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ORIGINAL ARTICLE
‘Excellent’ primary mathematics teachers’ espoused and enactedvalues of effective lessons
Chap Sam Lim • Liew Kee Kor
Accepted: 2 March 2012 / Published online: 16 March 2012
� FIZ Karlsruhe 2012
Abstract This paper reports a study that explored the
characteristics of mathematics lessons that were espoused
as effective by six ‘excellent’ mathematics teachers and
how they enacted their values in their classroom practice.
In this study, we define espoused values as values that we
want other people to believe we hold, and enacted values as
values that we actually practice. Qualitative data were
collected through video-recorded lesson observations (3
lessons for each teacher) and in-depth interviews with
teachers after each observation. At the end of the project,
stimulated-recall focus group interviews were used to
allow teachers to define the meaning of an effective
mathematics lesson as well as to recall and reflect on a
10-min edited video clip of one of their teaching lessons.
The findings showed that these teachers shared five com-
mon characteristics of effective mathematics lessons:
achieving teaching objectives; pupils’ cognitive develop-
ment; affective achievement of pupils; focus on low-
attaining pupils; and active participation of pupils in
mathematics activities. These values were espoused
explicitly as well as enacted in the lessons observed.
Keywords Effective lesson � Excellent/expert teacher �Primary mathematics teaching � Classroom practice �Espoused and enacted values
1 Introduction
This study was part of a larger, multinational research
program coordinated by the Third Wave Project. The larger
study was conducted by 12 teams in 11 different nations/
economies. The 11 nations/economies are Australia, the
Chinese mainland (2 teams from 2 different provinces),
Hong Kong, Japan, Macau, Malaysia, Singapore, Sweden,
Taiwan, Thailand, and the USA. Conducted over the years
2009–2011, the project aimed to optimize students’ math-
ematics learning experiences in school through the exam-
ination of effective lessons from the values perspective.
Considering the cognitive dimension as the first wave and
the affective dimension as the second wave, the value
perspective is considered as the third wave dimension for
understanding (mathematics) teaching and learning (see
Seah and Wong, this issue, for details).
In this study, we explored mathematics teachers’
espoused values of effective mathematics lessons and
examined through classroom observation the way they
enacted these values in practice. We investigated and
rationalized Ernest’s (1989) assertion that what was said
might not always be what would be done, and vice versa.
More importantly, we chose to observe mathematics les-
sons conducted by ‘‘excellent’’ primary school teachers in
Malaysia, because we assumed that these teachers would
deliver effective lessons more often. Also, we believed that
looking at effective lessons instead of effective teachers
alone might provide a more holistic classroom view, since
a classroom lesson involves both teacher and his/her
teaching, as well as other elements of the classroom such as
pupils’ participation, teachers’ use of resource materials,
etc.
The following section of this paper reviews the essential
literature on ‘Excellent Teachers’ in Malaysia, the socio-
C. S. Lim (&)
Universiti Sains Malaysia, Penang, Malaysia
e-mail: [email protected]
L. K. Kor
Universiti Teknologi MARA Malaysia, Kedah, Malaysia
e-mail: [email protected]
123
ZDM Mathematics Education (2012) 44:59–69
DOI 10.1007/s11858-012-0390-5
cultural perspective of classroom teaching, and distin-
guishes the differences between values and beliefs as well
as espoused versus enacted values. Section 5 describes in
detail the qualitative data collection method. The paper
then discusses the analysis of the results and ends with a
brief conclusion.
2 Background of the study
2.1 ‘Excellent Teacher’ in Malaysia
‘Excellent Teacher’ (‘‘Guru Cemerlang’’ in the Malay
language) in Malaysia is a promotion scheme introduced
by the Malaysian Ministry of Education in 1993 (Malay-
sian Ministry of Education n.d.). Under this scheme, the
award is conferred on a teacher who displays the following
characteristics:
(a) possesses high expertise, knowledge, and skills in his/
her subject area;
(b) exercises his/her duties and responsibilities in a
dedicated manner;
(c) is always motivated, particularly in the aspect of
teaching and learning;
(d) possesses at least 5 years of teaching experience.
Besides these four criteria, an ‘Excellent Teacher’ is
expected to possess competency in communication and
technology, contribute to academic research, and be highly
proactive and innovative in contributing toward upgrading
the quality of the nation’s education.
Teachers who are confident that they have achieved
these criteria can apply for the award. However, they will
have to go through a series of evaluations by their school
administrators, peers and pupils, as well as observations
and document inspection by the school inspectorates,
before the award is conferred.
