HOW PRIMARY TWO STUDENTS CONSTRUCT THEIR UNDERSTANDINGS OF THE CONNECTION

BETWEEN MULTIPLICATION AND ADDITION

CHTEN LEE SHING

A thesis submitted to Universiti Malaysia Sarawak

as partial fulfillment of the requirements of the Degree

Master of Science (Human Resource Development)

Faculty of Cognitive Science and Human Resource Development

UNIVERSITI MALAYSIA SARAWAK.

1999

Declaration

No portion of the work referred to in the dissertation has been submitted in support of an application for another degree of qualification of this or any other university or institution of higher Icarning.

This study was made possible with the help, guidance, and support of Inany people. I cxtetld my sincerest thilnks and apprcciatio~l to:

Dr. Abflng Ahmad Ridzitan as my sopervisor.

Pn. Sah;lsiah Husin and Mr. Philip Nuli Anding ror lhcir advise and support.

Prof. Madya Dr. Muhalued Awang, for ~ntroduc~ng ine to cognltive task analysis and the method of verbal

protocol analysis.

The principals and teachers of SRB St. Thomas, SRK Merpati Iepang and SRK En. Buyong, Kuching

for the permission ;lnd helps given lo conduct the study in the schools.

The principals and teachers of SRB Yee Ting and SRK Ulu Ranan. Kanowit for the permission and help to do the pilot study.

Very special appreciation: To my friends Felicia, Sia, Can, Tingang and Nirmala for their help,

comments & suggestions.

Especially to my ex-principal, Medam Judy Wong Liong Yung Ibr her encouragement and support to take up this course and all the lecturers in IAB especially

Dr. Quek Bong Cheang for his guidance.

Dedication

This disserlntion is dedicated to my father,

the late Chin Ah Tl~iam,

Iny mum. Siaw Cueh Hoon

&t i i y brothers and sisters.

This study was conducted to explore how Priinary Two stitdents zonstrilctivcly connect multiplicalion to addition. The sainplcs used in this study consisted o r45 Primnry Two stuclents, 15 of them horn each of the three primary schools choscn for the study. The sulqects were divided into 15 groups with three persons in a group. A set oi' six ~nathzrnatical activities was given to each group lo solve. Verbal protocol, which is a conitnon method in cognitive task analysis, was choscn to use in this study, whereby the verbal communication and explanation given by the subjects were recorded using a cassette tape recorder. The audio-recorded data collected was analysed and triangulated with the observation done by the rcscarchcr plus the written work of the subjects. The coding scheme used was similar to that used hy Hassehrock and Prictula (1992) in their analysis of liicdical rcasoning. Frotn thc analysis, eight conceptual operators were identified which consisted oi' data examination, data exploration, data explanalion, solution generation, lnathctnatical operation, cvalu:ktion, retrieval, and suminarisation. At the same time, six knowledge states were obtained too; they were number, operation, counting, grouping, row and column and niultiplication jump. The findings also indicated that students did constructively connect multiplication to addition and had their own strategies and short-cut to solve niultiplication problems such as using repeat addition, square and tlie multiplicalion of certain numbers like two, f ive and ten as a starting point to find answers. A simplii'ied mental model was developed basedon all the conceptual operators and knowledge states identified in this study. All the subjects could do multiplication excellenlly. However, both the audio and written data collected showed that some students had misconceptions about multiplication. Given a multiplication question. they (students) could not differentiate between the number to be repeated and the number of repetitions. Further research could be done on individual subject to develop individual mental model especially focussing on the mental model of those who could not get correct answers. The research could also be extended to Form 4 and 5 students in the learning of additional mathematics which is an important subject for higher learning in science and technology. The findings in this study though was only indicative, could be useful to teachers teaching mathematics in the primary schools.

K(rjian ini dijrrlr~nkan ur~t r~k rlleneroka I~ngnimona pelrrjar-pelujar l lnrjah Drru nlcnrhina />erlrrrbungan. nntrrm pordarubnrr detrg(m penaml~nhun. Sompcl yung dipilih dnlum knjinn irri terdiri rlari 45 orang pelujur I lar jah Dlrn dinmna 15 orarrg dari setirrp tiga buah sekolah rcrrrlaluh yrmg dipilih. Subjek-.srrl>jek ter.sc/~rrt dibrrhagi kepada kur~rl~ularr dengan tiga orang dalnm satlr krimpulan. Masuluh matemrrtik ynng terdiri dari 6 set aktiviti dihcri kepada sctiap kumpulrr~r rrntuk ~li.scle.srri. Krredah prr~tokoi lisan yang mer~ipakarl srrlah satu kaedah yang biasa digunakan rlalrirn ana1i.si.s kogrritif tugasan digunakan dalarrr kajiun irri. Segala komunikasi secara lisarl dun penjelusan )lung dihcri olch subjek-.sril>jek sernasn men,yele.saikan nrasalah matorratik dirakarlr dcngan menggunrrkrrn perukum krrset. Data yung diperoleh secara audio ke~nudiot~ dianu1i.si.s 1~er.sanra rlerrgnn r~roklrrnrrrt yong rliperoleh melului pemerhutirrn pengkaji semasa tnenjr~lankan sesi raknnran rlrrr i j~cgn kerja hern11i.s subjek-srrbjek. Skemu pengkodan yang digunakan rlaluni kajinn irri ialuh sama dengun yang digunakan oleh Ha.sse/~rock dun Prietula (1992) rlalatn kajian mereku. Urtri nna1isi.s tersebut, didapati lapan operator konseptual iaitu penyemukan data. pcnerokaun data, perrjelasan duta, penjanaan juwapan, f~/~era.si mutematik, penilaian. mengingut kenrbali dun rumusan telah diperc~leh. Padu masa yang sama, enani keudaan petlgerahuon dikcnalpu.sti juga yang terdiri dari nombor, operasi, penghitu~igrm, pengelumpokan, baris dun lajur dun lompatan pendaraban. Hasil kajiun juga menunjukkan pelujar rlupat membina perhubungan antnra pendaraban dengan penamhahan derlgan menggunakan cara dun strategi tersendiri seperti penggunaan penambahan berulang, kurrsa duu dun pendaraban yang melibatkan nombor-nombor tertentu seperti duo, lima dun sepuluh sebagai titik permulrran dalarn pcnyelesaian masalah. Satu model mental yang ringkus dibina dengun nrenggunakan semua operator konseptual dun kcadaaan pengetahuan yang diperoleh dari analisis. Semua subjek holch melakukan pendaraban dengan baik akan [etapi data audio dun kerja hertulis subjek menunjukkun tanggapan salah subjek-subjek tertentu tentang pendaraban. Kajian lanjutarr holeh dijalankan untuk mengkaji model mental individual subjek dengan memberi tumpuan kepada tanggapan salah. Dicadangkan juga kujian dilanjutkan kepada pelajur T4 dun T5 dalam pemhelajaran matematik tambahan. Hasil kajian ini walaupun merupakan kajian indikatg ia boleh membantu guru yang mengajar mutematik di sekolah rendah untuk meningkatkan lagi proses pengajaran-pembelajaran.

,

Tahlc aI'Contc~~ts

Declar:~tian Dedication Acknowledgcmcnts Ahstract Abstruk Table ol' Contcnts List of Tahlc List 111'Figurc

CHAPTER ONE

CHAPTER TWO

CHAPTER THREE

Pagc 11 , . . 111

iv v vi vii X

X I

THE PROBLEM AND ITS SETTING

Introduction Background of the Problem Stalclnent of the Problern Significance of the Study Purpose of the Study Objectives Scope and Limitation of Study Outline of thc Study

LITERATURE REVIEW

Introduction Constructivism Constructivism and mathematics Knowledge Information Processing Cognitive Task Analysis(CTA) Verbal Protocol Analysis Mental Model Problem Solving Summary

RESEARCH METHODOLOGY

Introduction Operational Definitions Description of Research Methodology Activity One Activity Two Activity Three Activity Four Activity Five Activity Six Subjects Data Collection

CHAPTER FOUR

4.5

4.6

4.7

CHAPTER FIVE

5.0

Talllc of Contcnts

Page

Data Processing and Analysis Analysing Conceptual operators, Knowledge states and Knowledge conslrocted in Solving Mathematical Prohlems Metllodologicnl Assumptions and Limitations Summary

FINDINGS OF THE STUDY

Introduction Knowledge Constructed by thc Students Discussion Activity One Discussion Activity Two Discussion Activity Thrce Discussion Activity Four Discussion Activity Five Discussion Activity Six Othcr observations Analysing the Conceptual Operators Conceptilal Operators Data Examination Data Exploration Data Explanation Solution Generation Mathematical Operation Evaluation Retrieval Summarisation Analysing the Knowledge States Knowledge States Number Operation Counting Grouping Row & column Multiplication lump Verbal protocol-coding scheme for solving mathematical problems Mental Model for mathematical problem solving Summary

SUMMARY, CONCLUSIONS AND RECOMMENDATIONS

Introduction

Page

BIBLIOGRAPHY

APPENDIX A

APPENDIX B

APPENDIX C

5.1 Sumtnary 5.2 Conclusions 5.3 Recom~nendations for the Study

ANALYSIS ON OPERATION. KNOWLEDGE & KNOWLEDGE CONSTRUCTED

Group I Group 2 Group 3 Group 4 Group 5 Group 6 Group 7 Group 8 Group 9 Group 10 Group 11 Group 12 Group 13 Group 14

MATHEMATICS ACTIVITY Activity I Activity 2 Activity 3 Activity 4 Activity 5 Activity 6

LETTERS RELATING TO THE APPROVAL OF THE STUDY

Table 2.1 Table 2.2

Table 3.1 Table 4.1

Table 4.2

Tablc 4.3 Table 4.4 Table 4.5 Tablc 4.6

Table 4.7 Table 4.8

Table 4.9

Table 4.10

Taxonomy of general and specific task analysis methods Cornnlonly uscd colnhini~tions o l task analysis dala collection and data representation tnelhods Schedule of Data Colleclion Analysis of related knowledge constructed by students in Activity I Analysis of related knowledge constructed by studcnts in Aclivity 2 Distribulion of groups huscd on numbcr of corrcct answer Dislribution of groups ilnswer c~~rrcctly bascd on questions Distribution of types of mistakes lnildc by suhjccts Analysis of related knowlcdgc constructed by studcnts in Activity 3 Distribution of question correctly answered based on group Analysis of related knowledge constructed by students in Activity 4 Analysis of related knowledge constructed by students in Activity 5 Analysis of related knowledge constructed by students in Activity 6

Page 13 13

Figure I Figure 2 Figure 3 Figure 4 Figure 5 Figurc 6

Figurc 7 Figure X

Three memory systems in processing input A simplified form of mental model Multiplication jump Multiplication jump Conceptual operations used by students Verbal protocol-coding scheme for mathematical problem solving Simplified mental model for matllematical problem solving Conceptual operators and knowledge states for solving mathematical problems

Page I I 15 19 3 1 39 44

List of Figure

Figure I Figure 2 Figure 3 Figure 4 Figure 5 Figure 6

Figure 7 Figure 8

Page Three memory systems in processing input I I A simplified Corm of mental modcl 15 Multiplication,ji~n~p 19 Multiplication jump 3 1 Conceptual operations uscd by students 39 Verbal protocol-coding schetne for mathetnatical problem 44 solving Simplified mental model for mathctniuical prohlc~n solving 45 Conceptual operators and knowledge states for solving 47 mathc~natical prol~lctns

CHAPTERONE

THE PROBLEM AND ITS SETTING

1.0 Introduction By entering into the new millennium or the year 2000, we are entering into a century so-called the "Third Wave" or the "Information revolution" age. According to Phillips (1997) we are going to face three major challenges in the new millennium. The first challenge is information explosion. Phillips states that the new information generated in the last 30 years was more than those generated in the last 500 years. Yet, in the new millennium, the amount of information is expected to double every two years. The second challenge will he the fast changes taking place in the workplace. It is expected that an individual would change about five to eight johs in his or her whole working life just because a job becomes obsolete within a very short period (Phillips, 1997).The third challenge is the impact of the mass media. With the help of information technology, we are able to access an unlimited amount of information. Looking at the speed and how this knowledge is transmitted and shared, it is necessary for an individual to cope with new information and this demands skills and knowledge. An individual must be able to self-censor this information and he selective to utilise relevant information to its maximum.

From the three challenges, we can foresee that both knowledge-intensive and technology- intensive processes are going to characterise the new millennium. Knowledge and technology are going to he powerful weapons for nations. In fact, any nation can create its own wealth by relying on the exchange of data, information, and knowledge. We can anticipate more and more information being generated, johs becoming obsolete and technology advancing and becoming more sophisticated. All these changes are going to take place in an exponential manner both in speed and quantity, in multi-direction and all the time (Awang Had bin Salleh, 1997).

Now, how are we teachers going to prepare our students to "survive" and to excel in the new millennium? Our primary school students are going to be adults at the turn of the first decade of the new millennium. This human resource is going to be the most precious resource we have. These are the people who are going to help our nation to actualise our Vision 2020 in the year 2020 as mentioned by our Prime Minister Dr. Mahathir Mohamad (1997:4) "...nothing is more important than the development of human resources... Our people are our ultimate resource." The importancc of human resource can not he further emphasised.

