Conhecimento dos professores sobre geometria nos anos iniciais do ensino fundamental: uma visão do estado da arteTeachers' knowledge of geometry in the early years of elementary school: a state of the art

ResumoNeste artigo, a partir de um estudo do estado da arte, analisa-se o tema conhecimento geométrico de professores dos anos iniciais em pesquisas em educação matemática realizadas no Brasil num intervalo de 19 anos, entre 2000 e 2019. A leitura de 31 estudos em nível de mestrado e doutorado apontam que entre os aportes teóricos que embasam o conhecimento geométrico do professor, destacam-se os de Shulman (1986, 1987) e Tardif (2002). Ao longo dos anos, esses modelos teóricos tornaram-se as principais referências para análise do conhecimento/saberes de professores. Em relação aos objetivos, a maioria dos estudos buscava analisar ou identificar como a formação em serviço ou formação continuada pode influenciar na mobilização de conhecimentos/saberes pelos professores. Embora o objeto de análise fosse praticamente o mesmo e os estudos utilizassem uma abordagem qualitativa, os procedimentos metodológicos eram diversificados, incluindo estudo de caso, pesquisa-ação e análise documental. Como instrumentos de coleta de dados destacaram-se diagnósticos; registros produzidos pelas participantes; diário de campo da pesquisadora; gravações em áudio e/ou vídeo e produção de sequências didáticas, entre outras. Observou-se uma tendência em coletar informações por meio de encontros formativos, oficinas e laboratórios de matemática, possivelmente para que os pesquisadores interviessem no desenvolvimento do conhecimento geométrico dos professores. Os resultados das pesquisas analisadas apontam para fragilidades no conhecimento conceitual e prático dos professores em relação à geometria. Também indicam que os processos formativos possibilitam mudanças no conhecimento conceitual e na prática educativa a partir da reflexão dessa prática e da construção de aprendizagens.Palavras-chave: Geometria, Conhecimento de professores, Educação Matemática.AbstractIn this article, based on a state of the art study, we analyse the theme of geometric knowledge of teachers of the early years in mathematics education in research carried out in Brazil between 2000 and 2019. The reading of 31 studies at the master’s and doctoral level points out that among the theoretical contributions that support the teacher’s geometric knowledge, Shulman’s (1986, 1987) and Tardif’s (2002) stand out. Over the years, these theoretical models have become the main references for the analysis of teachers’ knowledge/know-how. Regarding the objectives, most studies sought to analyse or identify how in-service education or continuing education can influence the teachers’ mobilisation of knowledge/know-how. Although the object of analysis was practically the same and the studies used a qualitative approach, the methodological procedures were diverse, including case study, action research, and documentary analysis. As instruments of data collection, we highlight the diagnoses; registers produced by the participants; researcher’s field diary; audio and/or video recordings and production of didactic sequences, among others. There was a tendency to collect information through formative meetings, workshops, and mathematics laboratories, possibly for researchers to intervene in the development of teachers’ geometric knowledge. The results of the studies analysed point to weaknesses in the teachers’ conceptual and practical knowledge of geometry. They also indicate that the education processes enable changes in conceptual knowledge and educational practice based on the reflection of this practice and the construction of learning.Keywords: Geometry, Teachers’ knowledge, Mathematics education.ResumenEn este artículo, basado en un estudio del estado del arte, se analiza el tema conocimiento geométrico de los docentes de los años iniciales en investigaciones en educación matemática realizadas en Brasil entre 2000 y 2019. La lectura de 31 estudios a nivel de maestría y doctorado señala que entre los aportes teóricos que sustentan el conocimiento geométrico del docente, se destacan los de Shulman (1986, 1987) y Tardif (2002). A lo largo de los años, estos modelos teóricos se han convertido en los principales referentes para el análisis del conocimiento docente. En cuanto a los objetivos, la mayoría de los estudios buscaban analizar o identificar cómo la formación en servicio o continua puede influir en la movilización de conocimientos por parte de los docentes. Aunque el objeto de análisis fue prácticamente el mismo y los estudios utilizaron un enfoque cualitativo, los procedimientos metodológicos fueron diversos, incluyendo el estudio de casos, la investigación-acción y el análisis de documentos. Como instrumentos de recolección de datos, se destacaron los diagnósticos; registros producidos por los participantes; diario de campo del investigador; grabaciones de audio y/o video y producción de secuencias didácticas, entre otros. Hubo una tendencia a recolectar información a través de reuniones formativas, talleres y laboratorios de matemáticas, posiblemente para que los investigadores intervinieran en el desarrollo del conocimiento geométrico de los docentes. Los resultados de las investigaciones analizadas apuntan a debilidades en los conocimientos conceptuales y prácticos de los docentes en relación con la geometría. También indican que los procesos de formación posibilitan cambios en el conocimiento conceptual y la práctica educativa a partir de la reflexión de esta práctica y la construcción de aprendizajes.Palabras clave: Geometría, Conocimiento de los profesores, Educación matemática.