2.2 Socio-cultural perspective of classroom teaching
Classroom teaching is a social–cultural activity that
involves close interaction between the teacher and the
pupils. de Abreu (2000) suggested the use of a socio-cul-
tural perspective that examines both macro and micro
contexts to investigate the social interactions in a class-
room. He referred to interactional settings consisting of
non-immediate activities such as mathematics practice at
home as the macro context, while the micro context
examines the immediate classroom interactions.
In this study, we draw upon literature featuring the
impact of teachers’ knowledge, beliefs, and values on their
classroom practices to examine the classroom interaction at
the micro level. In particular, we choose to adhere to the
framework developed by Fennema, Carpenter and Peterson
(1989, cited in Leu, 2005) that teachers’ knowledge,
beliefs, and values influence their instructional decisions,
cognition, behavior, and students’ learning. Subsequently,
we uphold Leu’s (2005) contention that, ‘‘…theoretically,
teachers utilize their knowledge and beliefs to make deci-
sions about what to teach and how to teach; and these
decisions are reflected in their instructional planning and
selection of learning activities and in their classroom
actions’’ (p. 177).
2.3 Beliefs and values
Beliefs and values are two affective constructs that are very
closely related. Kluckhohn (1962, cited in Seah 2003, p. 5)
attributed beliefs to the categories ‘true/false’ or ‘correct/
incorrect’ and values to ‘good/bad’. Raths et al. (1987)
offered a seven-criteria checklist to conceptualize values
which were subsequently condensed and categorized by
Bishop (1999) into four aspects: (1) the existence of
alternatives, (2) choices and choosing, (3) preferences, and
(4) consistency. Elements that do not satisfy these criteria
are classified as ‘beliefs’ or ‘attitudes’. According to
Clarkson and Bishop (1999), values are beliefs in action.
Kolberg (1984, cited in Bishop, 1999) contended that
‘‘aims or intended outcomes, means or teaching/learning
processes, and effects or actual outcomes’’ (p. 2) are three
key elements to reflect on when considering values.
In fact, teachers’ beliefs about the learning and teaching
of mathematics have been investigated in numerous
research studies (e.g., Barkatsas and Malone 2005; Nisbet
and Warren 2000; Pajares 1992; Perry et al. 1999). Also, it
is widely documented in literature that teachers’ beliefs
greatly influence their instructional practices and students’
mathematical thinking (e.g., Philipp 2007; Thompson
1992). Adding to these studies on teachers’ beliefs is a
gradual increase in the number of studies, which include
the influence of teachers’ values on their teaching practice.
These studies either focus on identifying and analyzing
teachers’ beliefs or values (Bishop et al. 2001; Lim and
Ernest 1997) or examine how teachers’ beliefs or values
are enacted in classroom practice (e.g., Bills and Husbands
2005; Leu 2005; Shimizu 2009). However, there are as yet
not many studies that examine how teachers enact what
they have espoused.
2.4 Espoused and enacted values
In this study, we define espoused values as values that we want
other people to believe we hold, and enacted values as values
that we actually practice. Espoused values are also considered
as public, whereas enacted values are implicit, which an
individual actually holds and uses in decision making.
60 C. S. Lim, L. K. Kor
123
Lim and Ernest (1997) pointed out that it was necessary
to address the intended plan, the implemented process, and
the attained outcomes when exploring values in mathe-
matics education. They recognized the importance of dis-
tinguishing between espoused and enacted beliefs of the
teacher. Ernest (1989) noted that some research studies on
beliefs have shown that what was said might not always be
what would be done, and vice versa. For instance, studies
by Cai and Wang (2010) showed that teachers’ espoused
beliefs were found to be not necessarily enacted in their
classroom practices.
According to Lim and Ernest (1997), enacted values are
harder to identify or to research into compared with
espoused values. Nevertheless, they argue that enacted
values are the more important, because these are values
that teachers communicate to their pupils through the
process of teaching and learning. In order to explore the
espoused and enacted values, we adopted Kolberg’s (1984,
cited in Bishop 1999) three key elements as a guide when
considering values in this study. The espoused values were
gauged at the intended (planning) level, while the enacted
values were gauged at the implemented (teaching process)
level, and finally we verified the espoused and enacted
values at the attained (outcomes) level.
2.5 Effective lessons
Teachers, teaching, and lessons are three inter-related
constructs. Recent literature on excellent teaching tends to
focus on effective teachers (e.g., Jahangiri and Mucciolo
2008; Lim 2009; Perry 2007; Walls et al. 2002; Wong
2007), effective teaching (e.g., Wilson et al. 2005; Wong
et al. 2009), or effective lessons (e.g., Kor et al. 2010; Perry
2007; Seah 2007; Wong 2007). Seemingly, there is an
association between high-quality instruction and effective
teachers. Muijs and Reynolds (2000) and Brophy (1996)
noted that teachers in high-quality classrooms were able to
apply multiple strategies in their teaching to produce
effective lessons that enhanced students’ cognitive and
affective experience.