Dr. Mahathir Mohamad proposed Vision 2020 in 1991. It consists of nine objectives or challenges that we Malaysians have to overcome in order to become a fully developed nation in our own mould by the year 2020. These nine challenges are multi-dimensional, comprising of the intellectual, developmental as well as spiritual dimensions. This is to ensure that in the process of becoming a developed nation we do not ignore any important or crucial aspects of development. To achieve our Vision 2020 in 20 years' time from now, we have to rely on our younger generation. Thus, to develop and educate this human resource is our first priority.

Education is not only important in providing knowledge and training to our people but it also equips them with the necessary information and skills to carry out their work as well as to guarantee a good life. The implementation of the Smart School is one of the seven flagships proposed to achieve Vision 2020. It hopes to produce citizens and workforce that our nation requires - a workforce that is well trained, flexible, creative and able to face the unknown.

1.1 Background of the Problem One of the nine challenges in our Vision 2020 is to create a scientific and technological society. We want to be innovative in science and technology. Instead of being followers, we want to be a world leader in both science and technology, a nation that possesses high technology and as a producer of technology (Sharifah Nor Puteh, 1997). However, Omar Abdul Rahman (1993) stated, currently, our country is facing a shortage of manpower to run and manage the national science and technology enterprise and to run industry productively and competitively.

The shortage of manpower is further worsened by the shortage of students studying in the pure science and mathematics stream as reported in Berita Ringkas Pendidikan Sarawak (1996). Schools are asked to produce more excellent students in diverse disciplines especially in science. mathematics, and technology. According to Shohtoku (1993:302), the education of boys and girls in mathematics and science at an early stage has a strong impact upon their aptitude for and interest in their choice of engineering and technology-related subjects at higher stages of education. It is reported by Shohtoku (1993) that the declining level in the achievement of mathematics in certain countries has became constraints in their efforts to compete internationally and even causes deterioration of their industrial strength and handicap their progress in research and development. From here, we can see the importance of mathematics and its influence in the progress of a nation.

Mathematics is a compulsory subject both in the primary and secondary school curriculum in Malaysia. Mathematics is needed and used everywhere in our daily life. We need it to operate the simplest tasks like paying our bills at the counter, to read time, and also to perform complicated tasks like running a computer programme and sending a rocket to outer space.

It is very interesting to note that even though mathematics is an interesting subject to most of our students, the maiority of them still has great fear for it especially during tests and examinations. According to Tham (1998), at the primary school or secondary school levels, many of our students have phobia with mathematics. Often, even our Form-Five students encounter difficulties in mastering basic mathematics. Most of our students do show and do have great interest in mathematics. However, interest without real understanding of what they have been taught is not going to help them obtain satisfactory result in mathematics. In fact, the performance of most of our students does not reflect the, interest that they have in this subject. It is not surprising that most of our students lose confidence in mathematics and eventually give up trying.

So now, why is mathematics so difficult to learn and understand? According to Mousley (1992), mathematics is still generally taken as a body of knowledge to be transmitted from one generation to the next by teachers rather than as understanding, which is constructed in the minds of learners. In Malaysia, the examination result is still the yardstick to determine the success of a school, its principal, and also its teachers. Teachers tend to rush through the syllabus to prepare students for examinations rather than spending more time to ensure that the students understand what is taught in the classroom. Drilling seems to be the common strategy used just to get a pass in the examination regardless of the interest of the students. The obtaining of correct answer is so much emphasised than the process of ohtaining it.

It is sad to see that we tend to produce students who may be able to reproduce or replicate what they are told in tlie same way they are told; but they will never be able to shape ideas or procedures lo their own use (Maloy. 1995). In other words, we produce students who simply "vomit out" wliatcver they were spoon-fed. That explains why many of our students seem to be

able to cope with their mathematics exercises in the classroom by applying algorithms that they had just learned. but fail to do so when new algorithms are introduced. During tests or examinations, they are confused and uncertain. This is because they are not sure of which algorithms to apply in order to find the answer to problems. In fact from this observation we can say that our students lack understanding what they are doing and have little comprehension of their meaning ( Bruning et al., 1995)

It is also found that traditional teaching method of mathematics is rigid. It is unattractive, lacks creativity, and consists of abstract presentations in the teaching-learning process. These are among the causes of poor mathematics performance among the Bumiputera students in Malaysia (Ravi all Munisamy et al. 1996). In the didactic teaching method, learning is passive and teacher-centred. The teaching is linear and the teacher is a sage on the stage and an ultimate authoritative dispenser of knowledge. The students have little or no choice of what they would like to learn. Skills are taught in isolation and the teacher decides the sequence of learning. The order these skills are presented may not necessarily match with the order in which the students need them. Mathematics is an abstract subject. T o be able to learn and understand it needs a high level of reasoning skill. Learning mathematics without understanding it results in students being unable to conceptualise the concept and this will prevent them from applying their mathematical skills to solve problems.

As such, teachers could try other teaching methods rather than rely solely on didactic teaching. The role of the teacher as a sage on the stage has to be replaced by guidance by the side. This makes learning an active process and students will learn by doing, the so-called hands-on and minds-on experience. Teaching is no more a linear-process. The needs and abilities of the students are taken into consideration. The teacher plays the role of facilitator, coach, and guide on the side. Skills would be taught in a relevant context and in the sequence needed by the students and not the teacher. The emphasis is on student-centred teaching where students learn things, as they need them, which are meaningful and relevant to them. A suitable alternative teaching method to go for is the constructivist teaching.

Constructivist teaching is looked upon as an approach that can replace didactic teaching. According to Robert Glaser, the director of the National Research Centre on Student Learning (NRCSL) housed at the University of Pittsburgh's Learning Research and Development Centre (LRDC), constrructivist teaching emphasises reasoning, the so-called fourth R. (cited in Maloy, 1995:l). The other three Rs are reading, writing, and calculating (arithmetic). Looking at the pace of how knowledge and the information are generated at the moment and the amount of information one is going to face, it is obvious that learning just how to read, write and calculate is definitely not enough. It is not enough for an individual just to be able to read a newspaper; fill a simple application form; write a business letter or perform simple mathematics accurately. In the new millennium, an individual is not going to spend his whole working life in the assembly line fixing pieces together following the self-explanatory routine instruction. The challenge and responsibility that an individual is going to face in real life, either personal or in the workplace, imposes greater intellectual demands and requires more problem-solving skills. A learner who has mastered just the three Rs would not be able to meet the demands of the job market or efficiently and effectively solve their daily problems.

So, i t is very crucial and important for our students to master the fourth Q the ability to reason actively wit11 new information and ideas. As stated by Maloy (1995:1), the Fourth R is [he ability to connect and galvanise disjointed bits of infbrmation and turn them to underslandiog. To do tl~nt, s tude~~ts must possess thinking skills, which are logical, rational, criticill, reflective,

and creative. They also need to have problem solving skills and most important of all be able to learn by constructing new knowledge on top of the knowledge they have possessed earlier.

1.2 Statement of the Probiem From the observation of the researcher as a student as well as a teacher, teachers still emphasise on memorisation in teaching mathematics. Students use rote learning to memorise the multiplication table. Drilling is another popular technique to help students to pass with minimum grade in the examinations. According to Good Br Grouws (cited in Kennedy & Tipps, 1994:14) teachers spend 70% of the time asking students to do individual work, very little time is used to find out how students reason out processes used to obtain answers.

Research in constructivism started in Malaysia since early 1980's. The main scope of research in constructivism focuses on the study of misconception and also on developing constmctivist- teaching strategies for the reconstruction of student knowledge of the misconception.

This study is trying to find out if Primary Two students construct their u~lderstandings of the connection between multiplication and addition when they are exposed to multiplication for the first time. The mathematical problems used are problems relating to addition and multiplication. Knowing how students relate multiplication to addition will help to enhance teaching-learning process of mathematics in primary schools.

1.3 Significance of the Study Reasoning by constructivist approach is a cognitive process where reasoning is done based on constructing new knowledge upon the old one. In primary schools, addition is taught before multiplication. Thus, we can say that students already possess knowledge of addition as prior knowledge before they learn multiplicat~on. By getting to know how students construct their knowledge relating multiplication to addition through solving prohlems, it can help teachers to understand how students think, why they make mistakes and what strategies and short cuts students use to make learning more interesting and meaningful.

To understand the cognitive process, verbal protocol is used which is a general method of cognitive task analysis. It enables us to find out the mental model, skills or operators, and knowledge states used to perform activities. The researcher will analyse the data, develop the verbal protocol-coding scheme, and later come out with the mental model of the subjects.

The knowledge obtained from the mental processes taking place in the mind of the students can enhance the teaching and learning process in schools. The knowledge is especially useful for new teachers and parents also. By knowing how our young children learn multiplication, it will l~clp our teachers Lo plan a better and more effective teaching strategy (Ausubel, 1968, cited in Sutlierland, 1992:79). Teachers will know better what their students need so teaching and learning become more relevant and meaningful. With that, teaching and learning process will be an interesting one, the school will be a happy place where the little ones look forward to go every day, and learning mathematics will no longer be a difficult and abstruct task.

1.4 Purpose of Study The purpose 01' the study is to explore how students constructively reason out the connection between inult~plication and addition while solving multiplication problems in mathematics. The ahility of the students to construct new knowledge that is tnultiplication on top of the knowledge of addition they had learnt earlier will make learning more meaningful and effective. The study also aims to lind out the conceptual operalors, knowledge states, verbal protocol coding scheme,

the mental model and knowledge constructed by students connecting multiplication to addition when solving mathematical problems.

1.5 Objectives This study has three main objectives. The first objective is to explore how students constructively reason out the connection between multiplication and addition. The second objective is to identify conceptual operators used and knowledge states possessed by students while solving mathematical problems. Finally, a general and simplified mental model is constructed based on the conceptual operators and the knowledge states identified.

1.6 Scope and Limitation of Study The findings in this study are limited to the Primary Two students from the three primary schools namely, Sekolah Rendah Bantuan St. Thomas, Sekolah Rendah Kebangsaan Merpati Jepang and Sekolah Rendah Kebangsaan Encik Buyong in Kuching Division.

The three schools were chosen due to their accessibility. Another reason for choosing these schools was to ensure verbal protocol could be carried out smoothly. In order to get a good verbal recording, students must be able to talk fluently and are not shy. It is expected that students from these three established town schools are more exposed to social interaction and can participate actively in the problem solving sessions.

1.7 Outline of the Study Chapter 2 will touch on literature review, which will review the concepts and ideas used in the study. The concept of constructivism, multiplication, addition, mathematics, and cognitive task analysis done by previous researchers will be discussed.

Chapter 3, on the other hand, will describe the methodology used. The chapter begins by discussing the selection of the mathematical activities followed by the procedure of getting entry into the three schools. A table on the actual schedule of data collection is also given. Discussion on how the study is conducted is presented too. By triangulating with the information obtained through observation done by the researcher while conducting the sessions and also the written work of the students, the audio-recorded data collected will be transcribed and analysed using the coding scheme similar to that used hy Hassebrock and Prietula (1992) in their analysis of medical reasoning. From the analysis, all the conceptual operators and knowledge states used and possessed by the students to solve the mathematical problems together with the knowledge constructed by students relating multiplication to addition will be identified.

In Chapter 4, discussion on the findings base on the verbal protocol analysis will be presented according to activity. The conceptual operators and knowledge states identified will be classified and categorised. Other relevant observations will be given and finally a general and a simplified mental model is constructed based on all the conceptual operators and knowledge states obatined.

Chapter 5 will summarise the findings of the study. Conclusions, implications, suggestions, and recommendations for further research will be included.

CHAPTER T W O

LITERATURE REVIEW

2.0 Introduction This chapter will talk about constructivism and its relation with ~nathematics. This will be followed by literature review on cognitive task analysis and verbal protocol analysis. Apart from that, knowledge states, conceptual operators, information processing, mental model and problem solving are also discussed in this chapter.

2.1 Constructivism It was reported by Maloy (1995) that the National Research Center on Student Learning in University of Pittsburgh had investigated the subject of reasoning, learning, understanding, and instruction for more than twenty years. Findings from study after study show that knowledge cannot be accumulated. Instead, knowledge must be constructed in order to be useful to the learners. In order to acquire knowledge, one has to construct it by oneself and that cannot be done through other people. The only way for one to construct new knowledge is by reasoning actively with new ideas and information. Reasoning, or the so-called fourth R, is the most important skill apart from reading, writing and calculating. It equips a person with skill to solve problems in the workplace and also to face the fast-changing unknown world. Maloy (1995:l) states that, without strong reasoning skills, learners will never be able to shape ideas or procedures to their own use.

According to Crowther (1997:2) "...basically defined, constructivism means that as we experience something new we internalise it through our past experiences or knowledge constructs we have previously established." Constructivism is an epistemology, a philosophy, and a theory of knowledge. The idea of constructivistn has been generated since 470-320 BC. The works of Socrates, Plato and Aristotle showed aspects of constructivist theory (Crowther, 1997). Crowther (1997) regards Piaget as the father of constructivism because his work provides the foundation for 111odern day constructivism. In Piaget's view, intelligence is made up of organisation of thoughts and the adaptation of new information. Being able to organise thoughts means a person has the ability to make sense and differentiate thoughts according to their importance and also has the ability to connect one thought with another. In the process of adapting new information, Berger (1978, cited in ~ rowthe r , 1997:4) says that one can either assimilate by adding the new information onto the cognitive organisation already possessed or otherwise one can adjust the intellectual organisation to the new information through accommodation.

