Geometry Lessons Research Articles

Implementation of problem-based learning in geometry lessons

The aim of this study is twofold. Firstly, it aims to examine the effects of the Problem-Based Learning (PBL) approach on students’ performance in the learning of geometry. Secondly, it seeks to gain insights from the students regarding the implementation of PBL in geometry lessons. The participants were 22 students from one Year 10 class in a co-educational secondary school in Brunei Darussalam. A mixed method design was employed with data collected from the pre-, post- and retention tests, and interviews. The findings from this study revealed positive influences on students’ performance in learning geometry as gain and retention of knowledge was observed. Meanwhile, mixed responses from the interviews implied that in terms of 1) learning attitudes, students favoured the idea of independent learning but some critiqued that the process of PBL might be time-consuming; 2) learning difficulties, some students struggled in assimilating information leading to poor decision- making; and 3) knowledge and skills, some students believed to have nurtured some skills such as communication and research skills.

Spatial Intelligence on Solving Three Dimensional Geometry Object Through Project Based Learning

One of the factors that cause students’ low academic achievements is their spatial intelligence. Spatial intelligence isvery important in the learning of geometry. Three-dimensional geometry object in the geometry lesson, according to the students’absorption capacity in the national examination is still in the low category. The purpose of this study is to describe the process andresult of students’ spatial intelligence through the learning model of Project Based Learning. The project based model uses theinitial problem in the planning of complex activities by allowing students the freedom to plan well-learning activities.Thisresearch uses descriptive analysis method. Data collection methods used to test. The subject of this research are twenty-sixstudents of VIII B.The data analyzed in this research is the test result sheet. The characteristics of the spatial visual intelligenceare (1) imagination, (2) conceptualizing, and (3) problem-solving. The results of this research showed that there are 9 students ofVIII B grade are capable to fulfill the three characteristics of spatial intelligence. There are 11 students have 2 characteristicsspatial intelligence, 4 students have 1 characteristics spatial intelligence, and 2 students who are not capable to fulfill thecharacteristics of spatial intelligent.

ИЗУЧЕНИЕ КОМПЬЮТЕРНОЙ ГРАФИКИ В СИСТЕМЕ ОБЩЕГО ОБРАЗОВАНИЯ

In this article the theoretical and the methodical basics of teaching computer graphics in the system of the general education are covered. First of all, it concerns lessons of informatics, but the computer graphics has serious opportunities for its using in the course of training other disciplines, in particular at geometry lessons. Use of vector graphics in geometry promotes activation of cognitive activity of pupils. Use of raster graphics also can make an essential contribution to development of a technique of geometry training. Raster graphics is fundamentals of computer graphics. When studying technology of raster editors it is impossible to ignore its value for understanding of other types of computer graphics, in particular fractal graphics. Studying raster technologies, it is also necessary to pay attention to transformation of formats of files as in the future the number of formats will only increase.Use of computer graphics allow children, even without art abilities, to feel as a certain creator, to create artistic images and gives big opportunities for self-realization. And the most important thing is the use of graphic opportunities of the computer allows to increase the interest of pupils in the lessons and to stir up their cognitive activity.

How are questions that students ask in high level mathematics classes linked to general giftedness?

This paper presents a part of a larger study, in which we asked “How are learning and teaching of mathematics at high level linked to students’ general giftedness?” We consider asking questions, especially student-generated questions, as indicators of quality of instructional interactions. In the part of the study presented in this paper, we explore instructional interactions in two high-school classes for mathematically promising students with specific focus on questions that students ask. The first class included generally gifted students (IQ ≥130) who were motivated to study mathematics at a high level (hereafter, a gifted class), and the second class included students characterized by high motivation regardless of their IQs (hereafter, motivation class). We analysed questions asked by the students during algebra and geometry lessons. Two types of questions are considered: elaboration and clarification. We found that students in a gifted class mostly asked elaboration questions, whereas students in a motivation class mostly asked clarification questions. We connect the revealed inclination to ask elaboration questions with intellectual curiosity that characterizes generally gifted students. Accordingly, we suggest that in classes of students who are motivated to study mathematics at high level, students who are generally gifted may create mathematical discourse of higher quality. We also argue that the identified differences in students’ questions observed in classes of different types are not only student-dependent (i.e. depend on the students’ levels of general giftedness) but can also be teacher-related and content-related.

IRE/F as a Cross-Curricular Collaborative Genre of Implicit Argumentation

We contend that the classroom-discourse routine of IRE/F (teacher initiation, student response, teacher evaluation/follow-up) is a genre of argumentation that is both collaborative and implicit because teachers and students cooperate during IRE/F exchanges not only to make and mirror knowledge claims but also to suppress justification for those claims. Analyzing consecutive IRE/F exchanges drawn from a first-grade geometry lesson, we propose that the implicit argumentation of IRE/F may conflict with students' learning of justification-centered forms of reasoning that are promoted in the cultural communities of academic disciplines. To highlight the argumentation and identification practices at work in IRE/F exchanges, and to expand teachers' options for responding to students, we offer the flexible heuristic WAIT (Welcome, Analyze, Include, Thank), which transforms the third IRE/F turn into an opportunity for explicitly analyzing students' argumentation practices and including students in disciplinary, school, and classroom communities.