Upaya Meningkatkan Pengenalan Geometri dengan Permainan Puzzle Bervariasi pada Kelompok B TK Muslimat Tahun Ajaran 2016/2017

Based on the data from the initial conditions before the classroom action research was obtained, it was obtained information that the percentage of learning completeness 53% had not reached the indicator. Thus, a study was conducted at Muslimat Kindergarten in Group B Academic Year 2016 / 2017. This study showed a significant increase in the introduction of geometry child group B TK Muslimat. This can be seen from the first cycle on the percentage of completeness 62% with an average observation (student activity) 68% (sufficient) and finally on cyclical II on the percentage of completeness 80% with an average observation (student activity) 82% ( well). Based on these data it can be concluded that varied puzzle games can improve geometry recognition in group B of the first semester in Muslimat Kindergarten in the 2016/2017 Academic Year. Suggestions that researchers can convey should be in the process of introducing geometry, teachers must create learning that is active, innovative, creative, effective, and fun. One of them through varied puzzle games, so that the goal of increasing geometry recognition can be achieved.

What Details Do Teachers Expect From Student Proofs? A Study of Proof Checking in Geometry

We investigated how secondary mathematics teachers check student geometry proofs. From video records of geometry teachers checking proofs, we conjectured that teachers have different expectations for details that follow from written statements than for details that are conveyed by diagrams. To test our conjectures, we randomly assigned 44 secondary mathematics teachers to 1 of 3 experiment groups (n & 13, n & 15, n & 16) in which they viewed and rated representations of instructional practice. Participants in each group viewed treatment or control versions of instructional scenarios and rated the appropriateness of the teachers' work in different segments of each scenario. We compared participants' ratings across and within experiment groups. We found that participants rated lower instruction that deviated from what we hypothesized to be their expectations, confirming our hypotheses.

Dynamic Approach to Teaching Geometry

Secondary geometry teachers from several urban school districts participated in a two-year professional development focused on integrating dynamic geometry into teaching. The chapter documents the positive impact of the professional development for teachers' Technological Pedagogical Content Knowledge (TPACK) development and their students' achievement in geometry through the use of the dynamic geometry approach. Instruments used to develop and assess teachers' TPACK included a Conjecturing-Proving Test, interviews and observation protocols. Participants' TPACK levels were identified using a TPACK Development Levels Assessment Rubric. Findings show that teachers' TPACK tended to remain within the three middle TPACK levels (accepting, adapting, and exploring). Recommendations and suggestions for future research are offered to those who implement school-based, mixed methods research studies involving technology.

Circular History

I would like to share with my fellow geometry teachers a real-life illustration of the importance of the adjective “equidistant” in the definition of a circle. This illustration is taken from Diary of the Dark Years, 1940-1944: Collaboration, Resistance, and Daily Life in Occupied Paris, by the French cultural critic, Jean Gúehenno. This was Gúehenno's diary entry for July 4, 1941:

Dynamic Approach to Teaching Geometry

Secondary geometry teachers from several urban school districts participated in a two-year professional development focused on integrating dynamic geometry into teaching. The chapter documents the positive impact of the professional development for teachers' Technological Pedagogical Content Knowledge (TPACK) development and their students' achievement in geometry through the use of the dynamic geometry approach. Instruments used to develop and assess teachers' TPACK included a Conjecturing-Proving Test, interviews and observation protocols. Participants' TPACK levels were identified using a TPACK Development Levels Assessment Rubric. Findings show that teachers' TPACK tended to remain within the three middle TPACK levels (accepting, adapting, and exploring). Recommendations and suggestions for future research are offered to those who implement school-based, mixed methods research studies involving technology.

Activities for Students: Inquiry into Fractals

The exotic images of fractals often pique the interest of high school mathematics students, and this interest presents an opportunity for geometry teachers to draw students into an investigation of transformations and patterns. By using a simple building block and fractals' self-imaging characteristic (as the figure grows, it retains the pattern established by the building block), teachers can bring construction of fractals into the high school geometry curriculum. The three activities described in this article engage students in constructing a fractal, searching a fractal for patterns, and using transformations to build different fractals. Students gain insight into patterns as their fractals grow; they flip or rotate fractal pieces by following a design and translating the pieces into place.