Adding to the above, Lim (2009) pointed out that
effective mathematics teaching was very much dependent
on one’s definition of mathematics. She opined that
teachers’ views on school mathematics and mathematics
learning determined how they taught mathematics. For
example, if effective mathematics learning was viewed as a
collection of formulae and procedures, then most probably
the teaching methods would emphasize drill and practice.
Conversely, if effective mathematics was viewed as
enquiring into a structure of concepts and relationships,
then teaching for conceptual understanding, problem
solving, and mathematical thinking was likely to take
place.
Hence, this study aimed to explore the espoused values
of ‘Excellent Teachers’ for effective mathematics lessons
and how they enacted these espoused values in their
classroom practice. It is envisaged that the findings will
provide good examples for practicing teachers, in particular
novice teachers who would like to know how to enhance
their mathematics teaching.
3 Purpose of the study
In the larger study, the aim was to identify the character-
istics of mathematics lessons regarded as effective by six
‘Excellent Teachers’ and their pupils, and the extent to
which these values (if they exist) might be similar across
different cultural groups. However, in this paper, we focus
on the characteristics that these teachers valued as effective
in mathematics lessons, and how they enacted what they
espoused as effective in their classroom teaching.
4 Participants in the study
The study involved six mathematics teachers selected from
the most recent list of ‘Excellent Teachers’ released by the
Ministry of Education in 2009. Table 1 provides brief
background information on the six teachers. Teachers G
and K were male Indian teachers who were teaching in two
different Tamil primary schools. Teachers C and L were
female Chinese teachers teaching in two different Chinese
primary schools. Teacher Z was a male Malay teacher,
Table 1 Respondents’ profile
Teacher Z R C L G K
Gender Male Female Female Female Male Male
Race Malay Malay Chinese Chinese Indian Indian
No. of years of teaching mathematics 30 10 30 31 11 9
Year awarded with ET 1999 2009 2001 2008 2008 2008
Type of primary school Malay Malay Chinese Chinese Tamil Tamil
ET Excellent Teacher
‘Excellent’ teachers’ values of effective lessons 61
123
while Teacher R was a female Malay teacher, teaching in
two different national primary schools. Among these
teachers, Z, C, and L were experienced teachers with more
than 30 years of teaching experience, while R, G, and K
were younger teachers and newly designated as ‘Excellent
Teacher’.
Besides these six ‘Excellent Teachers’, the larger study
also involved 36 pupils. However, in this paper, our anal-
ysis only focuses on data collected from the participating
teachers.
5 Methods of data collection
This study employed an interpretative research approach to
capture the espoused and enacted values of effective teach-
ing manifested in the mathematics lessons taught by the six
excellent mathematics teachers. Qualitative data were col-
lected through (1) video-recorded lesson observations; (2)
in-depth interviews with the teacher after each lesson; and
(3) stimulated-recall focus group teacher interviews.
5.1 Observation of classroom teaching
We used two video cameras to capture the classroom
scene. One focused on the teacher, while another focused
on the pupils. Each teacher was observed over the duration
of three lessons. These lessons were chosen by the
respective teachers and they were given sufficient time to
prepare the lessons on any topic from Grade 3 to Grade 6,
which they believed to be effective and would like the
research team to observe. Each lesson lasted about 50 min
to 1 h. Table 2 displays the grade level and topic of each
lesson taught by each teacher.
At the time of data collection, mathematics was supposed
to be taught using English as the medium of instruction in all
types of primary schools. However, to reduce the language
barrier, we allowed the participating teachers to speak and
communicate in the language that was most comfortable to
them in their classroom instruction as well as in the inter-
views. As a result, we observed that Malay language was
predominantly used in national schools, Mandarin in Chi-
nese primary schools, and Tamil in Tamil primary schools.
Also, teachers and pupils often code-switch between mother
tongue and Malay or English language.
5.2 Teacher interview after each lesson
Immediately after each lesson observation, the teacher was
asked to reflect on whether they felt that the objectives of
their lesson had been achieved. In addition, we asked the
teacher concerned to indicate the characteristics of a
teaching episode within the lesson that he or she perceived
as effective. In total, 18 lessons and 18 individual teacher
interviews were video recorded for analysis.
5.3 Stimulated-recall focus group teacher interview
After the three lesson observations, we conducted a
debriefing workshop. In this workshop, we used a stimu-
lated-recall interview technique (Busse and Borromeo Ferri
2003) to allow these teachers to reflect on their lessons.
Viewing the playback helped the teacher to recall what was
in his/her lesson. The playback was interrupted by the
interviewer (researcher) at certain moments to give the
teachers an opportunity to express their thoughts about and
make connections to the immediate scene.
