Výčet všech možných případů: Analýza žákovských prací

Annie Savard; Lucie DeBlois

doi:10.14712/18047106.44

Vol. 4 No. 1 (2013): sciED Research Reports

Enumerating All Possible Outcomes: An Analysis of Students’ Work

Annie Savard, Lucie DeBlois,

DOI: https://doi.org/10.14712/18047106.44

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Savard, A., & DeBlois, L. (2013). Enumerating All Possible Outcomes: An Analysis of Students’ Work. Scientia in Educatione, 4(1), 49-62. https://doi.org/10.14712/18047106.44

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Abstract

A variety of contexts in the learning of probability could provide opportunities for students to reason under uncertainty. This kind of reasoning could support students to develop critical thinking practices. This paper presents a study on how children in a grade four classroom developed strategies about enumeration of combinatorics using critical thinking. A preservice teacher taught those students and the analysis of their work showed the procedures they used. Then, these results emerge from a secondary analysis. The results suggest that students need opportunities to develop increasingly sophisticated methods of reasoning probabilistically. This study also suggests that critical thinking is both necessary to develop these modes of reasoning and is developed through this work.

References

BAILIN, S., CASE, R., COOMBS, J. R., DANIELS, L. B. Conceptualizing Critical Thinking. Journal of Curriculum Studies, 31(3), 1999, 285–302. Search in Google Scholar
BRIAND, J. Une expérience statistique et une premi`ere approche des lois du hasard au lycée par une confrontation avec une machine simple. Recherches en didactique des mathématiques, 25(2), 2005, 247–281. Search in Google Scholar
BROUSSEAU, G. Theory of didactical situations in mathematics. New York : Springer, 1997. Search in Google Scholar
BRUN, J., CONNE, F. Analyses didactiques de protocoles d’observation du déroulement de situations. Education et Recherches (3), 1990, 261–285. Search in Google Scholar
BRUNER, J. The Process of Education. Cambridge : Harvard University Press, 1960. Search in Google Scholar
BRUNER, J. Making Sense. The Child’s Construction of the World. London : Methuen, 1987. Search in Google Scholar
CARON, F. Splendeurs et miseres de l’enseignement des probabilités au primaire. Paper presented at the Colloque du Groupe des didacticiens des mathématiques du Québec (GDM) 2002: Continuités et ruptures entre les mathématiques enseignées au primaire et au secondaire, Université du Québec a Trois-Rivieres, 2004. Search in Google Scholar
DEBLOIS, L. 4 dizaines et 10 unités font 410, pourquoi? Trois-Rivi`eres: Editions Bande Didactique, 2001a. Search in Google Scholar
DEBLOIS, L. Un mod`ele d’interprétation des activités cognitives pour des él`eves qui éprouvent des difficultés d’apprentissage en mathématiques. Actes du colloque “Constructivismes: usages et perspectives en éducation”, vol. 2 Cédérom), 2001b, 565–573. Search in Google Scholar
DEBLOIS, L. Interpréter explicitement les productions des él`eves: une piste. . . Retrieved Éducation et francophonie, Volume XXXI (2), Document consulté le 16 juillet 2005, 2003. From http://www.acelf.ca/c/revue/revuehtml/31-2/08-deblois.html Search in Google Scholar
DEBLOIS, L., Maheux, J.-F. When Things Don’t Go Exactly as Planned: Leveraging from Student Teachers’ Insights to Adapted Interventions and Professional Practice. Paper presented at the The 15 th International Commission of Mathematical Instruction, Aguas De Lindoia, Brésil, 2005. Search in Google Scholar
DUVAL, R. Structure du raisonnement déductif et apprentissage de la démonstration. Educational Studies in Mathematics, 22, 1991, 233–261. Search in Google Scholar
ENGLISH, L. D. Combinatorics and the Development of Children’s Combinatorial Reasoning. In G. A. Jones (Ed.), Exploring Probability in School: Challenges for Teaching and Learning (Vol. Mathematics Education Library, 2005, p. 121–141). Search in Google Scholar
New York : Springer. Search in Google Scholar
FISCHBEIN, E., Schnarch, D. The Evolution with Age of Probabilistic, Intuitively Based Misconceptions. Journal of Research in Mathematics Education, 28(1), 1997, 96–105. Search in Google Scholar
GUILBERT, L. La pensée critique en science: présentation d’un mod`ele iconique en vue d’une définition opérationnelle. The journal of educational thought, 24(3), 1990, 195–218. Search in Google Scholar
HAWKINS, A. S., KAPADIA, R. Children’s Conceptions of Probability — A Psychological and Pedagogical Review. Educational Studies in Mathematics, 15, 1984, 349–377. Search in Google Scholar
JONES, G. A., THORNTON, C. A., LANGRALL, C. W., TARR, J. E. Understanding Students’ Probabilistic Reasoning Developing Mathematical Reasoning in Grades K-12 (Vol. 1999 Yearbook). Reston, Virginia : National Council of Teachers of Mathematics, 1999. Search in Google Scholar
KONOLD, C. Understanding Student’s Beliefs about Probability. In E. V. Glasersfeld (Ed.), Radical Constructivism in Mathematics Education, p. 139–156, Dordrecht : Kluwer academic publishers, 1991. Search in Google Scholar
LIPMAN, M. Thinking in Education (Second Edition). New York : Cambridge University Press, 2003. Search in Google Scholar
MAHER, C., POWELL, A., UPTEGROVE, E. B. Combinatorics and Reasoning: Representing, Justifying and Building Isomorphisms. New York : Springer, 2011. Search in Google Scholar
MAHER, C.,SRAN, M. K., YANKELEWITZ, D. Towers: Schemes, Stategies, and Arguments. In C. Maher, A. Powell, E. B. Uptegrove (Eds.), Combinatorics and Reasoning: Representing, Justifying and Building Isomorphisms. (Vol. Mathematics Education Library, p. 27–43. New York : Springer, 2011. Search in Google Scholar
MAHER, C., YANKELEWITZ, D. Representations as Tools for Building Arguments. In C. Maher, A. Powell, E. B. Uptegrove (Eds.), Combinatorics and Reasoning: Representing, Justifying and Building Isomorphisms. (Vol. Mathematics Education Library, p. 17–25). New York: Springer, 2011. Search in Google Scholar
Minist`ere de l’Éducation du Québec. Programme de formation de l’école québécoise. Enseignement primaire : Gouvernement du Québec, 2001. Search in Google Scholar
PAUL, R., ELDER, L. Critical Thinking. Tools for Taking Charge of you Learning and your Life. Upper Saddle River, NJ : Prentice Hall, 2001. Search in Google Scholar
PIAGET, J. La prise de conscience. Paris : Presses universitaires de France, 1974. Search in Google Scholar
PIAGET, J., INHELDER, B. La gen`ese de l’idée de hasard chez l’enfant, Deuxi`eme édition. Paris : Presses universitaires de France, 1974. Search in Google Scholar
SAVARD, A. Simulating the risk without gambling: Can student conceptions generate critical thinking about probability? Paper presented at the International Conference on Teaching Statistic (ICOTS 8), Ljubljana, Slovenia, 2010, July 5–9. Search in Google Scholar
SCHOLZ, R. W. Psychological Research in Probabilistic Understanding. In R. Kapadia, M. Borovcnik (Eds.), Chance Encount Probability in Education. p. 213–256. Dordrecht : Kluwer academic publishers, 1991. Search in Google Scholar
VERGNAUD, G. La théorie des champs conceptuels. Recherches en didactique des mathématiques, 10(2.3), 1990, 199–170. Search in Google Scholar

