Rethinking mathematics teachers’ professional knowledge for teaching probability from the perspective of probabilistic reasoning: A proposed framework
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1 Curriculum and Instruction Department, Faculty of Education, Tanta University, Tanta, EGYPT* Corresponding Author
Abstract
Probability signifies a mainstream strand in mathematics curricula. Nonetheless, many curricular documents prepared for teachers might not offer enough support. In such a situation, a further reflection on teachers’ professional knowledge for teaching probability is demanded; especially, from the perspective of probabilistic reasoning (PoPR) that is consistent with the need to pave the way for theories about mathematics education and cognitive psychology to consolidate achievements from each other. Accordingly, this study aims at conceptualizing mathematics teachers’ professional knowledge for teaching probability from the PoPR. The initial step towards this conceptualization started by inferring the fundamental entities of teachers’ professionalism through utilizing the mathematical knowledge for teaching model. Following this, three significant propositions were acknowledged. As a result, a conceptual framework was proposed, and a practical example was described. Such a description symbolizes a transition from emphasizing content knowledge towards highlighting teachers’ process knowledge, which may impact the development of probability education research.
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This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Article Type: Research Article
INT ELECT J MATH ED, Volume 17, Issue 3, August 2022, Article No: em0695
https://doi.org/10.29333/iejme/12145
Publication date: 05 Jun 2022
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Article Downloads: 1131
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Declaration of Conflict of Interest: No conflict of interest is declared by author(s).
Data sharing statement: Data supporting the findings and conclusions are available upon request from the corresponding author(s).
References
- Amir, G. S., & Williams, J. S. (1999). Cultural influences on children’s probabilistic thinking. Journal of Mathematical Behavior, 18(1), 85-107. https://doi.org/10.1016/S0732-3123(99)00018-8
- Ball, D. L., Lubienski, S. T., & Mewborn, D. S. (2001). Research on teaching mathematics: The unsolved problem of teachers’ mathematical knowledge. In V. Richardson (Ed.), Handbook of research on teaching (pp. 433-456). American Educational Research Association.
- Ball, D. L., Thames, M. H., & Phelps, G. (2008). Content knowledge for teaching. What makes it special? Journal of Teacher Education, 59(5), 389-407. https://doi.org/10.1177/0022487108324554
- Batanero, C. (2015). Understanding randomness: Challenges for research and teaching. In K. Krainer, & N. Vondrová (Eds.), Proceedings of CERME9 (pp. 34-49). European Society for Research in Mathematics Education.
- Batanero, C., & Sánchez, E. (2005). What is the nature of high school students’ conceptions and misconceptions about probability? In G. Jones (Eds.), Exploring probability in school: Challenges for teaching and learning (pp. 241-266). Springer. https://doi.org/10.1007/0-387-24530-8_11
- Batanero, C., Biehler, R., Maxara, C., Engel, J., & Vogel, M. (2005a). Using simulation to bridge teachers’ content and pedagogical knowledge in probability [Paper presentation]. The 15th ICMI Study Conference: The Professional Education and Development of Teachers of Mathematics. Aguas de Lindoia, Brazil.
- Batanero, C., Henry, M., & Parzysz, B. (2005b). The nature of chance and probability. In G. A. Jones (Ed.), Exploring probability in school: Challenges for teaching and learning (pp. 15-37). Springer. https://doi.org/10.1007/0-387-24530-8_2
- Batanero, C., Chernoff E. J., Engel, J., Lee, H. S., & Sánchez, E. (2016). Research on teaching and learning probability. Springer. https://doi.org/10.1007/978-3-319-31625-3_1
- Batanero, C., Contreras, J. M., Fernandes, J. A., & Ojeda, M. M. (2010). Paradoxical games as a didactic tool to train teachers in probability. In C. Reading (Ed.), Data and context in statistics education: Towards an evidence-based society: Proceedings of the 8th International Conference on Teaching Statistics. International Statistical Institute.
