Self-reported practices of probability teaching: �the use of the frequentist approach, �manipulatives and technological tools

Vincent Martin (Université de Sherbrooke)

Mathieu Thibault (Université du Québec en Outaouais)

Marianne Homier (Université de Sherbrooke)

To access the slideshow : monurl.ca/icme14prob

14^th International Congress on Mathematical Education

July 14^th 2021

Background

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Two-stage study to sketch a broad portrait of probability teaching within Québec’s primary and secondary schools

First stage of the study

• The preliminary results of the first stage were initially in the form of a statistical portrait of data collected from 626 respondents (Martin & Thibault, 2017).
• These respondents’ answers to an online questionnaire helped describe their perception of probability, the place of probability instruction in their classrooms, and the resources they used in this teaching.
• Inferential statistical analyses were then performed on the same data �(Martin et al., 2018; Martin et al., accepted), which turned up three main findings.
1. The respondents reported a significantly lower level of mathematical proficiency and instructional confidence in probability than in other areas of mathematics, both at the primary (6–12 years old) and secondary levels (12–17 years old)*;
2. The respondents who taught at the secondary level reported using the theoretical approach significantly more than those who taught at the primary level;
3. The primary teachers stated that they used significantly more frequently manipulatives to teach probability than the secondary teachers did.

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Second stage of the study

• Going a step beyond the questionnaire answers of this first stage, we conducted a qualitative analysis to gain in-depth understanding of certain self-reported teaching practices.
• In particular, to bring attention to the rich potential of probability teaching, we wanted to highlight practices that we considered to be exemplary.
• Thus, the research objective of the second stage is

to describe and more specifically understand examples of self-reported probability teaching practices considered to be exemplary, at the primary and secondary levels.

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Stage 2

Methodology

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Participants

• We selected 8 teachers who, in light of their answers to the online questionnaire, exhibited exemplary probability teaching practices according to 5 criteria:

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• They taught at various levels of primary and secondary school and all had at least 10 years of teaching experience.

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High sense of competence in probability and probability teaching

Relatively lengthy period of time devoted to this teaching in class

High frequency of using technological tools and manipulatives

Use of the different probabilistic approaches

Presentation of diversified examples of situations

Data collecting, processing and analysis

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• As a way to elaborate on some of their questionnaire answers and to discuss professional development avenues inspired by their experience, we conducted one-hour individual interviews with the participants.
• Audio recordings of the interviews were transcribed verbatim and a two-step content analysis (Bardin, 2013) was performed. Successive readings of the transcripts led to intuitive coding that suggested a thematic structure based on the themes that emerged from the teachers’ statements.
• A software (NVivo 12) was used to systematically code the transcripts according to this thematic structure. An in-depth consultation of coded excerpts for each theme allowed for identifying eleven themes that emerged from the statements of the eight selected teachers.

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Results and discussion

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Overview

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Eleven emerging themes

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The use of the frequentist approach �in probability teaching

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Use and role of the frequentist approach

• All mentioned this theme and reported using it in their teaching, �but what they said varied significantly from one person to another.
• For example, the way the situations are carried out varies in terms of the person running the trials (students or teacher), the size of the generated samples (between 10 and 16,000 trials) and the compilation method (individual, team, intra-class or inter-class) for the trials that were done.
• Questions remain about the true role of the frequentist approach in classrooms
• Is it used, as Thibault (2019) suggests, to observe a frequency that helps reveal a trend in long-term outcomes, in order to predict what might happen in the future and hence make an informed decision?
• Or is the frequentist approach used in class to bring a concrete (or playful) dimension to probability learning?

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Noise, time and sample size

• A few of the participants spoke of logistical aspects (noise and commotion) in implementing the frequentist approach in the classroom.
• We believe that these logistical aspects could constitute favourable arguments for using the theoretical approach, which can be implemented more quietly and less time-consuming than the frequentist approach.
• In a context where the teacher or students run only a few isolated trials without pooling the outcomes, the sample’s statistical power remains very limited and probably precludes inferences with a high level of confidence.
• Hence, what place is accorded to uncertainty and variability, which are invoked in the curriculum for primary and the beginning of secondary school? Is the law of large numbers given a place in instruction, even if it is not explicitly a subject of study in the curriculum?

