동적기하 환경에서 정사영에 관한 공간 능력 발현과 문제해결

This paper examined spatial visualisation ability as correlate of senior school students’ achievement in Physics in Sokoto State, Nigeria. The objectives of the study were to: (i) examine the relationship between senior school students’ spatial visualisation ability and their achievement in Physics; (ii) determine the relationship between senior school students’ spatial visualisation ability and their achievement in physics based on gender; (iii) assess the relationship between students’ spatial visualisation ability and their achievement in physics based on school type. This research adopted ex post facto research of the co-relational type. The population for the study were Senior Secondary School II (SSS II) students in Sokoto State. Proportional sampling technique was used in sample selection. Seven hundred and thirty-one (731) SSS II students, offered Physics in senior secondary schools across the three senatorial districts in Sokoto State, Nigeria, formed the sample for the study. Research instruments, employed to elicit data for the study, were: Students’ Spatial Ability Test (SSAT), and Physics Achievement Test (PAT). The instruments were validated by experts in science education, and Practicing Physics teachers in Sokoto, giving reliability coefficients 0.79 and 0.89 respectively. The data gathered were analysed using Pearson Product Moment Correlation Statistic and Z-test statistic, at .05 level of significance. The findings of the study were that: there was statistically significant relationship between students’ spatial visualisation ability and their achievement in Physics (r=0.32, p < .05); there was statistically significant difference in the strength of the relationship between students’ spatial visualisation ability and their achievement in Physics based on gender as the Zobs – value (-2.01) was outside ±1.96 boundary in favour of female students; and iii there was statistically significant difference in the strength of the relationship between spatial visualisation ability and achievement in Physics based on school type as the Zobs – value (-5.08) was outside ±1.96 boundary in favour of private schools. It was concluded, that students’ spatial visualisation ability positively predict their achievement in Physics. It was recommended, that students should be trained on spatial ability so as to be able to predict correctly their achievement in Physics

In this study, we considered dynamic geometry software (DGS) as the tool that mediates students’ strategies in solving and posing problems. The purpose of the present study was twofold. First, to understand the way in which students can solve problems in the setting of a dynamic geometry environment, and second, to investigate how DGS provides opportunities for posing new problems. Two mathematical problems were presented to six pre-service teachers with prior experience in dynamic geometry. Each student participated in two interview sessions which were video recorded. The results of the study showed that DGS, as a mediation tool, encouraged students to use in problem solving and posing the processes of modeling, conjecturing, experimenting and generalizing. Furthermore, we found that DGS can play a significant role in engendering problem solving and posing by bringing about surprise and cognitive conflict as students use the dragging and measuring facilities of the software.

This case study examined the metacognitive processes of a preservice teacher when solving a nonroutine geometry problem in a dynamic geometry environment. The main purpose of the study was to uncover and investigate patterns of metacognitive processes and to understand what circumstances, situations, and interactions in a dynamic geometry environment promoted metacognitive behaviors. An adaptation of Schoenfeld’s (1981) model of episodes and executive decisions in mathematics problem solving, and the theory of instrumentation (Rabardel, 2001) was used to identify patterns of metacognitive processes in a dynamic geometry environment. During different phases of problem solving the participant engaged in different metacognitive behaviors whereas the dynamic geometry software supported strategies that are available and/or not available on paper and pen. The effectiveness of solution paths was dependent on the presence of managerial decisions, and well-orchestrated usage of different resources, both knowledge and technology. However, the results of the study call to question to which extent engagement in metacognitive behaviors is necessarily desirable or productive.

Spatial ability is of great importance for successful work in various fields such as computer graphics, engineering, architecture and cartography. A number of studies have demonstrated that Descriptive geometry courses have the potential to develop spatial skills. The aim of this study is to investigate the effect of Descriptive geometry course with the current method of teaching on the spatial skills of students at the Faculty of Civil Engineering at the Technical University of Kosice. The study was conducted by a pretest and posttest method. The first grade students studying in the winter term of 2016 to 2017 educational year served as a sample for the study. A significant difference between the pretest and posttest scores was revealed by using the Wilcoxon signed rank test. The findings showed that Descriptive geometry course has a positive impact on development of students’ spatial skills. Although, the Descriptive geometry course provides numerous benefits, it is often regarded by students as one of the most difficult courses. The use of dynamic geometry software with the ability to represent three-dimensional space allows students to overcome the difficulties associated with the plot of this course. The future plans within Descriptive geometry teaching include integration of the dynamic geometry software GeoGebra into the educational process which should lead to the innovation and facilitation of the Descriptive geometry course. Keywords: Spatial ability, mathematics education, engineering education