- Batanero, C., Godino, J. D., & Roa, R. (2004). Training teachers to teach probability. Journal of Statistics Education, 12(1),1-15. https://doi.org/10.1080/10691898.2004.11910715
- Baumert, J., & Kunter, M. (2013). The COACTIV model of teachers’ professional competence. In M. Kunter, J. Baumert, W. Blum, U. Klusmann, S. Krauss, & M. Neubrand (Eds.), Cognitive activation in the mathematics classroom and professional competence of teachers (pp. 25-48). Springer. https://doi.org/10.1007/978-1-4614-5149-5_2
- Birel, G. K. (2017). The investigation of pre-service elementary mathematics teachers’ subject matter knowledge about probability. Mersin University Journal of the Faculty of Education, 13(1), 348-362. https://doi.org/10.17860/mersinefd.306023
- Borovcnik, M., & Peard, R. (1996). Probability. In A. J. Bishop, K. Clements, C. Keitel, J. Kilpatrick, & C. Laborde (Eds.), International handbook of mathematics education (pp. 239-288). Springer. https://doi.org/10.1007/978-94-009-1465-0_9
- Brase, G. L., Martinie, S., & Castillo-Garsow, C. (2014). Intuitive conceptions of probability and the development of basic math skills. In E. Chernoff, & B. Sriraman (Eds.) Probabilistic thinking. Advances in mathematics education. Springer. https://doi.org/10.1007/978-94-007-7155-0_10
- Brijlall, D. (2014). Exploring the pedagogical content knowledge for teaching probability in middle school: A South African case sudy. International Journal of Educational Science, 7(3), 719-726. https://doi.org/10.1080/09751122.2014.11890234
- Callingham, R., & Watson, J. (2011). Measuring levels of statistical pedagogical content knowledge. In C. Batanero, G. Burrill, & C. Reading (Eds.), Teaching statistics in school mathematics. Challenges for teaching and teacher education (pp. 283-293). Springer. https://doi.org/10.1007/978-94-007-1131-0_28
- Carranza, P., & Kuzniak, A. (2008). Duality of probability and statistics teaching in French education. In C. Batanero, G. Burrill, C. Reading, & A. Rossman (Eds.), Teaching statistics in school mathematics. Challenges for teaching and teacher education. Springer.
- CCSSI. (2010). Common core state standards for mathematics. National Governors Association for Best Practices and the Council of Chief State School Officers. https://files.eric.ed.gov/fulltext/EJ1105174.pdf
- Chassapis, D., & Chatzivasileiou, E. (2008). Socio-cultural influences on children's conceptions of chance and probability. In J. F. Matos, P. Valero, & K. Yasukawa (Eds.), Proceedings of the Fifth International Mathematics Education and Society Conference (pp. 197-206). Centro de Investigação em Educação, Universidade de Lisboa, Department of Education, Learning and Philosophy, Aalborg University.
- Chaput, B., Girard, J. C., & Henry, M. (2011). Frequentist approach: Modelling and simulation in statistics and probability teaching. In C. Batanero, G. Burrill, & C. Reading (Eds.), Teaching statistics in school mathematics. Challenges for teaching and teacher education. Springer. https://doi.org/10.1007/978-94-007-1131-0_12
- Chernoff, E. J., & Sriraman, B. (2014). Probabilistic thinking: Presenting plural perspectives. Springer. https://doi.org/10.1007/978-94-007-7155-0
- Chick, H. L., & Baker, M. (2005). Teaching elementary probability: Not leaving it to chance. In P.C. Clarkson, A. Downton, D. Gronn, M. Horne, A. McDonough, R. Pierce, & A. Roche (Eds.), Building connections: Theory, research and practice (pp. 233-240). MERGA.
- Chiesi, F., & Primi, C. (2014). The interplay among knowledge, cognitive abilities and thinking styles in probabilistic reasoning: A test of a model. In E. Chernoff, & B. Sriraman (Eds.), Probabilistic thinking. Advances in mathematics education (pp. 195-214). Springer. https://doi.org/10.1007/978-94-007-7155-0_11
- Contreras, J. M., Batanero, C., Díaz, C., & Fernandes, J. A. (2011). Prospective teachers’ common and specialized knowledge in a probability task. In Proceedings of the 7th Congress of the European Society for Research in Mathematics Education (pp. 766-775). University of Rzeszów & ERME.
