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THE ON EFFECT OF THE LOCUSMOTION AVAILABILITY OF PROBLEMSOLVING THE GEOMETER'S PERFORMANCE AND SKETCHPAD STRATEGIES STEPHANIE ROBINSON 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 ACKNOWLEDGEMENTS The completion document and the expert ences that precede d it have been possible only with the help many individual Kantowski have SThe been words a constant wisdom source of Dr. Mary energy Grace and inspiration throughout my graduate studi As chairperson committee , she has been a wonderful rol model both personally and professionally The experiences that gained during years of working and with Elroy Bolduc have provided a judi cioUS balance between tasks and people. Thanks is extended to Dr. James Algina his patient assistance in the data analy S1S gratitude ext ended other members committee and those who parti cipated at the defense dissertation John Gregory, . Charl Nelson, Eugene Todd. Throughout one doctoral studi the support Hlow doctoral students , th ose who have gone before those who will follow , is a driving force in the success comply etion program heartfelt thanks and endship to Dr. Katheryn Fouche . Thomasenia Adams, Sebastian Foti , Ms. Virginia Harder , Mrs . Juli professionally. Without the assistance and encouragement the teachers and staff at Gainesville High School who participated in my doctoral study, none of this would have been written I appreciate the help of Nancy Galloway, Paige Lado, Nanette Greene, Michelle Sanders, Jeff Kanipe, and Jeff Rung. wish to acknowledge the encouragement, support, and love of my family , who saw me through this experience: parents, Pauline and Harold Osgood , who never gave brother Carmen, and who sister; thought especially my would never daughter finish and Kalena get "real Finally , but foremost, wish thank my husband, Robert Robinson, love , patience, and understanding as I fulfill my dream, now our dream. TABLE ACKNOWLEDGEMENTS ABSTRACT OF CONTENTS * * * * JLJ . . . . . . VI CHAPTERS INTRODUCTION . . . . . 1 Purpose Rationa Outline Research Definit Summary of the Study le . o h i io] . . . . . . . . . f the Study . . . . 1 Questions . . . . 1 n of Key Terms . . . . 1 . . . . 1 REVIEW OF RELATED LITERATURE Cognition and Constructivist Spatial Visualization . Geometry . . . Summary . . . METHODOLOGY . . Theory . . 16 * . 24 * . . 35 . . . 42 * . . 44 Overview Res Sel The Loc Sel Pro Des Sta arc cti Spa sM cti edu rip ist the Study Ques of al V ion and S . uat liz lem cri S ion ati In pti' of the In Procedure * .. . 44 * S . . 45 Instruments . . 46 on Battery .. 46 ventory . . 50 on of Population Sample 54 * . . . 55 ructional Phase . 55 . . . 57 Summary S. . . . . . 58 ANALYSIS OF THE DATA . . . . 60 Research h Questions . . . . 6 . . . J15 page w SUMMARY AND CONCLUSIONS . a a a 82 Overview Results Limitati Implicat Summary of the Study . . . 82 and Discussion. . . . 83 ons . . . . . 93 ions and Recommendations . . 95 . . . . . 99 APPENDICES LOCUSMOTION INVENTORY PROBLEMS SCORE SHEET LOCUSMOTION INVENTORY CONSENT FORMS INSTRUCTIONAL ACTIVITY SHEETS INTERVIEW QUESTIONS LIST OF REFERENCES BIOGRAPHICAL SKETCH a a a . a a a a * Abstract of Dissertation Presented to the Graduate School the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy THE EFFECT OF THE AVAILABILITY OF THE GEOMETER'S SKETCHPAD ON LOCUSMOTION PROBLEMSOLVING PERFORMANCE AND STRATEGIES Stephanie O. December, Chairman: Robinson 1994 Mary Grace Kantowski Major Department: Instruction and Curriculum The purpose problemsolving this study was strategies used by to investigate geometry the students following instruction in a dynamic technological environment. The study explored the relationships among the students' spatial visualization ability mathematical ability, problemsolving strategies with without the availability of The Geometer's Sketchpad. Solution strategies were examined dynamic visualization as for tendency to related use drawings availability technology during locus of points problemsolving session. Three measures of including spatial ETS Card Rotations, visualization Cube were Comparisons, taken, and Paper Folding tests. A locusmotion problem inventory was administered. Participants for the study consisted 158 students were randomly assigned two groups for the Locus Motion availab Inventory: le. and th those who had the computer lose who did not. Individual software interviews were held with two students from each class, one from each group. availability the computer was significant factor for performance on LMI. Covariates spatial ability or mathematics variance. Teaching the achievement accounted specific skills for most of drawings motion on the computer resulted those strategies being used similarly students with and without the Sketchpad. Although this indicated in data analyses nonsignificant differences implications are positive. Results suggest that strategies learned with the technology are transferable paper and pencil situations, and that active important participation to successful instructional activities performance and use of strategies. Interview data revealed that students were able to solve problems successfully on the written LMI that they had previously missed. Students were able to go beyond the known problems solve novel questions with some discussion of the components of the question itself. Language and communication were critical to expanding the student's zone of proximal development. CHAPTER INTRODUCTION The technological society today demands fundamental restructuring the educational environment. The impact technology provides two main issues mathematics learning of education: mathematics the classroom the changing perspectives and the changing role of (Mathematical Sciences on the technology Education Board [MSEB], 1990) . Technology can assist the introduction, development, and reinforcement of mathematical concepts. the curriculum opportunities cognition and different computers for new calculators offer conceptualizations of opportunities sequences, learn new with a higher instruction and content, degree sophistication at every level. The overall goal of mathematics education students learn and experience power of mathematics (National Council of Teachers of Mathematics [NCTM], 1989). Understanding mathematical concepts the key to success power in mathematics. However, the results evaluations by the National Assessment Educational Progress (NAEP) indicate that performance by students foundations for mathematical understanding need to be strengthened. conceptually A curriculum that oriented is is active and recommended to assist students gaining the foundations and understanding (NCTM, 1989). environment skills of of exploration and discovery will inquiry help develop and problem solving. Technology may the key to unlocking the door revised mathematics curriculum and learning to a environment. the influences that shape mathematics education, technology stands out as the one with the greatest potential revolutionary impact" (MSEB, 1990, 22) The technological classroom must society adapt is a reality the changing the mathematics scientific workplace or fail to reach primary goals of providing power and opportunities the students of today world tomorrow. According to the constructivist perspective learning, students build their own interpretative frameworks for making world sense (Schoenfeld, the world, 1987a). In including the implementation mathematical of a dynamic technological environment that promotes visual and dynamic problemsolving strategies may influence curriculum , the instructional techniques, learning perspectives the mathematics classroom. Students construct their own knowledge by engaging in problem solving and actively students new through experiences interaction in the classroom. of previous knowledge with Providing multiple ways in which to approach mathematics learning and problem solving allows students to build knowledge from their own cognitive level. Alternative approaches and modes thinking need to be examined for their effectiveness facilitating understanding concept acquisition. The geometry inductive environment curriculum with potential deductive reasoning provides for problem solving and spatial a rich thinking. However, the traditional geometry classroom, buried in proof and structure, has been particularly conducive to an inquiry approach. A modified geometry curriculum may encourage a more versatile, integrated approach learning. The use of computers and dynamic modeling has drastically changed the nature of mathematics education algebra (Hershkowitz, 1990). Such a change is needed in geometry may be promoted by combining observations intuition with multiple representations in a dynamic environment. Purpose Study The purpose this study was investigate the problemsolving strategies used by geometry students following instruction in a dynamic technological environment. Specifically , the study explored the strategies with and Students were without instructed with the availability presentation technology. "hands experiences using a software product called The Geometer's Sketchpa d (Jackiw , 1993) . Specific lessons locusmotion concepts provided the basis exploring problemsolving performance strategies. Solution strategies were examined for tendency to use drawings and dynamic visualization as technology related to during the the availability problemsolving the session. Rationale To engage in mathematics is to participate activities of problem solving. Learners have to construct their own knowledge, through a process of reflection (Davis, Maher Noddings, 1990; Noddings, 1990; von Glaserfeld, 1987) process of purposeful activity (Krutetskii, 1976; Sowder, 1989). Metacognitive aspects of learning, such as selfmonitoring, pertinent explicit to efficient instruction regulation, construction the curricul evaluation, of knowledge um (Silver, are and require 1985) Constructivism emphasizes role of construction process as well as the awareness of that process and how to modify (Confrey , 1990). This study attempted investigate the effect of a specific learning environment the content and nature students' learning. future theories and practices (Balacheff et al., 1990; Sowder, 1989). The Cognitive Flexibility Theory learning , knowledge representation, and knowledge transfer acknowledges the ability to restructure one's knowledge spontaneously in many ways in response to changing situational demands (Spiro Jehng, 1990). Revisiting the same material from multiple perspectives the key to learning and transfer in complex domains (Spiro et al. , 1991) . Cognitive Flexibility combines the constructivist nature of learning patterns learning conceptual failure to examine dynamic perspectives (Spiro et al., cases 1991). from multiple Use of dynamic cases in this study added to the knowledge concerning students' response to multiple representations novel situations the complex domain of mathematics. Application of the Cognitive Flexibility Theory util izes the random access capability the computer to provide flexibility and pluralistic representations (Spiro Jehng, 1990) . Computer environments are an ideal tool to provide multiple representations in mathematics, allowing one shift from one representation to another to find the form most useful (Dreyfus, 1990; Fey, 1989; Kaput, 1987, 1989; Senechal, 1990; Sowder, 1989) . "Interactive technologies provide a means of intertwining multiple representations mathematical concepts and relationshipslike graphs and aspects computer environment need to be examined, including how knowledge affected by the environment and how students' cognitive behavior and constructs are modified (Balacheff et al., 1990). The interactive computer revolutionizing the study environment is also of shape and visualization within the study of mathematics (Senechal, 1990; Tall Thomas, 1989; Tillotson, 1984/1985). "Today the microcomputer increasing the range of presence stimulating aids in mathematics a great deal visualization classrooms enormously is also research and development in this area" (Bishop, 1989 , p. Spatial ability often a controversial subject when discussing mathematical performance and achievement. The definition of spatial ability the number of components factors involved, the relationship of spatial ability to problem solving and mathematical performance, ability to learn increase spatial skills are all topics that require continued research and investigation (Steen, 1990). Some researchers suggest that spatial ability unitary trait (Johnson & Meade, 1987; Moses, 1984) Others distinguish Tillotson, distinct c subject tc (Conner 1984/1985) , components. instruction Serbin or three Whether s , 1980; (Linn patial is also an Tartre, Petersen, ability issue. Many 1990; 1985) innate researchers 1977/1978; Tillotson, 1984/1985; Vinner, 1989). Furthermore, instruction is not only desirable, but necessary to improve spatial skills (Dreyfus, 1990; Krutetskii, 1976) . These topics will be explored further the literature review. Students skill need of spatial explicit visualization, instruction not only also the in the ability to monitor their own endeavors in the learning process (Dreyfus, 1990; Silver, 1985; von Glaserfeld, 1987). Students need to attend to the metacognitive aspects learning: selfmonitoring cognitive activity (Schoenfeld, regulation, 1989; and Silver, evaluation 1985). Constructivism emphasizes that students can learn improve their reasoning and problemsolving ability and become agents of their own learning (Paris Byrnes, 1989; Zimmerman, 1989) This investigation explores whether instruction solving can strategies influence awareness of in novel dynamic problem problem situations. Improvement intertwined with assessment the mode of mathematical instruction ability activity in progress. A change in assessment that more closely correlates becoming to instruction clearer that a representative process needed single of student in terms of product (NCTM static lev learning, e (Campione, , 1989). rel assessment dither i Brown, .n terms Connell, the current knowledge level the student, leading him or her into new mathematical territory (Campione et al., 1989; Ferrara, 1987/1988; Lampert, 1991) SThe difference between what a student can do alone and what he or she