Thc most well recognised principle of constructivistn is that learners actively construct their knowledge. Learncrs are no lnorc the passive receivers of knowledge from external sourccs such as teachers and textbooks. Knowledge is not a commodity to he transferred and transmitted to the leaners through com~nunication but it is a constructed entity of the learncrs. Leartlers do not absorb knowledge like sponge but are like growing trees, acquiring knowledge tl~rough their senses, through interacting with their environment by seeing, touching, hearing, smelling and tasting (Maloy, 1995).

The conception of constructivism fri~med by David Ausubel's theory of learning claims that learners can construct meaningful knowledge only in relation to their prior knowledge and experiences. This is in line with the view expressed by philosopher John Locke in the 17"' to

18"' centuries (cited in Crowther, 1997:3) saying that "no man's knowledge can go bcyond his experience."

Everyday we encounter countless new information and ideas. While confronted with new knowledge, we try to match it with our whole life experience stored in our memory. If the new knowledge matches with the knowledge and experience that we have already have, the knowledge is accepted automatically and it will integrate with old knowledge and experience. On the other hand, if the knowledge does not match and makes no sense, it gives rise to challenges and conflicts to the experiences and knowledge we already possess. It is said that we have met an obstacle. T o overcome this obstacle, we either use our past experience and knowledge to repair the mismatch or otherwise we figure out what we know and what we don't know and try to bridge or link between them. This is what Robert Glaser called the fourth R, reasoning skill (Maloy, 1995), which is the skill that enables us to construct new knowledge and thoroughly integrate it with our background knowledge. In other word, knowledge is used to build new knowledge. (Maloy, 1995:4).

When students encounter new knowledge in the classroom, they try to make sense of what is taught and from there, they build a picture of the world and knowledge of their own. The meaning they construct may be very different from what is intended by the teachers. Even among learners themselves, different meanings are attached to the same thing taught. Thus, it is very important that teachers take these differences of the students into consideration, since each student's knowledge is a unique complex of construction. The more a teacher recognises each student's understanding the more effective teaching is llkely to be. (Resnick,l989, Reynolds, 1992)

Looking at the trend in the new millennium, it is not surprising that constructivist schooling is very much encouraged by our Ministry of Education. According to Tham (1998) our Ministry of Education had spent considerable time and money to organise courses on constructivism for the teachers. Teachers who are concerned will be able to realist the importance of students using and work with the knowledge they acquire. They should find cffectivc ways to connect tlie new knowledge they are teaching with the knowledge of the students that they have gained from outside the school, that is, the knowledge that students already have. It is believed by Maloy (1995:7) that, such continuity will keep children's knowledge active, accessible, and useful to them.

In order to encourage better understanding of the students, teachers have to create situations for students to make their mental constructions, to support their own explanations, to evaluate, to communicate what they know and to apply what they have learned. Students should be given not only hands-on experience but also mind-on experience. In order to help the students to discover the principles and concept by themselves, teachers have to make surc that tlie content they are going to deliver is compatible with the current understanding of the students (Kennedy & Tipps, 1994).

Teachers can use thc following strategies to encourage students to use new knowledge learned in the classrclom: (Maloy, 1995)

a. Find out the students' existing knowledge; b. Allow students to learn through Fatniliar ways; c. Encourage students to have free discussions and reusoning; I . Use real life examples;

e Use language that students arc familiar with and understand; f. Provide rcal opportunitics lor students to practise what they have learned; and, g. Encourage studcnls to discover and invent.

It is very clear that an individual is going to face a greater intellectual demand both in private and in the workplace in the new millennium. Problem solving skill of a higher standard are required dealing with complex technologies, which are growing rapidly especially in the workplace. It is important for students to use their knowledge actively and learn to reflect and question what their texts are saying. While faced with a vast amount of information, they must bc able to verify the reliability of the information and to know where to look for information they lack. According to Maloy (l'995: 4) "...they should be able to apply a procedure, pursue a line of thought, abandon a task, or revise a problem-solving strategy." in order to excel in the new millennium.

2.2 Constructivism and Mathematics We often hear people say that mathematics is the study of numbers or the science of numbers. According to the Webster's New World encyclopaedia (1992:719), "...mathematics is the science of spatial and numerical relationships." It has its history dating back to the prehistoric time. It is believed that prehistoric human being knew how to count up to ten on their fingers. The study of mathematics was confined to numbers up to 500 BC in Egypt and Babylon. During the period of 500 BC to 300 AD, Greek mathematics was the main focus. T i e primary concern of Greece mathematics was geometry. With that, mathematics became the study of numbers and shapes. Until the middle of the seventeenth century, Newton and Leibniz invented calculus independently, one in England and the other one in Germany respectively. The scope of mathematics was Surthcr expanded to motion and change in calculus.

According to Dcvlin (1994:2), mathematicians studied "...the motion of thc planets and falling bodies on earth, the working of machinery, the flow of liquids, the expansion of gases, physical forces such as magnetism and electricity, flight, the growth of plants and animals, the spread of epidemics and the fluctuation of profits." From here, we can see that mathematicians study almost everything under the sun. With the different scopes of mathematics, it became difficult to find one common definition. Finally, more and more rnathematicians agreed that mathematics is the science of parterns.

In Fact, it is not difficult to realist that what mathematicians are investigating and examining are related to pattern, the numcrici~l patterns, patterns of shape, motion, behaviour, growth, and many others. What is very interesting is that (Delvin, 1994:3) "... those patterns can be either real or imagined, visual or mcntal, slaLic or dynamic, qualitative or quantitative, purely ulililarian or of little more than recreational intcrcsl." As a science of patterns, it gives us ;I way lo look ill the world, the physical, biological, sociological and cven the inner world in our minds and thoughts. It is also something that cxists everywhere and surrounds us all the time. There arc very few aspects of our lives that are not affected by it yet mathematics is an abstract, self' contained, self-consistent body of knowledge, which is completely divorced from the real world (Mosmrdini, 1988). Thc Fact that mathetnatics is abstract is because it is denoted in abstract symb(~lic nota~ions. A pcrson who is 11ot trained in the discipline is not able to understand its con~plexity and apprcciatc its hcauty. Mathematics can only come alive to those wlio can "sight- rc;~d" tlic sy~nhols and "sce" with t l ~ c "eyes of the mind". According lo Delvin (1994:4)." . . .M;~ll~e~n;~tics lives end hrealhcs it1 thc mind of the reader."

Most of the time, students show short-lived retention of what they had learnt. With the introduction of new algorithm, they forget the previous ones. Students need to have higher thinking abilities in order to reason and to conceptualise what is taught. Mathematical language is different from usual language and it has its own meaning which may be technical at times. It uses a lot of symbols, figures, formulae, signs, letters etc, which makes it even more difficult to grasp. Though mathematics is abstract, it is important because it teaches people to think and reason logically and rationally. Through logical thinking and reasoning, it enables learners to solve problems systematically and efficiently. It is a skill that is very much needed by a person to survive in the new millennium.

Mathematics cannot be absorbed in abstract form. It should be built based on the understanding of the students, and it should be introduced in stages depending on their experience (Jarko, 1990, cited in Tham, 1998:34). Mathematics is a hierarchical subject; the students need to master certain basic skills and concepts before the more difficult and advance mathematics is taught. Thus, it is obvious that students have to construct their new knowledge and concept by relating to conceptls they had learned earlier. So, the constructivist approach is one of the teaching methods, which is suitable to use to teach mathematics in schools.

Students start to learn mathematics from the very basic, that is starting from numbers and the fundamental operations like addition, subtraction, multiplication, and division. In Malaysia, children are first exposed formally to addition in kindergarten. As they proceed to primary school, addition is taught also but at a higher level in terms of difficulty and complexity. By the end of Primary One, the students have learned how to do addition with two digits within a hundred. Schools teach multiplication starting from Primary Two. In Primary Two, multiplication of two to five is taught in the first half of the year. Later, the multiplication of six to nine is introduced at the second half of the year. Thus, we can see that addition is taught before multiplication, that is, students have possessed the prior knowledge of addition before they learn the new knowledge of multiplication.

A very common practice used by teachers to ensure students remember multiplication is by asking the students to memorise the multiplication table. So, most of our students are drilled to memorise the multiplication table without making meaning of what they are reciting. Students who can memorise the multiplication table well can score high in examinations. However, rote learning is effective only for students who follow instructions closely and those who obey whatever the teacher says. With that, we are going to produce students who only know how to replicate what is taught without integrating their own personal opinions or ideas. This type of students may fail to compete in the international arena and may not be able to excel in the new millennium.

The use of constructivist approach in teaching mathematics is supported by the standards stated by the various national reports in United States in the 1980s. It was reported that mathematics classroom should be filled with students talking and thinking about mathematical concepts, discussing various alternatives to solve problems in different mathematical situations, make connections between concepts and knowledge and also communicating mathematically. Teachers on the other hand are expected to understand how students construct meaning and the difficulty they are facing in constructing new knowledge (Fuson, 1992).

2.3 Knowledge Generally, knowledge refers to mental facts, rules, procedures, skills and strategies required to perform tasks and activities. Knowledge acquisition is a fundamental component of intelligence. Cognitive psychologists have distinguished the types of knowledge stored in the memory into declarative knowledge and procedural knowledge (Bruning et al., 1995). Declarative knowledge is factual knowledge or "knowing what." It is related to recall things such as "The national capital of Malaysia is Kuala Lumpur" or "We had coffee this morning." It deals with objects, concepts, events, the relationship between objects with objects or concepts with concepts, and the interactions between them. Declarative knowledge is verbalizable which means it can be overtly declared or stated. Apart from being called the "what-knowledge", it is also sometimes known as explicit knowledge or the verbal knowledge." Declarative knowledge can be further divided into either conceptual knowledge or rule knowledge (Gordon, 1994)

On the other hand, procedural knowledge is the opposite; it is knowledge of "knowing how." By transforming information into action, it enables one to perform cognitive activities. Examples of procedural memory are like knowing how to operate a computer or riding a bicycle. Procedural knowledge is not necessarily a "high-order" knowledge. It can be simple and linked with declarative knowledge loosely. For instance, a young child who recalls her procedural knowledge to open a book or to unlock a door shows an example of loosely linked procedural knowledge with declarative knowledge. Procedural knowledge is usually "automated" whereby we do things without paying any conscious attention to what we are doing and we do not ask why we are doing it. For example, when we enter a class, we automatically look for a place to sit down and take out a pen and note pad to get ready for the lesson. Procedural knowledge is also known as implicit knowledge.

Researchers have discovered that students believe the nature of knowledge has a greal impact on their performance and also their critical thinking (Bruning et al., 1995). The distinction between declarative and procedural knowledge has its important appl~cation in the learning process of students. To learn about facts though is important, yet to be able to make use of what is learned is an achievement every student should aim for. Thus, teachers should combine declarative and procedural knowledge in the teaching-learning process. In fact, studies show that involvement of interaction hetween declarative or verbal knowledge and procedural aspects of knowledge result in successful adaptation in the learning process.

Both declarative or verbal knowledge and procedural knowledge are not independent inodes of functions, rather they supplement and enhance each other. For instance, in the process of solving prohlemns, the steps executed by the person arc guided by relevant declarative knowledge that is activated, and usually the acquiring and learning of a skill needs a person to rehearse statements or instruction in declarative form. Tulving and Squire (1972,1987 cited in Bruning el al., 199554) further categorises declarative memory into personal experience memory and general knowledge memory which labelled as episodic memory and semantic memory respectively.

2.4 Information Processing According to Hiiniachek (1990:190), information processing involves gathering and representing infonuation meaningfully (encoding); storing information, or retention; and recovering information when needed, or retrieval. The processes that take place in the br. C I I ~ ' are compotihle to that of a progwmmed computer except that a computer will only start to function allcr il is l'id wit11 data. To study learning, we have to look at three stages each located

separately in the memory system. The three stages are the sensory register, the working memory, and the long-term memory (Biggs &Moore, 1993).

a. Tlte sensory register. W e receive thousands of information through our five senses. No all the stimuli are relevant to us and we have to be selective. The function of the sensory register is like a shift; it retains only information that we are interested in. The duration where information is kept in the sensory register is very short, probably no more than a second.

b. The working memory has limited capacity to store information. This is where conscious thinking takes place. At any one time, we only attend to one major train of thought. In order to retain the information in the long-term memory, we have to process the information. We can either rehearse (repeating over and over), or by relating it to something we already know (coding). In working memory, information can be kept up to a minute.

c. Long-term memory. This is the place where all the processed information is stored. The information stored here will not be lost and can be recalled for a period as long as a lifetime.

The interconnection among the three memory systems is shown in Figure 1.

Input from environmen

Sensory register Working memory Long-term memory

Attending: very Processing: a more Storing: input now quick scanning of elaborate handling of processed and input for material to ensure available fbr recall importance (up to long-term retention (up to a lifetime) one second) (up to one minute)

Figure I: Three memory systems in processing input (Biggs &Moore. 1993:207)

2.5 Cognitive Task Analysis (CTA) Cognitive task analysis is an approach that is used to obtain data about cognitive skills that are embodied in the task. The usual mcthods used to collect data in CTA are interview, modelling techniques, and experimental procedures. It is especially suitable for tasks, which involve heavy problem solving, or decision-making components and also those, which require large amounts of knowledge to support performance or task that demand heavy mental workload.