- Danisman, S., & Tanisli, T. (2017). Examination of mathematics teachers’ pedagogical content knowledge of probability. Malaysian Online Journal of Educational Sciences, 5(2), 16-34.
- Dewey, J. (1964). The relation of theory to practice in education. In R. Archambault (Ed.). John Dewey on education. University of Chicago Press (Original work published in 1904).
- Díaz, C., & Batanero, C. (2009). University students’ knowledge and biases in conditional probability reasoning. International Electronic Journal of Mathematics Education, 4(3), 131-162. https://doi.org/10.29333/iejme/234
- 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. https://doi.org/10.29333/iejme/180
- Dollard, C. (2011). Preservice elementary teachers and the fundamentals of probability. Statistics Education Research Journal, 10(2), 27-47. https://doi.org/10.52041/serj.v10i2.346
- Dooley, T., & Gueudet, G. (Eds.). (2017). CERME10: Proceedings of the 10th Congress of the European Society for Research in Mathematics Education. DCU Institute of Education & ERME.
- Elbehary, S. G. A. (2021). Reasoning under uncertainty within the context of probability education: A case study of preservice mathematics teachers. Pythagoras, 42(1), a630. https://doi.org/10.4102/pythagoras.v42i1.630
- Even, R., & Kvatinsky, T. (2010). What mathematics do teachers with contrasting teaching approaches address in probability lessons? Educational Studies in Mathematics, 74, 207-222. https://doi.org/10.1007/s10649-010-9234-9
- Falk, R., & Konold, C. (1992). The psychology of learning probability. In F. S. Gordon, & S. P. Gordon (Eds.), Statistics for the twenty-first century (pp. 151-164). Mathematical Association of America.
- Fischbein, E. (1987). Intuition in science and mathematics. Reidel.
- Fischbein, E., & Gazit, A. (1984). Does the teaching of probability improve probabilistic intuitions? Educational Studies in Mathematics, 15, 1-24. https://doi.org/10.1007/BF00380436
- Fischbein, E., & Schnarch, D. (1997). The evolution with age of probabilistic, intuitively based misconceptions. Journal of Research in Science Teaching, 28(1), 96-105. https://doi.org/10.5951/jresematheduc.28.1.0096
- Franklin, C., Kader, G., Mewborn, D., Moreno, J., Peck, R., Perry, M., & Scheaffer, R. (2007). Guidelines for assessment and instruction in statistics education (GAISE) report: A pre-K-12 curriculum framework. American Statistical Association.
- Gal, I. (2005). Towards probability literacy for all citizens: Building blocks and instructional dilemas. In G. Jones (Ed.), Exploring probability in schools: Challenges for teaching and learning (pp. 39-63). Springer. https://doi.org/10.1007/0-387-24530-8_3
- Garfield, J., & Ben-Zvi, D. (2008). Developing students’ statistical reasoning: Connecting research and teaching practice. Springer.
- Gillard, E., Dooren, V. W., Schaeken, W., & Verschaffcl, L. (2009). Dual processes in psychology of mathematics education and cognitive psychology. Human Development, 52(2), 95-108. https://doi.org/10.1159/000202728
- Giordan, A. (1998). Apprendre. Ed. Belin.
- Giordan, A., & Pellaud, F. (2004). La place des conceptions dans la médiation de la chimie [The place of conceptions in the mediation of chemistry]. Numéro Spécial Médiation de la Chimie, L’actualité Chimique [Chemical Mediation Special Issue, Chemical News], Nov-Dec, 49-52.
- Godino, J. D., Batanero, C., & Cañizares, M. J. (1987). Azar y probabilidad. Fundamentos didácticos y propuestas curriculares [Chance and probability. Didactic foundations and curricular proposals]. Síntesis.
