Afinando o Foco em Matemática: Desenho, Implementação e Avaliação de Atividades MathTASK para a Formação de Professores de Matemática
Resumo
MathTASK é um programa de pesquisa e desenvolvimento que engaja professores de matemática em situações de sala de aula desafiadoras e altamente contextualizadas na forma de tarefas (chamadas de mathtasks). As respostas de professores às tarefas revelam seus discursos matemáticos e pedagógicos e abrem oportunidades para articular, considerar e reconstruir tais discursos. As tarefas foram usadas como instrumento de pesquisa e também de formação de professores e desenvolvimento profissional no Reino Unido, na Grécia e no Brasil. Neste artigo, apresentamos o programa MathTASK e um exemplo de mathtask. Em seguida, apresentamos um resumo dos construtos teóricos que emergiram na análise dos dados do programa MathTASK. Então, indicamos os princípios gerais do uso de mathtasks na pesquisa e formação de professores e damos quatro exemplos dos princípios, cada um dirigido a diferentes aspectos do ensino e da aprendizagem de matemática, e cada um desenvolvido tendo em mente níveis educacionais e contextos. Concluímos com observações sobre os benefícios de usar mathtasks como uma forma de estimular e facilitar reflexões de professores de matemática sobre suas práticas.
Referências
ARCAVI, Abraham. Developing and using symbol sense in mathematics. For the Learning of Mathematics, v. 14, n. 3, p. 42–47, 2005.
BALL, Deborah; THAMES, Mark; PHELPS, Geoffrey. Content knowledge for teaching. Journal of Teacher Education, v. 59, n. 5, p. 389–407, 2008.
BIEHLER, Rolf; KEMPEN, Leander. Students’ use of variables and examples in their transition from generic proof to formal proof. In: UBUZ, Behiye; HASER, Çiğdem; MARIOTTI, Maria-Alessandra (Eds.), Proceedings of the Eigth Congress of the European Mathematical Society for Research in Mathematics Education. Ankara, Turkey: Middle East Technical University and ERME, 2013. p. 86-95.
BIZA, Irene; NARDI, Elena. Scripting the experience of mathematics teaching: The value of student teacher participation in identifying and reflecting on critical classroom incidents. International Journal for Lesson and Learning Studies, v. 9, n. 1, p. 43-56, 2019.
BIZA, Irene; NARDI, Elena; JOEL, Gareth. Balancing classroom management with mathematical learning: Using practice-based task design in mathematics teacher education. Mathematics Teacher Education and Development, v. 17, n. 2, p. 182-198, 2015.
BIZA, Irene; NARDI, Elena; ZACHARIADES, Theodossios. Using tasks to explore teacher knowledge in situation-specific contexts. Journal of Mathematics Teacher Education, v. 10, p. 301-309, 2007.
BIZA, Irene; NARDI, Elena; ZACHARIADES, Theodossios. Teacher beliefs and the didactic contract on visualization. For the Learning of Mathematics, v. 29, n. 3, p. 31-36, 2009.
BIZA, Irene; NARDI, Elena; ZACHARIADES, Theodossios. Competences of mathematics teachers in diagnosing teaching situations and offering feedback to students: Specificity, consistency and reification of pedagogical and mathematical discourses. In: LEUDERS, Timo; LEUDERS, Juliane; PHILIPP, Kathleen. (Eds.). Diagnostic Competence of Mathematics Teachers. Unpacking a complex construct in teacher education and teacher practice, New York: Springer, 2018, p. 55-78.
BIZA, Irene; NARDI, Elena; ZACHARIADES, Theodossios. Utilizando tarefas de situação específica para explorar o conhecimento matemático e as crenças pedagógicas: Exemplos de Álgebra e Análise. In: ROGUE, Tiatiana; GIRALDO, Victor. (Eds.). O saber do professor de matemática: ultrapassando a dicotomia entre didática e conteúdo. Rio de Janeiro, Brazil: Editora Ciência Moderna Ltda. 2014, p. 329-378.
CARTER, Kath. What is a case? What is not a case? In: LUNDEBERG, Mary; LEVIN, Barbara; HARRINGTON, Helen (Eds.). Who learns what from cases and how? The research base for teaching and learning with cases. Mahwah NJ: Lawrence Erlbaum Associates, 1999.
CHEVALLARD, Yves. La transposition didactique [The didactic transposition]. Grenoble: La Pensée Sauvage Éditions, 1985.
CHRISTIANSEN, Bent; WALTER, Gail. Task and activity. In: CHRISTIANSEN, Bent; HOWSON, Geoffrey; OTTE, Michael (Eds.). Perspectives on mathematics education: Papers submitted by members of the Bacomet Group. Dordrecht: D. Reide, 1986, p. 243-307.