However, before showing the video clip, teachers were
asked to list the characteristics of the mathematics lessons
that they perceived as effective. After listing the character-
istics, the teachers were divided into three groups. Each
group consisted of two teachers from the same type of school
(Chinese, national, or Tamil), one researcher, and one
research assistant. Each group was then asked to view the
10-min video clip of one selected lesson of each teacher.
Viewing the video clip was aimed at helping the teachers to
recall and to reflect on what they had taught in the lesson, as
well as to identify and elaborate the reasons why certain
activities or parts of the lesson were valued as effective.
6 Data analysis
All the 18 observed lessons and 18 individual teacher
interviews were first transcribed and analyzed. We read
through the transcripts several times to identify the
Table 2 Grade level and topic
of each lesson taught by each
teacher
Teacher Grade of class taught Lesson 1 Lesson 2 Lesson 3
G Grade 6 Mixed operations Area Pie charts
K Grade 5 Percentages Mass Perimeter
C Grade 5 Multiplication Percentages Mass
L Grade 3 Volume Volume of liquid 3D shapes
Z Grade 4 Fractions Division Money
R Grade 4 Length Perimeter Volume of liquid
62 C. S. Lim, L. K. Kor
123
common themes that emerged from the data. From the
teacher interview transcripts, we identified a list of char-
acteristics that were espoused by the participating teachers
as effective. Likewise, we also identified another list of
common themes from the classroom observation tran-
scripts. However, the discussion of this paper focuses
mainly on analysis of data from the stimulated-recall focus
group teacher interview.
During this interview, two sets of data were collected.
First, teachers were asked to define what the characteristics
of an effective mathematics lesson were. The information
given was taken to be what the teachers espoused. From
their definitions, there emerged a list of common charac-
teristics. Some responses were explicitly expressed, for
example: ‘‘achieve learning objectives is one of the criteria
of an effective lesson’’ [Teacher K], so we categorized this
type of response according to the keywords as ‘achieving
learning objectives’. Likewise, other responses that contain
keywords—interest, enjoyment, or activity—were catego-
rized accordingly. However, some of the responses did not
contain clear-cut keywords. For these, we tried to interpret
them according to our understanding. For example, if a
teacher mentioned that ‘‘students can understand the
mathematical concepts that are taught by the teacher’’
[Teacher L], then we categorized it as ‘‘achieving teaching
objectives’’. The list of characteristics found in the teach-
ers’ definitions of an effective mathematics lesson is shown
in Table 3, together with examples of responses.
Second, teachers were asked to view a 10-min video clip
of one selected lesson of each teacher to identify and
elaborate the reasons why certain activities or parts of the
lesson were valued as effective. The elaboration and the
reasons given were used as the enacted values. Hence, in
the following discussion, we aimed to identify some
common characteristics in the list of definitions and link to
how the teachers enacted these characteristics in their
classroom practice.
7 Findings and discussion
7.1 Teachers’ espoused values of an effective
mathematics lesson
As discussed earlier in Sect. 6, we considered definitions
given by the teachers as their espoused values of an
effective mathematics lesson. Analysis of the teachers’
given definitions yielded a list of common characteristics
(Table 4).
Based on Table 4, three comparisons were made by type
of school, gender, and the number of years of teaching
experience.
Teachers from all three different types of schools
highlighted ‘‘effective mathematics lesson’’ to include
‘‘pupils’ cognitive development’’ and ‘‘active participa-
tion’’. However, there were no observed commonalities
when comparing by gender.
When comparing by seniority, the junior group with
\11 years of teaching experience (R, G, and K) appeared
to associate effectiveness with pupils’ cognitive develop-
ment, such as ‘‘pupils can answer question in class and in
exam’’ [Teacher R]. Perhaps, they were still very much
influenced by the definition given to them during their
teacher training or teacher education courses.
In comparison, the two most senior teachers (Z and C)
emphasized the affective achievement of the pupils. For
instance, Teacher Z defined an effective mathematics les-
son as ‘‘teaching and learning which can attract pupils’
interest or attention towards mathematics’’ while Teacher
C described it as:
Table 3 Teachers’ definitions
of an effective mathematics
lesson and examples of
responses
Category of characteristic Example of responses
1. Achieving teaching objectives R: ‘‘Achieve teaching and learning objectives’’
L: ‘‘Students can understand the mathematical concepts that are
taught by the teacher’’
2. Pupils’ cognitive development R: ‘‘Pupils can answer question in class and in exam’’
L: ‘‘Students are able to answer the questions correctly after the
lesson’’
3. Affective achievement of pupils—
interest and enjoyment
Z: ‘‘Teaching and learning which can attract pupils’ interest or
attention towards mathematics’’
C: ‘‘Students will become interested in mathematics’’
4. Focus on weaker pupils in
mathematics
C: ‘‘Weak students will improve after the lesson’’
5. Active participation L: ‘‘Students participate actively in classroom activities’’
C: ‘‘Lesson that students like to participate in’’
Z: ‘‘Activities that can cater to pupils’ interest towards math’’
K: ‘‘Must conduct group activities with clear instructions’’
‘Excellent’ teachers’ values of effective lessons 63
123
Students enjoy the lesson so much that they feel that
the time passes very fast or the duration of the
mathematics period is too short. When the lesson
ends, they say: ‘‘Why did the lesson end so fast?’’
