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Interface style, flow, and reflective cognition : issues in designing interactive multimedia mathematics… Sedighian, Kamran 1998

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Interface Style, Flow, and Reflective Cognition: Issues in Designing Interactive Multimedia Mathematics Learning Environments for Children by Kamran Sedighian B.Sc, Concordia University, 1985 M.Sc, McGill University, 1987 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF D o c t o r o f P h i l o s o p h y in T H E F A C U L T Y OF G R A D U A T E STUDIES (Department of Computer Science) We accept this thesis as conforming to the required standard The University of British Columbia February 1998 © Kamran Sedighian, 1998 In presenting this thesis/essay in partial fulfillment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Computer Science The University of British Columbia 2366 Main Mall Vancouver, BC Canada V6T 1Z4 Date: F**?™^ V, 1 11 * A b s t r a c t Many children find mathematics boring, irrelevant to their lives, and difficult to understand. These feelings are influenced by many factors. One of these factors is the learning environments in which children encounter mathematics. The National Council of Teachers of Mathematics recommends the use of interactive computer software in children's mathematics education. However, due to unique cognitive and affective needs, designing interactive software for children is complex and challenging. There is need for systematic interdisciplinary research to provide developers of educational software with sound design principles. The purpose of this dissertation is to explore four main inter-related issues: 1. Designers of educational software often use 'Direct Manipulation' and 'Command-Based' interface styles. What role does the user interface play in multimedia mathematics learning environments? How do different interface styles influence learning? 2. Formal understanding of mathematical concepts is important. How should the user interface be designed to support children's learning of explicit, formal mathematical concepts? 3. Reflection is crucial to deep understanding of mathematical concepts. How should a learning environment in general, and the interface in particular, be designed to afford 'reflective cognition'? 4. Designers know little about how to structure tasks to promote the optimal psychological experience of 'flow'. How should a multimedia learning environment be structured to be conducive to experiencing 'flow' in learning? What are some design elements that can make children's learning of mathematics fun and enjoyable? ii There are few, or no guidelines, for what constitutes effective human-computer interfaces for educational purposes. Due to a lack of proper interface design guidelines, designers of educational software for children often use the interaction styles that were originally designed for productivity tools. Recently, the casual use of such interaction styles for educational purposes has been questioned. This dissertation closely examines the issue of interface design for multimedia mathematics learning environments for children and makes recommendations for a new conception of interface manipulation styles resulting in more effective educational user interfaces. To structure a mathematics activity so that it combines the two elements of fun and formalism and affords reflective cognition is not an easy task. This dissertation examines a model of structuring mathematical activities for children to support their learning. It also examines a number of design features that help make the learning activity more enjoyable. This dissertation makes recommendations on how to design multimedia mathematics learning environments to address children's affective, cognitive, and pedagogical needs. Moreover, this research contributes to an increased understanding of how to design better game-based educational software. A few of the findings of this research are: 1. Interface design in educational software plays a crucial role in how learners interact with the educational content, and consequently how they acquire knowledge and what knowledge they acquire. The results showed significant achievement differences among students who used different interface styles. Interface techniques such as 'scaffolding' and gradual removal of visual feedback can promote reflective cognition and improve learning. 2. Direct manipulation graphical interfaces should be used with care in the context of interactive multimedia mathematics learning environments. The conventional interface design guideline calling for easier interaction and exertion of minimal cognitive load does not necessarily apply to educational environments. 3. By carefully taking into account children's cognitive and affective needs, the design can help children enjoy learning mathematics. 4. Inclusion of background music and visual aesthetics can make a learning activity more enjoyable. iii C o n t e n t s Abstract ii Contents iv List of Tables viii List of Figures x Acknowledgments xii Dedication xiv 1 Introduction 1 1.1 Children's Needs and the Design of Educational Software 3 1.2 Context and Initial Investigations 5 1.3 Selection of the Mathematical Topic 7 1.4 Need for a Theoretical Framework for Design 8 1.4.1 Flow and the Design of Educational Software 9 1.4.2 Human Cognition and the Design of Educational Software 11 1.5 Purpose of the Dissertation 13 1.6 Contributions of the Dissertation 14 1.7 Overview of the Dissertation 16 2 Background Research 19 2.1 Human-Computer Interaction 20 2.1.1 Design of HCI Artifacts 23 2.1.2 Usability of HCI Artifacts 24 2.1.3 Educational Software for Children 28 iv 2.1.4 Tools and Toys 31 2.1.5 Cognitive Artifacts: Partners in Cognition 32 2.1.6 User Interface Style: Implications for Learning 33 2.2 Learning: Related Issues 38 2.2.1 Nature of the Human Mind and Learning 38 2.2.2 Behavioral View of Learning 39 2.2.3 Cognitive View of Learning 40 2.2.4 Two Kinds of Cognition and Three Kinds of Learning ., 41 2.2.5 Constructivism 43 2.2.6 Construction versus Instruction 46 2.2.7 Experiential Learning 48 2.2.8 Situated Cognition and Social Aspects of Learning 50 2.2.9 Typology of Knowledge 53 2.2.10 Concept Learning 55 2.2.11 Symbols as Mental Interfaces for Concept Formation 59 2.2.12 Propositional versus Analogical Symbols 63 2.3 Motivation 65 2.3.1 Flow 67 2.3.2 Other Related Research 72 2.4 Electronic Games, Motivation, and Learning 73 3 Design Rationale 79 3.1 Challenge-Driven Learning Model: Promoting Flow in Learning 82 12 Activity Selection , 85 3.3 Game Issues to Consider 88 3.3.1 Operationalization of Activity 88 3.3.2 Sensory Stimuli: Engaging the Experiential Senses 92 3.3.3 Other Game Features 95 3.4 Epistemology at the Interface Level 96 3.4.1 Direct Concept Manipulation: Selected Manipulation Style '96 3.4.2 Selection of Symbolic Representations for D C M 102 3.4.3 Linking Experience to Concept Learning through D C M 103 3.4.4 Mouse Interaction Protocol: Further Support for D C M 109 3.4.5 Modifying D C M to Promote Reflective Cognition 113 3.4.6 Constraints: Further Promotion of Conscious Thought 123 3.5 Instructional Module 125 v 3.6 Super Tangrams: Integration of Design Issues 129 4 Research Method 137 4.1 Preliminary Evaluation of the Design 142 4.2 Summative Evaluation of the Design 145 4.2.1 Participants 145 4.2.2 Materials 146 4.2.3 Design of the Summative Evaluation 152 4.2.4 Sources of Data . 157 4.2.5 Research Setting 166 4.2.6 Procedures 168 5 Results and Discussion 173 5.1 Preliminary Evaluation 175 5.1.1 Children's Initial Reactions 175 5.1.2 Children's Descriptions of Their Learning 178 5.1.3 Summary and Discussion 182 5.2 Summative Evaluation 185 5.2.1 Achievement Results 185 5.2.1.1 Analysis of Overall Achievement Results 186 5.2.1.2 Fine-Grained Analysis of Achievement Results 192 5.2.1.3 Children's Perception of Their Learning 199 5.2.1.4 Summary and Discussion 203 5.2.2 Assessment of Reflective Cognition Affordance 205 5.2.3 Assessment of the Challenge-Driven Learning Model 214 5.2.4 Assessment of Sensory Stimuli 225 5.2.4.1 Background Music 226 5.2.4.2 Visual Aesthetics 228 5.2.5 Overall Affective Results 231 5.2.6 An Interview: Was Super Tangrams a Flow Activity? 238 6 Conclusions 247 6.1 Summary 247 6.2 Limitations of the Research 251 6.3 Implications, Recommendations, and Future Research 252 vi Bibliography 267 Appendix A: Transformation Geometry Test 293 Appendix B: Marking Scheme for the Transformation Geometry Test 315 Appendix C: Design Questionnaire 317 vii L i s t o f T a b l e s 4.1 Distribution of students among classes 145 4.2 Features of the six prototypes of Super Tangrams in the summative phase . . 152 4.3 List of participating groups by treatment 154 4.4 List of participating groups by class 155 5.1 Measures of central tendency and variability for the control group (N=20) for pre-posttest and pre-posttest change scores 187 5.2 Measures of central tendency and variability by group for pre-posttest and pre-posttest change scores 188 5.3 F table comparing instructional package by embellishments type treatment 190 groups 1 7 U 5.4 Adjusted and observed means for the groups 191 5.5 Post hoc comparisons among groups 191 5.6 F table comparing mediation versus no-mediation treatment groups 191 5.7 Students getting Questions 4 through 6 completely correct 193 5.8 Students getting both Questions 20 and 50 correct 193 5.9 Students getting Question 7 correct 195 5.10 Students getting Question 46 correct 196 5.11 Students getting Questions 7 through 12 completely correct 198 5.12 Students getting Questions 14 through 19 completely correct 198 5.13 Students getting Questions 26 through 31 completely correct 198 5.14 Children's perception of their learning after playing different prototypes of Super Tangrams 200 5.15 Children's responses regarding the level in which they learned most 201 5.16 Children's explanations regarding the level in which they thought they learned most ^02 5.17 Mean reflective indices for reflective cognition by group 207 viii 5.18 Mean reflective indices for each instructional package 208 5.19 Comments by the R students regarding need for reflective cognition 210 5.20 Responses to questions about Learn 210 5.21 Explanations given by the gF-E and gF+E students regarding why they thought they could have advanced through the game without the use of Learn 5.22 Average number of puzzles completed by each group 213 5.23 A child's perception of the Challenge-Driven Learning Model in Super Tangrams (Case 1) 217 5.24 A child's perception of the Challenge-Driven Learning Model in Super Tangrams (Case 2) 218 5.25 A child's perception of the Challenge-Driven Learning Model in Super Tangrams (Case 3) 219 5.26 A child's perception of the Challenge-Driven Learning Model in Super Tangrams (Case 4) 220 5.27 A child's perception of the Challenge-Driven Learning Model in Super Tangrams (Case 5) 221 5.28 A child's perception of the Challenge-Driven Learning Model in Super Tangrams (Case 6) 222 5.29 Explanations given by the gR-E, gR+E, and gR+E+M students regarding why they used Learn, and what they thought about Learn 223 5.30 Children's explanation for why they always had the music on 227 5.31 Children's explanation for why they did not always have the music on 227 5.32 Indices of students' overall affective response towards the different prototypes 232 5.33 Perception of how much mathematics was involved in each prototype by group 233 5.34 Written comments of children who used the R+E prototype of Super Tangrams 235 5.35 Written comments of children who used the R-E prototype of Super Tangrams 236 ix L i s t o f F i g u r e s 2.1 Structural dimensions underlying Kolb's model of experiential learning 49 2.2 Relationship between challenge and skill 70 3.1 Square from which tangrams pieces are created 86 3.2 Outline of a tangrams puzzle 86 3.3 Solution to a tangrams puzzle 86 3.4 Presentation of the tangrams activity on the screen 89 3.5 D O M versus D C M 101 3.6 Transformation representations: Left to right: translation, rotation, and reflection 1 0 3 3.7 Representation of translation in Super Tangrams 105 3.8 Representation of rotation in Super Tangrams 106 3.9 Representation of reflection in Super Tangrams 108 3.10 A puzzle with the square as the selected piece 1.13 3.11 The square as the selected piece along with the rotation representation 113 3.12 Translation: stage #1 116 3.13 Translation: stage #2 116 3.14 Translation: stage #3 116 3.15 Rotation: stage #1 119 3.16 Rotation: stage #2 119 3.17 Rotation: stage #3 119 3.18 Reflection: stage #1 121 3.19 Reflection: stage #2 121 3.20 Reflection: stage #3 121 3.21 Three stages of R D C M applied to the representation of rotation 123 x 3.22 Composite reflection, stage #1 125 3.23 Composite reflection, stage #2 125 3.24 Relationship between the game module and the instructional module 126 3.25 A screen of Super Tangrams 132 3.26 Opening screen of Learn 136 3.27 Cognitive Strategies Instruction screen 136 3.28 Menu for invoking the instructional screen for reflection 136 3.29 A Concept Explanation screen for rotation 136 3.30 A Concept Explanation screen for reflection 136 3.31 A Guided Interactive Practice screen for translation in Level 1 of the game . . 136 4.1 A snapshot of an R-E prototype screen 148 4.2 A snapshot of an I+E screen showing a horizontal flip 151 4.3 A nonequivalent pretest-posttest group design 153 4.4 Pretest-posttest group design for the summative evaluation phase of the study 1 5 6 4.5 An example of a grouped set of questions 161 4.6 Diagram of the computer room where the summative phase of the study was conducted 167 4.7 A group of students working with a prototype of Super Tangrams 167 4.8 A pair of students working together 169 5.1 Mean scores for pretests and posttests by group 187 5.2 Factorial design: 3 by 2 [(R vs. F vs. I) by (+E vs. -E)], and 1 by 2 [R+E by (mediation vs. no mediation)] 1^ 9 5.3 Percentage change in correct answers to rotation questions by group 195 5.4 Percentage change in correct answers to reflection questions by group 197 5.5 Children's liking of background music 226 5.6 Importance of colors and graphics for children 229 5.7 Children's liking of patterns and cartoons in puzzle pieces by group 230 6.1 Gradual conversion from a D C M to a command-based interface style 260 xi A c k n o w l e d g m e n t s Foremost, I would like to express my deepest sense of gratitude to my wife, my dearest, beloved Andishe. You have been my companion, best friend, and advisor - my everything. Words fail me to say how much I appreciate all your support. Very special gratitude goes to Dr. Maria Klawe, my supervisor. No one can hope and ask for a better supervisor. You were always there for me. You were friendly, kind, encouraging, fair, exact, insightful,.... Thank you Maria. I learned a tremendous amount from you. You will always be a part of my life. Special thanks also go to my supervisory committee - Drs. Ann Anderson, Susan Pirie, Miriam Reiner, Richard Rosenberg, and Marv Westrom - not only for guiding me to do my research, but also for being such good friends to me. Thank you Marv for always being there for me; I enjoyed our discussions together. Thank you Ann for being so kind to me from the very first time we met; thank you for teaching me what it means to really care for children. Thank you Miriam for having so much confidence in me; you taught me what it means to be a real scientist. Thank you Susan for broadening my understanding of education and research; I was very lucky to get to know you; I enjoyed your classes so much. Finally, thank you Richard; thank you for being supportive and broad-minded. ' A note of appreciation goes to Dr. Janet Kolodner of the Georgia Institute of Technology for being the External Examiner. I enjoyed your comments and the attention you had paid to details. I would also like to express my gratitude to Drs. Anna Kindler, xii David Lowe, and Janice Woodrow who served as University Examiners. Thank you for making my defense such an unforgettable experience. This research would have been impossible without the help of Ms. Eileen Phillips, Ms. Carolyn Dymond, Mr. Doug Super, Ms. Karen Bentley, Mr. Wolfgang Rothers, and all the other teachers whose students participated in my studies. Nick Harvey deserves much credit for helping me implement Super Tangrams. Last, but not least, a heartfelt appreciation goes to all the children who participated in my studies and who taught me so much about how to design child-centered interactive learning environments. This research was supported through funding from Natural Sciences and Engineering Research Council of Canada, Apple Canada, B.C. Advanced Systems Institute, Electronic Arts, and TeleLearning Network of the Centers of Excellence. I would like to thank the Vancouver School Board and all the schools who participated in my research. I would like to thank the faculty and staff of the Department of Computer Science for being so kind and helpful. Words fail me to thank Mr. Hooper Dunbar and Dr. Farzam Arbab enough; through the examples of their lives I was inspired to pursue an interdisciplinary Ph.D. They taught me how to serve the world of humanity. Finally, I owe everything to my parents, especially my father, who believed, ever since I was a child, that one day I would become a doctor. Well! I have a Ph.D. now! xiii dedicate this dissertation to my adorable daughter, Kimia, and a l l the children of the world. Regard man as a mine rich in gems of inestimable value. Education can, alone, cause it to reveal its treasures, and enable mankind to benefit therefrom. —Baha'u'llah xiv Chapter 1 I n t r o d u c t i o n I don't like math because it's boring. - Grade-6 boy Middle-school student, Vancouver, Canada Math is difficult. It's not fun. I like the arts. - Grade-6 girl Middle-school student, Vancouver, Canada I don't like math. We have to sit down and do it on a piece of paper. [It] doesn't have a meaning or a goal at the end. AH you do is exercises. It is really really really boring. I don't like doing it that way. - Grade-6 girl Middle-school student, Vancouver, Canada In the transition to adulthood, middle school students are forming lifelong values and skills. The decisions students make about what they will study and how they will learn can dramatically affect their future. Failure to study mathematics can close the doors to vocational-technical schools, college majors and careers - a loss of opportunity that happens most often to young women and minority students. Because many of the attitudes that affect these decisions are developed during the middle grades, it is crucial that conscious efforts be made to encourage all students, especially young women and minorities, to pursue mathematics. To this end, the curriculum must be interesting and relevant, must emphasize the usefulness of mathematics, and must foster a positive disposition toward mathematics. -National Council of Teachers of Mathematics [NCTM, 1989, p. 68] The above quotes from the three grade-6 students typify the attitude of a large number of middle-school children towards mathematics. They find it boring, irrelevant to their lives, 1 and difficult to understand. The attitudes expressed above are not ones that can be understood or analyzed through a "simple single-cause/single-effect linear progression" [Martinez & Martinez, 1996, p. 2]. These attitudes encompass feelings towards mathematics, both affective and cognitive, which weave together in a chain reaction [Martinez & Martinez, 1996]. These feelings originate from and are influenced by one's memories of past failures and successes, by one's peers, teachers, and parents, by the methods of teaching to which one has been subjected, by the type of mathematics to which one has been exposed, and by the learning environments in which one has encountered mathematics [Martinez & Martinez, 1996; Skemp, 1986]. The National Council of Teachers of Mathematics, in its Curriculum and Evaluation Standards for School Mathematics [NCTM, 1989], recommends the use of interactive computer software in children's mathematics education. However, designing software for children is complex and challenging because it is "difficult to appreciate the unique needs of children" [Soloway, 1996, p.l]. This is particularly true when the design is intended to produce motivating as well as instructionally effective interactive multimedia learning environments. Convergence of research findings and principles from several different fields is needed for effective design. Current knowledge in disciplines such as human-computer interaction (HCI), child psychology, learning, instruction, and mathematics education should be systematically integrated to provide designers of educational software with sound design principles. This requires an interdisciplinary research framework. 2 1.1 Children's Needs and the Design of Educational Software HCI seeks to produce user interfaces that facilitate and enrich human motivation, action, and experience, but to do so deliberately it must also incorporate means of understanding user interfaces in terms of human motivation, action, and experience. [Carroll, 1991b, p. 1] Strengthen the students' motivation to learn by building relationships between the content and objectives of the instruction and the learner's needs and desires. [Keller & Burkman, 1993, p. 10] 'Motivated' is a description we apply to behaviour which is directed towards satisfaction of some need. If we say that a certain piece of behaviour seems motiveless to us, we mean that we do not know, and cannot even guess, what need is satisfied by means of it. So questions about motives are usually, in disguise, questions about needs. [Skemp, 1986, pp. 123-4] One of the most important aspects of motivating children to learn mathematics is to understand their needs. To deal with the issue of motivation, one must ask: Why should children want to learn mathematics? Skemp [1986] states that there are two kinds of needs: innate and learned. Whereas the need to sleep is innate, Skemp [p. 124] suggests that "[m]athematics seems fairly obviously to be a learnt need . . . for satisfying other needs" such as understanding other subjects, creating new technology, or getting a job. Skemp [p. 124] points out that these goals "motivate many adults to mathematics, but [that] they are too remote to be applicable to the early years of school". This situation then must be rectified by placing children in environments in which the need for learning mathematics finds a tangible meaning. 3 Since the early 1980s electronic games have created a revolution in the lives of children. Electronic games play a major role in children's social life, consume a good portion of their time, and are part of their popular culture [Rheingold, 1983; Provenzo, 1991]. In sum, children are fascinated by electronic games [Malone, 1981a, 1981b, 1983a, 1983b; Norman, 1993]. For this reason, some researchers suggest the use of electronic games in motivating children to engage in learning activities [Klawe, 1994a, 1994b; Perkins, 1983; Papert, 1992; Norman, 1993; Noddings, 1985; Lepper & Malone, 1987; Malone & Lepper, 1987]. Therefore, with electronic games as contexts in which children encounter mathematics, the need for learning mathematics may find a tangible meaning as children want to be successful at playing the games. Silvern [1986] states that most electronic games that were designed for educational purposes in the early 1980s were of a drill-and-practice type. Silvern reports that these games had very simple and straightforward rules, such as guessing numbers and letters and plugging numbers into formulas. A similar assessment applies to most instructional mathematics games released over the last 10 years [Roblyer et al., 1997]. Roblyer et al. [1997] point out that most of these games have been designed to teach basic skills and lack a true problem-solving focus. These games generally satisfy children's affective and rote-learning needs; they do not establish environments in which children can tackle relatively complex mathematical concepts. Yet, this is what is needed if children are to become confident, "fearless" mathematicians and develop better attitudes towards mathematics [Martinez & Martinez, 1996; N C T M , 1989]. Indeed, since 1989, the National Council of Teachers of Mathematics has urged educators to "take advantage of 4 the expanding mathematical capabilities of middle school students to include more complex problem situations,... problems that demand extended effort to solve" [NCTM, 1989, p. 75]. 1.2 Context and Initial Investigations Consider what needs to be done and what you can do, and collaborate with others to implement the [NCTM] standards for the benefit of our students, as well as our social and economic future. [NCTM, 1989, p. 12] In late 1992, a research group called E-GEMS (Electronic Games for Education in Math and Science) was created. E-GEMS is an interdisciplinary research group whose mission is to investigate how to design and use interactive game-based learning environments to improve children's motivation and achievement in learning mathematics and science [Klawe & Phillips, 1995]. E-GEMS started its research studies in the summer of 1993 when members of the group observed thousands of children playing electronic games at a specially constructed exhibition in a science museum [Inkpen et al., 1994; Lawry etal., 1995]. In March of 1994, E-GEMS placed 4 L C III Macintosh computers in a grade 6/7 French immersion classroom at Elementary School A 1 , located in an upper-middle-class neighborhood of Vancouver, British Columbia, Canada. This class had 27 students who were approximately 11 to 13 years old. Many of these students were not particularly 1 All school names are to be kept anonymous. 5 interested in mathematics and found it boring, as evidenced by quotes cited at the beginning of this chapter. Initially, a few E-GEMS' prototype Hypercard™-based mathematics games as well as commercial games were installed in the class. I visited the classroom weekly for one to two hours for a period of slightly over one year. During these visits, I observed students playing the games and conducted class discussions with students about the educational content of these games: children's likes and dislikes, and what they thought they were learning. In addition, students were asked to keep a journal of their computer activities. I found that most students were often very difficult to satisfy; rarely did the majority of them consider a game "cool". Additionally, many games seemed to polarize students along gender lines - boys gathering around one game, while girls played with another game. In February and March of 1995,1 interviewed 25 of the children individually to learn what they expected from educational games, and what they considered to be attractive and engaging ways of learning mathematics. Some issues that were raised by the children included: 1. Many students seemed to find it difficult to study mathematics out of textbooks. One student, for instance, commented that it was difficult to interact with a textbook. 2. Many students strongly commented on how boring it was to study mathematics out of a textbook; in particular, several girls commented on the lack of nice colors and graphics in mathematics textbooks. 3. A majority of the students complained that doing exercises from textbooks did not provide them with a worthwhile goal to pursue, whereas computer games did. 6 4. Commenting on why they did not like the HyperCard™ games that we had installed for them, a number of the students mentioned the lack of attractive colors and graphics as some of the reasons. 5. A number of students commented on the importance of musical tunes in games, and how such tunes could act as memory cues for remembering the content of the games at a later time. Over one year of interacting with these children, observing them, interviewing them, and consulting with their teacher provided the context as well as the motivation for my future research work. 1.3 Selection of the Mathematical Topic Geometry has all but disappeared now that we have only shape and space, but I am sure that will have to come back, because it is so fundamental. [Mason, 1992, p. 24] The choice of the mathematical topic was an important decision in beginning my research on the design of interactive multimedia mathematics learning environments for children. Some mathematics researchers [Mason, 1992; Russell & Bologna, 1982] have noted that geometry is a neglected topic in the elementary school mathematics curriculum. This neglect is particularly true of transformation geometry. This observation was confirmed by my discussions with several teachers and educators. Many teachers either leave transformation geometry out of their curriculum or provide only a very cursory treatment of it towards the end of the school year. Nevertheless, the National Council of Teachers 7 of Mathematics [NCTM, 1989] emphasizes the importance of this branch of mathematics, and according to Crowley, a mathematics educator, [transformations] serve as powerful problem-solving tools. They link traditionally compartmentalized areas of mathematics. And they have applications in areas outside of mathematics. All these characteristics promote the learning of mathematics! [Crowley, 1995, p. 3] For these reasons, I chose two-dimensional transformational geometry as the mathematical context in which to conduct my research. My objective was to design a game-based interactive multimedia learning environment to assist children in moving beyond an informal and intuitive understanding of two-dimensional transformations, and to stimulate them to think about the more formal and sophisticated mathematics involved in transformation (motion) geometry. (The more specific objectives of the design are discussed in Chapter 3.) 1.4 Need for a Theoretical Framework for Design A number of designers of computer interface and instructional software [Carroll & Kellog, 1989; Duffy & Jonassen, 1992] have noted that "HCI artifacts" embody implicit, underlying theories of learning and inherent psychological claims without explicitly articulating these theories and claims. Sarama et al. [1996, p. 567], while discussing children's development of geometric ideas, state that software design can and should have an explicit theoretical and empirical foundation, beyond its genesis in someone's intuitive grasp of children's learning, and it should interact with the ongoing development of theory and research - reaching toward the ideal of testing a psychological theory of children's mathematical development by testing the software that reflects the objects and processes of this theory. 8 Carroll and Kellog [1989, p. 13] point out that relations between science and design are complex, and there is need for a "reciprocal relation between the articulation and rearticulation of a set of psychological claims and the iterations of design". In accordance with the above observations, prior to designing the transformation geometry software, a theoretical framework was sought, not only to guide the research, but also to ground the interpretation of the results of the research. To provide a background for the issues that this dissertation investigates, two integral elements of the theoretical framework of this research are introduced in this section. These and other elements of the framework are discussed in depth in Chapter 2. 1.4.1 Flow and the Design of Educational Software Whenever someone designs software that interacts with people, the effects of the design extend beyond the software itself to include the experiences that people will have in encountering and using that software. . . . The experience of a person who is interacting with a computer system is not limited to the cognitive aspects that have been explored in the mainstream human-computer interaction. As humans, we experience the world in aesthetic, affective, and emotional terms as well... . Designing for the full range of human experience may well be the theme for the next generation of discourse about software design. [Winograd, 1996, pp. xviii-xix] The most important social evolution within the computing professions would be to create a role for the software designer as the champion of the user experience. [Kapor, 1996, p. 1] Whenever the quality of human experience is at issue, flow becomes relevant.... jT]he most urgent applications of the flow model [are] in schools and on the job, where most people spend most of their lives - often in boredom or in states of uneasy anxiety. [Csikszentmihalyi, 1988a, pp. 12-14] 9 Since the 1960s, Csikszentmihalyi and his colleagues [Csikszentmihalyi, 1975; Csikszentmihalyi & Csikszentmihalyi, 1988; Csikszentmihalyi, 1990] have been studying the nature of happiness, joy, anxiety, and boredom. In their theoretical model of enjoyment, they describe the important construct of flow as an intrinsically motivating and rewarding state in which an individual derives peak or optimal experience from an activity. People from all walks of life have described how they feel when they are involved in an activity that is worth doing for its own sake. These activities produce order in people's consciousness, have a greater chance of being remembered, and people usually like to repeat them. In the flow state, people's minds are captured; they have intense and focused concentration; their experiential senses are engaged; they are immersed in the activity; and they are exhilarated. Many of these characteristics are important when learning non-trivial mathematical concepts (see [Skemp, 1986; Norman, 1993]), and yet, many children never experience mathematics in this way. Flow activities have certain structural characteristics in common, and these characteristics can be built into an environment by design [Csikszentmihalyi & Csikszentmihalyi, 1988; Norman, 1993]. Although the possibilities of the flow model are beginning to be applied to different domains [Csikszentmihalyi & Csikszentmihalyi, 1988], there is still much to be learned, particularly in the design of interactive educational environments. On this point, Norman's [1993] question appears especially important: "Is there some way of achieving this state of flow while learning?" [p. 32]. Consequently, 10 determining how to operationalize the flow construct in interactive multimedia learning environments calls for close examination. 1.4.2 Human Cognition and the Design of Educational Software [TJhe trick in teaching is to entice and motivate the students into excitement and interest in the topic, and then give them the proper tools to reflect; to explore, compare, and integrate; to form the proper conceptual structures. . . . The problem is to make the students want to do the hard work that is necessary for reflection. . . . Going back to the traditional classroom is not the answer. The fifty-minute class is not the answer. Ordinary people - you and I - cannot concentrate on a single topic for that long. Lectures are not the answer, no matter how good the lecturer. [Norman, 1993, pp. 30-39] As a general model, many researchers have identified two modes of cognition that appear central to "grasping" any incoming information [Bruner, 1986; Freire, 1973; Kolb, 1984; Norman, 1993; Piaget, 1970; Skemp, 1986]. These two modes have been given different names by different researchers, but they seem to refer to the same phenomena. For instance, Kolb [1984] refers to these modes as apprehension and comprehension; and Norman [1993] calls these two modes experiential and reflective cognition, respectively. The experiential mode is One in which people perceive and react to the events around them in a "seamless manner". The reflective mode is one in which people compare, make decisions, and induce order on the flow of incoming sensations in order to comprehend. These two modes co-exist on a continuum, and researchers emphasize the importance of a dialectic interaction or balanced tension that must exist between these two modes. Reflective cognition is essential in learning complex and abstract conceptual structures such as mathematics [Skemp, 1986; also see Dewey, 1933; Kolb, 1984], but it is hard 11 work [Norman, 1993]. Hayes and Broadbent [1988] suggest that effortful thought leads to explicit, insightful learning; whereas random, trial-and-error thought can cause passive aggregation of information and consequently intuitive, implicit knowledge. Mathematics is an explicit, formal knowledge system, and students need to develop competence in the use of its formal procedures [Noddings, 1985]. One of the significant learning difficulties in students is how their fragmented collection of implicit, intuitive knowledge of a topic can potentially interfere with textbook, formal understanding of the topic [diSessa, 1985, 1988]. The fragmented system of students' intuitive and naive knowledge, and their assumptions about knowledge and knowing may affect the concepts to which they pay attention [diSessa, 1985]. Some researchers [diSessa, 1988; Forman, 1988] have suggested that computers can provide a bridge from intuitive to formal knowledge. Forman suggests that reflection lies at the heart of this transition, and that representation of knowledge at the computer interface and interaction feedback influence the learner's ability to reflect and self-correct. Design plays a key factor in the mode of cognition in which the user engages [Norman, 1993; Salomon, 1979, 1993a; Winn, 1996; Resnick & Collins, 1996; Clark, 1996]. Norman [1993, p. 52] states that experiential designs "mediate between the mind and the world" and promote experiencing and acting upon the world in a seamless manner; whereas reflective designs "provide ways to modify and act upon representations". Norman expresses the concern that, at present, our multimedia educational learning environments and entertainment industry are mostly conducive to the experiential mode of cognition. He adds [pp. 31-41] that: 12 Motivated activity, whether experiential or reflective, can be challenging and rewarding. The mind is captured, the experience is exhilarating. . . . Successful learning . . . [however, best takes place] through a combination of experiential and reflective modes of cognition, optimal cognition, sustained andfocused upon the topics at hand, free of distracting interruptions. Multimedia for education must minimize the fluff and get the users working - working hard, not because they have to but because they want to. 1.5 Purpose of the Dissertation The problem posed to psychology and education is to design a series of experiences for students that will enable them to learn effectively and to motivate them to engage in the corresponding activities.... The more difficult problem, and the one that often leads to different prescriptions, is determining the desirable learning goals and the experiences that, if incorporated in the instructional design, will best enable students to achieve these goals. Of course, arriving at good designs is not a matter for philosophical debates; it requires empirical evidence about how people, and children in particular, actually learn, and what they learn from different educational experiences. [Anderson et al., 1995, p. unknown] Plato complained about the way Greek children were educated by praising the practical orientation of the way the Egyptians taught mathematics through games and activities. Recognition of the importance of doing things, of manipulating in order to develop a sense of some idea, and then gradually bringing sense-of to articulation (and hence abstracting it from the particular context), has been praised over and over again in mathematics education, through Dewey, Piaget, Bruner, Vygotsky & Davidov, Froebels, and Montessori, Gattegno and Skemp. . . . But doing is not in itself sufficient, for sense-making does not follow automatically from manipulation. [Mason, 1995, p. 10] As discussed previously, games are attractive to children and improve motivation for engagement in learning activities. "However, it is important to remember that instruction clothed in game format does not necessarily make the instruction effective" [Alessi & Trollip, 1991, p. 202]. The more difficult question, however, is: How should game-based interactive multimedia mathematics learning environments be designed so that they simultaneously are motivating, immerse children in complex problem-solving activities, 13 and are instructionally effective? In order to promote experiences of flow in learning for children, the design of interactive multimedia learning environments must take into account the characteristics of the flow model as well as models of human cognition. It is important to determine how to design environments in which children focus on mathematical concepts and are intensely engaged in them. The epistemological role of the user interface in providing a direct sense of engagement with mathematical concepts should be closely examined. The purpose of this dissertation is to explore four main inter-related issues: 1. What role does the user interface play in multimedia mathematics learning environments? How do different interface styles influence learning? 2. How should the user interface be designed to support children's learning of explicit, formal mathematical concepts? 3. How should a learning environment in general, and the interface in particular, be designed to afford reflective cognition? 4. How should a multimedia learning environment be structured to be conducive to experiencing flow in learning? What are some design elements that can make children's learning of mathematics fun and enjoyable? 1.6 Contributions of the Dissertation This dissertation contributes to research in two main areas, as well as exploring and raising a number of other issues. 14 Design of Effective Interfaces for Mathematics Education In many human-computer interaction and instructional technology books, research handbooks, and encyclopedias, there is either only a cursory mention of, or no guidelines for, what constitutes effective human-computer interfaces for educational purposes (e.g., see [Jonassen, 1996; Plomp & Ely, 1996; Carroll, 1991a; Bodker, 1991; Norman & Draper, 1986; Muldner & Reeves, 1997; Ashlund et al., 1993; Tauber, 1996; Edelson & Domeshek, 1996]). In most cases, due to a lack of proper interface design guidelines, designers of educational software for children have been consumers of the interaction styles that were originally designed for productivity tools. Yet, Frye and Soloway [1987] note that interface design for educational software was particularly important because it provided an entry to the content domain of learning and had to be sensitive to the general skill and developmental stage of the users. The most important revolution in interface design so far has been the conception and development of direct manipulation interfaces [Shneiderman, 1988, 1993; Norman & Draper, 1986] - i.e., interfaces that allow the user to manipulate objects on a screen with some kind of pointing device. However, recently a number of researchers have been questioning whether the direct manipulation style is effective for mathematics education or other educational purposes [Golightly, 1996; Hoist, 1996; Sedighian & Klawe, 1996a; Sedighian & Westrom, 1997; Svendsen, 1991]. This dissertation closely examines the issue of interface design for multimedia mathematics learning environments for children and makes recommendations for a new conception of interface manipulation styles resulting in more effective educational user interfaces. 15 Design of Enjoyable and Engaging Mathematics Software The trick is to marry the entertainment world's skill ofpresentation and of capturing the user's engagement with the educator's skills of reflective, in-depth analysis. [Norman, 1988, p. 39] It is remarkable how little scientific knowledge we have about the factors that underlie motivation, enjoyment and satisfaction. . . . That's the tradeoff between hard science, which requires things to be measured with precision, and soft science, which attempts to study those things for which measurement is difficult or impossible. As a result, we know little about how best to structure tasks and events so as to establish, maintain, and enhance the experience. [Norman, 1993, p. 32] To structure a mathematics activity so that it combines the two elements of fun and formalism and affords reflective cognition is not an easy task. This dissertation examines a model of structuring mathematical activities for children to support their learning. It also examines a number of design features that help make the learning activity more enjoyable. This dissertation makes recommendations on how to design multimedia mathematics learning environments to address children's affective, cognitive, and pedagogical needs. Moreover, this research contributes to an increased understanding of how to design better game-based educational software. 1.7 Overview of the Dissertation This dissertation includes six chapters. Chapter 2 discusses background research in human-computer interaction, learning, motivation, and electronic games. It provides a foundation and theoretical framework for the design rationale in the next chapter. Chapter 3 describes and discusses the design of an interactive multimedia mathematics learning 16 environment. The design addresses the research issues raised in this dissertation, and a careful examination of its rationale is essential for understanding the following chapters. Chapter 4 details the research method used to evaluate the pedagogical effectiveness of the design described in Chapter 3. Chapter 5 provides a detailed analysis and discussion of results of the research study. Finally, Chapter 6 includes conclusions drawn from this research, recommendations for design of more effective multimedia mathematics learning environments for children, and suggestions for future research. 17 18 Chapter 2 B a c k g r o u n d R e s e a r c h All engineering and design activities call for the management of tradeoffs. Real-world problems rarely have a correct solution of the kind that would be suitable for a mathematics problem or for a textbook exercise. The designer looks for creative solutions in a space of alternatives that is shaped by competing values and resource needs. In classical engineering disciplines, the tradeoffs can often be quantified. . . . In design disciplines, the tradeoffs are more difficult to identify and measure. The designer stands with one foot in the technology and one foot in the domain of human concerns, and these two worlds are not easily commensurable. .. . [SJofiware design draws on so many other disciplines: software engineering, software architecture, programming, human factors, graphic information design, art and aesthetics, sound production, psychology, and more. Software design, by the nature of what it aims to accomplish, sweeps into its scope all these interests, concerns, and disciplines. . . . [Software designers] need to be able to envision, to create, and to develop a representation of their vision that they can communicate. [Winograd, 1996, pp. xx, 297] Human-computer interaction is a discipline concerned with the design, evaluation, and implementation of interactive computing systems for human use and with the study of major phenomena surrounding them. [ACM SIGCHI, 1992, p. 5] Designing computational toys and environments for children is particularly challenging; children are not just short users. . . . [CJhildren are not typically designers of software, and professional designers are at some distance from their childhood experiences, making it especially difficult to appreciate the unique needs of children. . . . [CJhildren have clear cognitive development stages, and thus software needs to be age appropriate. [Soloway, 1996, p. 1] 19 Exploration of and research in determining how to design usable, useful, age appropriate, enticing, aesthetically pleasing, and educationally sound computational artifacts for children is complex. In order to conduct such research, it is necessary to integrate research findings from several domains, synthesize new ideas, and, highlight new research that needs to be done. Design models can provide a foundation for the development of design theories [Reigeluth, 1983]. Whereas instructional design theories are prescriptive (compromise-and decision-oriented), natural science theories (including theories of learning and cognition) are descriptive [Reigeluth, 1983; Simon, 1969]. Instructional and learning theories, however, are closely related and inform each other [Plomp & Ely, 1996]. This chapter has four main sections: human-computer interaction (HCI), learning, motivation, and electronic games. These review the contributing domains and provide a partial foundation and background leading to the design of an interactive multimedia mathematics learning environment for children described in Chapter 3. 2.1 Human-Computer Interaction Long [1989, p. 4] defines HCI as follows: "Human-computer interaction seeks to support interactions between humans and computers which make computers effective for performing work". A one-sentence definition, however, does not provide a full picture of the goals and scope of HCI. HCI is a complex and multi-disciplinary field. Booth [1988] states that many disciplines contribute to HCI's research and growth, and he names 20 disciplines such as ergonomics, software engineering, mathematics, cognitive psychology, artificial intelligence, computational linguistics, cognitive science, social psychology, organizational psychology, and sociology. Booth categorizes HCI research into five areas: 1) interactional hardware and software, 2) matching models, 3) task level analysis, 4) design and development, and 5) organizational impact. Booth [p. 17] explains the role of each of these areas as follows: HCI is the study and theory of the interaction between humans and complex technology [usually computers]. HCI is concerned with how current input and output technologies affect interaction and in what situations these technologies and techniques might be put to best use. It is also concerned with modeling the knowledge a user possesses and brings to bear on a task. HCI's interests at a task level are in the means by which the user's information needs and a system's information provision might be matched. In terms of design, HCI is the study of the design process, and its overall aim is to engineer a shift from system-centered to user-centered design. At an organizational level, the objective of HCI research is to suggest both design and implementation techniques that might prevent problems such as job deskilling and conflict between groups. While some researchers consider HCI as a field that provides "methods and metrics for evaluating the usability of computers", the computing industry perceives that "HCI must help guide the definition, invention, and introduction of new computing tools and environments" [Carroll, 1989, p. 48]. Norman [1987] suggests that HCI is not simply a discipline to study the interaction between humans and computers, but also a field for the application of cognitive science (i.e., cognitive engineering), as well as a research area for cognitive scientists to develop and test their models of human cognition. The cognitive perspective of HCI promotes a view in which the majority of human-computer interaction issues are of a cognitive nature [Carroll, 1987] (for instance, see [Card et al., 1983]). The cognitive approach has received 21 a great deal of attention within the HCI community, and researchers have developed various theoretical models, ranging from keystroke-level activities to user-level mental conceptualizations of tasks [Ackermann & Tauber, 1990; Card et al., 1983; Moran, 1981; Norman & Draper, 1986; Tauber & Ackermann, 1991; Payne & Green, 1986]. Despite the attention paid to the cognitive modeling approach, a growing number of researchers argue that the difficulty with this kind of modeling is the "grain of analysis" which is employed, and the extent to which a task or system can be broken down [Booth, 1988]. Booth [1988] states that a real-world, complex task may need thousands of rules to model the user's behavior, however, this does not seem to be ordinarily practical. Others criticize both the object and methods of mainstream HCI and argue against the exhaustive use of cognitive psychology and its research methods as the foundation of HCI [Kuutti & Bannon, 1993; Kuutti, 1995; Bannon & Bodker, 1991; Suchman, 1987; Bodker, 1989, 1991; Thomas & Kellog, 1989; Winograd & Flores, 1987]. They state that the cognitive approach does not address the contextual and social side of human-computer interaction. For instance, Winograd and Flores [1987] note that HCI designers should not treat a technological device in isolation, and simply have a junctional understanding of how it is used. They suggest that HCI designers need to understand the context within which the artifact fits, and the environment and all the other activities that surround it and go with it. Ultimately, the main goal of HCI should be to determine the user's "experience" in a holistic manner [Winograd, 1996]. 