. The mean, standard deviation, and range have been calculated for each instrument. Table 5 Entire Sample Posttest Means, Standard Deviations, and Ranges for All Instruments Instrument Mean S. D. Range Total Points Paper and Pencil Test 17.7 6.9 5 36.5 40 Computer Test 18.2 7.9 236 40 Card Rotation Test 98.7 39.4 5158 160 Paper Folding Test 7.1 4.5 3.820 20 Table 6 Experimental Group Posttest Means, Standard Deviations, and Ranges for All Instruments Instrument Mean S. D. Range Total Points Paper and Pencil Test 20.9 7 5 36.5 40 Computer Test 21.5 7.6 3.5 36 40 Card Rotation Test 102.5 38.9 3154 160 Paper Folding Test 7.5 4.2 1.3 16.3 20 Table 7 Control Group Posttest Means, Standard Deviations, and Ranges for All Instruments Instrument Mean S. D. Range Total Points Paper and Pencil Test 14.9 5.3 5.5 31 40 Computer Test 15.4 6.9 235 40 Card Rotation Test 95.6 39.5 5 158 160 Paper Folding Test 6.7 4.6 3.8 20 20 Statistical Analysis Initial differences were controlled for through the use of the Card Rotation Test (correlates strongly with Reflection/Rotation Instruments), the Paper Folding Test (determines visualization level), and the Language Assessment Battery (distinguishes between limited English proficient (LEP) and English proficient (EP) students). There were no statistically significant interactions between the students' level of visualization and the treatment based on the results of both the Dynamic and Static Reflection/Rotation Instruments (see Tables 8 and 9). There was no statistically significant interaction between students' English proficiency and the treatment based on the results of both the Dynamic and Static Reflection/Rotation Instruments (see Tables 8 and 9). Therefore, the following null hypotheses could not be rejected: Students' level of visualization will not interact with their instructional environment to affect their ability to identify and perform reflections and rotations in a (a) dynamic testing environment or a (b) static testing environment. Students' English proficiency will not interact with their instructional environment to affect their ability to identify and perform reflections and rotations in a (a) dynamic testing environment or a (b) static testing environment. Finding no significant interaction, the interaction terms were removed from the model. After removal of the interaction terms, students' instructional environment was a statistically significant variable on performance on both the dynamic and static Reflection/Rotation Instruments (see Tables 10 and 11). Visualization level was also a significant predictor of students' performance on the dependent measures (see Tables 10 and 11) with students at the high visualization level performing better than students at the medium visualization level, as expected. Students at the medium visualization level performed better than students at the low visualization level (see Table 12). Furthermore, it was determined that LEP students did not perform statistically significantly differently on either the dynamic Reflection/Rotation Table 8 Analysis of Covariance: Dynamic Reflection/Rotation Instrument Source DF SS F Ratio p Card Rotation Test 1 1385.75 35.12* 0.00 Visualization Level 2 550.41 6.97* 0.00 English Proficiency 1 65.00 1.65 0.20 Treatment 1 80.92 2.05 0.15 Treatment x Viz. Level 2 43.08 0.55 0.58 Treatment x Eng. Prof. 1 35.47 0.90 0.34 Model 8 5520.53 17.49* 0.00 Error 212 8365.73 Note: *significant for p=.01 Table 9 Analysis of Covariance: Static Reflection/Rotation Instrument Source DF SS F Ratio p Card Rotation Test 1 883.50 29.68* 0.00 Visualization Level 2 466.52 7.84* 0.00 English Proficiency 1 27.81 0.93 0.33 Treatment 1 103.44 3.47 0.06 Treatment x Viz. Level 2 111.83 1.88 0.16 Treatment x Eng. Prof. 1 6.78 0.23 0.63 Model 8 4412.75 18.53* 0.00 Error 212 6490.14 Note: *significant for p=.01 Instrument or the static Reflection/Rotation Instrument when compared to their EP peers (see Tables 10 and 11). Therefore, the following null hypothesis could be rejected: Students' (a) instructional environment and (b) visualization level will not affect their ability to identify and perform reflections and rotations in a (a) dynamic testing environment or a (b) static testing environment. The following null hypothesis could not be rejected: Students' English proficiency will not affect their ability to identify and perform reflections and rotations in a (a) dynamic testing environment or a (b) static testing environment. Table 10 Analysis of Covariance: Dynamic Reflection/Rotation Instrument Source DF SS F Ratio p Card Rotation Test 1 1425.00 36.26* 0.00 Visualization Level 2 1104.64 14.05* 0.00 English Proficiency 1 34.39 0.88 0.35 Treatment 1 1858.14 47.28* 0.00 Model 5 5437.17 27.67* 0.00 Error 215 8449.09 Note: *significant for p=.01 Table 11 Analysis of Covariance: Static Reflection/Rotation Instrument Source DF SS F Ratio p Card Rotation Test 1 870.03 29.07* 0.00 Visualization