can do with the assistance of a teacher or more capable peer (Rohrkemper zone , 1989) of proximal been development. termed by Vygotsky Placing the emphasis as the on the initial and possible learning states student, the environment with in which the constructivist learning is taking place, perspective of cognitive is consistent e change (Linn Songer, 1991) . The variety tasks offered this study introduced students to locus of points and extended the student's knowledge of problemsolving processes beyond the practiced level to the next dynamic level thinking. Although it has been proposed that knowledge constructed by the learner that mathematics is a complex domain, research needed to provide descriptions these theories applied to specific domains (Hiebert Wearne, 1991; Wearne Hiebert, 1988) According to Kantowski (1981), problemsolving proce sses depend as much on the type of problem as on style of solver. Krutetskii takes this idea one step further to state that specific content ability depends on instruction. illstructured domain of mathematics terms. A welldefined broad unit to be investigated in geometry such in global locus of use of dynamic technology a constructivist learning environment. content area locus of points was chosen because the subject independent of placement the curriculum (provided that sic concepts have been introduced) often left the geometry curriculum due time constraints. The topic locus of points enhanced a dynamic environment use of diagrams. The computer has the potential to provide fruitful environment exploring diagrams, conj ecturing searching patterns and generaliz nations (Bishop, 1989; Hershkowitz, 1990; Janvier, 1987; Lesh, 1987; Senechal, 1990; Steen, 1990) . Two issues are raised the use of calculators and computers in the classroom: understanding, finding and a balance problemsolving of conventional ability skill, that appropriate new technology discovering ways that technology ens entirely new approaches thinking about mathemati ideas problems (Fey, 1984). Emphasis geometry may shift to experimentation with shapes relations, inquiry building inherent strong geometric ir mathematics. tuition The and ins spirit tructional phase this study explored use of a relatively new teaching technique medium geometry assroom, providing some insight into ways curriculum might change and techniques of designing structuring new learning Many researchers agree that computer environments provide multiple representations , important flexibility thought problem solving (Dreyfus, 1990; Kaput, 1987; Sowder , 1989; Tall Thomas, 1989). However res earchers must concentrate environment on determining on what how effect students this learn learning (Balacheff 1990; Sowder 1989 Although Sowder (1989) referring study of algebra, geometry the same sentiment holds true: "Not much research been done evaluate programs what students or how learn programs interacting might be used with within the the context . curriculum" Outline Study Thi study was designed as a "cons tructivi teaching experiment" (Cobb Wood Yackel , 1990) investigate problemsolving following strategies instruction used in a dynamic geometry students technological environment to explore relation ships among students' spatial visualization ability , mathematical ability, problem solving Sketchpad strategies This with study without addressed availability some the needs research a domain specific content area locus points. Three independent measures spatial visualization al., Card Rotations, Cube locusmotion problem Participants students fro Comparisons, inventory for the study m seven geometr and Paper (LMI) consisted y classes Folding. was also administered. 158 geometry at one high school. All students participated in the instruction using The Geometer 's Sketchpad. The instructional phase included lessons that extended previous lessons on the Sketchpad introduced the concepts locus of points. Students were randomly assigned to two groups for the LocusMotion Inventory: available sess those who would have those who would not. ion followed a the computer The constructivist approach software problemsolving of creating "problematic environment that would elicit the student's adequate constructive endeavors" (Fischbein, 1990 , p. Data analysis had both a quantitative and a qualitative component. Research Questions The study addressed three questions of interest: Are correct response score (CRS), tendency to use drawings , and/or use of dynamic strategies (DS) that students utilize to answer locusmotion problems related students' spatial visualization index (SV) mathematics achievement (MA)? Does Sketchpad availability (SA) affect correct __ Are tendency to use drawings (DR) and/or use of dynamic strategies (DS) that students utili when attempting to solve locusmotion problems related students ' correct response score (CRS)? Definition of Key Terms Soatial visualization, or spatial ability, is defined this study as the ability to recognize the relationships among the elements manipulate mentally a given figural one or more of configuration, those parts. Spatial visualization index (SV) is operationalized the sum of the zscores, or standardized scores, the three tests in the Spatial Visualization Battery. Mathematics achievement (MA) score measured on a geometry semester examination given to all participants. Locusmotion inventory (LMI) is a set problems relating to locus of points. Nine problems were used the written portion LMI, five problems were available interviews. Correct response score (CRS) , or locusmotion problem solving performance, is the number of correct responses written LMI. Locusmotion problemsolvinq strategies, or methods of solutions, are scored in two ways: tendency to use drawing (DR) use of dynamic problemsolving Dynamic software, or dynamic technological environment, refers to The Geometer's Sketchpad (Jackiw , 1993) implemented on an local area network. Summary Mathematics education in the process of change, change, indicated within the Standards that recommends new curricula and modes instruction and provides mathematical power to students. innovations are proposed, planned, implemented, investigation into their feasibility and effectiveness is needed. Technology provides and one vehicle the methods employed This chapter presented for variations in the content addressed the mathematics the rationale classroom. a study designed to examine the effect instruction in a dynamic technological environment on performance and strategies employed in attempting locusmotion problems. The investigation included the exploration of relationships among the students' spatial visualization ability, mathematical without ability, and problemsolving the availability the Sketchpad strategies with during problem solving. The research questions pertaining to the study, well as the key definitions, have been outlined here. Chapter the questions contains a * . interest. review focusing literature on previous pertaining to research procedures utilized study found Chapter Chapter presents results data analy ses The findings their implications instruction and research are found Chapter CHAPTER REVIEW OF RELATED LITERATURE The theoretical premise that supports this study that the use of a dynamic technological environment, such The Geometer' Sketchpad , can enhance construction knowledge students influence problemsolving strategies using dynamic visualization skills The constructivist eory learning promotes the beli that students build their own knowledge through activity and experience Dynamic software can provide experien ces visualizing , conjecturing , and building frameworks within specific content domain. Experience will not only influence students' ability to perform success fully on the spec ific content problems activities , but use solving problems that are nove and perhaps more diffi cult The review literature presented thi chapter there fore , focuses on studi research pertaining cognition cons tructivist theory, spatial visualization, and curriculum instruction in geometry The connections between constructivism Cognitive Flexibility Theory are explored. Within t sec tions on cognition a d geometry, CoQnition and Constructivist Theory Mathematics educators generally agree with the basic tenet of constructivist theory that learners construct knowledge within an active environment. Krutetskii states that "mathematical abilities exist only a dynamic state, development; they are formed developed mathematical activity" (1976, brain is not passive, active , and engages in processes selection, interpretation, inference (Orton, 1987). Noddings (1990) summarizes views of constructivist theories follows: knowledge is constructed Mathematical knowledge process of is constructed, least in part, through reflective abstraction. There exist activated structures account explain the result cognitive processes of structures that are construction. for the construction; of cognitive activity These that they in roughly the way a computer program accounts for the output computer. 3. Cognitive development. P structures urposive under activity continual induces transformation existing structures. The environment presses the organism to adapt. Acknowledgement of constructivism as a cognitive position leads the adoption of methodological constructivism. Methodological constructivism in research develops methods of study consonant with assumption cognitive constructivism. b. teaching Pedagogical consonant w constructivism suggests methods ith cognitive constructivism. An example of how students build knowledge can seen in a study  Davis in a fifth parade class (Davis & Maher I __ pizza. two Detailed boys evinced thinking, investigation solution the constructing layers of conversation. solution the processes in layers two students interacted with each other and with various solution approaches , the successful solution emerged from use of concrete representation. students broke problem down into parts , the tendency was to look at the parts separately , not at the whole picture. The building concrete representations may have allowed the students concentrate on the parts , but still be aware of entire problem setting. "Put into simple terms, constructivism can be described as essentially a theory about the limits of human knowledge, a belief that all knowledge is necessarily a product our own cognitive acts" can be, and need (Confrey, to be, 1990, planned by the 108) . learner Cognitive acts or by instructor. supported by Following student i a five day interviews, study with Confrey one develop teacher, d a model teacher' constructivis instruction that supports m. Constructivist models pedagogical teaching recommend learning environments that promote process ses including acquisition of basic concepts, algorithmic skills, problem solving heuristics, and habits of reflective thinking (Davis, Maher Noddings, 1990) One le skills that must be developed flexibility One representational systems that they instinctively switch the most convenient representation to emphasize any given point solution process" (Lesh, Post, Behr, 1987, Successful problem solvers are able to build mathematical representations of problem situations relate problems together that have a similar structure (Schoenfeld, 1987a). Examining the skills that successful problem solver the gifted student possess reveals that their thinking process may be different (Dover Shore, 1991) . Dover Shore (1991) studied 11yearold students them schoolidentified gifted and 11 of average ability) focusing on setbreaking with the water jug problem. Results showed a three way flexibility interaction among giftedness, The gifted speed, students exhibited more and planning, more reflection , and more flexibility. Successful learners construct these metacognitive representations at representation for the same time problem. that Davis they build (1986) conducted three diverse studies that yielded similar results. One study involved a mathemati teacher solving a problem not in his musical area expertise. theme. A second study third study was a looked at taskbased interview a university calculus student working a quadratic equation. In each situation 'J representations were built: one 38) . metalevel, an observation of the method resolving the question. These and other metacognitive skills must be addressed directly and students taught to recognize and apply them in the proper situations (Campione, Brown, Connell, 1989; Hershkowitz, 1990; Janvier , 1987; Lester, 1989; Schoenfeld, 1987). Hembree (1992) found similar conclusions in a meta analysis of 487 reports and studies in problem solving. Results instructional methods on problem solving revealed that students who received instruction in problem solving and heuristics explicit had higher training. scores Effects of than did those with no classroomrelated conditions were also examined. Computer assisted problem instruction provided better results than did paper and pencil problem solving. Teaching the specific subskill of diagram drawing also showed positive results. "Taken together, these findings suggest that the dominant factor in problem solving less IQ than mental development" (Hembree, 1992, 268). However , Resnick (1989) offers a caution about metacognitive training: This points metacognitive to a fundamental training efforts problem with that focus certain attention on knowledge about problem solving rather than on guided and Such talk constrained practices efforts may about be more processes in doing problem solving likely to produce functions than ability to to perform them. Several studies cited by Campione et al. (1989) that given explicit instructions and did not attend their own processes included a aimed at p learning. learning phase, promoting transfer Campione et al. (1989) One study followed by Dynamic and Ferrara Ferrara dynamic tests, (1987/1988 (1987/1988 C assessment as described by ), provided information about the ability the student as well as the student' potential improvement. Assessment, current knowledge like i level instruction, should begin at the student and lead him the or her into new mathematical Ferrara, 1987/1988; territory Lampert (Camp , 1991) . lone et al., 1 The difference 989; between what a student can do alone and what he or she can do with the ass