CLARK-WILSON, Alison; NOSS, Richard. Hiccups within technology mediated lessons: A catalyst for mathematics teachers’ epistemological development. Research in Mathematics Education, v. 17, n. 2, p. 92-109, 2015.
COOPER, Jason. Mathematical discourse for teaching: A discursive framework for analyzing professional development. In: Proceedings of the Joint Meeting of PME. v. 38, p. 337–344, 2014.
DREHER, Anika; NOWINSKA, Edyta; KUNTZE, Sebastian. Awareness of dealing with multiple representations in the mathematics classroom - A study with teachers in Poland and Germany. In: LINDMEIER, Anke; HEINZE, Aiso. (Eds.). Proceedings of the 37th Conference of the International Group for the Psychology of Mathematics Education, v. 2. Kiel, Germany: PME, 2013, p. 249-256.
EPP, Susanna. Variables in mathematics education. In: International Congress on Tools for Teaching Logic. Springer, Berlin, Heidelberg, 2011, p. 54-61.
ERENS, Rolf; EICHLER, Andreas. Belief systems’ change - from preservice to trainee high school teachers on calculus. In LINDMEIER, Anke; HEINZE, Aiso. (Eds.). Proceedings of the 37th Conference of the International Group for the Psychology of Mathematics Education, v. 2, Kiel, Germany: PME, 2013. p. 281-288.
FREEMAN, James. Systematizing Toulmin’s warrants: An epistemic approach. Argumentation, v. 19, n. 3, p. 331–346, 2005.
GERVASONI, Ann; LINDENSKOV, Lena. Students with “special rights” for mathematics education. In: ATWEH, Bill, GRAVEN, Mellony; VALERO, Paola. (Eds.). Mapping equity and quality in mathematics education. New York, NY: Springer, 2011. p. 307–324.
GIRALDO, Victor; ROQUE, Tatiana. História e Tecnologia na Construção de um Ambiente Problemático para o Ensino de Matemática. In: GIRALDO, Victor; ROQUE, Tatiana (Org.), O Saber do Professor de Matemática: Ultrapassando a Dicotomia entre Didática e Conteúdo. Rio de Janeiro: Editora Ciência Moderna Ltda, 2014, p. 9-37.
GOODELL, Joanne-Elizabeth. Using Critical Incident Reflections: A Self-Study as a Mathematics Teacher Educator. Journal of Mathematics Teacher Education, v. 9, n. 3, p. 221-248, 2006.
GUEUDET, Ghislaine; BUTEAU, Chantalle; MESA, Vilma; MISFELDT, Morten. Instrumental and Documentational Approaches: From Technology Use to Documentation Systems in University Mathematics Education. Research in Mathematics Education, v. 16, n. 2, p. 139-155, 2014.
HEALY, Lulu; FERNANDES, Solange. The role of gestures in the mathematical practices of those who do not see with their eyes. Educational Studies in Mathematics, v. 77, p. 157-174, 2011.
HEALY, Lulu; POWELL, Arthur. Understanding and overcoming “disadvantage” in learning mathematics. In: CLEMENTS, McKenzie-Alexander; BISHOP, Alan; KEITEL, Christine; KILPATRICK, Jeremy; LEUNG, Frederick (Eds.). Third international handbook of mathematics education. NL: Springer, 2013. p. 69–100.
HERBST, Patricio; CHAZAN, Daniel. Exploring the practical rationality of mathematics teaching through conversations about videotaped episodes: The case of engaging students in proving. For the Learning of Mathematics, v. 23, n. 1, p. 2–14, 2003.
HOLE, Simon; MCENTEE, Grace. Reflection Is at the Heart of Practice. Educational Leadership, v. 56, n. 8, p. 34-37, 1999.
HILL, Heather; BALL, Deborah. Learning mathematics for teaching: Results from California's mathematics professional development institutes. Journal for Research in Mathematics Education, v. 35, n. 5, p. 330–351, 2004.
IANNONE, Paola; NARDI, Elena. On the pedagogical insight of mathematicians: ‘Interaction’and ‘transition from the concrete to the abstract’. The Journal of Mathematical Behavior, v. 24, n. 2, p. 191-215, 2005.
KAYALI, Lina. Mathematics teachers work with resources: Four cases of secondary teachers using technology. (Unpublished doctoral dissertation). University of East Anglia, Norwich, UK, 2019.
KAYALI, Lina; BIZA, Irene. “One of the beauties of Autograph is … that you don’t really have to think”: Integration of resources in mathematics teaching. In: DOOLEY, Therese; GUEUDET, Ghislaine (Eds.). Proceedings of the 10th Conference of European Research in Mathematics Education. Dublin: Dublin City University, 2017. p. 2406-2413.
KAYALI, Lina; BIZA, Irene. Scheming and Re-scheming: Secondary Mathematics Teachers’Use and Re-use of Resources. Digital Experiences in Mathematics Education, Online First, 2021. Disponível em: https://doi.org/10.1007/s40751-021-00091-x. Acesso em 3 de ago. 2021.