22 2.1.1 Design of HCI Artifacts Design is one of the most pervasive human activities. Many people have written about issues that deal with design (for instance, see [Carroll, 1991; Goel & Pirolli, 1992; Norman, 1988; Norman & Draper, 1986; Reigeluth, 1983, 1996; Simon, 1969]). "Software design sits at the crossroads of all the computer disciplines" [Association for Software Design: cited in Winograd, 1996, p. xv]. However, it is difficult to define what software design really entails [Winograd, 1996]. Design of software is not a thing but an activity [Winograd, 1996]; it is standing "with one foot in two worlds - the world of technology and the world of people and human purposes" and attempts to "bring the two together" [Kapor, 1996, p. 1]; it is conscious, yet not too rigid and formal, but "pervaded by intuition, tacit knowledge, and gut reaction" [Winograd, 1996, p. xx]; it is a cycle of "reflection in action" [Schon & Bennet, 1996]; it is more an art than a science whose "ultimate subject matter - human experience and subjective response - is inherently as changeable and unfathomable as the ocean" [Smith & Tabor, 1996, p. 37]; and, ultimately, it is striving to evoke "love" in the user for the object of design and determine the user's "whole experience" [Kapor, 1996; Liddle, 1996; Schon & Bennet, 1996; Winograd, 1996]. There are two main approaches to design: problem-driven and solution-driven [Visscher-Voerman & Plomp, 1996]. Visscher-Voerman and Plomp note that others refer to these approaches differently. Schon [1983] refers to them as objectivist and constructivist respectively, and Rowland et al. [1994] refer to them as rational and creative respectively. The problem-driven designer analyzes a given problem and 23 decomposes it into sub-problems until the underlying structure of the problem becomes visible and manageable. There is the assumption of a strong relationship between goals and means, and, therefore, design can take place through a sequence of linear activities. The solution-driven designer cannot rely solely on the general nature of scientific knowledge but must also use tacit knowledge and know-how [Schon, 1983; Winograd, 1996]. Design often takes place in ill-structured problem spaces [Rowland, 1990]. It goes through a process of "reflection in action" and "conversation with the situation" [Schon, 1983]. This is an iterative, cyclical process in which non-determined situations are constantly being shaped into determined situations through an evolving exploratory, action-reflection based framework. This is an approach in which issues and situations are continually reconceptualized by looking at them from different perspectives until the final design emerges. In instructional environments, design is especially complex and includes elements such as instructional systems design, message design, instructional strategies, and learner characteristics [Ely, 1996]. Instructional designers have devised a number of design models. These models essentially serve as process guides for designers of educational and training systems. Gustafson [1996, p. 27] uses the term instructional design to describe the complete process of design which includes: 1) analyzing what is to be taught/learned, 2) detennining how it is taught/learned, 3) conducting tryout and revision, and 4) assessing whether learners do learn. 24 2.1.2 Usability of HCI Artifacts Usability has always been at the center of the HCI debate. It is argued that the main focus of HCI is the design of more usable systems [Carroll, 1989]. Ironically, the term usability has been used with such diversity of meaning and purpose that it has become vague and ambiguous. Some regard usability as a subject of study within the field of HCI, whereas others approach all HCI issues from the point of view of usability. Some provide general characteristics of usability, and others provide more specific operational definitions. Traditionally, usability has been concerned with two issues: finding out and analyzing user requirements and evaluating human-computer systems [Booth, 1988]. Carroll [1989] suggests that many approaches to usability have only considered a small and non-generalizable part of HCI - in particular, quantitative measurement of a few system variables such as ease of use and ease of learning within a laboratory setting thereby providing few directives for the design of useful artifacts. In order to provide a more holistic view of usability, Eason [1984] proposes a framework that summarizes the variables affecting the overall usability of a system. In this framework, there are three independent variables: system function, user characteristics, and task characteristics. These combine and interact to determine the usability of the system. Each of these independent variables are dependent on other variables. These include: 1. system functions: i. task match [match between user's needs and the task], ii. ease of use, iii. ease of learning; 25 2. user characteristics: i. knowledge [user's choice of knowledge to be applied to the task], ii. discretion [user's option to use part of the system], iii. motivation [user's motivation to use the system]; 3. task characteristics: i. frequency [the number of times a task is performed], ii. openness [the degree of the modifiability of a task]. Eason states that many of these variables are excluded from "normal experimental paradigms." His overall definition of usability and how it should be evaluated is whether or not a system or program is used, not what laboratory measurements show. Ravden and Johnson [1989] suggest nine criteria for evaluating the effectiveness of user interfaces: 1. Visual clarity: Information displayed on the screen should be clear, well-organized, unambiguous, and easy to read. [p. 29] 2. Consistency: The way the system looks and works should be consistent at all times, [p. 30] 3. Compatibility: The way the system looks and works should be compatible with the user's conventions and expectations, [p. 32] 4. Informative feedback: Users should be given clear, informative feedback on where they are in the system, what actions they have taken, whether these actions have been successful, and what actions they should take next. [p. 33] 5. Explicitness: The way the system works and is structured should be clear to the user. [p. 35] 6. Appropriate functionality: The system should meet the needs and requirements of users carrying out their tasks, [p. 36] 7. Flexibility and control: The interface should be sufficiently flexible in structure, in the way information is presented, and in terms of what the user can do, to 26 suit the needs and requirements of all users, and allow them to feel in control of the system, [p. 37] 8. Error prevention and correction: The system should be designed to minimize the possibility of user error, with built-in facilities for detecting and handling those which do occur, [p. 39] 9. User guidance and support: Informative, easy-to-use and relevant guidance and support should be provided, both on the computer and in hard-copy document form, to help the user understand and use the system, [p. 40] The above criteria certainly provide useful guidelines for how to view usability and evaluate it. However, they seem to fail to address some important issues and cannot be applied across all application domains. The major problem with the proposed criteria is that they do not necessarily apply to the usability of educational artifacts. Usability of educational software can not simply be measured in terms of the operability of the software. It is important to have an integrated view of usability and educational issues. This is especially true now since the advent of sophisticated multimedia interfaces with generic interface packages may lure designers of educational multimedia into believing that it is easy to build effective educational software quickly and efficiently. Another problem is the implicit separation between functionality of the system and its interface. It seems that by "appropriate functionality", Ravden and Johnson refer to the functionality of the interface. Regarding the separation of the interface and the functionality of HCI systems, Bannon and Bodker [1991] express the concern that usability should be in the context of the work activities of the users, and that "good design comes from an empathy with the work process itself," and with a view for 27 "possibilities for individual and societal growth" [p. 249]. Carroll [1989] proposes a different view of usability: one in which "usability is seen as connecting the invention of HCI artifacts to user needs no less essentially than nerves connect organs and muscle tissue to sensory and motor brain centers;" one in which "HCI artifacts are not merely evaluated or described in terms of their usability," but rather, "they are conceived and created for usability" [p. 61]. In this view, functionality and usability are not artificially separated. Usability is viewed from an "ecological" standpoint and addresses "whole problems, whole situations, when they are still technologically current, when their resolution can still constructively affect the direction of technological evolution. Its principle goal is the discovery of design requirements, not the verification of hypothesized direct empirical contrasts or cognitive descriptions" [Carroll, 1989, p. 65]. Bannon [1986] stresses the importance of taking into account the whole situation so that HCI artifacts are not only usable but also useful. These views reinforce the concern that HCI artifacts for children, especially educational ones, should address their psycho-social needs in a holistic manner, both in terms of design and evaluation. 2.1.3 Educational Software for Children Although human-computer interaction (HCI) is a broad interdisciplinary field of study and encompasses many aspects of human-computer interaction, it has primarily been driven by the needs of adults working in commercial and industrial contexts. As a result, children and their needs have received little or no attention (for instance, see [Norman & Draper, 1986; Carroll, 1987, 1989, 1991; Bodker, 1991; Card et al., 1983; Long & 28 Whitefield, 1989; Tauber, 1996]). As more children start using computers, either as part of their classroom activities or at home, an examination of their needs becomes important. The discussions and arguments within HCI have traditionally been concerned with computing tools that are used to support and improve the performance of people in a work environment. Bodker [1991, p. 3], for instance, states that the main focus of her book is on "computer support for purposeful human work," and not on "games" and other casual uses of the computer. Although adults may treat games casually and not as purposeful work activity, children take electronic games very seriously and spend many hours of their time playing them [Provenzo, 1991; Inkpen et al., 1994; Lawry, et al., 1995]. Purposeful, in the context of using tools, is a relative concept very much related to motivation and need (for a discussion on the nature of need and motivation, see [Skemp, 1986]). Designs for children must satisfy requirements which are not necessarily the same as those for adults, and are often more complex [Berkovitz, 1994; Inkpen, 1997; Milligan & Murdock, 1996; Oosterholt et al., 1995; Piernot et al., 1995; Resnick et al., 1996; Schneider, 1996; Smith etal., 1996; Strommen, 1994; Soloway, 1996]. As certain studies on the impact of computing artifacts on children suggest [Turkle, 1984; Turkle & Papert, 1991], many children anthropomorphize computational objects and, while not equating them with human beings, because they do not have "emotions" and are machines, children consider computers as being "alive," able to "think," and having a personality. These findings provide an interesting new way of looking at child-computer interaction. Whereas a hammer is simply a physical artifact used to perform mechanical tasks better and in a 29 shorter time, computing artifacts are linguistic, interactive, communicative, and intelligible as well. These interpersonal dimensions imply that designers should pay close attention to the needs and goals of children using these artifacts, as well as their natural and preferred ways of communicating. The important point here is that for children's software to be effective, designers should take a child-centered view; software should be designed in a way that addresses children's cognitive, affective, and behavioral needs. This view is in conformity with lessons learned from study of computer tools - that designers must have a user- or human-centered view of technology, rather than a machine-centered view [Norman & Draper, 1986; Norman, 1993]. Much of the existing literature on children and computers focuses on conceptual aspects of software - how to help children develop thinking skills, learn a subject matter, or become better problem solvers [Klein, 1985; Underwood & Underwood, 1990; Forman & Pufall, 1988a; De Corte et al., 1992; Pea & Sheingold, 1987; Duffy & Jonassen, 1992; Duffy etal., 1993; Papert, 1980; Papert, 1980, 1992; Harel & Papert, 1991]. In general, mainstream HCI researchers have not conducted much research on how to design effective educational interaction (for instance, see [Norman & Draper, 1986; Carroll, 1987, 1989, 1991; Bodker, 1991; Cardet al., 1983; Long & Whitefield, 1989; Tauber, 1996]). Only a small group, most of whose research is very recent, have begun working in this area (for instance, see [Frye & Soloway, 1987; Soloway, 1996; Golightly, 1996; Hoist, 1996; Sedighian & Klawe, 1996a; Sedighian & Westrom, 1997; Svendsen, 1991; Inkpen, 1997]). It appears that in most cases, due to lack of proper interface design guidelines, designers 30 of educational software for children are consumers of the interaction styles that were originally designed for productivity tools. 2.1.4 Too ls and Toys Malone [1981a] makes a distinction between tools and toys; The main distinction is in the intended goal of using a system. He calls a system a tool if the goal of using it is to produce some external product. An example of a tool is a word processor, and the task is producing a well-written document. Users expect tools to be easy to understand, efficient, and reliable [Malone & Lepper, 1987; Norman, 1986]. If a tool is difficult and complex to use, users may become frustrated and not be able to perform the task efficiently. Malone [1981a] calls a system a toy if the goal of using it is for its own sake, with no external goal. An example of a toy is a computer game. Malone [1981a, p. 359] suggests that toys should "intentionally [be] made difficult to use, in order to enhance their challenge" and make them attractive as toys. The distinction between tools and toys has challenging design implications, especially in interactive game-based learning environments. In the case of computer games which are merely designed to provide children with an entertaining environment, the element of an embedded educational subject is missing. However, addition of this latter factor can make the design more difficult since a game-based learning environment can blur the distinction between tools and toys. "What elements of such system should be considered as tools and what elements as toys?" is not an easy question to answer. Is the user interface a tool whose external goal is to produce knowledge, or is it a toy with no external goals? In 31 short, it is difficult to determine whether interaction with such systems should be easy or challenging. 2.1.5 Cognitive Artifacts: Partners in Cognition A cognitive artifact is an artificial device designed to maintain, display, or operate upon information in order to serve a representational function. . . . Despite the enormous impact of artifacts upon human cognition, most of our scientific understanding is of the unaided mind: of memory, attention, perception, action, and thought, unaided by external devices. There is little understanding of the information-processing roles played by artifacts and how they interact with the information processing activities of their users. [Norman, 1991, p. 17] Some cognitive scientists have suggested a view of cognition that is distributed [Salomon, 1993a]. The extreme position of this view proposes that "cognitions" reside not simply in individuals' heads but in a distributed fashion among individuals, society, tools and artifacts. A moderate view states that tools and artifacts are partners in cognitive activities, and that the individual cannot be discounted [Nickerson, 1993; Perkins, 1993; Salomon et al., 1991; Salomon, 1993b; Norman, 1991, 1993]. Studies of cognitive performance show that thinking is "both enabled or constrained by tools and artifacts that share the load with human beings" [Resnick & Collins, 1996, pp. 50-51]. Some tools extend human thinking and allow one to think about things that one could not do unassisted, others can constrain thinking, forcing thought to follow a certain path of reasoning. Jonassen [1992, pp. 1-3] uses the term "cognitive tools" - tools that "extend cognitive functioning during learning" - and states that 32 Cognitive tools, if properly conceived and executed, should activate cognitive and metacognitive learning strategies. They are computationally based tools that complement and extend the mind. They engage generative processing of information. . .. They are knowledge construction tools - tools that extend the mind. Carroll and Kellog [1989, p. 7] note that all HCI artifacts inherently embody "multiple, distinct psychological claims". They [p. 8] provide an example of a tool with embodied psychological claims, Training Wheels (a reduced-function text editor): The key characteristics of this interface for the purpose of articulating its psychological claims is that the Training Wheels design "blocks" the consequences of problematic user selections. . . . Training Wheels embodies the claim that understanding real-world tasks is facilitated by filtering inappropriate goals. In designing HCI artifacts, especially educational ones, designers should carefully examine the underlying psychological claims and learning assumptions of their design. If these underlying assumptions are not examined properly, design choices may have negative influences on children and other users, constraining their thinking in subtle ways. 2.1.6 User Interface Style: Implications for Learning The conditions for the operational aspects that are given by the computer application are called the user interface. The user interface is the artifact-bound conditions for how actions can be done. Constituent parts of this interface can be conditions for both operations directed toward the artifact and for operations directed toward the real objects or subjects at different levels. [Bodker, 1991, p. 36] One of the breakthroughs in interface design has been the creation of Direct Manipulation (DM) interfaces. D M refers to interfaces that allow the user to manipulate objects on a screen with some kind of pointing device [Shneiderman, 1988, 1993]. D M stands in 33 contrast to a command-based syntax for manipulating objects. Some of the notions associated with D M interfaces include "icons", "windows", "easy", "explorable", and "learnable" [Norman et al., 1986, p. 489]. As a technique for controlling computer systems, D M is widely used in many application areas, including teaching/learning ones. The term "direct" has many implications which makes the concept of D M complex [Norman & Draper, 1986]. In discussing D M , Norman [1986, 1991] presents the notion of two gulfs (Gulf of Execution and Gulf of Evaluation) between the computer system and the user. "The gulf of execution refers to the difficulty of acting upon the environment (and how well the artifact supports those actions)" [Norman, 1991, p. 23]. "The gulf of evaluation refers to the difficulty of assessing the state of the environment (and how well the artifact supports the detection and interpretation of that state)" [Norman, 1991, p. 23]. These gulfs can either be bridged by bringing the system closer to the user, or the user closer to the system. Hutchins et al. [1986, p. 95] state that "the feeling of directness is inversely proportional to the amount of cognitive effort it takes to manipulate and evaluate a system". The goal is, therefore, to reduce the distance across these gulfs and thereby minimize the required cognitive effort. The research literature is unclear as to whether the minimization of cognitive load is only desirable for learning how to use a productivity tool, or whether it also applies to the design of educational interfaces. Recent HCI research in problem solving and learning suggests that computer interfaces with the lowest cognitive effort, particularly D M graphical user interfaces, are not the most educationally effective [Svendsen, 1991; Hoist, 1996; Golightly, 1996]. Studies by 34 Svendsen [1991] found that subjects who used a command-based interface learned more than those who used a D M interface. In Svendsen's study, participants made fewer errors, required fewer trials, and spent more time per trial. Hoist [1996] argues that increasing the users' cognitive load and requiring users to remember, evaluate, and design new courses of action puts them in a better position to make correct decisions. Golightly [1996, p.37] examined the notion of "Indirect Manipulation" and argues that "less direct interfaces cause the user to build a more verbalisable and transferable body of knowledge about the domain". The notion of "indirectness" in interface design is similar to the use of figurative speech in natural languages. While reading natural-language text, if the language is literal or direct, the reader does not have to invest much cognitive effort and can read passively [Holman & Harmon, 1992]. However, figurative language creates a process in which the reader has to participate and fill in the figurative gaps by comparing the known to the unknown [Holman & Harmon, 1992]. Figurative speech is indirect and requires more conscious effort and reflection, an important element of insightful learning [Hayes & Broadbent, 1988]. Payne [1991] suggests that the user interface can be treated as a language in that its vocabulary are actions that need to be performed such as pointing with a mouse. Payne notes, however, that advent of menu-driven systems reduced research in this area, but that interface language deserves research attention. This can be especially true with regard to interfaces in educational software in which the interface language can act as a powerful communicating mediator between the system and the learner. 35 While analyzing the different interface styles, it is important to note that D M is not entirely harmful. It has tradeoffs. One of the main advantages of D M is the notion of "Direct Engagement" [Hutchins et al., 1986]. D M interfaces put users into direct contact with a world of objects, eliminating the need for communication through an intermediary. (Direct engagement is an important element of the notion of 'flow', discussed in Section 2.3.1.) Users tend to find a D M interface more satisfying, whereas a command-based interface may be less motivating. On the other hand, a major disadvantage of D M is that it can support and amplify a user's naive or familiar understanding of a particular domain which may lose sight of the potential of this technology [Hutchins et al., 1986]. Therefore, interface designers are faced with the challenge of "providing . . . new ways [to think of and interact with a domain] and creating conditions that will make [users] feel direct and natural" [Hutchins et al., 1986, p. 118]. This is particularly true of educational user interfaces. In educational software, the interface can promote intuitive, automatic action or explicit, conscious action (for some examples of the role of the interface in general activities, see [Bodker, 1989; Carroll, 1991; Norman & Draper, 1986; Suchman, 1987; Winograd & Flores, 1986; Falzon, 1990]). Norman [1991], while reviewing the notion of "activity flow" in human-computer interaction [Bodker, 1989, 1991], states Automization of effort - and the resulting feeling of direct engagement - can occur where a consistent, cohesive activity flow is supported by the task, artifact, and environment. Interruptions and unexpected results break the activity flow, forcing conscious attention upon the task. For many activities, this "bringing to consciousness" is disruptive of efficient performance. . . . Automatic behavior is valuable in many skilled operations, for it permits the attention to be directed to one 36 area of concern while performing smoothly operations required for another area. . . . But at times, it might be valuable to force conscious attention to some aspect of performance by deliberately breaking the activity flow. A great deal of research is devoted to detennining mechanisms for automating productivity tools (e.g., word processors, spreadsheets, and drawing tools) so that users can perform a given task more smoothly. The aim of an interactive mathematics software is not automization of effort and performance, but rather conscious attention to and reflection on the embedded mathematical concepts. As pointed out in Section 1.6.1, "[i]nterface design for educational software has been accorded little study" [Frye & Soloway, 1987, p. 93]. Undoubtedly, the effective design of user interface^ for productivity tools is important. "Nonetheless, the interface for [educational] software is even more important than usual" [Frye & Soloway, 1987, p. 93]. The complexity involved in the design of interactive educational software is far greater and of a different nature. Not only should the user interface be easy to learn and work with, but also the system must engage the learner to consciously construct knowledge. For these reasons, the suggested guidelines and objectives developed by HCI researchers for the design of user interfaces may not all be appropriate for the design of interactive educational environments. Research is needed to determine how to design effective educational software. 37 2.2 Learning: Related Issues 2.2.1 Nature of the Human Mind and Learning There are many theories of learning and the mind (for example, see [Anderson, 1983, 1985; Bruner, 1966, 1973, 1986, 1990; Fetsinger, 1957; Gagne, 1970; Gardner, 1983, 1991; Guttenplan, 1995; Kolb, 1984; Lave, 1988; Margolis, 1987; Minsky, 1986; Newell & Simon, 1972;Ormrod, 1990; Paivio, 1986, 1991; Phillips & Solts, 1991; Piaget, 1952, 1954, 1970; Reber, 1993; Rogers, 1969; Rumelhart & Norman, 1978; Skinner, 1953; Sternberg, 1988; Vygotsky, 1978]). Some theories are philosophical, and others are psychological. There is overlap but no common definition of learning. Part of the reason for such diversity is that these theories are based on different metaphors and models of thought. Kuhn [1970] points out that scientific investigation is guided by tacit models and metaphors of which scientists are not always aware. Similar to this view, Pepper [1942] suggests that there exist "root metaphors" that guide our thinking and reasoning. This may be true of all research in the field of cognition and learning. Sternberg [1990, p. 3], addressing the problem with the diversity of theories about the nature and workings of the mind, states: Scientists are sometimes unaware of the exact nature of the metaphor underlying the research, and may even be unclear about the particular and limited set of questions that their metaphor generates. They may thus see their partial theories, which address only the questions generated by a single metaphor, as full theories of a phenomenon. Sternberg suggests that there are seven metaphors which underlie all these partial theories: geographic, computational, biological, epistemological, anthropological, sociological, and 38 systems. Being aware of the underlying metaphors of learning theories, including their pros and cons, can help HCI designers to select the appropriate theory or theories in their design - transcending a single metaphor and combining aspects of several of them that seem to have empirical support. Nickerson [1993, p. 259] observes: We seem to be creatures of extremes. We persist in seeing everything in either-or terms. In education, we contrast discovery learning with rote memorization, exploration with instruction, knowledge construction with information assimilation, domain knowledge with the ability to think, and we talk as though the student, the teacher, the system — whoever is in a position to make choices - must always choose one or the other. There is little middle ground, little recognition of the possibility that an effective educational process should include more than one mode or objective of learning or teaching. It may be important for designers to consider this observation when creating multimedia learning environments for children. 2.2.2 Behavioral View of Learning Behaviorists have examined observable instances of learning [Skinner, 1953]. The advantage of a behavioral approach is that data is accessible to other researchers, and findings can readily be replicated and validated, or not. However, the mind is a subjective and inaccessible entity. Consequently, Phillips and Soltis [1991] state that behaviorists are not able to explain or describe how new knowledge is acquired or constructed, and instead, they concentrate on new behaviors. Therefore, behavioral psychology is concerned with the expansion, control, and alteration of people's behavior, and not the ideas in their minds. Behaviorists concentrate on what is outside the head, rather than 39 what is inside it; their concern is changes in behavior, rather than changes in thought and mind [Phillips & Soltis, 1991]. 2.2.3 Cognitive View of Learning In contrast to behaviorism, cognitive psychologists are interested in unobservable factors of learning such as knowledge, meaning, feeling, and thought [Phillips & Soltis, 1991]. This means that there is a shift of focus from the object to the subject [Anderson, 1985; Bruner, 1973; Chipman et al., 1985; Eysenck & Keane, 1990; Norman, 1982; Piaget, 1954, 1970; Resnick, 1981; Rumelhart & McClelland, 1986; Segal et al , 1985]. One of the major findings of cognitive psychology has been that children do not simply absorb information, but that they are continually acting on, organizing, and constructing their own knowledge through repeated interaction with their environment, be it people or objects. The cognitive approach to learning also states that one of the most important factors in a learning process is what the learner brings to the learning situation [Phillips & Soltis; 1991; Ormrod, 1990]. Therefore, what the learner already knows determines to a great extent what he/she will learn in the new context [Resnick, 1981]. Cognitive psychologists theorize about different knowledge structures, and the processes through which these structures are created and modified [Eysenck & Keane, 1990]. Some of the topics they study include memory, perception, comprehension, and problem solving [ibid.]. They also study learning in a wide range of situations. Unlike behavioral psychologists who have general laws for learning, cognitive psychologists do not have a general theory of learning that accounts for everything [Phillips & Soltis, 1991; 40 Orrnrod, 1990]. There are a number of different traditions which have developed within this discipline [Sternberg, 1990; Eysenck & Keane, 1990]. One of these traditions uses an information processing metaphor to view human beings. The primary interest of this tradition is the scientific analysis of human mental processes and structures in order to understand human cognition and behavior. Much of the knowledge of the information processing model of cognition and behavior - especially the low-level findings such as symbol manipulation and memory constraints - may be of use in the design of human-computer systems. 2.2.4 Two Kinds of Cognition and Three Kinds of Learning Norman [1993] distinguishes between two kinds of cognition: experiential and reflective. He states that in order to create technological products that are appropriate for people, especially instructional ones, designers must understand how these modes of human cognition operate. "While the experiential mode of cognition can be practiced simply by experiencing it, reflection is more difficult" and "requires some structure and organization" [Norman, 1993, p. 16]. Rumelhart and Norman [Rumelhart & Norman, 1978; Norman, 1993] suggest that there are three kinds of learning: accretion, tuning, and restructuring. Accretion refers to the gathering and accumulation of facts and information. Tuning is the stage between novice performance and skilled performance in which a great deal of practice takes place until an activity such as reading, writing, or playing is carried out in a "subconscious," 41 "experiential" mode. "The difficult part of learning," however, "is forming the right conceptual structure" [Norman, 1993, p. 30]. Norman further states that: Restructuring is the hard part of learning, where new conceptual skills are required. And the trick in teaching is to entice and motivate the students into excitement and interest in the topic, and then to give them the proper tools to reflect; to explore, compare, and integrate; to form the proper conceptual structures. . . . Reflective [cognition] is essential for restructuring. The problem is to make the students want to do the hard work that is necessary for reflection. . . . [EJntertainment can provide the impetus for reflection. Once people are curious about the questions, then they are stimulated and willing to do the work involved in pursuing the answers. . . . Reflection is hard work, after all, but it can be pleasant where there is a reason for the effort. [Norman, 1993, pp. 30-31] The above notions are an echo of Skemp's [1986] description of the psychology of learning mathematics. He emphasizes the importance of reflective thought and suggests an approach that can be described as a tension-balancing cyclic process for experiencing mathematical concepts and reflecting upon them. Thinking is hard work. Once we have understood a mathematical process, it is a great advantage if we run through it on subsequent occasions without having to repeat every time (even though with greater fluency) the conceptual activities involved. If we are to make progress in mathematics it is, indeed, essential that the elementary processes become automatic, thus freeing our attention to concentrate on the new ideas which are being learnt - which in their turn must also become automatic. At any level, we can also distinguish between routine manipulations and problem-solving activity; and unless the former can be done with minimal attention, it is not possible to concentrate successfully on the difficulties. [Skemp, 1986,, pp. 82-83] Norman [1993] suggests that electronic games can. potentially be the tools that are conducive to a balance between the two modes of cognition: "An interesting example of how one can combine experiential cognition as a motivator with tools for reflective learning is provided by video arcade games" [p. 22]. 42 2.2.5 Constructivism Currently, one of the' most influential models of learning within the cognitive view is constructivism. The constructivist theory has many variations in the way it is understood, described, put into practice, and even labeled (for instance, see [Bruner, 1986; Cobb, 1996; diSessa, 1985, 1988, 1992; Forman & Pufall, 1988a; Glasersfeld, 1980, 1991; Butts & Brown, 1989; Fosnot, 1989; Harel & Papert, 1991; Kolb, 1984; Duffy & Jonassen, 1992]). However, the subject of constructivism is on how meaning is constructed, how something is understood, and what it means to understand it. In other words, the central focus of a constructivist theory of learning is knowledge and its acquisition - i.e., epistemology [Glasersfeld, 1980, 1991; Papert, 1992]. According to constructivism, knowledge acquisition is a dynamic, process-oriented activity at whose center the learner is actively engaged in building new mental structures and representations and modifying and replacing the old ones. This view argues that what we know in the world are the evolved and negotiated human interpretations of our experience of the world, or in other words we have a map of what reality lets us see and do [Glasersfeld, 1991]. This distinction has significant implications for the design of computer learning environments for children. Cobb [1996, p. 56] summarizes three instructional implications of constructivism as outlined by Glasersfeld [1989]: 1. priority should be given to the development of meaning and understanding rather than the training of behavior, 2. researchers and teachers should assume that students' actions are rational given the way that they currently make sense of things, and 43 3. students' errors and unanticipated responses should be viewed as occasions to learn about students' understanding. Summarizing the embodied properties of a constructivist model, Forman and Pufall [1988b] list three elements: epistemic conflict (or cognitive dissonance), self-reflection, and self-regulation. Epistemic conflict happens when one is introduced to a new idea or piece of knowledge that causes internal conflict. This internal conflict necessitates a new way of trunking about reality. The result of the latter is self-reflection. It is an opportunity to explicitly or consciously construct or transform one's way of representing reality. Finally, thoughts and concepts are restructured and regulated at a "higher" level of knowing - this being called self-regulation. Therefore, "conflict and self-reflection are, of necessity, conscious, whereas the last step, self-regulation, . . ., is in all likelihood unconscious in process and outcome" [p. 236]. C o n s t r u c t i v i s m a n d C o m p u t e r M i c r o w o r l d s Constructivism has had a tremendous impact on the way researchers in the field of computer-based learning environments approach the design of such systems [Forman & Pufall, 1988a; Duffy et al, 1993]. A successful application of constructivism has been computer microworlds for children [Papert, 1980]. The purpose of microworlds is to help children develop a clear and efficient understanding of a particular subject. Microworlds are artificial, self-contained software environments which embody a reduced set of principles and concepts from a domain of knowledge [Ginsburg & Zelman, 1988]. They are purposefully designed to be contextually fleshed out so that learners do not become 44 confused and disoriented by too much structural detail and complexity of the real-world version of the domain [Forman & Pufall, 1988a]. The first pioneers in this field were Papert and his colleagues, who developed the L O G O (turtle geometry) microworld [Papert, 1980]. Papert was convinced that microworlds could become "incubators of knowledge" for a diverse set of formal educational subjects, such as mathematics and physics. He also felt that children, as a result of interaction with these microworlds, would start walking on the "path of learning" by developing their own "transitional" theories and ideas about the particular subject, even if these ideas seemed wrong at first. Since then, a number of other researchers have been investigating the potential benefits of these kinds of learning environments in education, and how children interact with them [Forman & Pufall, 1988a; Harel & Papert, 1991; Biddlecomb & Whitmire, 1992]. Many researchers in the field of computer-based learning environments have concentrated on the design of microworlds to the exclusion of other forms of such environments. This may well distort or not be responsive to the learning needs of children. Although microworlds have added a lot of richness to the field of computer-based learning environments, they may fail to address the "motivational orientations" [Dweck, 1985] of all children. Additionally, many microworlds are discovery-based, exploratory environments. However, children have different learning styles and preferences, and some may not be good at discovery-based learning [Kolb, 1984; Keef, 1987; Lall & Lall, 1983; Schmeck, 1988]. For these reasons, the educational needs of children should be addressed through a varied and rich set of tools. Educational computer 45 games1 can play an important role in contributing to this need [Sedighian & Sedighian, 1996]. (Games are discussed later in this chapter.) 2.2.6 Construction versus Instruction There is an important difference between playing and practicing, doing an. activity and learning that activity. Just doing something does not necessarily lead to learning. This point is well understood in sports instruction. Coaches distinguish between unsupervised play and training. You could play for hundreds of hours and learn less than from half hour ofproperly supervised training. . . . The same is true whether the practice be of a sport, chess, or mathematical recreations. [Norman, 1993, p. 36] It is not just experience but its interpretation that is crucial. [Eraut, 1996, p. 2] That people discover knowledge is a myth. . . . Learning has been confused with development.. . . The biological metaphor of autonomous developmental growth has powerfully permeated our thinking about learning. [Novak & Gowin, 1984, pp. 4-26] Constructivism is sometimes interpreted to mean that students only learn through exploratory- or discovery-based activities, and that direct or expository instruction is not effective [Tamir, 1996]. This view is strongly challenged by Anderson et al. [1995]. They argue that if instruction simply means passive recording of information, then it is not effective. However, it is wrong to state that learning is not influenced by explicit instruction. In other words, instruction and construction are not located at antithetical poles. Direct instruction has been criticized and referred to as undesirable mainly because it is considered regimented, it is deemed authoritarian, it can promote fact accumulation at 1 Although some people consider LOGO a design game [Silvern, 1986], LOGO is not usually considered 46 the expense of thinking, and it promotes passivity in students [Rosenshine & Meister, 1996]. Rosenshine and Meister argue that there are applications of direct instruction which are superior to other forms of instruction, and that those who criticize direct instruction should be very specific about what practices they are criticizing. For instance, students who have received direct instruction in "cognitive strategies in reading, writing, and math" have significantly outperformed students in control groups [Rosenshine & Meister, 1996, pp. 360-361]. Anderson et al. [1995] report that direct instruction is sometimes quite effective in helping students learn better. This view seems to agree with Skemp's [1986] description of how mathematical concepts should be ordered and presented so that students are not confused or made anxious. It is reported [Tamir, 1996] that for most students discovery learning increases motivation, curiosity and interest. However, others [Anderson et al., 1995] argue that there is very little positive evidence to show that discovery learning is effective, and that, in fact, it is often inferior to other forms of learning. Resnick & Collins [1996, p. 50] summarized the situation as follows: For the most part, efforts to teach difficult concepts with more directness and clarity, or to train students in the metacognitive or problem-solving strategies that characterize expert thinking, have not worked. . . . On the other hand, in relatively unstructured 'discovery' or 'exposure' programs, the 'rich get richer' phenomenon was encountered repeatedly: that is, initially strong students often prospered, but the weaker ones did not and sometimes even lost ground when compared with students taught in more traditional supervised practice and memorization approaches. These views seem to agree with the findings of learning styles' researchers [Kolb, 1984; Keef, 1987; Lall & Lall, 1983; Schmeck, 1988]. a game. 47 2.2.7 Experiential Learning Experiential learning is all about providing opportunities for participants to make decisions (good or bad) and allowing them to experience the results of those decisions in a protected environment. Games and simulations contribute enormously to this philosophy and often provide the vehicle by which the experience can be gained. [Cudworth, 1996, p. 422] As a reaction to the existing educational system and the way children were educated, humanistic education [Valett, 1977] gave birth to experiential learning [Rogers, 1969], a view of cognition and learning which is influenced by several philosophical and psychological traditions of learning. This theory is elaborated by Kolb [1984] and emphasizes the central role that experience plays in the learning process. Some characteristics of experiential learning are [Kolb, 1984]: 1. Learning is a continuous process grounded in experience; 2. The process of learning requires the resolution of conflict between dialectically opposed modes of adaptation to the world; 3. Learning involves transactions between the person and the environment; and 4. Learning is the process of creating knowledge. A working definition of learning, thus, is: "Learning is the process whereby knowledge is created through the transformation of experience'''' [Kolb, 1984, p. 38]. This definition implies that learning is a process, not an outcome, and that personal knowledge is constructed. As a learner's experiences change, new knowledge is created. The distinction between learning and knowledge is important, since it has been at the center of many philosophical debates for centuries [Guttenplan, 1995]. Although learning and knowledge are closely coupled, they need be distinguished. 48 Novak and Go win [1984] suggest that learning refers to an individualized process in which the learner is engaged, but knowledge refers to the mental structures that are created in the learner's mind and can ideally be shared with others. Therefore, in the context of educational environments, one would expect different children to be engaged in different processes of learning, and thereby constructing different kinds of personal knowledge. Kolb states that the process of experiential learning has a four-stage spiral cycle involving four adaptive learning modes - concrete experience, reflective observation, abstract conceptualization, and active experimentation. This is shown in Figure 2.1. Conceptualization Figure 2.1 Structural dimensions underlying Kolb's model of experiential learning. (Adapted from [Kolb, 1984, p. 42]) 49 Kolb [1984, pp. 40-42] states that in this model concrete experience/abstract conceptualization and active experience/reflective observation are two distinct dimensions, each representing two dialectically opposed adaptive orientations. The structural bases of the learning process lie in the transactions among these four adaptive modes and the way in which the adaptive dialectics get resolved. . . . [TJhe abstract/concrete dialectic is one of prehension, representing two different and opposed processes of grasping or taking hold of experience in the world — either through reliance on conceptual interpretation and symbolic representation, a process I call comprehension, or through reliance on the tangible, felt qualities of immediate experience, what I call apprehension. The active/reflective dialectic, on the other hand, is one of transformation, representing two opposed ways of transforming that grasp or "figurative representation" of experience - either through internal reflection, a process I call intention, or active external manipulation of the external world, here called extension. One of the most important aspects of this theory is that it transcends tensions that exist between bipolar learning modes and views them as dialectics. This has important implications for the design of balanced multimedia learning environments. 2.2.8 Situated Cognition and Social Aspects of Learning Phillips and Soltis [1991] note that one of the problems with many cognitive, as well as behavioral, theories of learning is that the learner is portrayed as a lonely investigator outside of a social matrix. In these theories, the learner can interact with the environment, but there is no recognition of, or emphasis on, the fact that the learner almost always interacts with social groups within an environment. They state that, from early childhood, learners interact and communicate with their parents, brothers, sisters, friends, teachers, and fellow students. Children are stimulated by these people, get guidance from them, imitate them, and so on. The importance of the social nature of learning has been 50 emphasized by many researchers (for example, see [Bandura, 1970; Brown et al., 1989; Kolb, 1984; Lave, 1988; Vygotsky, 1978]). Kolb [1984, pp. 35-37], for instance, states: In experiential learning theory, the transactional relationship between the person and the environment is symbolized in the dual meanings of the term experience - one subjective and personal, referring to the person's internal state, . . ., and the other objective and environmental. . . . These two forms of experience interpenetrate and interrelate in very complex ways. . . . To understand learning, we must understand the nature and forms of human knowledge and the process whereby this knowledge is created. ... Knowledge is the result of the transaction between social knowledge and personal knowledge. The former, . . ., is the civilized objective accumulation of previous human cultural experience, whereas the latter is the accumulation of the individual person's subjective life experiences. Knowledge results from the transaction between these objective and subjective experiences in a process called learning. Situated cognition, or sometimes called situated learning, argues that learning takes place in and is a function of the context (both activity and culture) in which it is situated [Lave, 1988; Resnick, 1987]. In other words, cognition depends on situational affordances. Brown et al. [1989] emphasize the importance of "cognitive apprenticeship" and the notion of communities of practice and assert that there is need for a new epistemology. Situated cognition seems to be closely related to the theory of distributed cognitions [Salomon, 1993a]. The proponents of the situated cognition view argue that the major cause of the poor performance of students is that subjects are learned as out-of-context facts and isolated sub-skills. Decontexualized learning of subjects can result in development of knowledge representations in learners that tend to remain "inert" [Cognition and Technology Group at Vanderbilt, 1991; Brown et al., 1989]. Inert knowledge refers to knowledge that is not readily used by people when solving problems that really require the use of that 51 knowledge. Inert knowledge can usually be recalled when people are explicitly asked about it [Cognition and Technology Group at Vanderbilt, 1991]. Brown et al. [1989] suggest that to prevent the problem of inert knowledge instruction should be situated in the context of authentic experiences or activities that learners consider important. They further state that "[t]he activity and context in which knowledge is developed and deployed . . . is not separable from or ancillary to learning and cognition. Nor is it neutral. Rather, it is an integral part of what is learned" [ibid., p. 32]. In addition to providing a situated context for learning a subject, the activity or task can even influence the strategy and/or the style of the learning of the learner [Ramsden, 1988]. One way of promoting effective learning is to situate it in a problem-based activity [Barrows, 1986]. In this kind of activity, students learn while solving problems and reflecting on their activities. Constructivism and situated cognition are two closely related movements that have gained much popularity in educational circles. Similar to constructivism, some researchers take an extreme position in their interpretation of the situatedness of cognition. Anderson et al. [1995] argue against this extreme position and assert that the tightness of binding of learning to a context depends on the type of knowledge being acquired. They further point out that empirical research findings show that knowledge is more context bound when it is taught in a single context. Moreover, in answering claims for failure of knowledge to transfer, they argue that the kind of symbolic representation and degree of practice play an important role in the success of transfer, as well as where students' attention is directed during learning. 52 One of the important implications of situated cognition is that the social context within which children use a software program plays an important role in the effectiveness of the software. In other words, regardless of how well designed software may be, its eventual effectiveness is negotiated in the environment within which it is used. The classroom setting and how well computer programs are integrated within the curriculum can play a crucial role in their effectiveness. Teachers play an important role. Schoenfeld [1985] points out that students do not generally follow efficient paths to produce a solution, nor do they recognize whether a plan of action is not working or is not being carried out properly. Teachers can provide guidance to students and help them develop and monitor solution plans, as well as direct their attention to important ideas [Anderson, etal, 1995; Norman, 1993]. 