instance of a teacher or more capable peer (Rohrkemper, zone 1989) of proximal been development. termed by Vygotsky Schoenfeld (1978) describes as the this process learning: "One acquires higher order thinking skills exercising those skills ZPD with the help others and then internalizing those skills, that mastering as an individual those skills for which one, once, needed support" (1987b, 210) In a case study from "Computer as Lab Partner" project , Linn and Songer (1991 referred this difference as the range of possible cognitive changes. experiments, Students simulations, interacted with instruction. computerbased Results indicated that, with proper guidance and active engagement previously able to do. Focusing on the zone of proximal development Salomon (1989) fostered the speed conceptual produced similar results in a change. study c readingrelated metacognitive guidance with seventh grade students. three Seventyfour treatment groups w program with varying students were ith three ver levels randomly sons of of guidance assigned a computer varying cognitive levels of questions. Salomon reached conclusion that "computers can serve as tools that provide guidance in a child' zone of proximal development" (1989, 626). The studies Linn and Songer and by Salomon reflect similar cognitive constructs those found the Cognitive Flexibility Theory. The Cognitive Flexibility Theory learning, knowledge representation, and knowledge transfer provides for multiple representations of the same material in rearranged instructional sequences from different conceptual perspectives (Spiro Jehng, 1990). The goal to be able to apply independently the learned knowledge and processes new situations. An application this theory is seen in a project described by accessed Spiro and Jehng videodisc of (1990 in which provided a random foundational instruction scenebased materials. explorations Students engaged A cognitive fle themebased xibilit  Citizen Kane w Commentary was provided to expound upon thematic and symbolic contrasts within the scene. The goal was only to explore themes and symbols within Citizen Kane, also to demonstrate complex nature of literature provide students with active experiences with processing that complex knowledge and building new metacognitive skills. Many principles of Cognitive Flexibility Theory are also basis constructivism (Noddings, 1990; Spiro et al., 1991) . Students interpret knowledge, absorb Active participation and exploration by the student crucial. An environment of exploration allows construction of knowledge situations with the transfer ultimate goal of knowledge and skills SThe metacognitive as new pects constructing activity with knowledge learner important control for transfer. Purposive selfregulation builds new frameworks. Those activities must provide multiple representations Constructivist the concepts tenets outlined by within the domain. Paris and Byrnes (1989) are also reflected research by Spiro et (1990, 1991) . Instructional innovations that are taskoriented are needed to promote increased metacognition. Metacognitive skills are a necessary characteristic of good problem solvers, flexible thinking. According to Cognitive Flexibility Theory, instruction __~_ multiple However, representations of important the same concept to avoid or theme. compartmentalization information, narrowing the focus of the idea. Revisiting the same material multiple times, in multiple ways promotes flexible thinking by the student. Mathematics and problem solving are considered by many researchers (and many students!) to be complex or ill structured domains (Davis, 1986; Polya, 1957, Resnick, 1989). As a complex domain, mathematics can be approached from the Cognitive Flexibility theoretical basis of knowledge representation and transfer. Linear presentation not sufficient for understanding; multiple representations are necessary to provide a network interrelated ideas (Romberg, 1988; Spiro & Jehng, 1990). single conceptual perspective inadequate (Spiro et al., representation of dynamic transformations, movements is not 1991). "T situations incorrect he area involving < is particularly , just the g action, intriguing corresponds to domains where children encounter learning difficulties" (DufourJanvier , Bednarz, Belanger , 1987 , p. 122 "The dynamic and interactive media provided computer software make gaining (traditionally the mathematicia intuitive province of understanding professional interrelationships among graphic that the dynamic display of multiple representations valuable not only to discover the relationships, also learn skills of translating from one representation another, building 1989). consonant with , interpreting, "The computer the constructive perspective relating knowledge appears able (Resnick to offer qualitatively new thinking tools mainly through its graphic facilities the possibility synchronous representation different but related processes and situations" (Balacheff et al., 1990, 145) . Spatial Visualization Many studies have investigated whether spatial ability is a single t investigating spatial rait or a composite of gender ability differences in d Johnson and Meade traits. While developmental (1987) patterns considered spatial ability to be a unitary trait Moses (1977/1978) investigated the composition of spatial ability and came same conclusion. Moses described spatial ability as the ability perceive the essential relationships among the elements a given visual situation, to mentally manipulate one or more of these elements" (1977/1978 , p. The purpose of study conducted by Moses (1977/1978) spatial was ability fold: investigate to refine the definition relationships between 18) . instruction on spatial ability, problemsolving performance, and degree visuality. study fifthgrade students in four classes were given same battery tests measuring spatial ability and problem solving, prior to and following the instructional phase. The spatial tests were Punched Holes Test, Card Rotations Test, Form Board Test , Figure Rotations Test, Cube Comparisons Test. Problem Solving Inventory constructed by Moses was the sixth test. lessons The instructional involving two perceptions and some phase consisted threedimensio problemsolving tasks of nine weeks of nal geometric using visual solution processes. Correlation coefficients, factor analyses, and regression analyses were performed investigate the decomposability of spatial ability. Pearson productmoment correlation coefficients were used to analyze relationships among the variable es. Analyses of covariance were performed to analyze the effects instruction. Conclusions were reported follows: Spatial ability ability to manipulate to manipulate is not an enti parts fi decomposable int re figure and th gure. Moreover, :o the ie ability some of tests which have been classified as spatial tests can are be solved pure spatial in an analytic manner while other tests tests. 2. A] ability, .though it is spatial a good ability predictor is a general cognitive of problemsolving performance. . 3. An individual with high spatial ability will frequently not write down visual solution processes part of his solution. Problemsolving performance is best  edicted by C . .. ... w Spatial ability instruction will affect is a modifiable quantity, the spatial ability 1.e., an individual, affecting males more Instruction in certain than visual females. processes does significantly individual. affect the degree of visuality Instruction aided neither males of nor females on their degree of visuality scores. Instruction significantly performance of affect, in a in certain visual affect the general an individual; positive manner however, success processes does problemsolving it does on spatial problems . 9. Instruction does significantly affect the problemsolving performance of males nor females. Instruction has the same amount of effect on the problemsolving performance of both high and spatial 10. I spatial problems ability individuals. instruction significantly problems more (pp. affects success on than success on analytic 144154) Other researchers have distinguished two or three distinct components (Conner Serbin, 1980; Linn Petersen, 1985; Tartre, 1990; Tillotson, 1984/1985). Components have been termed by researchers as spatial orientation, spatial visualization, relations, and spatial I kinesthetic perception, imagery mental (Conner rotation, Serbin spatial , 1980; Linn Petersen, 1985; Tartre, 1990 ; Tillotson, 1984/198 "Spatial visualization is distinguished from spatial orientation tasks identifying what is to be moved; task suggests that all or part of a representation mentally moved altered, is considered a spatial visualization task" (Tartre, 1990 , p.217). Tartre continued describing spatial orientation as "those tasks that require that subject mentally readjust her or his perspective Tartre (1990) used these definitions as she explored role of spatial orientation component in the solution of mathematics problems with 57 tenthgrade students. The sample of students was chosen from those who scored the top or bottom third on the Gestalt Completion Test. According to Tartre this test was chosen because was best test to capture the essence of pure spatial thought. holistically That the tasks would be , it appeared unlikely that solved verbal analytic processes would contribute to subjects' solutions, and items directly required the structural organization visual information in order to make sense out of the partial pictures" (1990, 220) . problemsolving interview consisted 10 mathematics problems, geometric and nongeometric, Students that were asked could be to solve solved the p in more problems, than talking one way. aloud as they did. Interviews were recorded and later coded according following categories: Correct answer , Done like, Failure to break Mental movement, Misunderstood problem, Added marks, Drew picture , Drew relation, and Estimate error. Tartre concluded that "spatial orientation skill appears to be used specific and identifiable ways . accurately figure, estimating the demonstrating the approximate magnitude flexibility to change an unproductive mind , adding marks to show mathematical without help to a problem in which a visual framework was provided" (1990, . 227). However, like other researchers (Fennema & Tartre 1985 FerriniMundy, 1987; Kantowski , 1981; Krutetskii 1976; Moses, 1977/1978), Tartre (1990) noted that spatial skill , and other mathematical skills, may linked to more general thought patterns when making sense of new material and may be directly related the specific mathematical skill, test, or activity in progress. To discuss spatial ability in terms way in which individual solves or thinks through the problem basis of Krutetskii (1976) analysis of mathematical ability in general. Krutetskii (1976 described two different modes of thought: analyticlogical and visual pictorial. In his analysis, these modes of thinking corresponding abilities were interconnected with the mathematical activity in progress. Other researchers have reached a similar conclusion that type activity is an important factor (Fennema Tartre, 1985 FerriniMundy, 1987; Kantowski , 1981; Moses, 1977 /1978; Tartre, 1990). Thus , the issue the definition of spatial ability component traits may further confused as the activities thought processes in which students engage become more dynamic with of abilities, use one means of computer technology. psychological speaking characteristics person's activity. We must stress that in analyzing skills and habits as well as ability , we are analyzing activity" (Krutetskii, 1976, 71). Tartre summed up the discussion of the definition and components of spatial ability. Attempting to understand discuss something like spatial o intuitive The very disperses verbalize ceases to mental ac orientation skill, and nonverbal, i act of it. the reaching which like is by definition trying to grab smoke: to take hold It could be argued processes be spatial tivity. Any involved thinking. that any in spatial Spatial attempt t thinking skill use evidence about how manifested must be indirect, since we cannot get into people' heads see what they see in their mind' eye. Often, processes involved are not even understood by the people experiencing them. The resulting does set indirectness of limits on research it but should not in this area curtail spatial skills are important to mathematics, then researchers must specific roles mathematics. find ways to that spatial (1990, identify skills play and describe in doing 229) Although solvingsolving questioned the relationship of and mathematical , many researchers have spatial ability found skills also that to problem been a correlation does exist between visualization and mathematics (Battista, Wheat ley, Talsma, 1989; FerriniMundy, 1987; Tillotson, 1984/198 Usiskin, 1987; Vinner , 1989). Those researchers who are mathematical pursuing the convinced the direct achievement relationship of however spatial correlation , acknowledge skill with that and mathematics worthwhile (Fennema Tartre, 1985; Lean Clements, a~* a m. a  t SS IA Aa *f * . did not determine extent of mathematical giftedness, (1976, but did 315) determine Lean and type, Clements or cast expressed of mind" need additional investigation "before relationships between spatial ability and mathematical performance can be clarified" (1981, 277). Background review the study by Tartre (1990) previously discussed evinced beliefs by many researchers , such as Fennema and Sherman (1977), McGee (1979), Conner and Serbin (1985), that spatial skills are related to mathematics FerriniMundy differences based an learning achievement. investigation of achievement and spatial ability gender y of calculus students upon premise that there "a wellestablished finding well of male superiority as correlational on tests of logical spatial support ability, a relationship between mathematics performance and spatial ability" (1987 126) . Primary questions interest were gender differences the effects training program on calculus achievement and spatial ability. sample students included in three large groups and each four smaller groups. Over an eightweek period, treatment groups viewed six slidetape modules taped commentaries with a variety tasks situations with spatial