Last names of authors in order as on the paper
Good Mathematics Teaching from the
Perspective of Mexican High School Students
Gustavo Martínez Sierra
NationalPolytechnicInstitute. Mexico
This article reports on a qualitative investigation that identifies the characteristics of “good mathematics teaching” from the perspective of Mexican third-year high school students. For this purpose, the social representations, of “a good mathematics teacher” and a good “mathematics class”, were identified in a group of 67 students. In order to obtain information, a questionnaire was applied with open questions and focus groups with three or four students were organized. The answers to the questionnaire were analyzed by locating categories that consolidate a specific social representation.
Introduction
There is currently a growing interest in international research in relation to understanding the points of view of students and teachers regarding mathematics classes and the practices resulting from said views. This interest is based on the idea that classroom activity is a collaborative practice constructed with the participation of the teacher and the student, and cannot be split into teaching and learning.
Some of this research has been carried out in order to supplement research that has reported national norms and standards for student academic achievement and teaching practices in Australia, Germany, Japan and the US (Clarke, Keitel and Shimizu, 2006). This type of research is based on the view that the points of view of students, together with their practices, must be given the same priority as the research of points of view and practices of teachers. In this respect, Kaur (2008, p. 951) indicates that: "As learning is dependent upon the situations and circumstances in which it is engendered and the feelings these situations provoke in students, any attempt to improve mathematics teaching must take into account both teacher practice, student practice and their responses to each other’s practice".
Research that has focused on finding out the points of view of students and teachers in relation to mathematics classes includes research into what “good teaching”, a "good teacher”, a “good class", a "model class” or “effective teaching” is to them (Kaur, 2008, 2009; Li, 2011; Ngai-Ying, 2007; Pang, 2009; Seah & Wong, 2012; Shimizu, 2006; Yoshinori, 2009; Perry, 2007; Cai and Wang, 2010). These research papers have reported differences and similarities in the thoughts of teachers and students from different school levels and countries. The international project “Learner’s Perspective Study” (LPS) has focused significantly on analyzing classes given by competent teachers, according to the cultural context of each country, and not on the search for an average class. For LPS researchers, this decision appeals to what they think is interesting for a teacher: not so much knowing about a class that is supposedly average for one or several countries, but being informed about what very competent teachers do in their classes, in order to be able to have available successful strategies for their own classes (Clarke, Mesiti, Jablonka & Shimizu, 2006, p. 43).
Kaur (2008, 2009) carried out a series of studies in Singapore where 8th grade students were asked to describe the qualities of a "good mathematics class” and the “best mathematics teachers”. In those different studies, he found that students consider that a mathematics class is good when any of the following teacher characteristics are present (Kaur, 2009, pp. 343-346): the teacher “explained clearly the concepts and steps of procedures", "made complex knowledge easily assimilated through demonstrations, use of manipulatives, real-life examples", “reviewed past knowledge and introduced new knowledge”, "used student work/group presentations to give feedback to individuals or the whole class", “gave clear instructions, related to mathematical activities for in class and after class work”, “provided interesting activities for students to work on individually or in small groups” and “provided sufficient practice tasks for preparation towards examinations”. Descriptions by students of a good mathematics class include the teacher “moving from desk to desk” (Kaur, 2008, p. 960). Descriptions most frequently given of the best mathematics teachers were the following: “patient, understanding, caring/kind, good at mathematics, explains clearly, ensures students understand, and provides individual help”. Kaur (2009, p. 346) concluded that "good mathematics teaching in Singapore is student-focused but teacher-centered".
Other studies (Shimizu, 2006, 2009; Kaur, 2008, 2009) have been interested in comparing the practices and perspectives of students and teachers. In Japan, for example, Shimizu (2006) found discrepancies between the perceptions of students and teachers in relation to mathematics classes. He found that students identify significant events in classes differently to the teachers; and when they agree on a significant event, they do so for different reasons. In Shimizu (2009, p. 315-316), it is reported that Japanese students in public junior high schools in Tokyo identify a good class as being when ‘‘understanding’’ or ‘‘thinking’’ occur (“I can understand the topics to be learned”, the students say) and when there is a “whole class discussion”. Shimizu (2009, p. 317) concludes “that a “good” class is co-constructed classroom practices by the teacher and the students”.
In general, the research presented herein shows that the points of view of teachers and students regarding “good teaching", a “good teacher”, a “good class”, a “model class” or “effective teaching” vary in relation to multiple factors. Pang (2009) concludes, for example, “that good mathematics instruction may be perceived differently with regard to underlying social and cultural norms”. This is why various researchers have indicated the importance of carrying out comparative studies between different countries and social contexts. For example, the large international study reported in Clarke, Keitel & Shimizu (2006; p. 2) is based on the hypothesis that: “In the formulation of the Learner's Perspective Study that there might be sets of actions and associated attitudes, beliefs, and knowledge of students that might constitute culturally-specific coherent learner practices”.
In Mexico and Latin America, there has been no research into the points of view of students and mathematics teachers in relation to what constitutes “good teaching”. This research aims to start filling that gap that void by answering the following research question:
- From the perspective of High School students, what is “good mathematics teaching”?
There are many ways to carry out the conceptualization from the point of view of students and teachers. In this research, I have chosen to do this through social representations (Jodelet, 1986; Moscovici, 1976. Furthermore, due to the fact that a pilot study showed that the term “good mathematics teaching" was confusing to Mexican high school students; I decided to ask the following research questions in this paper:
- What social representations do a group of high school students possess in relation to a “good mathematics teacher”? and
- What social representations do a group of high school students possess in relation to a “good mathematics class”?