Students regularly struggle with mathematical tasks, particularly those concerning non-routine problems in geometry. Although educators would like for their learners to transfer their knowledge to non-routine and real-life situations, students run into a number of difficulties. The goal of this exploratory study was to analyze three participants’ problem solving processes in a dynamic geometry software (DGS), and therefore, gain insights about how DGS was used to support solving non-routine geometry problems. Here I viewed the DGS as a cognitive tool that can enhance and reorganize the problem solving process. The three participants were in different phases of their educational career in mathematics and/or mathematics education (bachelor, master, and doctoral student). Only one problem—the Land Boundary Problem—from the TIMSS video study will be discussed here. In this problem, the participants had to straighten a bent fence between two farmers’ land so that each farmer would keep the same amount of land. All three participants solved the problem, but used the same computer-based problem-solving tool differently. While a DGS allowed and supported some participants to discover new methods of thinking, and unanticipated ways of using it, it also inhibited the problem solving processes through development of tool-dependency by some. Its different use was dependent on the presence of managerial decisions, ability to manage different resources, and problem solving experience. Based on these findings, I make recommendations for technology-embedded problem solving with an emphasis on the importance of appropriate tool use in educational settings and offer some teaching methods that may be worthwhile for research.

Spatial orientation and spatial visualization are the foundation of students’ spatial ability. They assist students’ performance in learning mathematics, especially geometry. Considering its importance, the present study aims to design activities to help young learners developing their spatial orientation and spatial visualization ability. Photography activity was chosen as the context of the activity to guide and support the students. This is a design research study consisting of three phases: 1) preparation and designing 2) teaching experiment, and 3) retrospective analysis. The data is collected by tests and interview and qualitatively analyzed. We developed two photography activities to be tested. In the teaching experiments, 30 students of SD Laboratorium UNESA, Surabaya were involved. The results showed that the activities supported the development of students’ spatial orientation and spatial visualization indicated by students’ learning progresses, answers, and strategies when they solved the problems in the activities.

Spatial ability is the ability to generate, store, retrieve, and transform visual information to mentally represent a space and make sense of it. This ability is a critical facet of human cognition that affects knowledge acquisition, productivity, and workplace safety. Although having improved spatial ability is essential for safely navigating and perceiving a space on earth, it is more critical in altered environments of other planets and deep space, which may pose extreme and unfamiliar visuospatial conditions. Such conditions may range from microgravity settings with the misalignment of body and visual axes to a lack of landmark objects that offer spatial cues to perceive size, distance, and speed. These altered visuospatial conditions may pose challenges to human spatial cognitive processing, which assists humans in locating objects in space, perceiving them visually, and comprehending spatial relationships between the objects and surroundings. The main goal of this paper is to examine if eye-tracking data of gaze pattern can indicate whether such altered conditions may demand more mental efforts and attention. The key dimensions of spatial ability (i.e., spatial visualization, spatial relations, and spatial orientation) are examined under the three simulated conditions: (1) aligned body and visual axes (control group); (2) statically misaligned body and visual axes (experiment group I); and dynamically misaligned body and visual axes (experiment group II). The three conditions were simulated in Virtual Reality (VR) using Unity 3D game engine. Participants were recruited from Texas A&M University student population who wore HTC VIVE Head-Mounted Displays (HMDs) equipped with eye-tracking technology to work on three spatial tests to measure spatial visualization, orientation, and relations. The Purdue Spatial Visualization Test: Rotations (PSVT: R), the Mental Cutting Test (MCT), and the Perspective Taking Ability (PTA) test were used to evaluate the spatial visualization, spatial relations, and spatial orientation of 78 participants, respectively. For each test, gaze data was collected through Tobii eye-tracker integrated in the HTC Vive HMDs. Quick eye movements, known as saccades, were identified by analyzing raw eye-tracking data using the rate of change of gaze position over time as a measure of mental effort. The results showed that the mean number of saccades in MCT and PSVT: R tests was statistically larger in experiment group II than in the control group or experiment group I. However, PTA test data did not meet the required assumptions to compare the mean number of saccades in the three groups. The results suggest that spatial relations and visualization may require more mental effort under dynamically misaligned idiotropic and visual axes than aligned or statically misaligned idiotropic and visual axes. However, the data could not reveal whether spatial orientation requires more/less mental effort under aligned, statically misaligned, and dynamically misaligned idiotropic and visual axes. The results of this study are important to understand how altered visuospatial conditions impact spatial cognition and how simulation- or game-based training tools can be developed to train people in adapting to extreme or altered work environments and working more productively and safely.