2.2.9 Typology of Knowledge One of the central issues in learning is the typology of knowledge. Different authors have categorized the modes of knowledge in various ways [Farnham-Diggory, 1994; Sternberg, 1988; Gardner, 1983, 1991]. Some authors discuss two modes of knowledge which lie at opposite poles. For instance, Pepper [1942] discusses common sense versus refined knowledge; Noddings [1985] talks about intuitive versus formal knowledge; and Sternberg and Caruso [1985] talk about intuitive versus academic/scientific knowledge. All of these authors seem to classify knowledge as such "either by virtue of its acquisition or by virtue of its verification" [Sternberg & Caruso, 1985, p. 140]. Intuitive knowledge is at the level of personal opinion and does not have to hold under rigorous verification. "When 53 subjected to critical analysis, weaknesses are found - inconsistencies with accepted ideas, which make true assimilation to existing (and well-tried) principles impossible" [Skemp, 1986, p. 58]. Formal knowledge, on the other hand, has a social and academic dimension to it. It can be shared and goes beyond feelings, guesses, and personal opinions and has to be scientifically verifiable. Another classification of knowledge is that of skill or practical knowledge versus understanding. This classification focuses on whether knowledge should be considered as skills for doing something or as understanding of why something is done in a certain way [Glasersfeld, 1991; Skemp, 1986]. The former is generally concerned with observable actions, habit learning, rote memorization, speed of performance, right and wrong, and surface structures of concepts. In contrast, the latter is concerned with conceptual structures and operations, intelligent learning, and deep comprehension. A skilled person can perform amazingly well without understanding, and a person with good understanding may fail to perform well [Norman, 1982]. The acquisition of skills can generally be easily measured and evaluated. However, understanding is difficult to measure [Glasersfeld, 1991]. Although understanding is a complex psychological construct, and there is no exact "fit" or "match" to measure it, yet one can infer it by using indicators [ibid.]. Knowledge can also be considered from an organizational point of view. Knowledge can have simple organization or complex organization [Eysenck & Keane, 1990]. Simple knowledge organizations usually deal with knowledge structures such as object concepts (e.g., dog, bird), whereas complex knowledge organizations deal with knowledge 54 structures such as relational concepts, events, predictive structures, schemata, frames, and scripts. All types of knowledge are important. However, they require different modes of acquisition. Thus, when designing an interactive learning environment, it is important to know beforehand what kind of knowledge construction is intended, and design the learning environment accordingly. The most frequent use of computers for educational purposes has been for "drill and practice" [Ginsburg & Zelman, 1988]. The idea behind these programs is to reinforce a skill such as remembering information or practicing algorithmic procedures. Although such environments are beneficial and can lead to the acquisition of certain skills (e.g., multiplication and addition), they provide a narrow form of learning. Learning environments which promote reflective thought and intend to create complex knowledge structures, such as transformation geometry, require a different approach to design than environments which deal with development of skills. 2.2.10 Concept Learning The notion of concept is an unresolved issue in philosophy and psychology [Guttenplan, 1995]. Although the term is commonly used, it is not easily defined [Skemp, 1986]. Novak and Gowin [1984, p. 4] define a concept as "a regularity in events or objects designated by some label." They define an event as "anything that happens or can be made to happen," and an object as "anything that exists and can be observed." Skemp [1986, p. 21] elaborates issues related to this term as follows: 55 Abstracting is an activity by which we become aware of similarities . . . among our experiences. Classifying means collecting together our experiences on the basis of these similarities. An abstraction is some kind of lasting mental change, the result of abstracting, which enables us to recognize new experiences as having the similarities of an already formed class. Briefly, it is something learnt which enables us to classify; it is the defining property of a class. To distinguish between abstracting as an activity and abstraction as its end-product, we shall. .. call the latter a concept According to Skemp [1979, 1986], there are two kinds of concepts: primary concepts and secondary concepts. Primary concepts refer to those concepts which are "derived from our sensory and motor experiences" such as seeing a red object or tasting something sweet. Secondary concepts refer to those concepts which are derived from other concepts. Secondary concepts are usually complex knowledge structures. The following are a list of general ideas and principles related to the notion of concepts, and how they should be communicated. All of the following are cited from Skemp [1986]. 1. "A concept is a purely mental object - inaudible and invisible." [p. 64] 2. Concepts are closely linked to their names - these names being any symbol (e.g., auditory, visual or propositional). "A concept is an idea; the name of a concept is a sound, a mark on a paper, associated with it." [p. 22] 3. Concepts are closely linked to symbols, and they can be "perceived as a unity." [p. 65] 4. Concepts fit together and form conceptual structures or schemas. 5. "Concepts of a higher order than those which people already have cannot be communicated to them by definition but only by collecting together, for them to experience, suitable examples. . . . Definitions can thus be seen as a way of adding precision to the boundaries of a concept, once formed, and of stating explicitly its relation to other concepts." [p. 25] 56 Skemp argues that the way mathematics should be presented and taught must be fundamentally different from other natural and social sciences because the epistemology of mathematical concepts is different from other subjects. Mathematical concepts are very abstract and hierarchically dependent, and, therefore, their communication is more difficult than other concepts. He further states [1979, pp. 153-154] that one of the most commonly used, and also mis-used, ways of trying to [bring about the formation of new concepts in students J is by giving them a definition. . . . It is important for the communicator to know that definitions work only from higher to lower levels of abstraction or generality, and where the new concept can be formed by relating existing classes. . . . Communication succeeds at. . . low levels of abstraction more by luck than by knowledge. But in the teaching of subjects in which a major part of the learner's task is the formation of increasingly abstract concepts [such as mathematics], communication fails without either side knowing why. This happens particularly widely in the teaching of mathematics, and is one of the reasons why so many persons come to grief in trying to learn it. . . . [The communication of mathematics] requires a careful analysis by the communicator to find out exactly what [the] contributory concepts are, and what are the contributors to these, and so on. . . . The successful communication of mathematical concepts is so demanding a task that the more one begins to realize all that is involved, the better one understands why so many find it a problem subject. Regarding transition from an intuitive stage of understanding of mathematical concepts to a reflective stage, Skemp states [Skemp, 1986, pp. 62-63, bold added]: While learners are still at the intuitive stage they are largely dependent on the way material is presented to them. If the new concepts encountered are too far removed from any of their existing schemas, they may be unable to assimilate them, particularly if reconstruction is required, for this is largely dependent on reflection. . . . So the teacher of mathematics has a triple task: to fit the mathematical material to the state of development of the learners' mathematical schemas; to also fit the manner of presentation to the modes of thinking (intuitive and concrete reasoning only, or intuitive, concrete reasoning and also formal thinking) of which the pupils are capable; and, finally, to increase gradually the pupils' analytic abilities to the stage at which they no longer depend on their teacher to predigest the material for them. 57 The above process is an important element of the constructivist method of teaching and is referred to as scaffolding [Perkins, 1992; Rosenshine & Meister, 1996; Brown et al., 1989]. Scaffolds are temporary supports on which students can rely during initial stages of learning a subject. "The scaffolds are diminished as students learn the strategy and become independent" [Rosenshine & Meister, 1996, p. 361]. Indeed, "some instructional experiments suggest that, contrary to older theories in which prerequisites supposedly needed to be firmly in place before more complex [mathematical] problems were posed, it now seems possible to overcome weak preparation by engaging children in intellectually challenging problem-solving while providing supporting 'scaffolding'" [Resnick & Collins, 1996, p. 49]. Ausubel [1968, p. 153] prescribes that concepts have to be understood in ever greater degrees in a "progressively differentiated" manner, i.e., in terms of detail and specificity. Novak and Gowin [1984, p. 99] refer to this as meaningful learning and state that it is: a continuous process wherein new concepts gain greater meaning as new relationships (propositional links) are acquired. Thus concepts are never 'finally learned' but are always being learned, modified, and made more explicit and more inclusive as they become progressively more differentiated. Similar to this, Salomon [1981] discusses the importance of mental elaboration in learning. In his view elaboration implies a "progression of processing from surface to deep features" of the material being learned [p. 131]. Depth, in turn, is the amount of invested mental effort (AIME) in processing material and is composed of two elements: 1) number of mental elaborations performed, and 2) the degree to which these elaborations are non-automatic. AIME depends on both the message that a material communicates and the 58 person who processes the material. If the material becomes too familiar and automatic, the person processing the material may perceive it as not requiring great AIME. On the other hand, if the material communicates the message that it is unfamiliar and challenging, it may be perceived as calling for greater AIME leading to deeper learning. The notions of unity of symbols and concepts, scaffolds and scaffolding, progressive differentiation, and elaboration can have significant implications for the design of user interfaces of interactive mathematics learning environments. 2 . 2 . 1 1 Symbols as Mental Interfaces for Concept Formation A symbol can be a "sound, or something visible" [Skemp, 1986, p.65]; it can be "any object, movement, gesture, mark, or event," provided "it is taken to represent, denote, or express something beyond itself [Salomon, 1979, p. 29]. A symbol can stand for a concept, an event, an object, a process, or an idea and be connected to it. This connection is the meaning of the symbol, and without it the symbol is empty and meaningless [Skemp, 1986]. Through this connection, the symbol and what it represents are perceived as having unity of meaning. "These meanings may be simple, or highly abstract and with great interiority" [Skemp, 1979, p. 161]. As such, symbols can act as mental interfaces, or in other words accessible concepts, by which less accessible, abstract concepts can be activated and manipulated [Skemp, 1979]. This makes symbols of central importance to human cognition, communication, epistemology, and user interface design [Salomon, 1979, 1981; Guttenplan, 1995; Skemp, 1979, 1986; Vygotsky, 1962, 1978; Norman, 1991]. Some of the uses and functions of symbols are [Skemp, 1986, p. 64]: 59 1. commvinication, 2. recording knowledge, 3. the communication of new concepts, 4. making multiple classifications straightforward, 5. explanations, 6. making possible reflective activity, 7. helping to show structure, 8. making routine manipulation automatic, 9. recovering information and understanding, and 10. creative mental activity. Symbols are related to representations [Norman, 1991; Salomon, 1979]. There are two kinds of representations [Eysenck & Keane, 1990]: internal and external - the former referring to the cognitive mental representations of the learner, and the latter referring to the representation provided through an external medium. In epistemological discussions, the words symbol and external representation are used interchangeably and seem to be very close in meaning and application (for example, see [Schwartz, 1995]). In this dissertation, the two terms are also used interchangeably. There are many ways in which knowledge can be encoded and distributed [diSessa, 1979]. In the context of the present discussion, there are three disciplines from which to view symbolic representation: psychology, computer science, and education. From a psychological point of view, the question is: What are the internal representations or models of the mind for external concepts? From a computer science point of view, the 60 question is: How can a concept be represented through the computer interface so that it is consistent with the person's internal mental models? Finally, from an educational point of view, within the context of an interactive learning environment, the question is: How can a concept be represented at the interface level so that it helps the learner construct appropriate conceptual structures? A difference noted between experts and novices is the degree to which they are able to create a coherent, integrated, and unified internal representation of a concept [Sternberg, 1988]. Good external representations can greatly help in understanding concepts and solving problems [Sternberg, 1988]. External representations can aid the mind to reflect on abstract concepts and focus on higher-order thoughts [Norman, 1993; Skemp, 1979, 1986]. "The power of the unaided mind,'' Norman [1993] argues, "is highly overrated." He further adds that, "Without external aids, memory, thought, and reasoning are constrained" [p. 43]. Skemp [1986, p. 78] points out that: Making an idea conscious seems to be closely connected with associating it with a symbol. Just why this should be so is not yet known. Concepts are elusive and inaccessible objects, even to their possessors, and it may be that symbols (which are themselves primary concepts) are the most abstract kind of concept of which we can be clearly aware. . . . Once the association has been formed, the symbol seems to act as a combined label and handle, whereby we can select (from our memory store) and manipulate our concepts at will. It is largely by the use of symbols that we achieve voluntary control over our thoughts. diSessa [1988] argues that one of the difficulties in instructing children is to help them overcome their fragmented and naive mental models. Proper external symbolic representations thus seem to be crucial in overcoming the fragmented and naive mental 61 models of children and assisting them in developing more integrated, formal thought processes [diSessa, 1979, 1988; Pufall, 1988]. Different symbol systems and representations have different affordances [Norman, 1993; Salomon, 1979]. The term affordance was first used by Norman [1988, p. 9] to refer to "the perceived and actual properties of [a] thing, primarily those fundamental properties that determine just how the thing could possibly be used". Norman [1993] discusses how the representation of numbers can greatly influence how easy or hard it is to perform mathematical operations. Whereas it is easy to read Arabic numerals, they are not the best representation for counting because they are substitutive. In contrast, tally marks are additive. Affordance of representations, and in general child-computer interfaces, can play an important role in the effectiveness of educational learning environments. N o representation is better than another. Its suitability is determined by the task for which it is used and the instructional objectives of that task. A representation can reflect the surface or the depth of a concept. A t times the depth can be difficult to understand. However, in certain cases, especially educational environments, this depth of symbolic representation may be helpful in causing reflection [Skemp, 1986]. The difficulty in designing appropriate educational user interfaces is to represent the concept in such a way that it w i l l be of optimal use to the audience. Different media have different symbol systems and, therefore, different affordances [Salomon, 1979]. Salomon [1983] investigated children's viewing of television, and he found that children do not put much effort in learning from television, because they conceive it as being easy and therefore not requiring much mental effort. Therefore, while 62 watching television, even high-achieving children may fail learning. Salomon [1979, p. 217] argues that "different symbol systems represent different kinds of content," and, "even when representing the same content, differ with respect to the amount of mental translation from external symbol to internal mode that they require". He [1979, p. 217] adds that symbol systems differ with respect to the kinds of mental skills that they invoke in the process of knowledge extraction. To the extent that symbol systems call on quantitatively and qualitatively different mental skills, knowledge acquisition outcomes can be expected to vary respectively. The implications of these findings are quite important - in that HCI designers should not simply copy the symbol systems of media such as paper, particularly while designing multimedia learning environments. On a piece of paper, a representation is usually static. On the computer, however, representations can be both static and dynamic. Static representations usually convey the syntax of a problem, whereas dynamic representations can not only convey the syntax but also the deeper semantics of it. In many learning environments it is desirable to communicate concept-relevant characteristics both structurally and operationally. The selection of appropriate static and dynamic representations can play a crucial role in how conceptual meanings are conveyed. 2.2.12 Propositional versus Analogical Symbols Mathematics can be thought of as a language that must be meaningful if students are to communicate mathematically and apply mathematics productively. Communication plays an important role in helping children construct links between their informal, intuitive notions and the abstract language and symbolism of mathematics; it also 63 plays a key role in helping children make important connections among physical, pictorial, graphic, symbolic, verbal, and mental representations of mathematical ideas. [NCTM, 1989, p. 26] In general, symbolic representations are of two types: propositional and analogical [Eysenck & Keane, 1990; Paivio, 1986]. Analogical symbols are primarily visual (e.g., photographs, maps, and drawings), but they can also be auditory. Propositional symbols are more abstract, language-like representations (e.g., human languages, formal systems of mathematics, symbolic logic and computer languages). Paivio [1983, 1986, 1991] proposes the dual coding theory of learning. This theory suggests that humans have two cognitive sub-systems, one specialized for the processing of non-verbal representations and spatial events, and the other specialized for dealing with linguistic, sequential information. There is accumulating empirical evidence that memory for images is superior to memory for words or propositional symbols -"pictorial superiority effect" [Paivio, 1983]. Visual learning abilities develop prior to linguistic abilities, and "serve as the foundation" for the development of the latter one [Pettersson, 1996, p. 180]. Moreover, visual symbols represent meaning at an apparent, basic level [Pettersson, 1989]. Skemp [1986] states that although a visual symbol is more difficult to communicate, it provides a closer link to concepts, and is, therefore, a more individual, rather than collective, form of knowledge. Regarding the learning of geometry, he [1986, p. 94] points out that The resemblance of the geometric symbol to its concept has both advantages and disadvantages. An advantage is that it evokes well the properties of the concept. This is especially true when we represent visually several concepts together. The diagram 64 then brings into awareness the relationships between these concepts more clearly than does a verbal representation of the same concepts. . . . A disadvantage of the visual symbol is that it has to be drawn to be communicated. . . . Because it is a stage more concrete, we must do some of the abstracting ourselves. He adds that although "most geometrical communications begin with a diagram, they very soon change over to verbal-algebraic symbols," and that in the study of vectors, for example, "directed line segments are replaced by ordered pairs, triplets, or n-tuples of numbers." This is supported by Bruner [1966] and Piaget [1970] who suggest that, for children, a full understanding and internalization of abstract concepts is preceded by some form of concrete experience. 2.3 Motivation Motivation plays a central role in any learning activity [Csikszentmihalyi & Csikszentmihalyi, 1988; Dweck, 1986; Keller, 1996; Malone, 1981a; Skemp, 1986; Snow & Fair, 1987; Weiner, 1990]. It "influences both attention and maintenance processes" of learning [Tennyson, 1996, p. 55]. Broadly speaking, motivation refers to an external intervening process, and/or an internal state of a person, that induces him/her to action. Keller [1983, p. 3] defines motivation as the "magnitude and direction of behavior", and "it refers to the choices people make as to what experiences they will approach or avoid, and the degree of effort they will exert in that respect". Motivation is a very complex psychological construct and can not be examined in isolation. It results from the interaction of a large number of variables, such as attention, feeling, emotion, reward and feedback, anxiety, and need. In spite of the fact that there has been a great deal of research 65 on motivation and cognitive activities, most of these research studies have dealt with these two aspects of human activity as separate issues [Snow & Fair, 1987]. It is only in recent years that knowledge acquisition and motivation are being studied in the context of coherent, unified, and integrated learning activities [Snow & Farr, 1987]. Dweck [1986, p. 1046] reports that: Motivational processes have been shown to affect (a) how well children can deploy their existing skills and knowledge, (b) how well they acquire new skills and knowledge, and (c) how well they transfer these new skills and knowledge to novel situations. This approach does not deny individual differences in present skills and knowledge or in "native " ability or aptitude. It does suggest, however, that the use and growth of that ability can be appreciably influenced by motivational factors. Psychologists refer to two broad categories of motivation: intrinsic and extrinsic. Weiner [1990] points out that behavioral theories of learning emphasize extrinsic, reward-oriented motivation, whereas cognitive theories of learning focus on intrinsic, goal-oriented motivation. These two types of motivation refer to the factors on which human behavioral outcome depend. Intrinsic motivation usually stems from internal feelings of enjoyment and fulfillment and is not dependent on external rewards. Extrinsic motivation, in contrast, originates from outside of the person and is generated through such means as reward and/or punishment. There has been some debate over the negative impact of external rewards on a person's intrinsic motivation [Lepper & Greene, 1978]. Some researchers argue that external rewards have negative effects and can undermine people's intrinsic interests in a task. It is argued, for instance, that if a child who enjoys doing mathematics is rewarded for doing it (e.g., by buying him/her gifts), once the reward is discontinued, the child may 66 develop a negative attitude toward the activity and not spend as much time on mathematics as before. However, in a meta-analysis of ninety-six previously-conducted experimental studies that compared rewarded subjects to nonrewarded controls on 4 measures of intrinsic motivation, Cameron and Pierce [1994] report that their overall findings do not totally support this hypothesis and that external motivators have no detrimental effect on intrinsic motivation. The following is a brief outline of some of their conclusions: 1. Rewards are detrimental only under a highly specified set of circumstances. That is, when subjects are offered a tangible reward (expected) that is delivered regardless of level of performance, they spend less time on a task than control subjects once the reward is removed. The same condition has no effect on attitude [p. 395]. 2. Verbal praise and positive feedback enhance people's intrinsic interest [p. 397]. 3. Overall, the present review suggests that teachers have no reason to resist implementing incentive systems in the classroom [p. 397]. Therefore, it seems safe to assume that, similar to the tension between construction and instruction, what is important is to provide children with learning environments that have a dialectic balance between extrinsic and intrinsic motivational factors. 2.3.1 Flow Although there has been a great deal of research on motivation, yet there is "little scientific knowledge . . . about the factors that underlie motivation, enjoyment, and satisfaction.. . . 67 [W]e know little about how best to structure tasks and.events so as to establish, maintain, and enhance the [human] experience. Much of what we do know comes from the work of psychologists such as Csikszentmihalyi" [Norman, 1993, p. 32, italics added]. As discussed in Chapter 1, flow is an intrinsically motivating and rewarding state in which an individual derives optimal experience from an activity. Flow can be experienced through different activities such as seeing visual aesthetics, listening to music, learning, and playing games and sports [Csikszentmihalyi, 1990]. Flow may take place anywhere and at any time, as long as there is a match between a person's abilities and opportunities for action and the conditions of the environment [Csikszentmihalyi & Csikszentmihalyi, 1988]. Flow is more typical of interactions and conditions in which people participate voluntarily. "Games are obvious flow activities, and play is the flow experience par excellence" [Csikszentmihalyi, 1975, p. 37]. However, Csikszentmihalyi emphasizes that "playing a game does not guarantee that one is experiencing flow" [ibid.]. For an experience to become enjoyable certain conditions must be made possible - such as, "concentrated attention, clear goals, feedback, lack of distractions," and a balance between personal skills and the challenges of the situation [Csikszentmihalyi & Csikszentmihalyi, 1988, p. 85]. These conditions can be optimized and built into situations by design [ibid.]. While discussing the relationship between flow and learning, Norman [1993, p. 32] states: Is there some way of achieving this state of optimal flow while learning? Note that people are typically willing to exert great mental effort upon their recreational but not their educational activities. Yes, one is done for enjoyment, the other assigned as tasks or duties in the schoolroom or on the job, but the difference seems paradoxical, especially since many people will tell you that the educational work is more important. 68 The difference appears especially paradoxical if you simply compare what has to be done for each: The activities for recreation and education are essentially the same. Norman suggests that recreation, especially playing games, requires the "same kinds of learning, study, understanding, and practice as are required of any educational activity" [p. 32]. He notes that one of the reasons subjective feelings such as flow are not studied in the context of learning and education is because scientific studies have difficulty measuring them, and, at times, they are impossible to measure. The concept of flow has profound implications for how learning environments for children should be designed. As Norman argues, the current understanding of how to systematically build the conditions of flow into interactive learning environments is limited. These conditions are at best descriptive and generic; they are not prescriptive, and there is need for models that apply these conditions to different learning situations. The difficult task is to determine how to create these conditions. Beneath the wide differences in the situations reported [regarding flow] there runs a common theme: Regardless of gender, age, ethnic or cultural origin, enjoyment is the same everywhere, and it is made possible by the same configuration of subjective and objective conditions. [Csikszentmihalyi & Csikszentmihalyi, 1988, p. 85] Dimensions of Flow Flow activities have certain structural characteristics [Csikszentmihalyi & Csikszentmihalyi, 1988; Norman, 1993]. One of these characteristics is a balance between challenge and skill. Csikszentmihalyi [1988, p. 30] states that "optimal experience requires a balance between the challenge perceived in a given situation and the skills a person brings 69 to i t . . . . To remain in flow, one must increase the complexity of the activity by developing new skills and taking on new challenges. . . . Flow forces people to stretch themselves, to always take on another challenge, to improve on their abilities". Because of this "spiraling complexity" during the flow process, people discover new things. One of the most challenging tasks in producing sustainable, intrinsically motivating activities, then, is to keep the ratio between a person's capabilities and encountered challenge within a range which results in neither boredom and lack of fun nor worry and anxiety. When the challenge is greater than one's capabilities, one experiences worry and frustration; when one's skill is greater than the challenge, one experiences boredom [Csikszentmihalyi, 1975, p. 49], as seen in Figure 2.2. Low Skills High Figure 2.2 Relationship between challenge and skill. (Adapted from [Csikszentmihalyi, 1990, p. 74]) 70 Another universal characteristic of a flow experience can be described as "focused concentration," "immersion," and "merging of activity and awareness", which are typical of enjoyable activities [Csikszentmihalyi, 1988]. For this to happen, it is important to "avoid distractions and disruptions that intervene and destroy the subjective experience" [Norman, 1993, p. 35]. Norman [1993] suggests that "high-quality," "surround sound" increases the likelihood of being captured by an event. He [p. 34] adds that Another way to reduce distractions and increase concentration is to wear a headset. In any environment the event best captures attention when the sensory experience is maximized and distractions are minimized. Some other general characteristics of environments that are conducive to flow, summarized by Norman [1993, pp. 34-35], are listed below: • Provide a high intensity of interaction and feedback • Have specific goals and established procedures • Provide a sense of direct engagement, producing the feeling of directly experiencing the environment, directly working on the task • Provide appropriate tools that fit the user and task so well that they aid and do not distract 71 2.3.2 Other Related Research Malone [1981a], influenced by the work of Csikszentmihalyi and after observing children play with non-educational electronic games, proposed a framework for a theory of intrinsically motivating instructional environments. The framework is based on four categories: challenge, fantasy, control, and curiosity. The following is a brief outline of these categories. 1. Challenge: For an environment to be challenging, it has to provide meaningful goals. Moreover, the outcome of the game has to be uncertain. However, the self-esteem of the player should not be damaged. 2. Fantasy: A game can provide both intrinsic and extrinsic fantasies. However, Malone states that intrinsic fantasies are more interesting and more instructional than extrinsic ones. It is better if fantasies have a direct relationship to the embedded material. In addition, fantasies can have cognitive and/or emotional aspects. Gender plays an important role in whether or not certain fantasies are attractive. 3. Control: An environment should promote feelings of control in the player by being responsive and providing choice. 4. Curiosity: An environment should be "neither too complicated nor too simple with respect to the learner's existing knowledge. [It] should be novel and surprising, but not completely incomprehensible" [Malone, 1981a, p. 362]. Curiosity can be roused by both sensory and cognitive stimuli. Informative feedback can play an important role here. Keller [Keller, 1983, 1996; Keller & Kopp, 1987; Keller & Suzuki, 1988] also presented an instructional design model, the ARCS model, which synthesizes and 72 integrates a number of other motivational theories. The ARCS model has four components: attention; relevance, confidence, and satisfaction. It incorporates both intrinsic and extrinsic motivational factors. The components of the ARCS model can be implemented using many tactics [Keller, 1996]. In this model, the first requirement for motivating students is to attract their attention and sustain it over time. However, capturing their attention is not sufficient. The material or activity, moreover, must have relevance to their motives and goals. The next thing is to create positive expectancy for success, i.e., creating confidence in a balanced manner. When the first three conditions are met, the fourth component, satisfaction, is intended to sustain students' motivation through reinforcement and corrective feedback. 2.4 Electronic Games, Motivation, and Learning Many children like playing games and participate in them for extended periods of time without being conscious of time. As potential and valuable educational tools, play and games have been studied extensively [Avedon & Sutton-Smith, 1971; Shears & Bower, 1974; Piaget, 1951,1952]. The advent of electronic games, due to their interactive and dynamic nature, has given a new meaning to play. Electronic games are an important element of the social life of children and consume their time [Rheingold, 1983; Provenzo, 1991]. This has created a great deal of debate about electronic games, and much has been written about electronic games and the various roles they play in the lives of children [Dominick, 1984; Harris & Williams, 1984; Provenzo, 1992]. Some researchers focus on the motivational benefits of 73 these games [Braun & Giroux, 1989; Bowman, 1982; Silvern & Williamson, 1987; Selnow, 1984; Nowrock & Winner, 1983; Millar & Navarick, 1984; Lepper & Malone, 1987; Long & Long, 1984; Howard, 1983; Malone, 1981a, 1981b; Morlock et al., 1985]. Others analyze the effect of these games on cognitive skill acquisition such as trial-and-error, pattern recognition, rule generation, hypothesis testing, generalization, estimation, inductive thinking, and spatial skills [Gagnon, 1985; Greenfield, 1983, 1990; Silvern, 1986; Dorval & Pepin, 1986; Hawkins, 1986; Lowery & Knirk, 1982; Quinn, 1991; Quinn et al., 1993]. Still others investigate the social dimensions of these games [Kegan, 1983; Slaby, 1983; Leff, 1983; Brooks, 1983]. Many parents and teachers are alarmed at the amount of time children spend playing electronic games [McGinley, 1991; Upitis, 1994]. They wonder whether children should be engaged in other worthwhile activities instead. Furthermore, some claim that these games are addictive and waste children's time. Notwithstanding the controversy, there is no doubt that electronic games have a powerful motivating effect on children. Some researchers regard electronic games as potentially valuable educational tools [Kafai, 1995; Klawe, 1994a, 1994b; Lepper & Malone, 1987; Malone, 1981a, 1981b, 1983b; Malone & Lepper, 1987; Perkins, 1983; Papert, 1992; Norman, 1993; Rieber, 1996; Sedighian & Klawe, 1996b; Sedighian & Sedighian, 1996]. Regarding learning mathematics, Klawe [Klawe, 1994a, p. 14], for instance, states that "video games are an excellent vehicle to use to increase the exploration of mathematical concepts by children". A few approaches and models for the use of these games in the classroom have been proposed [David & Ball, 1986; Klawe & Phillips, 1995; Silvern, 1986; Malone, 1983b; Rheingold, 1983]. 74 As stated previously, most of the games that have been used for educational purposes have been drill and practice games, lacking a true problem solving focus. In a review of research on the effectiveness of games for educational purposes, Randel et al. [1992] reviewed a span of 28 years [from 1963 through 1991]. Despite the lack of focus on more sophisticated mathematical concepts, they report that "the use of games is superior to traditional classroom instruction for improving math achievement" [p. 269]. They add "that games/simulations are more interesting than traditional classroom instruction is [sic] both a basis for using them as well as a consistent finding. The greater interest in games holds true even when controls for initial novelty [Hawthorne effect] have been Used" [p. 270]. This finding seems to justify the use of games as motivating instructional environments. Clark [1983] states that one of the common sources of confounding in media research is the uncontrolled effects of the novelty of newer media that tends to disappear over time. However, according to what Randel et al. [1992] report, in the case of electronic games, novelty effect is not an uncontrolled effect. Alessi and Trollip [1991] categorize games into ten groups. However, they point out that these groups may overlap and play several roles simultaneously: 1. Adventure: "An adventure game is one in which the player assumes the role of a character in a situation about which little is known" [p. 172]. 2. Arcade: "Arcade-type games are similar to those found in an amusement arcade" [p. 174]. 3. Board: "Board games are often computerized versions of existing games" (e.g., chess) [p. 174]. 75 4. Card or Gambling: "These are generally characterized either by the existence of a large element of chance or by the use of money as a motivator" [p. 174], 5. Combat: These games "use combat or violent competition as their primary motivation" [p. 175]. 6. Logic: "Logic games are those that require the player to use logical problem solving to succeed" [p. 177]. 7. Psychomotor: "Psychomotor games are those that combine intellectual and motor skills" such as space, tennis and basketball games [p. 178]. 8. Role-Playing: "Role-playing games are those in which the student assumes the guise of a character and acts out that role" [p. 179]. 9. TV Quiz: These games "take the form of an ordinary television quiz game" [p. 180]. 10. Word: "Word games either teach about words or use words as the basis of the game" [p. 192]. Alessi and Trollip [1991, p. 202] conclude by stating that games are a powerful instructional tool if used appropriately. It is clear that they have a strong motivating influence on children and adults alike. However, it is important to remember that instruction clothed in game format does not necessarily make the instruction effective. It is not the game format itself that appeals to people, it is the challenge or enjoyment of a particular game. For a game to be successful, it must be enjoyable and satisfy your instructional requirements. Design Issues and Reflective and Experiential Flow Both concentrated experiential cognition and intense, focused reflective thought can be rewarding and be conducive to experiencing flow [Norman, 1993]. Norman [1993, p. 30] states that "[e]ntertainment exploits the experiential mode" of cognition, and that reflection "is not its major function". However, he suggests [ibid., p. 30] that 76 entertainment can provide the impetus for reflection. Once people are curious about the questions, then they are stimulated and willing to do the work involved in pursuing the answers. Norman [1993] notes that in many electronic games the experiential mode is driven by external events, thereby sustaining concentration. Norman adds [p. 35] that Games, especially action games, are stimulating and compelling because they are event-driven activities, always presenting some new challenge to the player, maintaining attention by continual new stimulation, new challenges. This is one of the powers of the experiential mode: The mind is externally driven, captured by the constant arrival of a barrage of sensory information. One of the problems with the kind of game described above is that many such games create this experiential situation using a time factor. By requiring the player to respond to the sensory information more and more quickly, the challenge in these games is increased. So as the player learns new skills, he/she must react to the challenges in the game faster. It is questionable whether such environments afford effective knowledge construction and learning. Indeed, research in behavioral and cognitive sciences [Fleming & Levie, 1993] shows that self-paced environments are more conducive to better knowledge acquisition and motivation. A fundamental research question is how to design games so that they afford a balance between experiential and reflective cognition, while still mamtaining the conditions in which optimal flow experiences can take place. Moreover, whereas we know how to build effective drill-and-practice games, much research is needed to determine how to design game-based learning environments - "learning environments" emphasizing the notion that 77 there is a difference between practicing what one knows and starting from a possible state of not-knowing to a state of formal, explicit knowing of abstract mathematical concepts. 78 Chapter 3 D e s i g n R a t i o n a l e This chapter describes the design of Super Tangrams, an interactive multimedia learning environment for middle-school children which assists students in learning two-dimensional transformation geometry. This chapter explains the main cognitive, affective, and epistemological issues that were considered in the design of Super Tangrams. This chapter follows a solution-driven approach rather than a problem-driven one (see Section 2.1.1). Through "reflection" on "conversation[s] with the [design] situation" [Schon, 1983], I attempt to bring the issues involved in the design of Super Tangrams to the rational plane in order to make them visible and accessible to others [Visscher-Voerman & Plomp, 1996; Winograd, 1996]. This chapter concentrates on the design itself, rather than on the genesis and the development process of that design. As such, this chapter is prescriptive and does not provide a complete chronological description of all the events that transpired in producing the design. Since design is pragmatic and goal-oriented and searches for optimality within certain imposed, situational boundaries [Reigeluth, 1996], this chapter identifies and presents alternative design methods which may be better than other existing alternatives. 79 Some of these alternatives are described in detail, and others are touched upon in merely outline form. The design of Super Tangrams was not merely rational, it was also creative and involved a cyclical, incremental development process and was continuously reconceptualized (see Section 2.1.1; also see [Rowland, 1990]). However, since the design took place in an ill-structured problem space, whose effect and outcome were probabilistic and difficult to know before full implementation, a cyclical development methodology was an appropriate approach. Some stages in the design process included extensive conversations with middle-school children to find out what they wanted to have in the design, talking to their teacher about what she thought students would like, testing prototypes of the design with both children and adults, making direct observations of how children interacted with the program, qualitative evaluations in naturalistic classroom settings, and finally running an experimental study with a large group of children to identify factors and elements of the design that were pedagogically effective. This chapter has six sections. Section 3.1 discusses the overarching model used to promote a flow experience while learning mathematics. The next four sections describe specific aspects and components of this model and ' discuss how these were operationalized in more detail. Section 3.6 presents an overall description of Super Tangrams as an integration of the issues and ideas discussed in the preceding four sections. (The reader may wish to view Section 3.6 before proceeding with the other sections.) 80 Before describing the program's design, a brief list of some of the instructional objectives of Super Tangrams, many of which are well beyond what is expected of middle-school students, is presented below. Super Tangrams is intended to help children 1. appreciate that a rotation is not merely turning a shape around its center, but involves setting an angle of rotation as well as setting a center of rotation anywhere on a two-dimensional plane; 2. realize that rotating a shape both turns and translates it; 3. understand that translation is not simply sliding a shape on a plane, but moving a shape along a straight line, in a certain direction, and by a certain amount; 4. understand that a translation arrow (vector) represents the distance and direction in which an object moves; 5. explore the relationships among different transformations and their equivalences; 6. understand the effect of reflection on symmetric and asymmetric shapes; 7. develop a sense of which transformation is more effective to use in a given situation; 8. develop a sense of visualization both in terms of how to put shapes together and in terms of what transformation or combination of transformations to apply to a shape to move it to a desired position; 9. realize that composite reflections are sufficient for performing all transformations, i.e., that the effect of any transformation can be achieved by an appropriate sequence of reflections. 81 3.1 Challenge-Driven Learning Model: Promoting Flow in Learning One of the important structural characteristics of a flow activity is a proper balance between challenge and skill (see Section 2.3.1). In the flow model, however, challenge and skill are generic parameters, and most researchers do not specify an approach or a model in which these parameters can be applied to the design of learning environments. Nevertheless, it seems possible to map these parameters onto learning situations. When these parameters are mapped to a mathematics learning environment, challenge and skill become a student's required mathematical capability and a student's current level of mathematical knowledge. However, learning is a process; consequently, a mechanism is required to assist students to construct new knowledge as new challenges are encountered. Through such a mechanism learning becomes a progressive dialectic process in which required knowledge (i.e., challenge) and constructed knowledge (i.e., skill) interact and are kept in balance. Keeping the balance between children's required mathematical capability and their current knowledge at any given point is the key to sustaining their interest in the learning activity and promoting the experience of flow in learning. One model that may be used to accomplish this dialectical interaction is a learning environment that has two complementary components: 1. a situated game activity, and 2. an embedded instructional module. In this model, the game module supports a constructivist, situated style of learning 82 (see Sections 2.2.5,2.2.7, and 2.2.8). The game is intended to provide an exploratory and discovery-based environment to motivate children to engage in mathematical activity. It is a context in which children "experience the enjoyment of goal-directed action" [Csikszentmihalyi, 1988b, p. 34]. The game has a series of well-defined sequential goals. These goals progressively become more challenging, thus requiring more mathematical knowledge for their achievement. This challenge is spread over a number of levels, not only to give children a sense of accomplishment but also to provide them with an indication of how much further they have to go to "beat the game" (as children themselves express it). The game is designed such that to advance through it, children must pay attention to finer and more specific conceptual details (i.e., "progressive differentiation"; see Section 2.2.10) and continuously increase and refine their understanding of the embedded mathematical domain to meet the challenges that arise in the course of play. This increase in the degree of challenge is directly proportional to the children's need for increased reflective thought in order to succeed in meeting the challenges (i.e., "mental elaboration"; see Section 2.2.10 and Figure 2.1). The embedded instructional module supports a more structured and directed style of learning. It is intended to allow children to refine and increase their mathematical knowledge via more formal explanations; that is, the instructional module should direct and guide children's explorations and interpretations in the experiential game environment according to more formal, shared knowledge (see Sections 2.2.6 and 2.2.9). This module does not automatically interrupt the flow of play based on pre-determined measures put in by the designer. Rather, the instructional component is invoked by children at will, any 83 time they desire. The reason for making the instructional module an on-demand activity is that most children do not like to learn mathematics unless it satisfies a need (see [Skemp, 1986; Sedighian & Sedighian, 1996]; also see [Carroll, 1990]). In order to satisfy the need for accomplishing the goals of the game, children have to meet the challenges presented to them. To meet these challenges children need to learn the mathematical concepts embedded in the game activity. Therefore, the instructional module is designed to be closely linked to the goals of the game and allow children to construct the knowledge required to advance through the game. However, this module is not like on-line help found in productivity tools; it contains general concepts that children need to understand and apply in the game context. To keep the ratio between challenge and knowledge construction in balance so as to promote the experience of flow while learning, the instructional component is adjusted to the knowledge requirements of where the player is in the game (see Section 2.2.10). Hence, as the game becomes more challenging and knowledge intensive, the instructional module likewise provides a more sophisticated conceptual presentation of the mathematical domain. To actualize this model, four important design issues have to be dealt with: 1. A situated activity must be selected to support the learning of the knowledge domain, i.e., transformation geometry. 2. The activity must be made operational as a computer-based, game-like activity. Furthermore, it must be decided how children will interact with the game. 3. It must be decided how children should interact with the underlying mathematical concepts embedded in the game. 84 4. A n d , finally, the instructional component must be designed to support knowledge construction in the game activity. The first issue plays an important role in determining the situational affordances of the activity (i.e., action opportunities) in terms o f learning the subject matter. The second issue plays a crucial role in children's perception of the activity. It is crucial that children think of it as being a fun game so that they are enticed to spend time with the software. The third issue plays a deciding role in how the symbolism of and interaction with the embedded concepts enable children to develop appropriate mental models o f the subject domain. The fourth issue helps children construct deeper knowledge and follow more efficient paths to solve the puzzles (as discussed before, according to [Schoenfeld, 1985]). 3.2 Activity Selection Problem situations that establish the need for new ideas and motivate students should serve as the context for mathematics in grades 5-8. Although a specific idea might be forgotten, the context in which it is learned can be remembered and the idea re-created. In developing the problem situations, teachers should emphasize the application of mathematics to real-world problems as well as to other settings relevant to middle school students. [ N C T M , 1989, p. 66] A n important aspect o f teaching a subject is the selection o f an appropriate supporting activity (see Section 2.2.8). In the case of Super Tangrams, an activity was selected so that not only could it support the learning of transformation geometry, but also it could be made into a game that would motivate children to engage in the learning activity. The rationale for how this activity was selected is discussed below. 