Technology Tips: Multiple Representations Using Dynamic Geometry

We introduce the problem of detecting a group of students from classroom videos. The problem requires the detection of students from different angles and the separation of the group from other groups in long videos (one to one and a half hours).We use multiple image representations to solve the problem. We use FM components to separate each group from background groups, AM-FM components for detecting the back-of-the-head, and YOLO for face detection. We use classroom videos from four different groups to validate our approach. Our use of multiple representations is shown to be significantly more accurate than the use of YOLO alone.

Cognitive dissonance represents someone’s cognition discrepancy on a particular concept and then encouraging the person to comprehend that specific concept better. Cognitive dissonance can be constructed through the use of multiple representations in chemistry learning. Chemistry learning’s nature always involves abstract and complex concepts. Therefore, students mostly face difficulty understanding chemical concepts so that they experience misconceptions. This study seeks to describes the syntax of cognitive dissonance combined with multiple representations to illustrate the implementations of multiple representation based cognitive dissonance strategies (MRCD) in reducing the students’ misconception within chemistry learning. It descriptively examined the articles related to cognitive dissonance and multiple representations. The obtained data were secondary in the form of relevant research articles. The data were analyzed using a descriptive qualitative technique. The study results suggest that multiple representation-based cognitive dissonances consist of a number of phases, namely the prerequisite knowledge, creation of cognitive dissonance, implementation of attained knowledge, and reflection. Each of those phases involves multiple representations. In the end, the multiple representations-based cognitive dissonance strategy that refers to syntax has been confirmed to effectively reduce students’ misconception and improve the learning quality. The multiple representation-based cognitive dissonance strategy should be adopted in chemistry learning since it facilitates students to construct abstract chemical knowledge. Further, this strategy can bring the utmost results if combine with the constructivist learning model.

The research question addressed by this study is whether or not students made effective use of dynamic geometry software to explore the validity of a geometrical statement. Particular attention was paid to the use of definitions and the types of justifications students employed. Interviews conducted with six secondary mathematics preservice teachers were video and audio taped and video output of the interviewee's Geometer's Sketchpad (GSP) work was recorded on video tape. The interviews were transcribed along with a synchronized record of the GSP work. The transcripts and videotapes were analysed and coded. Results indicated that reasoning based on prototypically constructed categories and difficulty correctly parsing mathematical statements, especially the ‘for all’ quantifier, interfered with the effective use of dynamic geometry software in justifying mathematical assertions. It may be that explicit instruction in how to make use of software programs to explore the validity of an assertion is needed.

This month, we bring you some “Technology Tips” that are related to probability and statistics. We begin with a contribution by Bishnu Naraine and Peiyi Zhao: an exploration that uses The Geometer's Sketchpad to discuss the broken-stick problem, which is also known as the spaghetti problem. This totally geometric solution nicely complements the analytic solution of the same problem, given in “The Spaghetti Problem Problem,” by Dennis Ippolito, which appeared on pages 422–26 of the May 2000 issue of the Mathematics Teacher. Next is a feature by Allen Holmes that discusses obtaining an explicit definition for a recursively defined sequence by using the built-in statistics regression ability of graphing calculators to model the equation of the sequence. Then a short program by John Jose for the TI-83 and TI-83 Plus graphing calculator helps investigate the law of large numbers by simulating the results of a simple two-dice experiment. Finally our “Surfing note” features a Web site that is devoted to statistics. It contains many resources for an elementary statistics course.

