The students’ and teachers’ perspectives of a good mathematics teaching can be analyzed from different theoretical framework. I chose social representations (Jodelet, 1986; Moscovici, 1976) because they are expressions of common sense knowledge (Berger and Luckmann, 1966). This choice emerge from the consideration that common sense knowledge is the most basic, primary, immediate knowledge of any individual as a member of a community, group or society, whose integration fundamentally depends on the existence of that knowledge. Common sense knowledge is the people’s certainty that phenomena are real and possess specific characteristics.
I assume reality as a social construction where people are social beings based on the reality in which they live but that also participate in the transformation of this reality. Although people create a particular vision of reality, this does not mean that it constitutes an individual process, instead its production is a social process that occurs in everyday life during interaction with others. According to Berger and Luckmann (1966), reality is an innate quality of the phenomena that we recognize as being independent of our own will; in other words we cannot make them disappear. Common sense knowledge is the knowledge that we construct during day-to-day relations, through models of thinking that we receive and transmit through tradition, education and communication, and which allows us to understand and explain facts and ideas that exist in our immediate world, since they provide us with a reference framework in order to know how to behave with other people.
Social representations establish a specific form of common sense knowledge, the specificity of which depends on the social nature of the process they produce.
They include the set of beliefs, knowledge and opinions produced and shared by individuals in the same group in relation to a particular specific social object (Guimelli, 1999). A social representation is a guide for people’s action in front of a specific social object. Therefore, the study of social representations is particularly
important since the way in which they are produced and transformed helps to understand human behavior. The representation operates as a system for the interpretation of the reality that governs the relationships of individuals with their physical and social environment, due to the fact that it establishes their behaviors or their practices. Social representations are guide for actions and social relations. In Abric’s opinion, social representations are a pre-decoding system of reality since it establishes a set of anticipations and expectations (Abric, 2004, p. 12).
In other words, social representation is practical knowledge. They give meaning, within incessant social movement, to events and activities that end up becoming commonplace to us and this knowledge forges evidence of our consensual reality, as it participates in the social construction of our reality (Jodelet, 1986, p. 473).
Consequently, social representations are characterized by their significant, shared character, where their genesis is composed of the interactions and their functions fulfill practical purposes and they are thus a socially created and shared knowledge for practical purposes that takes part in the construction of a shared reality for a social group and their function is to create behaviors and communication between individuals. Social representations are “cognitive systems” in which it is possible to recognize the presence of stereotypes, opinions, beliefs, values and norms that usually have a positive or negative behavioral orientation” (Araya, 2001, p. 11).
According with Gorgorió & Abreu (2009, p. 64) the notion of social representation may be easily applied to mathematical practices or mathematical learning. People interpret what happens around them as mathematical when it fits their image of what counts as mathematics. Teachers categorize their students as good, bad, or indifferent based on their images of what is entailed in learning mathematics.
The aim of this article is to explore the ways in which the notion of social representations can offer useful insights in understanding the students’ and teachers’ perspectives of a good mathematics teaching.
Methodology
Procedure
In order to obtain the data an open-ended questionnaire and focus group interviews were carried out. The purpose of these two techniques was to generate written and verbal discourse1, allowing us to find out the social representations.
1 In this paper the word discourse is used in a very broad sense to refer to narratives generated by students through the instruments used
The questionnaire was composed of open-ended questions to not limit the answers of the participants and to allow them to openly express their opinions, reducing the influence of the questionnaire to a minimum. Two questions were asked in order to discover the social representations of “good teaching”: 1) in your opinion, what characterizes a GOOD MATHEMATICS TEACHER? And 2) in your opinion, what characterizes a GOOD MATHEMATICS LESSON? In the questionnaire given to the students, the capital letters were used to emphasize the purpose of the representation of interest in each question. The questions asked in the focus group were the same as those in the questionnaire and the role of the interviewer was to ask for more specific information in relation to answers regarding the use and meaning of words or phrases used by the students. For this purpose, questions such as, in your opinion, what is a dynamic class?, why do you say that the lesson is boring?, what does it mean that a teacher knows how to explain?, etc.
Morgan (1996, p. 130) defines “the focus groups as a research technique that collects data through group interaction on a topic determined by the researcher”.