visualization and orientation. Control groups participated achievement or spatial visualization ability. There were interaction effects calculus achievement and the use visualization problems and a visualization significant solids solving solidofrevolution treatment of revolution" effect (FerriniMundy, 1987, 126) FerriniMundy (1987) suggested that significant training effects might have resulted a wider variety spatial tests had been used to measure spatial ability, since only the Space Relations Subtest , Form T the Differential Aptitude Test was used. There were indications that training may be more successful for women than for men. Calculus content and spatial visualization was also topic of consideration in a study by Vinner (1989) college students. The course was designed to emphasize the visual aspects of every algebraic concept theorem. Comparisons algebraic versus visual proofs indicated that students chose algebraic proofs even when drawings provided a visual proof upon examination. Vinner concluded that students believe that algebraic proofs are more mathematically acceptable, and memorization formulae and algorithms more successful in assessment situations. "Thus, seems that there is no research evidence that visual thinking is not needed success in higher mathematics" (Vinner, 1989 , p. 150) Presmeg (1986) reached analogous conclusions to those learner to excel in mathematical performance. An initial investigation by Presmeg 1985 of mathematical "stars" chosen by teachers revealed nonvisualisers. cognitive modes students who were that Presmeg researched , attitudes, they were predominantly the effect actions "visualisers. teacher upon high school thirteen teachers were grouped according to visuality their teaching seniors scored for their mathematical visuality, students chosen as visualisers were selected taskbased interviews. External factors , such as time constraints school testing procedures and teaching methods and textbooks that favor the nonvisualiser contributed the "preponderance of nonvisualisers amongst mathematical high achievers" ( hypothesized Presmeg, that 1986, visual 305). Presmeg teachers could teach also visual students more effectively than intermediate or nonvisual teachers. Regardless of the disputable nature of spatial ability relationship to mathematical achievement, increasing the awareness and ability of spatial visualization and spatial thinking may benefit students aiming them toward goal of mathematical power. In view fact the difficulties that most associated teachers with visual are unaware of processing mathematics, may be overcome, fact seems that likely that these difficulties increased teacher awareness of these issues could aid visualisers Although some studies have not reached significant results on the trainableness of spatial ability , BenChaim, Lappan, and Houang (1988) provided data that supported possibility that these skills can taught, can be learned. A sample of involving 1000 middle teachers D school anticipated students at in a three study sites of gender differences, spatial grade differences, visualization ability. effect of Immediately instruction before and after three week spatial visualization unit, students were administered the Middle Grades Mathematics Project (MGMP) Spatial different subsample Visualization Test, types of items. students an untimed A retention weeks test with test was given following the posttest. to a The results of training period were significant. "The most important result this investigation was that after the instruction intervention, middle school students, regardless sex , gained significantly from the training program spatial visualization tasks" (BenChaim et al., 1988, 66). indicated previous discussion the definition mathematical of spatial ability problem solving relationship and performance, Tillotson (1984/1985) described spatial ability as having least components, concluded that there is a significant relationship between spatial visualization and mathematics, investigating the nature of spatial visualization, correlation of spatial visualization to problemsolving performance, the effect instruction on spatial abilities. the study sixth grade students five asses two comparable schools were given same battery of four tests measuring spatial ability and problem solving prior to, following instructional phase. Control experimental groups were designated by school. The spatial tests were the Punched Holes Test, Card Rotations Test, Cube Comparisons Test. A Problem solving Inventory constructed by Tillotson (1984/198 was the fourth test. The instructional phase consisted ten weeks of lessons , one 45minute lesson per week, focusing activities designed to improve the student's perceptual skills. During the first and last weeks four tests were given. Correlation coefficients, factor analyses, and regression analyses were performed investigate decomposability correlation coe of spatial fficients we ability. re used Pearson to analyze productmoment relationships among the variables. Analyses of covariance were performed analyze effects of Conclusions were instruction. reported as follows: 1. Spatial ability. . 2. Spatial visualization not a visualization current mathematics curriculum single a skill taught Spatial visualization is a good predictor 3. Instruction affects performance on analytic problems differently than spatial 104) problems. (pp. 100 Supporting the inclusion of spatial skills the mathematics curriculum, Tillotson echoed the conclusions many other researchers (BenChaim , Lappan Houang, 1988; Dreyfus, 1990; Hembree , 1992; Krutetskii, 1976; Moses, 1977/1978; Presmeg, 1986; Vinner , 1989) Furthermore explicit instruction is not only desirable, necessary to improve spatial skills (Dreyfus , 1990; Krutetskii, 1976). Creating the proper environment also an important aspect. According to the constructivist , a goal the educator would be "the creation of a problematic environment that would elicit the student's adequate constructive endeavors" (Fischbein, 1990 , p. Geometry The correspondence of spatial ability geometry agreed upon by many researchers (Battista, Wheatley Talsma , 1989; FerriniMundy, 1987; Tillotson, 1984/1985; Usiskin, 1987; Vinner , 1989) . Preservice elementary teachers were relationship subjects of between a study that strategies used investigated in geometric problem solving two abilities, spatial visualization and formal reasoning (Battista, Wheatley , & Talsma, 1989). The study investigated following questions: problems ability related to the students' or formal reasoning spatial visualization ability? Is achievement in a geometry course preservice elementary teachers related to either the selection of effectiveness of problemsolving strategies utilized when attempting to problems? solve geometry 3. Do those elementary teachers who are successful at geometric problem solving utilize different strategies than those who are not successful? Five sections of preservice elementary teachers females, males) were administered a modified Purdue spatial visualization test, a modified version the Longeot test, investigatorconstructed geometry problemsolving test. effectiveness of type of the strategy was strategy used investigated. and Strategies included drawing , visualization, nonspatial. Percent use strategy and percent effective use of strategy scores were also given. One of the most interesting results was that although visualizati nonspatial effectively. was used more strategies, The more frequently than drawing the drawing useful strategy was strategies are and used more the ones used the most. Recognition this discrepancy may useful to students as they monitor their problemsolving processes. Battista indicated that "the balance between spatial logical ability likely to important factor geometry performance in general" (1990 , p. 48) The teachers. Focusing on different levels of geometry achievement and on gender differences, Battista (1990) tested five intact classes of high school geometry students. The variables that were visualization, logical reasoning, tested were knowledge of spatial geometry, geometric problemsolving strategies, the discrepancy between a student's spatial score and logical reasoning score, and use of correct drawings. Tests included Sheehan' version of the Longeot test of formal operations, Modified Purdue Spatial Visualization test, Cooperative Mathematics Test Geometry Part Form B, and the Geometric Problem Solving/Strategies test constructed by Battista. Strategies were classified as drawing, visualization without drawing, nonspatial, or none of the above. Intercorrelations between variables indicated following results (Battista, 1990) : Spatial visualization and logical reasoning were significantly related to both geometry achievement geometric problem solving for males females. Spatial visualization was significantly correlated with strategy variables of drawings and nonspatial strategies for males and with drawings and correct drawing females. Discrepancy score was significantly correlated with drawings, visualization without drawing s, and nonspatial students with a level of geometry achievement correlations were significantly higher between spatial visualization and geometric problem solving than between logical reasoning problem solving. Gender differences on different variables were also reported. Battista suggested that future research investigate interrelationships problemsolving between strategies "representational in geometry" schemes (1990, study also "suggests that instructional variables may critical factors in understanding interrelationships between variables, gender differences, and geometry learning" (1990, 59) An experimental verification of a method system of exercises developing spatial imagination was conducted by Vladimirskii (1971) to determine the role the diagram in mastering geometric material. The premise was that the basic task of geometry is to develop geometric thought, to apply theoretical knowledge in problem solving. Typically, study of geometry is mostly the memorization and reproduction of proofs, the generalization concepts. The diagram is the most often used visual aid, fully used enough to promote learning concepts. The experiments were conducted with sixth and ninth with cube the diagram. imaginations Preliminary use exercises of solid and dotte included moving d drawings. Resulting problem situations developed from the fact that book diagrams are too constrained, thus restricting the understanding the general concept. Particular relationships of the book diagram were taken to be essential features the diagram. Conclusions from the control experiments were: The diagram may be a hindrance as well as a help the reasoning process. the concept has been defined learned by the student, the properties cannot be transferred new material or problems. The two goals of imagination and foster the the exercise w formation ere to develop spatial of geometric concepts, the complexity exercises increasing with complexity the diagram. types tasks were recognition and composition of diagrams, explanations the geometric relationships. The experiment concentrated on the notion transformation: translation , rotation, reflection , and on shifting figures mentally without use of models. Results from the sixth grade indicated that graphic material can develop spatial imagination. Conclusions were that major goal should be to eliminate flaws in present methods Hershkowitz (1989) further examined acquisition basic geometry concepts in two experimental situations. Subjects were students in grades five, siX, and seven two schools, elementary teachers , both preservice inservice. process involved defining concept previously unknown partic ipants, and having them select or draw examples of the concept. The number critical attributes influenced the accuracy example choices. "There is a negative correlation between number to concept attributes and the mean success score the task" (1989, 67) . A single prototype shape was often envisioned by the participants. role of visualization is a complex process, according to Hershkowitz (1989) , in that one cannot form an image of a concept without visualizing its elements, that visualization made may constrict the correct image concept. The computer may be of benefit. "Visualization and visual processes have a very complex role in geometrical processes . that a dynamic interaction with a geometrical microworld S. contributes to visual flexibility. More work is needed to understand better the positive negative contributions visual processes" (Hershkowitz, 1990 Diagrams themselves do provide obstacles learner (Yerushalmy & Chazan, 1990; Zykova, 1969) Yerushalmy and 94) to research previously mentioned (Presmeg, 1986) , study Yerushalmy 1986 revealed that students using computer software called the Geometric Supposer used more diagrams a generalized of year test than did those students that did not have Supposer experience. Experience with computer images may increase utilization and usefulness of diagrams, and remove some the obstacles that they cause. From 1984 1988 the effect of the Supposer on students and teachers high school geometry classes was studied (Yerushalmy et al., 1990) In an inquiry approach to the teaching/learning process, data was gathered from six sources: classroom observations, student Supposer work, minutes of monthly teacher meetings, teacher interviews, teacher writing reflections, student clear materials interviews. creating good Guidelines inquiry problems resulted from the investigations. Inquiry teaching important for mathematics, difficulties" (Yerushalmy and presents et al., 1990, "challenges 242) to bring into the classroom. "The formulation inquiry problems will inquiry important approaches successful using development other toolbased of guided software environments in geometry .and in other domains" (Yerushalmy et al., 1990 , p. 242) Another Geometric study Supposer involving the indicated use of improved computer performance school honors students. Results indicated that there did exist an effect on problemsolving skills on geometry achievement from integrating Supposer activities into the curriculum every two weeks. Bishop restates one of these obstacles , "One problem geometry teaching generalized