Theoretical Framework
There are many ways of theoretically conceptualizing the perspective of students and teachers. This research has chosen to do so using social representations (Jodelet, 1986; Moscovici, 1976), which are conceptualized as an expression 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 said knowledge.
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 certainty that the phenomena are real and that they possess specific characteristics. 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 objective (Guimelli, 1999). A social representation allows guiding the people 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. It is a guide for action; they guide 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: 12).
In other words, social representation is practical knowledge. It gives 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: 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 are, thus a form of knowledge created socially and shared, with a practical purpose that takes part in the construction of a shared reality for a social group, the function of which 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: 11).
Data and methodology
For this research, we used a High School educational study center at the National Polytechnic Institute in Mexico City, these institutions are designed as centers for technical professional training and as a pre-university institute. We decided to work with a non-statistical sample of 67 fifth-semester students from a Physics and Mathematics education center in Mexico City, which offers technical specialty courses in computer sciences, industrial maintenance and plastics. The general curriculum in the field of mathematics is composed of seven courses with five hours each class per week: Algebra, Geometry, Trigonometry, Analytical Geometry, Differential Calculus, Integral Calculus, Probability and Statistics. When we carried out the field work for this research, the students were studying the final part of the Integral Calculus course.
Work with fifth-semester students we pretend to understand the social representation of students with a certain level of success at school, reflected in their decision to stay at the educational center and then understand the social representations associated with the institution, based on the hypothesis that part of their success is due to the internalization of the representations of the educational institution in which they had spent more than two years of their school life. In order to communicate with the students, they were told that the purpose of their participation as informants was to carry out an “opinion poll” relating to mathematics.
Thestudy was carried out with a qualitative focus and included two techniques: a questionnaire and interviews carried out in focus groups. The purpose of these techniques was to generate written and verbal discourse, allowing us to find out the social representation. We started off on the basis of the idea that language, through discourse, contributes to maintaining and reinforcing the construction of social and material reality, since the language used in everyday life continually provides essential objectifications and provides the order within which said objectifications make sense and in which everyday life makes sense to people (Berger and Luckmann, 1996). Thus, discourse is a privileged conveyor of the social representations that circulate in the symbolic universe of the students.
The questionnaire was composed of open questions so as not to limit the answers of the participants and in order to allow them to openly express their opinions, reducing to a minimum the influence of the questionnaire. Two questions were asked in order to discover the social representation of “good teaching”: 1) in your opinion, what is a GOOD MATHEMATICS TEACHER? And2) in your opinion, what is a GOOD MATHEMATICS CLASS? In the questionnaire given to the students, the capital letters were used to emphasize the purpose of the representation of interest in each question.
A focus group defines the set of people who meet in order to interact in a group interview, which is semi-structured and focused on a particular common, shared theme (Morgan, 1997). A focus group aims to ensure that the individuals selected by researchers discuss and elaborate, based on their personal experience, on a social theme or fact subject to research. In this research, 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, meaning of words and phrases used by the students. For this purpose, questions such as and what, in your opinion, is a dynamic class?, why do you say that the class is boring?, what does it mean that a teacher knows how to explain?, etc.
Both the questionnaires and the focus groups were carried in approximately hour and a half each sessions in a classroom allowing the students to gather at tables. During each session, they worked in groups of twelve to fifteen students with two interviewers, none of which were teachers of the students. The mechanics were as follows: 1) Individual application of the questionnaire, 2) Creation of focus groups of between three and four students, 3) Collectively answering the questionnaire, 4) Commenting and providing more specific information in relation to the answers with the interviewers. The second, third and fourth part were videotaped.
The students were identified with the labelsAn (with n being from 1 to 67). TheEnlabel identified either of the two interviewers in the focus groups (one of the interviewers was the writer of this paper). We used a diagonal line between two words to note that two words or phrases have the same meaning from the perspective of students. In addition to the above, square brackets were used to establish semantic equivalence between two phrases and between a phrase and a word. Thus, for example, daily/everyday indicates that for students the adjectives “daily” and "everyday” are equivalent, and apply / [put into practice] indicates a semantic equivalence between the word "apply" and the phrase "put into practice". Such equivalency of meanings was identified in the focus groups.
Analysis and findings
The constant comparison method, as well as the grounded theory approach (Glaser and Strauss, 1967) were used to identify the different social representations that were expressed through minimum reasoning (Singéry, 2001); which is a phrase that represents the global meaning that summarizes 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.
In order to classify social representations, I used the percent identified in the answers to the questionnaires. The arrangement does not intend to rank or generalize the results presented to other groups or sectors of students; it is only used to organize the data. The collected information in the focus groups assisted to clarifying the meaning of the words, phrases and notions of common sense knowledge used by the students. All of the social representations are related to each other. Most of the students gave at least one characteristic of a good teacher and a good class in their descriptions.
Generally, students believe that a good class depends on it being given by a good mathematics teacher. This results in the discourse of the students being more extensive and different with regards a good teacher and more similar and limited in relation to a good class. As a result, there were found more representations of a good teacher than of a good class.
The following shows the social representations, using the minimum reasoning, that I identified in the analysis of data from the questionnaire and the focus groups. It also contains examples that show the type of answers provided by students in the questionnaires in relation to what they think is the characteristics of a good mathematics class and a good mathematics teacher. You can notice that the percentages do not add up to 100% due to the fact that social representations are not mutually exclusive.