- Godino, J. D., Batanero, C., Roa, R., & Wilhelmi, M. R. (2008). Assessing and developing pedagogical content and statistical knowledge of primary school teachers through project work. In C. Batanero, G. Burrill, C. Reading, & A. Rossman (Eds.), Teaching statistics in school mathematics. Challenges for teaching and teacher education. Springer.
- Gras, R., & Totohasina, A. (1995). Chronologie et causalité, conceptions sources d’obstacles épistémologiques à la notion de probabilité conditionnelle [Chronology and causality, source conceptions of epistemological obstacles to the notion of conditional probability]. Recherche en Didactique des Mathématiques [Research in Didactics of Mathematics], 15(1), 49-55.
- Green, D. (1993). Ramdomness-a key concept. International Journal of Mathematical Education in Science and Technology, 24(6), 897-905. https://doi.org/10.1080/0020739930240615
- Grenon, V., Larose, F., Bourque, J., & Bédard, J. (2010). The impact of using pupils’ daily social practices as well as computerized simulators as a teaching medium on motivation and knowledge construction regarding probabilities among high school pupils. In C. Reading (Ed.), Data and context in statistics education: Towards an evidence-based society. International Statistical Institute.
- Gusmão, T., Santana, E., Cazorla, I., & Cajaraville, J. (2010). A semiotic analysis of “mônica’s random walk”: Activity to teach basic concepts of probability. In C. Reading (Ed.), Data and context in statistics education: Towards an evidence-based society. International Statistical Institute.
- Hacking, I. (1975). The emergence of probability. Cambridge University Press.
- Heitele, D. (1975). An epistemological view on fundamental stochastic ideas. Educational Studies in Mathematics, 6(2), 187-205. https://doi.org/10.1007/BF00302543
- Hokor, E. K. (2020). Pre-service teachers’ probabilistic reasoning in constructivist classroom. Pedagogical Research, 5(2), em0053. https://doi.org/10.29333/pr/7838
- Hurrell, D. P. (2013). What teachers need to know to teach mathematics: An argument for a reconceptualized model. Australian Journal of Teacher Education, 38(11), 54-64. https://doi.org/10.14221/ajte.2013v38n11.3
- Jones, G. A., Langrall, C. W., & Mooney, E. S. (2007). Research in probability: Responding to classroom realities. In F. K. Lester (Ed.), The second handbook of research on mathematics teaching and learning: A project of the National Council of Teachers of Mathematics (pp. 909-955). Age Publishing.
- Kaiser, G., Blömeke, S., König, J., Busse, A., Döhrmann, M., & Hoth, J. (2017). Professional competencies of (prospective) mathematics teachers-Cognitive versus situated approaches. Educational Studies in Mathematics, 94(2), 161-182. https://doi.org/10.1007/s10649-016-9724-5
- Kapadia, R., & Borovcnik, M. (2010). Reviewing and promoting research in probability education electronically. In C. Reading (Ed.), Data and context in statistics education: Towards an evidence-based society. International Statistical Institute.
- Kataoka, V. Y., Souza, A. A., Oliveira, A. C. S., Fernandes, F. M. O., Paranaíba, P. F., & Oliveira, M. S. (2008). Probability teaching in Brazilian basic education: Evaluation and intervention. In Proceedings of the 11th International Congress on Mathematical Education.
- Kataoka, V. Y., Trevethan, H. M. H., & Silva, C. B. (2010). Independence of events: An analysis of knowledge level in different groups of students. In C. Reading (Ed.), Data and context in statistics education: Towards an evidence-based society. International Statistical Institute.