present many that diagram. impossible is therefore diagrammatic examples to draw necessary of a geometric concept the learners are to be restricted specificity the diagram" (1983, 180) . Bishop (1989) Stewart (1990) further reinforced relationship of spatial visualization and geometry , and suggested that the dynamic visual images of computer graphics may development spatial skills. The obstacles, computer only potential spatial to resolve many visualization, these also processes of conjecturing, searching for patterns, and generalizations, that processes "doing mathematics" constructing knowledge (Bishop, 1989; Hershkowitz, 1990; Janvier , 1987; Lesh, 1987; Senechal, 1990; Steen, 1990) Summary review literature presented this chapter focused on studies and research pertaining to cognition and constructivist theory learning, students build new knowledge from prior knowledge experiences within environment active participation. research provided a strong basis investigations into student learning and computer environments within domain specific content areas. Spatial visualization geometry ese nt rich areas examination content, cognition, learning environment. The rese arch also indicate a need further examination of how students build knowl edge, specifically within an environment of dynamic technology available the mathematics classroom today. CHAPTER 3 METHODOLOGY Overview of the Study This chapter describes the research questions, the evaluation instruments, participants for the study. It outlines processes and procedures the design and implementation, The purpose of the data this analyses study was that were used. investigate the problemsolving strategies used by geometry students following instruction in a dynamic technological environment within a Yackel "constructivist , 1990) teaching Specifically experiment" study (Cobb, explored Wood the relationships among the students ' spatial visualization ability, mathematics achievement problemsolving strategies with and without the availability software called The Geometer's Sketchpad. Students were instructed with teacher presentation and the Sketchpad. Specific "hands on" lessons on experience locusmotion using concepts provided the basis exploring problemsolving strategies. The problemsolving session followed the constructivist approach creating "problematic environment that would  I I I r 1  investigated dynamic for the strategies tendency to use drawings in relationship to use the availability the technology during the problemsolving session with the LocusMotion Inventory (LMI). Research Questions The study addressed three questions of interest: Are correct response score (CRS), tendency use drawings , and/or use of dynamic strategies that students utilize to answer locusmotion problems related students' spatial visualization index (SV) mathematics achievement (MA)? Does Sketchpad availability (SA) affect correct response score (CRS) , tendency to use drawings (DR) or use of dynamic strategies Are tendency to use drawings (DR) and/or the use of dynamic strategies (DS) that students utilize when attempting to solve locusmotion problems related students' correct response score (CRS)? Research questions were investigated by more detailed statistical questions: the relationships or MA with CRS , DR, DS vary across the Sketchpad availability groups? relationships do vary across Sketchpad availability , what relationship of CRS, or DS Does Sketchpad availability affect CRS, or DS on the LocusMotion Inventory? the relationships of DR or DS with CRS vary across the Sketchpad availability groups? the relationships do not vary across Sketchpad availability what relationship of CRS with DR and Selection of Evaluation Instruments Measures of spatial visualization and mathematics achievement were taken to examine as predictor variables, and to provide blocking or matching variables statistical procedures, such as those used by Tall Thomas (1989) their study teaching algebra with computer. Three independent measures of spatial visualization were taken. Mathematics achievement data was taken from the first semester geometry examination. This curriculum based test given to all regular geometry students was created by Glencoe the Merrill Geometry text. locusmotion problem inventory was created administered specifically for this study. A description each instrument contained the next section. The Spatial Visualization Battery The Spatial Visualization Battery consisted three factor of Spatial Orientation are Card Rotations Test the Cube Comparisons Test. An additional test, Paper Folding, was used from the cognitive factor of Visualization which also includes Form Board and Surface Development Tests. scoring Since each schemes, three chosen the tests results were converted has different to standardized scores and summed a single factor called spatial visualization index. Standardized scores, zscores are corrected for the mean and scaled by the standard deviation the variable. They are useful when comparing combining different scoring schemes. Each these tests has been used alone conjunction with other tests in studies involving spatial ability by Moses (1977/1978) Tillotson (1984/1985) indicated in previous chapters, Tartre (1990) reported that these tests have also been cited as evidence spatial ability Conner Serbin (1980) Linn and Petersen (1985). Originally ETS considered three tests elements factor called spatial visualization. They were chosen this study based upon research to represent multiple parts of the dimensions spatial ability under consideration. Card Rotations Test The purpose this of problems test As recognize relationships among parts figure in order identify the figure when orientation is changed" (1984/1985, 42) . test has been identified with the dimen sion called spatial visualization or rotation (Moses, 1977/1978; Tartre, 1990; Tillotson, 1984/1985) The test is two parts 10 problems each. Each problem consists initial irregular figure on left followed by eight various representations of figure. The students decide whether each one of the eight figures the same as, or different from, one at left, mark a box each or D. A transformation by translation or rotation considered the same. transformation reflection is considered different. Students were given minutes each two parts test. score was number items answered correctly minus number answered incorrectly. Cube Comparisons Test This test displays cubes with three faces showing. purpose of test is to measure the student's ability to recognize parts of a given configuration (Tillotson, 1984/19 85) Although some analyses categorize this test as a measure of spatial visualization (Tartre , 1990; Tillotson , 1984/1985) , other analyses suggest that it may relate to a different factor Each problem contains drawings of two cubes that can be drawings of the same cube in a different orientation, must be different cubes altogether. The test contains two parts with twentyone pairs of cubes each. Students mark or D each pair. Students were given minutes each the two parts this test. The score was number items answered correctly minus the number items answered incorrectly. Paper Foldinq Test The purpose of this test, also known as the Punched Holes Test, to measure the student's ability to mentally manipulate a given spatial configuration into a different one. It also been identified with the dimension called spatial visualization or rotation (Moses, 1977/1978; Tartre, 1990; Tillotson, 1984/1095). In each problem, two to four figures represent a square piece paper being folded then punched with a hole. The student' task is to match that representation with the correct representation of unfolded piece of paper with hole(s) punched proper locations. The test contains two parts with 10 problems each. Five answer choices are provided each problem, only one is correct. Students were given minutes each two parts the test. The score was number items answered . a a i.I  * _ LocusMotion Problem Inventory Search problemsolving literature inventories, revealed several and many global specific to geometry or spatial orientation (Krutetskii, 1976; Moses, 1977/1978; Tillotson, 1984/1985). domainspecific content, However, such the criterion locusmotion to provide problems, caused difficulty specific to finding a the content and problemsolving applicable inventory to dynamic problem solving strategies. problemsolving inventory specific content researcher, locus incorporating of points was created by the locusmotion problems similar to those found in geometry textbooks. The content area locus of points was chosen because subject independent of placement in the curriculum (provided that basic concepts have been introduced) often left the geometry curriculum due time constraints. Studies Ferrara (1987/1988), Campione (1989), and Tillotson 84/1985) initial prompted following 30 problems: considerations prerequisite geometry content, varying level of difficulty, novelty of problems. This of problems was reviewed six current preservice mathematics educators, to assist devising the final light inventory of problems. following Each problem was questions: reviewed problem _ ...._ visualization techniques of motion or dynamism be useful solving the problem? Would the problem be especially interesting in an interview setting? Based upon the reviewer ratings, the set was reduced to nine problems the written LMI. least one problem from each difficulty level was selected. problems were able to be solved with pencil and paper only. Five problems were also chosen to be available students to solve in an interview setting See Appendix A problems interest the researcher was not only the performance on the problems, but also the strategies employed in reaching solutions. Problemsolving strategies identified and scored were tendency to use drawings and use of dynamic strategies, scored through observation, written work and computer drawings. Another means of ascertaining some measure of problemsolving process used by student of his was to ask the strategy student (Battista, 1990; or her Battista own assessment , Wheatley, Talsma, 1989) Each problem on the written portion consisted parts. students were asked to provide the solution problem, to describe solution thinking process involved. Data each problem consisted written work , including figures and printed copies sketches created on the computer students using The Geometer's Sketchpad. LocusMotion Inventory. score sheet for the LMI can be seen in Appendix B. Correct response score was awarded from 0 to increments of 0.25, based on amount success. Scores for tendency to use drawings were for no drawings, one, for two, for multiple drawings. Scores use of dynamic strategies were indication in words, or 1, or figures, depending use of on the motion, dynamic thought processes. Several scoring rubrics were used as models for the one created (Battista, 1990; Battista, Wheatley, Talsma, 1989; Fennema Tartre, 1985; Ferrara, 1987/1988; FerriniMundy , 1987; Krutetskii, 1976; Moses, 1977/1978; Presmeg, 1986; Tartre, 1990) . The scoring rubric also took into consideration some concerns addressed by Lean and Clements (1981) about types of questions and strategies required, use of incorrect solution attempts, and unwritten strategies. Although coming from very different perspectives, Spiro et al. (1991) Senechal (1990) images students suggested used. Ye taught more that data rushalmy with include Chazan the Geometric thinking process, number (1990) of different noted Supposer used including freehand that diagrams drawings, than those that had not been taught using the Supposer. Sixteen percent problems were coded additional rater interrater reliability verified (Ferrara, 1987/19 FerriniMundy, 1987; Wearne Hiebert, Taskbased interviews were conducted with students, the total sample, randomly selected within each class from each SA group. Davis (1986) described the taskbased interview as students solving specific problems talking aloud, and an interviewer observing those solutions with audio or video recording. Interviews have been used in many studies and recommended by many researchers to explore problem solving and thinking processes of students (Bishop, 1983; Confrey, 1990; Davis, 1983; Davis 1986; Davis & Maher, 1990; 1986; Ginsburg Tartre, et al., 1990; vo 1983; Krutetskii >n Glaserfeld, , 1976; 1987) The Presmeg, types interview strategies varied. Based upon previous research interview was designed to observe students in three situations: explaining and confirming the solutions problems completed on the LMI (Lean Clements, 1981) , attempting problems at next level of difficulty from written LMI, attempting problems available only interview (Campione et al., 1989; Ferrara, 1987/1988; Hoffer, 1983; Lampert, 1991; Rohrkemper, 1989). interviewer began with student' work from the written and proceeded to problems specifically designated taskbased interviews , depending upon level difficulty and success the student, exploring the activity development student within his (Vygotsky , 1978) or her As Krutetsk zone of ii (1976 proximal )and the aid of others. work was collected examination, as was transcribed audiotape of interview (Davis, 1986; Dover Shore, 1991; Resnick, 1989). Students were instructed talk as they worked, a technique used several studies and promoted within a variety assessment tools (Ginsburg et al., 1983; Huinker , 1993; Mashbits, 1975 Selection Description of Population Sample Participants for the study consisted geometry students from seven geometry classes at one high school. The seven classes were taught by four mathematics teachers interested in encouraging the use technology the mathematics classroom, specifically in using The Geometer's Sketchpad the geometry curriculum. The subjects were mixed gender , age, racial background in grades All students participated instruction using The Geometer's Sketchpad. The students were randomly assigned two groups. One group worked the LMI traditional paper pencil manner. Another group the computer software available as they solved problems. Two students from each class period were randomly chosen , one from each group, to participate in the interview sess ion. total students worked problems within an interview setting. 