Students’ spatial visualization ability is a key component in understanding geometry, yet many students exhibit suboptimal performance in this area. This study aimed to (1) evaluate the quality of the Geogebra-assisted 6E-IM learning model, (2) examine its effect on students' spatial visualization ability, and (3) describe students' ability levels after receiving the intervention. A sequential explanatory mixed-methods design was used. The independent variable was the Geogebra-assisted 6E-IM model, while the dependent variable was students’ spatial visualization ability. A total of 60 students participated, with 30 assigned to the experimental group and 30 to the control group. Data were collected through tests, interviews, and questionnaires. Quantitative results showed that students in the experimental group significantly outperformed the control group (p < 0.05), indicating a positive effect of the Geogebra-assisted 6E-IM model on spatial visualization ability. Qualitative analysis revealed that students with high ability met all spatial visualization indicators; those with moderate ability met two indicators, while those with low ability met only one. These findings demonstrate that the integration of Geogebra in the 6E-IM learning model can effectively enhance students’ spatial visualization skills in geometry. This study contributes to the development of innovative, technology-integrated instructional strategies in mathematics education.

Part I. Personal Reflections On Investigation, Discovery, and Proof: 1. Discovery and dissection of a geometric gem Douglas R. Hofstadter 2. The role and function of proof in dynamic geometry: some personal reflections Michael de Villiers 3. Dynamic geometry renews interest in an old problem Dan Bennett 4. Dynamic geometry as a bridge from Euclidean geometry to analysis Albert A. Cuoco and E. Paul Goldenberg Part II. Dynamic Geometry In The Classroom: 5. Dynamic visualization from middle school through College James Morrow 6. Geometer's sketchpad in the classroom Tim Garry 7. Students discovering geometry using dynamic geometry software Michael Keyton 8. Moving triangles 9. Experiences with geometer's sketchpad in the classroom Kathryn W. Boehm 10. Beyond elementary constructions: selected exercises Arnold Perham and Bernadette H. Perham 11. Interactive generation, manipulation, and application of loci Heinz Schumann 12. Calculus with dynamic geometry Catherine A. Gorini 13. Beginning geometry at college Tony Hampson 14. Identifying transformations by their orbits James M. Parks 15. Dynamic proofs that use similarities James King 16. Visualization of group theory concepts through dynamic geometry Doris Schattschneider 17. Using the geometer's sketchpad with preservice teachers Zhonghong Jian and Edwin McClintock 18. Fish in the pond: inquiry with dynamic geometry Fadia Harik Part III. Dynamic Visualization In History, Perception, Optics and Aerodynamics: 19. Drawing logarithmic curves with geometer's sketchpad: a method inspired by historical sources David Dennis and Jere Confrey 20. Lost in the funhouse: an application of dynamic projective geometry Susan Addington and Stuart Levy 21. The use of dynamic geometry software in teaching and research in optometry and vision science Benjamin T. Backus 22. Creating airfoils from circles: the Joukowski transformation John Olive Part IV. The Worlds Of Dynamic Geometry: Issues In Design And Use: 23. Drawing worlds: scripted exploration environments in the geometer's sketchpad R. Nicholas Jackiw 24. Dynamic geometry and declarative geometric programming Richard J. Allen and Laurent Trilling.

This study investigates the relationship between students’ spatial abilities and their ability to solve problems in physics, specifically in kinematics. The approach taken is to consider spatial ability not as single and undifferentiated, but composed of different components. The hypothesis is that different types of kinematics problems require different spatial abilities. Sixty undergraduate psychology students, who had not taken any physics courses at college level, took a battery of cognitive tests measuring different spatial skills, verbal ability and mechanical reasoning. In addition, students were presented with a series of kinematics problems by means of a written problem solving questionnaire. Analyses of students’ responses indicated that different types of kinematics problems require different cognitive skills. It was found, for instance, that extrapolating complex two-dimensional motion correlates significantly with spatial visualisation ability, whereas inferring direction of motion from a graph correlates with spatial orientation ability. However, performance on other types of kinematics problems (e.g., evaluating an object’s speed and some types of graph problems) do not correlate with spatial abilities, indicating that they may require mostly semantic knowledge of physics laws or mathematical reasoning.