- Kazak, S., & Pratt, D. (2017). Pre-service mathematics teachers’ use of probability models in making informal inferences about a chance game. Statistics Education Research Journal, 16(2), 287-304. https://doi.org/10.52041/serj.v16i2.193
- Kazima, M. (2007). Malawian students meaning for probability vocabulary. Educational Studies in Mathematics, 64(2), 169-189. https://doi.org/10.1007/s10649-006-9032-6
- Kleickmann, T., Richter, D., Kunter, M., Elsner, J., Besser, M., Krauss, S., & Baumert, J. (2013). Teachers’ content knowledge and pedagogical content knowledge: The role of structural differences in teacher education. Journal of Teacher Education, 64(1), 90-106. https://doi.org/10.1177/0022487112460398
- Konold, C. (1989). Informal conceptions of probability. Cognition and Instruction, 6(1), 59-98. https://doi.org/10.1207/s1532690xci0601_3
- Konold, C. (1991). Understanding student’s beliefs about probability. In E. V. Glasersfeld (Eds.), Radical constructivism in mathematics education (pp. 139-156). Kluwer Academic Publishers. https://doi.org/10.1007/0-306-47201-5_7
- Konold, C., Pollatsek, A., Well, A., Lohmeier, J., & Lipson, A. (1993). Inconsistencies in students’ reasoning about probability. Journal for Research in Mathematics Education, 24(5), 392-414. https://doi.org/10.5951/jresematheduc.24.5.0392
- Krainer, K., & Llinares, S. (2010). Mathematics teacher education. In P. Peterson, E. Baker, & B. McGaw (Eds.), International encyclopedia of education (pp. 702-705). Elsevier. https://doi.org/10.1016/B978-0-08-044894-7.00680-1
- Kunter, M., T. Kleickmann, U. Klusmann, & Richter, D. (2013). The development of tachers’ professional competence. In M. Kunter, J. Baumert, W. Blum, U. Klusmann, S. Krauss, & M. Neubrand (Eds.), Cognitive activation in the mathematics classroom and professional competence of teachers (pp. 63-77). Springer. https://doi.org/10.1007/978-1-4614-5149-5_4
- Kvatinsky, T., & Even, R. (2002). Framework for teacher knowledge and understanding about probability. In B. Phillips (Ed.), Proceedings of the 6th International Conference on Teaching Statistics. International Statistical Institute.
- Lecoutre, M. P. (1992). Cognitive models and problem spaces in “purely random” situations. Educational Studies in Mathematics, 23, 557-568. https://doi.org/10.1007/BF00540060
- Li, J., & Wisenbaker, J. M. (2008). Research and developments in the teaching and learning of probability and statistics. In M. Niss, & E. Emborg (Eds.), Proceedings of the 10th International Congress on Mathematical Education. Roskilde University.
- Liberman, V., & Tversky, A. (1996). Critical thinking (in Hebrew). The Open University of Israel.
- Lindley, D. (1994). Foundations. In G. Wright, & P. Ayton (Eds.), Subjective probability (pp. 3-15). John Wiley & Sons.
- Lopez, V., & Whitehead, D. (2013). Sampling data and data collection in qualitative research. In Z. Schneider, D. Whitehead, G. LoBiondo-Wood, & J. Habe (Eds.), Nursing and midwifery research: Methods and critical appraisal for evidence-based practice (pp. 124-140). Elsevier.
- Lysoe, K. O. (2008). Strengths and limitations of informal conceptions in introductory probability courses for future lower secondary teachers. In Proceedings of the 11th International Congress on Mathematical Education. Monterrey, México.
- Marks, R. (1990). Pedagogical content knowledge: From a mathematical case to a modified conception. Journal of Teacher Education, 41(3), 3-11. https://doi.org/10.1177/002248719004100302
- Martignon, L. (2014). Fostering children’s probabilistic reasoning and first elements of risk evaluation. In E. J. Chernoff, & B. Sriraman (Eds.), Probabilistic thinking, presenting plural perspectives (pp. 149-160). Springer. https://doi.org/10.1007/978-94-007-7155-0_9
- Nacarato, A. M., & Grando, R. C. (2014). The role of language in building probabilistic thinking. Statistics Education Research Journal, 13(2), 93-103. https://doi.org/10.52041/serj.v13i2.283
- NCTM. (2000). Principles and standards for school mathematics. National Council of Teachers of Mathematics. https://www.nctm.org/standards/
- Nisbett, R. E., Krantz, D. H., Jepson, C., & Kunda, Z. (1983). The use of statistical heuristics in everyday inductive reasoning. Psychological Review, 90(4), 339-363. https://doi.org/10.1037/0033-295X.90.4.339
- Otani, H., Fukuda, H., Tagashira, K., & Iwasaki, H. (2018). Effects of statistical words on the way students view data. In M. A. Sorto, A. White, & L. Guyot (Eds.), Looking back, looking forward. International Statistical Institute.
