Abstract

This The aim of this article is to examine the links and interdependencies between the teacher's piloting postures and the pupils' learning postures during the teaching of mathematics, particularly with regard to the solving of proportionality problems in 3^rd class at Bourbon secondary school on Reunion Island. This analysis will be based on the observation of teaching practices through video sequences and will be deepened through the use of ISA (Implicative Statistical Analysis) under HCIC-M_GK. We found that time management by the teacher generally leads pupils to adopt a 'first do' posture, whereas the latter often adopt a 'school first do' posture. What's more, when a student adopts a dogmatic posture, they often find themselves in a school posture too. Furthermore, teachers tend to give instructions when dealing with students' constraints. It is suggested that teachers encourage greater student involvement and foster an environment in which students take the initiative to work, thus facilitating the creation of a playful and creative environment in the classroom. It is also recommended to integrate a variety of pedagogical postures during teaching to better respond to the diverse needs of learners.

1. Introduction

The seventeen Sustainable Development Goals (SDG) defined by UNESCO aim to address the most pressing social, economic and environmental challenges between now and 2030. These goals cover a wide range of areas, including education (SDG-4). SDG-4 focuses on ensuring inclusive, equitable and quality education for all, while highlighting the crucial role of education in achieving the other sustainable development goals.

In France, the Ministry of National Education, Youth and Sport has updated and implemented the curriculum for cycle 4 from the start of the 2020 school year. The aim of this revision is to reinforce lessons on climate change, biodiversity and sustainable development. Given that Reunion is an overseas department of France, the curriculum at College Bourbon on Reunion Island has also been modified since the start of the 2020 school year to reflect these changes.

In this article, we will look specifically at the role of the teacher in the context of the SDG. The main objective is to investigate the interdependencies between the teacher's supportive postures and the pupils' study postures in the teaching of mathematics, in particular for solving proportionality problems in 3^rd class at Bourbon secondary school on Réunion Island, by analysing teaching practices. This study could highlight the need to train teachers to better understand pupils in order to adapt their teaching approaches, which would help to improve the effectiveness of the education system and guide public policy in the field of education. Postures or gestures are very important to teaching/learning in the classroom because the teacher's body speaks just like his or her voice ^{1 2 3 4 5 6}.

To this end, a filmed observation was carried out in a 3^rd class at Bourbon College on Reunion Island, focusing on the teaching of mathematics and, more specifically, the solving of proportionality problems. This observation is based on the theory of steering postures developed by ⁷. The teacher's steering postures and the pupils' study postures observed during all the class sessions were recorded in an observation grid^{1}. This observation grid was used to produce a binary table which will be analysed using Implicative Statistical Analysis (ISA), implemented with the HCIC-M_GK tool.

2. Materials and Methods

An observation of videos filmed in 3^rd classes at Bourbon College on Reunion Island, focusing on the teaching of mathematics and more specifically on solving proportionality problems, was carried out. This observation was carried out using the pedagogical theory of ⁷ as well as a number of informatics tools.

The theory detailed by ⁷, in particular on piloting postures, enables us to account for the diversity of ways in which the teacher conducts pupil activity during the class. It's an attitude adopted by the teacher to support his pupils (Pil₁,..., Pil₃):

• Pil₁: Manage constraints,

• Pil₂: Manage the situation time,

• Pil₃: Giving instructions.

In response to the teacher's piloting postures, ¹ identify six possible postures for pupils (Pel₁,...,Pel₆):

•Pel₁: "a school posture", characterising more the way in which the pupil tries above all to fit in with the expected school norms, tries to fit in with their teacher's expectations.

•Pel₂: "a first-doing posture", describing the way in which pupils throw themselves into the task without thinking too much, letting all sorts of ideas or solutions come to light without going into it any further.

•Pel₃: "a playful-creative posture", translating here that the pupil uses creativity to reinvent the task he has been given.

•Pel₄ : "a reflective posture", which is one that enables the pupil not only to be in the act, but to look back on this act, to "secondarise" it in order to understand its aims, failures and contributions.

•Pel₅: "a posture of refusal", designating the refusal to do, to learn, the refusal to conform, which is always an indicator to be taken very seriously. It often leads to identity and psycho-affective problems, and to symbolic or real violence suffered by pupils.

•Pel₆: "a dogmatic stance", which manifests an assertive lack of curiosity. The "I already know", the "my former teacher, my mother, etc. have already told me". In this case, the pupils are not involved in what they have to learn.

The digital tools used included YouTube to transcribe the videos, which lasted a total of 256 minutes, subdivided into 128 sequences of 2 minutes each. Excel was used to create an observation grid, while HCIC-M_GK was used to analyse the data collected (Table 1).

