Developing a Modelling Approach to Probability Using Computer-Based Simulations
The introduction of digital technology into secondary schools is ideally suited for supporting students as they manipulate and portray data in a range of different representations to draw inferences from it without relying on a classical understanding of probability theory. As a result, probability is overlooked from school curricula and is gradually becoming almost a non-existent topic. The aim of recent curricula (e.g. ACARA 2010) to support the parallel development of statistics and probability and then progressively build the links between them seems utopic since statistics prevails over probability in mathematics curricula. In this chapter, it is argued that it is worthwhile to consider an alternative approach for teaching probability—presenting probability as a modelling tool, which reflects the mindset of an expert when using probability to model random behaviour in real-world contexts.
How students articulated fundamental probabilistic concepts associated with the construction of univariate probability models when using probability to model random behaviour.
Students’ discussion as they engaged in exploring recently developed computer-based simulations which treat probability as a modelling tool.
In this chapter, references will be limited to students in Grades 6 to 9. Prodromou’s research studies address four research questions as follows: (1) How do middle school students use probability to model random behaviour in real-world contexts? (2) What connections do they build among fundamental probabilistic concepts when treating probability as a modelling tool? (3) How do they synthesize the modelling approach to probability with the use of distributions while concurrently making inferences about data? (4) What activities can be designed to support the proposed alternative approach for teaching probability?
The results of this study provide answers to the aforementioned research questions and suggest that the way students express the relationship between signal and noise is of importance while building models from the observation of a real situation. This relationship seems to have a particular importance in students’ abilities to build comprehensive models that link observed data with modelling distributions.
KeywordsModelling Probability Middle school Computer-based simulations Random behaviour Signal Noise Data
- Australian Curriculum, Assessment and Reporting Authority (2010). Australian Curriculum: Mathematics. Version 1.1. Online: http://www.acara.edu.au.
- Borovcnik, M. (2011). Strengthening the role of probability within statistics curricula. In C. Batanero, G. Burrill, & C. Reading (Eds.), Teaching statistics in school mathematics—challenges for teaching and teacher education, a joint ICMI/IASE study: the 18th ICMI study (pp. 71–83). New York: Springer. CrossRefGoogle Scholar
- Chaput, M., 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, a joint ICMI/IASE study: the 18th ICMI study (pp. 85–95). New York: Springer. CrossRefGoogle Scholar
- De Moivre, A. (1967/1756). The doctrine of chances: or a method of calculating the probabilities of events in play. New York: Chelsea (original work published in 1756, London: A. Millar). Google Scholar
- Henry, M. (2001). Autour de la modélisation en probabilités [Around modelling in probability]. Besancon: Presses Universitaires Franc-Comtoises. Google Scholar
- Hoerl, R. W., & Snee, R. D. (2001). Statistical thinking: improving business performance. Pacific Grove: Duxbury. Google Scholar
- Laplace, P. S. (1995/1814). Théorie analytique des probabilités [Analytical theory of probabilities]. Paris: Jacques Gabay (original work published 1814). Google Scholar
- Makar, K., & Rubin, A. (2009). A framework for thinking about informal statistical inference. Statistics Education Research Journal, 8(1), 82–105. Online: http://www.stat.auckland.ac.nz/~iase/serj/SERJ8(1)_Makar_Rubin.htm. Google Scholar
- von Mises, R. (1952/1928). Probability, statistics and truth (J. Neyman, O. Scholl, & E. Rabinovitch, Trans.). London: William Hodge and Company (original work published 1928). Google Scholar
- Moore, D. S. (1990). Uncertainty. In L. S. Steen (Ed.), On the shoulders of giants: new approaches to numeracy (pp. 95–137). Washington: National Academic Press. Google Scholar
- Pratt, D. (2011). Re-connecting probability and reasoning about data in secondary school teaching. Paper presented at 58th ISI World Statistics Congress, Dublin, Ireland. Online: http://isi2011.congressplanner.eu/pdfs/450478.pdf.
- Prodromou, T. (2007). Making connections between the two perspectives on distribution. In D. Pitta-Pantazi & G. Philippou (Eds.), Proceedings of the fifth congress of the European Society for Research in Mathematics Education, Larnaca, Cyprus (pp. 801–810). Online: http://ermeweb.free.fr/CERME5b. Google Scholar
- Prodromou, T. (2008). Connecting thinking about distribution. Unpublished doctoral dissertation, University of Warwick, Warwick, UK. Google Scholar
- Prodromou, T., & Pratt, D. (2006). The role of causality in the co-ordination of the two perspectives on distribution within a virtual simulation. Statistics Education Research Journal, 5(2), 69–88. Online: http://www.stat.auckland.ac.nz/~iase/serj/SERJ5%282%29_Prod_Pratt.pdf. Google Scholar
- Prodromou, T., & Pratt, D. (2009). Student’s causal explanations for distribution. In V. Durand-Guerrier, S. Soury-Lavergne, & F. Arzarello (Eds.), Proceedings of the sixth conference of the European Society for Research in Mathematics Education, Lyon, France (pp. 394–403). Google Scholar
- Renyi, A. (1992/1966). Calcul des probabilitiés [Probability calculus] (L. Félix, Trans.). Paris: Jacques Gabay (original work published 1966). Google Scholar
- Robson, C. (1993). Real world research. Oxford: Blackwell. Google Scholar
- Shaughnessy, J. M., & Ciancetta, M. (2001). Conflict between students’ personal theories and actual data: the spectre of variation. In Second round table conference on research on statistics teaching and learning, Armidale, New South Wales, Australia. Google Scholar
- Smith, T. M. F. (1999). Discussion. International Statistical Review, 67(3), 248–250. Google Scholar
- Sorto, M. A. (2006). Identifying content knowledge for teaching statistics. In A. Rossman & B. Chance (Eds.), Working cooperatively in statistics education. Proceedings of the seventh international conference on teaching statistics, Salvador, Brazil. Voorburg: International Statistical Institute [CDROM]. Online: http://www.stat.auckland.ac.nz/~iase/publications/17/C130.pdf. Google Scholar
- Tukey, J. W. (1977). Exploratory data analysis. Reading: Addison-Wesley. Google Scholar
- Wild, C., & Pfannkuch, M. (1999). Statistical thinking in empirical enquiry. International Statistical Review, 67, 223–266. Google Scholar