KIERAN, Caroline; FORMAN, Ellice; SFARD, Anna. Learning discourse: Discursive approaches to research in mathematics education. Kluwer Academic Publishers, 2002.
KONTOROVICH, Igor'. What can be ‘annoying’ about mathematical conventions? Analysing exchanges of mathematically competent discursants. In: JANKVIST, Ufe; VAN DEN HEUVEL-PANHUIZEN, Maria; VELDHUIS, Michiel. (Eds.), Proceedings of the Eleventh Congress of the European Society for Research in Mathematics Education. Utrecht, the Netherlands: Freudenthal Group & Freudenthal Institute, Utrecht University and ERME, 2019.
LAMPERT, Magdalene. How do teachers manage to teach? Perspectives on problems in practice. Harvard Educational Review, v. 55, n. 2, p. 178–194, 1985.
LEATHAM, Keith; PETERSON, Blake; STOCKERO, Shari; VAN ZOEST, Laura. Conceptualizing mathematically significant pedagogical opportunities to build on student thinking. Journal for Research in Mathematics Education, v. 46, n. 1, p. 88–124, 2015.
LEONT'EV, Aleksie Nikolaevich. Dieyatelinocti, soznaine, i lichynosti [Activity, consciousness, and personality]. Moskva: Politizdat, 1975.
MARKOVITS, Z.; SMITH, M.S. Cases as tools in mathematics teacher education. In: TIROSH, Dina; WOOD, Terry (Eds.), The international handbook of mathematics teacher education: Tools and Processes in Mathematics Teacher Education. Rotterdam: Sense Publishers, 2008, p. 39-65.
MASON, John; JOHNSTON-WILDER, Sue. Designing and using mathematical tasks. York, UK: QED Press, 2006.
MOUSTAPHA-CORRÊA, Bruna. Rumo a uma Postura Problematizadora na Formação de Professores de Matemática: Articulando Práticas Históricas e Práticas de Sala de Aula. Tese (Doutorado em Ensino e História da Matemática e da Física). Instituto de Matemática, Universidade Federal do Rio de Janeiro. Rio de Janeiro, p. 363. 2020.
MOUSTAPHA-CORRÊA, Bruna; BERNARDES, Aline; GIRALDO, Victor. Historical tasks to foster problematization. In: JANKVIST, Ufe; VAN DEN HEUVEL-PANHUIZEN, Maria; VELDHUIS, Michiel. (Eds.). Proceedings of the Eleventh Congress of the European Society for Research in Mathematics Education. Utrecht, the Netherlands: Freudenthal Group & Freudenthal Institute, Utrecht University and ERME, p. 2149-2157, 2019.
MOUSTAPHA-CORRÊA, Bruna; BERNARDES, Aline; GIRALDO, Victor; BIZA, Irene; NARDI, Elena. Problematizing mathematics and its pedagogy through teacher engagement with history-focused and classroom situation-specific tasks. Journal of Mathematical Behavior (Special Issue: Advances in Commognitive Research), v. 61, p. 1-10, 2021.
NARDI, Elena. Amongst mathematicians: Teaching and learning mathematics at university level. Springer Science & Business Media, 2008.
NARDI, Elena; BIZA, Irene; ZACHARIADES, Theodossios. Warran revisited: Integrating mathematics teachers pedagogical and epistemological considerations into Toulmin’s model of argumentation. Educational Studies in Mathematics, v. 79, n. 2, p. 157-173, 2012.
NARDI, Elena; HEALY, Lulu; BIZA, Irene. The CAPTeaM Project (Challenging Ableist Perspectives on the Teaching of Mathematics): A preliminary report. In: ADAMS, Gill. (Ed.) Proceedings of the British Society for Research into the Learning of Mathematics Day Conference, v. 35, n. 2, p. 51-57, 2015.
NARDI, Elena; HEALY, Lulu; BIZA, Irene. ‘Feeling’ the mathematics of disabled learners: Supporting teachers towards valuing, attuning, integrating and resignifying in an inclusive mathematics classroom. In: HUNTER, Roberta; CIVIL, Marta; HERBEL-EISENMANN, Beth; PLANAS, Núria; WAGNER, David (Eds.) Mathematical discourse that breaks barriers and creates space for marginalized learners. The Netherlands: SENSE Publications, 2017. p. 147-170.
NARDI, Elena; HEALY, Lulu; BIZA, Irene; FERNANDES, S.H.A.A. Challenging ableist perspectives on the teaching of mathematics through situation-specific tasks. In: CSÍKOS, Csaba; RAUSCH, Attila; SZITÁNYI, Judit (Eds.). Proceedings of the 40th Conference of the International Group for the Psychology of Mathematics Education (PME). Szeged, Hungary: PME, 2016. p. 347–354.
