Communication is present in every minute of our work and is the leitmotif of the discourse that gives meaning to mathematics. Several times I have wondered if communication that goes on in my classes is appropriated and if it is based on mathematical evidence that leads to the development of my students’ learning. The diversity of situations that may take place in a classroom over the established communication between students is huge, unpredictable and challenging. It is impossible for teachers to foresee the arguments’ sequence, and the students’ interventions, so it is important to adapt the existing framework in order to improve the situation and extend it conveniently. For students communication may mean a unique moment of sharing knowledge, learning and intellectual growth especially when this dynamic is a product of the interactions in small groups. So, students have several opportunities to explore ideas, help each other, to discuss strategies and argue mathematical meanings. But what happens when the group’s dynamic only develops around one member and if all the work depends on the person? Will it be a productive work when that member is in the role of a teacher, even though the person does not have the necessary experience in managing communication? In this article I present my investigation on how communication affects the group’s dynamic, development of its work and, therefore, the students’ learning. From the analysis of that episode we realize different communicational life of the two groups observed and how that difference influenced their work.
The Principles and Standards for School Mathematics (NCTM) notify that “in the classroom where students are encouraged to think and reason about mathematics, communication is an essential characteristic: at the same time they express orally and in writing their reasoning” (NCTM, 2000, 318). Therefore, teachers must implement in their classes a sense of community, so that students feel free to honestly and openly express their ideas without fear of being ridiculed. So, teachers should promote a climate of learning for all individual students to present and explain the strategy they used to solve a problem. Teachers should also use significant mathematical tasks and should let the students’ interventions to be evaluated through discussion in the class, thus enabling students to develop mathematical skills at different levels.
Working in small groups (as we can verify in this article) is valued by Ponte and Serrazina (1999) because it “enables students to expose their ideas, listen to their colleagues, questioning, discuss strategies and solutions, argue and criticize other arguments” (21). This type of work provides an opportunity for efficient mathematical communication, as well as teachers have an important role in helping to further the mathematicians objectives to all members of the group. According to the Principles and Standards for School Mathematics, teachers must resist when students’ try to make teachers think for them “and should” answer in a way that allows them to concentrate on thinking and reasoning, rather then acquisition of the right answer” (NCTM, 2000, 323).
According to Ponte and Serrazina (1999), “students learn in consequence of the activity that they develop and reflection on it” (3). Communication plays an important part in reflection and learning development when arguments are shared and mathematical concepts and processes discussed. The communication process generates mathematical meanings through their trading and use in a social interaction. According to the same authors, the three basic types of communication are exposition, questioning, and discussion.
In the first type, communication consists in interlocutors’ exposition of an idea. In the second one, the question presupposes that interlocutor put successive questions with an objective for others. In the third, the discussion involves the communication modes just mentioned, when it allows the interaction between different interlocutors, sharing ideas and questioning each other and, therefore, it is considered the most important mode of communication.
Ponte and Serrazina (1999) subdivided communication through questioning into three types of questions posed by teachers: (i) focus, (ii) confirmation, and (iii) inquiry. They considered focus questions “false” questions, with the aim of providing guidance to students so that they can complete the task. Teachers use the confirmation questions to make sure that students have got certain knowledge. The inquiry questions provide teachers with information that they have not and the authors consider them as the only real questions.
Students had some difficulty in the interpretation and understanding of the issue concerning the interpretation of the graph of the task, much more than in pure mathematical part of the activity. This raises the question of hierarchy of issues in terms of difficulty, i.e., would it not be preferable to start with the simplest task and, gradually make them more open and complex? In the classroom, the teacher led students through a process of shared communication to build and consolidate their mathematical thinking. She promoted a more detailed analysis of the issues and the formulation of explanations, different kinds of argument, and the reflection on the knowledge of students and on the ideas of others. The issues raised by the teacher were focus, confirmation and inquiry, used in the type of guidance that she wanted to give students. During the episodes she provided guidance to students so that they could reach certain knowledge; wanted to make sure that students fully understood the knowledge and expresses their doubts. Although the communication in the groups was dynamic, it was not shared by all its members. Two students, Ruben in group A and Renato in group B did not participate in the discussion. In subsequent conversation the teacher of the class indicated that Ruben is a repetitive student, introverted, and that he still did not feel well integrated in the classroom. Also Renato never participated in any activity; he was at risk and worked only when the teacher, during class, sat beside him and motivated him to work. Once the teacher finished individual support, he stopped working again. In developing the work done by both groups a communication involvement showed up, with the exception for these two students named before.
However, in group A all the suggestions/guidelines of Bruno, the leader of the group, were accepted thus making communication poor; there was no sharing of reasoning, knowledge and strategies. The communication in this group was basically unidirectional and Bruno, which is understandable, did not care about developing the learning of his colleagues but only wanted to transmit the knowledge he considered correct. In group B, it appeared that despite Tatiana being regarded as the leader, the students were involved in sharing strategies, questions and checking knowledge. Tatiana sought to promote dialogue in a confrontation of ideas and went on building knowledge of the group from the contributions of all. While no one can expect a student to have attained the required competency level of communication to promote the learning of colleagues, let us note that there were differences in dynamics in the groups resulting in a large part from different communication formats between their members.
This diversity of environments, communication, and learning takes place systematically in our classrooms. The way of talking and sensitivity of teachers is fundamental to the process of communication between students and is rewarded by its multiplicity. Teacher’s mediation is essential to develop work of a group in order to homogenize the discrepancies that inevitably occur within them. The combination of factors, such as knowledge that teachers have of their students, the activities they propose, the questions they pose, and the debate they promote about strategies, mathematical reasoning and meanings that favor building an effective and permanent educational success.
Analysis of these situations can enrich our teaching. The position of an “observer” allows a reflection on the different types of mathematical reasoning and approach. Apparently, communication in the classroom develops intuitively and spontaneously. But in reality there must be a coherent effort of teachers to enhance the communication and engage all students taking into account their specific features and characteristics.
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