Abstract
This paper describes how the notion of the strongly didactic contract can serve to characterize the teaching adopted to implement a task in probability. It is particularly focused on the reality of mathematical work performed by students and teachers. For this research, classroom sessions were developed in an in-service teacher training course designed (and adapted) according to the Japanese Lesson Study model. Through the combined use of the Theory of Didactical Situations (TDS) and the Theory of Mathematical Working Spaces (ThMWS), a coding of the sessions observed was developed. Based on this coding, different patterns emerged which gave each session a specific rhythm and identity from which it was possible to recognize and characterize different strongly didactic contracts. The study highlights the difference between the potential contracts intended by the teachers and those observed in practice. The tools, and especially the coding, developed for the study could be used for future research on instructional situations or in-service teacher training.
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Data availability
The data which were used for the analyses of the current study are not publicly available due to limitations derived by the Ministry of Education (which gave the permission for the study). However, the data are available from the corresponding author on reasonable request.
Notes
Augustin in charge of the implementation for group 2 did not follow the previous and common plan of the session and adopted a weakly didactic contract by ignoring students’ reactions (Masselin, 2020).
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Appendices
Appendix 1. Chronogram
The chronogram is a methodological tool which helps to visualize the chronology of interactions between the teacher and each group of students during a classroom session. The dates and durations of each of these interventions are given on a timeline.
Two types of events are identified and drawn differently on each group line. Dots represent a short (less than 30 s) teacher’s statement with no exchange with the group. A continuous line is marked on the group line when there exists a longer and significant interaction related to the mathematical content.
The different teacher’s interventions are linked by segments and the broken line shows the chronology and movement of the teacher from one group to another. The time left to students’ autonomous work is marked with dotted lines.
Lucie’s chronogram shows the nature of the teacher’s reorientations of the work in the different groups. Lucie gradually imposed (see ellipses) the binomial law as the only model for solving the task.
Lucie’s chronogram (Henriquez et al., 2022; Masselin, 2020).
Appendix 2
Identifying Emma’s pattern
Pattern | Time | Tasks assigned by the teacher |
|---|---|---|
N°1 | 16’ | Devolution 10:08 Emma: “Here is a discovery activity, I’ll give you the statement. Play five games with the dice on your desk. Count the number of times the hare wins or the turtle wins.” Students’ Action Groups of students play a series of games and obtain different results Synthesis and local institutionalization 10:23 Emma collects the results of each group 10:25Emma: local institutionalization “some find the hare; we haven't played enough games.” |
N°2 | 5’ | 10:25 Explanation of the rules of the game and justification of the simulation Emma rolls her rubber dice and obtains different faces of the dice At every throw, she asks the students who win to be sure they have understood the rules Action and results: the students answer Emma’s questions in successive throws When a six appears, Emma asks the class “Who thinks the hare won?” Who thinks the hare has a better chance of winning, based on your experiment?” Emma wrote on the blackboard “4 groups Tortoise and 2 groups Hare” One student says “More throws should be made, for example with a spreadsheet, as that could allow several hundred throws” Institutionalization. Emma wrote on the blackboard “Simulation, Spreadsheet, Scratch to throw several hundred throws” |
N°3 | 6’ | Devolution 10:28 Emma: “Using a spreadsheet or using Scratch, you can simulate several hundred throws, the computer can do it for you. We make a simulation. You’re going to go to the file… we chose Scratch.” Action Each group opens the simulation file Results They launch the software several times to obtain several game simulations Local institutionalization. No need |
N°4 | 29’ | Devolution 10:31 Emma “You log in, you see the table, you fill it in. What is the frequency of the turtle winning? It is the quotient of the number of games won by the turtle over the total number of games." 0:34 Emma: “Let’s go, you have to fill in the table and create the graph” Action: Groups work on their tables and graph Results Emma collects the groups’ productions (tables and graphs) 15-min student break |
N°5 | 29’ | 11:34 Concluding institutionalization |
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Kuzniak, A., Masselin, B. Strongly didactic contracts and mathematical work. Educ Stud Math 115, 289–312 (2024). https://doi.org/10.1007/s10649-023-10286-1
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DOI: https://doi.org/10.1007/s10649-023-10286-1