55 Procedures Prior to study the University of Florida' Institutional Review Board granted permission for the investigation take place. Students were informed and had to obtain parental permission (see Appendix C) participate study. The Spatial Visualization Battery was administered two occasions, before any exposure to the Sketchpad and following the session. A script explained the purpose and directions for the three parts the battery. First semester examination scores were provided by the teachers. The problems from the LocusMotion Inventory were presented the students to be worked within one 50minute class period. The classroom teachers administered the paper pencil session; researcher administered the computer sessi the on. LMI. A script was The used by interviews were conducted administrators by the to execute researcher, with Each audiorecording interview interview transcribed lasted approximately afterward. 20 minutes. Description Instructional Phase The effectiveness of instruction using technology can best be explored reality the context classroom in which (Fischbein , 1990; will Kulik occur Kulik, classes. Teachers used the software in a presentation mode several months. Students had several lessons "hands experience with software. The researcher presented lessons on locus of points. lessons were extension previous lessons on use of the Sketchpad , and lessons that introduced the concepts of locus of points. There were six lessons, each lasting one 50minute class period. Each was a combination of teacher demonstration, student discovery , and student experience with problems on the computer. Initial lessons presented the concept locus and the steps involved in solving locus problems, specifically applied to perpendicular bisectors, angle bisectors, triangles. Points concurrency such incenter , circumcenter, centroid, orthocenter were explored. Other lessons extended application locus to circles. applied Finally, to the the concept intersection locus figures, of points further was the intersections loci. The different researcher ces conducted all in results due lessons to eliminate teacher variability, was aware possibility that a threat validity may introduced by researcher influencing results. This was minimized by the use of specific presentation plans, written activities students with the computer. Seven orobiem sets anrd wnrkshpt nc rlir inr + i nct "r 'int i r nnn 1 Portions of lessons were tested during the spring 1993. The pilot was a series of lessons that introduced students to The Geometer's Sketchpad and explored intersection figures. lessons varied from "very guided" with specific instructions to open investigations guided only by presenting problem situations and questions. The purpose of pilot was to make an initial evaluation the clarity the activities and instructions. Two groups three students each worked on the activities with no teacher assistance or intervention. Each student provided observations and journal entries that discussed the instructions, activities, their involvement and attitude with the computer and The Geometer's Sketchpad. The results the pilot study were used revise the activities that introduced The Geometer Sketchpad to the students. Statistical Procedures Data analysis consisted three parts: descriptive statistics, qualitative quantitative st investigations. :atistical analyses Descriptive statistics were obtained question Review of st dealing with variables , appropriate U dies in spatial under statistical literature, abilities consideration. procedures For were each applied. especially those and mathematics, suggested several statistical procedures applicable to a similar study Tartre, 1990; Tillotson, 1984/1985). Statistical procedures for this study included correlation coefficients, analyses covariance, statistical and multiple regression analyses. approach The often problematic when dealing with educational 1985). In issues of addition instruction and to quantitative learning statistical (Schoenfeld, procedures, a qualitative component examined interview data describing the problemsolving processes of individual students. Summary This study was designed as a "constructivist teaching experiment" (Cobb, Wood Yackel, 1990) investigate problemsolving strategies used by geometry students following instruction a dynamic technological environment to explore visualization ability, relationships mathematical among the ability, students' and spatial problem solving strategies with and without technology. Three independent measures of spatial visualization were taken. They included ETS Card Rotations, Cube Comparisons , and Paper Folding tests. A locusmotion problem inventory was also administered. Participants for the study consisted geometry students from seven geometry classes one high school. students participated instruction using The Geometer's Sketchpad. The instructional phase 1 1 I * *  L J i I I assigned two groups: those who would have computer software available during the LMI, and those who would not. Data component. statistics analysis had both a Statistical , correlations quantitative and a procedures, , analyses o qualitative including descriptive f covariance, and multiple regression analyses , were used to explore the questions of interest in the study. results of data analyses are presented Chapter discussed Chapter CHAPTER ANALYSIS OF THE DATA This chapter contains the descriptive statistics variables under investigation and results of the data analyses pertinent to the questions this study. It also includes results the qualitative investigations observation and interview. This study was designed as a "constructivist teaching experiment" (Cobb, Wood Yackel, 1990) to investigate problemsolving following state' instruction gies in a used by geometry students dynamic technological environment to explore relationships among the students' spatial visualization index mathematics ability, correct response scores, tendency to use drawings, use of dynamic strategies Three with and without the Sketchpad independent measures of spatial available. visualization were taken at two different occasions. spatial visualization index included Card Rotations , Cube Comparisons, and Paper Folding tests. A locusmotion problem inventory (LMI was also administered. Participants for the study consisted geometry students from seven geometry classes at one high school. students Sketchpad. The instructional phase included six lessons that were an extension of previous lessons on use of the Sketchpad lessons that introduced the concepts locus of points. Students were assigned individually and at random to two groups for the LMI problemsolving session: those who would have the Sketchpad available, those who would not. The problemsolving session lasted one class period minutes. Individual interviews were held on the following day with two students from each class, one from each group. Interviews lasted approximately 20 minutes during which students worked locusmotion problems aloud. Research Questions The study addressed three questions interest: Are correct response score (CRS) , tendency to use drawings , and/or use of dynamic strategies (DS) that students utilize to answer locusmotion problems related students' spatial visualization index (SV) mathematics achievement (MA) Does Sketchpad availability (SA) affect correct response score (CRS) , tendency to use drawings (DR) or use of dynamic strategies (DS) Are tendency to use drawings (DR) and/or the use of dynamic strategies (.DS)  V *I I  students utilize when tha^t Results of Data Analysis Descriptive statistics are presented in Table 41, which includes the number means (M) , and standard deviations following variable Sketchpad a s: correct availability response s (SA) core for the CRS), tendency use drawings (DR) use of dynamic strategies (DS), spatial visualization index and mathematics achievement (MA). MA was measured by first semester geometry examination scores. total 60 was possible, the highest achieved was CRS, DR and DS were sums the measurements individual problems on the locusmotion inventory (LMI) Problem scores for CRS were awarded from increments of 0.25 based on amount success. Problem scores DR were for no drawings, one for multiple drawings. Problem scores were depending figures, use on the of motion, indication in words, or dynamic thought processes. was based on the composite zscores for the three measures taken time the LMI: Card Rotations, Cube Comparisons, between and Paper Folding Tests. two occurrences the spatial The correlations visualization tests were high; time therefore, were used the measures in analysis. taken closest Correlations between Table 41. Descriptive Statistics Sketchpad With and Without Variable Sketchpad Number Mean SD CRS With 78 1.82 1.50 Without 81 1.99 1.62 DR With 78 14.08 6.32 Without 81 15.92 6.12 DS With 78 3.45 2.02 Without 81 3.1 2.08 SV With 75 0.07 2.30 Without 90 0.28 2.37 MA With 75 42.38 7.97 Without 94 42.86 7.71 Although students were randomly assigned , for the group without the Sketchpad available for the LMI, the means were somewhat higher on matching variables SV and MA. The means were also higher for that group on Only the mean DS was higher for the group with Sketchpad available. Due scoring rubric for CRS allowing scores increments 0.25 the mean score low. reported in the following paragraphs. An alpha level .05 was used tests. the relationships of SV or MA with CRS, DS vary across the Sketchpad availability groups? The slope relationships of CRS, DR and DS with SV and MA were compared across Sketchpad availability groups by using multiple regression. results are reported Table indicate no variation over groups in the relationships. Table 42. Results Comparing Relationships Across Sketchpad Availability Groups Dependent Independent variable variable MSE F p CRS SV 2.10 2.36 .1265 MA 2.14 0.80 .3712 DR SV 36.52 0.68 .4123 MA 35.21 1.54 .2165 DS SV 3.54 0.76 .3894 MA 3.78 0.00 .9898 the relationships do vary across Sketchpad availability , what relationship of CRS, or DS with SV and MA? r L 1 relationship of MA and CRS, MA and DR and MA and DS did not vary across groups. Consequently the pooled relationships of SV with each of CRS, and DS as well as the pooled relationship between MA and each of CRS DR and DS were investigated using multiple regression. Results are shown in Table 43 For each of CRS, and DS there is a significant relationship with SV and MA .01. Table 43. Results of with Pooled Relationships CRS, DR, and DS of SV and MA Dependent Independent variable variable r MSE F p CRS SV .38 2.11 23.74 .0000 MA .37 2.14 22.22 .0000 DR SV .30 36.44 11.17 .0011 MA .34 35.34 15.05 .0002 DS SV .38 3.54 24.02 .0000 MA .28 3.75 12.27 .0006 Does Sketchpad availability affect CRS , DR, or DS the LocusMotion Inventory? investigate the question , for each of CRS DR and an analysis of covariance (ANCOVA) was conducted with SV MA as covariates. Results are reported Table and  I I 9 A t LL_ 1_ L 1 11 1  * I t < Table 44. Summary ANOVA Table for CRS, and DS Dependent variable Source df SS MS F p CRS SA 1 1.64 1.64 0.82 0.3673 Error 139 279.36 2.01 DR SA 1 109.46 109.46 3.12 0.0797 Error 139 4883.9 35.14 DS SA 1 2.89 2.89 0.85 0.3595 Error 139 474.51 3.41 the relationships of or DS with CRS vary across the Sketchpad availability of groups? slope of relationships of CRS with DR and DS were compared across Sketchpad availability groups by using multiple regression. results are reported in Table indicate no variation across groups in the relationships. Table 45. Results Comparing Relationships Sketchpad Availability Groups. Across Dependent Independent variable variable MSE F p CRS DR 1.15 1.60 .2077 the relationships do vary across Sketchpad availability , what relationship of DR and DS? The analyses relevant to question indicated relationships of CRS with DR and CRS with DS did not vary across Sketchpad availability groups. Consequently the pooled relationship between DR and as well as the pooled relationship multiple CRS there between DS and regression. is a CRS were Results are shown significant investigated in Table relationship with using 46. DR and the level. Table 46. Results of Relationships of CRS with DR and DS. Dependent Independent variable variable r MSE F p CRS DR .73 1.15 176.6 .0000 DS .78 0.95 234.6 .0000 Other Findings For the group with the Sketchpad available , the number of Sketchpad drawings (NSD) was also scored. instances which figures were not saved nor mentioned a score of zero was awarded. Data indicates that students did use Sketchpad when was available. mean number figures SA_ I a M.  A  A 1 rF A .