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[1] BAILIN, S., CASE, R., COOMBS, J. R., DANIELS, L. B. Conceptualizing Critical Thinking. Journal of Curriculum Studies, 31(3), 1999, 285–302. Search in Google Scholar

[2] BRIAND, J. Une expérience statistique et une premi`ere approche des lois du hasard au lycée par une confrontation avec une machine simple. Recherches en didactique des mathématiques, 25(2), 2005, 247–281. Search in Google Scholar

[3] BROUSSEAU, G. Theory of didactical situations in mathematics. New York : Springer, 1997. Search in Google Scholar

[4] BRUN, J., CONNE, F. Analyses didactiques de protocoles d’observation du déroulement de situations. Education et Recherches (3), 1990, 261–285. Search in Google Scholar

[5] BRUNER, J. The Process of Education. Cambridge : Harvard University Press, 1960. Search in Google Scholar

[6] BRUNER, J. Making Sense. The Child’s Construction of the World. London : Methuen, 1987. Search in Google Scholar

[7] CARON, F. Splendeurs et miseres de l’enseignement des probabilités au primaire. Paper presented at the Colloque du Groupe des didacticiens des mathématiques du Québec (GDM) 2002: Continuités et ruptures entre les mathématiques enseignées au primaire et au secondaire, Université du Québec a Trois-Rivieres, 2004. Search in Google Scholar

[8] DEBLOIS, L. 4 dizaines et 10 unités font 410, pourquoi? Trois-Rivi`eres: Editions Bande Didactique, 2001a. Search in Google Scholar

[9] DEBLOIS, L. Un mod`ele d’interprétation des activités cognitives pour des él`eves qui éprouvent des difficultés d’apprentissage en mathématiques. Actes du colloque “Constructivismes: usages et perspectives en éducation”, vol. 2 Cédérom), 2001b, 565–573. Search in Google Scholar