The teacher's piloting postures and the students' study postures observed during all the sessions of the course are recorded in the observation grid in order to produce a binary table (e.g. Table 1) which will be used for subsequent analysis.

The data collected is processed and studied using the theory of Implicative Statistical Analysis (ISA) ⁸, which is a non-symmetrical method of analysing data that crosses subjects or objects with variables of different types (in our case Boolean). The technique of extending Hierarchical Classification Implicative and Cohesive - HCIC ⁹ called Hierarchical Classification Implicative and Cohesive based on M_GK (HCIC-M_GK) ¹⁰ is applied to the analysis of these data collected to deal with the numerical and graphical problems required by the use of ISA. This application is also used to classify M_GK-valid rules according to the cohesion index based to M_GK ¹¹ in order to obtain meta-rules to facilitate interpretations of the analysis results.

Let () be a binary context (e.g. Table 1). Consider the set of sequences and the set of items (Postures) (e.g. Table 1). The association rules between two Boolean variables are defined by analysing the contingency table obtained by crossing them (e.g. Table 2).

Mathematical modelling: Consider a discrete finite probability space () such that for any event in . Let us note the set of sequences, over which we have measured Bernoulli random variables, and let be the set of items . For all , for all is an application from E to and , where . Any non-empty part of will be called a motif of . So for the motif and. For the motifs and , the number of transactions that use both and . Agree to be the logical negation of a motif . The real number will be called the of the motif noted ¹².

A probabilistic interest measure is a real function of ) such that for any association rule , with , the value of is computed from the four quantities , , and Finally, for two motifs (or items) and in a binary context, the measure of interest M_GK is defined by

Theoretical research in ^{13 14} allowed us to develop an M_GK-valid association rule extraction algorithm ¹¹.

• The extraction of association rules is based on the M_GK measure of interest, which detects a possible non-symmetrical relationship between two of the variables, the validation of the extracted rules is done with respect to the favorable component which is implicative and the critical value ) having a relationship with the χ² statistic of independence or dependence of degree of freedom 1 at the risk threshold α chosen by ourselves such that, with, . In our case, α=10%, which gives the critical value of χ² equal to 2.7 ;

• The value of the support according to such that is generally small ¹⁴. It is therefore essential to normalise this value ¹¹, and we denote it by, with

• An implicative graph, denoted G, consists of a finite set of variables representing the vertices of the graph, and edges associated with the normalised support value of valid rules. For ease of interpretation and to highlight meaningful relationships, only implications with a support value are retained; alternatively, the implicative tendency of over is preferred to neutrality ¹⁵.

• The value of is used to establish the value of cohesion between two items, denoted ¹¹ with:

• The hierarchical classification implicative and cohesive method according to the (HCIC-M_GK) measure of interest is based on the cohesion ¹³.

3. Results and Discussions

Table 3 and Table 4 above highlight the most common postures among the participants. It appears that the pupils' postures, in particular the school posture and the first-doing posture, are the most common, with rates of 81.25% and 64.84% respectively. In addition, teachers were observed to adopt mainly the posture of giving instructions, with an occurrence rate of 58.59%. Less frequent postures, with rates of less than 50%, were classified as rare, notably the second reflective posture, the dogmatic posture, the management of constraints, the management of situation time, the playful-creative posture and the refusal posture.

Table 4 below summaries the frequency of these postures, which are ranked according to the average number of occurrences. The teacher and the students all have postures classified as rare, while the very frequent ones are those of the students.

Table 5 examines the relationships between the different postures adopted by teachers and students. We note that there is a likely correlation between teacher time management and student behaviour, with students being more inclined to act without thinking too much (first-do posture). In addition, students who adopt a dogmatic posture often seem to find themselves in a scholastic posture, which suggests a possible influence of the students' attitude on their engagement in the learning process.

The G graph shown in Figure 1 has 6 vertices and 3 edges, generated with implication supports greater than 0.5 on valid rules according to M_GK. The vertices Pil₁ (Manage constraints), Pil₂ (Manage situation time) and Pel₆ (Dogmatic posture) are all connected to their respective counterparts, Pil₃ (Give instructions), Pel₂ (First do posture) and Pel₁ (School posture), with normalised support values of 0.671, 0.822 and 0.787 respectively. This means that "time management by the teacher generally leads students to adopt the first-doing posture. Secondly, students who adopt the dogmatic posture are often in the academic posture. Finally, the management of constraints often leads the teacher to give instructions to the pupils".

Table 6 and Table 7 highlight several significant cohesions between the different postures observed. In particular, a strong link was observed between the dogmatic posture and the academic posture, as well as between the time management postures and the first-do posture. On the other hand, the correlation between the management of constraints and the playful-creative stance seems to be weaker.