Level 2 607.92 10.16* 0.00 English Proficiency 1 32.64 1.09 0.30 Treatment 1 2021.52 67.55* 0.00 Model 5 4289.21 28.67* 0.00 Error 221 10902.89 Note: *significant for p.01 Table 12 Means by Visualization Level on Reflection/Rotation Instruments Visualization Level Mean for Dynamic Test Mean for Static Test High Visualization 21.65 20.22 Medium Visualization 18.59 18.57 Low Visualization 14.49 14.76 After the researcher controlled for initial differences on pretest scores, the students who received the dynamic treatment performed significantly better on the Card Rotation Test at the .05 level than their peers who did not receive the dynamic treatment (see Table 13). There was no statistically significant difference between the Treatment and Control Groups on the Paper Folding Test (see Table 14). Once again, LEP students did not perform statistically significantly differently than their EP peers on either of the dependent measures (see Tables 13 and 14). Therefore, the following null hypothesis could be rejected: Students' instructional environment will not affect their twodimensional visualization. The following null hypotheses could not be rejected: Students' instructional environment will not affect their threedimensional visualization. Students' English proficiency will not affect their (a) two dimensional visualization or (b) threedimensional visualization. Table 13 Analysis of Covariance: Card Rotation Test Source DF SS F Ratio p Card Rotation Pretest 1 213366.74 403.00* 0.00 English Proficiency 1 368.71 0.70 0.40 Treatment 1 3078.71 5.81* 0.02 Model 3 227300.83 143.10* 0.00 Error 227 120185.47 Note: *significant for p=.05 Table 14 Analysis of Covariance: Paper Folding Test Source DF SS F Ratio p Paper Folding Pretest 1 2117.31 192.32* 0.00 English Proficiency 1 0.25 0.02 0.88 Treatment 1 26.38 2.40 0.12 Model 3 2166.02 65.58* 0.00 Error 224 2466.08 Note: 'significant for p=.01 Limitations of the Study The present study had several limitations including the fact that the researcher instructed the experimental group throughout the study. The decision to teach the treatment group was based on previous studies (Pleet, 1990; Rosebery, Warren, & Conant, 1992; Warren & Rosebery, 1993; and Warren & Rosebery, 1992) in which recommendations were made to insure that teachers using computers in lab settings have excellent command of the software as well as the pedagogical methodology required in computer labs. The student scores on the Language Assessment Battery (LAB) were supplied by the students' middle school. The students were not tested simultaneously but were tested throughout the previous school year and during the first week of the current school year. Students who were tested during the previous school year may have increased their English language proficiency to the extent that they would not have fallen below the sixtieth percentile on the LAB. This may have altered the classification of some students as limited English proficient. The students in the experimental group were videotaped by an individual other than their teacher during the first class period spent in the computer lab. This limitation was removed directly following that session due to the belief held by the researcher that the video taping was effecting the students' performance. Students seemed to be distracted by the camera when they believed it was focused on them. A stationary, unmanned camcorder was used for all subsequent sessions in the computer lab. The stationary camcorder did not seem to elicit a similar reaction however some students were aware of its presence. The computer and paper and pencil instruments were similar in objectives and content. However, they were not equivalent forms so they could not be compared statistically. Because of the nature of the computer test, it was helpful if students were unsure of directions of positive and negative angles. The movement involved with the computer test did not assist the students to proper solutions if they did not already possess substantial knowledge of the concepts of reflection and rotation. The paper and pencil test was not any more helpful than the computer test if students did not understand the content* it did, however, allow students to turn and fold the paper to aid in visualization of solutions. Equivalent forms of the computer and paper and pencil tests would be necessary for an objective analysis of the tests' comparative instructional value. The sample consisted of all nine heterogeneously grouped eighthgrade mathematics classes in the middle school. In order to include four classes in the treatment group, the researcher was required to teach one class that was not originally on the researchers' schedule. The other three classes had been taught by the researcher from the first day of school, whereas the fourth class was added after the students had been in school for four days. This class was the least cooperative of the four