85 One of the activities people have enjoyed for centuries has been solving problems dealing with dissection puzzles in the plane and space [O'Daffer & Clemens, 1976]. Tangrams puzzles are a children's activity similar to jigsaw puzzles, which many adults also enjoy. One version of tangrams, Chinese Tangrams, is usually presented as a set of seven two-dimensional geometric figures (two small triangles, a medium triangle, two large triangles, a square, and a parallelogram) that must be assembled, by moving the pieces together, into a larger shape. The pieces, which the Chinese Tangrams is comprised of, can be cut from a single large square. Figure 3.1 depicts this square and its seven cut pieces. The angles in all these pieces have only three values: 45, 90, or 135 degrees. Figure 3.1 Figure 3.2 Figure 3.3 Square from which Outline of a tangrams Solution to a tangrams tangrams pieces are puzzle. puzzle. created. Chinese Tangrams is a popular game and can be found in many game and activity books (for example, see [Loyd, 1968; Read, 1965; Moscovich, 1984; Pappas, 1989, 1991]). The puzzle is presented by providing an outline, such as Figure 3.2, in which players are required to arrange the seven pieces so that they fit the outline, as in Figure 3.3. Many interesting shapes can be formed with these seven basic pieces. Read [1965] 86 has compiled 330 different possible outlines that can be made with the tangrams pieces. Pappas [1991], however, reports that over 1600 designs can be created. This gives Chinese Tangrams its richness and provides for an entertaining, yet intellectually challenging activity. Besides being entertaining and intellectually challenging, some mathematics educators suggest that tangrams can provide numerous worthwhile mathematical experiences for children, particularly in learning transformation geometry [Buell et al., 1978; Rahim & Sawada, 1989; Russell & Bologna, 1982]. While trying to solve these puzzles, children are constantly engaged in moving the pieces about, sliding, turning, and flipping them. In other words, the tangrams activity affords motion geometry in a very natural way. Solving tangrams puzzles puts children in a situation in which they are immersed in geometric problem solving and reasoning - immersion being an important aspect of learning according to Dewey (reported in [Cognition and Technology Group at Vanderbilt, 1991]). In addition to the reasons mentioned above, some tangrams puzzles are moderately complex problems requiring extended effort to solve and can help children develop a sense of geometric visualization as well as reasoning. The National Council of Teachers of Mathematics [NCTM, 1989, p. 75] suggests that "the mathematics curriculum should engage students in some problems that demand extended effort to solve... . For grades 5-8 an important criterion of problems is that they be interesting to students". Furthermore, "problem solving should be the central focus of the mathematics curriculum," and, "as such, it is a primary goal of all mathematics instruction and an integral part of all 87 mathematical activity". Therefore, "problem solving is not a distinct topic but a process that should permeate the entire program and provide the context in which concepts and skills can be learned". Greeno [1978] presents a typology of problems, and how they can be solved. He specifies three types of problems: 1. problems of inducing structure, 2. problems of transformation, and 3. problems of arrangement. Tangrams present a blend of transformation and arrangement tasks. Greeno [p. 241] states that "[t]he skills needed for simple problems in transformation involve skill in planning based on a method called means-end analysis", and arrangement problems require skills in composition, "a process of constructive search, where the problem solver is required to find the solution in a search space but must also know how to generate the possibilities that generate the search space". These problem solving skills are required in a wide range of mathematics learning situations and make Chinese Tangrams a worthwhile activity for learning transformation geometry. 3.3 Game Issues to Consider 3.3.1 Operationalization of Activity Since the game element is intended as a motivational environment to entice children into a learning situation and engage them in the mathematical activity, the game activity itself 88 should not impose too much cognitive load on children, in terms of usability. Moreover, it should not interfere with the main purpose of the program, which is learning transformation geometry. In other words, playing of the game should be an experiential activity, whereas learning transformation geometry should be a reflective one. This strategy is supported by the research evidence that in problem-solving environments in which means-ends analysis is required, a large amount of cognitive processing capacity has to be devoted to solving the problem, thereby leaving fewer cognitive resources for conceptual schema acquisition [Sweller, 1988]. One of the first things to decide is how the activity is presented on the computer screen. In Super Tangrams, each puzzle is presented as a given outline with the geometric shapes (pieces) placed around it, as in Figure 3.4. As will be seen later, this strategy allows the geometric shapes to be placed at locations that can create educationally desirable situations. w Figure 3.4 Presentation of the tangrams activity on the screen. 89 In Figure 3.4, the speckled area represents the outline. Since there is more than one shape, there needs to be a mechanism by which the shapes are selected individually to be transformed. In the figure, the black triangle, in contrast to the gray shapes, is the selected shape. In addition to the seven shapes, there are three operations, translation, rotation, and reflection. Whereas there is no need for awareness of different operations when the activity is performed using physical manipulatives, in the computer medium, once a shape is selected, a child must be able to specify a transformation operation to perform on the shape. After selection of the operation, the parameters for the operator must be determined, either implicitly through acting upon the selected shape, or explicitly through setting those parameters and then applying them to the selected operand. The design of Chinese Tangrams poses some complex design choices. Due to space, only some of these are discussed here. When solving tangrams puzzles using physical manipulatives, children need not worry about how accurate they are when arranging the shapes to fit a given outline. That is, they can leave physical gaps between the adjacent shapes, can overlap them slightly, and can place portions of some shapes outside the boundaries of the outline. Since the physical medium provides only visual feedback, it is the player who decides when a puzzle is completed. In the computer-based game, the program must decide if all the shapes are in their appropriate place. This means that students have to exercise a great deal of accuracy in positioning the shapes. This, however, may detract children from 90 having a sense of intense engagement, a characteristic of children's games, and may repel them altogether. In the design of Super Tangrams, measures are taken so that the program accepts inaccurate placements by some margin. To reduce children's cognitive load and make the game easier, the program has a Snap-in-Place feature. Using this feature, the program automatically compensates for the inaccurate placement of a shape whose location in a puzzle's outline is part of a correct solution, snaps it into its intended position, displays a happy-face icon on the selected shape, and provides a click (snap) auditory feedback sound, indicating the correctness of the position for the selected shape. Another complexity arises in the case of puzzles that have multiple solutions. In the case of Chinese Tangrams, some puzzles have as many as 124 solutions. Given the possible permutations for arranging the shapes to fit an outline, Super Tangrams hypothesizes a possible solution the player may have in mind. As children go down different search paths trying to solve a puzzle, the hypothesis is dynamically and seamlessly re-adjusted so as not to distract and disturb children. This is crucial since it can be a deciding factor in whether children will like the program or not, thereby influencing their learning experience. Many games use some kind of a timing mechanism to create a sense of excitement and challenge. Super Tangrams is a "logic game" (see Section 2.4) within which children are engaged in learning mathematical concepts. A time factor in this game may encourage children to rush through the game thereby engaging more in experiential rather than reflective cognition. Therefore, Super Tangrams does not use time as a limiting factor. Children are provided with a self-paced learning environment in which they have as much 91 time as they need to solve any given puzzle. Three other noteworthy features include: Hint, Undo, and Reconfigure. The Hint feature, when invoked, tells children where a certain shape may fit in the outline. This gives children a chance to concentrate on application of appropriate transformations, rather than spending time trying to determine the position of the shapes. In effect, this is intended to encourage a shift of the children's cognitive resources from concentrating on solving the puzzles to concentrating on determining what transformations to use to solve the puzzles, and how to use the given transformations. The Undo feature is intended to enable children to retract and perform a transformation move anew, thereby gaining confidence in how it can be done. The Reconfigure feature allows children to start a puzzle afresh. To preserve the seriousness of the activity, the recorded number of moves made for the given puzzle is not set to zero. This feature is intended to allow children to exit partially solved puzzles and, so to speak, messy situations. 3.3.2 Sensory Stimuli: Engaging the Experiential Senses To enhance the learning experience and promote experiential flow, Super Tangrams engages children's senses using two types of stimuli: auditory and visual (see Section 2.4). Auditory Gross [1995] states that one of the reasons some people do not enjoy learning is that it can often take place in an uncongenial environment. Some people like to have background 92 music when working or studying in order to relax or to concentrate better. Csikszentmihalyi [1990] suggests that music can actually help organize the mind, reduce psychic entropy, ward off boredom and anxiety, and even induce flow experiences. Moreover, Norman [1993] suggests that surround sound and music can affect one's whole experience and help one concentrate and be immersed in an activity. Fabe [1996] suggests that music can reach into the preverbal realm of listeners, engage their emotive response, and become inextricably bound up with their affective and perceptual tendencies towards" its source. Background music is not usually incorporated into interactive learning environments. This may partly be due to the concern that music might distract the learner and interfere with the learning task. Contrary to this view, several studies have found that music can have a positive effect on on-task-performance and does not necessarily interfere with learning tasks [Davidson & Powell, 1986; Thaut & de l'Etoile, 1993; Wolfe, 1982]. Super Tangrams includes background music, and every puzzle plays a new musical piece. Unlike the repetitious tunes used in many video games, the background music in Super Tangrams is only popular, instrumental music without lyrics. An attempt has been made to include music which seems to have energy and matches the mood of the different puzzles. For instance, if a puzzle is difficult, the music is mystical. Super Tangrams allows children to turn the music up, down, or off altogether. Super Tangrams also includes sound effects intended to enhance children's overall experience and knowledge construction. Each transformation has a unique sound effect associated with it - a whooshing sound for the animation of translation, a ticking sound 93 for changing the angle of rotation and its animation, and a projectile followed by a thump sound for reflection. (These transformations and how they are implemented are discussed later in this chapter.) These sound effects are intended not only to act as motivational features but also to combine with graphical animations and establish associative memory cues for remembering the mathematical concepts. In addition to these, there are feedback sounds for when a transformation goes outside the bounds of the screen, when one presses on a disabled object, and so on. V i s u a l A e s t h e t i c s Research shows that color can have an effect on people's emotions and their aesthetic experience [Valdez& Mehrabian, 1994; Pettersson, 1989; Chapman, 1993; Misanchuk & Schwier, 1995; Zentall, 1986]. Color can even influence an individual's physiology such as blood pressure and brain wave patterns [Horton, 1991]. Color can be used to induce certain behavioral outcomes. In general, vivid colors such as bright reds and yellows can promote idea generation and activity, whereas darker colors may evoke feelings of anger [Sanders & McCormick, 1987]. Moreover, for girls visual imagery is very important [Rogers, 1995]. Ih short, visual aesthetics can induce experiences of beauty and flow and play a subtle role in shaping a user's overall feelings towards an artifact [McMahon, 1996; also see Csikszentmihalyi, 1990]. To produce Super Tangrams, close to one thousand hours were spent on crafting an artifact which would be visually pleasing. The background environment of the screen uses bright reds and blues to evoke feelings of excitement. As the game progresses, these colors 94 change so as to create a sense of expectation and movement - an aspect of flow activities (see Norman's [1993] description of how the constant arrival of sensory information captures the experiential pole of cognition). For instance, towards the end of the program children enter a "twilight zone" where they hear a mystical voice welcoming them to the zone. This zone belongs to the "masters" of transformation geometry and the background screen is black, dotted with stars. In addition to the background screen, each puzzle employs different patterns as the background color of the puzzle shapes. At the beginning of the game, these shapes mostly consist of children's cartoons. Towards the end, especially in the twilight zone, most of the patterns consist of very exotic fractal images. All in all, an effort has been made to create a journey through a world of colors and patterns as children advance through the program. 3.3.3 Other Game Features Super Tangrams includes a number of other embellishments which are intended to enhance its level of fun. Some of these include: 1. A scoring system - the harder a puzzle is to solve, the higher its maximum score. 2. A par system - each puzzle can be solved using a minimum, average, or maximum number of moves, and children's scores depend on the number of moves they make; this feature has a two-fold purpose: to introduce an element of challenge and to encourage children to reflect more before making moves in the game. 3. A verbal feedback/reward system - depending on the number of moves children make to solve a puzzle, they receive from very positive (e.g., "You are a genius!") to less positive (e.g., "Try harder next time") comments. 95 3.4 Epistemology at the Interface Level Once some of the game issues are decided upon, the designer needs to look into the epistemological issues involved in the design of the artifact. An artifact's interface embodies underlying theories of learning and communicates the language of the system (see Sections 2.1.5 and 2.1.6). Therefore, before designing the interface for a learning environment, it is important to ask some questions: How does epistemology take place at the interface level? How should the shapes be manipulated? Should children be allowed to manipulate the tangram shapes directly? Or should they manipulate and move the shapes indirectly through typed-in commands or propositions? How should the transformation concepts be represented, and how should they be operationalized? What should the sequence of mouse clicks (or actions) be? These issues are addressed in this section. 3.4.1 Direct Concept Manipulation: Selected Manipulation Style In designing the tangrams activity, it was important to devise an interface style for manipulating the geometrio shapes. This interface manipulation style needed to support the development of an explicit and formal understanding of the three transformation geometry concepts. To provide a concrete example of what is intended by explicit, formal understanding, consider the concept of rotation. At an intuitive level, by grade 6, most children know that turning a two dimensional object involves a change in its orientation. When they are asked about this concept, they usually demonstrate it by physically turning their hand. However, for most children, this intuitive knowledge may not encompass a formal understanding of the rotation concept which involves other 96 abstractions such as an angle of rotation and a center of rotation. Moreover, this knowledge may not include an understanding that a shape may rotate about a center outside of its own perimeter, or that rotation is composed of both an in-place turn and a slide. In designing Super Tangrams, three possible manipulation styles were considered which are discussed below. Command-Based Manipulation One possibility is to have a command-based interface (can also be called a propositional or algebraic interface) which is linguistic in nature. For instance, children can be required to specify the parameters of the arc of rotation by typing in an (x , y) pair as the coordinates of the center of rotation and a number as the angle of rotation. Or to give another example, translation can be described using a number pair such as (1, 3), meaning a mapping of each point of the plane to another point by going 1 to the right and 3 up [Graening & Nibbelink, 1978]. However, requiring this from children implies certain assumptions on the part of the designer. These notations require children to have some degree of formal linguistic knowledge of the subject. As discussed in Chapter 2, linguistic notations are more abstract than other forms of representation since they have no direct relationship to the perceptual modalities of the objects to which they refer (see [Eysenck & Keane, 1990]). It is unrealistic to assume children having this type of linguistic knowledge, as many children likely start working in the environment with virtually no propositional or other prior knowledge of the topic's formalism. As noted in Chapter 2, learning theorists (such as [Piaget 1970; Bruner 1966; Skemp, 97 1986; Kolb 1984]) suggest that a better internalization of an abstract concept occurs when it follows concrete experience. Tall and Thomas [1991] argue that traditional approaches to mathematics put too much emphasis on algebraic and logical mathematics, and not much on visual mathematics, leading to a narrow algebraic understanding of mathematics. There is also growing research suggesting that children should be exposed to visual imagery while learning mathematical concepts [Dawe, 1993]. Moreover, visual processing and recall are superior to linguistic processing, and the latter is built on the foundation of the former (see Section 2.2.12). In addition to these cognitive issues, commands or propositions may leave a negative affective impact on children at initial stages of their entry into a knowledge domain (see Section 2.1.6). Consequently, as suggested by research in cognitive psychology, a linguistic input manipulation seems to be more effective if it follows a pictorial manipulation. Although this may not be true of all knowledge domains, the nature of the content at hand (i.e., transformation geometry) suggests that a pictorial symbol system (symbol as defined by [Salomon, 1979]) may initially make transformation geometry more accessible to children (see [Skemp, 1986]). Direct Object Manipulation (DOM) Another possibility is to have a D M interface in which children manipulate the pictorial representations of the geometric shapes (or the objects in the environment) directly -henceforth referred to as Direct Object Manipulation (DOM). That is, children can click on the shape itself to perform a transformation on it. For instance, to move a shape from position A to B, children can move the mouse cursor to position A, click on the shape, 98 drag the shape to position B, and release the mouse. This form of manipulation of the tangrams shapes is similar to their physical manipulation by hand in which a piece is grabbed and moved. While playing the tangrams game, children's main goal is to move the pieces to desired target locations. Therefore, interacting with a D O M interface style is both natural and intuitive. Indeed, there are a number of tangrams programs on the market which provide this type of interface manipulation style. (A prototype of Super Tangrams including this type of interface style is described in Section 4.2.2.) As discussed before, the use of direct manipulation interfaces in educational software has been questioned by researchers (see Section 2.1.6). One of the potential shortcomings of the D O M interface, as described above, is that children interact with the learning environment in an automatic and intuitive manner. That is, the interface minimizes the distance across the gulfs of execution and evaluation for them, and therefore may not demand that children consciously interact with or pay attention to the underlying transformation concepts (supported by Salomon's discussion of symbol systems and their interaction with cognition and learning [Salomon, 1979, 1981]). Another potential problem is that, as Hutchins et al. [1986] point out with regard to D M interfaces, D O M may reinforce children's naive and misconceived understandings of the transformation concepts - a hypothesis consistent with the notion of cognitive tools either extending or constraining users' conceptual understanding (see Section 2.1.5). Direct Concept Manipulation An attempt to present pupils solely with objects as the focus of mathematics, . . ., is to miscue fundamentally and in consequence contribute to the common failure to function 99 mathematically with fluency. Mathematicians use and produce symbols, endlessly. . . . Symbols become objects, the things of mathematics themselves. Ironically, manipulation is about touch, whereas symbols are predominantly about sight, resulting in mathematics being in the grip of a powerful mixed metaphor. [Pimm, 1995, p. 185] In a mathematical learning environment the main goal is to learn particular mathematical concepts. Therefore, the learning system should ideally provide an environment that immerses children in direct engagement with the mathematical concepts. It is necessary, as far as possible, to enable children to interact with and think in terms of the concepts being learned rather than the objects that the concepts act upon. So, it logically follows that children's focus should be shifted from manipulating the objects directly (DOM) to manipulating the concepts directly (Direct Concept Manipulation, henceforth DCM). However, since, unlike objects, concepts are abstract entities, this manipulation must be directed at the referents to those concepts - that is, interface elements that externally represent the structural and semantic properties of the concepts, or in other words their symbolic abstractions. Therefore, in the case of rotation, for instance, children should be engaged in figuring out angles and centers of rotation for moving shapes rather than simply turning a shape and then dragging it to a location. A shift from D O M to D C M has certain design implications. One implication is that whereas an ideal D O M interface implicitly tries to reduce the semantic distance between the output and the user's mental model (i.e., intuitive understanding), a D C M interface should invite the user to gradually cross this distance and adapt to the formal symbol system of the medium representing the concepts. In most D O M interfaces, manipulation 100 of an object triggers some implicit function behind the scene whose effect is seen by the user (see Figure 3.5). In a D C M interface, manipulation is intended to take place at an explicit level of conceptual representation (see Figure 3.5). Another implication involves the epistemological characteristics of objects and concepts. In most cases, objects represent concrete entities, and consequently require shallow, surface understanding. In contrast, concepts are more abstract entities and have depth (see [Eysenck & Keane, 1996]; also see primary and secondary concepts in [Skemp, 1986]). Thus, concepts have to be understood in ever greater degrees in a "progressively differentiated" manner, i.e., in terms of detail and specificity (see Section 2.2.10). Object Object Implicit Representation If Implicit Referent Concept Representation Referent • Concept UJ Manipulation D O M Manipulation D C M Figure 3.5 D O M versus D C M . Another implication is that symbolic referents to concepts have to be selected carefully (see Section 2.2.11). As stated above, since concepts are abstract entities, D C M 101 involves manipulation of objects representing the linguistic and educational aspects of concepts. Since suitable symbolic representations are very important for understanding and communicating concepts [Guttenplan, 1995; Polya, 1957; Forman, 1988; Norman, 1993; Salomon, 1979; Gibson & Salvendy, 1989], the choice of these external interface representations plays an important role in the educational effectiveness of the D C M metaphor. 3.4.2 Selection of Symbolic Representations for DCM To teach transformation geometry concepts, appropriate representations are needed to allow children to visually and concretely (in a human-computer sense) interact with the formal language of these concepts. Transformation geometry is a knowledge domain which is more pictorial and visual than linguistic. For many years geometry textbooks have used a set of static pictorial representations to facilitate the teaching of transformation geometry concepts (for some examples, see [Beberman et al., 1971; Nowlan & Washburn, 1975; Graening & Nibbelink, 1978; Wells et al., 1978; Hirsch et al., 1984]). Figure 3.6 depicts some typical representations of translation, rotation and reflection. 102 Figure 3.6 Transformation representations: Left to right: translation, rotation, and reflection. The above pictorial representations were considered and selected for use in the game. The next step was to operationalize them. 3.4.3 Linking Experience to Concept Learning through D C M Transformation geometry concepts incorporate causal as well as interrelated properties. For instance, changing the length of a translation arrow causes a corresponding change in an object's final destination, and translation and rotation can each be accomplished by using a sequence of reflections. The operationalization of the representations, therefore, should naturally reflect these properties. Moreover, these representations should provide children with "predictive knowledge structures" [Eysenck & Keane, 1990, p. 249] which they can manipulate and, thereby develop an understanding of the underlying functional consistency of these concepts -consistency being an important criterion for an appreciation of the beauty of mathematics [Skemp, 1986]. Manipulation of these representations through the computer should also make the dynamic aspect of transformation concepts cognitively more accessible than the static, paper-bound version. 103 As discussed above, since many children may begin using the program without prior knowledge of transformation geometry, it seems desirable to have children initially manipulate these representations at a more concrete, experiential level so that they develop a sense for the structural abstractions of the concepts as well as their underlying functional language. Three examples from Super Tangrams are presented to demonstrate the implementation of the concepts of translation, rotation, and reflection. Translation A translation vector (or arrow) shows the distance and direction of the translation of each point of an image. Changing the length or direction of the translation vector affects the destination of the translation image. Figure 3.7 shows a pictorial representation of translation in Super Tangrams. This figure depicts how children initially encounter the concept of translation. The black triangle is the image to be moved, and the gray, speckled triangle is the target area. The outline triangle, called the "ghost image", shows the destination that results if the current translation vector is applied. The vector has three mouse-sensitive handles (or areas) at the tail, midpoint, and head. The length and direction of the vector can be changed by moving (dragging) either the head or the tail, and these actions move the ghost image. The entire vector can be moved by dragging the midpoint, but this does not change the position of the ghost image. The midpoint handle is intended to help children discover (through active experimentation) that a change in the position of the vector (without changing its length or direction) does not have any effect on the relational positions of the black image and its ghost image. The ghost image is intended to 104 act as a "scaffold" (see Section 2.2.10) to help children observe the consequences of manipulating the translation vector. Application of the translation vector causes an analog translation animation of the black image towards its ghost image, thereby reinforcing the cause and effect relationship. i 3 I „ 1 m t imag b^b^b^b^b^b^b^b^b^bv Gho^ .•$.„. g_ Slide g_ 3. r lil He d 1 i ; k ; — t 1 s 1 K 1 t t 2 1 Figure 3.7 Representation of translation in Super Tangrams. Rotation A rotation can be described by an arc whose center signifies the center of rotation and whose angle shows the direction and magnitude of the rotation (see Figure 3.8). The direction of a rotation is either clockwise or counterclockwise, and its magnitude is correspondingly shown in either negative or positive degrees. Changing the center or angle of rotation affects the destination of the rotation image. Figure 3.8 depicts how children initially encounter the concept of rotation in Super Tangrams. As in the case of 105 translation, the black triangle is the image to be moved, the gray, speckled triangle is the target area, and the ghost image shows the destination that results if the current rotation arc is applied. I -0 Wk Ce'n'te r 01 Ih ; arc 9 (J 3. if rota ion Headc fthea e 1 3 1 1 t i 1 3 Figure 3.8 Representation of rotation in Super Tangrams. The arc has two mouse-sensitive handles: the head of the arc and the center of the arc. The handle at the head of the arc allows children to set the angle of rotation. Children can rotate this handle in either a clockwise or a counter-clockwise direction. Changing the angle of rotation does not affect the position of the center of rotation, but it changes the direction and position of the ghost image. The handle at the center of the arc allows children to adjust the position of the center of rotation. Dragging the center of the arc does not affect the direction of the ghost image, but it changes the position of the ghost image and shrinks or enlarges the arc. The numerical value next to the center of the arc shows the 106 magnitude of the angle of rotation in either negative or positive numbers, depending on whether the direction of the rotation is clockwise or counter-clockwise, respectively. This number is intended to provide an algebraic association with the pictorial representation of rotation. Application of the rotation arc causes an analog motion of the black image along the perimeter of the arc towards its ghost image with the point of the image connected to the tail of the arc ending up at the head of the arc. The design of this manipulation style for rotation is intended to assist children to see the underlying relationship between rotation and translation. Reflection A reflection in a plane is described by a symmetry line (or line of reflection) which indicates how far each point on an image is transformed as well as the direction of reflection. Changing the position or orientation of this line affects the destination of the reflected image. Figure 3.9 shows a pictorial representation of reflection. This figure depicts how children initially encounter the concept of reflection in Super Tangrams. As in the previous cases with translation and rotation, the black triangle is the image to be moved, the gray, speckled triangle is the target area, and the ghost image shows the destination that would result if the current reflection line was applied. The thick vertical line shows the reflection line, and the line connecting the lower vertex of the black image to its corresponding ghost image is the perpendicular bisector. The arc shows the angle of the reflection line with respect to the horizontal line. Inclusion of this angle in the 107 interface representation is intended to assist children to discover the relationship between rotation and reflection. I S 1ft- -; J J j . „ u i e o f refl action Dra j hand e y / — — J • HOT zuiitd Uue ,„ „ ffijgfe 5 / / V n 1 p j | V Han ni i i e a t r i "the-ar cad rpe'ri'di iiflar fc 1 sector 1 j • ! V i • t e i 1 1 5 1 !; t s Figure 3.9 Representation of reflection in Super Tangrams. The reflection line has two mouse-sensitive handles. The handle, which in this figure is seen in the target area, allows children to drag the line of reflection in any direction on the screen plane. Dragging this handle does not affect the orientation of the ghost image, but it changes the position of the ghost image moving it closer or farther from the black triangle. The handle at the head of the arc allows children to change the angle of the line of reflection. They can rotate this handle in either a clockwise or a counter-clockwise direction. Changing the angle of reflection does not affect the position of the drag handle, but it changes the orientation of the line of reflection, as well as the orientation and position of the ghost image. The numerical value next to the center of the arc shows the 108 magnitude of the angle of the line of reflection in either negative or positive numbers, depending on whether the direction of turning the line of reflection is clockwise or counter-clockwise, respectively. This number is intended to help children discover the relationship between the concepts of reflection and rotation. Application of the reflection line causes an analog motion of the black image flipping over the line and landing on its ghost image. 3.4.4 Mouse Interaction Protocol: Further Support for DCM This section discusses the role that the mouse interaction protocol can play in children's formation of correct concepts as well as their liking of the program. In this document, mouse interaction protocol refers to the series of mouse clicks (or actions) that children must perform to move a shape - i.e., the number of clicks, their order, and the screen locations that are mouse sensitive. The number of possible interaction protocols, even for a small learning environment, can be far too many to analyze. It is, therefore, beyond the scope of this dissertation to provide a detailed analysis of this issue. However, the following example may convey what some of the general issues are that should be considered when designing a learning environment for children. Consider the case of rotating a shape, assuming that the shape has somehow been selected to be rotated. Figure 3.10 depicts a puzzle where the square is selected. Let us look at some possible ways of rotating this shape. One possible way is for children to click on the Turn button, move the mouse to the play area, and then click somewhere in the play area for the arc of rotation to appear. 109 Now the question is: Where should the arc appear? Should the arc appear at the mouse location or not? If yes, at which point of the arc should the mouse cursor be positioned: the handle for the center of rotation or the handle for the angle of rotation? Should children be allowed to move the mouse and simultaneously adjust the arc while the mouse is depressed? Should they first have to let go of the mouse and then adjust the settings of the arc? Should the play area remain mouse sensitive so that children can click somewhere else for the arc to reappear there? In the initial stages of the development of Super Tangrams, the interaction protocol for rotation was to first click on the Turn button, and then click in the play area. An arc of rotation (set at a predetermined value) would then appear with its center at the mouse position. The center of rotation would follow the position of the mouse while the mouse button remained depressed. Once the mouse button was released, children could adjust the center and angle of rotation by manipulating the two handles. If the mouse was clicked at any location other than the two handles, the arc would be repositioned with its center at that point. From a design perspective, it is important to note that this interaction protocol was not successful. After clicking on the rotate button, children tended to click in the play area and drag the mouse in circular motions expecting the shape to rotate around itself. Instead the arc would move around the screen with the mouse resulting in confusion and frustration of the children. Further investigation led to certain conclusions. Part of the problem arose from children's intuitive and kinesthetic understanding of rotation. They intuitively thought 110 that by rotating what was under the mouse, the selected shape would turn. Another reason for the confusion was children's lack of familiarity with the more formal concept of rotation and its formal representation. Whereas for adults who knew transformation geometry the design looked smart and elegant, for children it did not make sense. After all, they had never seen such a representation. How could they know it had something to do with rotation? After several phases of modification and testing, it was discovered that, the mouse interaction protocol still afforded and promoted confusion. The main causes of confusion were: 1) the representation was live at the instant it appeared (i.e., the center of rotation moved in response to mouse movements until the button was released) so children had no opportunity to view all aspects of the representation before inadvertently manipulating it; and, 2) the whole play area was mouse-sensitive so children would fail to focus on the representation handles. The interaction protocol did not direct children's attention towards the main elements of the symbolic interface representation of the concept, namely, the center and angle of rotation. It became clear that the interaction protocol was very closely linked to the style of manipulation. That is, in the example given above, although the intended manipulation style was D C M , the mouse interaction protocol did not support it. In fact, it was intuitively more supportive of a D O M manipulation style. In a manipulation style which promotes concept understanding, the mouse interaction protocol must draw children's attention towards the conceptual elements of the representation structure on the screen. I l l One way to resolve the confusion with respect to the transformation structures and direct children's attention towards the conceptual elements of these structures is to do the following. As soon as the player clicks on a transformation button the formal representation appears in the play area with a predetermined setting and at a predetermined location (see Figure 3.11). This allows some lag time for children to see the interface representation first. In addition, the only mouse-sensitive spots in the play area are the representation's handles. Since, in the case of rotation, there are only two mouse sensitive spots on the screen, children have no choice but to interact with only those two points. In their first encounter with the representation, they can click all over the screen, but they soon realize that they can only operate on two spots. In addition, the first time children encounter the representation, the two handles are blinking. This is intended to visually draw their attention to the two handles on the arc and reduce uncertainty as to what to do next, or where to click next. The same process of analysis was needed for the other two transformations, making sure that the interaction protocol supported D C M and directed children's attention towards the concept referents rather than the shapes. Many tests were performed to ensure that the protocol did not detract from the game-like quality of the environment. Due to space, this issue is not discussed in greater length. However, the importance of the need to pay attention to this aspect of the design of interactive multimedia learning environments for children should not be underestimated. 112 3.4.5 Modifying DCM to Promote Reflective Cognition The D C M interface style is intended to allow children to manipulate the formal representations of transformation concepts and acquire a more explicit and formal understanding of transformation geometry. Initially the D C M interface structures may seem unfamiliar to children requiring exertion of mental effort to discover the underlying functionality of the structures. However, continuous manipulation of these structures can make them seem "familiar" to children, and therefore communicate the message that children need not invest much mental effort to use the concepts. Salomon [1981] discusses the relationship between "the amount of invested mental effort" (AIME) and learning - the more AIME, the deeper the learning. He suggests that AIME depends on the perception of task requirements, "which in turn is influenced by past experiences and on learners' perceived self-efficacy" [p. 138]. He goes on to add that it is "critical to provide learners with experiences that change their perception of certain stimuli and increase their AIME" [p. 138]. If learners perceive a situation as "familiar and 113 undemanding", they process the messages communicated by the situation "mindlessly". Transformation geometry concepts have depth and need to be understood through reflection and in a progressively differentiated manner (see Section 2.2.10). Induction of "epistemic conflict" into the learner's mental schemas can lead to reflection and restructuring of the learner's conceptual structures (see Section 2.2.5). Super Tangrams accomplishes this epistemic conflict by gradually removing cognitive scaffolds (i.e., process of scaffolding). That is, to create a sense of unfamiliarity, components of each transformation representation are gradually removed (gradual visual feedback reduction). The following examples illustrate how this interface strategy is implemented and applied to the concepts of translation, rotation, and reflection. Translation Figures 3.12 to 3.14 demonstrate the visual removal sequence (scaffolding process) as applied to the concept of translation. In Figure 3.12, children encounter the most generalized notion o f translation without having to be concerned with the structural complexity or details of the concept. That is, they can simply drag the head or tail of the translation arrow, experientially observing the change in the position of the ghost image until it reaches the desired position. In this process children need not consciously know or reflect on what the length and direction of the translation arrow should be to move a shape to a desired location, although some children may choose to do so. Translating a shape, while the ghost image exists, does not require of children the investment of great mental effort [Salomon, 1981]. The ghost image, while allowing children to become 114 familiar with the concept of translation, makes it possible for children to process the symbol system at a shallow level, without needing to deeply elaborate its meaning in their minds. Figure 3.13 depicts the next stage in the induction of epistemic conflict into the learner's mental schemas or the scaffolding process. Here, the ghost image has been removed from the representation. This removal of the ghost image communicates two instructional messages to the learner: 1) that the learner is in unfamiliar territory and needs to exert fresh mental effort to figure out how to operate in the environment [Salomon, 1981], and, 2) that the existing representation is sufficient for accomplishing the goals of the game. At this stage, children can no longer depend on the visual feedback provided by the ghost image to know how to translate a shape to a desired location. To achieve a desired translation, children do not necessarily need to know what the length of the arrow should be. However, they must know what the direction of the arrow should be and where to place the head and tail of the arrow, as seen in Figure 3.13. To accomplish the above task requires investment of time and mental effort on the part of the learner. In trying to solve the puzzles in the game, not paying attention to the interface messages and simply using trial-and-error to translate the puzzle shapes can lead to frustration and prolonged effort - a consequence which many children may not be willing to accept. . 115 I " [ 1 1 r (— 1 1 — H i 1 V 1 jiHba Ii, I « -I j . -! I - I-j 1 » i i i 4 4 I I j ! S 5 •< ijB 11 lk 15 lfr F i g u r e 3 .12 T r a n s l a t i o n : s tage #1. F i g u r e 3.13 T r a n s l a t i o n : s tage #2. F i g u r e 3 .14 T r a n s l a t i o n : s tage #3. 116 Figure 3.14 shows the third stage in the scaffolding process. Here, the tail of the translation arrow is fixed to a predetermined coordinate position, and the middle and tail handles no longer exist. As in the previous stage, children must know what the direction of the arrow should be. However, unlike the previous stage, they must also know the exact length of the translation arrow and how to set it. As a result, children must use the background grid to carefully calculate the displacement the shape must make along the x-y coordinates, if they did not already do so in the previous two stages. This latter step of using the coordinates may be considered as the beginning of conscious, formal algebraic thought. Rotation Figures 3.15 to 3.17 demonstrate the visual removal sequence as applied to the concept of rotation. In Figure 3.15, children encounter the most generalized notion of rotation without needing to worry about the structural subtleties and details of the rotation concept. That is, they can adjust the angle of rotation, experientially observing the change in the position of the ghost image until it assumes the desired orientation; afterwards, they can adjust the center of rotation until the ghost image reaches the desired target location. In this process, children develop a notion of the existence of the two sub-concepts of angle and center of rotation that are involved in rotating a shape. However, as in the case of translation, they need not consciously know or reflect on what the angle of rotation should be, or how to determine the location of the center of rotation. Rotating a 117 shape, while the ghost image exists, does not require children to invest much mental effort to "elaborate" and understand at any depth the meanings embedded in this complex and abstract mathematical concept. Figure 3.16 depicts the next stage in the induction of epistemic conflict into the learner's mental schemas. Here, the ghost image has disappeared. As in the case of translation, children need to decipher the intended meaning of the instructional message. They can no longer rely on the visual feedback of the ghost image. At this stage, to rotate a shape to a desired location, children need to reflectively visualize and/or determine the required angle of rotation. Therefore, they must pay attention to the numerical value of the angle of rotation (i.e., the numerical tag displayed near the center of rotation). Once this angle is determined and set, the proper setting of the center of rotation is not very difficult. Upon rotation, the shape's vertex, which is connected to the tail of the arc, is transformed to where the head of the arc is. Children can use this visual information to experientially drag the center of rotation until the head of the arc reaches the desired target position. 118 IB W K ffi IN Figure 3.15 Rotation: stage #1. Figure 3.16 Rotation: stage #2. Figure 3.17 Rotation: stage #3. 119 In Figure 3.17, not only has the ghost image disappeared, but also the arc of rotation itself has faded. Now, children have to determine both the angle and center of rotation reflectively. In this figure, two pieces of information still remain: a numerical feedback indicating the current angle of the arc, and a line symbolizing the radius of the arc connecting a vertex of the selected shape to the center o f rotation. The position of the handle for adjusting the angle of rotation is moved closer to the center in order not to easily reveal the head o f the arc o f rotation. Determining where a selected shape is supposed to move, what the angle of rotation should be, and where the center of rotation should be placed are not trivial tasks and require hard work and a great deal of reflective mathematical thinking and means-ends analysis reasoning. Rotating shapes to desired target positions is very difficult at this stage, and even adults with substantial knowledge of transformation geometry may find it mentally taxing and quite challenging. (See Figure 3.17; try solving this puzzle using the rotation operation only.) 120 Figure 3.18 Reflection: stage #1. Figure 3.19 Reflection: stage #2. i ia_J— i M ie j K - 1 »- —1 7-9~~ i -p or'" f t t *—ms S ! j . — i— i~ — i — • i i i ! t 4 t 1 H i rte ti it i!s M Figure 3.20 Reflection: stage #3. 121 Reflection Figures 3.18 to 3.20 demonstrate the visual removal sequence as applied to the concept of reflection. These stages are very similar to those for translation and rotation. In Figure 3.18, children encounter the most generalized notion of reflection without needing to consider the details of the concept of reflection. In Figure 3.19 the ghost image is removed, and children can only rely on the visual information provided by the bisector line. In Figure 3.20, both the ghost image and the bisector line have disappeared. Consequently, as in the cases of translation and rotation, children encounter highly cognitively demanding scenarios in which a great deal of reflective thought and reasoning may be required. Reflective Direct Concept Manipulation (RDCM) In summary, the interface strategy above is an extension of the D C M interface style and can be referred to as Reflective Direct Concept Manipulation (RDCM). R D C M is intended to create a spiral process of action-reflection (see Section 2.2.7) in which children are moved through the stages of familiar to unfamiliar. This strategy gradually puts more cognitive responsibility on children, requiring greater AIME and "deeper mental elaboration" to progressively differentiate conceptual details of each transformation function (see [Salomon, 1979, 1981] for a discussion of shallow versus deep symbol processing). Additionally, this three-stage strategy, by having visual scaffolds at first, may leave a "cognitive residue" [Salomon, 1992] in children's minds, which by their subsequent removal trigger a need for reconstruction of the scaffolds in the 122 mind of the children (view Figure 3.21 from left to right). Figure 3.21 Three stages of R D C M applied to the representation of rotation. 3.4.6 Constraints: Further Promotion of Conscious Thought The application of the R D C M interface style is one approach for increasing reflective cognition. Other strategies can be employed to further promote conscious reasoning and thought. One of the objectives of Super Tangrams is to promote an understanding of the relationships among the different transformation concepts. Another objective is to help children discover the more subtle phenomena tacitly embedded in these knowledge organizations. For instance, rotating a shape both turns it and translates it. Although the D C M implementation of rotation provides a visual demonstration of these co-occurring phenomena, for some children this may not be readily apparent. Therefore, if children are given the option to use a combination of rotation and translation concurrently, they may 123 first use rotation to adjust the orientation of a given shape and afterwards simply use translation to move the shape to a desired location. All along they may not notice the relationship between these two concepts - i.e., that one can turn and slide a shape in place by using rotation alone. Providing children with only a subset of the three transformations (while the rest are disabled) creates new knowledge construction situations which may not easily be afforded otherwise. Situations created using such constraints are intended to put more demand on children to go beyond their comfort zone of knowledge and process the deeper aspects of the transformation concepts. The above strategy can be made more effective by arranging the puzzle shapes in ways which create educationally desirable situations. For instance, composite reflection is an important and difficult concept when learning transformation geometry. It refers to the transformation resulting from a sequence of reflections - reflections in parallel mirrors or reflections in intersecting mirrors. Using composite reflection, any transformation can be accomplished (e.g., a translation through twice the distance between two parallel reflections). If children are given only one transformation tool (e.g., reflection), the puzzle shapes can be arranged so that the puzzle can be solved only through the application of a sequence of reflections. Such a situation can highlight the relationship between reflection and the other two transformations in ways that may not be achievable otherwise. 