Transforming Middle School Geometry Instruction

A recent report on the teaching of geometry by a committee of the Mathematical Association (The Teaching of Geometry in Schools) suggests that a coherent geometrical system could be erected on a foundation other than that of Euclid’s axioms and postulates. Indeed, a cogent argument is advanced for the replacement of the principle of superposition and the parallel-postulate by the “principles of congruence and similarity” which are regarded as the expressions of more profound beliefs. The success of the recommendation, to substitute the principle of congruence for the principle of superposition, is undoubted, and the comparative reluctance to adopt the twin “principle of similarity” may be due, in part, to an obscurity in the treatment of parallels with this principle as basis.

In recent years a good deal of emphasis has been placed on the study of geometry in the elementary school. Modern textbook series give more attention to the study of geometry than more traditional series. Curriculum committees such as the Cambridge Conference (Educational Services Inc. 1963; Education Development Center 1969[b]) and the Ontario Institute for Studies in Education (1967) have presented arguments favoring the study of geometry in the elementary school. Also, many articles dealing with geometry (Brydegaard and Inskeep 1970) employ “what to teach,” “how to teach,” or “why teach” approaches. The intent of this review is to determine what recent research has to say about geometry instruction in the elementary school. The research described here has been classified in related sections.

Research has documented the prevalence of lessons characterised by homework check, followed by teacher lecture and demonstration, followed in turn, by learner practice sequence of classroom instructional activities in our classrooms. This sequence of classroom activities does not allow for the development of sound mathematics practices and mathematical proficiency. Meanwhile, curriculum reforms in South Africa as well as in other parts of the world recommend classroom activities where teachers create opportunities for, listen to and extend learners. This paper presents a sequence of activities to be used in the teaching of geometry and surface areas of solid shapes in a grade 8 classroom. The sequence portrays the teaching of these concepts as an investigative rather than instructive process.

This paper analysed gifted middle students' conception of the definitions of point and line and the uses of definitions in proving. The findings of this paper suggest that the concept of mathematical definitions is very unnatural to students, therefore teachers and textbooks need to explain explicitly the characteristics of mathematical definitions which are different from dictionary definitions using common sense. Also introducing undefined terms in middle school geometry would give students a critical chance to deal with the transition from dictionary definitions to mathematical definitions.

Using national longitudinal data, the authors investigated factors related to student learning or growth in statistics, algebra, and geometry in middle school and high school. The authors used hierarchical linear models to model variation in student rate of growth with factors associated with student characteristics and instructional and environmental factors. In addition, the authors designed the study to identify factors that differentially affect student growth at different levels of secondary school (middle school vs. high school) and for different mathematical content areas (statistics, algebra, geometry). Results indicated substantial growth in all 3 content areas in both middle school and high school. Factors related to student learning were identified and found to differ by level of secondary school and content area.

The purpose of this study is to investigate if information and communications technology (ICT) helps improve students’ achievement in geometry regarding solids’ nets. Our research compares the achievement of the students in the experimental group taught using our ICT intervention to students in the control group taught by traditional teaching methodology. In particular, we designed a teaching intervention with educational activities, in which we incorporated Augmented Reality (AR) technology to test whether geometry teaching is enhanced. The study dealt with primary school students of fourth grade, who were divided into two groups (experimental and control). Students in both groups were pre-tested and post-tested for their achievement in geometry. The results of the study indicated that teaching and learning through ICT is an interactive process for students at primary school and has a positive effect on learning geometry as compared to the traditional teaching method.