The main purpose of CTA is to determine the cognitive structures and processes underlying problem solving. It provides insight into the cognitive processes in the mind. The knowledge structures of the task allow us to see how knowledge is organised and concepts are interrelated. CTA identifies skills needed for the task performance of the task. What is very unique about CTA is it can be used to determine mental models in task performance that was never found in any traditional methods. Another important feature of CTA is it also attempts to identify information-processing strategies. Usually a person doing the same problem could come out with several strategies to reach a similar solution. As categorised by Redding (1993, there are three aspects in CTA: determination of cognitive structures, recognising skills needed to perform a task and analyse the mental models.

The findings obtained from CTA help us to understand how a task is learned: how to speed up learning progress; to find out optimal performance knowledge and types of skills needed to perform a task. Such findings are especially useful to train a novice to become an expert and to improve teaching-learning process in schools.

However. CTA cannot provide all the necessary data about the mental process, so we need behavioural method to supplement data. Some methods for conduciing CTA are (Gordon, 1994):

a. Structural interviews; b. Verbal protocol analysis; c. Critical decision method; d. Conceptual graph analysis; and, e. Means and Gott's cognitive task analysis

There are three categories of task analysis methods:

a. General methods h r data collcction; b. General methods for data representation; and, c. Specific task analysis methods.

Though the firs1 iwo categories are generic mcthods, the most conlmonly used method is the third one, which is the speciiic analysis method. Table 2.1 shows the taxonomy of general and specific task analysis and Table 2.2 shows combinations of task i~nalysis data collection and datn representation methods commonly used.

General Methods for Data Collection General Methods for Data Representation

Documentation and equipment analysis Unstructured interviews Structured interviews Group interviews Sorting and rating Questionnaires Verbal protocol analysis Observation Task simulation with questions

- List and outline - Matrix (cross-tabulation table) - Structural network - Hierarchical network - Flow chart - Timeline chart

Table 2.1: Taxonomy of General and specific task analysis methods (Gordon, 1994)

Tnble 2.2: Commonly used combinations of task analysis data collection and data representation ~nelhods (Gordon, 1994)

Questionnaire Verbal Protocol

Ohscrvation Task Simulation with Questions

*

* *

*

*

*

*

* *

*

*

* *

*

*

From the table above, we can see that the data rcprescntation of verbal protocol can be in the form of list and outline, hierarchical network or flow chart.

2.6 Verbal Protocol Analysis Protocol analysis uses verbal report as data. According to Newell and Simon (1972, ctted in Luger et al., (1994:119]) "...the distinct advantages of using the verbal report data is that it occurred in the context of ongoing problem solving behaviour." With that, it allows the convergence between what was said with what was actually done by the subjects. There are a lot of pros and cons about this so-called direct research method and many people still have misconceptions about it. By giving the same problem to a number of subjects, data collected allowed Newell and Simon to see the invariant aspects of the solution process and avoid idiosyncrasies in an individual's protocol (Luger et al., 1994). Simon and Kaplan (1989:21) stress the importance of protocol analysis by quoting

... Its importance stems from the fact that it is one of the few methods in cognitive science that eathers data with sufficient temnoral densitv to test models that account for behaviour nearly - second by second (but not millisecond by m;llisecond)

Many methods can be used to generate verbal reports such as by asking questions, reporting mental processes, and asking the subjects to talk or think aloud. Dunker first introduced the thinking-aloud method in 1945 (cited in Muhamed Awang. 1992:95). Although data obtained from thinking-aloud method is not the isomorphic mental operation, it does provide a direct overall view or at least a partial view of the strategy used in problem-solving which cannot be obtain from any other methods (Hayes, 1968; Newell & Simon, 1972; cited in Muhamed Awang, 1992:95). This is further supported by Redding (1995) stating that, the talk- or think- aloud method is said to he able to provide the most accurate records about mental processes on their nature as well as sequence. Thus, protocol analysis is particularly useful for investigating problem solving and mental models (cited in Anding, 1997:16).

Verbal protocol can be carried out either concurrently, retrospectively, or prospectively. However, sometimes it is difficult to conduct concurrent protocols if the event takes place very fast or is too cognitively demanding (Hoffman, 1987), so retrospective protocol will be the other alternative (Ericsson & Simon, 1993).

We can do concurrent protocol by asking subjects to perform some tasks such as problem solving and at the same time asking the subject to think aloud. The verbal report is usually recorded on tape, later transcribed, coded from the transcript, and then analysed. Sometimes, retrospective protocol is done after concurrent protocol as a supplement to provide the missing ~nformation or to fill the gaps in concurrent protocol. While analysing the retrospective protocol data, caution should be taken because the subjects may reconstruct cvelits that did not actually occur while performing the task (Simon & Kaplan, 1989).

According to Lugcr et al. (1994:121), "...most cognitive psychologists now believe that many of the mental processes that influence our hehaviour are not accessible to consciousness." It is not likely that self-report procedures can reveal all the mental processes taking place in the subjects while performing the tasks. To overcome the shortcomings, Luger ct al. (1994) suggests not lo use this method at all, or treating it as a rich source hypothesis and later evaluating by using other techniques, or improving on the verbal report procedures. We can improve the verbal report procedures by monitor~ng the eye movements of the subjects. These eye tnove~nents are used as tool to support the verbal report.

2.7 Mental Model According to Qin & Simoil (1995:40S), "...By mental models we mean the structure of the subjects' knowledge about the world." A mental model is more stable, decper, systematic, and general than the images formed for a specific task in short-term memory. It provides a source of inforination. They are the bread and butter of cognitive psychology (Rogers, 1992). Johnson- Laird (1983) cmploys "mental model" for what a person prefers, and Newell and Simon (1972) refer to a mental model as a problem representation. According to Rogers (1992), mental model is the implementation of the differing knowledge bases enabling the operator actively to gather information, make inferences, anticipate outcomes and make plans for future decision-making. It is the internal construction of the external world to certain aspects, it can be manipulated to make predictions and inferences.

T o summarise, we can say that mental model is a functional abstraction about a job which provides a deductive framework for prohlem solving (Ryder & Redding, 1993). It contains and integrates conceptual knowledge, procedural knowledge, decision-making skills for reasoning, and strategies for problem solving. Mental model has become an important tool among training analysts and also in the system interface community. Its usefulness can be extended to school to enhance the teaching and learning process. As teachers, we have to accept the fact that a person may possess several mental models regarding a single concept or problem. Each mental model represents a different vlew.

Furthermorc, it is essential for a teacher to consider students' mental models, which are relevant to the knowledge going to be delivered while planning the lesson. With that, the teaching- learning process will be more effective, relevant, and meaningful. Similarly, in the preparation of computer aided software programmes for the Smart School, teachers and system interface designers can use the concept of mental model to design various simple and usable mental models for students who possess different abilities. Figure 2 is a simplified mental model that was used by Seamstcr et al. (1993).

Mental model

. Working Memory Where knowledge is used or processed in order to perform cognitive tasks

. Switching Mechanism Factors that call for a dif'erent set of procedurcs and strategies in long-tern~ memory to be used.

. Long-Tcmm Memory Where knowledge is stored

Figure 2: A simplified forin of mentill model (Seatnster et a]., 1993)

IS

2.8 Problem solving Problem solving is a mental activity of higher level. According to Newell and Simon (l972), the first step in problem solving is to determine the problem space and the next step is to determine a strategy to solve it. In ordcr to solve problems, one has to reason and by doing so, thc solver actually is expanding progressively his knowledge of the problem situation, continuing until he discovers the solution.

According to Hunt, (1994), at each step in reasoning a problem solver has to develop a mental construclion of the logical situation, and then react to the features of that mental construction. Johnson-Laird (1983) recalls this mental construction as a mental model. The complexity of a mental model is determined by the capacities of the immediate memory.

During problem solving, the solver actually is comparing his or her current state to the goal state and at the same time trying to find wayls to eliminate the difference between the two states. If the problem shows certain familiarity then solvers will rely to a great extent on previously memorised solution schemata. This is a short-cut through the problem space to lead to solution. According to Hunt (19941, properly applied schemata transfer the information-processing burden from immediate memory, where the human solver is weak, to long-term memory, where the problem solver is strong.

2.9 Summary This chapter discussed the use of constructivist approach in the teaching of mathematics and why CTA is used in this study. Mental model in problem solving is also presented here. In the following chaplcr, research methodology will be discussed which includes procedures of data colleclion and how analysis is carried out on the data collected.

CHAPTER THREE

RESEARCH METHODOLOGY . . ~.

3.0 Introduction This chapter describes the method used to collect and analyse data. Verbal protocol is used to collect data in this study. The audio-recorded data collected will be transcribed and later analysed referring to the coding scheme used by Hassebrock and Prietula (1992) in their analysis of medical reasoning. From the coding scheme obtained, knowledge constructed conceptual operators and knowledge states used by the students in the solving of mathematics problems are identified.

3.1 Operational Definitions In order to analyse the verbal protocols, we have to identify and interpret correctly a few entities as stated below.

(1) Knowledge states are units of knowledge used by the students while solving mathematical problems.

(2) Conceptual operator is the cognitive operator that generates knowledge required by the students to solve the mathematical problems.

(3) Addition is one of the four basic operations in mathematics. It is stated in the Academic Press Dictionary of Science and Technology (1992:40) that addition is the process of combining two or more numbers to obtain one equivalent quantity, which is the sum; the binary operation is denoted by the plus sign, +.

(4) According to Academic Press Dictionary of Science and Technology, (1992:1427) multiplication is an operation that, for integers, consists of the addition of a quantity a (the multiplicand) to itself as many times an there are units in another quantity b (the multiplier) to obtain a third quantity c (the product), that is an operator of repeated addition. This is symbolised by such forms as

( 5 ) Constructivistn. Constructivism is the study of how we construct our understanding and meaning through our experience and do not passively receive from the environment.

(6) Primary Two students are the second year students studying in primary schools. All the Primary Two students follow the same mathematics syllabus.

3.2 Description of Research methodology Verbal protocol was used in this study to obtain data. Six sets of mathe~neticnl activities consisting of addition and multiplication questions were given to the subjects to do. Verbal cominonication, which included explanations given by students while solving the mathematical problems, were recorded using a tape recorder. The verbal report data was then consequently

counting by research subjects, the researcher employed the use of observation technique and by relerring to the written work of the research subjects. The transcript is presented in Buhasa Ma1u)~siu. Analysis would focus on the knowledge constructed, conceptual operators and knowledge states involved in the problem-solving process.

A coding scheme on the mental processes of the students was developed and later adapted and used to determine the mental model of the students in solving mathematical problems. The knowledge states and the conceptual operators were then used to develop a coding scheme, which was then used to represent the mental model of the students. The coding scheme used in this study is similar to that used by Hassebrock and Prietula (1992) in the analysis of medical reasoning.

This study began by selecting mathematical activities. Mathematics books used in schools and mathematical activities downloaded from the Internet were the sources used. Nine activities were chosen and adapted based on the level of complexity and their relevance to a Primary-Two student.

A 2-hour discussion session was held between the researcher, the supervisor, and a mathematics teacher before the final choice of the mathematical problems was made. After going through all the activities and taking the time constraints and relevance of the activities into consideration, finally a set of six activities were finally chosen (Appendix B). Three activities were discarded because the problems emphasised mainly on memorisation and were redundant. The selected activities were arranged according to increasing levels of difficulty. A brief discussion on the activities is given below.

3.2.1 Activity One Activity 1 is made up of two parts. Part 1 consists of nine simple addition questions like 9+4, l o t 2 ... etc. Part 2 consists of nine simple multiplication questions. Example: 6x7, 3x9 ... etc. T h ~ s activity is sort of a "warming up" exercise to find out how good students are in both addition and multiplication, at the same time this act~vity also serve to build up students' confidence.

3.2.2 Activity Two The purpose of this activity is to find out if the students can relate multiplication to multiplication jump (repeating jump) or in other words connecting multiplication to repeating addition. While doing multiplication in Activity One, students may relate it to repeating addition. This activity enables us to see if students really know and are able to identify the number ofjumps and number of steps in a jump.

From herc, we can find out if students rcally understand the concept of repeating addition, that is, given 3x5, the students know 5 is the number to repeat and 3 represents the number of rcpetitions. Students are also asked to write a malhernatical sentence lo represent the ~nultiplication jump. Figure 3 below shows the example used in Activity 2.

2 + 2 + 2 Multiplication jump Mathematical sentence: 3x2=2+2+2=6

Figure 3: Multiplication jump

3.2.3 Activity Three Similar to Activity 2, Activity 3 is another activity used to see if students really understand repeating addition. In this activity, the approach used is grouping. Given a question, students are asked to group dots to represent the multiplication. For example, in order to show 7x2=14 correctly, it should consist of seven groups with two itemsldots in each group. An example of the above grouping is shown below.

3.2.4 Activity Four In Activity 4, an example, which is the predecessor to the question is given before every question. For example, 6x2=12 is given before the question 6x3. The purpose of setting questions in this way is to see whether students will make use of the predecessor to find the answer to the actual questions by adding.

3.2.5 Activity Five In this activity, students are tested on their abilities to split a number into repeating units and then summarise the repeating units into a mathematical sentence. For example:

With the six boxes given, students have to determine what number should be used as the repeating unit and later write a mathematical sentence to represent the repeating units.

3.2.6 Activity Six In this last activity, students are given 12 pieces of wooden blocks to construct a building, which must have equal number of rooms in each floor. From the building constructed, students are asked to relate the number of rooms in the building, which is 12, either using addition or multiplication. This activity tests the ability of the students to relate rows and columns to multiplication. For a building which consists of 4 floors and 3 rooms in each floor, the total number of rooms for the building can be presented as 3+3+3+3 (4 rows), 4+4+4 (3 columns). Then the repeating addition can be further related to 4x3 and 3x4 respectively.