[10] DEBLOIS, L. Interpréter explicitement les productions des él`eves: une piste. . . Retrieved Éducation et francophonie, Volume XXXI (2), Document consulté le 16 juillet 2005, 2003. From http://www.acelf.ca/c/revue/revuehtml/31-2/08-deblois.html Search in Google Scholar

[11] DEBLOIS, L., Maheux, J.-F. When Things Don’t Go Exactly as Planned: Leveraging from Student Teachers’ Insights to Adapted Interventions and Professional Practice. Paper presented at the The 15 th International Commission of Mathematical Instruction, Aguas De Lindoia, Brésil, 2005. Search in Google Scholar

[12] DUVAL, R. Structure du raisonnement déductif et apprentissage de la démonstration. Educational Studies in Mathematics, 22, 1991, 233–261. Search in Google Scholar

[13] ENGLISH, L. D. Combinatorics and the Development of Children’s Combinatorial Reasoning. In G. A. Jones (Ed.), Exploring Probability in School: Challenges for Teaching and Learning (Vol. Mathematics Education Library, 2005, p. 121–141). Search in Google Scholar

[14] New York : Springer. Search in Google Scholar

[15] FISCHBEIN, E., Schnarch, D. The Evolution with Age of Probabilistic, Intuitively Based Misconceptions. Journal of Research in Mathematics Education, 28(1), 1997, 96–105. Search in Google Scholar

[16] GUILBERT, L. La pensée critique en science: présentation d’un mod`ele iconique en vue d’une définition opérationnelle. The journal of educational thought, 24(3), 1990, 195–218. Search in Google Scholar

[17] HAWKINS, A. S., KAPADIA, R. Children’s Conceptions of Probability — A Psychological and Pedagogical Review. Educational Studies in Mathematics, 15, 1984, 349–377. Search in Google Scholar

[18] JONES, G. A., THORNTON, C. A., LANGRALL, C. W., TARR, J. E. Understanding Students’ Probabilistic Reasoning Developing Mathematical Reasoning in Grades K-12 (Vol. 1999 Yearbook). Reston, Virginia : National Council of Teachers of Mathematics, 1999. Search in Google Scholar

[19] KONOLD, C. Understanding Student’s Beliefs about Probability. In E. V. Glasersfeld (Ed.), Radical Constructivism in Mathematics Education, p. 139–156, Dordrecht : Kluwer academic publishers, 1991. Search in Google Scholar

[20] LIPMAN, M. Thinking in Education (Second Edition). New York : Cambridge University Press, 2003. Search in Google Scholar

[21] MAHER, C., POWELL, A., UPTEGROVE, E. B. Combinatorics and Reasoning: Representing, Justifying and Building Isomorphisms. New York : Springer, 2011. Search in Google Scholar

[22] MAHER, C.,SRAN, M. K., YANKELEWITZ, D. Towers: Schemes, Stategies, and Arguments. In C. Maher, A. Powell, E. B. Uptegrove (Eds.), Combinatorics and Reasoning: Representing, Justifying and Building Isomorphisms. (Vol. Mathematics Education Library, p. 27–43. New York : Springer, 2011. Search in Google Scholar

[23] MAHER, C., YANKELEWITZ, D. Representations as Tools for Building Arguments. In C. Maher, A. Powell, E. B. Uptegrove (Eds.), Combinatorics and Reasoning: Representing, Justifying and Building Isomorphisms. (Vol. Mathematics Education Library, p. 17–25). New York: Springer, 2011. Search in Google Scholar

[24] Minist`ere de l’Éducation du Québec. Programme de formation de l’école québécoise. Enseignement primaire : Gouvernement du Québec, 2001. Search in Google Scholar

[25] PAUL, R., ELDER, L. Critical Thinking. Tools for Taking Charge of you Learning and your Life. Upper Saddle River, NJ : Prentice Hall, 2001. Search in Google Scholar

[26] PIAGET, J. La prise de conscience. Paris : Presses universitaires de France, 1974. Search in Google Scholar

[27] PIAGET, J., INHELDER, B. La gen`ese de l’idée de hasard chez l’enfant, Deuxi`eme édition. Paris : Presses universitaires de France, 1974. Search in Google Scholar

[28] SAVARD, A. Simulating the risk without gambling: Can student conceptions generate critical thinking about probability? Paper presented at the International Conference on Teaching Statistic (ICOTS 8), Ljubljana, Slovenia, 2010, July 5–9. Search in Google Scholar

[29] SCHOLZ, R. W. Psychological Research in Probabilistic Understanding. In R. Kapadia, M. Borovcnik (Eds.), Chance Encount Probability in Education. p. 213–256. Dordrecht : Kluwer academic publishers, 1991. Search in Google Scholar

[30] VERGNAUD, G. La théorie des champs conceptuels. Recherches en didactique des mathématiques, 10(2.3), 1990, 199–170. Search in Google Scholar

Scientia in educatione

Enumerating All Possible Outcomes: An Analysis of Students’ Work

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References

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