classes possibly due to the students' belief that they were being taught by a "substitute". The students labeled as advanced in mathematics were taught in homogeneously grouped classes and were thus excluded from the study. The advanced group were those in the top quartile of the mathematics students in the middle school. These students may have benefitted from the treatment differently from the non advanced students. Finally, the classes were intact and nonrandomly assigned to treatment and control groups. This is not uncommon among studies conducted in school settings. CHAPTER V QUALITATIVE RESULTS Methodology The students' computer projects, observations of students working on activities or their computer projects, and the data gathered from student responses on the computer and paper and pencil Reflection/Rotation Instruments afforded the researcher valuable insights into students' understandings and misunderstandings of reflection and rotation and related concepts. The students' computer projects were helpful in assessing students' ability to measure distance and angle, negotiate the meaning of positive and negative directionality, and use the properties of a circle as they apply to measurement. Observations were conducted throughout the study of students in the Treatment Group. The observations were audiotaped and videotaped through the use of a stationary audio/video camera set up in the corner of the lab and left running for the duration of each lesson. The tapes were viewed and portions involving students working on activities or their computer projects that might reveal insight were transcribed and interpreted. The purposes of the observations were to reach a better understanding of how students interacted together using The Geometer's Sketchpad to construct the concepts of reflection and rotation and to determine whether the dynamic computer environment is an effective instructional setting for this construction to take place. An additional purpose of the observations was to gather insight into visualization strategies used by students as they performed rotations and reflections using the computer. Of particular interest was LEP students' collaboration as appropriate English vocabulary was developed for reflections and rotations. The data gathered from student responses on the Reflection/Rotation Instruments provided evidence of student's actual success or lack of success with recognizing and performing reflections and rotations. The activities and the computer project were part of the treatment therefore the Control Group did not take part in either. The Reflection/Rotation Instruments were administered to both Treatment and Control Groups. A short questionnaire was administered to a subsample of the treatment group in order to determine whether or not the experimental version of The Geometer's Sketchpad was helpful in students' acquisition of the mathematical vocabulary involved in reflections and rotations. The experimental version of the Sketchpad used for this study incorporated icons, or pictorial representations, as clues to appropriate menu commands. The commands under transformations were coupled with symbols designed to provide context clues for the associated vocabulary. Computer Projects A major goal of the computer project was to have the students explore and discover the properties of reflection and rotation. The students, working in pairs, were to use up to four screens (on The Geometer's Sketchpad) to demonstrate properties of reflection and rotation. The students were required to use drawings, color, at least one circle, measurement, and written explanation to illustrate the properties they had discovered. The project requirements were designed to influence the students' use of multiple representations to communicate about various properties of reflection and rotation. The multiple representations used by the students supplied evidence of the depth of the students' conceptual knowledge of the properties of reflections and rotation. In figure 1, the property that an object and its image after reflection will be equidistant from the mirror line was represented using measurement, drawing, and writing. Various uses of the circle were also observed. Figure 2 illustrates two students' use of circles to verify the relationship between dimensions of a flag and its images after rotation, the circles also show that the flags are all the same distance from the center point, however, the students do not call attention to this fact. Figure 3 exhibits the circles' use as a measure of distance from the center to any point on the circle as equidistant. Figure 4 displays the circle as part of a drawing. Some students defined the same term in various ways, for instance, the direction of positive and negative angles in figure 5. Other students discovered properties that far exceeded the expectations of the researcher as in figure 6. REFLECTIONS THE FLAGS ARE ALWAYS THE SAME DISTANCE AWAY FROM EACH OTHER. Distance(J to E1) = 4.00 ca Distance(J' to El) = 4.90 cm Distance(L to N) = 3.32 cr Distance(L' to N) = 3.32 cm Distance(B1 to K) 4.83 cm Distance(K' to BI) = 4.83 cm Distance(B1 to K) = 4.83 cm Distance(K' to B1) = 4.83 cm THE FLAGS ARE ALWAYS THE SAME SIZE,SHAPE, COLOR,AND FACE THE OPPOSITE DIRECTION FROM EACH OTHER Figure 1 Equidistance Including Students' Written Comments ROTATIONS 450 w i135o ALWAYS THE SAME DISTANCE FROM THE CENTER POINT. Distance(D1 to Fl) 1.98 1800 THE CIRCLE SHOWS THAT THE FLAGS ARE THE SAME DISTANCE FROM TOP TO BOTTOM. 