Using Of Information Technologies To Improve The Spatial Understanding Of Students

It is possible to improve the development of students’ spatial thinking with various information technologies. We can improve students understanding with three dimensional models, created in the CAD program and special processing it in PowerPoint in the program, together with drafting in three planes. Gradually solving tasks in planes and explaining it in three dimensional models, the solution of concrete task is much clearer, because it is possible to look a situation together in planes and 3D models. The students of three faculties were polled that cognize the students’ opinions about teachers teaching methods in descriptive geometry lessons and analyze spatial understanding of students. A possibility to use information technologies to improve a descriptive geometry lectures have been researched in this work. The most important information about working out methodological materials has been generalized in discussion part. The results of research show that lectures with information technologies using improve students’ understanding of solved tasks during the lectures.

“It’s Understandable Enough, Right?” The Natural Accountability of a Mathematics Lesson

ABSTRACTA fundamental problem for all educational reform efforts is one of specifying in practical terms just what the reform requires of teachers and students. In addressing it, educational designers have introduced the term “accountability” to characterize something missing in classroom discourse. This term, however, held special significance in the writings of the American sociologist Harold Garfinkel. Livingston applied Garfinkel’s treatment of accountability to understanding the lived-work of doing mathematics, particularly proving work. We examine an 8th-grade student’s presentation during a Japanese geometry lesson as a proof-account. Within it we see elements of both classical and natural accountability placed on display.

On the Robustness of the Quizzing Effect under Real Teaching Conditions

Lab experiments and field studies showed that studying improves short-term memory, whereas active retrieval improves long-term memory – a phenomenon known as the quizzing (or testing) effect. In a quasi-experimental field study with four elementary school classes (18 < n < 22) and a pretest – posttest (5 min after the intervention) – posttest (6 weeks later) design, we tested the robustness of the quizzing effect under real conditions involving verbal teacher-student interactions in geometry lessons on symmetry in elementary schoolers. Results showed that re-studying compared to active retrieval (quizzing) enhanced learning more when measured directly after the lessons. This pattern revered when knowledge was measured six weeks later, demonstrating that the quizzing effect was robust. Moreover, long-term memory generally increased after six weeks. Limitations of the quasi-experimental approach are discussed.

Untangling Geometric Ideas

This geometry lesson uses the work of abstract artist Wassily Kandinsky as a springboard and is intended to promote the conceptual understanding of mathematics through problem solving, group cooperation, mathematical negotiations, and dialogue.

What Pre-service Teacher Student Said Arranging and Publishing Blog Page

In this study, aim was to provide pre-service mathematics student teachers with a new learning and teaching experience while they were playing two roles as learners and teachers within a arranging and publishing blog environment. The question of study was to investigate how a blog-mathematics course affected the ideas pre-service mathematics student teachers as a result of their experiences within that environment. The study showed that participants were: Willing toimplement blog construction activities about geometry lessons for their students; Willing to develop new methood sands trategies for conceptual geometry lessons; and to gain new perspectives on developing a blog as a teacher; Able to discover the importance of good team communication and appreciate developing a product for their students.Willing to uses ocial soft ware (wikipedia, wiktionaryetc.)It was concluded that the mathematic thinking processmay be improved as a result of preaparing a blogpage in mathematics teaching.Thus by considering the positive effects of blog page instruction, wesuggest that thephylosophical aspects and dynamics behind instructional technology should be clearly understood by teachers. Therefore educational faculties need to perform more research in this field.

THE EFFECT OF ORIGAMI-BASED INSTRUCTION ON SPATIAL VISUALIZATION, GEOMETRY ACHIEVEMENT, AND GEOMETRIC REASONING

This research study examined the effect of origami-based geometry instruction on spatial visualization, geometry achievement, and geometric reasoning of tenth-grade students in Turkey. The sample (n = 184) was chosen from a tenth-grade population of a public high school in Turkey. It was a quasi-experimental pretest/posttest design. A control group (94 students) received regular instruction during a geometry unit in a tenth-grade classroom whereas an experimental group (90 students) received origami-based instruction for 4 weeks. The Spatial Visualization Test (SVT) was used to measure students’ spatial visualization ability in this study. The SVT consists of the Card Rotation Test, Cube Comparison Test, and the Paper Folding Test which were originally developed by Ekstrom, French, Harman & Derman (1976) and translated into Turkish by Delialioglu (1996). Besides, Geometry Achievement Test and Geometric Reasoning Test were developed by the researcher to measure geometry achievement level and geometric reasoning level of participants in key aspects of triangles. All tests had versions of pretest and posttest, and these tests were administered to both groups. A repeated-measures Analysis of Variance was used on each test scores to analyze data. The results indicated that origami-based instruction had significant effect on all dependent variables (spatial visualization, geometry achievement, and geometric reasoning). This suggested that origami might be integrated into high school geometry lessons to make geometry learning more effective.