Teacher C elaborated further in the stimulated-recall
focus group teacher interviews:
…to see that your lesson is effective, my way is make
them [the pupils] more interested in mathemat-
ics…let them be involved more, not only we teach
and they listen, let them have the chance to do on
their own.
This feedback could imply that as the teachers gain
more experience, their focus of ‘‘effectiveness’’ shifts to
affective outcomes rather than cognitive. Another conjec-
ture that can be made from teachers Z and C’s focus on
pupils’ affective outcome is that although mathematics is a
cognitive subject, having pupils achieve the learning
objectives is not enough to sustain their learning. However,
if pupils are interested and inspired to have a passion for
learning mathematics, then they will continue to learn
mathematics with full interest.
Observation and inference from the teachers’ responses
support Thompson’s (1992, cited in Perry et al. 1999)
finding that it is teachers’ teaching experience rather than
their training that influences their beliefs about mathe-
matics and mathematics teaching.
7.2 How teachers enacted the espoused values
of an effective mathematics lesson
To examine how these teachers enacted what they had
espoused as characteristics of an effective mathematics
lesson, we asked each teacher to review one of their edited
lessons and then identify and elaborate the reasons why
certain activities or parts of the lesson were valued as
effective. In the following sections, we discuss with
examples each espoused value listed in Table 4.
7.2.1 Achieving teaching objectives
Teacher L defined two teaching objectives for her selected
lesson on two-dimensional (2D) shapes as: (1) to
understand and use the vocabulary related to 2D shapes;
and (2) to describe the features/characteristics of 2D shapes
and to name 2D shapes. To achieve these objectives,
Teacher L explained her teaching steps as follows.
First, she showed some cards with various 2D shapes to
her pupils and asked a few questions such as ‘‘What is the
name of this shape?’’, ‘‘How many sides?’’, ‘‘How many
angles’’, as shown in the following video transcript:
L: This shape has how many sides and how many
angles?
Pupil: One, two… (The pupil held up a card of pentagon
shape and counted the number of sides with his fingers.)
L: OK, speak louder.
Pupil: One, two, three, four, five. (The pupil counted the
number of sides with his fingers touching the sides of the
pentagon.)
L: How many sides are there?
Pupil: Five.
L: Five sides. Well done.
L: How many angles?
Pupil: One, two, three, four, five. (The pupil counted the
number of angles, at the same time touching the angles
of the pentagon.)
L: OK, how many are there?
Pupil: Five.
L: Five sides and five angles.
(Translated from Chinese, SJKC, Teacher L, 3rd lesson
video clip)
As highlighted by Teacher L in the focus group inter-
view, she used questioning techniques to guide the pupils
to list the characteristics of the 2D shapes. In addition, she
stressed that pupils must be given ample opportunities to
observe the shapes, to count the number of sides and angles
(the characteristics), and only then name the shapes. She
believed that this process of concretizing the concepts
would help the students to understand and remember better
the characteristics without rote memorizing.
7.2.2 Pupils’ cognitive development
Four teachers (R, L, G, and K) espoused their definitions of
an effective mathematics lesson related to pupils’ cognitive
Table 4 Comparison of characteristics of an effective mathematics lesson given by the six teachers
Characteristics of an effective mathematics lesson Z R C L G K
1. Achieving teaching objectives 4 4
2. Pupils’ cognitive development 4 4 4 4
3. Affective achievement of pupils—interest and enjoyment 4 4
4. Focus on weaker pupils in mathematics 4 4
5. Active participation 4 4 4 4
64 C. S. Lim, L. K. Kor
123
development. To enact this value, Teacher K explained that
he would ask pupils to demonstrate their solutions to a
problem on the blackboard in front of the class. Then, he
led the whole class to review the solution written by the
particular pupil. After that, he pasted a pre-prepared solu-
tion card on the blackboard. According to him, this would
allow pupils to compare the teacher’s method of problem
solving with that of the pupils. If the pupils had done it
incorrectly, the teacher could correct their solution steps
immediately. In the long run, this approach would ensure
that pupils would be able to answer examination questions
successfully.