* J nrt _ 1 A 1 regression analysis with NSD as dependent variable indicate independent that variable and the relationship was significant, Further visualization F(1,74) = 2.11, analysis of index .1443. tests within offered some additional spatial findings. three tests that comprised the spatial visualization index were given on two occasions, before exposure the Sketchpad following the LMI sess ion. Although there was a high correlation between the score there were differences in the means that should be noted. Mean scores the Card Rotations test (CR) increased from 98.65 115.79 , an improvement of 17.1 points. the Cube Comparison test (CC) , mean scores increased from 12.55 16.65, an improvement of points. The means scores the Paper Folding test improved points, from 10.46. Additional analyses examined the relationship of individual tests, measured at time of the LMI, the scores to determine any test had a more significant relationship to CRS, or DS Since there was influence of multiple SA with regressions with on either CRS, these and DS variables respectively were implemented, each time with CRpost, CCpost, PFpost independent variables. Results are presented Table 47. Table Effect of Spatial Visualization Subtests on CRS, and DS. Dependent variable Source r MSE F p CRS CRpost .19 2.45 4.28 .0402 CCpost .37 2.18 22.94 .0000 PFpost .32 2.21 17.59 .0000 DR CRpost .19 39.58 2.69 .1034 CCpost .26 38.24 8.03 .0052 PFpost .31 36.09 13.24 .0004 DS CRpost .23 4'.14 7.10 .0085 CCpost .35 3.83 19.47 .0000 PFpost .30 3.88 14.51 .0002 Results the multiple regression analyses indicate that CCpost had the greatest influence on CRS and DS both .01. PFpost had the greatest influence on DR also .01. CRpost had least influence on CRS, Interview Results This study also had a qualitative component. Two I r qm 1 the written LMI had been determined prior to the interview, and three or four problems had been selected discussion during the interview. In most cases two of problems were from the written LMI one chosen at next difficulty level from a of problems not seen by the student before. Students solved problems aloud, with varying assistance Results from the analyses of researcher. variance comparing the means of interviewees on CRS, , and DS with those students that took only the written LMI indicated that there were no significant differences interviewed were in the means. not different from those students not being interviewed for the research questions. A main purpose of interview component the study was to explore the ability the student within his or her zone of proximal discover what development student has Vygotsky, learned, 1978) what he to is capable learning with the aid researcher (Krutetskii, 1976). Students worked problems from the written again with cases, students varying were able degrees to solve assistance. problems successfully on the written LMI that they had previously missed. Ten interview students were also able work least one problem that they had not seen before which had a difficulty level higher than one missed on the Patterns, or causes, failure were also of interest, since they & Jehng, 19 suggest ways 90). In many to expand a cases, student's students "zone" indicated (Spiro that understanding the problem caused biggest obstacle. . when you read the problemit hard understand , but when you went over way easy to comprehend. R: So the words were first stumbling block? Yes. didn't know really what it was asking for. didn't know how to do number three. What I was instead R: So did trying to make d of sixty fee these aren't S. see them read feet t apart. terrible are the fro problem wrong m race mark they? out . I just have a hard time getting the question the words. knew what I had to draw didn't know how to explain the locus. didn't understand. . Is that asking what will make it consist one point? didn't understand this one very well at all. S10: really didn't understand the question. I'm not sure times the words what they within the are asking. problems caused difficulty. Meanings terms , such as plane M, tangent, isosceles a circle that contains two given points , were often not known. Students would ask specifically , "what does this mean?" Plane M confused me. haven't learned much about planes. R: What was hard about the question? What gave you : I Another such lack obstacle was as distance r, of real or radius use variable measurements, Students commented numbers or specific directions to make on the the question understandable. Can You solve this don't have any numbers don't know the radius? so you don't know what to do. This hard to understand 'cause doesn't say where S14: they Point are. P is anywhere? Although one problemsolving heuristic think similar problem, students often related problems exactly to those they had previously experienced and did not take note of differences in the question or conditions. This student made use of heuristic in a positive manner: Matter fact very same figure as first one with isos celes triangle. Remembering a similar problem exactly like one on written LMI test, was a hindrance student to another indicated student. on two occasions that problem was like one already solved, or that remembered the solution. In both cases, there were distinct conditions that changed the question and solution. It was interesting to note the extent to which students were aware of what caused obstacles with problems. Students were often aware that they did not understand question. As can be seen the quotations above, they were less aware that they had not noted differences problem situations when similar to those already experienced. The need for understandable materials problems was obvious did not from the interpret investigations. In many problem correctly. cases, When t students :he conditions were clarified, or the correct figure drawn, the student was able to proceed to a correct solution. The wording the problems was critical the correct figure and the correct solution. three instances students made valid interpretations of the question different from interpretation instances the researcher. the students produced a two of correct the three solution, but way was expressed was not clear to researcher. Communication was critical from the students' researcher' perspective. An emphasis the mathematics classroom on communication important to understanding, instruction, assessment (NCTM, 1989) . That was clearly evident these interviews. Another purpose interview was to explore the use the computer, potential for use. Students were asked a series survey questions at beginning problemsolving twofold: session to put see student at Appendix E). ease The with purpose was interviewer, to explore student attitude and beliefs about more mechanical and boring. instances, students noted the technical aspects technology obstacle. In all instances , students indicated that everyone should learn something about computers. Stated benefits of the computer included accuracy figures faster and measurements, drawings, precision, easier and ability to move calculations, figures while seeing changes simultaneously. available figures Students for the written and measurements that did not have interview sessi the Sketchpad ons indicated that they would have used the computer been available, had the compute perhaps would have done r available appreciated better. Those capability that ies. Well, took the test in the classroom. could have done better it had been on the computer Because would have do much better than my visualized drawings. it better. could have can put a point more locus, sca probably a midpoint than mine. could have 'cause used it would them knew how to read the question correctly. you had had computer would have used Yeah. Would it have helped? can move stuff around like circle. think it helped learn think. It may have helped since it's more accurate easier to just a kind o line see. can see you draw yourself a bother to say this an angle bisector that helps. is what It's want and highlight everything. R: So the technical part may not be worth th ie effort. you think When we were the compute learning, r helped yo it did make in any way? things clearer see, cause sometimes you sketched it out then you started seeing things and make sense. that made other things change the way you might solve? S9: your you You It might have. own then think can it yourself used it did. things it makes the computer it a on almost you it and learn ask. easier. every problem, right? S13: Yeah. Could you have done these you hadn't had computer? S13: It would have been a longer. just can't draw. assessment should be, interview sessions themselves were a learning experience. Students were able to solve problems with ass instance. Students were able share the i their opinions instruction, and knowledge about the content area, technology. I Did actually am glad that you went over learn something? Yeah. just very well. never really try to have listen, read been able then can learn math understand it myself. You knew some of the answers, said you were afraid. had ideas. That always happens to me though won't put it down. won't want to embarrass myself something. was making it harder than was. Are these hard? S10: Yeah. Why are they hard? They make you think. Attitudes and beliefs, as well and knowledge and success on the content. were i mnrnvPd with rnmmlln I i n tnn 76 Summary This used study geometry investigated students problem following solving instruction strategies a dynamic technological environment explored the relationships between students spatial visualization index, mathematics ability, correct res ponse scores, tendency use drawings, use of dynamic strategy ies with without Sketchpad available The study addressed three questions of interest Are the correct response score (CRS), tendency use drawings DR) , and/or use of dynamic strategies that students utilize students answer ' spatial locusmotion visual zation problems index related or mathemati achievement (MA)? Does Sketchpad availability (SA) affect correct res ponse score CRS), tendency use drawings (DR) , or use of dynamic strategies (DS) Are the tendency use drawings and/ use of dynamic attempting strategic to solve that locusmotion students problems utili related when to the students ' correct response score (CRS)? Research questions were inves tigated more detail statistical stions outlined below. Do the relationships of SV or MA with CRS, , or DS vary across Sketchpad availability groups relationships not vary across Sketchpad availability, what the relationship CRS, or DS with SV and Does Sketchpad availability affect CRS, or DS LocusMotion Inventory? Do the relationships of DR or DS with vary across the Sketchpad availability of groups? If the relationships do not vary across Sketchpad availability, what s the relation ship of CRS DR and Results of both the quantitative and qualitative components of the study have been described detail chapter and are summarized this sec tion. Initial analy ses indicated that none of the relationships of spatial visualization index with correct response score , tendency use drawings , or use of dynamic strategies varied across Sketchpad availability. Similarly none of the relationships of mathemati achievement with correct dynamic response score, strategies varie tendency across use groups drawings, Results or use or of multiple session analyses indicated that each of the variabi correct res ponse score , tendency use drawings , and use dynamic spatial strategies visual had zation a significant index relationship and mathematics with achievement To investigate whether Sketchpad availability ects correct response score, tendency use drawings, and use dynamic strategies , an analy S1S covariance (ANCOVA) was conducted each variable with spatial visual nation index and mathemati achievement as covariates. Results indicated that correct Sketchpad response availability score does not , tendency significantly use drawings affect use dynamic strategies. analy ses concerning relationships correct res pons e score with tendency use drawings with use dynamic c strategies indicated that none of the relationships varied across Sketchpad availability groups Results multiple regression analy ses indicate ed that correct response score there is a significant relationship with both tendency use drawings and use of dynamic strategies. For students with the Sketchpad available , th ere was no significant relationship between the number computer created use drawings correct Sketchpad when was response score. available Subj Students ects did that computer available during the problem solving session created a mean of 4 figures on the comput er. students computer that had drawings the for computer one or more available or 94% create problems Further visualization anal index of the offered tests some a within additional spatial findings. three were tests given exposure written that on two to the comprised occasions Sketch session. !pad and Although the spatial before the immediately there was visualization students following index any the a high correlation between scores, there were differences the means. Additional analyses examined the relationships the individual tests the LMI scores to determine any test had a more significant relationship to correct response score , tendency to use drawings, or use of dynamic strategies. Results of the multiple regression analyses indicate that Cube Comparisons test had the greatest influence on strategies. correct response Paper score Folding test had use of dynamic the greatest influence on tendency to use drawings. Card Rotations test had least influence on correct response score, tendency to use drawings, use of dynamic strategies. Analysis of interview data revealed that cases students were able to solve problems successfully on the written LMI that they previously missed. Sketchpad solutions availability did not make a to interview problems. difference In many cases the students indicated that understanding the problem caused the biggest obstacle. times the words within problems caused difficulty. Another obstacle was use of variable measurements, such as distance r, or radius Students commented on the lack of real numbers or specific directions to make the question understandable. Although one problemsolving heuristic think those they had previously experienced take note of differences Students in the question or conditions. were aware of what caused obstacles with problems. Students were often aware that they did not understand the question They commented on the fact that they did not understand the question or did not know the definitions of terms. They were less aware that they not n those oted differences already in problem situations when similar experienced. The need for understandable materials inquiry problems cases, s When the drawn, was obvious students did not conditions student from the investigations. interpret were clarified, was able to proceed In many problem correctly. or the correct to a figure correct solution. The wording problems was critical correct figure the correct solution. Another purpose of interview was to explore use the computer potential for use. Stated benefits the computer included accuracy figures measurements, precision, easier calculations faster drawings, ability to move figures while seeing changes in the figures and measurements simultaneously. Students that did not have the Sketchpad available for the written interview sessions indicated that they would have used the computer it been available, perhaps would The interview sessions themselves were a learning experience. Students were able to solve problems with assistance. knowledge Students were able about technology. to share the content area, Attitudes and beliefs, their opinions and instruction, as well the as performance were improved with communication between learner and instructor. CHAPTER SUMMARY AND CONCLUSIONS Overview of Study The purpose of problemsolving this study was strategies used investigate geometry the