GAN TECK
HOCK, SU HOWE
YONG, AHMED
SULEMAN
Jabatan Matematik, IPGKKB
abstracts
This paper shares some of our insights from a
collaborative project with a Mathematics teacher from an Orang Asli school in
the state of Kelantan. The main aims of
the project are: (a) identifying the learning needs of the Orang Asli pupils
in Mathematics; and (b) developing
indigenous pedagogy for Mathematics that matches the learning needs of these
pupils. This paper focuses on the use of
manipulatives on a class of Year Two pupils’ learning of Mathematics. Specifically, this study found that although
initially they were ‘reserved’ in their expression of emotions, the Orang Asli
children are capable of learning Mathematics in an activelearning environment. However, there seems to be a need to address
the issue of ‘readiness for active learning’ in order to increase their
efficiency in learning. On the question
of how to help these pupils to learn Mathematics, this study found that when
modelling a mathematically task to these children, the task need to be broken
into smaller tasks and each smaller task need to be modelled directly in the
exact sequence to the pupils.
Furthermore, the modelling of such tasks need to be carried out in a
consistent manner and care should be taken to avoid asking the pupils to handle
more than one ‘mental scheme’ at a time.
Finally, this paper concludes that education for Orang Asli children
should focus on their ability rather than their inability.
Introduction
The Universal Declaration of Human
Rights (United Nation, 1948) states that education is
a basic right of every child. In the
recent Education for All movement, UNESCO reaffirms the importance of
addressing the issue of discrimination in the education of disadvantaged
children, including indigenous children (UNESCO, n.d.). Thus, upgrading the quality of education
provided by the Orang Asli schools has always been a major concern of educators
in Malaysia (Bahagian Pendidikan Guru, 2004).
This study is a small effort towards increasing Orang Asli children’s
access to quality education in Mathematics.
The Project
Aims
This study is a collaborative project between three school
teachers and six Institut Pendidikan Guru
(IPG) lecturers to improve the teaching and learning of Mathematics in an Orang
Asli school in the state of Kelantan. The main aims of this project are: (a) identifying
the learning needs of the Orang Asli pupils in Mathematics; and (b) developing
indigenous pedagogy for Mathematics that matches the learning needs of these pupils. As such, the project involved a wide range of
instructional practices in the classrooms.
However, this paper only focuses on describing the effects of handson
manipulatives on a class of Year Two Orang Asli pupils’ learning of Mathematics. In addition, it also shares some critical
lessons we have learnt from interacting with the pupils.
The School
The Orang Asli school involved in this project is
located about 130 km from Kota Bharu and it is accessible by tarred road. However, the school is about 15 km off the
main road. There are 12 trained teachers
and seven supporting staff under the management of one headmaster. The school has six classes of pupils, from
Year One to Year Six, for the year 2011.
All the pupils are from the Jahai subethnic group of Orang Asli. The official enrolment is 129 pupils of which
119 of them are registered as hostel borders.
However, in reality, majority of the boarders will still go back to their
own house in the nearby village everyday.
Only those who stay far away from the school will stay in the school
hostel.
The Instructional Team
The instructional team involved with the study
described in this paper consists of one Year Two Mathematics teacher and three Mathematics
lecturers. The teacher is an experienced
Mathematics teacher who has 30 years of teaching experience. All three lecturers are experienced
Mathematics lecturers. Each of us has 32
years, 28 years and 11 years of experience in teaching Mathematics.
Implementation of the Project
This project started in February 2011 and was initially
intended to end in October 2011.
However, due to unforeseen circumstances, the teacher was only able to
attend one planning meeting held in the IPG, while the lecturer team was only able
to make three visits to the school. The
first visit was a oneday visit for the lecturer team to get to know the
teachers as well as the school environment.
The second visit lasted two days while the third visit lasted three
days. During the second and third school
visits, the instructional team worked together to explore ways to help the Year
Two Jahai pupils learn Mathematics. The
collaborative tasks of the team include (a)
determining instructional goals, (b) planning instructional activities, (c)
preparing instructional materials, (d) exploring instructional actions, (e)
observing effects of the actions, and (e) analyzing as well as evaluating the
effects of the actions.
Instructional Challenges
During the first planning meeting, we (the lecturer
team) were told by the teachers that, generally the Jahai pupils in the school did
not show much interest in classroom learning as they considered formal
education less useful in preparing them for a more comfortable life in the future
(Meeting Note, 20 Apr 2011).