Finally, the hierarchical classification of the valid rules made it possible to identify 3 meta-rules (e.g. Table 8 and Figure 2) relevant to understanding the dynamics observed in class:

• R(1): (Pil₂⇒Pel₂): The teacher's management of time generally leads students to adopt the first-to-do posture.

• R(2) (Pel₆⇒Pel₁): Students adopting a dogmatic posture are also often in a schooling posture.

• R(3): (Pil_1⇒Pil₃): When the teacher manages the constraints pupils face in class, they tend to give instructions systematically.

Based on the results obtained by HCIC-M_GK, several points can be identified as avenues for discussion in order to be able to give a suggestion for improving the teaching and learning environment in the classroom.

Encouraging reflection: According to Table 4, it seems that the second reflective posture is one of the rare postures. By promoting a second reflective posture in students, teachers can encourage more intentional and reflective practice. This could be achieved by encouraging teachers to participate in communities of practice where they can reflect on their teaching experiences, share ideas and receive constructive feedback from their peers.

For students, it is important to create opportunities for critical thinking in the curriculum. This can include activities such as class discussions, reflective journals and research projects that encourage students to question, analyse and evaluate information.

Time management and student engagement: Effective classroom time management is essential to maximise learning time (R(1)). For example, in sequence 121 out of 128, the teacher gives the pupils a very specific amount of time to work: “[…] You're allowed to work in groups. We'll do this for 6 minutes […] You make up the chart […] I say 6 minutes”. The pupils immediately open the discussion with their classmates so as not to waste any time (e.g. Figure 3).

Teachers could benefit from lesson-planning strategies that allow them to cover material effectively while leaving sufficient time for interactive and reflective activities.

To encourage deeper student engagement, teachers could incorporate strategies such as differentiated instruction, group discussions and collaborative projects that allow students to actively apply their knowledge and skills.

Creating a playful and creative environment:

Compared with Table 4, it would appear that the playful and creative stance adopted by pupils is one of the rarest.Yet incorporating playful and creative activities into the curriculum can help to capture students' attention and encourage their engagement in learning ⁷. This can include educational games, art projects, simulations and problem-solving activities that allow students to learn in a fun and interactive way.

Teachers could work with arts education specialists and games designers to develop activities that are appropriate to the content of the curriculum and the needs of the students.

Encouraging student-teacher interaction: To encourage more dynamic interaction in the classroom, teachers could adopt practices such as open questioning, constructive feedback and small group discussions. This enables students to take an active part in their learning and to develop social and cognitive skills.

Teachers could also create an inclusive and safe classroom climate where students feel comfortable asking questions, sharing ideas and taking intellectual risks.

Diversification of teaching approaches: By incorporating a variety of teaching postures, teachers can better respond to the diverse needs of learners. For example, the second reflective posture encourages teachers to reflect on their own practice and adjust their teaching methods to suit the individual needs of their pupils. Similarly, the playful-creative posture can make learning more attractive by incorporating interactive activities and creative approaches.

Professional training courses could be offered to teachers to raise their awareness of the different pedagogical postures and encourage them to experiment with them in their day-to-day teaching.

4. Conclusion

Analysis of the relationships and interdependencies between teachers' control postures and pupils' study postures when solving proportionality problems revealed several significant aspects that can influence the effectiveness of teaching practices. Teacher time management plays a crucial role in the way students approach tasks. Students tend to adopt a 'first-do' posture under strict time management, which can limit their critical thinking and deep engagement in the learning process. Teachers are therefore advised to strike a balance between time management and encouraging deeper reflection. In addition, the dogmatic posture of students, often correlated with a scholastic posture, suggests that some students may feel constrained by overly rigid pedagogical approaches. To remedy this, teachers should promote more flexible and interactive learning environments, where students can explore and express their ideas creatively and autonomously. The results also show that teachers tend to give instructions when dealing with student constraints.

This highlights the importance of developing teaching strategies that encourage students' autonomy and their ability to manage their own constraints, rather than being constantly dependent on teacher direction. To improve the teaching and learning environment, it is crucial to encourage critical thinking, manage time effectively while actively involving students, create a playful and creative environment, and foster dynamic interaction between students and teachers. Diversifying teaching styles and training teachers in these various approaches can also help to better meet learners' needs. In conclusion, a balanced and diversified pedagogical approach that takes into account the different postures of teachers and students can greatly enhance the educational experience and contribute to achieving the sustainable development goals for education. This study highlights the importance of adapting teaching practices to encourage students' active engagement, critical thinking and creativity.

Notes

{1}. observation grid inspired by the one developed by Hella Feki, Christelle Juguet, Véronique Crosson and Guillaume Jourcin, november 2019, https://padlet.com/monsieurjourcin/accompagnement-des-professeurs-n-o-recrut-s-locaux-xces3tkkk4ch/wish/417653551

References