Similarly, Teacher G who strongly stressed cognitive
outcomes, such as ‘Pupils can answer the questions cor-
rectly’ or ‘Achieve the learning objective of the lesson’,
was observed to enact this value by ‘‘Always write down
the formulae, so that students can answer the questions
correctly’’. Teacher G also emphasized the importance of
writing the unit of measurement, since students were
required to write down the unit clearly in their examina-
tion. The following transcript of the video clip illustrates
the situation:
G: What is the area?
Class: 36 cm.
G: Square centimeters! Say again the answer.
Class: 36 cm2.
G: How to write square centimeters? Can anybody help
me?
G: Yes, Nanyenmi.
(The pupil came out and wrote the answer on the
blackboard.)
G: Is this correct?
Class: Correct.
G: Very good.
(SJKT, Teacher G, 2nd lesson video clip)
Furthermore, to ensure that his pupils could answer
correctly, Teacher G always encouraged his pupils to use
the fastest way to get a solution to a problem using a
shortcut where possible, or practice the easiest way to get
the correct answer, for example practice cancellation of
fractions. Therefore, his lessons appeared to be very
structured, procedural, and systematic. He always began
his lesson by giving a formula, working out the solution,
and then writing down the correct units.
7.2.3 Affective achievement of pupils: interest
and enjoyment
As we noticed from the comparison in Table 4, the two
senior teachers, C and Z, explicitly emphasized the affec-
tive achievement of pupils as one important characteristic
of an effective mathematics lesson. For them, an effective
mathematics lesson should be able to attract pupils’ interest
and attention toward learning mathematics, as well as
enjoying the lesson so much that they forgot about time.
To fulfill the espoused values of interest and enjoyment,
Teacher C stressed the importance of letting pupils expe-
rience the activity, such as weighing the given products by
themselves and recording the mass, as this was part of a
real-life experience that would help to create pupils’ desire
to learn more. For instance, in the lesson on mass, she
pasted a number of cards on the blackboard, on which were
written the names and mass of various products such as
sugar (1 kg) and oats (500 g). Then she posed some
problems, such as ‘‘The mass of sugar is how many times
the mass of the oats?’’ She encouraged her pupils to give
their answers freely. She wrote down each answer given by
her pupils on the blackboard as illustrated in the following
transcripts:
C: The mass of sugar is how many times the mass of the
oats?
Pupil 1: Four times.
C: Four times. (Teacher wrote the answer given by the
pupil on the blackboard.)
Pupil 2: Two times. (Teacher wrote the answer given by
the pupil on the blackboard.)
Pupil 3: Five hundred times. (Teacher wrote the answer
given by the pupil on the blackboard.)
Class: Haha… (pupils laugh)
C: Don’t laugh at him. You come. (Teacher asked
another pupil to give her answer.)
Pupil 4: Two times.
C: OK, two times.
(Translated from Chinese, SJKC, Teacher C, 3rd lesson
video clip)
Even though there was a pupil who gave a ridiculous
answer, ‘‘500 times’’, and the whole class laughed, Teacher
C still accepted the answer and continued to encourage her
pupils to give their answers. In this way, Teacher C created
an enjoyable classroom atmosphere for learning.
She also encouraged her pupils to pose their own
problems based on some given information. For example,
in the same lesson, she asked one pupil to pose a new
problem based on the information given:
Pupil 1: The mass of the red bean is green bean… (pause
for 2 s)
L: The mass of…Pupil 1: …is how many times the mass of the green
bean?
L: OK, good.
L: You know the answer? (Teacher asked another pupil
to answer the question.)
Pupil 2: Two times.
‘Excellent’ teachers’ values of effective lessons 65
123
L: Two times, is that right?
Class: Yes.
(Translated from Chinese, SJKC, Teacher C, 3rd lesson
video clip)
Teacher C believed that allowing pupils to express their
thoughts freely would not only enhance their interest in
learning, but also promote their creativity.
7.2.4 Focus on weaker pupils in mathematics
Both teachers, C and K, espoused that a mathematics les-
son would be considered as effective if not only the good
and average pupils, but also the weak pupils, can improve
after the lesson. Perhaps they were teaching pupils from
low-performing classes, as they appeared to show much
concern about the learning of the weak pupils. They talked
explicitly about how they applied different approaches to
teach this type of pupil. They differentiated their pupils in
terms of academic ability in determining the type of
teaching approach, or the number and difficulty level of
questions given.
For instance, Teacher C would give easier questions to
weaker pupils, explaining her reason as:
I give easy questions first, to build up their confi-
dence. Oh, I know already, if I immediately gave
them difficult ones, they could not do them. So usu-
ally I start from easy and then more difficult, then
give a more challenging question to let them think.