students following instruction in a dynamic technological environment. Specifically study explored relationships among the students' spatial visualization ability, mathematical ability, and problemsolving strategies with and without technology. Students were instructed with teacher presentation and "hands experience Sketchpad. using a software Specific product called lessons on locusmotion he Geometer's concepts provided Methods basis exploring problemsolving solutions were examined level strategies. drawing dynamic visualization used relationship availability technology during the problemsolving session. Three independent measures of spatial visualization were taken two different occasions. spatial visualization index included ETS Card Rotations, Cube for the study consisted geometry students from seven geometry classes at one high school. All students participated in the instruction using The Geometer's Sketchpad. The instructional phase included lessons that were an extension previous lessons on use of the Sketchpad lessons that introduced concepts locus of points. Students were ass signed individually at random two groups for the problemsolving session: those who did have the Sketchpad available, those who did not. The problemsolving session session lasted one class period 50 minutes. Individual interviews were held on the following day with two students from each class , one from each group. Interviews lasted approximately 20 minutes, which students worked problemsolving session locusmotion problems followed aloud. the constructivist The approach of creating "problematic environment that student's adequate constructive endeavors" would elicit (Fischbein, 1990, Data analysis had both a quantitative and qualitative component. Results Discussion The theoretical premise that supports this study that use of a dynamic technological environment, such The Geometer's Sketchpad, can enhance construction constructivist theory learning promotes the belief that students build knowledge through activity experience. Dynamic software can provide experiences conjecturing, building in visual frameworks within a zing, specific content domain. The study addressed three questions of interest: Are use drawings the correct (DR), response and/or use of score dynamic (CRS) , tendency to strategies (DS) that students utilize to answer locusmotion problems related the students' spatial visualization index (SV) mathematics achievement (MA)? Does Sketchpad availability (SA) affect correct response score (CRS), tendency to use drawings (DR) , or use of dynamic strategies (DS) Are tendency to use drawings (DR) and/or the use of dynamic strategies (DS) that students utilize when attempting to solve locusmotion problems related students' correct Research response questions were score (CRS)? investigated by more detailed statistical questions outlined below. relationships or MA with CRS, DS vary across the Sketchpad availability groups? relationships do vary across Sketchpad availability, what relationship of CRS, or DS with SV and MA? the relationships of DR or DS with CRS vary across the Sketchpad availability of groups? the relationships do not vary across Sketchpad availability Initial what is the analyses relationship of indicated that CRS and DR and DS? none relationships of spatial visualization index with correct response score, tendency to use drawings, or use of dynamic strategies varied across Sketchpad availability. Similarly none the relationships of mathematics achievement with correct response score, tendency to use drawings, or use dynamic strategies varied across groups Results of multiple regression analyses indicated that each the variables drawings, of correct and use of response score, te dynamic strategies ndency to use significant relationship with spatial visualization index mathematics achievement. To investigate whether Sketchpad availability affects correct response score , tendency to use drawings, use of dynamic strategies, conducted each an analysis of variable with covariance spatial (ANCOVA) visualization was index and mathematics achievement as covariates. Results indicated affect that Sketchpad correct response availability score does , tendency to significantly use drawings, use of dynamic strategies. The availability computer during the problem II_ _C _ the covariate accounted spatial for most of ability or variance. mathematicc Although achievement specific to the geometric content locus of points, these findings supplement those by Moses (1977/1978) and Presmeg (1986) that spatial ability is a good predictor of problemsolving performance. Moses (1977/1978) found that students with high spatial ability However often do , in this write down study, visual solution processes. students were encouraged to write down solutions, make drawings, and describe visual solution processes. Both spatial visualization and mathematics achievement were significantly related tendency to use drawings. Additional findings included trends seen in the subtests spatial visualization index. Although study did not specifically address the components of spatial ability nor whether is a modifiable quantity , descriptive statistics warrants c provide some comment here an interesting d, perhaps, information that further research. three tests that comprised spatial visualization index were given on two occasions, before any exposure the Sketchpad and following the session. Although there was a high correlation between scores, there were differences means that should be noted. Mean scores on the Card Rotations test (CR) increased from 98.65 to 115.79, improvement 4.1 points. The means scores on the Paper Folding (PF) test improved points, from 7.99 10.46. Although there were differences in the means of the subtests on two occasions, without a control group that did not use the Sketchpad at all, there can be no direct comparison. However , the fact that the mean scores on all subtests overall index score increased suggests that spatial ability a modifiable quantity , as indicated the s 1988; earch the Hembree, literature 1992; Moses, (BenChaim 1977/1978; Lappan Tillotson, Houang, 1984/1985; Vinner, 1989) Additional analyses examined relationship the individual tests, measured time of the LMI, scores to determine any test had a more significant relationship to correct response scores , tendency to use drawings, multiple or use of dynamic strategies. regression analyses indicated Results of that the Cube Comparisons test the greatest influence on correct response score .37) use of dynamic strategies .35) Paper Folding test had the greatest influence on tendency to use drawings .31) . Card Rotations test had the least influence on correct response score, tendency to use drawings, tests, use Paper dynamic Folding strategies. Cube Comparisons involve three three dimensional figures or motion Whereas, Card Rotation Spatial Orientation, the Paper Folding test as one the measures of Visualization. The relationships among the variables were also investigated. The analyses concerning the relationships of correct response score with tendency to use drawings, and with use of dynamic strategies indicated that neither the relationships varied across Sketchpad availability groups. Results of multiple regression analyses indicated that correct response score there a significant relationship with both tendency to use drawings use of dynamic strategies. Students extensively used drawings and dynamic strategies. Both variables were highly correlated to correct response scores, indicating that they were effective strategies locus motion problem solving. As reflected to successful pro in research, blem solving flexible thinking (Davis,1986; Dover related Shore, 1991; Schoenfeld, 1987; Spiro Jehng, 1990) . Flexibility indicated by the tendency to use drawings Based on the scoring rubric, on the average each student drew more than one figure each problem. Research and experience indicate that lack of multiple drawings is often a limitation Results to success suggest in mathematics that experience with (Vladimirskii, computer 1971). images may increase utilization and usefulness of diagrams. The of rigidity fixedness associated with images (Yerushalmy & Chazan, 1990; Zykova, 1969). Students did use the Sketchpad when was available. Subjects solving that had session computer. the computer created a mean of the students available during the 4.3 that had problem figures on the computer available, or 94% created computer drawings one or more problems. Having learned content with the Sketchpad available, they students did not have imitated it available its capabilities for the even when problemsolving session Students used such language as, traced the locus I" "I imagined the motion, " etc. that indicated that they envisioned motion as they would have on computer. Students also commented that they would have been more accurate with solutions their drawings, they and more correct had had the Sketchpad their available. Since the results indicated that both groups had similar correct response scores, it appears that the drawings dynamic strategies produced similar results. In an environment in which assessment may be different from instruction, these results suggest that problemsolving strategies learned in one environment may transfer to others. computer this carried study , the strategies over to paper employed and pencil with the problems. The levels problems. Although the difficulty levels the problems chosen for the LMI were agreed upon by reviewer ability students may have been overestimated. Although there was a correlation between spatial visualization and correct response score .38), the correlation between strategies utili and correct response score was significantly higher DS). The findings agree with those of other researchers that instruction and practice in a specific content area are more dominant factor for problem solving than prior knowledge and skill (Hembree, 1992 Kantowski, 1981; Krutetskii, 1976). Interview concerning the sessions use provided additional of drawings findings the availability computer that supports prior research conclusions that computer potential conjecturing, searching to resolve many for patterns, obstacles generalizations (Bishop, Senechal, 1989; 1990; Hershkowitz Steen, , 1990; 1990) . Janvier, Students 1987; that Lesh, did not 1987; have the Sketchpad available for the written interview sessions indicated that they would have used computer had Thos it been available, e that had the computer perhaps would have done available appreciated better. its capabilities. Benefits c the computer included accuracy changes figures and measurements simultaneously. providing students with "hands on" experiences with these characteristics, the development of spatial orientation skill as described by Tartre (1990) may be accelerated. Tartre concluded that "spatial orientation skill appears be used in specific and identifiable ways . accurately estimating the approximate magnitude of a figure, demonstrating the flexibility to change an unproductive mind set, adding marks to show mathematical relationships, mentally moving or assessing the size shape of part figure, and getting the correct answer without help to a problem in which a visual framework was provided" (1990 , p. A main purpose interview component the study was to explore the ability student within his or zone of proximal development (Vygotsky , 1978) and discover what student has learned what he capable learning with the aid the researcher (Krutetskii, 1976) . the interview session, students were successful problems with varying degrees assistance from investigator. Students were able to go beyond the known problems solve those novel questions with some discussion components the question itself. Once students understood the question, they demonstrated ability to change an unproductive mind solutions. Sketchpad availability make a difference solutions interview problems. Patterns, or causes, failure were also of interest, since they suggest ways to expand a student' zone. Student comments indeed indicated the most that first phase of important, the most critical problem solving (Polya, 1957). Understanding the problem was the biggest hindrance to successful that solutions. understanding the In many cases, problem caused the students biggest indicated obstacle. Understanding problem may have been curtailed lack an initial were able within th to draw the eir grasps. figure. Once figure, t Assisting students understood and he solution was often students well in building that initial visual framework indeed a critical step in the problemsolving process communication were most (Polya, important 1957). Language to expanding the student's zone of proximal development. Students problems. were aware of what caused Students were often aware that obstacles they with did understand the question. As can be seen in the quotations in Chapter students commented on the fact that they understand the question or did not know the definitions terms. Anecdotal evidence found in students' descriptions of how they solved problems that indicates that students previous problems and the new ones was often a hindrance. Students are accustomed to repeating on assessments the same kind of problems experienced during instruction and practice. The need comprehensible materials problems was obvious from the investigations. In many cases, students did interpret problem correctly. When the conditions were clarified, or the correct figure drawn student was able to proceed to a correct solution. The wording the correct problems w solution. ras critical Communication the correct figure was critical from the students' and researcher's perspective. An emphasis mathematics classroom on communication important to understanding, instruction, and assessment (NCTM, 1989) Limitations This study contained limitations that affect generalizability results. Participants consisted high school geometry students in one high school with four teachers. The domain was contentspecific to locus points. limitation of scoring the LMI lies the use selfreport documentation of how students solved problems, drawing conclusions from the figures and written solutions provided. Since there were no common standardized tests that could be used for matching, common 