This definition has three essential components. First, it clearly states that focus groups are a research method devoted to data collection. Second, it locates the interaction in a group discussion as the source of the data. Third, it acknowledges the researcher’s active role in creating the group discussion for data collection purposes. A focus group is a group of people who are asked about their perceptions, opinions, beliefs, and attitudes towards a product, service, concept, advertisement, idea, or packaging. Questions are asked in an interactive group setting where participants are free to talk with other group members. A focus group technique aims at the individuals selected by researchers to discuss and elaborate, based on their personal experience, on a social theme or fact subject to research. Focus groups is an appropriate method for data collection when one is interested in social representations because they are based on communication and it is the heart of the theory of social representations (Kitzinger, Markova &
Kalampalikis, 2004).
Both, the questionnaires and the focus groups, were carried out after school in approximately hour and a half in a classroom, this allowed the students to gather at tables. We worked in sets of students with two interviewers, none of which were teachers of the students (two sets of twelve students, two sets of fifteen students and a set of thirteen students). The procedure were as follows:
1) Individual application of the questionnaire, 2) Creation of four focus groups of between three and four students as students themselves decided, 3) Collective answering the questionnaire in each focus group, 4) Commenting and providing
more specific information in relation to the answers with the interviewers. The second, third and fourth part were videotaped.
Participants and context
The IPN (National Polytechnic Institute for its acronym in Spanish) is a public institution that provides free or very low cost studies in Mexico City, from high school to postgraduate level in the area of science and technology. The CECYT (Centre for Science and Technology Studies for its acronym in Spanish) are part of the education offered by high school level of IPN dedicated to the training of technicians. The participants were a non-statistical sample of 67 fifth-semester students, 16 to 18 years. It is important in focus group research that participants have some form of homogeneity. Therefore it was decided that the participants were students enrolled in the same school in the same math class with the same mathematics teacher.
Data analysis
The general strategy for this analysis was a constant comparative method (Strauss, 1987; Glaser and Strauss, 1967), which permitted the categories to emerge from the data.
The data was analyzed using constant comparison method, as well as the grounded theory approach (Glaser and Strauss, 1967; Strauss & Corbin, 1990). The steps in the constant comparison method of analysis are: 1) Begin collecting data, 2) Look for key issues, recurrent events, or activities in the data that become categories for focus, 3) Collect more data that provide many incidents of the categories of focus with an eye to seeing the diversity of the dimensions under the categories, 4) Write about the categories that you are exploring, attempting to describe and account for all the incidents you have in your data while continually searching for new incidents, 5) Work with the data and emerging model to discover basic social processes and relationships, and 6) Engage in sampling, coding, and writing as the analysis focuses on the core categories. The encoding was performed by assigning a key sentence to each of the responses to the questionnaires or focus group interviews according to the main idea of each paragraph. Following the constant comparison method these codes were grouped into categories. Each category was interpreted as a social representation and was expressed through a minimal expression (Singéry, 2001); which is a phrase that represents the global meaning and condenses the form in which subjects capture the represented object: what this object is for them and their position in relation to that reconstruction. This global meaning is constructed by the researcher and is the result of the entire
content of the representation, the point of reference on the basis of which the set of dimensions and cognitions is organized.
The students were identified with the labels An (with n being from 1 to 67). The En label identified either of the two interviewers in the focus groups (one of the interviewers was the author of this paper). I used a diagonal line between two words to note that two words have the same meaning from the perspective of students. In addition to the above, square brackets were used when two phrases or a phrase and a word have the same meaning. Thus, for example, daily/everyday indicates that for students the adjectives ‘daily’ and ‘everyday’ are equivalent, and apply/[put into practice] indicates the same meaning between the word ‘apply’
and the phrase ‘put into practice’. Such equivalency of meanings was identified in the focus groups.
The next sections show the social representations of a good mathematics teacher and the social representations of a good mathematic lesson that I identified in the analysis of data from the questionnaire and the focus groups. They also contain examples of student responses (provided in their responses to open-ended questionnaire). In the first social representation (A good teacher has knowledge and knows how to [teach it]/[transmit it]/[explain it]) I have also placed some examples of dialogues conducted in focus group interviews that allowed me the interpretation and subsequently built the categories identified as social representations.