During the first school visit, we observed that the
Mathematics lesson for the Year Two class was conducted with the teacher
spending time explaining and showing the use of tally marks to find basic
addition facts such as [6+2=8] on the white board as shown in Figure 1.
6 + 2 = 8

Figure 1. Using tally marks to find basic addition
facts.
In later part of the lesson, the teacher spent a
significant amount of time guiding the pupils practised addition individually. In
this practice session, the pupil concerned was first asked to pick two numeral
cards and put them on an additionsentence frame written on an A4 paper to form
an addition sentence such as ( 3 ) + ( 2 ) = (
). Then, the teacher guided the
pupils to put the corresponding number of pencils below each of the addends. Apparently, the pencils were used as counters
and the pupil was guided to count all the pencils in order to get the addition
fact (Video Recording, 28 Feb 2011).
Overall, this lesson seemed to suggest to us that the pupils were highly
dependent on the teacher’s instruction in order to perform the addition with
pencils as counters.
In another lesson on the first day of the second visit,
although the teacher tried very hard to guide the pupils to use their own
fingers to perform subtraction individually, we observed that the pupils were rather
passive in performing the task. It was quite
difficult to expect them to perform the subtraction proactively without the
teacher’s instruction. Again, they were
very dependent on the teacher’s instruction to perform the subtraction tasks with
the use of their own fingers (Gan’s Observation Note, 15 May 2011). Later, when the pupils seemed to have
difficulty in understanding mathematical sentences written with numerals and
symbol such as 54 – 2 = ( ), the
teacher tried to use drawing on the whiteboard as shown in Figure 2 to explain
the takeaway process of subtraction.
However, the drawing did not seem to help the pupils connect the
abstract mathematical sentence with its process.
Figure 2. Drawing used to show 54 – 2.
From the observations on these two lessons, it
appeared to us that (a) these pupils were highly dependent on their teacher,
and (b) the tallymark drawing did not seem to help them visualise the abstract
mathematical ideas. Thus, we need to
think of ways to make them more independent in learning and we also need to
explore the use of suitable concrete materials to help them learn Mathematics.
Instructional Actions and the Effects
Two
major instructional actions that the team decided to explore were (a) using
handson manipulatives to promote active learning and help the pupils visualise
abstract ideas in Mathematics, and (b) using peer interaction to reduce
dependency on teacher in their learning.
In addition, we also employed the ActionReflectionModelling (ARM) approach
(Dickerson et al., 2011) as the framework of our instructional explorations.
Exploration I
Manipulatives
are concrete objects useful in helping children to learn abstract ideas in
Mathematics (Tipps, Johnson & Kennedy, 2011). Children are only able to learn symbolic and
formal representations of mathematical ideas after they understand the meaning
of the ideas through the use of concrete representations (Reys, Lindquist,
Lambdin & Smith, 2009). As such, the
instructional team decided to explore the use of suitable base10 materials such
as the Dienes’ blocks to help the Year Two pupils to learn the idea of place
value in the second day of the second visit.
The ten and oneblocks and numeral cards were used to help the pupils established
the connection between the symbolic and concrete representations of 2digit
numbers such as 34 as shown in Figure 3.
34

Figure 3. Symbolic and concrete representations of
thirtyfour.
There were eight pupils in the class on that day. Su started the lesson by introducing the ten
and oneblocks to the pupils. Then, he showed
them how the blocks could be used to represent several 2digit number. Later, the pupils were divided into two small
groups to have more practice using the blocks to represent numbers. Su and the teacher each guided four pupils in
the small groups and Gan was the observer paying attention to the effect of the
use of the manipulatives. Enough sets of
Dienes' blocks were prepared to ensure that every pupil had the opportunity to
use the blocks.
The pupils appeared to respond positively to the use
of these blocks. They were attracted by
the blocks and were more active than before.
However, Su felt that somehow the pupils did not seem to show any
‘excitement’ although they were actively manipulating the blocks. It appeared that these pupils were very
reserved in their expression of emotion while learning in the classroom. Gan observed that some pupils were able to
show the correct representations for certain numbers but were generally not as
‘energetic’ as expected until a later stage when Su tried to make the pupils
competing among themselves to show the correct representations of 2digit
numbers using the blocks. Suddenly, the
classroom became very lively with the pupils showing more urgency and tried
their best in getting the answers (Gan’s Observation Note, 16 May 2011). Apparently, there is a need to find ways to
get these ‘reserved’ pupils to express their emotion freely during classroom
activities.
