Selasa, 28 Ogos 2012


Jabatan Matematik, IPGKKB
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 active-learning 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.
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
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 hands-on 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 sub-ethnic 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 one-day 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 addition-sentence 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 take-away 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 tally-mark 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 hands-on 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 Action-Reflection-Modelling (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 base-10 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 one-blocks and numeral cards were used to help the pupils established the connection between the symbolic and concrete representations of 2-digit numbers such as 34 as shown in Figure 3. 

Figure 3.  Symbolic and concrete representations of thirty-four.
There were eight pupils in the class on that day.  Su started the lesson by introducing the ten- and one-blocks to the pupils.  Then, he showed them how the blocks could be used to represent several 2-digit 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 2-digit 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 hands-on manipulative further.  Marbles and three plastics containers were used by the teacher to help the pupils learned basic facts of addition with the count-on 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 count-all 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 count-on 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 count-on strategy but some pupils were very much dependent on the count-all 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 hands-on 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 count-on 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 count-on strategy comfortably while his partner, B, was having difficulty with the strategy.  Then, Ahmed asked boy A to help boy B in using the count-on strategy.  However, boy A just took over the marbles and used them to find the basic facts with the count-on 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 hands-on 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 count-on 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 count-on 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 count-all strategy although he had successfully using the count-on 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 count-all mental scheme had disturbed his previous count-on mental scheme and hence caused his reversal of mental scheme.
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 hands-on 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 in-depth 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 active-learning 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.
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!
Bahagian Pendidikan Guru. (Jan 2004). Kertas makluman Projek Model Persiangan-Salinatan Bahagian Pendidikan Guru. Unpublished circular distributed to all Malaysian teachers’ training colleges.
Dickerson, C.; Binti Abdullah, A.; Bowtell,, J.; Graham, S., Jarvis, J.; Levy, R.; Lui, L. N.; Rawlings, B.; Su, H. Y.; Tan, B. M.; Thomas, K. & Warren, V. (2011).  Learning together through international collaboration in teacher education in Malaysia.  Report of a project to develop a Bachelor of Education (Honours) in Primary Mathematics.  Hertfordshire, Great Britain: University of Hertfordshire Press.
Gan, T. H. & Miheal, P. P. (2007). Fulfilling Children’s Learning Needs: How to help Penan pupils to learn Mathematics. Miri, Sarawak: Institut Perguruan Sarawak.
Hafiz Ismail. (2011). Life through my eyes: A teacher’s little steps towards perfection. Kuala Lumpur: MPH Publishing.
Reys, R., Lindquist, M. M., Lambdin, D. V. & Smith, N. L. (2009).  Helping children learn Mathematics. 9th ed.  Hoboken, NJ: John Wiley & Sons.
Tipps, S., Johnson, A. & Kennedy, L. M. (2011). Guiding children’s learning of mathematics. 12th ed. USA: Wadsworth Cengage Learning.
UNESCO. (n.d.). Ten things you need to know about Education for All. Available online at:
United Nations. (1948). The universal declaration of human rights.  Available online at:

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