POTARI, Despina; PSYCHARIS, Giorgos. Prospective Mathematics Teacher Argumentation While Interpreting Classroom Incidents. In: STRUTCHENS, Marilyn; HUANG, Rongjin; POTARI, Despina; LOSANO, Leticia (Eds.), Educating Prospective Secondary Mathematics Teachers. ICME-13 Monographs: Springer, 2018, p. 169-187.
PULTORAK, Edward. Facilitating reflective thought in novice teachers. Journal of Teacher Education, v. 44, p. 288–295, 1993.
ROQUE, Tatiana. História da Matemática: Uma visão crítica, desfazendo mitos e lendas. Rio de Janeiro: Zahar, 2012.
ROWLAND, Tim; THWAITES, Anne; JARED, Libby. Triggers of contingency in mathematics teaching. Research in Mathematics Education, v. 17, n. 2, p. 74-91, 2015.
SFARD, Anna. Thinking as communicating. Human development, the growth of discourse, and mathematizing. New York, NY: Cambridge University Press, 2008.
SHULMAN, Lee. Toward a pedagogy of cases. In: SHULMAN, Judith (Ed.), Case methods in teacher education. New York, Teachers College Press, 1992. p. 1–29.
SHULMAN, Lee. Those who understand: Knowledge growth in teaching. Educational Researcher, v. 15, n. 2, p. 4–14, 1986.
SHULMAN, Lee. Knowledge and teaching: Foundations of the new reform. Harvard Educational Review, v. 57, n. 1, p. 1–22, 1987.
SKEMP, Richard. Relational understanding and instrumental understanding. Mathematics Teaching, v. 77, p. 20– 26, 1976.
SKOTT, Jeppe. The emerging practices of a novice teacher: The roles of his school mathematics images. Journal of Mathematics Teacher Education, v. 4, n. 1, p. 3-28, 2001.
SPEER, Natasha. Issues of methods and theory in the study of mathematics teachers’ professed and attributed beliefs. Educational Studies in Mathematics, v. 58, n. 3, p. 361–391, 2005.
THOMA, Athina. Transition to university mathematical discourses: A commognitive analysis of first year examination tasks, lecturers’ perspectives on assessment and students’ examination scripts. (Unpublished doctoral dissertation). University of East Anglia, Norwich, UK, 2018.
THOMA, Athina; NARDI, Elena. Discursive shifts from school to university mathematics and lecturer assessment practices: Commognitive conflicts regarding variables. In: Dooley, Therese; Gueudet, Ghislaine (Eds.). Proceedings of the Tenth Congress of the European Mathematical Society for Research in Mathematics Education. Dublin, Ireland: DCU Institute of Education and ERME, 2017. p. 2266-2273.
THOMA, Athina; NARDI, Elena. Transition from School to University Mathematics: Manifestations of Unresolved Commognitive Conflict in First Year Students’ Examination Scripts. International Journal of Research in Undergraduate Mathematics Education, v. 4, n. 1, p. 161-180, 2018.
THOMPSON, Alba. Teachers’ beliefs and conceptions: A synthesis of the research. In: GROUWS, Douglas (Ed.), Handbook of research on mathematics teaching and learning. New York: Macmillan, 1992. p. 122–127.
TIROSH, Dina; WOOD, Terry. (Eds.). The International Handbook of Mathematics Teacher Education. v. 2. Rotterdam: Sense publishers, 2009.
TOULMIN, Stephen. The uses of argument. Cambridge, UK: Cambridge University Press, 1958.
TRIPP, David. Critical incidents in teaching: Developing professional judgement (Classic ed.). London; New York: Routledge, 2012.
TURNER, Fay; ROWLAND, Tim. The Knowledge Quartet as an Organizing Framework for Developing and Deepening Teachers’ Mathematics Knowledge. In: ROWLAND, Tim; RUTHVEN, Keneth. (Eds.). Mathematical Knowledge in Teaching. London and New York: Springer, 2011. p 195-212.
ZASLAVSKY, Orit; SULLIVAN, Peter. (Eds.). Constructing knowledge for teaching: Secondary mathematics tasks to enhance prospective and practicing teacher learning. New-York: Springer, 2011.
ZASLAVSKY, Orit; WATSON, Anne; MASON, John. Special issue: The nature and role of tasks in mathematics teachers' education. Journal of Mathematics Teacher Education, v. 10, p. 4-6, 2007.
ZAZKIS, Rina; SINCLAIR, Natalie; LILJEDAHL, Peter. Lesson play in mathematics education: A tool for research and professional development. New-York: Springer, 2013.
ZAZKIS, Rina. Divisors and quotients: Acknowledging polysemy. For the learning of mathematics, v. 18, n. 3, p. 27-30, 1998.
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