3.3 Subjects The subjects in this study are the Primary Two students from primary schools. Fifteen students were selected from each of the three primary schools in Kuching District. Thus, a total of 45 students were involved in this study. All the 45 students were put into 15 groups, which meant there were five groups of students from each school. With each session roughly lasting approximately 1 hour 15 minutes, a total 17 hours 45 minutes of verbal protocol was recorded and collected.

The Primary Two students were chosen for this study because they had studied addition in Primary One and also in kindergarten. They have just been exposed to multiplication for the first time in the present year, where multiplication from one to five was introduced in the first semester and then multiplication from six to nine in the second semester. Thus, this is the most suitable stage to see how students who have just newly learned multiplication can reason out the connection between multiplication and addition.

Three primary schools were chosen for the reason that they are situated in the town and are long established. The schools are Sekolah Rendah Bantuan (SRB) St. Thomas, Sekolah Rendah Kebangsaan (SRK) Merpati Jepang and Sekolah Rendah Kebangsaan (SRK) Encik Buyong. SRB St. Thomas is a mission school with over six hundred male students. Both SRK Merpati Jepang and SRK En. Buyong are national type co-ed primary schools with a population of more than eight hundred and one thousand four hundred forty-nine students, respectively. The three schools share a common feature that is, the language of instruction used is the national language, Bahasa Malaysia. -The difference between SRB St. Thomas and the other two schools is that SRB St. Thomas is a semi-aided school whereas the other two schools are fully aided schools.

The class mathematics teachers of the schools were given the responsibility to choose the subjects as they know the students better. Fifteen students are chosen from each school. The students were put into groups of three to do the mathematical activities together. If a student were to do the activities alone, he or she would feel very tense, shy, and self-conscious. On the other hand, by putting two students together, they might have difficulty to come to an agreement if there was a disagreement. By having three in a group, it was assumed that the number was suitable whereby the students would talk and argue more freely, at the same time an agreement could be reached.

However, there is a possibility that certain students might dominate the whole session. In order to avoid that, the researcher sets two conditions in the choice of subjects. The first condition was that students must be able to talk freely, actively, and fluently in Bahasu Muluysiu. Secondly, the three students chosen to be in a group have to have more or less the same ability in mathematics. All the three of them could be from the high, medium or low level in terms of ability.

3.4 Data Collection Upon receiving the approval letter from the Buhugian Perancangun Dun Penyelidikan Dasur Pcndidikun (BPPDPIEPRD), permission to conduct the study in the related schools was obtained from the State Education Department of Sarawak (refer Appendix C). The schools were conlacted first through telephone call and a brief outline of the study was given to the principal and also to fix the date of first visit.

During the first visit, a brief explanation was given to the principal and the mathematics teachers about the purpose of the study and the procedure of data collection. Teachers were told about the conditions on how to choose the students. Opinions were shared and activities to be used were shown to teachers for comments and further improvement.

Altogether, there were 45 students. They were divided into 15 groups. Thus, there were 15 sessions of recordings with five sessions in each school. The researcher carried out two sessions, one before recess and another one after recess during every visit made to a school. It took roughly two to three days to finish all the five sessions in one school. The dates and time for data collection were suggested to the principals for their approval. Table 3.1 shows the schedule for data collection to all the three schools.

Table 3.1 Schedule of Data Collection

The students were called from the class either by the teacher or by the researcher. The schools decided the place where the researcher was going to conduct the session.

Before the students were asked to do the activities, the researcher began by telling them why they were called. They were assured that right or wrong answer was not important. What was important was that they must talk. They could discuss, argue, and negotiate to reach a solution. Students were asked about their background, ambition, their favourite subject, and their performance in mathematics in order to make them feel relaxed thus gradually forgetting the presence of the tape recorder, Then they were asked to start the activities.

The researcher chipped in once in a while to ask them to go on talking. Sometimes after a particular problem was solved, the subject was asked to explain how helshe reached a solution and why helshe solved it in such a way. After going through about half of the activities, the researcher allowed the students to go for a break in the company of the researcher. The activities were resumed after a 15-20 minutes break.

During the sessions, the researcher recorded observations that were related and relevant to the study and which can help to supplement information that cannot be recorded by tape recorder such as using fingers in counting. The written work of the subjects were collected and would be referred for unrecorded information which was not verbalised by the subjects.

3.5 Data Processing and Analysis The audio recording was converted into typewritten transcript. The transcript was segmented and encoded at the completion of a question or idea or at the completion of sentences or clauses or even pauses. Only knowledge that was relevant for the solving of mathematical problems would be analysed. The analysis focused on the knowledge states and the conceptual operators. At the same time, the transcript was also used to determine if students would construct their own knowledge to connect addition to multiplication while solving the mathematical questions.

3.5.1 Analysing Conceptual Operators, Knowledge States And Knowledge Constructed by the students in solving mathematical problems

The transcript was analysed based on operation (conceptual operators), knowledge (knowledge states), and knowledge constructed. In the coding scheme, the transcript was segmented according to conversation. What the researcher said was put in parentheses and was not coded. Only those said by students was coded (not in parentheses). An example of a portion of the coding scheme is shown below followed by explanation how the coding was done. Relevant information was underlined and only those information underlined was coded.

Transcript (Jadi sama dengan inilah, ada satu kumpulan dua bintang, dua bintang, dua bintang. Kalau enam darab tiga?) b... eh, lauan helas ... hitung lagi, satu lagi, dua, dua, dua, tiga ... (Kalau enam darab satu?) Isi satu saia dalam

(Enam darab tiga, lapan bela?. Sekarang dengan tujuh darab lima. ..lepas itu tujuh darab enam.) Tiga bclas.. .salah. ..eh.. . (la ~nesti lebih.. .) Emmt ~uluh dun (Empat darab lima.) Tok empat puluh dun (Very fast. Sekarang tengok aktivili ini, adik dikehendaki isi twang-ruang .. .) Maca~n ini ya, ruang-ruaog

:

