GOOD MATHEMATICS TEACHING FROM THE
PERSPECTIVE OF MEXICAN HIGH SCHOOL STUDENTS
Gustavo Martínez Sierra
National Polytechnic Institute. 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. The information collected in the focus groups contributed to shedding
light on the meaning of the words, phrases and notions of common sense knowledge
used by the students..
INTRODUCTION
There is currently a growing interest in international research in relation to
understanding the views 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 research is based on the hypothesis that the views and practices
of students must have the same priority as the ones from 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 focused on students’ and teachers’ perspectives of mathematics classes
includes their meanings of “good teaching”, “good teacher”, “good class”, “model
class” or “effective teaching”. (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
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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 views 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
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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 views 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 a “good
mathematics teaching”?
There are many ways to carry out the conceptualization from the students and the
teachers’ views. In this research, I have chosen to do this through social
representations. A social representation is a stock of values, ideas, beliefs, and
practices that are shared among the members of groups and communities (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 does a group of high school students possess
in relation to a “good mathematics teacher”? and
 What social representations does a group of high school students possess
in relation to a “good mathematics class”?
THEORETICAL FRAMEWORK
The students’ and teachers’ perspectives of a good mathematics teaching can be
analysed from different theoretical framework. I choose to use 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 said knowledge. Common sense
knowledge is the people’s certainty that phenomena are real and that they posses
specific characteristics.
I assume reality as a social construction where people are social beings based on the
reality in which they live but they 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
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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 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, the participants are High School students’ at the National
Polytechnic Institute in Mexico City. We decided to work with a non-statistical
sample of 67 fifth-semester students based on the hypothesis that part of their at
school 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
The study was carried out with a qualitative focus and included two techniques: a
questionnaire and interviews carried out in focus groups of three and four students.
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
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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.
We used an open questionnaire because we wanted the participants to express their
opinions openly and reduce the influence of the researcher to the minimum. Two
questions were asked in order to discover the social representation of “good
teaching”: 1) in your opinion, what characterizes a GOOD MATHEMATICS
TEACHER? And 2) in your opinion, what characterizes 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 is a group of people 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 (Morgan, 1997). A focus group aims that the
individuals selected by researchers discuss and elaborate, based on their personal
experience, on a social theme or fact subject to research. Focus groups are an
appropriate method data collection when one is interested social representations
because they are based on communication and it is the heart of the theory of social
representations (Kitzinger, Markova & Kalampalikis, 2004).
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, the interviewer asked questions such as “In your opinion,
what is a dynamic class?, Why did you say that the class is boring? What does it
mean that a teacher knows how to explain?
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, two sets of fifteen students and a set of
thirteen students). The mechanics 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.
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 is 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
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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.
The social representations of a good mathematics teacher
A good teacher has knowledge and knows how to teach it/transmit it/explain it (43%)
Someone who perfectly masters mathematics and is able to easily explain it to someone
else (A47).
A teacher who knows how to clearly express their knowledge in front of the class (A7).
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Someone who […] has a command of the subject so that they can teach it well (A13).
Someone who gives you the knowledge in order to be able to solve exercises using a good
method (A4)
A good teacher explains step by step and does not leave doubts (26%)
Someone who knows how to explain something to you as many times as necessary,
someone who is patient, who knows the subject and masters it (A3).
Someone who provides good examples and provides a detailed explanation of each case,
someone who not only knows the subject you are going to study, but lots of subjects so
that they can answer any question (A53).
Someone who provides a detailed explanation of every problem and clearly answers
questions (A51).
Someone who does not leave doubts in the minds of their students (A11).
A good teacher is patient/ comprehensive (16%)
Someone you can talk to when you have a question about their class (A48)
Someone who is patient, accessible, understanding (A22).
Someone who gives their class in a patient, tolerant way, with an efficient level of
preparation (A30).
Someone who teaches the class in a friendly way, with pleasure, with dedication... who
masters the subject so that they can teach it well (A13)
A good teacher does not make classes tedious/boring and teaches in an
easy/simple/clear way (16%)
Someone who makes an effort so that students understand and doesn’t make the classes
difficult (A50).