124 Figure 3.22 Figure 3.23 Composite reflection, stage #1. Composite reflection, stage #2. Figure 3.22 depicts the initial arrangement of a puzzle where only reflection can be used. Figures 3.22 and 3.23 demonstrate one of the possible ways of moving the selected triangle from its initial position to its target location in the puzzle (i.e., where the ghost image is placed in Figure 3.23) using two intersecting reflections. As Grayson [1995] reports, even grade 10 students in England struggle with this form of reflection. In solving such puzzles, children have to visualize an in-between reflection position from which they can transform the selected shape to its desired destination. In the case of the rhombus, positioning may require up to three consecutive reflections to reach the desired target location. 3.5 Instructional Module In order to actualize the Challenge-Driven Learning Model, three of the four design elements have so far been discussed. The first element, selection of the tangrams puzzles, 125 provided for a situated activity in which transformation geometry could be performed. The second element, the operationalization of the activity into a game, provided for a motivating environment to entice children into learning mathematics. The third element, epistemology at the interface level, dealt with issues of concept manipulation, concept representation, and concept elaboration. This third element provided for strategies that would continually increase the mathematical challenge, thus causing epistemic conflict and the need for reflective thought and restructuring of mental schemas. The fourth element, the embedded instructional module, is described in this section. Figure 3.24 shows a pictorial representation of the intended relationship between the game module and the instructional module. • mm u 3 w S I . SB e CD Instructional Module F l o w ( \ r \ Game Game Module Module ^ ) o o cu .0* Instructional Module Game Module o .£< u Instructional Module Increase i n M a t h e m a t i c a l C h a l l e n g e Figure 3.24 Relationship between the game module and the instructional module. 126 The double-headed arrow between the game module and the instructional module represents the reciprocal relationship between these two modules. As the need to overcome mathematical challenges in the game arises, children can go to the instructional module to discover what the deficiencies in their conceptual structures are and to fine-tune current knowledge and construct new knowledge. As children interact with the instructional module, it, in turn, communicates to them the ideas that need to be noticed, as well as how to view the mathematical concepts in the game. The game and instructional modules support each other and help children's knowledge construction in a reciprocal manner: the game influences what should be explored in the instructional module, and the instructional module influences what concepts should be paid attention to in the game to be more successful at advancing through the game. This reciprocity can lead to mental restructuring and formation of "proper" (i.e., conventional, desired, and/or formal) conceptual structures - i.e., newly acquired mathematical skills and knowledge. As the game advances, once this mental restructuring takes place, the learner can respond to the challenges that arise in the game and remain in flow. (See Salomon [1981] for an in-depth discussion of reciprocal-interactionist view of communication and learning; also see Norman [1993] for a discussion of restructuring and the need for having tools for exploration, comparison, integration, and formation of proper conceptual knowledge.). To provide a tool in which children can explore and form proper conceptual knowledge, the instructional module consists of three interrelated components: Concept Explanation, Guided Interactive Practice, and Cognitive Strategies Instruction. Children control whether to invoke the instructional module or not. Once a child invokes the 127 instructional module, he or she can spend as much time as desired interacting with the module. Children have different learning preferences and needs in terms of the way the instructional material is presented to them [Keef, 1987], so the instructional module presents all concepts using pictures, text and voice simultaneously. Concept Explanation This component provides explanations of transformation geometry concepts similar to ones found in mathematics textbooks. Children can activate pop-up windows containing diagrams and text describing transformation geometry concepts, such as the definition of a translation arrow and properties of rotation and reflection. Diagrams and text are accompanied by voice explanations of the displayed information. Children can invoke Concept Explanation windows any number of times and exit them at any time, even while a concept is being explained. Guided Interactive Practice This component serves as a coach for children. Children can activate sound icons that tell them what the interface representations in the game signify, how these interface representations work, and what issues are important to play the game efficiently. Simultaneously, while the sound icons are active, the Guided Interactive Practice component provides an interactive environment in which children can explore and practice the interface representations which are used in the game. The following is an example of an explanation of the translation arrow to which children can listen while simultaneously 128 operating on its concept representation on the screen: In the game, the translation arrow is shown as seen here. It has three sensitive points: the green dot at its head, the red dot at its tail, and the blue one in the middle. To change its length and direction, you can click on the green dot or the red dot and drag it. To move the arrow itself you can use the blue one. Try this now! Children can stay in this interactive practice component for as long as they like without losing any points in the game. Cognitive Strategies Instruction Rosenshine and Meister [1996] report that teaching students cognitive strategies has a significant positive effect on their performance in a given task. The Cognitive Strategies Instruction component provides children with tactics on how to use their mental resources to solve tangrams puzzles more effectively. Children can activate sound icons that tell them what to do and what not to do when solving tangrams puzzles. The following excerpt provides a sample strategy to which children can listen: For every puzzle, before you do anything, first look at the puzzle very carefully. And then try to see in your head where you think each piece would fit. This will improve your power of visualization, and, as you solve more puzzles, this will be easy to do. 3.6 Super Tangrams: Integration of Design Issues So far, Super Tangrams has been described in a piecemeal manner to highlight its specific design features. This section provides an integrated description of Super Tangrams and summarizes many of the points in earlier sections. 129 Super Tangrams is intended to be an artifact that is attractive to children, that engages them in the transformation geometry activity, that is conducive to a both reflective and experiential cognition, that acts as a cognitive tool for children to construct explicit, formal knowledge of transformation geometry, that is conducive to experiencing flow in learning, and that promotes a sense of enjoyment for those engaged in the activity. The Super Tangrams screen (see Figure 3.25) is divided into five main areas: play area, buttons panel, information panel, score board, and music volume control panel. The play area has a coarse grid in the background. To transform a shape, children must first select the shape by clicking on it. The selected piece has a unique pattern. Then, they select an operation (Slide, Turn, or Flip) from the buttons panel. This selection causes a transformation representation and a ghost image of the selected shape to appear in the play area. Rather than moving the shape, children directly manipulate the handles on the transformation representation. When a child has the representation parameters set as desired, he or she clicks the GO button and the selected shape moves into the ghost image spot. The UNDO button reverses the last committed transformation. Super Tangrams has 40 tangrams puzzles which are divided among and spread over three levels: 14 puzzles in Level 1,11 puzzles in Level 2, and 15 puzzles in Level 3. In order to allow children to interact with the transformation geometry concepts in a progressively differentiated manner while causing cognitive dissonance, the components of interface representations are gradually removed over these three levels of the program, as described in the previous sections. Children progress through these puzzles (and levels) sequentially. Level 1 starts with relatively easy puzzles to solve. It is intended to 130 help children to be successful at the game initially and learn the basic operations of both the game and the interface representations. Level 2 becomes more difficult requiring children to pay greater attention to the underlying mathematical concepts. Level 3 is very complex and requires quite a sophisticated understanding of transformation geometry. In Level 1, children can see a ghost image of the shape being transformed. The first 8 puzzles in this level are considered easiest to solve. These puzzles are selected and arranged to help children familiarize themselves with the new concepts. In three cases, single-shape puzzles are used so that children can learn the basic operations of translation, rotation and reflection without needing to worry about solving complex tangrams puzzles. However, towards the end of the first level, both the puzzles and the sequence of operations needed to solve them become more complex. For instance, in Puzzles 13 and 14, composite reflections are introduced in which an understanding of the unique features of reflection, as well as some knowledge of symmetric and asymmetric shapes, is supposed to develop (see Figure 3.25). This is accomplished by restricting the accessibility of translation and rotation to force children to explore alternative ways of solving the puzzles. The initial positions of the geometric shapes in each puzzle are carefully chosen to create different degrees of challenge. 131 Figure 3.25 A screen of Super Tangrams. In Level 2, children do not see the ghost image anymore. This means that for each transformation they need to understand additional new concepts. For instance, whereas in Level 1 many children in their rush to play the game may fail to understand the role that the translation arrow plays in transforming a shape, in Level 2 they cannot advance through the game until they discover the relationship that exists between the direction of the arrow and where a shape ends up - even if this discovery is initially only tacit. Similarly, for rotation, children quickly realize that unless they calculate rotation angles carefully, it may take them an extended period to rotate a shape to a desired target location. Since determining the appropriate setting of angles can be difficult for children 132 (especially if they have not learned about angles previously), the possibility of their advancement through the game is reduced unless they start using the instructional module. In Level 3, children need to know all the formal concepts related to the three transformations to solve the puzzles. Transforming any shape requires an understanding of how to use the coordinates in the play area. For instance, to translate a shape, children have to count the grid spaces on the screen to set the length and direction of the translation arrow; to rotate a shape, not only do they have to determine the angle of rotation, but also they have to determine approximately where the center of rotation should be placed. Similarly, to reflect a shape, not only do they have to visualize a reflected form of a shape, but also they have to determine the exact location and placement of the line of reflection, including its angle and distance from the shape being reflected. Upon starting Super Tangrams, a musical theme is played and children hear a "Hello", welcoming them to the program. Every puzzle has a name and a maximum score. This maximum score depends on the level of difficulty of the puzzles. Each puzzle can be solved using a minimum number of transformations or moves (called par). A child's score for a given puzzle is a function of the number of transformations he/she uses to solve the puzzle. A child who meets the minimum par set for a puzzle gets the maximum score assigned to that puzzle. Use of extra transformations costs the child some points and lowers the score. The obtained score for a finished puzzle is added to a total score and accumulates as a child advances through the game. Puzzles in Level 3 have much higher 133 maximum scores than those in Level 1. A timer shows elapsed time for every puzzle; however, the amount of elapsed time does not affect the children's score. Puzzle shapes are decorated with colorful patterns and popular cartoon characters. The patterns of puzzle shapes vary from one puzzle to the next. Moreover, the colors and patterns of the play area and other panels change from one level to the next. Popular music is played in the background while the students solve each puzzle. Different puzzles have different pieces of background music. As long as a child has not finished a puzzle, a piece of background music is replayed. The volume of the background music can be adjusted up, down, or off altogether. Each transformation has a unique sound effect associated with it. Upon completion of each puzzle, a decorated window pops up. This window contains an elaborated cartoon character holding a sign containing an encouraging, positive comment in a variety of flashy colors and patterns as well as the score obtained for solving the puzzle. This sign is accompanied by a human voice stating the comment that is written on the sign. Some of the typical comments include "Fantastic!," "Super Job," "Wow!!," "Simply Splendid," "Well Done!," and "Awesome". Upon exiting the game, children hear an echoing "G' Bye!" voice. Upon restarting the program, children can either go to the last puzzle they reached the previous time, or they can choose one of the earlier puzzles. To assist children to overcome the challenges faced in the game and stay in flow, Super Tangrams provides them with an instructional module (called Learn). The presentation of concepts in Learn is closely tied not only to the mathematical concepts in the game but also to the goals of the game. Learn can be invoked (by clicking on the Learn 134 button) at any point during a playing session. Figure 3.26 shows the opening screen of Learn where children can click on the left-hand-side menu items to access either the "Tips on How to Play" or one of the transformation items. Upon invoking the instructional module, the background music, if it is already on, stops playing. Background music resumes playing upon returning to the game module. Clicking on the "Tips on How to Play" item invokes the Cognitive Strategies Instruction component, as discussed previously (see Figure 3.27). Clicking on any of the SLIDE, TURN, and FLIP items takes children to screens (e.g., see Figure 3.28 for the reflection transformation) where they can invoke the Direct Concept Explanation and Guided Interactive Practice components of the instructional module. Figures 3.29 and 3.30 show two Direct Concept Explanation screens. Figure 3.31 shows a Guided Interactive Practice screen. The interface representation of the mathematical concepts as well as the sound icons in the Guided Interactive Practice component of Learn change for each level of the game to correspond with the knowledge requirements of the game at that - stage. In addition, the sound icons do not simply explain mathematical concepts in a dry manner; in some instances, children are encouraged to learn a concept by statements such as: "Level 3 is for grand masters of Super Tangrams. It is a tough level; so do not become frustrated easily. Since you have reached Level 3,1 assume you must have mastered your angles in Level 2. Now listen to sound icon number 2." 135 Figure 3.26 Opening screen of Learn. 1 | gj||||||||| : p f f s i f &£ „_, ". L' mm S 3 ESI. / ^ B i l ^ i N M m i i n i i s c i a K ! \m s i Figure 3.28 Menu for invoking the instructional screen for reflection. Figure 3.27 Cognitive Strategies Instruction screen. " •3 is When you look at a rotated shape, all its lines have 11 turned by the amount of the angle of rotation. This figure shows the number of each line. | Ik i i 6*; .,. ... * 1 1 r — •= i , a Figure 3.29 A Concept Explanation screen for rotation. Figure 3.30 A Concept Explanation screen for reflection. Figure 3.31 A Guided Interactive Practice screen for translation in Level 1 of the game. 136 Chapter 4 Research Method Traditionally, the method for evaluating the usefulness of electronic learning tools has been to conduct experimental, quantitative studies that measure students' knowledge of a subject before (pre) and after (post) using such tools. Following these measures, statistical analyses are performed and generalizations on the relative effectiveness of the tool are made. Some researchers argue, however, that the results which are gathered from this kind of investigation are not sufficient for a better understanding of how computer-based learning environments should be designed (e.g., [Neuman, 1989]). One of the reasons for this insufficiency is the lack of an in-depth understanding of why something is effective and/or ineffective, and how it can be improved. For instance, Neuman [1989], a researcher in the field of computer-based instruction and learning, voices the concern that it is difficult to know and predict what the effects of system components will be on learning. Neuman suggestes that researchers need to go beyond general effectiveness measures, probe the minds of users, and examine their perceptions of and feelings about these educational tools. The evaluation and assessment of the usability of HCI artifacts has also been influenced and dominated by the research methods of experimental psychology [Carroll, 137 1989]. These evaluations primarily aim at gathering quantitative data on user performance indicators such as error rates, the amount of time it takes to learn a system, the number of keystrokes required to perform a task, and so on. However, HCI, evaluated by these methods, has been criticized and questioned. A psychologically motivated evaluation may not be able to penetrate into and explain more subtle human-computer interaction factors [Bannon & Bodker, 1991; Carroll, 1989; Kuutti, 1995]. In an analysis and review of the evolution of the field of HCI, Carroll [1989, p. 51] states that: [I]t may be more important to know how people approach a task, or how they feel about their performance, than it is to know how quickly or successfully they perform. . . . The focus on quantitative differences inclined investigators to focus on the simplest of performance measures. . . . Such work could not answer the underlying "why" questions that motivated human factors evaluation in the first place; it could not provide the depth of understanding necessary to help guide the design of new software techniques and applications. A number of researchers [Guba & Lincoln, 1982; Patton, 1990; Peshkin, 1993] argue that, despite its rigor and success in natural scientific inquiry, scientific-rationalistic theory, and experimental investigation and quantitative data gathering have weaknesses and can not adequately address all educational research concerns. Moreover, these researchers note that quantitative methodology is not necessarily the only, or best approach for examining some aspects of human cognition, such as beliefs, thoughts, perceptions, and motives. This inadequacy, however, does not imply that quantitative methods should be abandoned altogether [Schumacher & McMillan, 1993; Patton, 1990]. Each research method has its strengths and weaknesses. Patton [1990] suggestes that since there are no perfect methodologies to investigate reality, it is the duty of the 138 researcher to reject 'methodological orthodoxy' and instead, make appropriate pragmatic decisions about the choice of research methodology - a choice dependent on the purpose of the inquiry, the question(s) being investigated, and the resources available (also see [Salomon, 1991]). Patton further suggested that researchers should not be bound by a specific research method, but mix different aspects of different methodologies, as deemed appropriate for the study. Another paradigm in research methodology is the qualitative method. In brief outline: Qualitative methods permit the evaluator to study selected issues in depth and detail. Approaching fieldwork without being constrained by predetermined categories of analysis contributes to the depth, openness, and detail of qualitative inquiry. Quantitative methods, on the other hand, require the use of standardized measures so that the varying perspectives and experiences ofpeople can be fit into a limited number of predetermined response categories to which numbers are assigned. . . . The advantage of a quantitative approach is that it's possible to measure the reactions of a great many people to a limited set of questions, thus facilitating comparison and statistical aggregation of the data. This gives a broad, generalizable set of findings presented succinctly and parsimoniously. By contrast, qualitative methods typically produce a wealth of detailed information about a much smaller number ofpeople and cases. This increases understanding of the cases and situations studied but reduces generalizability. [Patton, 1990, pp. 13-14] Patton [1990, p. 44] further suggestes that Qualitative methods are particularly oriented toward exploration, discovery, and inductive logic. Moreover, Unlike quantitative research, which assumes that phenomena must be broken into their component parts - variables - in order to be studied and understood, qualitative research seeks to understand the ways in which the parts come together to form a whole, a whole that is greater than the sum of those parts. [Whitt, 1991, p. 407] Qualitative research is concerned with understanding the social phenomenon from the participants' perspective. 139 [Schumacher & McMillan, 1993, p. 373] Qualitative methods are considered to be superior to other research methods for achieving in-depth understanding of. . . complex processes, such as learning or change. .. . Studies of process ask how something happens and portray the dynamics of action and change, including the perceptions, experiences, and interactions ofpeople involved in the process. [Whitt, 1991, p. 409] Levine [1996] suggests that there is no single, simple path for an evaluator to follow, and that meaningful evaluation should use "an eclectic approach to evaluation grounded within a defined conceptual framework" [p. 266]. Levine [1996, p. 266] adds that an eclectic approach requires the integration of various methodologies (experimental, quasi-naturalistic, and naturalistic) to be implemented in different educational settings (artificial or laboratory and authentic or classroom), in order to gather significant and relevant data (quantitative and qualitative, process-based and produce-based [sic], segmented or holistic), for different purposes (formative, summative, . . .), while using different instruments for data collection (observations, tests, questionnaires, etc.) from different users. The main questions addressed by this research are: 1. Is Super Tangrams usable in terms of its mathematical content and satisfying children's affective needs? 2. Does Super Tangrams help children learn formal concepts of transformation geometry? 3. Is Super Tangrams' Reflective Direct Concept Manipulation interface style effective in promoting reflective cognition and better learning? Would a Direct Object Manipulation interface style accomplish the same goals? 4. What are children's perceptions of the Challenge-Driven Learning Model implemented in Super Tangrams? Does the model promote experiences of flow in learning? 140 5. Does Super Tangrams satisfy children's psychological needs and promote enj oyment while learning mathematics? 6. Do specific design features (e.g., background music) of Super Tangrams influence children's attitudes towards the program? If so, how and why? 7. Levels 2 and 3 in Super Tangrams contain mathematical challenges that extend beyond what children normally study in grade-6 mathematics. Does the high degree of mathematical content and difficulty detract from Super Tangrams' game-like feeling or make children want to avoid engaging in the activity? To investigate these questions, a multi-method research design was used. The research design included different types of data-collection instruments and procedures: children's written journals, a transformation geometry achievement test, a design questionnaire, direct observations, and individual interviews with children. The elements of the design as well as its overall effectiveness were investigated both qualitatively and quantitatively. 'Cross-method triangulation' was performed to find regularities in data, situations, and methods to determine if the same patterns kept recurring (for a discussion of 'triangulation', see [Schumacher & McMillan, 1993, p. 498]). The research had a preliminary evaluation phase and a summative evaluation phase. In the preliminary phase, a prototype of Super Tangrams, similar to the design described in Chapter 3, was evaluated in practice using qualitative ethnographic methods in a naturalistic classroom setting. This evaluation provided a non-formal preliminary assessment of the usability of Super Tangrams in terms of its mathematical content and children's reactions towards its design, helped identify design deficiencies leading to fine-tuning the system, and helped clarify the design of the summative evaluatioa phase. In the 141 summative phase, Super Tangrams was evaluated quantitatively and qualitatively to determine its instructional and motivational effectiveness. In terms of learning and design features, the quantitative portion of this phase was intended to provide a comparison of the main design along with five variations of Super Tangrams. (The main design of Super Tangrams was described in Chapter 3.) This comparison was based on relative measures of the main design's pedagogical effectiveness and measures intended to provide an understanding of "the origin of effects rather than the effects themselves" [Levine, 1996, p. 266]. The qualitative portion of this phase was intended to capture children's voices, provide explanations for some of the 'hows' and 'whys' of the experimental findings, and cross-validate the quantitative findings. The remainder of this chapter consists of two main sections which describe the preliminary and summative evaluation phases. 4.1 Preliminary Evaluation of the Design In late October of 1995, a prototype of Super Tangrams was introduced to a grade 6/7 French immersion classroom at Elementary School A in Vancouver, British Columbia. This class had participated in the initial investigations of this research during the previous school year (see Section 1.2). The prototype of Super Tangrams that was used resembled 142 the one described in Chapter 3, but without the instructional module.1 The class had 25 students, half of whom I knew from the previous year. Students' knowledge of transformation geometry was minimal, according to their teacher. Moreover, a majority of children were not fond of mathematics, as evidenced by remarks such as "Math is boring!". I visited the classroom weekly. Each visit was for a period of one to two hours. During these visits children played Super Tangrams while I recorded observations of their interactions with the program. In each session, the teacher divided the students into two groups. One group worked on the computers for 15 to 20 minutes while the other group continued with their regular mathematics activities. The groups would then switch places. During a group's use of the computers, three students were allocated to each computer. Students took turns controlling the mouse, but most of the time they consulted each other on how to solve the puzzles. All students were given small journals and were asked by their teacher to record their interactions with Super Tangrams. In their journals, children described what they felt they learned from the program, and what their feelings and opinions were about Super Tangrams.2 After three weeks, based on on-site observations, a few modifications were made to Super Tangrams to improve its mouse interaction protocol. Shortly thereafter, in mid-November, Super Tangrams was reinstalled in the classroom. 1 This module was still under construction and testing. 2 The practice of writing comments in personal journals started from the time that E - G E M S began conducting research in this classroom. Children were required by their teacher to record all their computer activities in their E - G E M S ' journals. 143 Two weeks later, as children advanced through the puzzles in Super Tangrams and reached Level 2 where the ghost image disappeared, many students started experiencing difficulties with the program. I hypothesized that the difficulties children were experiencing were due to the lack of a mechanism (e.g., an embedded instructional module) to help children cope with the mathematical challenges of the game. It was decided to investigate whether adult mediation would assist children in coping with Super Tangrams. Twenty-one students were left on their own to continue using Super Tangrams. Four students (one grade-6 boy, one grade-6 girl, one grade-7 boy, and one grade-7 girl), considered by their teacher as low or marginal achievers in mathematics, were selected to receive adult mediation3 while using Super Tangrams. This mediation was done once a week for a period of 20 minutes and consisted of cognitive strategies guidance on how to solve the puzzles in the game. This phase of the study was crucial in understanding how to design the embedded instructional module so that it would perform some of the duties of an adult mediator.4 Additionally, I decided to test the effectiveness of the embedded instructional module compared with an adult mediator, as described in the next section. All students regularly played the game once a week until late February, 1996. 3 I was the mediator, and sessions were videotaped. 4 Initially this module was called "Help" and was mostly text based. The students at School A would ignore it because they did not like to read text, and it did not seem "cool" to use help when playing a game. Later on, this module was renamed "Learn". 144 4.2 Summative Evaluation of the Design 4.2.1 Participants One hundred and sixteen grade-6 students (11- and 12-year olds) from Elementary School B, located in an upper-middle-class neighborhood of Vancouver, British Columbia, participated in the summative phase of the study. These students were from five different classes and represented the entire body of grade-6 students at the school. Table 4.1 shows the distribution of students among classes. Class # N Grade Female Male 1 29 6 12 17 2 15 6/7 1 9 6 3 15 5/61 8 7 4 29 6 15 14 5 28 6 14 14 Total 116 2 58 48 1 Only grade-6 students participated in the study. 2 Initially, there were 118 students. But soon after the students were assigned to different conditions and the study had started, two students from classes #1 and #5 dropped out - one left the country and one became sick, reducing the participants to 116. Table 4.1 Distribution of students among classes. Participating students originated from a mix of cultural backgrounds, but most were Caucasian. None of the participants had used Super Tangrams before. About half of the students at School B had participated in an E-GEMS study the year before [Super et al, 1996]. All students were able to use computers, as attested by their teachers. The school had a computer laboratory with different types of educational software that all students used during their computer sessions. As would be expected, students were observed to 145 have a wide spectrum of mathematical knowledge and abilities. Four children in Class #1 were considered special-needs students (Attention Deficit Disorder [ADD]). One teacher recommended exclusion of these four children from the study, but in consultation with the students' other teacher, it was decided to include the entire class in the study. Classes #2 and #3 were mixed-grade classes; only grade-6 students from these classes participated in the study. In addition to the five classes outlined above, a sixth grade-6 class served as a control group. The control group was included to determine if a repetition of the measures would influence children's knowledge of transformation geometry, and to gauge the stability of the Transformation Geometry Test (described later in this Chapter). The control group was a grade-6 class from Elementary School A. (The other grade-6 class was the French immersion class that had participated in the two-year on-going investigative phase and preliminary evaluation phase.) The size of the control group was 20 students (8 females and 12 males). Approximately three-fourths of the class were ESL students from Taiwan, Hong Kong, and Singapore. 4.2.2 Materials To obtain comparative measures of Super Tangrams' effectiveness, six different prototypes of the software were designed. Prototypes varied in terms of two factors: instructional package and embellishments. Instructional package refers to design features that were intended to address the learning needs of children. Embellishments refer to design features that were intended to enhance an instructional package with respect to fun 146 and amusement. There were three instructional packages: Reflective (R), Formal (F), and Intuitive (I). Instructional packages differed in terms of features such as their interface styles and embedded instructional modules. Embellishments included auditory, visual, and reward features such as background music; sound effects; graphics, colorful patterns, and cartoons; encouraging comments; and a scoring system. The six prototypes that were designed resulted from either adding embellishments (+E) to each instructional package (e.g., R+E) or not (e.g., R-E). The six prototypes were: R+E, R-E, F+E, F-E, I+E, and I-E. All six prototypes were called Super Tangrams, so as not to inadvertently bias participant children towards any particular prototype. The six prototypes are described in detail below. R+E (Reflective Instructional Package with Embellishments) This prototype was identical to the Super Tangrams design described in Chapter 3. That is, its instructional package was the full implementation of the Challenge-Driven Learning Model (see Section 3.1) including all embellishments. This prototype was intended to put children in direct interaction with explicit, formal representations of transformation geometry concepts, as well as to promote and afford reflective cognition. It used an R D C M (Reflective Direct Concept Manipulation) interface style (see Section 3.4). That is, geometric shapes were transformed by manipulating the symbolic representations of the transformation concepts, and there was gradual removal of representation components. Additionally, R+E placed constraints on children's use of the transformations. The instructional module of this prototype had three levels. This 147 prototype included 40 sequentially-arranged puzzles and was the main design whose effectiveness was compared with the other five prototypes. R - E (Reflective Instructional Package without Embellishments) In terms of the instructional package, this prototype was identical to R+E, but without the embellishments. Puzzle pieces had a plain-light-brown color, and their color did not vary across different puzzles. The colors and patterns of the play area and other panels were all black or gray (see Figure 4.1). Programming of the non-colored version of the instructional module (Learn) was not completed in time to be included in the study, so the instructional module in this prototype was identical to that of R+E (i.e., it was colorful and included sound effects). Figure 4.1 A snapshot of an R-E prototype screen. 148 F+E (Formal Instructional Package with Embellishments) This prototype was intended to put children in direct interaction with explicit, formal representations of transformation geometry concepts. It used a D C M (Direct Concept Manipulation) interface style (see Section 3.4). That is, geometric shapes were transformed by manipulating the symbolic representations of the transformation concepts. The F+E prototype did not include design features intended to promote reflective cognition. Whereas R+E had 3 levels over which the visual feedback of the interface was gradually reduced, the F+E prototype did not include any levels. That is, the interface remained the same throughout the game, and it was identical to the interface style of R+E in Level 1 (i.e., the ghost image was always present). This prototype did not place constraints on children's use of the transformations (i.e., transformation operators were never disabled and were concurrently accessible to children in all puzzles). The F+E prototype did not have a par mechanism, so children's scores were not dependent on the number of moves they made to solve the puzzles. The score for each puzzle depended on the level of difficulty of the tangrams puzzle. There was no Hint or Snap-in-Place feature (cf. Section 3.3.1), as it was thought that these would make the game too easy to play. The instructional module was identical to the one included in R+E, but since there was only one level, the contents of the Guided Interactive Practice component remained constant throughout the entire game. This prototype included 83 sequentially-arranged puzzles. 149 F-E (Formal Instructional Package without Embellishments) In terms of the instructional package, this prototype was identical to F+E, but without embellishments. Similar to the R-E prototype, puzzle pieces had a plain-light-brown color, and their color did not vary across different puzzles. The colors and patterns of the play area and other panels were all black or gray (see Figure 4.1). I+E (Intuitive Instructional Package with Embellishments) This prototype was intended to provide children with an environment in which they could solve tangrams puzzles intuitively. Transformation geometry was an implicit part of the game. The I+E prototype was not designed to support formal understanding of transformation geometry, nor was it intended to promote reflective cognition with respect to the embedded mathematical topic. I+E used a D O M (Direct Object Manipulation) interface style (see Section 3.4). That is, geometric shapes were transformed by directly manipulating the shapes themselves. In terms of embellishments and total number of puzzles, the I+E and F+E prototypes were identical. In I+E, the desired outline was presented as a rusty red area in the centre of the screen, with the available shapes scattered around the periphery (see Figure 4.2). 150 Figure 4.2 A snapshot of an I+E screen showing a horizontal flip. Buttons on the side allowed children to select drag, clockwise rotate, counterclockwise rotate, horizontal flip, or vertical flip mode. The mode chosen was indicated by the mouse cursor. In drag mode, the user could simply drag the shape to the desired location. In rotate mode, clicking on a shape caused the shape to turn 22 .5° 5 (clockwise or counter-clockwise). In flip mode, clicking on any shape caused it to flip over (horizontally or vertically). The shapes did not lock into place, but behaved much like cardboard pieces. The sound effects for the transformations were different from those of R+E and F+E prototypes - drag produced a hissing sound, every 22.5° clockwise or counter-clockwise turn produced a ticking sound, and horizontal and vertical flips produced a "boing" sound. Since in I+E transformation representations were not explicit and formal, it did not include an instructional module. 5 The resolution of the rotation, 22.5°, was the largest setting that permitted all of the required variations to solve the given puzzles. 151 I-E (Intuitive Instructional Package without Embellishments) In terms of the instructional package, this prototype was identical to I+E, but without embellishments. Similar to the R-E and F-E prototypes, puzzle pieces had a plain-light-brown color, and their color did not vary across different puzzles. The colors and patterns of the play area and other panels were all black or gray. Table 4.2 summarizes the features of the six prototypes of Super Tangrams described above. Prototype Interface Style Math Challenge Increases Enforces Constraints Includes Hint & Snap-in-Place Feature Includes Instructional Module Number of Puzzles Includes Embellishments R+E1 RDCM yes yes yes yes (3 levels) 40 yes R-E RDCM yes yes yes yes (3 levels) 40 no F+E DCM no no no yes (1 level) 83 yes F-E DCM no no no yes (1 level) 83 no I+E DOM no no no no 83 yes I-E DOM no no no no 83 no R+E was the main design of Super Tangrams described in Chapter 3. Table 4.2 Features of the six prototypes of Super Tangrams in the summative evaluation phase. 4.2.3 Design of the Summative Evaluation The summative evaluation was designed to serve several purposes. In terms of children's knowledge acquisition, the evaluation was intended . 1. to determine which prototypes of Super Tangrams were the most effective; 2. to determine the effect of embellishments on children's learning; and, 152 3. to investigate whether the instructional module was able to support children's knowledge construction process, or whether adult mediation was also needed. In terms of motivation and children's attitude towards, and perception of, the design, the evaluation was intended to investigate 1. children's affective response to the R+E prototype and its design features; 2. whether the R prototypes (i.e., R+E and R-E) afforded reflective cognition as intended; 3. children's reactions, opinions and feelings about the Challenge-Driven Learning Model; 4. what children thought about design features such as embellishments and the instructional module; and, 5. whether the R+E prototype afforded flow experiences in learning. To investigate children's knowledge acquisition, a "quasi-experimental nonequivalent pretest-posttest group design" was chosen [Schumacher & McMillan, 1993, p. 316]. Figure 4.3 shows a diagram of this design. The quasi-experimental nonequivalent pretest-posttest group design is "very prevalent" in education research [Schumacher & McMillan, 1993, p. 316]. In this design, intact, already established groups of subjects are used for research studies. The groups receive a pretest, treatment, and posttest. For this research, seven groups were established - six for the different prototype conditions, and one for the mediated condition. Table 4.3 provides a list of all these groups. 153 Group Pretest Treatment Posttest Gl > PR ^ TI ^ PO G2 '. > PR . 5> T2 => PO Gn ! > PR : > Tn : > PO Figure 4.3 A nonequivalent pretest-posttest group design. Group Treatment gR+E+M R+E prototype + adult mediation gR+E R+E prototype gR-E R-E prototype gF+E F+E prototype gF-E F-E prototype gl+E I+E prototype gl-E I-E prototype Table 4.3 List of participating groups by treatment. Since this research involved an extended period of working with school children, ethical and logistic considerations made it necessary to conduct this phase of the study with entire classes. Of the seven groups, three were the same as Classes #1, #2, and #3. The other four groups were established by randomly splitting Classes #4 and #5 into two groups each. Afterwards, the seven groups were randomly assigned to the seven different treatment conditions. Since classes were nearly equivalent on background characteristics, it was reasonable to assume that "the selection differences probably would not account for" later achievement results [Schumacher & McMillan, 1993, p. 318]. Furthermore, to compensate for the lack of total randomization, analysis of covariance was used (using the pretest scores as the covariate) to adjust the groups statistically [Schumacher & McMillan, 1993]. Table 4.4 lists the groups according to their treatment, the number of 154 students in each group, and the class from which they were drawn. Group N Class gR+E+M* 29 1 gR+E 15 2 gR-E 15 3 gF+E 14 4 gF-E 13 5 gl+E 15 4 gl-E 15 5.. Because of its size, this group was divided into two subgroups, which received the same treatment. Table 4.4 List of participating groups by class. Students in each group wrote a transformation geometry test (described later in Section 4.2.4). Each group then received ten separate 35- to 40-minute sessions of treatment held on consecutive school days during students' regular mathematics periods. After receiving the tenth treatment, each group was administered the same transformation geometry test as posttest. The independent variables were the seven treatments. The dependent variable was knowledge of transformation geometry, from which statistical inferences were made with regard to the effectiveness of the different treatments. To investigate motivational issues (e.g., enjoyment) and children's reactions towards the design features (e.g., embellishments), immediately following the transformation geometry posttest, students in each group answered questions in a design questionnaire (described later in Section 4.2.4). Students' answers provided quantitative and qualitative indicators permitting assessment of children's feelings and opinions about the design prototypes and their features. 155 This portion of the study took ten school days (two weeks) for each class. The gR-E, gR+E, and gR+E+M groups received the treatments during the first two weeks, and the gF-E, gF+E, gl-E, and gl+E groups received the treatments during the following two weeks. Figure 4.4 summarizes the conditions of the design, where T G T stands for the transformation geometry test, and DQ stands for the design questionnaire. Time Group Pretest Treatment Posttest gR+E+M T G T — - R+E + Med — ^ TGT, DQ gR+E TGT — • R+E — • TGT, DQ gR-E T G T • R-E ^ TGT, DQ gF+E . TGT — "~ F+E m— TGT, DQ gF-E : TGT — • F-E — • TGT, DQ gl+E TGT — • I+E — *~ TGT, DQ gl-E TGT — ^ I-E — *~ TGT, DQ Figure 4.4 Pretest-posttest group design for the summative evaluation phase of the study. Shortly after writing T G T and DQ, I interviewed 20% of students from each experimental group. Hypotheses Before the summative evaluation phase, the following results were expected: 1. It was hypothesized that the R prototypes (i.e., R+E and R-E) would afford more reflective thought and result in more effective learning than the F (i.e., F+E and F-E) and I prototypes (i.e., I+E and I-E). It was also hypothesized that the F prototypes would be more conducive to learning than the I prototypes. 156 2. It was thought that the high degree of mathematical challenge involved in solving the tangrams puzzles in the R prototypes might detract from their game-like feeling and consequently have a negative effect on children's overall motivation or their attitudes towards the R prototypes (especially in Level 2 and 3). This concern was particularly true of the R-E prototype which did not include embellishments. 3. It was not clear whether the instructional module would be able to support children's knowledge construction process so that they would experience flow in learning. 4. It was expected that the group who received adult mediation (i.e., gR+E+M) would gain more knowledge than the group which only had the instructional module (i.e., gR+E). 5. Based on previous interactions with other children (see Section 1.2), it was expected that the prototypes that included embellishments would be liked more than the others. 6. It was not clear what the effect of embellishments would be on children's learning of transformation geometry. That is, the embellishments might improve student learning, or they might distract children and have a negative effect on learning. 4.2.4 Sources of Data Five sources of data were used to evaluate the effectiveness of different treatments and design features. Descriptions of these data sources are provided below. Transformation Geometry Test (TGT) The primary purpose of T G T was to provide a comparative measure of students' overall 157 understanding of transformation geometry, particularly their formal, explicit knowledge of transformation geometry, rather than their intuitive, implicit knowledge of the concepts. The secondary purpose of T G T was to provide measures of students' finer-grain understanding of some of the specific concepts outlined in Chapter 3 (e.g., composite reflection). These measures permitted comparison of the relative effectiveness of the different design prototypes. In the process of developing TGT, a number of standardized provincial mathematics achievement tests, appropriate for students in grades 4 to 8, were examined, but none met the requirements of this research. Students using the F and R prototypes were hypothesized to achieve a greater knowledge of transformation geometry than the available standardized tests measured. Moreover, standardized tests did not include questions that ranged over the entire spectrum of concepts covered by the R prototypes; nor did they include questions that would permit comparison and differentiation of the features of the different prototypes. Consequently, a special paper-and-pencil test was constructed.6 Questions in T G T were designed and presented in a format that would be understandable to anyone learning transformation geometry, not just to the users of the different prototypes of Super Tangrams. T G T was initially developed and modified through pilot-testing with two different classes.7 TGT's content and construct were validated and approved by two mathematics education experts in the Curriculum Studies 6 TGT was administered to 40 grade-9 and -10 students from one of the schools in the city of Richmond, BC, and the students managed to achieve a mean score of only 41% - an indication that the test was not easy, especially for grade-6 students. 158 Department at the University of British Columbia. To assess TGT's reliability, a coefficient of stability was calculated by applying the Pearson product-moment correlation coefficient to the pretest and posttest scores of the control group. A high reliability coefficient was obtained with r = 0.88 (see [Schumacher & McMillan, 1993] for a discussion of stability). This coefficient suggested a strong positive relationship between the scores on the two tests. Since the pretest and posttest were administered six weeks apart, it was assumed that consistency in scores was not influenced by memory or practice effects (see [Schumacher & McMillan, 1993, p. 229]). T G T was also assessed for its internal reliability. Split-half reliability coefficient on the pretest for the entire body of the students at School B was .78. The .78 coefficient, considering that the respondents were children, shows that the test had internal reliability and provided a fairly reliable measure of students' performance. T G T contained 51 questions with varying degrees of difficulty, ranging from very easy to very difficult8 (see Appendix A). Of the 51 questions, 49 were multiple-choice questions, and two required drawing solutions on paper. The questions were conceptually divided among four categories: translation questions (11 items in total9), rotation and angle questions (7 and 8 items each10), reflection questions (17 items in total11), and successive transformation questions (8 items in total12). 7 One of these classes was the grade 6/7 class at School A, and the other was a grade 5/6 class at another elementary school in Vancouver. 8 An example of a 'very easy' item is Question 1 which can be found in grade-4 textbooks; an example of a 'very difficult' item is Question 46 which requires knowledge of composite reflection and is usually taught in grades 10 or 11 [Grayson, 1995]. 9 These included Questions 1, 3, 4, 5, 6, 15, 20, 29, 49, 50, and 51. 1 0 These included Questions 7, 9, 14, 25, 28, 41, 48; and 21, 22, 23, 24, 33, 34, 35, and 36. 