The ability of a student to obtain, analyze, measure and compare the many instances of a mathematical proposition in a dynamic geometry environment gives him/her opportunities to make conjectures and test a proposition. These roles for dynamic geometry software are widely acknowledged as having the potential to enrich the teaching of geometry. Furthermore, Hoyles and Jones (1998) claim that dynamic geometry, supported by ‘‘what if’’ and ‘‘what if not’’ questions, has the potential to promote links between empirical and deductive reasoning. A classic set of problems in Euclidean plane geometry consists of finding the path of a point that is subject to given constraints. Its systematic study goes back to a lost work (in two books) of Apollonius of Perga, Plane loci (Botana and Valcarce 2003). Except for the simplest loci, such as lines, circles and perhaps the conics, this subject is usually avoided in most geometry texts. This is due to the common difficulties faced when mentally visualizing various objects with different movements. With the emergence of dynamic geometry software, the locus problems have attracted new interest from researchers. Using the locus generation features of these software one can easily obtain the locus of a point under some constraints. Although there are some minor differences, different dynamic geometry software, behave in a similar way when obtaining loci. In general, two objects must be selected: the first one, usually a point called the driving point or mover, is bound to a path, whereas the other, the locus point, must depend somehow on the first one. Since the element

Spatial cognitive processing is a fundamental aspect of human cognition, influencing our comprehension of spatial environments. Researchers have defined spatial ability in various ways, encompassing skills such as generating, visualizing, memorizing, and transforming visual information. Despite the diversity in definitions, there is a shared understanding that spatial ability is an inherent skill aiding individuals in tasks requiring visual and spatial acumen. One of the dimensions of spatial ability is spatial visualization that governs our day-to-day activities of staying and working in and navigating through space. One of the factors that could impact our spatial visualization ability is the alignment of visual and body axis that is maintained on earth due to gravitational cues. However, such cues are not available in micro-gravity environments that exist aboard the International Space Station (ISS). It is imperative to understand if human spatial visualization is impacted by such conditions to determine safety and productivity risks. In this paper, we present results of our research examining if the non-alignment of body and visual frame of reference (FOR) affects spatial visualization ability. We administered the Purdue Spatial Visualization Test: Visualization of Rotation (PSVT:R) to measure the spatial visualization ability of 230 participants. The PSVT:R assesses an individual's capacity to mentally rotate 3D objects. Participants matched the rotated view of a test object to a provided example, evaluating spatial visualization skills and cognitive abilities. The study included three test conditions, one control and two experimental conditions simulated in Virtual Reality (VR) using Unity 3D game engine. The control condition (C1) had the body axis and the visual FOR aligned just like a space on earth. The experiment conditions E1 and E2 depicted a micro-gravity environment to simulate statically and dynamically non-aligned visual and body axes, respectively. Participants sat in a swivel chair and wore HTC Vive Pro Eye headsets to experience the three conditions. Results consistently indicated a significant difference between response time (RT) and accuracy of participants’ responses under the three study conditions. Moreover, a negative correlation was found between the response time and accuracy, which implied a trade-off between response time and accuracy—a common phenomenon where individuals may prioritize speed over precision or vice versa. Our findings support the existence of a relationship between response time and accuracy, characterized by a significant difference and a weak correlation. The Bland-Altman analysis offered additional insights, emphasizing the variability in this relationship. In the C1 condition, the correlation coefficient was -0.1902, suggesting a weak tendency for accuracy to slightly decrease as reaction time increases. Similarly, the E1 condition exhibited a negative correlation of -0.2333, indicating a weak but negative trend of decreased accuracy with longer reaction times. In the E2 condition, the correlation coefficient was -0.1049, suggesting a mild decrease in accuracy as reaction time increased. Overall, the consistent negative correlations across all conditions imply a general pattern: participants with longer reaction times may exhibit slightly lower accuracy, and vice versa. Results also showed that the non-alignment of visual and body axes impact spatial visualization ability.