(Translated from Chinese, SJKC, Teacher C interview after
2nd lesson)
Besides the difficulty level of questions given, the
number of questions was also different for the academi-
cally better pupils and the weaker pupils. Teacher K said,
‘‘For example, when giving an exercise, for good pupils
I’m giving 10 questions. For the weak pupils, I modify to
three or four’’ (SJKT, Teacher K interview after 3rd
lesson).
Teacher C also added that some weak pupils tended not
to finish their homework at home. For these pupils:
I will try to get them to finish their work in class, also
not giving too much homework. Unlike the academ-
ically good pupils, I want them to practice a lot. [But]
the effect can be counterproductive [for weak pupils].
(Translated from Chinese, SJKC, Teacher C interview after
1st lesson)
Therefore, she tried to ensure that these pupils finished
their homework in class, by not giving them too much to
do. She reasoned that weak pupils could not be compared
to the better classes—if we wanted them to do more, it
might worsen the outcome.
In addition, Teacher C tended to be stricter and more
demanding with her academically better pupils. For
example, she would want from the academically better
pupils that ‘‘the work must be neat, lines must be drawn,
writing must be neat and tidy, unit must be perfect’’
(Translated from Chinese, SJKC, Teacher C interview after
3rd lesson).
However, Teacher C would treat the weaker pupils with
more patience, care, and encouragement, such that:
He is already very slow, and you harshly demanding
of him is pitiful. I gave him, he mastered, then slowly
step by step: writing must improve slowly, must use
ruler to draw lines, must go slow… All these must be
stressed, just that sometimes they don’t do them.
When they don’t I will keep on talking to him. [If] he
still does not take action, I will help him to draw, I
will use a red pen to draw, then he will not be like
that again—next time he will draw it himself.
(Translated from Chinese, SJKC, Teacher C interview after
3rd lesson).
In brief, we observed that to enact their espoused value
on weak pupils, these teachers opted for different approa-
ches of teaching and giving exercises to their weak pupils,
as well as showering these pupils with extra personal care
and motivation.
7.2.5 Active participation
For both teachers from the Chinese primary school, C and
L, an effective mathematics lesson should be a lesson in
which ‘‘pupils like to participate’’ (Teacher C) or ‘‘pupils
participate actively in classroom activities’’ (Teacher L).
For this to happen, both teachers involved their pupils’
active participation in the following ways.
1. Active student engagement through mediating materials
As we have discussed elsewhere (see Lim and Kor,
2010), the teachers seemed to play the role of a conductor
with their pupils as the members of an orchestra. To engage
pupils actively, Teacher C was observed to prepare a suf-
ficient number of cards or materials that would allow every
pupil to have an equal opportunity to participate in each
learning activity. For example, Teacher C gave each pupil
a small piece of grid card with 100 squares in it (Fig. 1).
Every pupil was asked to color the area which represented
a given fraction, for example 2/5 or 7/25.
Teacher Z had created an innovative tool, which he
called a ‘‘talking board’’ (see Fig. 2). The talking board is a
piece of 2 feet 9 1 foot cardboard painted with blackboard
paint on both sides. Teacher Z called it a talking board,
because it allowed the pupils to ‘‘talk’’ to the teacher and
other pupils through displaying their solution steps on the
66 C. S. Lim, L. K. Kor
123
board. Since each pupil was provided with a talking board,
we observed that every pupil was busily engaged in solving
problems given by their teacher. After the pupils had
completed their exercises, the teacher carefully selected
some pupils’ work and displayed it in front of the class.
The teacher then asked the pupils to compare the answers
to spot any mistakes. Hence, every pupil was constantly
engaged and actively participated in the class activity.
2. Group activities
Group activity is another way of encouraging pupils to
be actively involved. However, as we were reminded by
Teacher K, the teacher must make sure that every pupil
participates, as some pupils could be dominating the group
while others might be too passive. He explained that he
would assign a different role to each pupil, such as leader,
presenter, or note taker. When presenting their solution, he
would ask the whole group to come to the front. Figure 3
depicts this scenario.
To further ensure active participation, he awarded scores
to each pupil based on his/her contribution in the group
activity.
3. Questioning during lesson development
When Teacher L was explaining the concept of two-
dimensional shapes, she invited her pupils to participate in
the teaching and learning process through oral questioning
of the whole class, as well as inviting some pupils to come
to the board to write down the answer. The following
transcript illustrates the scene:
L: This shape has how many sides? (Teacher L pointed
to a semicircle.)
Class: Two.
L: Two. One is straight and one is curved. How many
vertices does it have?
Class: None.
L: No vertex.
L: OK, I let him (a pupil) try. Please help me write
down. A hexagon has how many sides and how many
vertices? Write the number of sides first then follow with
the number of vertices. (Teacher asked a pupil to write
the answer on the blackboard.)
