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ORIGINAL ARTICLE ‘Excellent’ primary mathematics teachers’ espoused and enacted values of effective lessons Chap Sam Lim Liew Kee Kor Accepted: 2 March 2012 / Published online: 16 March 2012 Ó 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 Á Excellent/expert teacher Á Primary mathematics teaching Á Classroom practice Á 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

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Page 1: ‘Excellent’ primary mathematics teachers’ espoused and enacted values of effective lessons

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

Page 2: ‘Excellent’ primary mathematics teachers’ espoused and enacted values of effective lessons

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

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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

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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

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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

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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

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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

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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

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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

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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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