- Papaieronymou, I. (2009). Recommended knowledge of probability for secondary mathematics teachers. In Proceedings of the 6th Congress of the European Society for Research in Mathematics Education. Lyon, France.
- Paul, M., & Hlanganipai, N. (2014). The nature of misconceptions and cognitive obstacles faced by secondary school mathematics students in understanding probability: A case study of selected Polokwane secondary schools. Mathematical Journal of Social Sciences, 5(8), 446-455.
- PCMI. (2017). The importance of teaching probability. Park City Mathematics Institute. https://projects.ias.edu/pcmi/hstp/sum2017/int/briefs/ImportanceofTeachingProbability.pdf
- Pfannkuch, M. (2011). The role of context in developing informal statistical inferential reasoning: A classroom study. Mathematical Thinking and Learning, 13(1-2), 27-46. https://doi.org/10.1080/10986065.2011.538302
- Pfannkuch, M., & Wild, C. (2004). Towards an understanding of statistical thinking. In D. Ben-Zvi, & J. Garfield (Eds.), The challenge of developing statistical literacy, reasoning, and thinking (pp. 17-46). Kluwer Academic Publishers. https://doi.org/10.1007/1-4020-2278-6_2
- Pratt, D. (2005). How do teachers foster students’ understanding of empirical and theoretical probability? In G. A. Jones (Ed.), Exploring probability in school: Challenges for teaching a learning (pp. 171-189). Kluwer Academic Publishers. https://doi.org/10.1007/0-387-24530-8_8
- Prodromou, T. (2012). Connecting experimental probability and theoretical probability. ZDM-The International Journal on Mathematics Education, 44(7), 855-868. https://doi.org/10.1007/s11858-012-0469-z
- Savard, A. (2010). Simulating the risk without gambling: Can student conceptions generate critical thinking about probability? [Paper presentation]. The International Conference on Teaching Statistic. Ljubljana, Slovenia.
- Savard, A. (2014). Developing probabilistic thinking: What about people’s conceptions? In E. J. Chernoff, & B. Sriraman (Eds.), Probabilistic thinking: Presenting plural perspectives (pp. 283-298). Springer. https://doi.org/10.1007/978-94-007-7155-0_15
- Sharma, S. (2016). Probability from a socio-cultural perspective. Statistics Education Research Journal, 15(2),126-144. https://doi.org/10.52041/serj.v15i2.244
- Shaughnessy, J. M. (1977). Misconceptions of probability: An experiment with a small group, activitybased, model building approach to introductory probability at the college level. Educational Studies in Mathematics, 8, 295-316. https://doi.org/10.1007/BF00385927
- Shaughnessy, J. M. (1992). Research in probability and statistics: Reflections and directions. In D. A. Grouws (Eds.), Handbook of research on mathematics teaching and learning (pp. 465-494). Macmillan.
- Shulman, L. (1986). Those who understand: Knowledge growth in teaching. Educational Researcher, 15(2), 4-14. https://doi.org/10.3102/0013189X015002004
- Shulman, L. (1987). Knowledge and teaching: Foundations of the new reform. Harvard Educational Review, 57(1), 1-22. https://doi.org/10.17763/haer.57.1.j463w79r56455411
- Skoumpourdi, C., & Kalavassis, F. (2003). Didactic materials used in probabilistic activities [Paper presentation]. The CIEAEM 55. Poland.