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The use of manipulatives �in probability teaching

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Using typical or unusual manipulatives

• Participants mentioned using typical manipulatives (e.g. dice, coins, cards).
• Some participants also reported using manipulatives we considered as unusual (Martin & Thibault, 2019), which, among other things, are everyday life items (e.g. Smarties, padlocks, non-regular dice, styrofoam cup, etc.).
• These unusual manipulatives often seem to be used as a way to exploit the frequentist approach, based on experimentation, as a lead-in to probabilistic situations for which a theoretical approach would be impossible (Parzysz, 2011)
• Participants also reported using unusual manipulatives for different reasons:
• to enable students to visualize a situation;
• to run trials (as a necessary condition for the frequentist approach);
• to support students’ probabilistic reasoning (in order to demonstrate a solution).

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Different ways of using a coffee cup...

• These different uses are illustrated by the different ways that two participants use a coffee cup.
• One participant uses it to prepare students to visualize a geometric probability:

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• The other participant uses a coffee cup to introduce a frequentist approach and run a number of trials in order to make links with theoretical probability:

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[In grade 8], I talk about the Tim Hortons cup, because there’s nothing more annoying than lifting the flap on my coffee cup and realizing that I’m drinking at the seam. So, I ask them: “What are the odds that this seam will be in my coffee cup opening?” Obviously, they are destabilized, because they don’t yet know how to calculate that.

When I teach [probability], it’s always during the Tim Hortons “Roll Up the Rim” contest, in [grade 7 and 8]. It’s funny because one year, I arrived each morning with a coffee, and I talked about how it claimed I had a one in four chance [of winning], so we tested out the theoretical probability versus frequentist probability. After four coffees I hadn’t won anything, so why wasn’t it working?

The use of technological tools �in probability teaching

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Diversity of technological tools and various uses

• There appears to be widespread agreement on the added value of technological tools. However, the rationales that would support the idea of added value are generally not explained.
• They all mentioned using a simulator. Some mentioned the use of other technological tools, including calculators, spreadsheets, programming tools, videos, interactive whiteboards and websites.
• The teachers also stated that they used these tools themselves most of the time; few reported that students use them directly.
• This being said, Pea (1987) makes a distinction between technological tools’ powers of amplification (doing more, faster) and reorganization (doing things differently, or something else).

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Intentions behind the use of technological tools

• The intentions behind the use of these technological tools vary:
• to reveal a general trend in the context of the frequentist approach, as a way for students to infer a theoretical probability or to show that a player loses;
• to be more efficient, i.e., among other things to gain time, to be more precise, to collect experiment results or to avoid the noise caused by experimentation;
• to “do things differently”, in particular to stimulate new types of situations or to observe the effects of the parameters in a calculation.
• These various intentions reported by the participants regarding the use of technological tools in probability instruction are consistent with Thibault (2019), who notes several potential roles that such tools can play in this context.

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Concluding remarks

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Next stage...

For the next stage of the project, we want to get closer to the real practices of teachers to support their professional development with regard to the teaching of probabilities. To do this, we plan, among other things, to resort

• to participatory research “‘with’ rather than ‘on’ practitioners” (Desgagné et al., 2001, p. 33, our translation), not only to continue to produce new scholarly knowledge on probability teaching, but also to more directly stimulate teacher professional development;
• to a research approach (didactic clinic of activity) rooted in the everyday teaching activity of teachers and which makes it possible to jointly reflect on new achievable possibilities linked to professional objects targeted in this real context of probability teaching.

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Self-reported practices of probability teaching: �the use of the frequentist approach, �manipulatives and technological tools

Vincent Martin : Vincent.Martin@USherbrooke.ca

Mathieu Thibault : Mathieu.Thibault@uqo.ca

Marianne Homier : Marianne.Homier@USherbrooke.ca

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To access the slideshow : monurl.ca/icme14prob

Thank you for your interest ! We look forward to continue the discussions.

References

Bardin, L. (2013). L’analyse de contenu. Paris: Presses universitaires de France.