Contents: Preface. Part I: Why Teach Geometry? E.P. Goldenberg, A.A. Cuoco, J. Mark, A Role for Geometry in General Education. K. Gravemeijer, From a Different Perspective: Building on Students' Informal Knowledge. D. Chazan, M. Yerushalmy, Charting a Course for Secondary Geometry. R.L. Devaney, Chaos in the Classroom. Part II: Studies of Conceptual Development. J. Pegg, G. Davey, Interpreting Student Understanding in Geometry: A Synthesis of Two Models. R. Lehrer, M. Jenkins, H. Osana, Longitudinal Study of Children's Reasoning About Space and Geometry. R. Lehrer, C. Jacobson, G. Thoyre, V. Kemeny, D. Danneker, J. Horvath, S. Gance, M. Koehler, Developing Understanding of Space and Geometry in the Primary Grades. D.H. Clements, M.T. Battista, J. Sarama, Development of Geometric and Measurement Ideas. M.T. Battista, D.H. Clements, Students' Understanding of Three-Dimensional Cube Arrays: Findings From a Research and Curriculum Development Project. J.A. Middleton, R. Corbett, Sixth Grade Students' Conceptions of Stability in Engineering Contexts. K. Raghavan, M.L. Sartoris, R. Glaser, Interconnecting Science and Mathematics Concepts: Area and Volume. D. Dennis, J. Confrey, Geometric Curve-Drawing Devices as an Alternative Approach to Analytic Geometry: An Analysis of the Methods, Voice, and Epistemology of a High School Senior. K.R. Koedinger, Conjecturing and Argumentation in High School Geometry Students. Part III: Defining a New Semantics of Space: Computers, Software, and the Electronic World. E.P. Goldenberg, A.A. Cuoco, What Is Dynamic Geometry? M. de Villiers, An Alternative Approach to Proof in Dynamic Geometry. J. Olive, Opportunities to Explore and Integrate Mathematics With the Geometer's Sketchpad. D.L. Watt, Mapping the Classroom Using CAD Program: Geometry as Applied Mathematics. L. Zech, N.J. Vye, J.D. Bransford, S.R. Goldman, B.J. Barron, D.L. Schwartz, R. Kisst-Hackett, C. Mayfield-Stewart, The Cognition and Technology Group at Vanderbilt, An Introduction to Geometry Through Anchored Instruction. K.A. Renninger, S.A. Weimar, E.A. Klotz, Teachers and Students Investigating and Communicating About Geometry: The Math Forum.

Mathematical problem solving in Basic General Education remains a pedagogical challenge closely related to conceptual understanding, representation, and logical reasoning. This study aimed to analyze the influence of visual thinking and the use of multiple representations on the development of mathematical problem-solving skills at this educational level. A quantitative approach with a quasi-experimental pretest–posttest design was applied to a sample of 120 fifth-, sixth-, and seventh-grade students from public urban schools. The intervention was based on the systematic use of diagrams, schemas, and visual representations within active learning methodologies. The results revealed statistically significant improvements in problem interpretation, procedure selection, and solution justification, as well as an increase in high-performance levels. The study concludes that visual thinking and multiple representations are effective instructional strategies for fostering meaningful and functional mathematical learning in Basic General Education.

Students organize scientific knowledge and reason about issues in the Earth sciences by manipulating internally-constructed mental models and socially-constructed, expressed, conceptual models. The Earth sciences, which focus on the study of complex, dynamic, Earth systems, may present unique cognitive difficulties to students in their development of authentic, accurate expressed conceptual models of these systems. This pilot study came about as we were seeking to construct inquiry modules to assist undergraduate students as they developed an understanding of eutrophication along the coastal margin, a good example of a complex, dynamic, environmental process. The modules we developed coupled the use of physical models and information technology (IT)-based multiple representations with an inquiry-based learning environment that allowed our students to develop and test their conceptual models based on available evidence and to solve authentic, complex, and ill-constrained problems. Our hypothesis was that the quality of students' conceptual models would predict their performance on inquiry modules, and that students' prior knowledge (measured by number of previous courses in geology) would mediate the strength of the relationship between students' model expression and their inquiry performance. Statistical results of this study indicated such a relationship existed only among students in the high prior knowledge group. In the light of our findings, we make recommendations for pedagogical accommodations to improve all undergraduates' abilities to understand complex, dynamic, environmental systems, with a particular emphasis on students who have lower levels of prior knowledge.

In previous chapters, we discussed the use of visualization from the perspective of the different tasks involved, and the different representation formats possible. We now attempt to tie the tasks and representations together. In Section 18.1, we first discuss experimental evidence supporting the use of multiple representations, and then discuss how optimization systems can support multiple representations (Section 18.2). Finally, we discuss how these representations can be built and maintained by multiple users perhaps working in multiple locations at different points in time (Section 18.3).KeywordsMultiple RepresentationGroup Decision Support SystemHomogeneous ApproachMathematical Programming FormulationElectronic Meeting SystemThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Studies indicate that the use of multiple representations in teaching helps students become better problem solvers. We report on a study to investigate students’ difficulties with multiple representations. We conducted teaching/learning interviews with 20 students in a first semester calculus‐based physics course. Each student was interviewed four times during the semester, each time after they had completed an exam in class. During these interviews students were first asked to solve a problem they had seen on the exam, followed by problems that differed in context and type of representation from the exam problem. Students were provided verbal scaffolding to solve the new problems. We discuss the common difficulties that students encountered when attempting to transfer their problem solving skills across problems in different representations.