Exploration II
In the third visit, we tried to investigate the effect
of handson manipulative further. Marbles
and three plastics containers were used by the teacher to help the pupils
learned basic facts of addition with the counton strategy. There were nine pupils in the class for the
first lesson. The teacher started the lesson
by first writing the question [5+3] on the white board. He then counted five marbles into the first plastic
container, followed by three marbles into the second container. After doing so, he used the countall
strategy by taking all the marbles from the first container and counted the
marbles again into the third container.
Then, he took all the marbles from the second container and continued
counting the marbles into the third container.
As the teacher counted the marbles each time, he also asked the pupils
to count along with him. The modelling
was repeated for another two rounds.
However, in the third round, instead of counting all the marbles, the
teacher used the counton strategy to get the total number of marbles. In all these rounds, all the pupils were able
to count as the teacher manipulated the marbles (Gan’s Observation Note, 5 July
2011).
After three rounds of teacher’s modelling, the pupils
were given the opportunity to practise addition with the marbles and containers
in three pairs and one group of three.
Each pupil was given a question card such as 6 + 2 = ( ) and they were supposed to use the marbles
and three containers shared within the pair or group to obtain the answer. Some pupils were able to get the answer by
counton strategy but some pupils were very much dependent on the countall
strategy.
In the next day lesson, we tried to promote peer
interaction by encouraging the pupils to help and check their partners’ answers
in pairs. Marbles and containers were
still used to obtain the basic facts of addition. In doing so, we hoped the pupils would be
more independent in their learning with the manipulatives. Three interesting observations were made
during this lesson.
Observation I. The first observation was made when a girl
was trying to find the basic fact of
[5 + 6]. After putting five and six
marbles into the first and second containers respectively, the girl paused and
thought for a while. Then, she took the
bigger number (i.e. 6) of marbles instead of the first number (i.e. 5) and then
continued to count on five marbles to obtain the answer (Gan’s Observation
Note, 6 July 2011). This seems to
suggest that opportunity to interact with handson manipulatives in pairs has
provided the time and space for the pupil to ‘reflect in action’ during
learning. Apparently, this reflection
has helped the girl to realise that it is easier to start with the bigger
number instead of the first number in finding basic addition facts with the
counton strategy.
Observation II. The second observation was made when a pair
of boys was using the marbles and containers to find basic addition facts. One boy, A, was able to use the counton
strategy comfortably while his partner, B, was having difficulty with the
strategy. Then, Ahmed asked boy A to
help boy B in using the counton strategy.
However, boy A just took over the marbles and used them to find the
basic facts with the counton strategy himself without teaching boy B. Apparently, boy A was unsure of what he
should do in order to help his partner to learn the strategy. Seeing so, Ahmed told boy A that he did not
want boy B to just find the answer. Then,
he showed the act of teaching to boy A and told boy A that he should help his
partner to learn. However, boy A was
still not sure of what Ahmed wanted him to do.
Then, Ahmed asked him to stand at where Ahmed was standing in order to
teach boy B. This simple act of asking boy A to stand had apparently helped him
to understand the expected teaching role and he proceeded to teach boy B the
handson strategy successfully. This
seems to suggest that we need to model any expected behaviour as exactly as
possible in order for the boy to understand what is expected from him. In addition, given the opportunity and with
the help of the marbles, the two boys were able to correct their own mistakes and
helped each other to learn (Gan Observation Note, 6 July 2011).
Observation III. The third observation also involved boy A and
boy B. After several rounds of using the
marbles and containers, boy B had successfully used the counton strategy to
find basic addition facts. However, his
mastery of the strategy was not as strong as boy A. In one of the round, after putting all the
marbles from the first container into the third container, he then took all the
marbles from the second container and continued counting the marbles into the third
container with the counton strategy.
Nevertheless, he made a mistake in his counting and ended up getting the
wrong answer. Seeing him making the
mistake, Ahmed tried to correct his mistake by asking him to count all the
marbles in the third container. This
move of asking boy B to count all the marble again seemed to be a wrong
move. In the next round of question, boy
B reverted back to the countall strategy although he had successfully using
the counton strategy several times (Ahmed’s Observation Note, 6 July 2011). It appeared that boy B could only focus on
one mental scheme at a time. When he was
asked to count all the marbles, this countall mental scheme had disturbed his previous
counton mental scheme and hence caused his reversal of mental scheme.