90* Figure 2 Multiple Uses of Circles Including Students' Written Comments THE CIRCLES ARE SHOWING THAT THE OBJECTS ARE T'iIE SAME DISTANCE FROM THE CENTER POINT ALL AROUND. Figure 3 Circle as Measure of Distance Including Students' Written Comments Este dibujo es otro ejemplo de REFLECION. joy Ao Figure 4 Circle as Part of Drawing Including Students' Written Comments t af t: This is a good way to explain rotationrbecause it makes iteasier to find the results to your problem and will also help you leam rotations faster and better. MAIN POINT The drawing is showing you the rotations of the MAIN flag point It shows that the 4 negative stations go clockwise and that the positive rotations go counter dock wise. This drawing also shows that if you draw a circle with center at rotation the circle will touch the flag at the same spot as the other flags. 900 .900 When you go this way it equals the negative rotation! 180* When you go this 180o way t equals the positive rotation! Figure 5 Positive and Negative Angles Including Students' Written Comments These quadralatrals are 45 degrees apart. We fit 8 quadralatrals in 360 degrees Figure 6 Repeated Rotations Including Students' Written Comments Observations The Treatment Group was audiotaped and videotaped through the use of a stationary audio/video camera set up in the corner of the lab and left running for the duration of each lesson. The tapes were viewed and portions were transcribed and interpreted. The purpose of the observations was to reach a better understanding of how students interacted together using The Geometer's Sketchpad to construct the concepts of reflection and rotation in order to determine whether the dynamic computer environment is an effective instructional setting for this construction to take place. An additional purpose of the observations was to gather insight into visualization strategies used by students as they performed rotations and reflections using the computer. Of particular interest was LEP students' collaboration as appropriate English vocabulary was developed for reflections and rotations. Each of the following observations was based on the aforementioned audio/video recordings taken while students were working in pairs on activities or their computer projects using The Geometer's Sketchpad. Student conversations dealt with conjecturing about reflections and rotations or meeting the requirements of the assigned computer project. The requirements of the computer project included the use of four screens or less to illustrate the properties of reflection and rotation. The students were to illustrate the properties both pictorially as well as verbally. Line segments, circles, measures, and color were all necessary components of the project. A brief description of the setting for each observation precedes the associated transcription. The captured computer screen resulting from the students' collaboration follows the transcription of each conversation leading to the completion of a computer project. Observations from Activities Shortly after the class had begun to practice conjecturing with reflections, two students called the teacher to their computer. According to the students, they kept choosing incorrect locations of images after reflection and they did not understand why their conjectures were wrong. Together, the teacher and the students determined that the conjectures were actually correct but the procedures to test the conjectures were incorrect. The students were not marking the predetermined mirror line before reflecting the figure. In this case, the students were questioning what had previously been considered as the "authority" when they could not make sense of the "solutions". Once students had learned how to perform reflections and rotations using the Sketchpad and had played the reflection and rotation games, the students were directed to work together conjecturing about positions of images after reflection and rotation. When conjecturing about reflections, one student was to choose the location of a figure and construct a mirror line, the other student was to draw an outline of the figure in the location he/she believed the image would lie after the transformation had been performed. The following observation took place while two students were 
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