1 1 These included Questions 2, 8, 10, 13, 16, 17, 26, 27, 37, 38, 39, 40, 43, 44, 45, 46, and 47. 1 2 These included Questions 11, 12, 18, 19, 30, 31, 32, and 42. 159 Some sets of questions in T G T were grouped together to provide indicators of students' understanding, of the relationships among and equivalences between the different transformations. Figure 4.5 displays an example of a grouped set of questions. The logic for grouping questions was that a correct answer to all o f them would provide a better indicator of students' understanding o f the relationship among the different transformations. Grouping questions also acted to reduce the possibility that students answered a question by chance. 1 3 In addition to groupings o f questions, some questions were used as repeated measures for gauging students' understanding of specific concepts. For instance, Questions 20 and 50 (see Appendix A ) dealt with the same aspect of the concept of translation. Answering both questions would increase the reliability of measuring students' formal understanding of the vector notation o f translation. 1 4 1 3 Similar groupings include Questions 4-6, 7-12, 14-19, 21-24, 33-36, 37-42, 43-45, and 49-51 (see appendix A). . . 1 4 Other examples of repeated measure questions included test items 25 and 48; 38, 46, and 47. 160 I want to move this shape from position R to position B as shown here. Can I do this with Can I do this with Can I do this with Can I do this with Can I do this with Can I do this with Figure 4.5 An example of a grouped set of questions. Questions were weighted so that those which required a more sophisticated understanding of formal operations were assigned higher marks. The lowest mark for a question was zero for an "I don't know" response, an unanswered question, or a wrong answer. The highest mark for a question was an eight for the two questions that required drawing solutions on paper. Appendix B lists the assignment of marks to the different questions. Yes No I don't know 1 « > • • • Yes No I don't know • • • Yes No I don't know 2 turns? {J Q \J Yes No I don't know 2ni»5' • • • Yes No I don't know 1 flip and 1 slide? [ | | [ | | Yes No I don't know 1 turn and 1 slide? [ [ [ [ 161 Design Questionnaire (DQ) The purpose of DQ (see Appendix C) was: 1. to investigate children's reactions towards different design features such as the background music and the instructional module; 2. to assess children's affective response towards R+E; 3. to measure children's affective response towards R+E compared to the other prototypes; 4. to assess the extent to which R+E and other prototypes afforded reflective cognition; and, 5. to assess the Challenge-Driven Learning Model, as implemented in the R prototypes, and to investigate flow in learning while children used R+E. DQ consisted of Likert scale and Semantic-Differential scale questions (see [Schumacher & McMillan, 1993, pp. 244-246]). Questions required a range of responses from factual (e.g., "Did you always have the music on?") to perceptual (e.g., "How much did you like having background music in Super Tangrams?"). Based on verbal and written comments made by the children who participated in the preliminary evaluation phase of the study, some questions were phrased in a strong tone, using words such as, "love" and "hate" or "best" and "worst", in an attempt to reveal children's real feelings towards different issues. DQ was designed to collect both quantitative and qualitative data. The quantitative and qualitative data were intended to cross-validate each other. Many scaled questions terminated with an "explain why" prompt inviting open-ended written comments. This was the main source of qualitative data, which captured children's opinions and feelings and added more depth to their quantitative responses. 162 To assess design issues, children's responses were either counted or ranked on a 5-point Likert scale in which the strongly positive responses were assigned a value of 5, and the strongly negative responses were assigned a value of 1. In some cases, instead of asking children one question, DQ contained several questions asking about an issue from different perspectives (e.g., Questions 7, 8, and 28 deal with the instructional module). Children's related responses were combined to make a more robust overall measure of each issue. For instance, to assess affective response of children and affordance of reflective cognition, two overall indices were used. To assess children's affective response towards the design, nine questions in DQ were used to provide an affective index (i.e., Questions 5, 6, 13, 14, 16, 17, 35, 38, and 41). These questions dealt with whether children liked the program and its features, and whether they enjoyed their learning experience. To assess the extent to which the design afforded reflective cognition three questions in DQ were used to provide a reflective index (i.e., Questions 11, 12, and 24). Besides the reflective index, children's quantitative and qualitative answers with respect to the instructional module were used to assess affordance of reflective cognition. To assess the Challenge-Driven Learning Model and investigate flow in learning, children's answers to all questions in DQ were intended to provide holistic indicators for how children perceived the model and whether the learning experience was enjoyable (see Chapter 5). DQ had several versions. Variation among the questionnaires depended on the prototype that children used. For example, students whose prototypes included background music answered related questions, whereas other students whose prototypes 163 did not include background music did not. Versions of DQ varied only in terms of the number of the questions they contained. Versions intended for groups which used the R+E prototype contained the largest number of questions, and the version for the I prototype contained the fewest number of questions. Variations in DQ introduced differences in the time required to fill in the answers; however, the practical significance of these procedural differences was considered to be unimportant. Appendix C includes the lengthiest version of DQ. Videotape Recordings of Interviews Videotape recordings were made of memory-recall interviews with 20% of the students from each group. The interviews were intended to provide 1. a mechanism for cross-validating the experimental findings (see [Schumacher & McMillan, 1993, p. 498] for "cross-method triangulation"); 2. an examination of children's opinions about the different prototypes; and, 3. qualitative data in which children's thoughts and feelings were expressed holistically. To cross-validate the experimental results, students' tests and questionnaires were used to ask children open-ended questions bearing on their answers. Students were also shown several design prototypes, other than the one they had used during the experimental phase of the study, and asked to compare the different prototypes of Super Tangrams with what they had used. Typical questions that were asked during the interviews included: 164 1. You say that you think it is a good idea to have Learn in Super Tangrams. Doesn't it bother you that you have to interrupt playing the game to go to Learn? 2. Was Super Tangrams more a "thinking" game, or was it more a "guessing" game? How much of your time do you think was spent thinking about how to use the transformations? 3. You say that "Compared to other educational games you have played," you loved Super Tangrams. Why do you say this? What does it mean to love something? How did you come to love it? And so on. 4. You say that Super Tangrams has made you like motion geometry. Why? What did you think about motion geometry before? How do you feel about other math topics? And so on. 5. In an educational game, which one of the following is the most important thing to you? I mean which one would you rank as the most important thing you want to get out of it? Then the next, and so on. So, the items are: music, nice colors and graphics, learning, sound effects, a scoring system, and challenge. 6. Now you have seen these different prototypes of the game. Which one do you think you would like most? Why? Which one do you think would need more thinking? Why? Which one do you think you would learn more from? Why?. And so on. Direct Observations As children interacted with the different prototypes, written notes were made of their overall patterns of use and verbal comments. Particular attention was paid to children's use of the instructional module, their interaction with the different interface styles, how hard they had to work to solve the puzzles, what design features attracted their attention, 165 and whether at any stage during the study children lost interest in the activity. As in the case of interviews, direct-observation data provided a mechanism for cross-validation of different data sources. Log Files Logs of students' interaction with the prototypes were stored by each prototype of Super Tangrams. The programs stored data log files of the number of moves needed to solve each puzzle, the number of puzzles solved, the transformations selected, and the time taken to solve each puzzle. 4.2.5 Research Setting The summative phase of the research was conducted in a temporary computer room set up in a very small room15 (see Figure 4.6 for dimensions) adjacent to a grade 4/5 classroom at School B. A partition wall separated the research room from the classroom. Eight Macintosh Performa computers were arranged on three long tables, with two or three computers on each table. Figure 4.6 shows the setup of the room. 1 5 This was the only room that was available in the school to conduct the study. 166 2.5 meters j C o m p u t e r • C o m p u t e r I C o m p u t e r \ i C o m p u t e r I C o m p u t e r • Bench • T a b l e s C h a i r , P a r t i t i o n Figure 4.6 Diagram of the computer room where the summative phase of the study was conducted. Figure 4.7 A group of students working with a prototype of Super Tangrams. 167 All sessions of the study for all the groups as well as the interviews were conducted in this computer room. The students sat tightly next to one another (see Figure 4.7). No more than 16 students were in the computer room at any given time. 4.2.6 Procedures Consent forms including a description of the research project and the software were distributed among all students in each classroom. All children were required to obtain consent from their parents to participate in the study.16 Children were told that they were not obliged to participate in the investigation, and they had the option to drop out any time they wished during the course of the study. All participating teachers (iricluding the control group) agreed not to teach any topics related to geometry or transformation geometry concepts for a period of six weeks after the administration of the pretest. Students and teachers were told that different groups would use different prototypes of an educational software. To prevent possible biases towards the prototypes, neither teachers nor students were informed about how the prototypes differed. I visited each classroom and, in the presence of their teacher, asked students to write TGT, giving them 40 minutes to complete the task.17 Children were told that the results of the test would not affect their mathematics grade at school. Children were explicitly asked not to write the test by guessing answers to questions. The participants were told that the purpose of the study was to assist researchers understand how to design better 1 6 The students in the control group only needed consent to write the transformation geometry test. 1 7 Administration time of these tests varied depending on the time each class had its mathematics period. The gR-E group wrote the test at 9:00AM; the gR+E group at about 10:30AM; the gF-E and gl-E groups at 9:00AM; the 168 educational software for children. All groups completed the pretest in the last week of April, 1996. Teachers were asked to pair their students according to any criteria they saw fit. These pairs were assigned to work on computers together.18 All computers were equipped with two sets of headphones to enable students to individually hear the audio output of their programs. The headphones served a dual function: they provided students with a surround sound effect, and they prevented the audio output of several computers from becoming disruptive. Figures 4.8 shows two students working together during the study. Figure 4.8 A pair of students working together. gR+E+M, gF+E, and gl+E groups at 1:00PM; and the control group at 9:00AM. Administration time for the posttest was the same as the pretest. 8 This was mainly because of logistic reasons as well as the fact that in many computer labs children are paired to work together. Therefore, the setup was similar to what children were accustomed to doing. 169 I tried to maintain a casual, non-threatening environment during the entire study. As long as children's actions did not interrupt the work of others, they were free to ask questions from one another, change partners (which only 3 students did), or do anything that was within the bounds of healthy participation (the gl+E group, for instance, decided to turn the lights off while working on the computers.). During the first session, all groups were given a brief overview of what the goal and the rules of the program were and how to operate it. Thereafter, all non-mediated groups, when asking for adult assistance, were advised that answers to their questions could be found in the program. Students in the mediated group were allowed to ask me any question they wished. If students encountered any difficulties or problems, they could raise their hands, and I was available to answer procedural questions related to the operation of the program, but did not provide direct instruction on test-related content. The availability of a mediator encouraged some children to seek expediency in solving the puzzles. In order to get ahead of fellow students, in terms of number of puzzles solved and accumulated score, some children asked questions whose answers were readily accessible in the program. However, due to the large number of students and lack of sufficient time, I could not answer all questions and provide all children with individualized attention. I responded more to the needs of students who seemed less confident in their ability to solve puzzles on their own. The day after each group received its final treatment, in the presence of the teacher, and in students' usual mathematics classroom, students from each group completed T G T 170 as a posttest. Once again, I reminded the students not to guess at answers. In addition to and immediately after the posttest, children completed D Q . 1 9 A few days after each group wrote TGT and DQ, 20% of the students from the group were selected. Selections were made based on my perception of children's thoughtfulness and ability to communicate. Selectees were informed of their selection and were asked for permission to be individually interviewed. Before each interview, the purpose of the interview was briefly described. Interviews were conducted in the same room in which children had received their treatments. In order to make children feel at ease, a conscious effort was made to keep the tone of the interviews conversational and friendly. Interviews were videotaped, and each interview was 20 to 30 minutes in duration. (NOTE: Throughout the remainder of this document a number of phrases are used interchangeably: R prototypes refers to the R+E and R-E prototypes which included the R instructional package; R groups/students refer to all groups/students whose prototypes included the R instructional package. The same rule applies to the F and I prototypes and groups.) 1 9 There was a difference in the length of time between when each group did their pretests and their posttests. This was due to the logistics of the study and agreement with the teachers. The students in the gR-E, gR+E, and gR+E+M groups started their treatments two weeks prior to the other groups. Therefore, they finished two weeks earlier and wrote their posttests two weeks earlier. However, since for all the groups the posttests were administered the day after their last session, this two week gap does not seem to have significant practical implications in terms of the results of the study. 171 172 Chapter 5 Results and Discussion This chapter reports the results of the evaluative phases of this research. The results were derived and interpreted in the context of a 'triangulated' research methodology. Evaluation was conducted through an eclectic approach in which children's test scores, written comments, questionnaires, interviews, and researcher's observations were analyzed and interpreted by grounding them within the conceptual research framework discussed in Chapters 1, 2, and 3 (see [Levine, 1996] for a discussion of the 'eclectic approach to evaluation' of educational software). Throughout this chapter, an effort is made to cross-validate the different findings so as to reduce the amount of noise in the data. The results are presented in two major sections: preliminary evaluation and summative evaluation. Each section is divided into several subsections addressing the research issues discussed in Chapter 4. At the teginning of each section, the purpose of that section is briefly discussed. Before substantiating and reporting the findings of this research in the following sections, the major findings are reported here. Six major findings resulted from the two evaluative phases: 173 1. Students who used the R prototypes performed significantly better than the students who used the F and I prototypes. 2. Students who used the F prototypes performed significantly better than the students who used the I prototypes. 3. Addition of embellishments to an instructional package (R, F, or I) neither diminished nor increased achievement results within the respective instructional package treatment groups (i.e., embellishments did not interact with children's learning of mathematics); however, they did add affective flavor to the learning experience. 4. The R prototypes afforded a high degree of reflective cognition. Moreover, relative to the F and I prototypes, the R prototypes demanded the investment of a higher degree of mental effort. 5. The Challenge-Driven Learning Model was effective in assisting children learn transformation geometry concepts and enjoy their learning. The inclusion of the instructional module seemed to be able to support children's knowledge construction. Addition of adult mediation, in the case of the R+E prototype, did not affect student's mean achievement performance. 6. Considering its mathematically-challenging and knowledge-intensive content, children responded quite positively towards the R+E prototype. The results provided suggestive evidence that many children who used the R+E prototype experienced 'flow' in learning. (NOTE: This chapter quotes extensively from children's written and verbal comments. All written comments are verbatim and preserve children's own spellings.) 174 5.1 Preliminary Evaluation The purpose of the preliminary evaluation phase was to obtain an early assessment of the usability of Super Tangrams in terms of its mathematical content and children's reactions towards its design, help fine-tune the design, and provide a basis for conducting the summative evaluation. This section is organized into three subsections. First, children's initial reactions to Super Tangrams are reported. Second, children's descriptions of their learning are provided. Third, the preliminary findings are summarized and followed by a brief discussion of my observations of the children. 5.1.1 Children's Initial Reactions All children's journal entries were read in their entirety. Children's references to how they reacted towards Super Tangrams and their perceptions of it were highlighted. These highlighted references were further separated to distinguish the general comments (e.g., "I think this game is neat.") from the more specific comments (e.g., "I liked the graphics and sound effects."). These samples were reread and both typical as well as unique comments were chosen to be used as exemplars in this report. None of the following responses were prompted.1 1 See Section 4.1 for how journal entries were recorded by children. 175 Children's Comments 1. A grade-6 girl, who said she disliked mathematics, wrote: I thought that ST [Super Tangrams] was a great game. It was really interesting and it certainly did make math fun! But, at the beginning, it was difficult to catch on because to rotate or even to slide a shape was complicated, and there were a lot of steps to memorize! I liked the designes on the shapes and I liked turning the shapes because it was a challenge, and because watching all the angles flip by was neat. I never did catch on to that dum angle stuff, because there were so many, and there were special names for particular angles. The only one I recall is the 'right angle'. ST is good for learning angles, because if a kid played it often enough, the angles would eventually just imprint into his or her memory. It was also educational because one could learn geometry on ST, because to fill the shapes into the spaces, you have to know the shape of the form you 're working with and what it would [be] like, rotated in all directions. All in all, ST is a wonderful program, exciting, educational, and interesting are the elements a program needs to make it good, and in my point of view ST does contain all of these! 2. A grade-7 girl, considered by her teacher as a low achiever in mathematics, wrote: / think this game is superb. It is sooooooooo much fun. I love the music and how they use cartoons for pieces of the puzzle. ... Tarn learning a lot about geometry but this game makes it much more fun and inter sting. 3. A grade-6 boy wrote: When I was in grade 2 math was fun then in grade 3 math was not Jun until grade six with Super Tangrams. Its a fun way of learning math, and a fun way to do math. 4. A grade-7 boy, considered by his teacher as a marginal achiever in mathematics, wrote: / think that super tangrams is one of the best games that I ever played. . . . The music gives it a nice rythem to work on. I think that this game is a good way to learn motion giomatry instead of reading it from a text book for a lot of ways. One you can not move thing in a text book. 2 a text book does not talk to you. 3 [ends here] 5. A grade-6 girl wrote: 176 I think this game is neat. I liked the graphic and sound effects. The point of the game is simple but as the game progresses it gets harder. This game makes math (geometry) seem more fun and less boring. I liked this games because you 're playing and learning at the same time. 6. A grade-7 girl, considered a high achiever by her teacher, wrote: I thought this game was sooo fun I loved the graphics, the color, the music and the idea! I learned how to turn & flip the shapes to fit in the picture. 7. A grade-7 girl, who said she disliked mathematics, wrote: It was really fun. I learned the real way to do a flip. But I thought it was a bit hard. But it was fun.... 7 really like the game. 8. A grade-6 girl wrote: I like this game. ... It's little difficult for me. So I have to practice more. 9. A grade-7 boy wrote: I think that this game is the best game we have as a class. I like how you cane flip, rotate and move the pesie [pieces]. I also like the music too. 10. A grade-6 boy wrote: Super Tangrams was very educational. You have to use your brains alot. Of the 25 children in the preliminary trial, none expressed any dislike for the program. The children had played several other games since March, 1994, but Super Tangrams was the first game for which a unanimously positive response with no difference along gender lines was observed. In a previous year, many of the students in this classroom had reacted negatively towards other prototype games with mathematical content that E-GEMS had introduced to the class (comments such as "I don't like a game if there is math in it."). However, despite the obvious mathematical content of Super Tangrams, every child 177 wrote strongly positive comments about it in their journals, and almost all students indicated that they appreciated learning mathematics in a "fun" way. 5.1.2 Children's Descriptions of Their Learning As in the case of children's reactions towards Super Tangrams, their written comments and perceptions about their own learning were analyzed. All children's journal entries were read in their entirety, and children's references to what they thought they learned from Super Tangrams were highlighted. These highlighted references were further separated to distinguish the general comments (e.g., "I learned about flip today.") from the more explicit references to mathematics learning. Children were reminded several times by their teacher and I to record what they thought they were learning from the program. However, many of them stated that they found it difficult to write about their learning. Children's journal entries contained more general comments than specific ones. This section presents children's comments which explicitly referred to learning of transformation geometry concepts. Both unique as well as typical comments are reported: four cases from non-mediated students, and three cases from mediated students. The comments were written when a majority of children were completing Level 1, and some had started Level 2 (see Chapter 3 for a description of the levels). That is, these comments indicate children's thoughts at the end of Level 1 and are not representative of the concepts that children can learn in Levels 2 and 3. 178 Non-Mediated Children Casel A grade-7 girl wrote: In Super Tangrams I learned more about rotating shapes. I learned that when you rotate something it goes along the curved arch. I also learned that the + sign beside the angle number means counter-clockwise and the - sign is clockwise. I learned more about the center of rotation. . . . 7 learned that the circle is called the arc of rotation, and when you rotate somthing it goes along that line. Case2 A grade-7 boy wrote: I learned from the wale2 the [that] you cane [can] move a pice into its spot by only fliping it two tims [times] our [or] once but the rofbis [rhombus] can be flip once, two tims our [or] three tims. I learned that if you just stop and think you will find out where it gose [goes]. I learned that to reflect a image into the cored spote you must calculate the angl between 90° and its angel (I) then divide that number in hafe [half] and add that to 90° then with the number you get that is the angel you need to get the image into its cored spote (2). I learned that the denstence [distance] [end here, but is followed by a diagram] Case 3 A grade-6 girl wrote: In my experience of playing ST [Super Tangrams] I learned that you can use turn once instead of using turn then slide. You can also use flip twice to get a shape anywear. 2 This refers to the last puzzle in Level 1, called "Whale", where the player is only allowed to use reflection to solve the puzzle. The shapes are arranged in such a way that the minimum number of transformations to solve this puzzle is 15; that is, 2 reflections for each of the 6 symmetric shapes, and 3 for the asymmetric parallelogram shape. 179 Case 4 A grade-6 girl wrote: / learnt what the arc of rotation is and the center of rotation. I also learnt what the radius is. I learned how to do two rotations in one move and that the happy face appears every time a shape is in the correct spot. I learnt that there are only three levels. . . . I learned about refelction today. I learned that triangles always take two reflections. Media ted C h i l d r e n Case 5 The following extended excerpts are from the journal entries of one of the mediated grade-6 students recorded over a period of two months, playing 40 minutes per week (15 to 20 minutes of mediated play, and 15 to 20 minutes of non-mediated play; overall, slightly more than four hours). Some of these comments were written prior to him receiving any mediation. This student was described by his teacher as quite weak in all subject areas, and as someone who had difficulty thinking and reasoning about abstract concepts. In his initial encounter with Super Tangrams, he verbally commented that it was very difficult for him, but that it was cool, and he needed to work harder. After his first encounter with Super Tangrams, he wrote: It was really cool. . . . The music, graphics were really cool. . . I learned that even though geometry is boring if you add something to it people will really like it alot. A few weeks later he wrote: I enjoyed how you can't use flip or turn and only use slide on some puzzles. I enjoyed it because people can use the turn move. The turn move is that you can turn and place the object where you want it in one move. . . . 180 On the first day of Super Tangrams I though ftj that the game was hard, but then I though ft] it was pretty cool. . . . Level [2] was really good even though we didn't get past the first puzzle. I liked puzzle number one [in Level 2J. I learned how to find the right angle without the ghost image, which is all the lines have to be parrarel to where you 're going to put the object(s). And each point has to go in the right place. It's really fun playing the game!... I found out that if you 're on a negative angle and you go to 0° and you turn one more you '11 get a positive angle. I really enjoy doing this! Even thogh I've loved math the game just makes me do alot of math. I've learned that with turn you can rotate and translate in one move. . . . I've spoke to some of my friends and they love the game, alot of them have learned alot just like myself. Even if the puzzles are hard I still like it. Some other things I learned about were the arc of rotation, angle of rotation and center of rotation. The arc of rotation is that arrow shows how much you 're going to rotate the object. The first time I saw an arc of rotation arrow was in this game. The angle of rotation shows how much you've turned or fliped your object3. The center of rotation is the green dot on the radius, . . . If the object can't go in you must use two moves if the point of the object is in its place only turn to angle and leave it.4 And, finally, his last comment was: Now I can visulize angles really good in my head. Every move someone makes I think in my head and tell the person if its wrong or right. Case 6 A grade-7 girl wrote: I'm learning a lot about geometry. Such as angles like 270° is the same as -90°. . . . 7 learned that the arc of rotation show's how far and how much to turn it. I can see the angles in my head how far it should go. I don't really know why it's just that I can see it's just something I can see now. 3 It appears that he is referring to both the arc of rotation and the arc used in the reflection representation. 4 He seems to be referring to cases where the length of the radius of the arc is 0 , and he has to overlap the green and red dots on top of each other. 181 Case 7 A grade-7 boy wrote: Today I learned how to rotait an image without a ghost image. I had not learned to do this till I played this game. Most if not all of the thing in this game I did not know ontill I played this game. I can now gess angles without a protracter. 5.1.3 Summary and Discussion This phase of the study, although short in the amount of time that children spent with the game, suggested that, Super Tangrams met preliminary usability requirements. In terms of its affective impact, children's journal comments as well as direct observations suggested strongly positive reactions towards the program and its design features. Children were particularly fond of Super Tangrams' background music and colorful patterns and cartoons on puzzle pieces. One of the girls commented that she was in love with the flip operation because of its sound effect, and she kept reproducing the sound and enacting the animation of reflection in the program. In terms of its mathematical content, Super Tangrams seemed to have a sound presentation of the topic. Children's written comments as well as their comments while they were playing the game suggested that Super Tangrams fulfilled its intended objectives (see beginning of Chapter 3). My observations suggested that sometimes children's conception of their learning was somewhat naive and exaggerated, yet the excitement that this sense of learning generated motivated them to continue to like the topic and learn it. Indeed, some of their descriptions (above) suggest the beginnings of formal knowledge of some difficult transformation concepts, such as composite reflection (Cases 3 and 4), arc of rotation 182 (Cases 5 and 6), and positive and negative angles (Case 1). One of the noteworthy aspects of this phase of the study was the impact Super Tangrams had on students with a history of low achievement in mathematics. In particular, it seemed that playing Super Tangrams increased children's confidence in their ability to do mathematics. For instance, comments of the student presented as Case 5, when viewed holistically, seem to demonstrate how he responded to Super Tangrams both emotionally and cognitively. His first comment seems to indicate that the program initially helped him to have a positive attitude towards learning the topic. Because of this liking, he seemed to be inclined to work harder. Later on, as he gradually started learning some of the concepts, he not only explained them but expressed his liking of them. Additionally, he discovered some abstract concepts, such as the relationship between positive and negative angles on a circular abstraction rather than a linear one. He demonstrated the use of formal vocabulary in describing transformation geometry concepts, such as "arc of rotation" and "center of rotation". Last, but not least, was the degree of his confidence in how he could "visualize" and "think in [his] head". This confidence can also be observed in other students' comments (e.g., Cases 6 and 7). (Children's comments in Section 5.2 will substantiate this further.) Another interesting observation during this phase of the study was how Super Tangrams captivated a few students who reportedly had little interest in regular class activities. For instance, according to his teacher, the boy quoted in Case 7 had very poor writing skills and showed little initiative towards general classroom activities. However, through interaction with him, I was surprised by his ability to quickly grasp difficult 183 mathematical concepts and by the degree to which he was captivated and motivated by Super Tangrams. His teacher once told me that this boy had forgotten appointments with the principal of the school, but, on the days that I was to visit their class, he kept, asking his teacher when I would arrive so that he could play Super Tangrams. This boy along with the grade-7 mediated girl, who at the teginning of the research were considered as marginal and low achievers, were, subsequent to exposure to Super Tangrams, put into grade-8 mathematics. As a result, their teacher surprisingly asked me: "What have you done to them? They keep asking for more math ever since they have been doing this game!" During the preliminary evaluation phase of Super Tangrams, although most children were very excited about the program in Level 1, as they reached Level 2 they started growing frustrated. Although while in Level 1 children kept asking when they would get to a more challenging level, they had difficulty coping with the knowledge-intensive challenges in Level 2. In other words, the level of mathematical challenge exceeded children's mathematical skills. The only students who coped with Levels 2 and 3 were the mediated students. Based on this observation, I hypothesized that children needed a mediating mechanism to help them advance through the game. This observation played a crucial role in how the embedded instructional module was designed (see Chapter 3) and later evaluated in the summative evaluation phase (see Chapter 4). 184 5.2 Summative Evaluation The sumrnative evaluation was intended 1. to determine which prototypes of Super Tangrams were the most effective; 2. to determine the effect of embellishments on children's learning; 3. to investigate whether the instructional module was able to support children's knowledge construction process, or whether adult mediation was also needed; 4. to investigate whether the R prototypes (i.e., R+E and R-E) afforded reflective cognition as intended; 5. to investigate children's reactions, opinions and feelings about the Challenge-Driven Learning Model; 6. to investigate what children thought about the inclusion or lack of sensory stimuli in the prototypes; 7. to investigate children's affective response to the R+E prototype and its design features; and, 8. to investigate whether the R+E prototype afforded flow experiences in learning. The summative evaluation findings are reported in six subsections. The first subsection deals with the first three issues in the above list. The second through sixth subsections deal with the fourth through eighth issues in the above list respectively. 5.2.1 Achievement Results This section has four subsections: 1) analysis of overall achievement results, 2) fine-grained analysis of achievement results, 3) children's perception of their learning, and 4) 185 summary and discussion. In the first subsection, an overall analysis of the achievement results is performed and the results are reported both descriptively and inferentially. In the second subsection, a fine-grained analysis of the achievement results is performed and the results are reported along the conceptual dimensions of transformation geometry. In the third subsection, an analysis of children's design questionnaires is performed and the results pertaining to children's perception of their own learning are reported. Finally, in the fourth subsection, the results in the previous subsections are summarized and discussed. 5.2.1.1 Analysis of Overall Achievement Results To determine if repeating TGT influenced children's knowledge of transformation geometry on the test, a paired samples Mest (two-tailed) was used to compare the pre-and posttest scores for the control group. No statistically significant differences were found (/(19) = ,62; p > .05). Table 5.1 tabulates the measures of variability and central tendency for the control group. Since the control group sample came from another school (i.e., another population), to avoid confounding effects, the control group was not used in comparisons with the other conditions. 186 Tests Measures (pre-post change) Central Tendency Variability Mean% Median% SD Range pretest posttest (A) 45.1 41.8 44.0 39.3 -1.1 -2.6 13.3 16.9 8.0 26.0 - 76.8 23.7 - 80.8 -14.2 - 10.2 Table 5.1 Measures of central tendency and variability for the control group (N=20) for pre-posttest and pre-posttest change scores. At a descriptive level, Figure 5.1 shows the mean pre- and posttest T G T scores for each of the seven treatment groups. gl-E gl+E gF-E gF+E gR-E gR+E gR+E+M * pretest • posttest i i i 1 i i 1 1 • 1 i i i 1 i 1 1 1 i 1 0 5 1 0 1 5 2 0 2 5 3 0 3 5 4 0 4 5 5 0 5 5 6 0 6 5 7 0 7 5 8 0 8 5 9 0 9 5 1 0 0 Mean Scores Figure 5.1 Mean scores for pretests and posttests by group. As Figure 5.1 suggests mean pretest scores across all the groups were at about the same level. However, the figure clearly shows large gains for the R treatment groups (i.e., gR+E+M, gR+E, and gR-E). When the mean pre- and posttest scores are compared, the R groups evidence marked difference between their pretests and their posttests. The F groups (i.e., gF+E and gF-E) show some improvement from pretests to posttests. The I 187 groups (i.e., gl+E and gl-E) do not show improvement from pretests to posttests. In fact, the I groups' posttest scores decreased relative to their pretest scores. Data points used to plot Figure 5.1 are listed in Table 5.2. Table 5.2 shows the measures of variability and central tendency for each of the different treatment groups. Tests Measures (pre-post change) Central Tendency Variability Group N Mean% Median% SD Range pretest 21.8 22.6 13.5 0.0 - 57.1 gR+E+M 29 posttest 75.6 85.8 19.5 40.7 - 100.0 (A) 53.8 63.2 19.2 19.8 - 91.0 pretest 25.0 25.2 15.0 1.1 - 59.8 gR+E 15 posttest 76.1 86.2 20.6 41.2 - 100.0 (A) 51.1 61.0 23.1 14.2 - 89,6 pretest 37.1 37.9 14.7 7.9 - 70.5 gR-E 15 posttest 74.9 71.2 14.6 54.8 - 96.0 (A) 37.8 33.3 17.5 2.4 - 70.2 pretest 22.3 16.7 16.8 0.0 - 45.8 gF+E 14 posttest 38.4 37.3 15.3 14.1 - 67.2 (A) 16.1 20.6 19.3 -15.3-46.3 pretest 25.4 26.1 16.8 1.1 - 61.1 gF-E 13 posttest 44.8 46.9 14.6 20.9 - 71.8 (A) 19.4 20.8 16.7 2.8 - 60.5 pretest 22.5 22.0 10.0 8.5 - 40.1 gl+E 15 posttest 18.9 18.0 10.3 4.5 - 39.5 (A) -3.7 -4.0 11.1 -27.1 - 13.6 pretest 29.9 29.9 15.4 3.4 - 59.2 gl-E 15 posttest 22.2 22.6 13.1 1.1 - 52.5 (A) -7.6 -7.3 10.9 -33.3-18.1 Table 5.2 Measures of central tendency and variability by group for pre-posttest and pre-posttest change scores. The results in Table 5.2 indicate that whereas the distribution of scores for all F and I groups on the posttest are relatively normal (i.e., mean = median), the distribution of scores for two of the R groups (i.e., gR+E and gR+E+M) are negatively skewed (i.e., 188 mean < median); that is, in these groups, a majority of the students showed superior knowledge rather than a'few students improving dramatically. Beyond a descriptive level, at an inferential level, the pre- and posttest achievement results of the seven treatment groups were analyzed using the two factorial designs depicted in Figure 5.2. The design on the left of the figure entails a 3 by 2 factorial analysis with instructional package (R vs. F vs. I) as one factor and embellishments (+E vs. -E) as another factor. This design permitted an examination of the effect of the instructional package as well as the effect of embellishments on children's knowledge acquisition. The design on the right of the figure includes a separate comparison of the achievement results of the gR+E+M group with the gR+E group to determine the effect of adult mediation on children's knowledge acquisition. Instructional Package Embellishments R+E F+E I+E R-E F-E I-E Mediation R+E R+E + Med Figure 5.2 Factorial design: 3 by 2 [(R vs. F vs. I) by (+E vs. -E)], and 1 by 2 [R+E by (mediation vs. no mediation)]. Pre- and posttest achievement results of six of the groups (gR+E, gR-E, gF+E, gF-E, gl+E, and gl-E) were examined in the 3 by 2 factorial analysis of covariance (ANCOVA) design with instructional package as one factor and embellishments as another factor. The transformation geometry posttest scores were the dependent variable and the pretest 189 scores served as the covariate. The hypothesis tested, stated in the null form, was that there were no mean achievement score differences among the different groups on T G T , given pretest scores as the covariate. As displayed in Table 5.3, there was a significant main effect of instructional package among the groups (F(2, 80) = 104.09; p < .05); and, the effect of embellishments on achievement was not significant (F(l, 80) < \ ; p » .05). Furthermore, there was no interaction effect between instructional package and embellishments (F(2, 80) = 1.03;/? > .05). Source SS DF MS F P Intercept 24998.20 1 24998.20 124.39 .000 Regression 2327.41 1 2327.41 11.58 .001 IP1 41836.21 2 20918.10 104.09 .000 E 2 .67 1 .67 .00 .954 IP*E 414.78 2 207.39 1.03 .361 Residual 16076.86 80 200.96 Total 64812.37 86 753.63 ' Instructional Package. 2 Embellishments. Table 5.3 F table comparing instructional package by embellishments type treatment groups. Table 5.4 shows the observed and adjusted means for the six groups. Given a significant instructional package result, using the adjusted means, Tukey's HSD post hoc tests were performed to compare the means for the different instructional package conditions. Four significant differences emerged among the groups: gR+E differed significantly from gF+E; gR-E differed significantly from gF-E; gF+E differed significantly from gl+E; and, gF-E differed significantly from gl-E. The results of this analysis are summarized in Table 5.5. 190 Group Obs. Mean Adj. Mean gR+E 76.03 76.03 gR-E 74.92 71.28 gF+E 38.38 40.08 gF-E 44.82 45.40 gI+E 18.87 20.50 S l - E 22.25 21.22 Table 5.4 Adjusted and observed means for the groups. Between Groups Q P Difference gR+E vs. gF+E 9.84 < .05 Significant gR-E vs. gF-E 6.80 < .05 Significant gF+E vs. gl+E 5.24 < .05 Significant gF-E vs. gl-E 6.36 < .05 Significant Table 5.5 Post hoc comparisons among groups. An 1 by 2 analysis of covariance using pretest scores as the covariate was performed to test the null hypothesis that, within the R+E conditions, the mean scores of the gR+E+M and gR+E groups were equal. As shown in Table 5.6, no significant main effect was found (F(l, 41) = .02; p > .05), showing that the addition of adult mediation did not affect students' mean level of performance. Source SS DF MS F P Regression 1488.56 1 1488.56 4.05 .051 Main Effects 8.79 1 8.79 .02 .878 Residual 15074.75 41 367.68 Total 16564.75 43 385.23 Table 5.6 F table comparing mediation versus no-mediation treatment groups. 191 5.2.1.2 Fine-Grained Analysis of Achievement Results The overall analysis of the pre- and posttest scores found a statistically significant difference between the adjusted mean posttest scores of the R groups and the F groups. Moreover, the results found a statistically significant difference between the adjusted mean posttest scores of the F groups and the I groups. The overall analysis used weighted question scores to obtain the results. In this section, achievement results are obtained and reported by using different marking schemes (see Section 4.2.4). As will be demonstrated in this section, frequency counts of the number of children who answered individual pre-and posttest questions revealed other more fine-grained patterns in the data that support the previous overall findings and may be attributed to the type of prototype that students used. Translation Table 5.9 displays the results from T G T that dealt with questions on translation. Questions 4 through 6 asked children if a particular transformation of a shape could be performed with "1 slide" (question 4), "2 slides" (question 5), and "47 slides" (question 6). If a student answered all three questions correctly, he/she received a score of 1. If any of the answers to any of the three questions was wrong, the student received a score of 0. The key finding, shown in Table 5.7, is that children's pretest results are initially identical for all groups - few, if any, students could answer the questions correctly. On the posttest, children in the I groups did not improve. Students in the F groups, on the other hand, improved their performance only modestly over pretest levels. However, 192 students in the R groups significantly improved their pretest performance. Nearly 75% of the R students answered the questions correctly on the posttest; whereas, slightly less than 50% of the F students answered the questions correctly on the posttest, and less than 10% of the I students answered the questions correctly on the posttest. This pattern of results (i.e., R > F > I) was replicated on other questions, as will be shown. Group N Pretest Posttest (A/n)% gR+E+M 29 2 24 76 gR+E 15 0 12 80 gR-E 15 0 9 60 gF+E 14 0 6 43 gF-E 13 0 7 54 gl+E 15 0 2 . 13 gl-E 15 2 0 -13 Table 5.7 Students getting Questions 4 through 6 completely correct. Table 5.8 displays the results from grouping together questions 20 and 50 in TGT. If a student answered both questions correctly, he/she received a score of 1. If any of the answers to any of the 2 questions was wrong, the student received a score of 0. Group N Pretest Posttest (A/n)% gR+E+M 29 0 17 59 gR+E 15 0 6 40 gR-E 15 1 7 40 gF+E 14 0 1 7 gF-E 13 1 1 0 gl+E 15 0 0 0 gl-E 15 0 0 0 Table 5.8 Students getting both Questions 20 and 50 correct. 193 Questions 20 and 50 are also translation questions for which children are required to know that the vector for moving one shape to another can be placed anywhere independent of the object's location. A noteworthy point is that of 116 students who wrote the pretest, only 2 students answered both questions correctly. However, on the posttests only the R groups improved. Of the 30 students in the R groups who answered both questions 20 and 50 correctly, 28 had reached or completed the third stage of the R D C M representation of the translation concept in Level 3 (where the tail of the translation arrow has a fixed coordinate; see Figure 3.14). Only 2 students who had not completed the third stage answered both questions correctly. Rotation Table 5.9 displays the results from T G T that dealt with Question 7. This question asked children if a particular transformation of a shape could be performed with "1 turn" , a relatively simple rotation question (see Appendix A). The results for Question 7 show almost the same pattern of results found for the translation items, except for one difference: the I groups' scores decreased for this question. 194 Group N Pretest Posttest (A/n)% gR+E+M 29 9 27 62 gR+E 15 8 14 40 gR-E 15 8 14 40 gF+E 14 5 6 7 gF-E 13 5 7 15 gl+E 15 6 2 -11 gl-E 15 6 1 -45 Table 5.9 Students getting Question 7 correct. Figure 5.3 shows the percentage change between the pre- and posttests in the total number of rotation questions that the different groups answered correctly. This figure shows the same pattern of results found for Question 7, above. gl-E > g l - * \ gF-E ; gF+E ; gR-E I gR+E; gR+E+M ; - 2 5 - 2 0 - 1 5 - 1 0 1 0 1 5 2 0 2 5 3 0 3 5 4 0 4 5 5 0 5 5 6 0 Percentage Change 6 5 Figure 5.3 Percentage change in correct answers to rotation questions by group. Reflection Table 5.10 displays the results from TGT that dealt with Question 46. Question 46 asked children if a particular transformation of a shape could be performed with composite 195 reflections in intersecting mirrors, one of the most difficult questions on the test (see Section 4.2.4). Slightly more than 75% of the students in the R groups correctly answered the question on the posttest. Group N Pretest Posttest (A/n)% gR+E+M 29 0 18 62 gR+E 15 3 10 47 gR-E 15 6 13 47 gF+E 14 0 2 14 gF-E 13 2 2 0 gl+E 15 0 1 7 gl-E 15 2 0 -13 Table 5.10 Students getting Question 46 correct. Figure 5.4 shows the percentage change between the pre- and posttests in the total number of reflection questions that the different groups answered correctly. This figure shows a pattern of results similar to that found for rotation questions. 196 gl-E , [ Percentage Change Figure 5.4 Percentage change in correct answers to reflection questions by group. Equivalence Questions As stated in Chapter 4, some sets of questions in T G T were grouped together to provide indicators of students' depth of understanding of the relationships and equivalences among the different transformations. Tables 5.11, 5.12, and 5.13 display the results from T G T that dealt with grouped Questions 7 through 12, 14 through 19, and 26 through 31, respectively. For any grouping, if a student answered all questions correctly, he/she received a score of 1. If any of the answers to any of the questions was wrong, the student received a score of 0. 197 Group N Pretest Posttest (A/n)% gR+E+M 29 0 9 31 gR+E 15 0 5 33 gR-E 15 0 7 47 gF+E 14 0 0 0 gF-E 13 0 2 15 gl+E 15 0 0 0 gl-E 15 0 0 0 Table 5.11 Students getting Questions 7 through 12 completely correct. Group N Pretest Posttest (A/n)% gR+E+M 29 0 9 31 gR+E 15 0 5 33 gR-E 15 0 4 27 gF+E 14 0 0 0 gF-E 13 0 1 7 gl+E 15 0 0 0 gl-E 15 0 0 0 Table 5.12 Students getting Questions 14 through 19 completely correct. Group N Pretest Posttest (A/n)% gR+E+M 29 0 17 59 gR+E 15 0 12 80 gR-E 15 0 6 40 gF+E 14 0 3 21 gF-E 13 0 1 8 £l+E 15 0 0 0 gl-E 15 1 0 -7 Table 5.13 Students getting Questions 26 through 31 completely correct. A key finding, shown in Tables 5.11, 5.12, and 5.13, was that children's pretest results showed the same patterns for all groups. Of 116 students, only one student managed to answer one set of grouped questions correctly on the pretest. On the posttest, none of 198 the children in the I groups improved on grouped questions. Overall, only 8.5% percent of the children in the F groups improved their performance on grouped questions, while 42.5% of the children in the R groups improved their performance on grouped questions. Despite the fact that the scoring scheme used to obtain these findings was strictly dichotomous, and as a consequence quite narrow and binding; nonetheless, more children in the R groups than the F groups evidenced depth of understanding of the relationships and equivalences for the different transformations. 5.2.1.3 Children's Perception of Their Learning Children responded to DQ (see Appendix C) questions after they wrote their T G T posttest. Children's responses to questions in DQ dealing with their perception of how much they thought they had learned after playing Super Tangrams were analyzed. Table 5.14 displays the means for individual questions (i.e., Questions 15, 1,2, and 3), and an overall mean for Questions 1,2, and 3. Responses were rated on a 5-point Likert scale in which "A" responses on the scale were assigned a value of 5, and " E " responses were assigned a value of 1. 