Knowledge Constructed

Relate 6x3 to 6 groups with 3 items in each group

Relate 6x1 to 6 groups with I item in each group

Operation

Solution generation: Deduce Retrieval: Recall

Solution generation: Deduce

Retrieval: Rccall

Knowledge

Number Grouping Multiplication

Number Grouping

Multiplication

Transcript Sik tahu.. .tambahbah. ..dua belas, kosong ah. ..dua belas darab -...Oh! dua belas darab kosonc, kosoog, Dapatlah dua belas darab kosong, kosong.. . dua belas, satu. Dua belas darab satu ... dua belas darab enam ... dua belas darab satu ...dua belas darab tiga, dua belas kali. Aku tahu, aku tahu, tok dua belas tok, darab satu. Eh, dua belas darab satu, dua belas. Tokkan dua belas darab satu ... dua belas tambah. dua ouluh emnatbah, befullah. . .dm belas darab dua belas, seratus empat empat.. .Dah, cikgu. (Dah? Sudah setuiu?)

~~~

Mathematical Addition Relate 12 to 12x1=12 operation: Multiply Relate 24 to 12+12

Addition Counting Number

Dah (Ok dua puluh empat ... ) Dua, emnat, enam, lauan. senuluh, dua belas ... eh sik danat, satu, dua, tiga, empat, empat, emoat. ernoat. ~ . . . u d a h . . . t o k dua puluh, tok lima, na ia enam belas.. .- belas, emoat, dua ~ u l u h . Dua puluh, empat, dua puluh empat, betullah, betul.. .dah tadi, satu. dua, tica, emuat. lima. enam. Tok enam darab emnac. Dua puluh empat lagi. Empat, enam, lapan, sepuluh ... Dua, empat, la pan... eh. enam, lapan ... dua puluh, dua puluh empat. Cunlah. Tok hujung macam itu

Relate 24 to 4+4+4+4+4+4=6x4

Solution generation: Trial &error Evaluation: Disconfirm Mathematical operation: Add & count Summarisation: Summarise

LLl.E~1...eh,[21 l aaan 6cla.s ... hitung logi, sutu lagi, duo, duo, dua, tiga.. .

Secment I

[ I ] Tigu Explanation: Six groups of stars (2 each) were presented to show 6x2. When students were asked the answer to 6x3, they suggested having &(riga) slars in a group. They generated the solution by doing deduction from the groups presented. T h e operator used is Solution eeneration: Deduce T h e knowledge used is Number and Grouoing.

121 lapan Oelas Explanation: When scudents were asked what the answer to 6x3 was, they mentioned =(lapan beltrs). T h e students recalled the answer from their memory. T h e operator involved was Retrieval: Recall The knowledge possessed in order to recall the answer was Multiulication. T h e knowledge c o n s t r u ~ t e d is relatine 6x3 to 6 erouos with 3 items in each rrouu.

Seenient 2

Isi satu saia dalam.

Explanation: With 6x2, we have two stars in a group, then for 6x3 we have three stars. When asked about 6x1, the students suggested one star (is; satu saja dalam). Again, the students deduced the answer from the previous portion. The operator used was Solution peneration: Deduce. The knowledge used was Number and Groudng.

Segment 3

Sik tahu ... tambahbah ... dua belas, kosong ah ... dua belas tantbah kosong ... Oh! [Ildua belas durab kosonu, kosons Dapaflah dua belas darab kosong, kosong ... dua belas, satu. Dua belus darab satu ... dua belas darab enam ... dua belas darab satu ... dua belas darab tiga, dua belas kali. [2]Aku tahu. aku tahu, tok dua belas tok, dua belos darab satu. Eh, [3]dua bclas darab satu dua helas. Tokkan dua bclas darab satu ... [4]dua belas tambah, dua nuluh empathah. -...dun belas darab dua belas, seratus onpat empat ... Dah cikgu

[ I ] Pr [3] Twelve times zero is zero (duo belas darab kosong, kosong) and twelve times one is twelve (dua belus darab satu dua belas). Explanation: the students were recalling the multiplication of twelve. The operator used was Retrieval: Recall The knowledge possessed by the students was Multiulication.

[Z] I know, I know, this is twelve (Aku tohu, nku rahu, tok dua belas tok. ..) Explanation: students suddenly saw the answer The operator used was Solution ecneration: Trieeer. The knowledge used was m. 141 Twelve nlus.. . twenty four (dun belas tambah ... duapuluh empafbah). Explanation: the students were adding twelve to twleve. The operator used is Mathematical oneration: Add. The knowledge used was Addition. The knowledee constructed is relating 24 to 12+12.

[I lI)ucr. <,tllrlnt, enam, laurm. .srpuluh, dnn bclas ... eh [2]- satu, duo, tign, enpat, entpat, [3]ent11at, entoat. lapan ... udah ... tok duu puluh, tok lima, nu ia enani belus ... enant belas. entoat, duo puluh. duo auluh, fijur, duo puluh four, betullah, betul ... dah tadi. [4]srrtu, dua, tipa. emuat, lima, ennm. Tok enam darab emtxzt. Dua puluh empat lagi. [S]Entoat, enam, laoan, seoulrrh ... dua, entr~at, lanun ... eh, enant, lauan ... dua r>uluh, dua t~uluh empat. Cunlah. Tok hujunfi mucant itu

I I] Two. four, six. eirrht, ten, twelve ... (Dua, enlpat, enam, lupun, sepuluh, dua helas) Explanation: the studcnts were counting. Tlie operators used were Mathernnticnl operation: Count und Solution eeneration: Trial &error. Thc knowledge used was number and countinx.

[2] ...Cannot (sik dapat). Explanation: students discard the possibility of an alternative The operator used was Evaluation: Disconfirm

[31 Four. four. eieht ... sixteen. four, twentv: twentv. four, twentv four ...- empat, lapan ... enam belas, empat, dua puluh, dua puluh, four, dua puluh four) Explanation: students were doing counting and addition at the same time. The operators used were Mathematical oneration: Count and add. The knowledge used was Counting & addition.

[41 One. two, three. four, five, six ... (satu, dua, tiga, empat, lima, enam). Explanation: students were operating counting. e he operator used was ~a thdmat i ca i o~era t io i : Count. The knowledge used was Counting and number.

[51 Four. six, eieht. ten ... two, four, eieht ... (Empat, enam, lapan, sepuluh ... dua, empat, inpan ... enam, lapan ... dua puluh, dun puluh empat). Ex~lanation: students were trving with various alternatives to reach a solution. . - The operator used was Solution eeneration: Trial &error. The knowledge used was Countine and number

After doing the coding, the different operators and knowledge states were identified and listed. Then, all the different operators and the knowledge states were categorised and used to develop a coding scheme and later to develop a mental model. Knowledge constructed was also identified and presented according to activity.

3.6 Methodological Assumptions and Limitations Three assumptions can be made about verbal protocol analysis. It is assumed that students did express their knowledge and reasoning while doing the activities but they can only report something that they are attending to at a particular time. There are instances whereby at certain stages, the performance would become automatic and will not be reported resulting in the loss of data.

Another assumption is that verbal protocol allows a complete insight into how the students do the mathematical activities. It is also assumed that the same cognitive processes that bring about the overt responses and behaviour cause verbalisation. These assumptions (Redding 1995) imply that:

a. Information can only be reported if it is attended to or is being thought about or looked at;

b. Since only information that is attended to can be reported, as task performance becomes more automatic, the intermediate steps of a process become unavailable fbr verbal report;

c. Verbalisation of task performance is an added load to mental processing and acts to divide attention so that performance wily be inhibited; and.

d. Instructions on what to report have a direct effect on verbalisation performance

According to Ericsson and Simon (1993), thougl~t structure that comes into attention in the liiind also serves at the same time as the input to the processes for verbnlisation. Thus, there is nn

elapse of substantial time before the subject begins to talk even if the oral code for the structure is directly rctrievable.

3.7 Summary This chapter discussed how constructed knowledge, conceptual operators, and knowledge states were analysed. A brief discussion was given on the activities used in this study. The procedure of collecting data and the methodology used in analysing data was also described. A few assumptions about verbal protocol analysis were listed also.

CHAPTER FOUR

FINDINGS O F T H E STUDY

4.0 Introduction This chapter presents the findings of the study in two parts. The first part of the findings shows how students connect multiplication to addition while solving the mathematical problems. The second part of the findings discusses thc verbal protocol-coding scheme used and also a mental model developed from the coding scheme. The findings for the first part are presented according to activity. The detail analysis of the verbal protocol is presented in Appendix A.

4.1 Knowledge constructed by the students Six activities were given to students to solve. The students had to "think-aloud" while solving the problems and the whole session was recorded using a cassette tape recorder. The verbal protocol was transcribed and analysed and knowledge constructcd by the students was identified. In this study, only knowledge constructed related to multiplication was discussed.

Table 4.1 Analysis of related knowledee constructed by students in Activitv 1

I l l C) Recall multiplication table: 5 + l O + 15 4 2 0 4 2 5 4 3 0 d) Recall square and then do addition: Use squarc of 5 as a 1

Activity 1

I I I groui and then sum up 9+9+9=27 1

Problem 5x6

I

Knowledge C o n s t r e a) Repeat addition: add 5 for 6 times, 5+5+5+5+5+5=30 b) Repeat addition by doing successive addition of 5: 5+5=10,

10+5=15.15+5=20.20+5=25.25+5=30

I starting point, then add 5 to 5x5=25 to gct 30 3x9 I a) Groupinr and addition: Take 3 groups with 9 items in each

4x4 b) Kepiat addition: ~ d d - 3 for 9 times, 3+3+3+3+3+3+3+3+3=27 a) Repeat addition: Add 4 for 4 times, 4+4+4+4=16 b) Repeat addition by doing successive additions of 4: 4+4=8,

8+4=12.12+4=16

8x5

c) Multiplication and addition: Break down 4x4 into 4x2=8 and 4x2=8, then sum up 818 to get 16

d) Multiplication and addition: Usc 4x3=12 as a starting point, then add 4 to 12to get 16

a) Repeat addition: Add 8 for 5 timcs, 8+8+8+8+8=40 h) Repeat addition: Add 5 for 8 times, 5+5+5+5+5+5+5+5=40

I point, then add 7 I” 49 to get 56/ Repcat addition: Add 6 for six times, 6+6+6+6+6+6=36

SC qoest~o~~s were not in the activity but they were asked for further clarification andcl”l”l‘ccrI,cI,I purposes

Prohle”l5x7

‘x2

‘x6*3x2*

,x3*

‘XII*

5X8*

3x7*

7x.5*

7x8*

5x6*

Knowledge Constructeda) Reneat addition: Add 6 for 7 times, 6+6+6+6+6+6+6=42 -b) Recall square and then do subtraction: Use square of 7 as a

starting point, 7x7=49. then subtract 7 from 49 to get 42.(6x7=49-7=42)

c) Multiolication and addition: Break down 6x7 into 3x7=21 and3x7=21, then sum up 21+21 to get 42

d) Multiolication and addition: Break down 6x7 into 2x7~14,2x7=14 and 2x7=14 then sum up 14+14+14 to get 42

e) Recall multinlication table and then do addition: Use 35 as astarting point, then add 7 to 35 to get 42

t) Reoeat addition by doing successive additions: 7+7=14,14+7=21,21+7=28,28+7=35,35+7=42

g) Recall souare and then do addition: USC square of 6, 6x6~36,as a starting point, then add 6 to 36 to get 42

a) Rcneat addition: Add 7 for two times, 7+7=14b) Repeat addition by doing successive additions: 2+2=4, 4+2=6,

6+2=8,8+2=10,10+2=12,12+2=14c) Recall multi&ation table and then do addition: Use 12 as a

staring point, then add 2 to 12 to get 14Repeat addition: Add 9 for 6 times, 9+9+9+9+9+9=54Recall multiolication table and the” do addition: Use 2x12=24 as aStarting point, then add 2 to 24 to get 26Groupinc and addition: Take 4 groups with 3 items in each groupand sum up, 3+3+3+3=12Recall multiplication table then do addition: Use 9x10~90 as astarting point, then add 9 to 90 to get 99Recall the multiolication:6312--f18--f24--f30336342--f48,Recall multiolication table:7-+14+21+28+35-+42--f49-+56Recall multiplication table:..~

(5-+10-+15+20+25330335Recall square and then do addition: Use square of 7 as a starting

4.1.1 Discussion Activity OneActivity I c”ns$s of two parts. First part consisted of 9 simple additiorl questions and thesecond part 9 wnple multiplication questions like 1x1, 5x6, 3x9...etc. Though the questionswere direct, students came out with various ways to solve them too. Students did not face anyproblem doing both the addition & multiplication. The various strategies used by students tosolve the addition questions were discussed under other observations in scctiotl 4.2. ‘Tbc mostc”o~n~“n nWh”tl oscd hy students to solve the multiplication questioos wx repcat addition. Thesludcnts ~~CSCINC~ the rcpcat addition process in different ways,showed rcpcat addition in the following ways:

For example, in 5x6, students

il. WI addition: S-tS+S-+S+S+5=30

2K

b. Kcocat addition by successive add~t~ons: 5+5=10,10+5=15,15+5=20,20+5=25,25+5=30 c. Recall multinlication table: 5 -+ lo-+ 15 - + 2 0 3 2 5 3 3 0 d. Recall square and then do addition: Use square of 5, which is 5x5=25, as a starting point and

then add 5 to 25 to get 30

Part (a) is the usual way of doing repeat addition. However, the repeating addition is more suitable for 6x5 rather than 5x6, which should be 6+6+6+6+6.

In (b), students were doing successive addition by adding the 5s one after another.

In (c), students used their fingerslhands to indicate 5, 10, 15 ... etc until they got 30. We can say that students were operating counting and addition at the same time. It was observed that students seldom made mistake in doing counting and addition with five. At the same time, we can also say that students were recalling the multiples of five.

In (d), students recalled the square of 5 and used it as a starting point to get 5x6. 5x5 is 25 and then students added 5 to 25 to get 30. Students claimed that it was easier to remember a square regardless of any number. Apart from that, students also stated that multiplication of 2, 5 and 10 were easy to remember too. This is further discussed in Section 4.2.

Apart from doing repeat addition, students also used grouping to find the answers. For 3x9, they drew three circles (groups) with 9 items in each circle and then they summed up all the items, 9+9+9, to get 27.

Apart from the usual ways of doing repeating addition. Students show a few creative ways to find 6x7 and 4x4 respectively.

a. They broke down 6x7 inlo 3x7 and 3x7. The answer for 3x7 is 21, so from there they added 21 to 21 to get 42.

b. They broke down 6x7 into 2x7, 2x7 and 2x7. Then they added up 14+14+14 to get 42. They could even summarise 14+14+14 as 3x14=42.

c . To find 6x7, they recalled 5x7=35, which was a multiple of 5, then they added 7 to 35 to get 42.

d. To find 6x7, they recalled the square of 6, 6x6=36. then they added 6 to 36 to gct 42. e. T o find 6x7, they recalled the square of 7,7x7=49, then they subtracted 7 from 49 to get 42. f. T o find 4x4, they broke down 4x4 into 2x4 and 2x4, which was equal to eight each. Then

they added 8 to 8 to get 16. g. To find 4x4, they recalled 4x3=12, then they added 4 to 12 to get 16.

Tablc 4.2 Analvsis of related knowledge constructetl bv students in Activitv 2

Activity 1 Problern I Knowledge Constructed 2 ( 5x3 / a) Multi~lication iumn: Jumping from 0 - 3 5 + 10f 15. 3 . .

multiplication jumps with five steps in each jump. b) R e ~ e a t addition: Add 1 for 15 times.

1+1+1+1+1+1+1+1+1+1+1+1+1+1+1=15 c) Repeat addition: Add 5 for three times. 5+5+5=15 d) Repeat addition: Add 3 for five times. 3+3+3+3+3=15 e) Multi~lication i u m ~ : Jumping from 0 + 3 3 6 4 9 -3 12 + 15. 1

2x7 I a) Repeat addition by doing successive additions of 2: 2+2=4. 4+2=6,6+2=8,8+2=10,10+2=12,12+2=14

b) Reoeat addition: Add 2 for seven times. 2+2+2+2+2+2+2=14 c) Repeat addition: Add 7 for two times. 7+7=14 d) Multi~lication iumo: Jumping from 0 + 7 3 14. Two

multiplication jumps with seven steps in each jump e) Addition: 14=7+3+4 a) Reoeat addition: Add 4 for four times. 4+4+4+4=16 b) Addition: 4x4=5+5+6=16 c) Multi~lication iumn: Jumping from 0 -3 4 + 8 -+ 12 + 16. Four

multiplication jumps with four steps in each jump d) Repeat addition: Add 2 for eight times. 2+2+2+2+2+2+2+2=16 a) Multi~lication i u m ~ : Jumping from 0 -+ 3 -+ 6 -+9 -+ 12 + 15.

Five multiplication jumps with three steps in each jump b) Repeat addition: Add 3 for five times. 3+3+3+3+3=15 c) Reneat addition: Add 1 for 15 times.

1+1+1+1+1+1+1+1+1+1+I+1+1+1+1=15 d) Reveat addition: Add 5 for three times. 5+5+5=15 e ) Multinlication iume: Jumping from 5 4 I 0 3 15. Three

/ multiplication jumps with five steps in each jump 10x1 I a) Reneat addition: Add I for ten times. 1+1+1+I+1+1+1+I+1+1=10

b)-: 10as 10+O C) Reoeat addition: Add 2 for five times. 2+2+2+2+2=10 d) Reneat addition: Add 5 for two times. 5+5=10

4.1.2 Discussion Activity Two Questions in Activity 2 were about multiplication jump. If the students could draw the jumping arrows correctly, it meant that they understood the concept of multiplicntion, bemuse they could identify which number is the number to repeat and which number indicates numbcr of repetitions. Figure 4 helow shows the multiplication junips for 3x2. 3x2 could he interpreted as ,jumping two steps for three times.