Summary form only given. Assessment of student learning requires the use of techniques for measuring student achievement. Assessment is more than a collection of techniques, however. It is a systematic process that plays a significant role in effective teaching. It begins with the identification of learning goals, monitors the progress students make toward those goals, and ends with a judgment concerning the extent to which those goals have been attained [1]. Assessment can be classified in terms of their functional role in classroom instruction as follows: placement, formative, diagnostic and summative assessment [2]. In the literature, the integration of technological tools into the assessment process is expressed by the "e-assessment" concept [3,4,5]. Assessment using ICT has come to be known as e-assessment, which includes the entire assessment process, from designing assignments to storing the results with the help of ICT [5]. The most of the researches related to the e-assessment focused on how to transfer the examinations in paper-pencil environment to the electronic environment [4,5]. In addition to those studies, in the research about the integration of computer assisted assessment tools to the mathematics instruction, the participant pre-service teachers use these tools in favor of the formative assessment [6]. In a similar study, pre-service teachers prepared lessons and home works using technological tools such as dynamic geometry software and thus it is shown that those tools can be used in the process of assessment [7]. In the literature, there is numerous studies concerning technology supported mathematics instruction [8,9, I 0, 11]. But none of them is related to the assessment dimension of the mathematics teaching. In particular, it is highly recommended to conduct researches about the use of technology in that dimension [6,12,13]. The aim of this study is to determine some necessary points of assessment of computer based mathematics instruction. As this aim is related to a very large domain of research, this study focuses on only three open-ended geometry problems that can be solved in dynamic geometry environments. Case study is considered as the methodology of the research. The reason is that a case study investigates a contemporary phenomenon in depth and within its real-life context when the boundaries between phenomenon and context are not clearly evident [14]. 30 pre-service mathematics teachers have participated to the study and have divided randomly into three groups. In each group, every pre-service teacher worked individually on one of the three problems during 30 minutes. At the same time, they explained their approaches to solve problems by filling out a paper. In addition, a screen capture program is used to grab the video of what is seen on the computer screen. Meanwhile the researchers observed pre-service teachers' work on computers. The findings of the study are revealed via the analysis of each video, of their comparison with the written explanations and of the researchers' observations. Based on the findings, the most significant result of the study is that the screen captures should be definitely used for the assessment in dynamic geometry environments; because all of pre-service teachers' work can only be seen in those videos. The final version of the saved file doesn't represent the whole tries that one pre-service teacher does while dealing with problems. Furthermore a rubric that depends on the time, the number of tools, the relation between the drawing and the construction must be absolutely used to give a score.

Spatial abilities are divided into two, namely spatial orientation and spatial visualization. In the field of engineering, especially Industrial Engineering, spatial abilities are usually used in the design process. The use of spatial orientation abilities is influenced by mental activity. Thus, mental fatigue will influence the application of spatial orientation abilities. Mental fatigue among students is usually influenced by academic workload. This research was conducted to determine the effect of academic workload and AX-CPT treatment on the spatial abilities of industrial engineering students. This research involved 40 respondents who were divided into two groups. The research was carried out with mental fatigue engineering using the AX-Continuous Performance Test for 20 minutes for one group of respondents before taking a spatial ability measurement test using the Spatial Orientation Test (SOT) while the other group was not given the AX-Continuous Performance Test. After taking the test, respondents will fill out a survey regarding academic workload. The test was carried out using multiple linear regression with a p-value of 0.814 so there was no influence from the academic workload and AX-CPT treatment on the spatial abilities of industrial engineering students

Investigating how learners appropriate technological tools while engaging in mathematical tasks provides an insight into how learners’ tool-mediated activities shape their mathematical knowledge. The theory of instrumental genesis (Lonchamp in International Journal of Computer-Supported Collaborative Learning, 7(2), 211–237, 2012; Rabardel and Beguin in Theoretical Issues in Ergonomics Science, 6(5), 429–461, 2005) explains how learners appropriate technological tools and accomplish tasks while interacting with these tools. In this study, we investigate how teachers appropriate a collaborative, dynamic geometry environment, the Virtual Math Teams with GeoGebra (VMTwG), and how their appropriation shapes their geometrical knowledge. The environment contains a chat panel and multiuser version of GeoGebra (a dynamic geometry software) that teachers share. The participants are seven middle and high school mathematics teachers who participated in a 15-week professional development course in which they collaborated synchronously in VMTwG to solve geometrical tasks. We analyzed the work of one team of two high school teachers. Our analysis shows that the teachers appropriated collaboratively technological and mathematical tools, which supported their mathematical problem solving. These tools enabled them to manipulate and explore geometric objects and to notice and discuss relations among them. Our study contributes an analysis of longitudinal data to understand the co-emergence of learners’ collaborative appropriation of digital technologies and their construction of mathematical knowledge. This study suggests further research to understand how to orchestrate learners’ appropriation of collaborative technological tools in which instructors are not contemporaneously present during learners’ problem-solving activity.