Pupil 1: (Pupil wrote his answer on the blackboard.)
L: OK, one more. (Teacher paused to choose another
pupil to come out and write the answer. This was
because many pupils raised their hands wanting to
volunteer.)
L: OK, Lixuan. Faster.
Pupil 2: (Pupil wrote her answer on the blackboard.)
L: OK.
(Translated from Chinese, SJKC, Teacher L, 3rd lesson
video clip)
In a similar manner, Teacher C liked to challenge her
pupils with a lot of ‘‘why?’’ questions so as to get her
pupils to think and to involve actively. For instance,
Teacher (T): How do we represent � as a percentage?
Pupils (P) gave various answers such as 20/100; 50/100;
2/100. Later they decided to shout the answer as 50/100.
T: Why is it 50/100? I did not cut the card into 100
pieces.
P: � is 50, so multiply by 50.
T: Why multiply by 50?
P: So as to become 100.
T: Why do we need to make it 100?
(Translated from Chinese, SJKC, Teacher C, 3rd lesson
video clip)
Teacher C elaborated in the interview that ‘‘I want them
to think for themselves why certain things must be done in
certain ways.’’ She also emphasized that she preferred
pupils to understand the concepts rather than just memorize
the procedures.
Fig. 1 Hundred squares grid
card
Fig. 2 Talking board
Fig. 3 Group activity
‘Excellent’ teachers’ values of effective lessons 67
123
Analysis of the interviews and lesson observations of the
teacher participants on the whole showed that there was a
coherency between what the teachers said they valued and
that observed as being valued. The result of this study thus
provides a different response to Ernest’s (1989) contention
that what is said might not always be what will be done,
and vice versa. A focus on values in this study has enabled
us to avoid the difficulties associated with the inconsis-
tency in espoused and enacted beliefs reported by Ernest
(1989). The coherency also fits well into one of Bishop’s
(1999) four aspects of considering values, that is,
‘consistency’.
8 Conclusion
In this paper, we analyzed the definition of an effective
mathematics lesson given by six ‘Excellent Teachers’ of
what we considered as their espoused values. We then
analyzed further how they enacted (or not) these values in
their actual lessons.
The findings show that these teachers shared five com-
mon characteristics of an effective mathematics lesson:
achieving teaching objectives; pupils’ cognitive develop-
ment; affective achievement of pupils; focus on low-
attaining pupils; and active participation of pupils in
mathematics activities. These values were espoused
explicitly as well as enacted in the lessons observed, which
implicitly reflected the characteristics of an effective
mathematics lesson that they had defined earlier (see
Table 3).
The results also showed that teachers with more years of
experience in teaching espoused and enacted the value of
‘‘effectiveness’’ to affective outcome (such as, interest and
enjoyment in the lesson) rather than cognitive achievement
(such as, pupils can answer the questions correctly). As
such, we conclude that it is the actual teaching experience
rather than the formal training in teacher preparation that
may have more profound impact on teachers’ values about
teaching a good mathematics lesson.
In this study, we defined enacted values as values that
the teachers actually practice in the classroom. These val-
ues were implicit and extracted from two major sources:
teacher interviews and classroom observations. Since these
data were gathered from two different sources—teachers
and researchers—it is possible to receive two different
claims when judging the enacted values. These two groups
might have different frameworks and understanding of the
constructs, as well as holding different values. We
acknowledge that this is a limitation of our study and we
agree that there is a need to have synergy between what the
teachers say is valued and what the researchers judge as
enacted in the classroom. For this aspect, we triangulated
the data collected from the classroom observations (images
of characteristics of effective lessons as claimed by the
researchers to have been enacted) with the teacher inter-
view transcripts (as claimed by the teachers to have valued)
to check for consistency when analyzing the data so as to
minimize these discrepancies.
In highlighting the classroom practice of the six excel-
lent mathematics teachers, we hope to inspire more pri-
mary school teachers, in particular mathematics teachers,
to internalize the characteristics of quality teaching. In
particular, the results of this study could provide good
examples for novice teachers who are seeking to enhance
their mathematics teaching.
As teaching is a complex task, guidelines alone are
insufficient for teachers to visualize how to deliver an
effective lesson. We attest that role modeling the high-
quality classroom instruction of excellent teachers could be
an alternative and effective way to disseminate good
classroom practices to all teachers. Actual scenes captured
of events in a high-impact classroom can be treated as a
quick and succinct way to impart inherent knowledge of
good practices to classroom teachers. In this study, this
knowledge takes the form of consistent valuing of qualities
that facilitate student learning.
Acknowledgments The study reported in this paper was made
possible by the generous support from the Universiti Sains Malaysia
Research Grant (Account No.: 1001/PGURU/811117).
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