Comparisson On The Effectiveness Of Cooperative Learning Model Of Numbered Heads Together (NHT) And Team Assisted Individualization(TAI) In Geometry

The study is an experiment research which aims at examining (1) the implementation of cooperative learning model of TAI (Team Assisted Individualization) type and NHT (Numbered Head Together) type; (2) the effectiveness of learning model of TAI type based on the aspects of learning outcomes, students’ activity, and students’ response; (3) the effectiveness of learning model of NHT type based on the aspects of learning outcomes, students’ activity, and students’ response; and (4) the comparison on the effectiveness of cooperative learning model of TAI type and NHT type in Geometry of grade VII students at SMPN 4 Sungguminasa. The study was conducted in grade VII at SMPN 4 Sungguminasa in Gowa on the second semester of academic year 2012/2013. Samples were selected randomly and obtained two classes out of thirteen classes. The chosen classes implemented the TAI type and the other one implemented the NHT type in Geometry lesson. The effectiveness of learning which consisted of thee aspect, namely learning outcomes of students, learning activity of students, and students’ response were examined. The result of study reveal that (1) the implementation of cooperative learning model of TAI type in geometry was conducted well with the mean score 3.60; (2) the implementation of cooperative learning model of TAI type in Geometry was effective based on the aspects of a students’ learning outcomes with the mean score of post-test greater than the pre-test by 79.47 ≥ KKM and students’ mastery of the lesson was 88.89% ≥ 85%, b) students’ learning outcomes with the implementation of activity 3.24, and c) students’ response with the mean score 3.57 (positive category); (3) the implementation of cooperative learning model of NHT type was conducted well with the mean score of the implementation 3.69; (4) the implementation of cooperative learning model of NHT type was effective based on the aspect of a) students’ learning outcomes with the mean score of post-test was greater than pre-test by 78.64 ≥ KKM and students’ mastery of the lesson was 86.11% ≥ 85%, b) students’ response with the mean score 3.46; (5) based on the set effective criteria, it is confirmed that the geometry lesson by determining the circumstance and area of square and triangle and used in solving the problem on the implementation of cooperative learning model of TAI type is better than NHT type of grade VII students at SMP where the three aspects, namely students’ learning outcomes, students’ activity, and students’ response were better than the TAI type.

High school students’ perceptions of geometrical proofs proving and refuting geometrical claims of the ‘for every … ’ and ‘there exists’ type

Proving and refuting mathematical claims constitute a significant element in the development of deductive thinking. These issues are mainly studied during geometry lessons and very little (if at all) in lessons of other mathematical disciplines. This study deals with high school students’ perceptions of proofs in the geometry. The study explores whether students know when to use a deductive proof and when an example is sufficient for proving or refuting geometrical claims. The findings indicate that in cases of simple claims, the students corroborate them by using a deductive proof. However, when the claim is more complex, the students tend to present both a proof and an example. Moreover, they are unsure whether using an example can constitute a method for proving a mathematical claim, believing that in mathematics everything must be proven. They believe that examples are used merely for illustration purposes rather than as a means of convincing. The research conclusions support the need for deepening and developing the students’ distinction between cases where examples are insufficient and cases where an example is sufficient for proving a claim.

Cultivating inquiry about space in a middle school mathematics classroom

During 46 lessons in Euclidean geometry, sixth-grade students (ages 11, 12) were initiated in the mathematical practice of inquiry. Teachers supported inquiry by soliciting student questions and orienting students to related mathematical habits-of-mind such as generalizing, developing relations, and seeking invariants in light of change, to sustain investigations of their questions. When earlier and later phases of instruction were compared, student questions reflected an increasing disposition to seek generalization and to explore mathematical relations, forms of thinking valued by the discipline. Less prevalent were questions directed toward search for invariants in light of change. But when they were posed, questions about change tended to be oriented toward generalizing and establishing relations among mathematical objects and properties. As instruction proceeded, students developed an aesthetic that emphasized the value of questions oriented toward the collective pursuit of knowledge. Post-instructional interviews revealed that students experienced the forms of inquiry and investigation cultivated in the classroom as self-expressive.