This study examines the impact of implementing multiple representation method in responding to the challenge of learning science specifically in relation to enhancing students’ reasoning skill. Pre-experimental method was applied to obtain research data from participants consisted of 8th grader students at a private junior high school in Bandung, Indonesia. The effect on reasoning skill was studied from students’ arguments in answering question related to respiratory system. The arguments were then analyzed by the completeness of argument component, level of argument, and quality of argument referring to Toulmin’s argumentation pattern (TAP). The findings show that students experienced significant improvement in reasoning skill expressed in their arguments. The improvement was evident in the higher argument level and better argument quality achieved by students after the implementation of multiple representation method. In relation to quality of argument, the overall quality of argument shifted from the domination of weak arguments to the domination of strong arguments. Additionally, the use of multiple representation in learning assists students to be able to use various forms of information such as graphics, visual images, tables, scientific formula, and written expression in explaining concepts.

I have always used concrete marupulatives, such as marshmallows and toothpicks, to create models for my geometry and discrete-mathematics courses. These models have come in handy when discussing volume, introducing the 4-cube, or illustrating isomorphic or bipartite graphs. However, after discovering what a dynamic geometry–software package could do for geometry teaching, which has been well documented by research (e.g., Battista and Clements [1995]), I realized that this type of technology also had much to offer for teaching graph theory in my discrete-mathematics course. Although this article discusses The Geometer's Sketchpad 3 (Jackiw 1995), any software that can draw, label, and drag figures can be substituted for Sketchpad.

Dynamic geometry tools (e.g., Cinderella, Geometer's Sketchpad, Cabri, Eukleides) visualise geometric objects, allow interactive work, and link formal, axiomatic nature of geometry (most often — Euclidean) with its standard models (e.g., Cartesian model) and corresponding illustrations. These tools are used in teaching and studying geometry, some of them also for producing digital illustrations. The common experience is that dynamic geometry tools significantly help students to acquire knowledge about geometric objects. However, despite the fact that geometry is an axiomatic theory, most (if not all) of these tools concentrate only on concrete models of some geometric constructions and not on their abstract properties — their properties in deductive terms. The user can vary some initial objects and parameters and test if some property holds in all checked cases, but this still does not mean that the given property is valid.

We selected a sixth grade student to participate in this study. Prior to remedial instruction, the subject did not understand the concept of proportionality and used ratio formulas excessively to solve problems without any meaning on it. In addition, he could not employ multiple representations to explain his solutions. Using an electronic whiteboard and Geometer’s Sketchpad (GSP) to design the dynamic multiple representations for proportional scenarios in remedial instruction induced the subject has better understanding of proportions as well as a better grasp of the linear relationship between two variables. In the final test, the subject was able to use multiple representations to solve problems and check his calculations, therefore increasing problem solving accuracy.

Summary PCLogo and Geometer's Sketchpad are powerful tools that may be used by mathematics teachers who want to integrate technology with geometry instruction in the elementary classroom. The purpose of this study was to examine the usefulness of PCLogo and Geometer's Sketchpad to stimulate thinking about geometric concepts in elementary age children. We used a collective case study design that included four cases, two girls (ages 8 and 10) and two boys (ages 10 and 11). All participants were trained to use PCLogo and Geometer's Sketchpad to construct geometric shapes and measure the attributes of the shapes. After the training, participants used these technologies as tools to stimulate thinking about geometric concepts. As a result of our observations of children's performances as they reasoned about geometric concepts, we developed a learning model for teaching children about geometry.

As a kind of mathematical geometry education software, dynamic geometry software is an application software that can interactively create and manipulate geometric objects. Its vivid image changes and colorful geometric objects provide teachers with an efficient Geometry teaching tool. Comparing and analyzing the existing geometry software, we found that there is a curve smoothness problem in its trajectory drawing. In this article, we propose the uniform point selection method based on the principle of the trajectory generation geometric method. Under the premise of ensuring rendering efficiency, extensive case studies have shown that the smoothness of the trajectory curve generated by this method has been improved.