The aim of this paper is to explore a framework for articulating the relationship between teachers' practices and students' self-efficacy in teaching and learning of Euclidean Geometry (RTP&SS-EG) in high school.The effective learning of Mathematics, especially Euclidean Geometry by students, is directly influenced by teachers' confidence and competence.The RTP&SS-EG framework was developed based on the gap in literature revealed through international studies, indicating a direct relationship between teachers with a higher level of self-efficacy, and students' achievement.However, the researchers did not come across national studies promoting the relationship between the teachers' practices and students' self-efficacy on the learning of Euclidean Geometry in high school.The question asked was: To what extent is the relationship between the teachers' practices and the students' self-efficacy in teaching and learning of Euclidean Geometry in High school?Geometry Students Self-Efficacy (SGS-EQ) and Geometry Teacher Self-Efficacy questionnaire (GTS-EQ) were utilized.The participants were 81 Mathematics teachers, and 390 Grade 10-12 students from the Mpumalanga and North West provinces in South Africa.Reliability for the entire scale of both instruments computed in terms of the internal consistency was found to be .83.To establish the content validity of the scores from both instruments, the exploratory factor analysis was computed using varimax rotation.The four-factor solution accounting for 43.03% of the total variance was obtained from 36 items of the GTS-EQ.A five-factor solution for 45.37% of the total explained variance was obtained from the 46 items of the SGS-EQ.In RTP&SS-EG, three education theories, constructivism, social constructivism and social cognitivism, policies, as well as teacher and student self-efficacy principles enjoin the practice enacted from the relation between the teachers' practices and the students' self-efficacy in the learning of Euclidean Geometry.The findings revealed that the learning value associated with the relationship between teachers' practices and students' self-efficacy may be integrated into the everyday preparations of teaching-learning plans in Euclidean Geometry.It is critical that policymakers, curriculum developers, teachers and students make use of this model to ensure that self-efficacy of both teachers and students is taken into consideration in Euclidean Geometry.

Este trabalho é um recorte de uma pesquisa realizada no Programa de Pós-Graduação em Educação em Ciências e Matemática da Universidade Federal de Goiás (UFG). Foram realizados o mapeamento e a análise das pesquisas publicadas entre 1987 e 2017, que abordam o ensino de geometria plana no ensino médio com o uso de softwares de geometria dinâmica como recurso didático. Para tanto, os procedimentos metodológicos do Estado do Conhecimento nos guiaram no planejamento, na organização, análise e apresentação dos dados encontrados. O campo empírico baseou-se nos bancos de dados das bibliotecas de instituições que possuem mestrado e doutorado relacionados à Educação Matemática e no Banco de Teses e Dissertações (BDTD) da CAPES. Definiu-se como marco inicial o ano de 1987, por ser o ano de criação do 1º software de geometria dinâmica e 2017 como marco final devido ao de que o levantamento foi realizado em 2018. Os resultados indicam o Geogebra como o software mais utilizado nas experiências com o ambiente dinâmico. Outro aspecto que emerge dos dados coletados são influências da teoria dos registros de representação semiótica e da didática francesa. As perspectivas são permeadas por uma discussão que valoriza o “aprender a aprender” e a formação para o mercado de trabalho, focando nos aspectos técnicos em detrimento da construção de conceitos geométricos.

Teaching Secondary and Middle School Mathematics combines the latest developments in research, standards, and technology with a vibrant writing style to help teachers prepare for the excitement and challenges of teaching secondary and middle school mathematics today. In the fully revised fifth edition, scholar and mathematics educator Daniel Brahier invites teachers to investigate the nature of the mathematics curriculum and reflect on research-based "best practices" as they define and sharpen their own personal teaching styles. The fifth edition has been updated and expanded with a particular emphasis on the continued impact of the Common Core State Standards for Mathematics and NCTM’s just-released Principles to Actions, as well as increased attention to teaching with technology, classroom management, and differentiated instruction. Features include: A full new Chapter 7 on selection and use of specific tools and technology combined with "Spotlight on Technology" features throughout clearly illustrate the practical aspects of how technology can be used for teaching or professional development. Foundational Chapters 1 and 2 on the practices and principles of mathematics education have been revised to build directly on Common Core State Standards for Mathematics and Principles to Actions, with additional references to both documents throughout all chapters. A new Chapter 4 focuses on the use of standards in writing objectives and organizing lesson plan resources while an updated Chapter 5 details each step of the lesson planning process. A fully revised Chapter 12 provides new information on teaching diverse populations and outlines specific details and suggestions for classroom management for mathematics teachers. Classroom Dialogues" features draws on the author’s 35-year experience as an educator to present real-world teacher-student conversations about specific mathematical problems or ideas "How Would You React?" features prepares future teachers for real-life scenarios by engaging them in common classroom situations and offering tried-and-true solutions. With more than 60 practical, classroom-tested teaching ideas, sample lesson and activities, Teaching Secondary and Middle School Mathematics combines the best of theory and practice to provide clear descriptions of what it takes to be an effective teacher of mathematics.

