Developing Secondary School Teachers’ Didactic–Mathematical Knowledge about Probability
DOI:
https://doi.org/10.17921/2176-5634.2019v12n2p134-144Resumo
Abstract
A formative experience oriented towards development of didactic-mathematical knowledge on probability and associated notions with mathematics teachers working of Fundamental Education in Brazil is described. Theoretical tools from the “onto-semiotic approach of mathematical knowledge and instruction” are used to design and analyze the formative experience. The phases of a didactic design based on this theoretical framework are shown composing the main thread of the developed experience. In the same way, the categories of common, advanced and specialized knowledge, from the mathematics teacher’s didactic-mathematical knowledge model, are applied. The formative model designed, especially the sequence of proposed activities and their a priori analysis, is a contribution allowing to support and educate adequately mathematics teachers on the specific issue of probability and its didactic.
Keywords: Teacher Education. Probability. Didactic-Mathematical Knowledge. Onto-Semiotic Approach.
Resumo
Descreve-se uma experiência formativa orientada para o desenvolvimento do conhecimento didático-matemático sobre probabilidade e noções associadas com professores de matemática que atuam no Ensino Fundamental no Brasil. Ferramentas teóricas da “abordagem ontossemiótica do conhecimento e instrução matemática” são usadas para o desenho e analise sa experiência formativa. As fases de um desenho didático baseado neste referencial teórico são apresentadas como o fio condutor da experiência desenvolvida. Da mesma forma, as categorias de conhecimento comum, avançado e especializado, do modelo de conhecimento didático-matemático do professor de matemática, são aplicadas. O modelo formativo desenhado, especialmente a sequência das atividades propostas e sua análise a priori, é uma contribuição que permite apoiar e formar adequadamente os professores de matemática sobre a questão específica da probabilidade e sua didática.
Palavras-chave: Formação Docente. Probabilidade. Conhecimento Didático-Matemático. Abordagem Ontossemiótica.
Referências
Ball, D. L., Hill, H. H., & Bass, H. (2005). Knowing Mathematics for teaching: Who knows mathematics well enough to teach third grade, and how can we decide? American Educator, Fall, 14-46.
Batanero, C., & Díaz, C. (2012). Training school teachers to teach probability: reflections and challenges. Chilean Journal of Statistics, 3 (1), 3-13.
Batanero, C., Henry, M., & Parzysz, B. (2005).The nature of chance and probability. In G. A. Jones (Ed.), Exploring probability in school: Challenges for teaching and learning (pp.16-42). New York, USA: Springer. DOI http://dx.doi.org/10.1007/0-387-24530-8_2
Batanero, C., Gómez, E., Gea, M. M., & Contreras, J. M. (2014). Assessing and developing prospective teachers’ understanding of random sequences. In U. Sproesser, S. Wessolowski, & C. Wörn (Eds.), Daten, Zufall und der Rest der Welt – Didaktische Perspektiven zur anwendungsbezogenen Mathematik, (pp.1-11). Heidelberg: Springer Fachmedien Wiesbaden. DOI http://dx.doi.org/10.1007/978-3-658-04669-9_1
Contreras, J. M., Batanero, C., Díaz, C., & Arteaga, P. (2013). Evaluación de la falacia del eje temporal en futuros profesores de educación secundaria. Acta Scientiae, 14 (3), 346-362.
Cordani, L. K., & Wechsler, S. (2006). Teaching independence and exchangeability. In A. Rossman, & B. Chance (Eds.), Proceedings of the Seventh International Conference on Teaching Statistics. Salvador, Brasil: International Association for Statistics Education. [CD-ROM].
Díaz, C., & De La Fuente, I. (2007). Assessing students’ difficulties with conditional probability and bayesian reasoning. International Electronic Journal of Mathematics Education, 2 (3), 128-148.
Falk, R. (1986). Conditional probabilities: insights and difficulties. In Proceedings of the Second International Conference on Teaching Statistics.Victoria, Canada: International Association for Statistical Education. Retrieved from
http://iase-web.org/Conference_Proceedings.php?p=ICOTS_2_1986
Fischbein, E. (1993). The interaction between the formal, the algorithmic and the intuitive components in a mathematical activity. In R. Biehler, R. W. Scholz, R. Straser, & B. Winkelmann, (Eds.), Didactics of Mathematics as a Scientific Discipline, (pp. 231-245). Netherlands, Dordrecht: Kluwer.
Font, V., Godino, J. D., & Gallardo, J. (2013). The emergence of objects from mathematical practices. Educational Studies in Mathematics, 82(1),97-124. doi: http://dx.doi.org/10.1007/s10649-012-9411-0
Gal, I. (2005). Towards “probability literacy” for all citizens: building blocks and instructional dilemmas. In G. A. Jones, (Ed.), Exploring probability in school: Challenges for teaching and learning.(pp.39-63). New York, USA: Springer. doi http://dx.doi.org/10.1007/0-387-24530-8_3
Garfield, J. (2006). Assessment resource tools for improving statistical thinking - Artist. Retrieved from https://apps3.cehd.umn.edu/artist/
Godino, J. D. (2009). Categorías de análisis de los conocimientos del profesor de matemáticas. Unión Revista Iberoamericana de Educación Matemática, 20, 13-31.