Multiplication Jump - Mathematical sentence: 3x2=2+2+2=6

Figure 4: Multiplication Jump

In this activity, it was observed that all the groups could do the multiplication correctly that is they got all the multiplication correct. However, when students were asked to draw multiplication jump and summarise the multiplication jump into a mathematical sentence, it was observed that certain groups did the correct interpretation of the questions whereas some other groups gave interpretation that was opposite. This was observed in all the questions except for 4x4. For 4x4, both the number of steps to jump and the number of jumps are the same that is four. So, it is not possible to tell the difference. For other questions, like 5x3, 2x7, 3x5 and 10x1 it was observed that some students had misconception about multiplication. For example, 5x3, w h ~ c h should be interpreted as jumping three steps five times yet was interpreted as jumping five steps three times by certain groups.

Out of the 14 groups, only one (7.1%) group could answer Activity 2, that 1s all the 5 questions correctly. Two (21.4%) groups managed to answer four questions correctly whereas four (28.6%) groups could answer correctly three & two questions respectively. Finally, two (14.3%) groups of students could answer only one question correctly. The result is summarised in Table 4.3

Table 4.3 Distribution of grouos based on number of correct answer

Number of questions correctly Number of group(%) answered

5 (All correct) 4 (I mistake) 3 (2 mistakes) 2 (3 mistakes) 1 (4 mistakes) 0 (All wrong)

~ ~

1 (7.1%) 3 (21.4%) 4 (28.6%) 4 (28.6%) 2 (14.3%) 0 (0%)

Note. Total number of groups=14

The number of groups that could answer correctly according to questions is shown in Table 4.4

Table 4.4 Distrihution of rrrouos answer correctly based on fluestions

Question Number of groups that can 7% answer correctly

5x3 4 28.6

From the result shown in Table 4.4, it can be wid 4x4 is the question that most of the groups could answer correctly. This coincidentally was due to the fact that 4x4 is a square where there is no difference in term of number of jumps and the number of steps in a jump. Though question (1) and (4) are Inverse of each other and were purposely set to see if the students could differentiate between them, some students still made mistake without relating and comparrng the two.

Out of the 31 mistakes, 19 (61.3%) cases were due to wrong interpretation made. For example, students interpreted 5x3 as three multiplication jumps with five steps in each jump instead of five multiplication jumps with three steps in each jump. In nine (29%) of the cases, students drew the multiplication jumps correctly but the mathematical sentence they wrote was the inverse or opposite of what they had drawn. Two very interesting and creative mathematical sentences were observed where they expressed 4x4 as 5+5+6 and 10x1 as 10+0 respectively. Two (6.5%) cases were due to not being able to do the multiplication jumps correctly but thc mathematical sentences written were correct. Finally the remaining one (3.2%) case was considered wrong in both the multiplication jumps and also the mathematical sentence written. The rcsult is su~nmarised in Table 4.5.

Table 4.5 Distribution of tvpes of mistakes made by subiects

Type of mistake Number of ceses % Inverse multiplication jump & mathematical 19 61.3 sentence Multiplication jump correct but mathematical 9 29.0 sentence wrong Multiplication jump wrong hut mathematical 2 6.5 sentence correct Both multiplication jump & mathematical sentence 1 3.2 wrong

Total 31 I 00

Table 4.6 Analvsis of related knowledge constructed bv students in Activitv 3

4.1.3 Discussion Activity Three In Activity 3, students were given a multiplication question and asked to group dots to represent the multiplication. Similar to Activity 2, if students could group the dots correctly it meant they understood the concept of multiplication. That is given a multiplication, they knew which number indicated number of groupings and which number indicated number of dots in a group.

As in Activity 2, students did not have any problem in doing the multiplication. However, certain groups did showed misconceptions about multiplication in the grouping process. For example, 9x4 should be interpreted as nine groups with four dotslitems in each group but certain groups interpreted it as four groups with nine dotslitems in each group. Table 4.7 shows the number of questions correctly answered versus number of groups.

Knowledge Constructed a) Growing: Taking 7x2 as 7 groups with 2 items in each group b) Grouping and counting: 2 +4 + 6 + 8 + 10 + 12 + 14 c) Grouuing: Taking 7x2 as 2 groups with 7 items in each group d) Repeat addition: Add 7 for two times, 7+7 =I4 e) Repeat addition: Add 2 for seven times, 2+2+2+2+2+2+2=14 a) Grouping: Taking 2x7 as 2 groups with 7 items in each group b) Repeat addition: Add 2 for seven times. 2+2+2+2+2+2+2=14 a) Grouping and counting:

4+8+12+16+20+24+28+32+36 b) Repeat addition: Add 9 for four times, 9+9+9+9=36 c) Repeat addition: Add 4 for nine times, 4+4+4+4+4+4+4+4+4=36 d) Grouuing: Taking 9x4 as 4 groups with 9 items in each group e) Repeat addition by doing successive additions: 9+9=18, 18+9=27,

27+9=36 f ) Grouping: Taking 9x4 as 9 groups with 4 items in each group g) Recall multiulication table and then do addition: Recall 3x9=27

and use it as a starting point, then add 9 to 27 to get 36 (from Activity 1)

h) Grouping and counting: 9 + 18 + 27 + 36 a) Grouping: Taking 8x5 as 8 groups with 5 items in each group b) Grouping and counting:

5 + I 0 + 1 5 + 2 0 + 2 5 ~ 3 0 + 3 5 + 4 0 c) Grouting: Taking 8x5 as 5 groups with 8 items in each group a) Grouping: Taking 3x3 as 3 groups with 3 items in each group b) Groutling and counting: 3 + 6 + 9 b) Repeat addition: Add 3 for three times, 3+3+3=9 a) Grouping: Taking 6x4 as 6 groups with 4 items in each group b) Grouping and counting: 4 + 8 + 12 + 16 + 20 + 24 c) Grouuing: Taking 6x4 as 4 groups with 6 items in each group d) Repeat addition: Add 4 for six times, 4+4+4+4+4+4=24 e) Repeat addition: Add 6 for four times, 6+6+6+6=24

Activity 3

Problem 7x2

2x7

9x4

8x5

3x3

6x4

All the 23 mistakes made were due to misinterpretation in doing the grouping. For example: 7x2 should be taken as drawing 7 groups with 2 dotslitems in each group but it was presentcd the other way round.

Table 4.7 Distribution of question correctly answered based on gl-oun

Number of questions Number of mistakes N % correctly answered

6 No mistake 5 35.7 5 1 mistake 2 14.3 4 2 mistakes 2 14.3 3 3 mistakes 1 7.1 2 4 mistakes 3 21.5 1 5 mistakes 1 7.1 0 all wrong 0 0

W . N = 1 4

Table 4.8 Analvsis of related knowledge constructed bv students in Activity 4

4.1.4 Discussion Activity Four In Activity 4, an example, which is a predecessor to the question, was given before the question. For example, 6x2=12 was given before the question 6x3. Out of the 14 groups, only three groups made use of the predecessor to find the answer for the question. For the rest of the groups, they preferred to use their own method to find the answer. When asked whether there was any relationship between the predecessor and the question followed, four groups even claimed that there was none. Students preferred to recall the multiplication because questions chosen were very simple and easy so they did not need to do much calculation. On the whole, students could do all the multiplication correctly.

Activity 4

Problem 6x3

2x6

7x6

6x7

7x3

Knowledge Constructed a) Reneat addition: Add 6 for 3 times, 6+6+6=18 b) Grou~inq: Taking 6x3 as 6 groups with 3 items in each group c) Recall multi~lication and then do addition: Take 6x2=12 as a starting

point, then add 6 to 12 to get 18 d) Recall multiplication table: 6 + 12 -+ 18 Recall example and then do inversion: Taking 6x2=12 from the example given, then inverse 6x2 to 2x6 to get 12 a) Repeat addition: Add 6 for seven times. 6+6+6+6+6+6+6 =42 b) Recall multiplication table: 12 + 18 + 24 3 30 3 36 3 42 c) Addition: Taking 7x5=35 from the example given, then add 7 to 35

to get 42 c) Recall multiplication table: 7 -+ 14 3 21 -+ 35 -+ 42 d) Recall sauare and then do addition: Taking square of 6, 6x6=36, as a

starting point, then add 6 to 36 to get 42 Recall multinlication table and then do inversion: Inverse 6x7 to 7x6 to get 42 Take 7x2= 14 and then add 7 to 14 to get 2 1

Table 4.9 Analvsis of related knowledge constructed bv students in Activitv 5

2.1.5. Discussion Activity Five The students did not face much difficulty in solving questions in this activity. They could get the repeating unit/s for all the questions correctly but it was observed again that certain groups showed misconceptions about multiplication by summarising the repeating units in an inverse order in the mathematical sentence. For example, for the repeating units of 1 + 1 + 1 +1+1 +I +1+1+ +1+1+1 +I, some of them summarised it as 12x1 and others took it as 1x12. Actually out of the 61 mistakes made, 42 (68.9%) were due to writing the inverse mathematical sentence and the remaining 19 (3 1.1 YO) were cases of writing wrong mathematical sentences. This result shows that students were not able to identify which was a repeating unit and which number indicated the number of repeating units in a mathematical sentence.

Activity 5

Problem 12

12

12

12

12

12

24 24

24 24

P

24 24

Knowledge Constructed a) Summarising 1+1+1+1+1+1+1+1+1+1+l+laslxl2 b) Summarising 1+1+1+1+1+1+1+1+1+1+I+las 12x1 a) Summarising 2+2+2+2+2+2 as 6x2 b) Summarising 2+2+2+2+2+2 as 2x6 a) Summarising 3+3+3+3=4x3 b) Summarising 3+3+3+3=3x4 a) Repeat addition: 4+4+4 b) Summarising 4+4+4=4x3 a) Repeat addition: 6+6 b) Summarising 6+6 as 2x6 a) Summarising 12 as 1 x12 b) Summarising 1 2 as 12x 1 a) Summarising 4+4+4+4+4+4 as 4x6 a) Recall and inversion: Recall 4x6 and then inverse it to 6x4 and

then relate 6x4=6+6+6+6 a) Summarising 12+12 as 2x12 a) Summarising 3+3+3+3+3+3+3+3 as 8x3 fi

a) Summarising 8+8+8 as 3x8 a) Summarising 2+2+2+2+2+2+2+2+2+2+2+2 as 12x2

Table 4.10 Analysis of related knowledge constructed bv students in Activity 6

4.1.6 Discussion Activity Six In this last activity, the students were given 12 wooden blocks to construct a building. Every floor must possess the same number of rooms. After the completion of the building, the students were asked to relate the columns and rows to twelve. All of them could recognise the number of columns and rows and from there, they constructed mathematical sentences either by using addition or multiplication.

Activity 6

For example, a building made up of six floors of two rooms each was related to 6+6, 2+2+2+2+2+2 by using addition, and 6x2 and 2x6 by using multiplication. However, students were not able to relate correctly the addition to its corresponding multiplication that is students are not able to relate 6+6 to 2x6 and 2+2+2+2+2+2 to 6x2. It was observed that they could relate better 6+6 to 2+2+2+2+2+2. After they had suggested 6+6, they could see better 2+2+2+2+2+2 and similarly after they had come out with 6x2 then they inversed it to 2x6.

Another observation was that students had the ability to compartmentalise the rows and columns. For example, a building with 12 rooms in a row or column can be cornpartmentalising into units made up of two rooms, three rooms, four rooms, and even six rooms. With that, students suggested 2+2+2+2+2+2,3+3+3+3,4+4+4, and 6+6 as the answers related to 12.

Problem 12

4.2 Other observations 1. Given a pair of numbers to add, students could do the addition flexibly. They could either

add the second number to the first number or vice versa. For most of the cases, the students preferred to place the bigger number of the two in the heartjheadlmouth and the remaining number in the fingers, regardless of whether it was the first number or the second number. From there, they started counting from the number which they placed in the heartjheadlmouth and continue till all the fingers indicating the second number were used up. This way of counting is called counting-on.

Knowledge Constructed a) Repeat addition: 4+4+4 and summarise as 3x4 b) Repeat addition: 3+3+3+3 and summarise as 4x3 c) Counting and addition: 3 + 6 + 9 + 12 d) Repeat addition: 1+1+1+1+1+1+1+1+1+I+l+l and summarise as

1x12 e) Repeat addition: 1+1+1+1+1+1+1+1+1+1+l+l and summarise as

12x1 f) Repeat addition: 2+2+2+2+2+2 and summarise as 6x2 g) Reveat addition by doing successive addition 2+2=4,4+2=6,6+2=8,

8+2=10,10+2=12 h) Repeat addition: 6+6 and summarise as 2x6 i) Reveat addition: 6+6 and summarise as 6x2

At other times, all the students preferred to use counting-up while doing the addition. For example: while adding 7 and 4, a student svarted counting from one, two, three, four, five.. . up to eleven instead of counting starting fiom either 7 or 4 like the others.

There was another case where a student placed 4 in his head, and he stretched out altogether ten fingers. He started counting from four, five, six.. . till eleven, which meant he only used up seven fingers although he showed ten fingers initially.

Students showed a tendency to place the first number of the two in the headkeartlmouth regardless of whether the number was bigger or smaller than the second number. There was another short cut to do thc addition. When adding 7 to 4, the student took three from four and added onto seven to make a total of ten, then he added the remainder to ten to make eleven.

2. Students claimed to like multiplication of certain numbers and said it was easy to remember. They favoured squares of any number and multiplication of 2, 5, and 10 as in the cases below.

a. To find the answer to 6x7, they took 7x7=49 (square) then they subtracted 7 from 49 to get 42.

b. To find the answer to 6x7, they took 7x5=35 (multiple of five) then they added 7 to 35 to get 42.

c. To find the answer to 6x7, they took 6x6=36 (square) then added 6 to 36 to get 42. d. To find the answer to 7x8, they took 7x7=49 (square) then added 7 to 49 to get 56. e. To find the answer to get 7x2, they used 2x6=12 (multiple of two) then they added 2 to 12

to get 14. f. To find the answer to 9x1 1, they took 9x10=90 (multiple of ten) then they added 9 to 90 to

get 99. g. To find the answer to 13x2, they took 2x12=24 (multiple of two) then they added 2 to 24 to

get 26.

3. Students were confused when they had to draw the multiplication jump. Sometimes they went for convenience rather than followed logic. Although they knew that 3 x 5 was the inverse or opposite of 5 x 3 they could not differentiate the multiplication jump or the repeating units of both.

It was observed that they drew the multiplication jump accordingly but the mathematical sentence they wrote was the opposite or it was ii-relevant. For example, they showed 3 multiplication jumps of 5 steps each for 3x5 but the mathematical sentence they wrote was 3, 3, 3, 3, 3 instead of 5, 5, 5. Some students even wrote the same mathematical sentence for both 3 x 5 and5 x 3 a s 5 , 5 , 5 .

This showed that although students knew how to use repeating addition while doing the multiplication, some of them did not really know which was the repeating unit and which was the unit to repeat. This observation was again found in Activity Three, Four and Six.

4. Students claimed that zero is nothing (tidak adu apa-apa). Some said that zero did not have a value (ni1ui)lnumber.

5. One of the students claimed that while doing division, the bigger number must be in front and the smaller number at the back.

6. Students showed very high determination. They used trial and error with different alternatives until they obtained the solution to a particular problem. For example in Activity

Five, they started trying with two, three, four ... etc, gradually increasing the number until they struck the correct one.

One very striking observation was that students would try again with the same numbers that were already used up in the previous portions. This showed that they did not have the concept of eliminating the alternatives that were already used up.

7. While counting the repeating units, most of the students could do the mental calculation for the first few units instantly, but they had to count manually for the remainder of the repeating units to get the total.

For example, to sum up a series of repeating unit of fours, 4+4+4+4+4+4, they could do four, four, eight, twelve then most of them would start to count thirteen, fourteen, fifteen, sixteen, seventeen, eighteen, nineteen, twenty, twenty one, twenty two, twenty three, twenty four. This implied that the students had mastered simple addition with small numbers but not with big numbers.

8. While asked to look at the building consisting of just one row or one column of twelve blocks, the students claimed that they could see repeating units of two, three, four, and six by grouping them mentally.