199 Question Number Mean Group 15' gR+E+M gR+E gR-E gF+E gF-E gl+E 4.6 4.6 3.8 4.2 3.7 4.3 4.1 4.4 4.3 4.2 3.7 3.7 3.0 3.3 4.5 4.2 4.1 3.6 3.9 3.1 3.2 4.6 4.1 4.0 3.9 3.6 2.9 3.0 4.5 4.2 4.1 3.7 3.7 3.0 3.2 • ' How new were the math concepts in Super Tangrams for you? A) everything was new to me; B) most things were new and some things were review; C) half was new and half was review; D) some things were new and most things were review; E) nothing was new to me. 2 Compared to what you knew about turn before playing Super Tangrams, how much have you learned about turn now that you have played the game? A) I have learned so much that I can't believe it; B) I have learned quite a bit; C) 1 have learned some; D) 1 have learned very little; E) I have hot learned anything at all. 3 Compared to what you knew about flip before playing Super Tangrams, how much have you learned about flip now that you have played the game? A) I have learned so much that I can't believe it; B) I have learned quite a bit; C) I have learned some; D) I have learned very little; E) I have not learned anything at all. 4 Compared to what you knew about slide before playing Super Tangrams, how much have you learned about slide now that you have played the game? A) I have learned so much that I can't believe it; B) I have learned quite a bit; C) I have learned some; D) I have learned very little; E) I have not learned anything at all. 5 Mean score for Questions 1, 2 and 3. . Table 5.14 Children's perception of their learning after playing different prototypes of Super Tangrams. Mean scores for the R groups show that children who used the R instructional package perceived that they learned "quite a bit" (all mean scores are greater than 4). It is noteworthy that the children's perception of their own learning coincides with their achievement results shown in the previous sections. That is, relatively speaking, the R students felt that they had learned more than the F students, and the F students felt that they had learned more than the I students. Table 5.15 displays the R children's responses to Question 22, regarding the level in which they thought they learned most. 200 Group Level gR+E+M gR+E gR-E Total (Total/59)% Level 1 5 4 2 11 18.6 Level 2 10 4 7 21 35.6 Level 3 14 7 6 27 45.8 Q# 22: Of the 3 levels in Super Tangrams, in which level did you leam most about slide, turn and flip? A) Level 1; B) Level 2; C) Level 3; Explain why: Table 5.15 Children's responses regarding the level in which they learned most. Of the 59 children, 18.6% thought they learned most in Level 1 of the game; 35.6% thought they learned most in Level 2; and 45.8% thought they learned most in Level 3. Table 5.16 lists some of children's typical explanations as to why they responded to Question 22 as they did. In considering these explanations, it is of interest to note that, although less than 20% of the R students thought they learned most in Level 1, the Level-1 puzzles in the R instructional package are still more difficult to solve than the ones in the F instructional package since in the F instructional package none of the transformations are ever disabled (see Section 4.2.2). 201 Level Group Explanation 3 gR+E+M Because it's harder and in real life you don't get a ghost image. 3 gR+E+M Because you couldn't use the ghost image any more so you really had to know yer stuff. 3 gR+E+M It challenged you by not giving you the guiding lines. 3 gR+E+M Because you have to do everything by having no ghost image. 3 gR+E+M Most challenging! 3 gR+E It didn't show the shape so I had to learn to make one in my head. 3 gR+E it is challenging and hard. 3 gR+E In level 3 I learned turn the most. 3 gR-E It got harder so I had to use learn more often. 3 gR-E / learned most in level 3 because they have better explanation. 3 gR-E because level 3 had the most challenging problems. 2 gR+E+M You couldn't just get the ghost image in the spot but it still showed around where it would go. 2 gR+E+M Because that was when the harder geometry began. 2 gR+E+M Because it is in between the other levels. 2 gR+E+M Because I think is the most challenging of all. Because the math is harder than better 1. I think Level 3 is too different for me. 2 gR+E because you had to learn where the shadows were. 2 gR+E Because it didn't use a ghost image, but taught me those things with all the indications. 2 gR+E There were no ghost images, making me use learn. 2 gR-E because it helped me get prepared for the tough level 3. 2 gR-E / was stuck for the longest in level 2. 1 gR+E+M because since I was learning the math for the first time I learnt the most on the first level. gR+E+M Because it was eazyer. gR+E+M because it showed you slide turn and flip and you could choose and learn different skills. gR+E+M because there is the ghost image and the angle of rotation. gR+E / learned the most because it gave the shape. gR+E Level I taught me the basic concepts which helped me understand everything. gR-E because it was introducing the concept to me. gR-E Because you learn what a slide, turn, and flip are. Table 5.16 Children's explanations regarding the level in which they thought they learned most. 202 5.2.1.4 Summary and Discussion The seven treatment groups' achievement results were analyzed. Four main findings resulted: 1. The instructional package factor did not interact with the embellishments factor. That is, addition of embellishments to an instructional package (R, F, or I) neither diminished nor increased achievement results within the respective instructional package treatment groups. 2. R students performed significantly better than F and I students. 3. F students performed significantly better than I students. 4. Addition of adult mediation, in the case of the R+E prototype, did not affect students' mean achievement performance. The findings coincided with children's perception of their own learning from the different instructional packages. Furthermore, R students' explanations regarding the level of the game in which they thought they learned most indicated that the gradual removal of visual feedback was effective in helping children learn the concepts. (Results presented in later sections will substantiate this further.) On-site observations and post-hoc interviews provided added depth in understanding the above findings. While observing the different groups, I noticed that the shift from a D O M interface style to a D C M interface style had an immediate effect on children's conceptions of transformation geometry. Based on their comments and strategies for solving the puzzles, children using the D C M style seemed to clearly understand that translation is along a straight line. In contrast, some students who used the D O M 203 interface style seemed to conceive of translation as a "drag" motion, moving curvilinearly in a two dimensional plane. One of the reasons for this conception was the difference in operational affordances of the two interfaces. The D O M style allowed children to drag an object in any direction they desired. In contrast, the D C M style afforded a lag between the time children adjusted the parameters of a transformation and the time they clicked on the GO button. This lag time provided children with the opportunity to observe and discuss what was taking place on the screen. Differences in understanding between children from these two groups were also observed with respect to the concepts of rotation and reflection. Addition of gradual visual feedback reduction to the D C M interface style, resulting in the R D C M style, added much to the latter's efficacy to support conceptual development of transformation geometry in children. Children commented (see Table 5.16) that not having the "ghost image" made the game "harder" and "challenging", so they had to make the image in their "head", "learn where the shadows were", and "use learn [the instructional module] more often". More than 80% of the children stated that they learned most in Levels 2 and 3. Inclusion of the ghost image in Level 1, however, helped them learn the "basic concepts". The addition of constraints to the D C M interface style, resulting in the R instructional package, also added to the D C M style's effectiveness. The most immediate effect was observed in Level 1, which was almost identical to the D C M interface style, except that children could not use all three transformations. To do a rotation in an F prototype, many children would perform two operations, a "Turn", adjusting the angle of 204 rotation, followed by a "Slide". This, in some cases, meant that children never realized that they could accomplish this task using only one rotation. In contrast, children using the R prototypes discovered and became aware that "Turn" accomplished this task in one transformation. Some students excitedly articulated the power of "Turn" when they discovered this feature. In the post-hoc interviews, the R students who were shown an F prototype immediately grasped how the prototype worked. They observed that the F activities seemed much easier than their own tasks and had the same reaction to the I prototypes. Children stated that the I prototypes were very easy compared with the R prototypes. In comparison, the F students found the R activities more difficult, and the I activities easier. The I students had difficulty understanding what to do in the R activities, and felt that the F activities were more difficult than their I tasks. These findings suggest that the knowledge of children who used the more concept-focused interface styles encompassed the knowledge of children who used the less concept-focused interface styles. (For an example, see Section 5.2.6.) 5.2.2 Assessment of Reflective Cognition Affordance To assess the extent to which the R prototypes (i.e., R+E and R-E) afforded reflective cognition, Norman's [1993] description of this psychological construct was used. (It is also possible to use Salomon's [1981] construct, 'amount of invested mental effort', discussed in Chapter 3.) As discussed previously, reflective cognition refers to conscious, purposeful thought that is directed at a problem in order to understand it and form 205 integrated conceptual structures. Reflective cognition is 'conceptually driven', requires 'concentration', requires a comparison of one's knowledge with other sources of knowledge so as to 'restructure' one's understanding, uses 'questioning and examination as tools' for reflection, is ' s low' and time consuming, sometimes requires 'periods of quiet' and 'hard work' , and is the opposite of event-driven, reactionary cognition. Children's responses (both quantitative judgments and qualitative comments) to questions in D Q as well as the program's log files (see Section 4.2.4) were examined to assess the extent to which the prototypes afforded reflective cognition according to its description above. Three indicators were examined: children's own perception of how hard they had to think to solve the puzzles; children's need for or use of the instructional module (Learn) to compare their own knowledge with another source of knowledge to restructure their understanding; and, number o f puzzles that each group solved - solving fewer puzzles indicating a need for more time to reflect to solve the puzzles. The latter indicator was deemed appropriate, since during the entire length of the study, children were never observed to be off task. During all ten sessions, children were always engaged in solving the puzzles. Children's Perception To examine evidence for reflective thought, three questions in D Q that probed for children's perception of reflective cognition were combined to form a reflective index. Table 5.17 displays the mean ratings for each question, and an overall, mean reflective index for each group. A s in previous cases, the question scores ranged from 5, for 206 'strongly agree' responses, to 1, for 'strongly disagree' responses. (Values for Question 12 were reversed to enable comparison with the other questions. That is, a "strongly agree" response was assigned a value of 1, and an "strongly disagree" response was assigned a value of 5.) Question Number Group ll' 122 243 Mean Reflective Index gR+E+M 4.6 4.2 4.4 4.4 gR+E 4.3 4.1 4.2 4.2 gR-E 4.1 3.6 4.5 4.1 gF+E 3.5 3.3 N/A 4 3.4 gF-E 3.3 3.1 N/A 3.2 gl+E 3.2 2.9 N/A 3.1 gl-E 2.8 2.4 N/A 2.6 ' Most of the time, I had to really think about how to move a piece. I couldn't just move a piece without much thinking. A) strongly agree, B) agree, C) undecided, D) disagree, E) strongly disagree. 2 It was easier and faster to solve the puzzles if I guessed, instead of calculated, each move. A) strongly agree, B) agree, C) undecided, D) disagree, E) strongly disagree. 31 think that not having the ghost image challenged me to think hard. A) strongly agree, B) agree, C) undecided, D) disagree, E) strongly disagree. 4 "N/A" indicates that this question is not applicable to this group since their prototype did not have this design feature. Table 5.17 Mean reflective indices for reflective cognition by group. Table 5.18 displays mean reflective indices for each instructional package. These indices were obtained by calculating the mean scores of all the above questions for each instructional package. Such an index was deemed appropriate since, as previously observed, inclusion of embellishments, as a factor, did not interact with learning (see Section 5.2.1). 207 Instructional Package Mean Reflective Index R 4.2 F 3.3 I 2.8 Table 5.18 Mean reflective indices for each instructional package. Children's quantitative judgments suggest that students in the R groups perceived their prototypes to require hard thinking (mean reflective index greater than 4). Moreover, relative to the F and I instructional packages, the R instructional package afforded a higher degree of reflective cognition (4.2 versus 3.3 and 2.8). These reflective indices gain added significance when one notes that, unlike the F and I instructional packages, the R instructional package included a Hint button as well as the Snap-in-Place feature (see Section 3.3.1). The Hint button, when used, showed children where to place a shape in an outline, and the Snap-in-Place feature displayed a happy-face icon on a transformed shape whose placement in a puzzle's outline was part of a correct solution. These two features greatly reduced the game's 'cognitive load'. As a result, children could focus on deciding how to move a shape rather than what shape to put where and whether its placement was correct or not. An analysis of the interaction log files (see Section 4.2.4) of the R groups found that children used the Hint button frequently. For instance, in the case of the gR-E group, some children used the Hint button more than 40 times to solve puzzles that had only 7 shapes. The other two groups (gR+E and gR+E+M) used the Hint button at least 4 times per puzzle on average. My observations of children's interaction with the R+E prototype suggested that these two groups used the Hint button fewer than the gR-E group because 208 they were cautious to avoid losing too many points by using the Hint feature. Nonetheless, the use of the Hint button in all three groups is significant since in tangrams puzzles, once two or three shapes are put in the right place in the outline, it is rather trivial to know where the rest of the shapes should go. The quantitative reflective index, indicating that the R prototypes afforded more reflective cognition than the F and I prototypes, is validated by an analysis of children's qualitative comments. Reflective cognition is a psychological construct. In their written comments and explanations in DQ, children did not explicitly refer to a need for 'reflection' to solve the puzzles. Instead children used words such as 'challenging' and 'hard'. A few comments from some children who explicitly referred to the need for concentration, thinking, understanding, and figuring out are listed in Table 5.19. These comments were selected to show variations of children's comments. Children's written explanations are shown in italic characters; statements directly related to reflective cognition are presented in bold face. (More comments related to this issue from children are provided in Section 5.2.4.) 209 Stem' Response I learned most in Level 3 because I learned most in Level 3 because I used Learn I learned most in Level 3 because I learned most in Level 3 because I learned most in Level 3 because I used Learn I learned most in Level 3 because I used Learn because I did not always have the music on I did not always have the music on Compared to other educational math programs I have use, I think Super Tangrams is better than most it was the hardest so you really had to think, it really made you think. because when I was very frusterated in the game I wanted to figure out what I was doing wrong. it didn't show the shape so I had to learn to make one in my head, it challenged you by not giving you the guiding lines [i.e., removed visual feedback]. it got hard, and I pressed learn more and learned better, to try and understand some of the concepts, you have to figure eveything out for yourself. I needed to understand how to do turn. Some time I just had to concentrate hard. Sometimes we need to concentrate on the math, because you have to think alot. 1 Reconstruction of the question that prompted children's explanation. 2 Relevant comments by children are highlighted in bold face. Table 5.19 Comments by the R students regarding need for reflective cognition. Use of Learn Table 5.20 summarizes the results of the children's responses to questions in DQ that specifically dealt with the instructional module, Learn. Question Number 2S2 Group Yes% No% Yes% No% gR+E+M 93 7 263 . 67 gR+E 100 0 13 87 gR-E 100 0 33 67 gF+E 0 100 100 0 gF-E 69 31 92 0 Did you ever use Learn in Super Tangrams? A) Yes, B) No; Explain why: 2 Do you think you could have progressed through the game without using Learn at all? A) Yes, B) No; Explain why: 'Some percentages do not add up to 100%. This is because some students wrote that they were undecided. Table 5.20 Responses to questions about Learn. 210 Table 5.20 shows that among the three R groups, almost all students used Learn (Question 7). The single student in the gR+E+M group who did not use Learn commented that she learned from her "mistakes". The majority (74%) of the R students did not think they could have progressed through the game without the use of Learn (Question 28). By contrast, the majority of the students in the F groups did not use Learn (Question 7). Overall, 33% of the F students reported using the instructional module, and almost all (96%) of them thought that they could have advanced through the game without the use of Learn (Question 28). It is of interest to note that the majority of the 69% of the students in the gF-E group, who said they used Learn, stated that they went into Learn: "to see what was in it," to "hear sounds", because "it had cool sound effects," because "it made cool music," and so on. Considering that the F-E prototype did not include embellishments, children's qualitative comments explain why the quantitative index is higher than the gF+E group. Very few students from the F groups said that they used the instructional module for learning, or that they needed to use it. This is evident from the overall response of the gF+E and gF-E students to Question 28 in which 96% of them thought that they could have advanced through the game without the use of Learn, and that they did not need it. Sample comments of the F students responding to Question 28 are listed in Table 5.21. As will be seen in the upcoming Section 5.2.3, these comments exhibit a marked contrast with the comments of the R students regarding Learn. 211 Group Sex Response g F - E M Becaues D. and I did. F Because I understood everything already. F Because I caught on easily. M / only use Learn for fun. I don 7 know undecided, someone else might not know how to play. Because once we got the hang of it we didn 't need any help. M M F It was kind of simple. gF+E F / did progress through the game without using Learn. F because we didn't use it and we had progress. M We could figure it out ourselves. F We just didn't seem to need it and we didn't pay attention to it being there. M Because it was easy. F What is Learn? Sex: M=Male, F=Female. Table 5.21 Explanations given by the gF-E and gF+E students regarding why they thought they could have advanced through the game without the use of Learn. Number of Puzzles Solved Table 5.22 shows the average number of puzzles solved by each treatment group. The information in the table was distilled from each prototype's stored interaction log files. On the average, the F and I groups solved more than twice as many puzzles as the R groups. 212 Group Average Number of Puzzles Solved gR+E+M 33 gR+E 31 gR-E' 35 gF+E 68 gF-e 70 gl+E 78 gl-E 77 Table 5.22 Average number of puzzles completed by each group. Considering the availability of the Hint feature in the R prototypes and the overall achievement results of the groups, it is safe to state that the relatively low number of puzzles that the R students solved can not be contributed to their lack of ability to solve these puzzles, but rather the degree of mental effort that was required to solve the puzzles. Summary and Discussion The extent to which the R prototypes afforded reflective cognition and required investment of mental effort was assessed using three indicators: children's own perception of how hard they had to think to solve the puzzles; children's need for or use of Learn to advance through the game; and, number of puzzles that each group solved. The results for all three indicators suggested that the R prototypes (i.e., the R instructional package) afforded a high degree of reflective cognition. Moreover, relative to the F and I prototypes, the R prototypes demanded the investment of a higher degree of mental effort. These findings were corroborated by direct observations and post-hoc interviews. During the study it was observed that the I and F groups' activities were characterized by 213 considerable moving of the puzzle shapes. The R group's activities were characterized by more pausing, reflecting, and thinking. As children moved to Levels 2 and 3, the progressive differentiation of the concepts, as promoted by the gradual removal of visual feedback, required children to engage in progressively greater degrees of reflection and reasoning. Children paid attention to details and specifics of the concepts they were manipulating. In their interviews, some children reported that in Levels 2 and 3 they had to think hard to "picture" the ghost image in their "mind" to be able to perform the transformations. This requirement for a greater depth of conceptual knowledge encouraged many children to use the instructional module to try to understand the mathematical concepts and avoid making unnecessary and costly moves. This observation is further supported by children's comments presented in the next section. 5.2.3 Assessment of the Challenge-Driven Learning Model The Challenge-Drive Learning Model, as implemented in the R instructional package, was described in Chapter 3. Results bearing on the educational effectiveness of the R prototypes with respect to children's achievement scores and reflective cognition have already been presented. In this section, children's written comments are used to assess the effectiveness of the Challenge-Driven Learning Model. Before describing how the qualitative data were analyzed, the model is summarized. The Challenge-Driven Learning Model contained two components: a situated game activity and an embedded instructional module. The application of this model was intended to create a learning environment in which advancing through the game activity 214 maintained a dynamic balance between children's existing knowledge and required mathematical knowledge. The game activity was designed to progressively increase knowledge-intensive challenges (which were closely tied to the interface style). Increase in challenge was intended to cause epistemic conflict (or cognitive dissonance) and thereby create a need or desire for conceptual restmcturing of the children's existing knowledge structures. The embedded instructional module then provided a tool for helping children compare and resolve the conflict to advance through the game so that 'flow' in learning could be maintained. Unlike the analysis and presentation of children's comments in the previous sections, a more holistic case study method was used to analyze children's comments and to assess the Challenge-Driven Learning Model. Fifty-nine children used the R instructional package. All 59 children's written comments in their design questionnaires were examined carefully. Comments that suggested a progressive increase in challenge, epistemic conflict, need for mental restructuring, reciprocity between the game activity and Learn, and movement or flow in learning were highlighted and linked together. Questionnaires which made general references to "challenge", "hard", and Learn "helped" were discarded. Of the rest of the questionnaires that seemed to refer to the indicators of the learning model above, six cases were selected and are presented in this section. Each case contains most of the child's comments that are related to the model. (Comments regarding other design features such as the background music and graphics are not included in the cases.) After presentation of the holistic cases, to provide a better sense of how the majority of children interacted with and felt about the model, an extended list of children's 215 comments with regard to the instructional module are presented. These comments further substantiate the findings reported in the previous section. The comments of children, both for the six holistic cases and the extended list, are presented in Tables 5.23 to 5.29. For the six holistic cases, the stems of the questions that prompted children's responses are included in the tables to facilitate understanding of the comments. Children's written explanations are shown in italic characters; statements that directly reinforce a challenge-driven model are presented in bold face. 216 Case 1 Table 5.23 lists the comments and responses of a boy in the gR-E group. This student's pretest score was 38.8% (nearly equal to the mean for that group), and his posttest score was 92.1 % (nearly 2 standard deviations above the mean for his group). Stem I strongly agree that I like to learn math from computer games like Super Tangrams because I strongly agree that I liked the way motion geometry was presented in Super Tangrams because Compared to other educational math programs I have used, I think Super Tangrams is the best because Compared to other educational games I have played, I liked Super Tangrams because Super Tangrams has helped me love motion geometry because Compared to other ways of learning math, Super Tangrams was much more fun because I liked Level 2 I learned most in Level 3 I used Learn because I think it is good to have Learn because I do not think I could progress through the game without using Learn at all because Response It makes learning fun and interesting. it seems to work your way up to a challenge. It is a lot of fun because it is challenging. the levels were a bit too easy at the beggining otherwise I would"ve loved it. it presents the fun side of math. It didn't time you or put you under stress. it wasn't easy or too hard, it really made you think. Without learn I wouldn 't have been able to figure out turn. Without learn people wouldn't figure anything out. Learn helped me understand the math. Table 5.23 A child's perception of the Challenge-Driven Learning Model in Super Tangrams (Case 1). 217 Case 2 Table 5.24 lists the comments and responses of one of the weakest students in the gR+E+M group. (According to her teacher, this students had Attention Deficit Disorder). This student's pretest score was 1.1% (nearly one and a half standard deviations below the mean score of her group), and her posttest score was 49.6% (one and a third standard deviations below the mean for her group). Stem Response 1 strongly agree that I like to learn math from because it Makes Math More interesting. computer games like Super Tangrams I agree that I liked the way motion geometry was I found it hard to understand but it was fun. presented in Super Tangrams Compared to other educational math programs I have because it's Much Moor interesting. used, I think Super Tangrams is the best Compared to other educational games I have played, it was hard to understand but fun. I liked Super Tangrams because Super Tangrams has helped me like motion geometry Motion geomitry is fun. because Compared to other ways of learning math, Super it had interesting ways of teaching. Tangrams was much more fun because 1 liked Level / it was the easiest. I learned most in Level 3 it was a challenge becase there was No goset imeges. I used Learn because I did not understand somethings. I think it is good to have Learn so people can understand it. I do not think I could progress through the game becase I had to understand things. without using Learn at all Table 5.24 A child's perception of the Challenge-Driven Learning Model in Super Tangrams (Case 2). 218 Case 3 Table 5.25 lists the comments and responses of a boy in the gR+E group; this student's pretest score was 39.9% (one standard deviation above the mean for that group), and his posttest score was 100% (more than one standard deviation, and at the ceiling of the scores of that group). Stem Response I strongly agree that I like to learn math from computer games like Super Tangrams I strongly agree that I liked the way motion geometry was presented in Super Tangrams Compared to other educational math programs 1 have used, 1 think Super Tangrams is better than most Compared to other educational games I have played, I loved Super Tangrams because Super Tangrams has helped me love motion geometry because Compared to other ways of learning math, Super Tangrams was much more fun because I liked Level 3 I learned most in Level 3 I used Learn I think it is good to have the Learn module I do not think 1 could progress through the game without using Learn at all Because it makes learning fun instead of boring stuff on paper. Because it showed how they moved. Because other ones don't use enough math or others make it boring by making all math. I liked it because it wasn't boring. It used to be hard but now it is easy, it wasn't boring. It makes me learn more. It didn't show the shape so I had to learn to make one in my head. Because I didn't know how to use flips, slides, turns without the shape outline. So you don't get frusterated but can learn and get farther. Because I didn't know anything about flip, slide, turns before. Table 5.25 A child's perception of the Challenge-Driven Learning Model in Super Tangrams (Case 3). 219 C a s e 4 Table 5.26 lists the comments and responses of a girl in the gR+E+M group; this student's pretest score was 19% (nearly at the mean of that group), and her posttest score was 87.9% (nearly half standard deviation above the mean of that group). Stem I strongly agree that I like to learn math from computer games like Super Tangrams because I strongly agree that I liked the way motion geometry was presented in Super Tangrams because Compared to other educational games I have played, I liked Super Tangrams because Super Tangrams has helped me love motion geometry because I liked all the levels because 1 learned most in Level 2 because I used Learn because I think it is good to have Learn because I do not think I could progress through the game without using Learn at all Response / didn't know anything about turn, flips, and slides before. It makes it easier to understand and funner. It made learning motion geometry fun. It was fun and I learned a lot from it. It is fun moving shapes around and the game made it way funner. You also don't just have a whole bunch of numbers. They gradually made it harder so that when you begun you could learn. If they always did a ghost image, it would be too easy and if you always did it like on level 3 it would be too hard. You couldn't just get the ghost image in the spot but it still showed around where it would go. I wanted to test to see if I was using the arrows right without losing points from the game and if I didn't understand I'd listen to the sound icons. If you don't understand how to use slides, flips, and turns it shows you. I need to look at practice to understand where the shape would end up if I put it in a certain place. Table 5.26 A child's perception of the Challenge-Driven Learning Model in Super Tangrams (Case 4). 220 Case 5 Table 5.27 lists the comments and responses of a girl in the gR+E group; this student's pretest score was 26.0% (nearly at the mean of that group), and her posttest score was 95.5% (nearly one standard deviation above the mean of that group). Stem I strongly agree that 1 like to leam math from computer games like Super Tangrams because Compared to other educational math programs I have used, I think Super Tangrams is the best because Compared to other educational games I have played, I loved Super Tangrams because Super Tangrams has helped me love motion geometry because I liked Level 3 most I learned most in Level 3 because 1 used Learn because I think it is good to have Learn because I do not think I could progress through the game without using Leam at all Response Learning math from games is so much funer and more interesting than learning it on paper. Other games I've played are boring and dull but Super Tangrams isn't, sometimes it's frustrating but never boring. I learned so much from it and I never got bored. It was fun to play. It's much more interesting and I understand it better now. I liked all the levels but level three was most challenging. Some of the things in levels I and 2 I already knew but level 3 was all new to me. When we first came to the turn where there was no shadow thing it confused us and learn helped us learn the angles. It teaches you what steps you have to go through to get the answer. I would have never completely understood motion geometry without it. Table 5.27 A child's perception of the Challenge-Driven Learning Model in Super Tangrams (Case 5). 221 Case 6 Table 5.28 lists the comments and responses of a girl in the gR+E+M group; this student's pretest score was 25.2% (nearly a third of standard deviation above the mean of that group), and his posttest score was 91.5% (nearly one standard deviation above the mean of that group). Stem I strongly agree that I like to learn math from computer games like Super Tangrams because I strongly agree that I liked the way motion geometry was presented in Super Tangrams Compared to other educational math programs 1 have used, I think Super Tangrams is the best because Compared to other educational games I have played, I loved Super Tangrams because Super Tangrams has helped me love motion geometry because I liked all the levels because I learned most in Level 3 because I used Learn because I think it is good to have Learn because I do not think I could progress through the game without using Learn at all Response Super Tangrams is fun and it teaches you in small steps and is better than book work. The presentation of Super Tangrams was good because every thing was easy at first and then it got harder. It's really fun and I recommend it to be used in teaching motion geometry. It's fun and you never get bored of it. with Super Tangrams you look forward to the next puzzle. There all fun but level one was the easist. you have to figure everything out for yourselve. I went to learn to see. how to rotate. Because if your not sure of something you can go there and learn it. you can practise there when you need to brush up your skills. Table 5.28 A child's perception of the Challenge-Driven Learning Model in Super Tangrams (Case 6). 222 Children's Comments about Learn Comments about Learn Because it will help me along my way. learn helped me pass the game. Because it helped me solve the puzzle. Because it helps you understand the concepts of the game. Because learn gets you to understand better. Because when we didn't understand something it would help us quite a bit. Because if it didn't most people would not understand the game and then would hot be having fun. Because 1 didn't understand what I was doing until I went to learn. I also think that the sound icons were very useful. Because if you didn't understand something, you would be stuck, if there wasn't learn. because if you don't understand things you can go to learn and practice and then try it out on the puzzles. Because if you get confused then you just go to learn. because when you got stuck you could just go to learn and learn of to flip or slide etc. Because when I always get struck. I always go to learn before asking the teacher. In higher levels it gets a bit confusing. Sometimes I struggle so I use learn. because when I was very frustrated in the game I wanted to figure out what I was doing wrong. [My partner] and me were frustrated in level two so weyoused learn. I wouldn't have the fogest on how to do the math. Because sometimes you needed interactive help to let you pass a puzzle. because if you didn't [use Learn] you wouldn't have a clue of what to do. So you can practice if your not quite sure what to do. Because sometimes you can't do it with out learning how to do it. Because when you need to know something learn button is right there on the screen. I used learn because I needed to know more info for my puzzles. because if you do not know something it will tell you how to do what you want to do in the game, without learn I wouldn't be able to find out how to do the puzzles. Because without it I cbuldn 't figure out what to do. To get more knowledge and it doesn't take away points if you use too much flip. Because you could learn what to do at the puzzles after learning it instead of guessing in every puzzle. Because it teaches you alot. Because with all the knowledge about the turn, flip, and slide helped me. because I did not know how to turn it in level 3 for a while, now I know how to turn it in level 3. I did because you don't lose points learning. I had never learned the concepts at school and learn explained them. I was learning motion geometry for the first time and I didn't know anything. . . . // you don't understand things you can go to learn and practice and then try it out on the puzzles. . . . I needed to learn so much. Because you have to learn the method of flip, turn, slide. Table 5.29 Explanations given by the gR-E, gR+E, and gR+E+M students regarding why they used Learn, and what they thought about Learn. 223 Summary and Discussion The Challenge-Driven Learning Model, as implemented in the R prototypes, was assessed. Six cases of children's holistic comments as well as an extended list of children's written explanations of why they used the embedded instructional module were presented. Some children commented that they enjoyed "learning math from games"; they considered the game "a lot of fun because it was challenging"; they liked how the game "teaches . . . in small steps" and that it "gradually" "work[ed] up to a challenge"; they experienced epistemic conflict when "there was no shadow thing [i.e., the ghost image]"; they had to "really . . . think"; they resolved their epistemic conflict by using the game's embedded instructional module to get explanations of the concepts and to "understand better", otherwise "people wouldn't figure anything out"; the instructional module was "needed" to "helped [them] pass the game" and "solve the puzzle[s]" so as to experience flow in learning and avoid "confusion", "frustrat[ion]", "struggle", "get[ting] stuck", and "los[ing] points" in the game (cf. Tables 5.20 and 5.21). The log files showed that all pairs of children had used Learn extensively.5 The log files showed that some students spent more than 20 minutes of a 35- to 40-minute session in Learn, sometimes accessing (i.e., listening) a particular sound icon in the Guided Interactive Practice component of Learn several times before they went back to the game. This pattern was particularly noticeable in Levels 2 and 3. Children's written comments 5 The log files showed that all pairs had used Learn. But in their design questionnaires, 58 of 59 children reported that they had used it. 224 contained many statements appreciating Learn, sometime even stating that they liked the game because it had Learn in it. These findings were corroborated by direct observations and post-hoc interviews. M y observations suggested that children derived enjoyment and self-confidence from being able to meet the mathematical challenges and advancing through the game.6 In one o f the interviews, a girl from the gR+E group stated: I haven't played very many math games before but the ones that I've played they're just, they do very easy stuff and they're not very challenging, or else they're too challenging. And most of them they don't really gradually work up to being very hard and they don 7 have the music in the background or the pictures and stuff - those make it more interesting. .. . [For me] the two most important things [in Super Tangrams] were learning and challenge, but probably challenge would be the most, then learning. These one are mostly the important ones, the ones where you actually learn something. But the music and the graphics and the sound effects, those just help it, they just make it more fun. . . . [AJfteryou finish the first 2 levels, what do you have left to do? So then the third level is good being there, because once you finally get to it, even if it takes like a month, then you can work on that. . . . If things are too easy they 're boring. A conceptually-grounded interpretation o f the overall pattern o f the comments, in conjunction with the achievement results presented in a previous section, clearly suggests that the model was effective in assisting children learn transformation geometry concepts and enjoy their learning. (This finding is further substantiated by the results in Section 5.2.6 in which an interview with a child is presented.) 5.2.4 Assessment of Sensory Stimuli The rationale for the inclusion of sensory stimuli in Super Tangrams was discussed in Chapter 3. To determine i f the inclusion of auditory and visual stimuli in Super Tangrams 6 See [Csikszentmihalyi, 1990] for a discussion on the "delights of science" and why scientists • who are engaged in non-challenging problems are not motivated to do research, and vice versa. 225 played an important role in satisfying children's experiential needs, children's quantitative and qualitative responses to questions in DQ that dealt with these issues were examined as primary sources of data. The results are reported below. 5.2.4.1 Background Music Of the 116 children who participated in the study, 73 of them used prototypes that included energetic background music (i.e., children who used the embellished prototypes). These children wore headphones contributing towards a sense of surround sound. Figure 5.5 shows a frequency count of children's responses to how much they liked "having background music in Super Tangrams" (Question 38). Slightly more than 86% of the children either "loved" or "liked" having background music in the program. Only 5% of the children "disliked" background music. L O L I SS D H Lik ing Liking: LO=Love, LI=Like, SS=So-so, ENDislike, H=Hate. Figure 5.5 Children's liking of background music. 226 Children were also asked whether or not they had the music on at all times, and were asked to explain their answer. Of the 73 children, 69% of them responded that they had the music on during their entire interaction with Super Tangrams, and only 31% did not always have the music on. Some typical explanations as to why children did or did not have the music on all the time are quoted in Tables 5.30 and 5.31. Response Because it got me into it, and if I didn't have the music, it would be boring. Because it helps me think. I can only work with noese. Because I always work hard with music. Because it won't be boring when you are thinking. It set the mood. Because it sort of help me relax. Because I always get excited when the music is on. It makes us think better. Makes the game exciting. Because it was awsome. It was entertaining. Because in normal math their is not music. Table 5.30 Children's explanation for why they always had the music on. Response _ _ Some time I just had to concentrate hard. Because it confused you when you where trying to think. It distracted me sometimes. After about 5 minutes it got really annoying. Sometimes it was annoying. Table 5.31 Children's explanation for why they did not always have the music on. Children's reactions to the background music was very intense. Many children, in their written comments referred to how much they liked Super Tangrams because it had 227 background music (cf. Section 5.1.1). A number of students would increase the volume to the maximum level and engage in solving the puzzles. The following are just two quotes from the children's questionnaires regarding the background music.7 (Also see Section 5.2.5.) [Compared to other educational math programs I have used, I think Super Tangrams is the best] Because you get to listen to music while your playing and other games don't have music. I loved it [Super Tangrams] because it had good music and you have fun while you are learning. It is interesting to note that many computer games have sound and some type of background tune. However, students reported that it was not just any music they were interested in; they liked the background in Super Tangrams because it was "cool" popular music. 5.2.4.2 Visual Aesthetics Of the 116 students in the study, 73 of them used prototypes that had very colorful backgrounds (vivid and textured blues, reds, and yellows), eye-catching graphics (multi-colored fractal images), and fun cartoon pictures on puzzle pieces (i.e., children who used the embellished prototypes). Forty-three of the students used prototypes that had black and gray backgrounds and used light brown and dark blue colored puzzle pieces (i.e., children who used the unembellished prototypes). Figure 5.6 shows the responses of all participant children to how important it is to have nice colors and graphics in a program (Question 37). Of the 116 children, 73% of 7 See [Csikszentmihalyi, 1990, pp. 108-113] for a discussion of experiencing flow through listening to music. 228 them considered nice colors and graphics either "very important" or "important". Comparatively, only 8% of children did not consider colors and graphics important. VI I SS NI NAAI Importance Importance: VI=Very important, I=Important, SS=So-so, NI=Not important, NAAI=Not at all important. Figure 5.6 Importance of colors and graphics for children. Figure 5.7 shows a count of the children's responses about how much they liked the colors and patterns of the puzzle pieces. More than 97% of children who used the colorful prototypes either "loved" or "liked" having colorful patterns and cartoons. In comparison, only 40% of children who used the unembellished prototypes were satisfied with plain brown and blue colors, and 21% either "disliked" or "hated" the colors and patterns of the puzzle pieces. 229 70% i mm 60%. l 50%. 1 40% • 1 " T'; • Colorful 30%. • ™1„ . M • Not Colorful 20% -1 ' ' - - f i 10% - 1 0% -L O LI SS D H Liking Liking: LOLove , LI=Like, SS=So-so, D=Dislike, H=Hate. Figvire 5.7 Children's liking of patterns and cartoons in puzzle pieces by group. The distribution of results for the colorful group is skewed. A large majority of these students either loved or liked having bright, colorful backgrounds and multi-colored, fun puzzle pieces. In comparison, the distribution of results for the non-colorful group is almost normal. On average, children thought that light brown and dark blue colored puzzle pieces were "so-so". During the study, I observed that some students who used embellished prototypes expectantly waited to see what the patterns of the next puzzle would be. Several times children exclaimed, "Look at that picture man!", and later commented that "It was fun to see the pieces in each level".8 In the questionnaires, many children commented about the program's graphics. Students made comments such as "[Nice colors and graphics] maikes the program catch your eye", "colorful pictures. made learning fun", "[I loved the program] because it had good graphics", "[Super Tangrams is the best educational math 8 See [Csikszentmihalyi, 1990, pp. 106-108] for a discussion of aesthetic flow experiences. 230 program] because the other math has [an insulting word] graphics and sound effects", "[the] designs were cool", and "[the] pictures were comical". While observing children whose prototypes were unembellished, I noticed some children complaining about the lack of nice graphics and sound. For this reason, these children never came to like the program, and I felt that the only reason they continued with the rest of the study was to be with the rest of their classmates. 5.2.5 Overall Affective Results To investigate children's affective response to the R+E prototype, nine questions from D Q (see Section 4.2.4) were used to construct an affective index. To assess children's affective response towards R+E compared to the other prototypes, these nine questions were also used to obtain children's affective response towards the other five prototypes. Table 5.32 displays the means for each individual question as well as an overall, mean affective index for each group. Responses were rated on a 5-point Likert scale in which the strongly positive responses on the scale were assigned a value of 5, and the strongly negative responses were assigned a value of 1. 231 Question Number Group 5' 62 133 144 16s if 357 38s 4? Mean Affective Index gR+E+M 4.8 4.5 4.6 4.6 4.4 4.7 4.7 4.8 4.4 4.6 gR+E 4.5 4.3 4.1 4.3 4.0 4.5 4.3 4.3 4.2 4.3 gR-E 4.6 4.1 4.1 3.9 4.6 4.6 3.4 N/A N/A 4.2 gF+E 4.5 4.3 4.1 4.3 4.0 4.6 4.9 4.4 4.4 4.4 gF-E 4.2 4.1 3.8 3.9 4.0 4.2 2.6 N/A N/A 3.8 gl+E 4.5 4.1 4.1 4.4 3.7 4.7 4.6 4.8 4.5 4.4 gl-E 4.5 4.2 4.3 4.4 3.9 4.6 3.5 N/A N/A 42_ I like to learn math from computer games like Super Tangrams. A) strongly agree, B) agree, C) undecided, D) disagree, E) strongly disagree. 21 like the way motion geometry was presented in Super Tangrams. A) strongly agree, B) agree, C) undecided, D) disagree, E) strongly disagree. 3 Compared to other educational math programs you have used, what do you think about Super Tangrams? A) it is the best, B) it is better than most, C) it is neither better nor worse, D) it is worse than most, E) it is the worst. 4 Compared to other educational games you have played, how much did you like playing Super Tangrams? A) loved it, B) liked it, C) so-so, D) disliked it, E) hated it. 5 How much has Super Tangrams helped you like motion geometry? A) has made me love motion geometry, B) has made me like motion geometry, C) hasn't done anything for me, D) has made me dislike motion geometry, E) has made me hate motion geometry. 6 Compared to other ways of learning math, how much fun was learning math through Super Tangrams? A) much more fun than other ways, B) somewhat more fun than other ways, C) just as fun as other ways, D) somewhat less fun than other ways, E) much less fun than other ways. 7 How much did you like the colors and patterns of the puzzle pieces in Super Tangrams? A) loved it, B) liked it, C) so-so, D) disliked it, E) hated it. 8 How much did you like having background music in Super Tangrams? A) loved it, B) liked it, C) so-so, D) disliked it, E) hated it. 9 How much did you like having sound effects in Super Tangrams? A) loved it, B) liked it, C) so-so, D) disliked it, E) hated it. Table 5.32 Indices of students' overall affective response towards the different prototypes. The mean affective indices for the gR+E+M and gR+E groups (4.6 and 4.3 out of 5) clearly suggest a strong positive affective response towards the main design of Super Tangrams described in Chapter 3 (i.e., the R+E prototype). Table 5.33 displays mean indices for each group, as well as each group's perception of how much mathematics they thought was involved in their prototype. 232 Question Number Group / Mean Affective Index gR+E+M 4.4 4.6 gR+E 4.2 4.3 gR-E 4.3 4.2 gF+E 3.7 4.4 gF-E 3.5 3.8 gl+E 3.1 4.4 gl-E 3.3 4.2 11 though that Super Tangrams was full of math. A) strongly agree; B) agree; C) undecided; D) disagree; E) strongly disagree. Table 5.33 Perception of how much mathematics was involved in each prototype by group. The indices suggest that students who used the R prototypes perceived them to be "full of math". (For more information about children's perception of the degree of mathematical challenge involved in the R prototypes, see Sections 5.2.2 and 5.2.3.) In comparison, the indices suggest that students who used the I prototypes were undecided. These results suggest that being mathematically challenging and knowledge-intensive did not make children like the R prototypes less than the F and I prototypes. Moreover, the R D C M and D C M interfaces, being more difficult to manipulate (or execute) compared with the D O M interface9 (see Section 2.1.6), did not have a negative effect on children's liking of the R+E prototype. The mean affective index provides a quantitative assessment of the R+E prototype. Mean affective indices for other prototypes also provides a relative, comparative measure of R+E's overall effectiveness. Most of the questions in Table 5.32 terminated with open-ended questions that asked children to "explain why" they had responded to the 9 That is, it was much easier to perform the task of solving the tangrams puzzles using the DOM interface style than the DCM or RDCM interface styles. See, for instance, Comment #1 made by one of the children in the preliminary evaluation phase: "to rotate or even to slide a shape was complicated, and there were a lot of steps to memorize!". 