Abstract
This text proposes a probabilistic problem aimed at teaching probability through problem solving in the first 2 years of junior high school. This problem’s adaptability in terms of its level of complexity and the ease of adjusting it to students’ level of mathematical development makes it highly versatile.
As we will see, the problem is rich owing to the connection it draws between the theoretical and frequentist approaches via its consideration of the characteristics of probability and probabilistic thinking. Indeed, even if connecting these two approaches is desirable in the context of teaching probability, such a connection represents a significant challenge for teachers (Martin V, Theis L. Can J Sci Math Technol Educ 16(4):1–14, 2016). This likely explains why these two approaches are rarely addressed in their multiplicity or complementarity (Caron F. Splendeurs et misères de l’enseignement des probabilités au primaire. In: Proceedings of the « Colloque annuel du Groupe de didactique des mathématiques du Québec », Trois-Rivières, Québec, 2002; Nilsson P, Eckert A. Interactive experimentation in probability—opportunities, challenges and needs of research. In Proceedings of the thirteenth international congress on mathematical education, Hamburg, Germany, 2016).
Finally, the text will conclude with a discussion of the use of this problem for teaching probability by reviewing the teacher’s challenges in managing the problem, as well as its mathematical potential and the favourable context it represents for students deemed to be in difficulty in mathematics.
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Notes
- 1.
In this text, rather than speaking in terms of a “problem situation” and “resolving a problem situation,” we will use the expressions “problem” and “problem solving” in order to establish correspondence between the world of Francophone research in didactique des mathématiques, on one hand, and the world of Anglophone research on mathematics education, on the other. We are aware of the important semantic nuances between these terms, but given that this is not the topic of the text, we have chosen not to further address the issue here.
- 2.
Probabilistic conceptions, which appear as probabilistic reasoning, are associated with the notions of chance, probability and relationships to randomness. The presence of these conceptions can lead to difficulties with learning or teaching probability. However, rather than labelling them as misconceptions, following on Savard (2008), we choose to refer to them as probabilistic conceptions. This designation, which is neutral with respect to such reasoning, appears more accurate to us in that these conceptions are fairly common, and can develop, weaken, be modified or evolve with experience (Savard 2014).
- 3.
The problem we propose here was originally developed in the context of doctoral studies by Martin (2014). The author’s doctoral thesis afforded the possibility of studying the didactical methods used by two 5th–6th grade elementary teachers (students 11–12 years old) to teach probability.
- 4.
The original version of the roulette as used by Martin (2014) was a colour version with blue, red and yellow angular sectors; however, in the present text, these have been respectively replaced by grey, white and black angular sectors.
- 5.
Unlike other types of dice, with a four-sided die, the outcome is not given by the top face, but rather by the sum of the upper vertex of the four-sided die, with each side being numbered 1– 4. This type of die has 3 numbers on each face, i.e., one at each angle. When the die is cast on a flat surface, it is the upward facing vertex that indicates the outcome, with the same number showing on each bordering angle.
- 6.
It is worth noting that the problem, like the PFEQ education program for elementary and junior high school in Quebec, entirely sets aside the subjective approach and focuses exclusively on the two probabilistic approaches.
- 7.
The choice to use four-sided rather than six-sided dice is based on two main arguments. First, the smaller number of possible outcomes associated with the pair of four-sided dice leads to a faster stabilization of relative frequency for the results obtained through trials. Second, the task of obtaining a certain sum of outcomes with two six-sided dice has frequently been used in textbooks and instructional research, whereas conducting the same task with a pair of four-sided dice is much less frequent. This ensures that the odds of winning associated with the different sums of possible outcomes are not known from the outset.
- 8.
This law states that the higher the number of trials, the closer the probability (arising from the trend observed based on the frequency of outcomes) should come to the theoretical probability (Bernouilli,1713, in Borovcnik and Peard 1996).
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Additional Suggestions for Further Reading
Martin, V., & Mai Huy, K. (2015). Une réflexion didactique sur des activités pour penser l’enseignement-apprentissage des probabilités et des statistiques à l’école primaire. Bulletin de l’AMQ, 55(3), 50–67.
Martin, V., & Theis, L. (2016). A study of the teaching of probability to students judged or not with learning difficulties in mathematics in regular elementary classes in Québec. In Proceedings of the thirteenth International Congress on Mathematical Education, Hamburg, Germany.
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Martin, V., Oliveira, I., Theis, L. (2018). Teaching Probability in Junior High School Through Problem Solving: Construction and Analysis of a Probabilistic Problem. In: Kajander, A., Holm, J., Chernoff, E. (eds) Teaching and Learning Secondary School Mathematics. Advances in Mathematics Education. Springer, Cham. https://doi.org/10.1007/978-3-319-92390-1_31
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