Someone who does not get tired of explaining and does so in a clear way so that the
subjects do not seem so complicated (A52).
A teacher who does not get irritated and helps you, someone who explains the subjects to
be taught with pears and apples [basic explanation for small children] (A43).
A good teacher arouses interest in students (7 %)
Someone who arouses my interest to learn and to solve problems (A45).
Someone who teaches their class in such a way that students are interested (A38).
Someone who makes students interested in the subject, as well as allowing them to
understand and resolve it (A35).
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The social representations of a good mathematics class
A good class is dynamic / [is not boring] and you learn (67%)
When a teachers manages to teach you something, without making it boring or
monotonous (A22).
In which you are not bored and the knowledge you learn stays with you (A4).
When time goes quickly as, if that is the case, the class is not boring, which means that
you are making an effort or you understand the subject being discussed (A50).
In a good mathematics class, there are [good explanations]/ [step-by-step
explanations]/ [you are not left with any doubts] (26 %)
A class in which there are no doubts at the end because all doubts were resolved during
said class (A19).
When all doubts are resolved (A18).
When time is utilized to resolve all doubts in relation to the subject and the teacher
provides support (A25).
When everything is understood and there are no doubts at the end (A32)
In a good mathematics class, there are lots of exercises and student participation
(22%)
A good explanation of the subject with examples explained by the teacher teaching the
subject and exercises to answer, reviewing answers in order to see that we are doing
them correctly (A3).
When the entire classroom is very attentive, solving exercises in relation to the subject and
participating in the class (A15).
An interactive class where students solve exercises on the blackboard and the teacher
corrects mistakes and resolves doubts that arise (A64).
A class that is enriched with exercises and lots of participation from my classmates (A1).
CONCLUSIONS
This paper studied the points of view of Mexican high school students in relation to
good mathematics teaching, through the theoretical and methodological proposal of
social representations. The results reported herein will allow us to better understand
how students perceive life in the classroom, how they feel about their learning and
their perceptions of how learning can be improved. This data may be useful in the
current search to involve students and motivate them to study mathematics.
The conclusions of this research study have similarities and differences with the
conclusions of other studies that have examined what students consider a
mathematics class to be and what makes a good mathematics teacher. One of the
most common findings is that students emphasize that good teaching has much to do
with how the teacher explains things, as has been found in other studies (Kaur, 2008,
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2009; Shimizu, 2009; Murray, 2011). The explanation is valued by students because
they want to understand the process and concepts they are taught. Students consider a
good teacher to be someone with knowledge who knows how to transmit it, provide
step-by-step explanations and leave any doubts. The metaphor on which this concept
is based on the consideration of knowledge or the ability to solve problems as a
material asset or possession that one person may pass on to another, by explaining, as
if was an object or thing. So the students that took part in the study believe that
teaching is the same as giving and that good teaching depends on giving a good
explanation provide step-by-step explanations and leave any doubts. For the students
a good teacher is patient/comprehensive and does not make classes tedious/boring,
they teach in a simple fun way and arouse interest in students. In a good mathematics
class, you don’t get bored and you learn. There are good, varied explanations and lots
of exercises and student participation.
The social representations reported herein demonstrate the desires and expectations
of students in relation to the teaching of mathematics. They also demonstrate the
values that have been socially constructed around what happens at school; values
which represent the most appreciated and important aspects by students because they
are everything that is subject to their preference or choice. This is why understanding
the students’ value in relation to mathematics teaching allows us to draw closer to
levels of mathematical well-being (Bishop, 2012). In particular, in the group of
students whose representations are reported herein, I found that the desire of not
being bored significantly predominates, which indicates a high level of dissatisfaction
in the classroom experience. The fact that a limited number of students indicate that
not being bored is based on a positive attitude on their part shows us that, for
students, the idea that mathematics classes are boring is part of their nature and can
only be beaten with a good mathematics teacher capable of attracting their attention.
In my opinion, this research is a primary overview of the problem of good
mathematics teaching from the perspective of Mexican high school students and
teachers. In future researches, I would recommend asking the opinion of teachers
from Mexico and other Latin American countries in relation to the good teaching of
mathematics and contrasting those opinions with the opinions of their students.
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