Metacognitive support promotes an effective use of instructional resources in intelligent tutoring

We tested whether the provision of metacognitive knowledge on how to cope with the complexity of a learning environment improved learning. In an experimental setting, high-school students (N = 60) worked through a computer-based geometry lesson either with or without metacognitive support in the form of a cue card. This cue card encouraged students to use instructional resources in the learning environment (i.e., textual and graphic representations and different help facilities) more strategically. During learning, the learners' gaze and log-file data were recorded. The metacognitive support made learning more efficient (i.e., less learning time without impairing outcomes). In addition, low-prior knowledge students developed deeper conceptual understanding. The effects on learning outcomes were mediated by reducing the non-strategic use of help facilities. Our findings suggest that a lack of metacognitive conditional knowledge (i.e., in which situation to use which help facility) can account for learning difficulty in computer-based learning environments.

Primary School The 6th, 7th And 8th Grade Students’ Perceptions On Line Segment, Linearity, Ray And Angle Concepts

Geometry is established on the undefined and defined concepts. Misunderstanding of the basic concepts in geometry lesson, students will prevent to comprehend on the understanding of other geometry issues exactly. The purpose of this study is to determine the primary six, seven and eight grade students’ the levels of perception and their misconceptions related to the line segment, linearity, ray and angle concepts (defined concepts) which are the basic concepts Interview protocol including semi-structured essay type questions was formed (GP) in the light of expert opinions of GP under Elementary Mathematics Program framework. The interview protocol was asked students’ responses about line segment, linearity, ray and angle. Students’ opinions are classified according to their levels of perception. Themes were determined for each level of perception and each theme were divided sub-themes according to the students’ relationship concepts. The descriptive analysis was used for data analysis. Sampling consists of eight primary schools in which each six, seven and eight grade classrooms randomly including 461 of students. Survey method is carried out as a research method. As a result, the students have been identified some difficulties on which the comprehension of the basic geometry concepts such as line segment, linearity, ray and angle. In addition, these concepts are confused with each other and they have some misconceptions.

Arts Integration in Teacher Preparation: Teaching the Teachers

This classroom study focused on modeling a hands-on approach for understanding classroom applications of multiple intelligence theory through arts-based integration. Thirty-five preservice teachers enrolled in Educational Psychology classes participated in an interdisciplinary geometry lesson modeling Artful Learning™, experiencing an arts- based pedagogical approach in the lesson. Students identified and described geometric concepts and relationships and photographed geometrical elements authentically on campus as part of the model’s original creation. Assessment of lesson objectives revealed that students appreciated arts-based pedagogy, but had difficulty translating theory into practice when creating their original lesson plans. Discussion includes reflective responses of preservice teachers to inquiry and arts-based classroom instruction for enhancing student understanding, as well as implications for integrating art pedagogy in professional practices.

Using Dynamic Software in Teaching of the Symmetry in Analytic Geometry: The Case of GeoGebra

With the advent of technology in our time, many researches have been conducted and much software has been developed regarding the usability of computers in mathematics teaching. By using GeoGebra which is dynamic mathematics software, this study has aimed to prepare materials about symmetry which is one of the subjects of analytic geometry lesson in secondary education. For that purpose, materials have been formed, which can assist students in taking symmetry of point and line in accordance with coordinate axes, origin, y=x and y=-x lines by paying attention to the directions on the given worksheet, and as a result, internalizing the basic logic of the concept of symmetry.

Effects of different ratios of worked solution steps and problem solving opportunities on cognitive load and learning outcomes

A crucial challenge for instructional designers is to determine the amount of support that is most beneficial for learning. This experiment investigated effects of different ratios of worked solution steps (high assistance) and to-be-solved problem steps (low assistance) on cognitive skill acquisition in geometry. High-school students ( N = 125) worked on a geometry lesson in a Cognitive Tutor under five different ratios (from zero worked steps and five to-be-solved steps to four worked steps and one to-be-solved step). Effects on cognitive load and learning outcomes were assessed. We expected the effectiveness of different ratios to vary with the type of learning outcomes (i.e., procedural vs. conceptual knowledge) and the difficulty of the to-be-learned principles. Results showed that for procedural knowledge (but not for conceptual knowledge) problem solving alone was most beneficial for the acquisition of procedural knowledge related to an easy principle. For a difficult principle, no ratio of worked steps and problem solving showed an advantage over another. Problem solving induced more extraneous load than studying worked examples. Thus, in determining optimal amounts of guidance type of knowledge and difficulty of the single to-be learned knowledge chunks should be considered.

Mathematical Exploration: Trapezoids to Triangles

Pattern blocks are often used in math class during geometry lessons to support concepts of shape, angle, tessellations, and symmetry. They are also used with fraction study (e.g., Lanius 2007; NCTM 2009a). Using pattern blocks to construct other larger geometric shapes can help students understand the relationships among various shapes. Algebraic relationships and functions emerge when students count blocks and measure figures. The activities in this article support the development of such geometric vocabulary as perimeter, area, sequence, and nonstandard measurement.