- Smith, J. P., diSessa, A. A., & Roschelle, J. (1993). Misconceptions reconceived: A constructivist analysis of knowledge in transition. The Journal of the Learning Sciences, 3(2), 115-163. https://doi.org/10.1207/s15327809jls0302_1
- Stohl, H. (2005). Probability in teacher education and development. In G. A. Jones (Ed.), Exploring probability in school (pp. 345- 366). Springer. https://doi.org/10.1007/0-387-24530-8_15
- Theis, L., & Savard, A. (2010). Linking probability to real-world situations: How do teachers make use of the mathematical potential of simulations programs? In C. Reading (Ed.), Data and context in statistics education: Towards an evidence-based society. International Statistical Institute.
- Thomas, D. (2006). A general inductive approach for analysing qualitative evaluation data. American Journal of Evaluation, 27(2), 237-246. https://doi.org/10.1177/1098214005283748
- Torres, G. E., & Contreras, J. M. (2014). Meanings of probability in Spanish curriculum for primary school. In K. Makar, De. B. Sousa, & R. Gould (Eds.), Proceedings of the 9th International Conference on Teaching Statistics. International Statistical Institute.
- Torres, G. E., Batanero, C., Diaz, C., & Contreras, J. M. (2016). Developing a questionnaire to assess the probability content knowledge of prospective primary school teachers. Statistics Education Research Journal, 15(2), 197-215. https://doi.org/10.52041/serj.v15i2.248
- Tsakiridou, H., & Vavyla, E. (2015). Probability concepts in primary school. American Journal of Educational Research, 3(4), 535-540. https://doi.org/10.12691/education-3-4-21
- Tversky, A., & Kahneman, D. (1974). Judgment under uncertainty: Heuristics and biases. Science, 185(4157), 1124-1131. https://doi.org/10.1126/science.185.4157.1124
- Van Dooren, W. (2014). Probabilistic thinking: Analyses from a psychological perspective. In E. Chernoff, & B. Sriraman (Eds.), Probabilistic thinking: Advances in mathematics education (pp. 123-126). Springer. https://doi.org/10.1007/978-94-007-7155-0_7
- Vosniadou, S., & Verschaffel, L. (2004). The conceptual change approach to mathematics learning and teaching. Learning and Instruction, 14(5), 445-548. https://doi.org/10.1016/j.learninstruc.2004.06.014
How to cite this article
APA
Elbehary, S. G. A. (2022). Rethinking mathematics teachers’ professional knowledge for teaching probability from the perspective of probabilistic reasoning: A proposed framework. International Electronic Journal of Mathematics Education, 17(3), em0695. https://doi.org/10.29333/iejme/12145
Vancouver
Elbehary SGA. Rethinking mathematics teachers’ professional knowledge for teaching probability from the perspective of probabilistic reasoning: A proposed framework. INT ELECT J MATH ED. 2022;17(3):em0695. https://doi.org/10.29333/iejme/12145
AMA
Elbehary SGA. Rethinking mathematics teachers’ professional knowledge for teaching probability from the perspective of probabilistic reasoning: A proposed framework. INT ELECT J MATH ED. 2022;17(3), em0695. https://doi.org/10.29333/iejme/12145
Chicago
Elbehary, Samah Gamal Ahmed. "Rethinking mathematics teachers’ professional knowledge for teaching probability from the perspective of probabilistic reasoning: A proposed framework". International Electronic Journal of Mathematics Education 2022 17 no. 3 (2022): em0695. https://doi.org/10.29333/iejme/12145
Harvard
Elbehary, S. G. A. (2022). Rethinking mathematics teachers’ professional knowledge for teaching probability from the perspective of probabilistic reasoning: A proposed framework. International Electronic Journal of Mathematics Education, 17(3), em0695. https://doi.org/10.29333/iejme/12145
MLA
Elbehary, Samah Gamal Ahmed "Rethinking mathematics teachers’ professional knowledge for teaching probability from the perspective of probabilistic reasoning: A proposed framework". International Electronic Journal of Mathematics Education, vol. 17, no. 3, 2022, em0695. https://doi.org/10.29333/iejme/12145