At the beginning of the twentieth century, David Eugene Smith wrote a book about teaching geometry. Although it was written for secondary school geometry teachers, his preface describing why he wrote the book is applicable for all grades. He said that his book “… stands for vitalizing geometry in every legitimate way; for improving the subject matter in such a manner as not to destroy the pupil's interest; for so teaching geometry as to make it appeal to pupils …” (Smith 1911, p. iv). At the beginning of the twenty-first century, the Principles and Standards for School Mathematics (NCTM 2000) gives us guidance in revitalizing geometry for elementary school students. In this article, we share some of its messages for teachers of prekindergarten through grade 5.

Present day instruction in geometry is ineffective. Results of the fourth mathematics assessment of the National Assessment of Educational Progress (NAEP) (Brown et al. 1988) indicate that fewer than half the eleventh-grade students who had taken geometry could apply the Pythagorean theorem in a routine problem and that fewer than a third of these students could find the perimeter of a rhombus drawn on grid paper. Eleventh-grade students who had taken geometry performed only slightly better on spatialvisualization tasks than eleventh-grade students who had not taken geometry.

This work proposes a classification of geometric figures studied in a systematic course in school geometry. Classification of geometric shapes is said to be an important aspect of teaching geometry to enable students to understand and analyze different shapes; identify similarities and differences between figures; recognize common features and differences between figures belonging to the same class or type; systematize and organize knowledge; reveal patterns and relationships between figures with similar properties, which provides the basis for subsequent discoveries; derive new properties and theorems; solve spatial geometric problems. New areas of spatial perception (virtual space, multidimensional space, digital technologies, including cloud, etc.), technologies and industries are increasingly emerging. In such changes, the relevance of the science of geometry does not disappear, but rather increases. This is due to the fact that geometric knowledge and geometric thinking give a person the basis for solving various problems in theoretical (geometric) and practical (physical) spaces. However, despite this, geometry is considered a difficult subject to perceive and understand.

This study presents an analysis of whether geometry teachers create opportunities for student discussion when engaging students in proving as revealed through the use of a multimedia survey instrument. We presented 42 secondary mathematics teachers with 8 multimedia narratives set in the situation of doing proofs in high school geometry and asked them to choose what they would do next, presenting options that included a normative instructional action which closed off discussion and less typical actions that encouraged student discussion. Our analysis provides insight into the professional obligations that teachers use to justify their departure from the norm in order to encourage student talk as well as the background variables that are associated with such decisions. We found that while secondary mathematics teachers frequently chose to promote discussion in their classrooms, the rationale that they chose for this decision and the reasons they might choose not to immediately encourage discussion differed according to the amount of experience they had teaching geometry. We use these differences to illustrate how the professional obligations can be used to better understand how teacher decisions are rationalized at the level of the instructional situation.

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.

The purpose of this study is to determine the influence of selected personal and professional demographic characteristics on the critical thinking abilities of teachers in a parish in Southwest Louisiana. The study is a correlational design using a descriptive survey technique with questions for the data collection. The examiner looked at attribute independent variables – characteristics that a subject has before a study begins such as gender, age, race, highest level of education completed, and years of teaching experience. Three hundred and twenty-four teachers in twelve public schools (three high schools, four middle schools, and five elementary schools) participated in the research. The examiner hypothesized that critical thinking is a by-product of higher-level post-graduate degrees. However, this theory was not proven in the study. There were few findings that showed relationships to the independent variables. One finding was that Caucasians had higher critical thinking scores than other races. Another finding was that social studies teachers at the middle school level had higher scores on the Watson Glazer Critical Thinking Appraisal Short Form Test sub-scale of “Interpretation” than did middle school teachers in other content areas. The researcher found that 51 males (15.8%) and 272 (84.2%) females participated in the study, showing a gender disparity among teachers who participated in the study. The researcher concluded that high school mathematics teachers had higher critical thinking skills in three sub-scales than other high school teachers in other areas. This conclusion was based on the finding showing that high school mathematics teachers had higher scores on the WGCTA sub-scales of Deduction, Interpretation, and Overall Scales than high school teachers who did not identify mathematics as a primary content area of teaching. The researcher recommends that additional research should be done to confirm or disprove the finding that math content influences critical thinking. The researcher also recommends that additional research studies should concentrate on public school teachers of algebra versus teachers of geometry versus teachers of calculus.