Desgagné, S., Bednarz, N., Couture, C., Poirier, L., & Lebuis, P. (2001). L’approche collaborative de recherche en éducation : un nouveau rapport à établir entre recherche et formation. Revue des sciences de l’éducation, 27(1), 33-64. https://doi.org/10.7202/000305ar

Martin, V., & Thibault, M. (2017). Enquête sur les pratiques déclarées d’enseignement des probabilités au primaire et au secondaire au Québec : esquisse d’un portrait statistique. In A. Adihou, J. Giroux, A. Savard, & K. Mai Huy (Eds.), Données, variabilité et tendances vers le futur. Actes de colloque du Groupe de didactique des mathématiques du Québec (pp. 179-195). Montréal. https://www.researchgate.net/publication/324261468

Martin, V., Thibault, M., & Roy, N. (2018). A survey of teachers’ self-reported practices of probability teaching in primary and secondary school levels in Québec. In M. A. Sorto, A. White, & L. Guyot (Eds.), Proceedings of the Tenth International Conference on Teaching Statistics (ICOTS10). Kyoto: International Statistical Institute. https://www.researchgate.net/publication/326316737

Martin, V., Thibault, M., & Roy, N. (accepted) Pratiques déclarées d’enseignement des probabilités : enquête auprès de personnes enseignantes du primaire et secondaire au Québec. Canadian Journal of Science, Mathematics and Technology Education, 37 pages.

Martin, V., Thibault, M., & Theis, L. (2019). Introduction. L’enseignement des premiers concepts de probabilités. In V. Martin, M. Thibault, & L. Theis (Eds.), Enseigner les premiers concepts de probabilités : un monde de possibilités ! (pp. 1-16). Québec: Presses de l’Université du Québec. https://www.researchgate.net/publication/335840174

Parzysz, B. (2011). Quelques questions didactiques de la statistique et des probabilités. Annales de didactique et de sciences cognitives, 16, 127-147. https://mathinfo.unistra.fr/websites/math-info/irem/Publications/Annales_didactique/vol_16/adsc16-2011_006.pdf

Pea, R. D. (1987). Cognitive technologies for mathematics education. In A.H. Schoenfeld (Ed.), Cognitive science and mathematics education (pp. 89-122). New Jersey: Lawrence Erlbaum Associates, Inc.

Thibault, M. (2019). Le recours à des simulateurs pour l’enseignement des probabilités. In V. Martin, M. Thibault, & L. Theis (Eds.), Enseigner les premiers concepts de probabilités : un monde de possibilités ! (pp. 169-192). Québec, Presses de l’Université du Québec. https://www.researchgate.net/publication/335877602

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Abstract

The first stage of a study in Québec enabled us to draw up a statistical portrait of probability teaching practices self-reported by 626 teachers at the primary and secondary levels. For the second stage of the study, discussed here, we wanted to elaborate on some of the questionnaire answers and to discuss professional development avenues inspired by the teachers’ experience.

We conducted one-hour individual interviews with eight teachers who had taken part in the first stage and whose self-reported teaching practices were considered to be exemplary. By means of a content analysis of their discourse, we explore issues surrounding certain self-reported probability teaching practices through examples related to the use of the frequentist approach, manipulatives and technological tools.

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Appendices

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General observations on the convergent themes

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The eight consulted teachers appeared convinced of the social usefulness of probability. 

The discourse of all the participants was very homogenous on the theoretical approach, both in describing and explaining the types of tasks given to students in this context. 

Several traditional tasks were also mentioned by the participants (e.g. sum of two dice). 

Most of the interviewed teachers struggled to determine the drivers that could promote a culture of professional development, dialogue, discussion and mutual assistance between peers in this context.

The taught probabilistic concepts emerging from the teachers’ discourse generally appear to be consistent with the curriculum, and certain recurrent difficulties were perceived by most of the participants, in the areas of arithmetic and probability.

General observations on three (of six) divergent themes

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In the participants’ discourse, the subjective approach does not appear to be as well-known as the two other probabilistic approaches. Some reported being unfamiliar �with the subjective approach, and the descriptions of the participants who reported �using it in their instruction varied from one person to another.

The discourse of all the teachers consulted (except one) suggests that probabilistic approaches are connected in their instruction, but the probabilistic approaches involved and how this connection is made varies depending on the teacher.

The participants reported using a diversity of unusual tasks in their probability instruction, for example Dan Meyer’s “Three-Act Math Tasks”, situations where a Tim Hortons cup is used to run trials or to contextualize a geometric probability situation, a situation where students are led to invent games of chance, etc.