Godino, J. D., Batanero, C., Font, V., Giacomone, B. (2016). Articulando conocimientos y competencias del profesor de matemáticas: el modelo CCDM. In: Fernández, C. et al. (Ed.). Investigación en Educación Matemática XX, (pp288-297). Málaga: SEIEM.
Godino, J. D., Batanero, C., Rivas, H., & Arteaga, P. (2013). Componentes e indicadores de idoneidad de programas de formación de profesores en didáctica de las matemáticas. Revemat – Revista Eletrônica de Educação Matemática, 8(1),46-74. doi http://dx.doi.org/10.5007/1981-1322.2013v8n1p46
Godino, J. D., Ortiz, J. J., Roa, R. & Wilhelmi, M.R. (2011). Models for statistical pedagogical knowledge. In C. Batanero, G. Burrill, & C. Reading. Teaching statistics in school mathematics-challenges for teaching and teacher education: a joint ICMI/IASE Study.(pp.271-282). Berlin, Alemanha: Springer.
Godino, J. D., Rivas, H., Arteaga, P., Lasa, A., & Wilhelmi, M. R. (2014). Ingeniería didáctica basada en el enfoque ontológico - semiótico del conocimiento y la instrucción matemáticos. Recherches en Didactique des Mathématiques, 34 (2/3), 167-200.
Gómez, E. (2014). Evaluación y desarrollo del conocimiento matemático para la enseñanza de la probabilidad en futuros profesores de educación primaria. (Tese doctoral) Facultad de Ciencias de la Educación, Departamento de Didáctica de la Matemática. Universidad de Granada.
Gras, R., & Totohasina, A. (1995). Chronologie et causalité, conceptions sourcesd’ obstacles épistémologiques à la notion de probabilité conditionnelle. Educacional Studies in Mathematics, 28(4), 337-363. doi http://dx.doi.org/10.1007/bf01274078
Green, D. R. (1983). A survey of probability concepts in 3000 pupils aged 11-16 years. In D. R. Grey, P. Holmes, V. Barnett & G. M. Constable, (Eds.), Proceedings of the First International Conference on Teaching Statistics. (pp.766-783). Sheffield, UK: Teaching Statistics Trust.
Hill, H. C., Ball, D. L., & Schilling, S. G. (2008). Unpacking pedagogical content knowledge: Conceptualizing and measuring teachers’ topic-specific knowledge of students. Journal for Research in Mathematics Education, 39 (4), 372-400.
Ives, S. E. (2009). Learning to teach probability: Relationships among preservice teachers’ beliefs and orientations, content knowledge, and pedagogical content knowledge of probability. (Doctor ship Thesis) Faculty of North Carolina State University.
Kataoka, V. Y., Souza, A. A., Oliveira, A. de C. S., Fernandes, F., Paranaíba, P., & Oliveira, M. S. (2008).Probability Teaching in Brazilian Basic Education: Evaluation and Intervention. Anais do ICME 11, Monterrey, Mexico. Retrieved from http://tsg.icme11.org/tsg/show/14
Lecoutre, M. (1992).Cognitive models and problem spaces in "purely random" situations. Educational Studies in Mathematics, 23 (6), 557-568. doi http://dx.doi.org/10.1007/BF00540060
Brasil. MEC-SEF. (1998). Parâmetros curriculares nacionais: Matemática, 5ª a 8ª série. Brasília, Brasil: Ministério de Educação e Cultura – Secretaria de Ensino Fundamental.
Mohr, M.J. (2008). Mathematics knowledge for teaching: The case of preservice teachers. In G. Kulm, (Ed.), Teacher knowledge and practice in middle grades mathematics. (pp.19-43). Rotterdam, The Netherlands: Sense Publishers.
Nunes, T., Bryant, P., Evans, D., Gottardis, L., & Terlektsi, M. (2012). Teaching primary school children about probability. Teacher handbook. Education Department, Oxford University.
Pietropaolo, R. C., Campos, T. M. M., Felisberto de Carvalho, J. I., & Teixeira, P. (2013). Um estudo sobre os conhecimentos necessários ao professor para ensinar noções concernentes à probabilidade nos anos iniciais. Anais do IV Seminário do Observatório da Educação da CAPES. Brasil. Retrived from http://seminarios.capes.gov.br/observatorio-da-educacao/resumos/108-educacao-basica
Pietropaolo, R. C., Campos, T. M. M., & Silva, A. F. G. (2012). Formação continuada de professores de matemática da Educação Básica em um contexto de implementação de inovações curriculares. Ensino de Ciências e da Matemática, 8 (2), 377-399.
Sánchez, E. (2000). Investigaciones didácticas sobre el concepto de eventos independientes em probabilidad. Recherches en Didactique des Mathématics, 2(3), 305-330.
Shulman, L. S. (1987). Knowledge and teaching: Foundations of the new reform. Harvard Educational Review, 57(1),1-22. doi http://dx.doi.org/10.17763/haer.57.1.j463w79r56455411
Tversky, A., & Kahneman, D. (1982). Causal schemas in judgment under uncertainty. In D. Kahneman, P.Slovic, & A. Tversky. Judgement under uncertainty: Heuristics and biases (pp.117-128). Cambridge, Inglaterra: Cambridge University Press. doi http://dx.doi.org/10.1017/CBO9780511809477