Discussion
During all the school visits, we observed that the
Jahai pupils are happy and joyful kids when they are outside the classroom
during recess time. However, initial
observations in the classroom showed that they were very much reserved in
expressing their emotion and were quite passive physically. In the three lessons with the use of
manipulatives where we ensured every pupil had the opportunity to use them in
handson activities, we began to see smiles on their faces in the classroom. They were able to interact more freely with
the manipulatives as well as their peers.
This seems to suggest that the use of manipulatives and peer interaction
is able to attract the Jahai pupils’ interests in learning Mathematics and
bring joy to their formal classroom experience.
Our journey in understanding how to help these Jahai
pupils in their formal education is still at its infant stage. This paper was written based on our
experience interacting directly with the Jahai pupils in four Mathematics
lessons. With this limited experience,
we had observed some encouraging effects of using manipulatives to learn
mathematics among these pupils. However,
it was not as easy as we expected. Much
more need to be explored in order to develop more effective way to help these
pupils learn. We believe we should be
able to bring these Jahai pupils to a greater height in their learning of
Mathematics if we are able to have an indepth understanding of their unique learning
needs. One of these needs is their
readiness for active learning. Besides
getting familiar with the manipulatives, we also realised that these pupils
need to understand their expected roles in active learning in order to learn
efficiently in an activelearning classroom.
This understanding is difficult to achieve by simply telling them the
roles. In fact, there seems to be a
critical need to provide more opportunities for them to experience active
learning in a less formal environment. Many
cases of classroom teaching involving indigenous children have also indicated
that a broad base of informal experience with manipulatives is crucial for
these children to learn Mathematics efficiently (Hafiz, 2011; Gan & Miheal,
2007). Without a reasonable amount of
informal experience, it will be difficult for indigenous children to formalise
the mathematical ideas to be learnt. Therefore, future teaching involving
indigenous children need to allocate sufficient time for these children to get
familiar with manipulatives that will be used in their learning of
Mathematics. Our ideas of achieving this
goal are (a) allowing them to play with the manipulatives at the initial stage,
and (b) getting them to make manipulatives from familiar objects from their
environment.
On the question of how to help indigenous pupils learn
mathematics through the use of manipulatives, we would suggest the ARM approach
to be used in the following way:
(a) Modelling: Break down the mathematical task into smaller
tasks. Teacher uses manipulatives to
model each smaller task directly in the exact sequence to the pupils.
(b) Action:
Provide opportunity for every pupil to use the manipulatives to do the task. Pupils need to do this repeatedly in order to
strengthen their mastery of the task.
(c) Reflection: Put the pupils in small
groups and get them to check what their peers are doing. We strongly encourage peer interaction in
pairs because it maximizes pupils’ opportunities of interaction.
When modelling any mathematical strategy to these
indigenous pupils, we should be cautious of the possible interfering effects if
more than one strategy is presented together.
The finding from this project seems to suggest that a newer mental
scheme will tend to replace an earlier scheme when these children are learning
Mathematics. Thus, it is very important
to be consistent in the presentation of mathematical strategy to be learnt in
order to promote effective learning. As
such, the sequence, pace and time of modelling the strategy need to be
considered carefully based on the instructional goals.
Conclusion
Before
embarked on this exploratory journey, the picture painted for us on the Jahai
pupils was indeed focused very much on their disabilities in the formal classrooms. With such picture, we made a noble wish of providing
equal access to good education for these indigenous children. As we built up our courage to sail through
this short journey, we realised that it was not helpful to just sat and prayed
for a ‘big’ solution that hopefully could solve all the problems. Instead, we should be realistic and try our
best to respond to each instructional challenge that arises. Then, we will be able to take a step forward
each time a challenge comes by. We
believe that the cumulative effect of this continuous effort of taking one
small step at a time will indeed be magical in bringing desirable changes to
the education of indigenous children! Although
these Jahai pupils come from a very unique cultural background, we see no
difference between them and children from other communities in Malaysia in term
of their ability to learn. Although they
might have certain disabilities in learning, we also see that they do have a
lot of other abilities in learning.
Hence, we strongly believe that education for indigenous children should
focus on their ability rather than disability!
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