233 questions as they did. Table 5.34 presents a sample of the written comments of the children who used the R+E prototype. To prepare this table, all children's responses to DQ were collected, categorized according to their stem questions, and read in their entirety. Within each category, children's responses were further classified based on their commonality (e.g., satisfaction from learning, enjoyment of music, and comparison with textbooks). As discussed in Chapter 4, children's qualitative comments are intended to add more depth to their quantitative responses. To provide a general sense of why the R+E groups' affective indices were high (4.6 and 4.3 out of 5), Table 5.34 presents unique (e.g., "Because most don't really focus on the concept . . .") as well as typical (e.g., "Because there was music . . .") explanations of children with regard to the listed questions. 234 # Group Q' R2 Sex3 Comment 1 gR+E+M 5 A M It's so fun! Why do boring Math!? 2 5 A F Because it is easier and funner to place flip lines, or other lines after you play it is easier to picture in your head. 3 5 A M Because you could learn or/and master motion geometry and you could learn a lot, and also it is fun. 4 6 A F Because there was music and every thing was different. 5 6 B M Because it gradually got harder and made you use your knoledge. 6 6 A F because it was much easier to picture flips, turns, or slides especially since I was learning as I went along. 7 13 A M It taught me so much. 8 13 A F Because it got challenging, good graphics, rad music, was not confusing, and I learned alot. 9 13 A M Because most don't really focus on the concept or the skill it's trying to teach. 10 14 B F Because it taught me lots and was fun. 11 14 A M You get to show your ability to do math. 12 14 A M Because it has all the accesaries a kid would love. 13 16 A F I don 7 know but it just make me feel I love it. 14 16 B F It showed me how fun it is. 15 16 A F Because the way it is made, makes it fun and interesting. 16 17 A M Hand not moving much. Learn geometry from playing game. 17 17 A F There was no teacher to talk and talk so you only have 5 min to work. 18 17 A F Because you work hard and have fun and take your time. 19 gR+E 5 A M Because you get your own teacher with learn button. 20 5 A F I like it because you get so in to the game because its so fun you don't want to stop. Also the way the game is put together. 21 5 A M Because it makes learning fun instead of boring stuff on paper. 22 6 A F The graphics and sound effects made learning motion geometry fun and easier to learn. 23 6 B M I agree because it showed it [motion geometry] clearly, and it was easy to learn from. It showed how they moved. 24 6 A F The scores and learn were great. The actual design of the game was updated and original. . 25 13 A M Because other ones don't use enough math or others make it boring by making all math. 26 14 B F I don't know. I just enjoyed it. 27 14 A F I learned so much from it and I never got bored. It was fun to play. 28 16 A M It used to be hard but now it is easy. 29 16 B F Because I first thought it was boring and hard but once I played it, it became fun. 30 17 A F It's much clearer than an explanation from a teacher. 31 17 A M Because it's more fun playing game of math then to write pages of questions. 'Q=Question number in DQ; 2R=Children's Responses (i.e., A, B, C, D, or E); and 3Sex: M=Male, F=Female. Table 5.34 Written comments of children who used the R+E prototype of Super Tangrams. 235 Considering the results reported in Section 5.2.4, although R-E did not include embellishments, it received a strong positive affective response from children (4.2 out of 5). Table 5.35 presents a sample of children's qualitative comments so as to provide a general explanation for why the gR-E group's affective index was high. # Q' R2 Sex3 Comment 1 5 B F because it is alot more fun than learning from a text book and you learn so much better. 2 5 B M Because its games involve using math. 3 6 B M because it was fun and challenging. 4 6 C M Because I would like it better if it was also a bit of a game. Most educational games I have played are more fun and you only have to do bits of math to go on and not just math. 5 6 A M It was lots of fun and tought you all sort of math. 6 13 C M Because I have used better and worst programs. 7 13 B F Because it more fun than most of them but you still learn quite a bit. 8 13 B M it couldyouse just a little more graphics and interactive stuff. 9 14 B F Because it was using more math. 10 14 A M it was much more challenging. 11 14 C F Because it is like review the math again. It got the same things not very exciting and no music. 12 16 B F I liked it because now I understand it better. 13 16 B F It is getting harder and harder so you really want to keep on doing it. 14 16 A M On the first test I hated it but on the post test I loved it. 15 17 A F because it gets boring listning to a teacher talk or from a textbook and it is harder to understand from a teacher or a textbook. 16 17 A M It didn't time you or put you under stress. 17 17 C F no sound except to learn, not much exciting. 'Q=Question number in DQ; 2R=Children's Responses (i.e., A, B, C, D, or E); and JSex: M=Male, F=Female. Table 5.35 Written comments of children who used the R-E prototype of Super Tangrams. Children's comments suggest that most liked this version because it was "challenging" and taught them mathematics (see Comments #3, #7, #9, #10, #12, and #13 in Table 5.35). In contrast to the children who used the R+E prototype, some R-E children asked for more 236 excitement and embellishments (see Comments #4, #6, #8, #11, and #17 in Table 5.35; cf. Comments #4, #8, #12, #24 in Table 5.34 and Comments #2, #4, #5, #6, and #9 in Section 5.1.1). Analysis of all 15 students' comments in DQ suggested that one of the main reasons for a high affective index for the R-E prototype, besides it being a game, may have been the degree of challenge involved in the prototype. There were 16 explicit references to the word "challenging" in children's questionnaires and a number of implicit references to challenge (e.g., Comment #13 in Table 5.35). Summary and Discussion Children's affective responses towards the different prototypes were analyzed. Children indicated strong positive affective response towards the main prototype, R+E, as well as other prototypes. Being knowledge-intensive and "full of math" did not make children like R less than the F and I prototypes. Moreover, difficulty of interface manipulation did not have a negative effect on children's liking of the prototypes. Despite the stand-alone positive affective response that each prototype received, an emergent effect was observed during post-hoc interviews when children compared their own prototype with the ones they had not seen previously. Students who were shown prototypes with the same instructional package but differing in embellishments stated that they preferred the prototype with embellishments. Some students generally stated that embellishments would add "flavor" to the learning experience and make it "more fun". (See the interview in Section 5.2.6.) As noted above, the mean affective index for R-E was high (4.2 out of a 5-poirit scale). 237 In post-hoc interviews, the R-E students who saw the R+E prototype stated that they preferred the latter. The> students who in their questionnaires had stated that they "liked" Super Tangrams, commented that had they used R+E, they would have stated that they "loved" it because of the inclusion of embellishments. However, when the same children saw the F+E and I+E prototypes, they stated that, compared to these, they preferred R-E because it was more challenging and they learned more. Ordinarily, a designer might expect that the mathematical difficulty of R-E would have a negative effect on children's liking of this prototype compared to the other two which even include embellishments. The preference for a challenging and instructional environment over embellishments is a significant finding, which while not conclusive due to the limited number of students who were interviewed, suggests why R-E students' overall affective mean score was high. 5.2.6 An Interview: Was Super Tangrams a Flow Activity? The 'flow' experience [Csikszentmihalyi, 1975; Csikszentmihalyi & Csikszentmihalyi, 1988; Csikszentmihalyi, 1990] is a complex psychological construct that was described earlier in terms of a number of its indicators (see Sections 1.3 and 2.3.1). Some of these indicators included focused concentration, immersion, a sense of exhilaration and enjoyment, captivation, a sense of accomplishment, a sense of enlightenment or ordered consciousness, a sense of having had a rewarding experience, and a desire to repeat the experience. Furthermore, flow experiences are recalled easier. Previous sections have presented the results of this study so as to highlight specific features. However, children experienced each prototype as a whole. So far, the results, if 238 viewed collectively, may suggest that many children who used the R+E prototype experienced it as a 'flow' activity (cognitively as well as emotionally). However, the piecemeal description of children's reactions did not permit a holistic assessment of whether children who used the R+E prototype experienced it as a flow activity. To provide an integrated and holistic summary of the results in the previous sections from the perspective of one of the participant children, this section presents an extended sample of one child's experience. Of the 44 children who used R+E, nine (= 20% of children) were interviewed individually, three from the gR+E group and six from the gR+E+M group. Each videotaped interview was viewed several times. In these interviews, although all nine children expressed more or less the same opinions, some children seemed more articulate than others. Due to space, only one interview is presented in this section which seemed typical of the views of the majority of the students who used the R+E prototype. In this extended interview, the researcher used the child's design questionnaire to read his responses and comments back to him, and asked him why he answered as he did. The student's pretest achievement score was 1.1% (which was nearly one and a half standard deviations below the mean of that group), and his posttest score was 78% (which was about the same as the mean of that group). As seen below, this interview provides strong suggestive evidence, that the child experienced Super Tangrams as a flow activity. Indeed, almost all indicators of the optimal psychological experience of flow, as described by Csikszentmihalyi, can be found in this interview. In the interview, the child 239 1. indicated that he loved and enjoyed his experience in learning mathematics with Super Tangrams; 2. indicated that the experience was rewarding, fun, and challenging; 3. stated that he wanted to keep on doing the activity and would repeat the experience; 4. suggested he was immersed in the activity; 5. stated that he had to concentrate and engage in reflective cognition; and, 6. stated that he felt enlightened after understanding the topic. In the excerpts below "S:" stands for "student, and "R." stands for "researcher". The child's explanations are shown in italic characters, and some of the statements that reinforce the above observations are presented in bold face. R: You said you've learned so much that you can't believe it. 1 0 What made you think that you had learned so much? S: Because I could just do it after I played it, and it was soooo jun that you wanted to learn more. You just wanted to keep on playing and playing and playing and learning more. And it was fun cuz it wasn't boring, it was fun when you were learning it. R: Did you use the game while you were doing the second test? S: Yah. R: How? S: Well, cuz I kind of visualized what I'd be doing in the game and then I'd put it down on the paper; cuz I'd kind of visualize the screen and what everything would be like, so I knew what I was gonna do. His answer to Questions 1, 2, and 3 in DQ. 240 R: You say it's made you love motion geometry." How did it make you love motion geometry? S: Cuz it was Soooo FUN! You just didn't want to stop. You just wanted to keep on playing. R: Do you feel this passion for other school subjects? S: Sometimes maybe in art, but otherwise no. R: How about other math subjects? S: NOT AT ALL! R: Why not? S: They 're all so boring. They make you want to fall asleep! R: What did you think about Learn? S: Well it really helped you understand the concepts of it, and also when it had where you could do it yourself [referring to the Guided Interactive Practice component of the instructional module], that really helped so you didn't lose points in the game and so you didn't mess up in the game. So you could actually try and do it before you actually got it. Most of the times when I was trying to do it on Learn then it made sense, cuz you can click on the buttons [sound icons] and they tell you what to do and then when you try and do that, you can just learn it. R: Doesn't it bother you that you have to go out of the game [into Learn] and then come back? S: Well, sometimes, but it wouldn't be much fun ifyou didn't understand it.12 R: Why is understanding so important to you, because you keep repeating here [in DQ] so many times that it was important to "understand"? S: Well, cuz if you didn't understand it, then you 'djust be guessing a lot of the time and you wouldn't be really understanding it and well getting into it, like you know you're doing it and you know you're improving at it, and you know it's getting harder and you 're getting better. " His answer to Question 16 in DQ. 1 2 Need for order in consciousness. 241 R: You say you loved Level 3. 1 3 Why did you love Level 3? S: It's because in Level 1 it was like really fun, but there was not much of a challenge in it because you had the ghost image and you had the arc of rotation and all that, and so you could see where you were going. In this one [Level 3] you really need to know your stuff. You need to know what you were doing before you did it or else you could never finish it. R: Why is challenge so important for you [based on his comments in DQ]? S: It makes it more exciting. It makes it so you 're not just breezing through the levels so you're actually trying really hard. It's just, it's ... I can't really explain it! It's just... fun! R: So for you there's a relationship between fun and challenge? S: Yah! because if the game's so easy, if the game's really easy, then there's not much point. And, well, sometimes in math if questions are really easy, usually you'd think that it would make you happy, but it's kind of boring! R: Isn't too much challenge frustrating? S: Sometimes it can be but after you understand it, you're really enlightened and you really have fun with it. R: You say here the "minimum par" was very important for you. 1 4 S: Yah! So you weren't just doing all these dumb moves. You really need to concentrate on where you were doing it so you could get it in, cuz the points was good cuz it was more of a challenge and so you wanted to get more points so you needed to do your best. R: When you were playing the game, how much of your time was spent guessing, how much of it was spent thinking hard?15 S: I'd say like 90% of it was thinking hard. . . NO! 98% of it was thinking hard and sometimes, just sometimes if you couldn't really get it, then once in a while you guessed. 1 3 His answer to Question 21 in DQ. 1 4 His answer to Question 31 in DQ. 1 5 This question is intended to cross-validate his responses to Questions 11 and 12 in DQ. 242 R : D i d n ' t it bother y o u that y o u had to think so hard? S: Not really! Cuz it was rewarding afterwards when you got your points and that you were ahead and you got to the next level. R : Regarding the color pieces and graphics, again y o u say y o u loved those . 1 6 W h y ? S: Because well they're just fun. Instead ofjust boring old colors, you used really fun things. .. and it was fun to see the new pieces in each level [referring to the change ofpatterns from one puzzle to the next].17 R: G i v e n an educational game, which one o f these elements are the most important for you? N i c e graphics, learning, sound effects, music , score, par, and challenge. W h i c h one is the most important thing for you? S: Well, to me it's more a tie between challenge, learning and nice graphics. R : T e l l me what' the most important, then the next, and the next. S: Well, probably the challenge, learning, nice graphics, score, par, music, then sound effects. But all of them were great! So it's not like they're bad, the lowest ones. They're still great! But the ones at the top are like Great great! They 're like the king of them all! R: I 'm gonna show y o u a couple o f other versions [of the game], and y o u tell me what y o u think about these. S: OK R: O n e version w o u l d be like this [ R - E ] . There's no music, no par, no score; everything else is the same. T h e colors are like this. T h e y never change. S: It's depressing! R: Depressing! W h y ? S: Well, with music and everything it kind of dresses it up. It's like wearing your casual clothes and putting on a tuxedo. It's those little differences that make it nice and enjoyable.18 1 6 His answers to Questions 35 and 36 in DQ. 1 7 Refer to [Csikszentmihalyi, 1990], Section: 'Flow through the senses: The joys of seeing'. 1 8 Refer to [Csikszentmihalyi, 1990], Section: 'Flow of music'. 243 R: In your version when you click this [trying to adjust the angle of rotation], you hear a sound. Do this and tell me whether it feels different or not, whether that's important [this version had no sound effects]. S: / / feels different cuz you can usually hear it going CLICK CLICK CLICK and so you, it's just you can kinda sense when you 're going down or when you 're going up, or if you made one too many going down. Like if you knew, if you knew the angle that you were going to get at, and then you really didn't look - because you thought you were just moving down one, and you didn't really look and you went straight to GO, then you wouldn't really know. R: So that sound effect gives you a feeling? S: Yah, of what you are doing. R: How about a version like this one [I+E]? It has nice colors - more than yours, lots of colors, lots of patterns, but the way you move pieces is like this - you always have all of these [all the transformations are available]. S: That's easy, that's no challenge at all. Our class could"ve finished that in one period! R: Would you play with something like this? S: Maybe once or twice, but no! I like ours; it was nice and challenging AND fun. R: Which one would you think you would learn more from? This one [I+E] or yours [R+E]? S.Ours. R: Why? S: This one [I+E], you just move the piece around and you're just kind of. . . you know where your move . . . all you need to know is where you 're gonna move the piece. There's no skill to it or anything. R: How about something like this [F+E]? This is another version. This one is very much like Level 1 of your version. It has 83 puzzles. There is no par, you always have all of these [all three transformation buttons are available at all times]; none of them are disabled and it's always like Level 1 of yours so this is how you move things. What do you think of something like this? There is no Level 2 or 3. 244 S: Still too easy! It's good but it is not as good as our one. R: So which one do you think you would learn more from? S: Our one. R:Why? S: Because in our one [R+E] it gets, you don't have as much luxuries as this [F+E]. Like you need in Level 3 [see Chapter 3], with slide it has here, so you need to count the squares and the spaces that it is from; you don't just slide it right into place. You need for the turn . . . you see it has this [the entire arc of rotation, plus the ghost image] but in our one it just has this little line [referring to the representation of rotation in Level 3 of R+E; see Chapter 3] and this little line, so you need to know your angles and you need to know where it's gonna end up. R: Would you play with something like this [F+E]? S: Maybe for a bit, but it would be kind of easy. R: Do you think you're going to play it again? S: Of course! But there should be something after you pass it, you can put it onto a harder skill level, so like say by Level 2 it's like Level 3 on the first one, and by Level 3 it's like really hard, and so once you pass it on the normal one, then it gets harder, then you can keep on making it harder. You can make like 3 hardness levels. . . . In Nintendo you 're having Jun right? and a lot of the games they 're just like shooting up and stuff and you 're not learning much, you 're just having Jun. But in this [R+E] you're having fun AND you're learning. So well it would be like this - [shows with hands] this here would be Nintendo, and here would be Super Tangrams [shows with other hand right above the first]. R: Why do you have such a passion for this one? S: Well, I've probably learned more in this period of time than I've learned in this period of time in most other subjects. Maybe they've come really close but not really, I haven't learned as much. . . . I didn't know anything when I started. R: Does that motivate you to learn more? S: Yeah! r 245 R: Do you think that as a result of playing this it has given you confidence to do well on other math subjects? S: Yah. R: Why? S: Well, if you try and do it you can make it fun, and you can learn a lot in a really short time, cuz we only spent like two weeks on this and now I like know practically everything about motion geometry. R: You say this is much better than learning math in other ways.19 Why? S: It's just like I've said before. It's sooooo fun!! And you just wanna keep on playing and playing and you don't wanna stop. You could just sit there for the rest of your life! There's so many levels, you could just sit there for like a week and play straight It's Sooo FUN! 1 9 His answer to Question 17 in DQ. 246 Chapter 6 Conclusions This chapter includes three main sections. The first section summarizes the contents of the previous five chapters. The second section briefly enumerates the limitations of this research. The last section discusses the implications of this research, makes recommendations for design of more effective multimedia mathematics learning environments for children, and makes suggestions for future research. 6.1 Summary The purpose of this dissertation was to explore four main inter-related issues: 1. What role does the user interface play in multimedia mathematics learning environments? How do different interface styles influence learning? 2. How should the user interface be designed to support children's learning of explicit, formal mathematical concepts? 3. How should a learning environment in general, and the interface in particular, be designed to afford reflective cognition? 4. How should a multimedia learning environment be structured to be conducive to experiencing flow in learning? What are some design elements that can make children's learning of mathematics fun and enjoyable? 247 To explore these issues, a computer-based learning environment (Super Tangrams) was designed which was aimed at assisting middle-school children in learning two-dimensional transformation geometry. Super Tangrams used an overall design model, the Challenge-Driven Learning Model consisting of a situated, progressively challenging game activity coupled with an embedded instructional module. Implementation of the model involved four major design issues: 1. selection of an activity that could both support learning of transformation geometry and be developed into an engaging game activity; 2. operationalization of the activity into a seamless game environment and inclusion of embellishments intended to enhance the activity's fun and amusement; 3. conceptualization and operationalization of an interface style to support an explicit and formal understanding of transformation geometry, and to promote reflective cognition; and, 4. conceptualization and creation of an instructional module to assist children in learning transformation geometry so that they can cope with the mathematical challenges in the game and experience flow in learning. The Chinese Tangrams puzzle activity was selected and made into a computer-based activity to create the situated game. A number of embellishments were added to the game to make it a more enjoyable activity. To assist children in acquiring a more explicit and formal understanding of transformation geometry, the Direct Concept Manipulation (DCM) interface style was conceptualized and implemented. To promote reflective cognition so that children developed a deeper understanding of each transformation concept, the Reflective Direct Concept Manipulation (RDCM) interface style was 248 developed in which the visual feedback of the interface representations were gradually reduced. To further promote reflective cognition and help children discover the relationships among the transformations, a constraining mechanism was implemented which provided children with only a subset of the three transformations (the rest were disabled). The instructional module was an on-demand activity, and its content changed so as to assist children in constructing mathematical knowledge and coping with the mathematical challenges in the game. To evaluate the effect and interaction of the above design features, an eclectic approach to evaluation (qualitative and quantitative methods that included tests, questionnaires, observations, and interviews) was used. Evaluation was conducted in two phases: a preliminary evaluation and a summative evaluation. During the preliminary phase, a prototype of Super Tangrams was evaluated in a naturalistic classroom setting to assess its usability in terms of the program's mathematical content and children's reactions towards its design. During the summative phase, the main prototype of Super Tangrams (R+E) was compared with five other prototypes which varied in terms of two factors: instructional package (R, F, and I) and embellishments (+E and -E). The results of the studies were interpreted within the conceptually-grounded framework defined in Chapters 1 and 2. Main Findings Children's test scores and their perceptions of their own learning suggested that the Challenge-Driven Learning Model was effective in assisting children learn transformation 249 geometry concepts (see Section 5.2.1). The instructional module was liked by children and seemed essential in supporting their knowledge construction process (see Section 5.2.3). Children enjoyed 'learning math from [the] game" and considered it "a lot of fun because it was challenging" (see Section 5.2.3; also see Section 5.2.5). They liked how the game gradually became challenging, and they needed the instructional module to help them pass the game and not get stuck (see Section 5.2.3; cf. Section 5.2.2). Considering Super Tangrams' mathematically-challenging and knowledge-intensive content, children responded quite positively towards it (see Section 5.2.5). The results provided suggestive evidence that many children who used the program experienced flow in learning (see Sections 5.2.3, 5.2.5, and 5.2.6). Inclusion or exclusion of embellishments in different prototypes of Super Tangrams did not affect children's knowledge acquisition of transformation geometry (see Section 5.2.1.1). However, embellishments added affective flavor to the learning experience and were conducive to children's enjoyment of the learning activity (see Sections 5.1, 5.2.3, and 5.2.6). This was especially true of features such as background music and visual aesthetics (see Section 5.2.4). The Reflective (R) instructional package afforded a high degree of reflective cognition (see Section 5.2.2). Moreover, relative to the Formal (F) and Intuitive (I) instructional packages, the R package demanded the investment of more mental effort to solve the puzzles. Children's comments, their quantitative judgments, direct observations, and post-hoc interviews all suggested that children who used the R D C M interface style had to progressively think harder and use the instructional module more often to solve the 250 puzzles (cf. Sections 5.2.2 and 5.2.3). Children who used the F instructional package performed significantly better than children who used the I instructional package (see Section 5.2.1). The D C M interface style (used in the F package) afforded children to think more formally about the transformation concepts than the D O M interface style (used in the I package) did (see Section 5.2.1.4). In fact, the D O M interface style seemed to hinder children's formal understanding of the concepts (see Section 5.2.1.2). Children who used the R instructional package performed significantly better than children who used the F instructional package (see Section 5.2.1). The R D C M interface style as well as the constraints used in the R package afforded children to have a deeper understanding of the transformation concepts than the D C M interface style did (see Section 5.2.1.4; also see children's comments in Sections 5.2.1.3, 5.2.3, and 5.6). 6.2 Limitations of the Research It is important to note that the generalizability of this study's findings and implications is influenced by the following characteristics: 1. Students who participated in this research (i.e., investigative, preliminary, and summative) were from schools in upper-middle-class neighborhoods of Vancouver, and therefore the results may not be expected to generalize to other children populations. However, preliminary findings from scaled-down studies at other schools seem to support the ones reported here [Pimm, 1997]. 2. Al l participant students knew that they were taking part in a research study, and therefore they might have changed their behavior. 251 3. The experimenter (who was also the designer of the prototypes) was present during all study sessions, and therefore the experimenter may have unintentionally affected children's responding. 4. During the summative evaluation, students used the prototypes in pairs, and therefore the results may not be expected to generalize beyond dyads. 5. The summative evaluation of the prototypes was conducted in a very small room which was not a typical laboratory or classroom setting, and therefore the findings may not generalize to other contexts. The close proximity among the students might have contributed to the intense feelings of excitement expressed by most of the students. 6. The frequency of playing periods during the summative evaluation (i.e., 10 consecutive school days) is unusual in terms of current access patterns in schools, and therefore the results might have been affected by this. 7. Participants were grade-6 and grade-7 students, and therefore the results may not generalize to students in other grades. 8. The results may not generalize beyond transformation geometry and the tangrams activity. 6.3 Implications, Recommendations, and Future Research The research findings have implications for the design of effective human-computer interaction for educational software. Some design implications and recommendations as well as suggestions for future research are discussed below. 252 Interface Style in Educational Software At a general level, this research demonstrates that interface style in educational software plays a crucial role in how learners interact with the educational content, and consequently how they acquire knowledge and what knowledge they acquire. This research confirmed the findings of studies in cognitive performance (see Section 2.1.5) that tools can influence human cognition to think along a particular path. While the R D C M and D C M interface styles aided children to think about and visualize abstract and formal transformation geometry concepts, the D O M interface style did not seem to do so. In fact, D O M seemed to have supported and amplified children's naive and familiar understandings of transformation geometry concepts (as discussed theoretically in Sections 2.1.5 and 2.1.6 and demonstrated by the results reported in Section 5.2.1). This research brings into question the conventional interface design assumption that easier interaction or the need for less cognitive load is preferred. The minimal cognitive load assumption is premised on the notion that since computer users have limited cognitive resources to perform a given task, the supporting tools for performing the task should consume as few mental resources as possible, leaving the remaining cognitive resources for accomplishing the task at hand. The above assumption has been used to justify the implementation of direct manipulation graphical user interfaces. However, .this assumption only seems to apply to productivity tools; it may not completely extend to educational artifacts. In the design prototypes in this research, the D O M interface style, both in terms of conceptual familiarity and interaction, was easier for children to understand and manipulate than both the D C M and R D C M styles. Nonetheless, in terms 253 of educational effectiveness, D C M and R D C M were superior to D O M . In fact, a game-based educational artifact may consist of both toys and tools (see Section 2.1.4). One might say that components of the interface which are integral parts of the toy (e.g., R D C M in Super Tangrams) should be challenging; however, components of the interface which are integral parts of the tool (e.g., instructional module or sound icons) should be easy to use and understand. In educational software, the design guideline regarding ease of use may be formulated differently. In educational software, HCI designs should aim at reducing learners' cognitive load for performing non-content-related tasks so as to enable learners to allocate more cognitive resources to understand the educational content. The findings of this research imply that, unlike productivity tools, when mathematical content is embedded in a task (e.g., Super Tangrams), interaction with the system should not necessarily be made easier. Indeed, ease of interaction may unintentionally communicate the message that the user need not invest much mental effort to perform the task and plan his/her actions with care. More research is needed to determine the extent to which these hold true for mathematics as a whole and other educational subjects. Direct Concept Manipulation Direct Manipulation (DM) interfaces were discussed in Chapters 2 and 3. Hutchins et al. [1986] outline different aspects of "directness" which are important in discussing the implications of Direct Concept Manipulation (DCM). They [pp. 94-101] state: An interface introduces distance to the extent there are gulfs between a person's goals and knowledge and the level of description provided by the systems with which the 254 person must deal. These are the Gulf of Execution and the Gulf of Evaluation. . . . The Gulf of Execution is bridged by making the commands and mechanisms of the system match the thoughts and goals of the user as much as possible. The Gulf of Evaluation is bridged by making the output displays present a good Conceptual Model of the system that is readily perceived, interpreted, and evaluated. The goal in both cases is to minimize cognitive effort.. . . The more of the gulf spanned by the interface, the less distance need be bridged by the efforts of the user.. . . Semantic distance in the gulf of execution reflects how much of the required structure is provided by the system and how much by the user. The more that the user must provide, the greater the distance to be bridged. To bring out the full potential of D M interfaces, Hutchins et al. [1986, p. 118] suggest that the challenge of interface designers is to "provid[e] . . . new ways [to think of and interact with a domain] and creat[e] conditions that will make [users] feel direct and natural". Directness is characterized in terms of "distance" and "direct engagement". Distance refers to the "gulfs" that must be spanned. Direct engagement refers to the feeling that results when the user is directly engaged with control of the interface objects. Accordingly, to produce the feeling of directness in users, the challenge of D M interface designers is to design systems that maximize engagement and minimize distance. One of the most important contributions of this dissertation is in conceptualizing and providing a new DM construct, i.e., D C M (see Section 3.4.1). Results from this research suggest that D M graphical interfaces should be used with care in the context of interactive multimedia mathematics learning environments. The D M construct can unintentionally be misapplied or misused. In many concept-centered learning environments, manipulation may be directed towards objects. Based on the results of this research, it can safely be stated that, in learning transformation geometry concepts, direct manipulation of objects (or shapes) was the main educational deficiency of the I prototypes. In contrast, the 255 D C M interface style helped learners explicitly and directly interact with formal interface representations of the embedded concepts and acquire more transformation geometry knowledge. These findings imply that in a concept-centered environment which uses a D M interface style, manipulation should be directed towards concept representations rather than objects. In the D C M construct, the two parameters of D M interfaces (directness and engagement) are still present, however, with some modifications in their conceptualization and characterization. In D C M interfaces, it is still desirable to produce a feeling of direct engagement for users. However, the user is in direct control of graphical representations of concepts rather than objects. In terms of spanning the gulfs of execution and evaluation, it is still desirable to keep the distance across the gulf of evaluation to a minimum so that learners can readily perceive and interpret the results of their actions. However, it does not seem desirable for the interface to fully span the gulf of execution for the learner. In a concept-centered learning environment, the D C M interface must place a semantic distance across the gulf of execution for the learner to bridge. That is, the learner must exert effort to learn the conceptual semantics of the interface. Rather than the system conforming to and supporting how the learner already thinks, the learner must conform to the conceptual model of the system. Knowing that they are involved in a learning activity, learners already expect to encounter this distance. Consequently, the learning distance does not diminish the feeling of "directness" for the learner. The way D C M interfaces are operationalized plays an important role in their instructional effectiveness. However, operationalization of D C M interfaces is not 256 straightforward. In operationalizing concept-centered interactive mathematics learning environments, D C M designers must carefully examine concept teaclimgAearning literature in mathematics education to determine: 1. what symbolic representations to use to facilitate development of proper mental schemas, (and, if need be, what type of scaffold to use as part of the representation so that the initial encounter with a concept is not beyond learners' "zone of proximal development" [Vygotsky, 1978]); 2. what elements of the symbolic representation of a concept to allow to be manipulated (i.e., what handles to manipulate), and in what order; and, 3. what type of mouse interaction protocol to implement to direct learners attention towards the essential aspects of a concept. When designing D C M interfaces, it is important to note that not all concepts can be treated the same, and small changes in design can have unpredictable effects on how a concept is conceived. In this research, transformation geometry naturally lent itself to D C M . Nevertheless, a great deal of work was involved in determining how to operationalize each transformation concept so that its mathematical properties were properly reflected through the interface. For instance, the interface representation of the translation arrow had handles at its head, tail, and middle. The arrow could have been operationalized differently by implementing two handles: one to control an angle to adjust the direction of the arrow, and one to adjust the length of the arrow. However, this implementation might have made it difficult for children to discover the positional independence of the vector. Another example is the inclusion of the ghost image as an instructional scaffold. If the ghost image was not included, for instance, it might have been 257 very difficult for children to readily observe the relationship between manipulating the concept representations and the resulting transformational effects. This research has explored a very limited aspect of the potential that D C M interfaces may offer in teaching abstract mathematical concepts. A great deal of research is needed before appropriate design guidelines can be developed for D C M interfaces. For instance, in the case of numbers and operations like addition and subtraction, it is not clear how D C M lends itself to presenting these concepts. Further investigation is needed to determine how to apply the D C M interface style to different concepts. Reflective Direct Concept Manipulation There are many ways in which reflection can be promoted (e.g., asking questions). One of the methods that this research used was a strategy in which the elements of the interface representations of concepts were gradually removed. Removal of visual feedback might be considered as a logical extension of D C M interfaces. As Skemp [1986] points out most mathematical concepts are abstract and secondary concepts, and their understanding requires reflection. Consequently, a D C M interface requires a mechanism to promote reflective cognition so that learners develop conceptual insight. Once the D C M interface is in place, it must draw the learner's attention towards a deeper understanding of a concept by making the representation of the concept more abstract. A well designed system must constantly and in a stepwise manner readjust the distance across the gulf of execution. However, it is important for the distance created in each step to be manageable for the learner to span. The reduction of 258 feedback must be such that learners can use their current knowledge to discover and determine how to bridge the new gulf of execution at each step. Stated differently, as the visual scaffolds are removed (i.e., process of scaffolding), the learner should be presented with new representations communicating a sense of unfamiliarity and a need for a greater "amount of invested mental effort" (AIME) or "mental elaboration" (see [Salomon, 1981]). This process, however, should be performed in a careful manner so that by further elaboration the learner is able to discover the "progressive differentiation" (see [Novak & Gowin, 1984]) of a concept in terms of its specifics and details. In the R prototypes, the last stage of abstraction (Level 3) still included pictorial representations of transformation concepts. An important future research goal is to determine how to design an interface for learning mathematics so that pictorial representations gradually convert to algebraic notations (i.e., command-based interface). For instance, in the case of the rotation concept, after Level 3, there could exist a Level 4 in which the arc of rotation disappeared altogether. In Level 4 children would have to specify the parameters of the arc of rotation by typing in an (x , y) pair as the coordinates of the center of rotation and a number as the angle of rotation (see Figure 6.1). From this study, it does not seem likely that children would react to such an interface negatively since they might view a command-based interface in Level 4 as part of the challenge of the game. A visual D C M interface that gradually converts to an algebraic notation may satisfy the educational belief that algebraic knowledge should be built on the foundation of visual knowledge (see Chapter 3). Additionally, since feedback reduction from highly visual to algebraic abstraction is gradual, the command-based stage of the' interface may 259 not have a negative affective result on children. Further research is needed to explore visual-to-algebraic interfaces. An effective visual-to-algebraic interface style may well provide a timely solution for DM's current inappropriateness for some educational topics. Concrete Experiential Interaction Abstract Reflective Interaction Step 1 +292.5° Center (x , y) = (? , 7) Angle = ? Step 2 Step 3 Step 4 Visual D C M Interlace Visual Feedback Reduction Command-Based Algebraic Interface Figure 6.1 Gradual conversion from a D C M to a command-based interface style. Constraints Another mechanism to promote reflection was the use of constraints. The constraints feature is conceptually orthogonal to the D C M interface style. That is, constraints are a general mechanism and can play an important role in any learning environment (not just, in conjunction with DCM) to stimulate an understanding of the limitations and relationships of concepts. In the R prototypes, constraints served to highlight the relationships among the different transformation geometry concepts. However, the choice of constraints must be specific to the mathematical concepts in a learning environment. Therefore, designers 260 must examine concept teaching/learning literature in mathematics education to determine what subsets of the interface should be disabled and under what conditions. In this research, due to logistical reasons, R D C M and constraints were combined in one instructional package. Future research should separate these two strategies to determine the effect of each on learning and concept formation. Background Music and Aesthetics Based on children's achievement results, no evidence was found to suggest that background music interfered with children's learning (cf. R-E and R+E results). On the contrary, the majority of students reported that background music made learning of the mathematics "fun" and enjoyable and helped them "relax" and "think better". Only a few students commented that they sometimes needed to turn the music off to be able to "concentrate", and only a couple of students found it "annoying" altogether. To satisfy all children, it seems reasonable to incorporate background music in interactive multimedia mathematics learning environments. However, it is essential that children be given the option to turn the music up, down, or off altogether. By providing these options, children can choose what satisfies their own particular learning styles and psychological needs - whether it be to have the music off altogether, have it on sometimes, or have it on all the time. Since children may have different musical tastes, it would be helpful to provide children with a facility that permits children to select the background music either from a set of different types of music or load in their own music. 261 The results show that a large majority of the students either loved or liked having bright, colorful backgrounds and multi-patterned puzzle pieces in the prototypes they used. Comments such as "[nice colors and graphics] makes the program catch your eye", "colorful pictures made learning fun", and "[I loved the program] because it had good graphics" suggest that inclusion of visual aesthetics in learning environments may be very important to some children. The issue of beauty and aesthetics in the design of interactive learning environments is one that should be taken seriously and investigated further, especially in environments designed for children. To use a food metaphor, the preparation and design of the educational content of interactive learning environments constitutes its nutritional part, but inclusion of embellishments can act as the spice that changes the flavor of the instructional package. Challenge-Driven Learning Model and Flow in Learning Two of the main researchers whose work influenced this study are Norman [Norman, 1993] and Csikszentmihalyi [Csikszentmihalyi & Csikszentmihalyi, 1988]. Norman suggested that the trick in teaching was to entice learners into excitement about a subject so that they want to do the hard work involved in reflection and deep learning, and then give them the proper tools to reflect, explore, compare and form proper conceptual knowledge. Csikszentmihalyi called for the urgent application of the flow model to educational settings and suggested that the structural characteristics of this model could be built into an activity by design. 262 The Challenge-Driven Learning Model, as implemented in Super Tangrams (see Chapter 3), was conceptualized primarily to address the above issues. The game activity helped to entice children into excitement about transformation geometry. It provided an environment in which children explored the topic. The user interface acted as a tool to promote reflection on transformation geometry concepts. The instructional module provided children with a tool to form proper, formal conceptual knowledge. The reciprocal relationship between the game activity and the instructional module helped keep a balance between children's perception of the mathematical challenge in the game and their mathematical skills, an important characteristic of flow in learning. Engagement of experiential senses using auditory and visual stimuli contributed towards enhancing the learners' enjoyment of the learning activity. Children reacted very positively towards the implementation of the Challenge-Driven Learning Model. They appreciated the gradual increase in the mathematical challenge in the game and equated it with fun. Surprisingly, even some students who had difficulty understanding the mathematical concepts stated that they "found it hard to understand but it was fun". Many children wanted to repeat the experience and found it rewarding. Children perceived the instructional module as indispensable for making progress in the game and understanding the mathematical concepts. Not only did children enjoy, the learning activity, but also they dramatically improved in their posttest achievement scores. The effectiveness of the Challenge-Driven Learning Model in terms of both knowledge construction and enjoyment of learning mathematics suggests that the model should be examined closely. The generic nature of the model permits it to be generalized 263 and adapted for designing other interactive multimedia mathematics learning environments for children. The Challenge-Driven Learning Model that was conceptualized scratched the surface of the possibilities of how to apply Norman's and Csikszentmihalyi's psychological prescriptions to the design of interactive multimedia mathematics leairiing environments for children. We know very little about how to design educational activities that are conducive to flow experiences, and even less about how to scientifically measure such experiences. A great deal of research on the Challenge-Driven Learning Model is needed to improve it. Some specific suggestions for future research are: 1. Activity Selection: Research is needed to determine which activities promote concept development for different mathematical topics. Such research would allow designers of interactive multimedia mathematics learning environments to select educationally suitable mathematical activities for children. 2. Game Activity: a) Research is needed to develop guidelines on how to minimize children's focus on the game activity, and gradually shift it towards the mathematical activity. b) Research is needed to determine what types of embellishments enhance children's enjoyment of the learning activity without distracting them from the main goal of the activity - i.e., learning mathematics. c) Research is needed to investigate other navigational strategies besides the sequential navigation used in Super Tangrams. Skemp [1986] suggests that teaching mathematics requires a careful analysis of the sequence in which mathematical concepts are communicated. An important research question is: Can children be put into an environment which has random access navigation and provides children with the option of selecting what they want to learn? 264 3. Instructional module: a) Research is needed to investigate how to increase the coupling between the game module and the instructional module. For instance, should the instructional module address the mathematical challenge of each puzzle individually, or should it be level-based (as in Super Tangrams)? b) Research is needed to investigate how to encourage children to use the instructional module more often. As this research demonstrates, design of effective interactive multimedia mathematics learning environments for children is complex and must take into account a broad range of cognitive, affective, aesthetic, and emotional issues. 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