4.3 Analysing the Conceptual Operators In this study the verbal protocol was analysed using a coding scheme based on the coding scheme by Hassebrock and Prietula (1992) to identify the operators that generated the relevant knowledge used while solving the mathematical problems.

4.3.1 Conceptual Operators The protocol representation in the coding scheme identified the conceptual operations used to generate states of mathematical knowledge used by the students to solve the mathematical problems.

Both the coding scheme and the conceptual operators used were similar to those used by Hassebrock and Preitula (1992). Figure 5 gives the set of conceptual operators used by the students to solve the mathematical problems.

Data Examination - a. Read b. Examine

a . V b. Locate

Data exploration - c. Elaborate 2. Identify 3. Extract

Data explanation - a. Explain

a. Trigger Solution generation - b, ~ ~ l ~ t ~

c. Trial and error

a. Add b. Multiply

Mathematical operation --+ C. d. Divide e. Count f. Group g. Multiplication jump

a. Confirmation b. Disconfirmation

a. Recognize Retrieval - b. ~ ~ ~ ~ l l

Summarisation - a, Summarise

Figure 5: Conceptual operations used by students

Data Examination When given a set of problems to solve, it is important that the students understand the requirement of the activities before they solve the problems. For certain problems, the students had to read the example provided and try to understand it in order to solve the problems.

a. Read. The students read'the instructions, examples, and questions before they tried to solve problems.

b. Exumine. While reading the instruction, examples and problems, the students examine the instruction, examples and problems trying to determine what the problems required. Students also examined examples given and used them as a guideline to solve the problems. Examination on data was necessary so that after the situation had been recognised and interpreted, subsequent operations could be carried out.

Data Exploration The students conducted a more detailed interpretation of the data previously examined.

a. Examine. Three sub operators were chosen here to further interpret the data there are compare, identify, and extract. Students compared data with the norm in terms of difficulty, they also compared number size and quantity too. For example, students claimed that multiplication tables of certain numbers were easier to memorise.

Compure. Given a pair of numbers to multiply, students seemed to favour certain numbers. For example: 7 x 2 and 2 x 7. Students knew the answer for both questions was fourteen yet if they were asked to present in terms of multiplication jumps or by repeating addition they preferred 2 x 7. This was because 2 x 7 = 7 + 7 was simpler and shorter and 7 x 2 = 2 + 2 + 2 + 2 + 2 + 2 + 2 was said to be longer and tedious.

Identify. Given a pair of numbers to add together, the students had to identify the number to add, number to start counting. For multiplication, the students needed to identify the number to repeat/group/jump and also the number of repetitions/groupings/multiplication jumps.

Extract. The students had to extract important points and facts from the instructions or examples given. These points and facts were later used as a guideline to solve problems.

b. Lacate. When doing addition, students preferred to locate numbers as separate entities. For 9 + 4, usually the students would locate one of the numbers either in the heart or mouth or brain, and the other number in the fingers. Only then, they started to do the counting or addition.

c. Elaborate. Students were asked to elaborate further on process taking place while they were solving problems.

Data Explanation The students explained the process or the reasons for doing certain operations.

Solution Generation Students used a few methods to generate solutions for the problems.

a. Trigger. Sometimes when the students were stuck and did not know how to go about solving the problems, a hint and suggestion from friends helps to trigger their thoughts to come out with the solution.

6. Relate. Students generated solution by relating the current problem to their experience or something that they could remember very well. For example: they related 7x6 to 7x7 (=49) which is a square of 7. They obtained the answer for 7x6 by subtracting 7 from 49 or (49-7) to get 42. If they were asked to find 6x2 they preferred to relate it to 2x6 by just reversing it because it was easier to remember 2x6 than 6x2. Given a building made of wooden blocks, the students were able to relate the columns and rows to multiplication. For example: a building of four floors with three rooms in each floor would have a total of 3x4=12 rooms.

c. Trial und error. Students used trial and error most of the time especially while solving problems in Activity 5 & 6. They would try with the smallest number first and verified its relevance. If it failed, then they would try with another alternative by gradually increasing the value of the alternative until they obtained the answer.

d. Deduce. Students used deduction to generate solutions. When adding 5+6, they claimed that the answer was 11 because5+5 is 10, so 5+6 must be 11. Then in Activity 5, it happened that the answer to part (a) and (b) was 1 and 2 respectively so the students deduced that the answer to part (c) would be 3 which coincidentally was true also.

Mathematical Operation While solving the mathematical problems in the activities, students used the following mathematical operations, add, multiply, group, count, subtract, divide and multiplication jump.

a. Add. To add is to join, to unite, to combine two or more groups of objects or to sum up. For example: to add 5 and 5 we get 10.

b. Multiply. To multiply is to take a given quantity or number a given number of times. It can be considered also as to increase a certain quantity or number a number of times. For example: 3x2 means to increase 2 for 3 times to get 6.

c. Group. To group is to collect or place objects together. In this study, it is necessary to group equal items in each group. For example, to express 2x7 by grouping is to have two groups made up of 7 items in each group.

d. Count. To count is to say in order for the purpose of finding out the total quantity. We can count one by one that is the proceeding number is always one size bigger that the preceding number. It was observed that students could do counting 2 by 2 and 5 by 5 too.

e. Subtract. To subtract means to take away or minus.

Divide. To divide is to distribute equally.

g. Multiplication jump. To do a multiplication jump is to repeat jumping and each jump is made up of a certain number of steps. For example: 3x5 is taken as 3 multiplication jumps with 3 steps in each jump.

Evaluation It is important to evaluate data.

a. Confirm. By checking the steps and the calculation involved the problems, students were able to self-evaluate themselves. A data is confirmed if it was said to be correct.

b. Disconfirm. A data is said to be disconfirmed if it was not correct and would subsequently be rejected.

Retrievul An operator was said to be a retrieval operator if students had to recall or retrieve information or knowledge from the long-term memory.

a. Recognisa. While reading the instructions or examining data students recognise things and itcms that they knew. For example, operations involved in the problems such as addition and multiplication. Given a building made up of wooden blocks, the students were able to recognise columns and rows.

b. Recall. Given a problem, sometimes students could just recall mentally answer to the problem without any hesitation. This indicated that students were performing the mental calculation or they retrieved from their memory things they had memorised. Recalling was especially obvious in doing multiplication.

Summarisation Students summarised repeating units and also multiplication jumps into a mathematical sentence. In doing so, the students showed the ability to simplify parts into a single representation and also the ability to rccognise pattern. For example, given a repeating units of one like 1+I +1+1+1+1+1+1+1+1+l+l, students were able to recognise one is the repeating units and there were twelve of them, so it could be simplified as 12x1.

4.4 Analysing the Knowledge States The relevant knowledge used by the students while solving the mathematical problems was analysed, identified, and categorised.

4.4.1 Knowledge States To solve mathematical problem dealing with addition and multiplication, it was essential for students to possess certain mathematical knowledge. The mathematical knowledge identified from the verbal protocol was:

a. Number b. Operations - addition, multiplication, subtraction, division, c. Counting d. Grouping e. Column and row f. Multiplication jump

Number Number is a word or symbol used to express how many. For example, 10 is the symbol for ten. A number indicates a quantity of units and is numeral. The numbers involved in all the activities in this study are positive whole numbers including zero.

Operation The operations used by the students in the verbal protocol included addition (+), multiplication (x), subtraction (-) and division (i ) that represent the basic operations in all mathematics. To add means to combine two or more numbers to obtain one equivalent sum. For example, to add three to nine we get twelve as the sum, 3+9 = 12. The resulting quantity is bigger than either one of the numbers. Twelve is bigger than either three or nine.

To multiply, in its simplest way, is a process to operate repeating addition of an integer in a specified number of times. For exiimple, 3x2 can be interpreted as adding 2 to itself three times, 3x2=2+2+2=6.

Subtraction is another operation whereby to subtract means to take away a certain quantity from a number. For example, to subtract seven from 10 will remains 3, (10-7=3).

To divide means to distribute equally a quantity among a group. For example: to divide twelve by three we get four, 12 + 3=4.

Counting In this study, to operate counting is to name numbers in order. Usually the students counted numbers in the order of 1,2,3.. . each succeeding number being greater by 1 than its predecessor. Nevertheless, students did counting in other orders also such as 2,4,6 ... 3,6,9 ... 5,10,15 ... etc based on the situation required. By counting, students were able to add up either one by one or by units or groups, in order to get a total.

There are two types of counting, counting-up and counting-on (Bruer,1993). To count-up means to count starting from one, two, and three.. . In counting-on, students pick a larger number of the two addends, and count up from there.

Grouping To operate grouping is to group, assemble, or gather a number of items or objects to form a unit or group.

Row & column Row & column here relate to the buildings constructed using wooden blocks. Row means the number of floors in a building whereas column is the number of blocks/rooms in each floor of the building.

Multiplication Jump To do a multiplication jump is to draw arrows on the number line to indicate the number of repetitions for a multiplication. For example, 3x4 can be represented by three jumps whereby each jump is made up of 4 or as 4 + 8 + 12.

4.5 Verbal Protocol Coding Scheme for Solving Mathematical Problems The verbal protocol-coding scheme for solving mathematical problem basically has two protocol representations, that is, knowledge state and conceptual operators. In this study, six knowledge states and seven conceptual operators were identified. However, the result of this study should be treated as indicative and not representative as this study was done on 45 students from three schools, 15 from each school. This study used only audio-recorded data.

Figure 6 gives the coding scheme for the verbal protocol of the mathematical problem solving.

Protocol Representation I 1. Knowledge States

1 6. Multiplication jump I

1. Number 2. Operation 1. Addition

2. Multiplication 3. Subtraction 4. Division

3. Counting 4. Grouping

I 5. Row & column

2. Conceptual Operator 1. Data Examination 1. Read 2. Examine

2. Data Exploration 1. ~xamine' 2 . 3. Elaborate

4. Solution generation 1. Trigger 2. Relate 3. Trial and Error

1.C0mpwe 2. Identify 3. Extract

I

5. Mathematical operation I . Addition 2. Multiplication 3. Subtraction 4. Division 5. Counting 6. Grouping

1 7. Multiplication jump

3. Data Explanation 1. Explain

6. Evaluation 1 .Cofirmation 2.Disconfrimation

7. Retrieval 1. Recognise 2. Recall

8. Summarisation 1. Summarise

Figure 6: Vcrbal Protocol Coding Scheme for Mathematical Problem Solving

4.6 Mental Model for Mathematical Problem Solving The following mental m nod el as shown in Figure 7 was devised l'ro~n the findings of the study. The simplified mental model represents the knowledge i~nd the cognitive operations used by the students while they were solving the ~nathe~natical problems. This ~nental niodel is only

indicative and not representative. The content of this mental model is derived from the verbal protocol-coding scheme obtained.

Number

KNOWLEDGE STATES

olumn and Row

CONCEPTUAL OPERATORS

Figure 7: Simplified mental model for Mathematical Problem Solving

4.7 Summary This chapter present the main findings of this study. Findings are presented in two parts. The

first part focused on the discussion on the knowledge constructed by students followed by discussion on the conceptual operators and knowledge states identified f ~ o m the protocol analysis. Protocol-coding scheme and mental model developed from the findings were presented also.

CHAPTER FIVE

SUMMARY, CONCLUSIONS AND RECOMMENDATIONS

5.0 Introduction This chapter presents the overall summary of the whole study, conclusions, and recommendations.

5.1 Summary Solving mathematical problems is a mental process that is not observable. T o understand unobservable work processes, the researcher chose to observe verbal behaviour. Cognitive task analysis is commonly used to understand learning processes and cognitive processes related to problem solving. In this study, verbal protocol analysis, which is one of the general methods used in cognitive task analysis is used.

This study is centred on how Primary Two students connect addition to multiplication through constructivism in solving mathematical problems. By knowing how students relate multiplication to addition will help to enhance learning of mathematics in primary schools. The protocol coding scheme in this study is similar to that used by Hassabrock and Prietula (1992) in their analysis of medical reasoning.

The objectives of this study are to find out how students reason out the connection between addition and multiplication and to develop a mental model for mathematical problem solving. This involves analysing the verbal protocol and developing a verbal protocol-coding scheme. Similar to the verbal protocol coding scheme used by Hassehrock and Prietula (1992), two basic types of protocol representations or semantic elements, that is the knowledge states and the conceptual operators are thc main concerns of'this study apart from the constructed knowledge. From the coding scheme, a mental model is developed. This mental model is the mental representation of the students in solving the mathematical problems. Types of knowledge, information processing theory, mathematics, and constrcutivism were also discussed i n this study.

Tlie main terms in this study are constructivism, knowledge states, and conccptual operators. Constructivism is a theory of knowledge, which claims that learners actively construct their knowledge, which they acquire through their senses. Knowledge state is the unit of knowledge required by the students to generate knowledge while solving the mathematical problems. The conceptual operators are operators that operate on the new knowledge or incormation received and which may be integrated with old knowledge and expcrience possessed by the learners. From there, a new sct of knowledge is constructed.

In this study, students in groups of three were given a set of six activities to do. Students were asked to "think aloud" while solving the ~nathe~natical proble~ns and data was recorded audibly. This audio-recorded data was transcribed and analysed together with the obscrvation done by the researcher and also the written work of the subjects. The transcript was then coded and the knowledge states and the conceptual operators identified. Frotn thc analysis, a inental inodel WEIS developed. Construction of knowledge was also identified to see how students construct knowledge and connect multiplication lo addition while solving the problems.

Form the findings of this study, students do connect multiplication to addition through constructivism in various ways. The main strategies used by the students to solve the mathematical problems are as follow:

a. Students related multiplication to repeat additions. For example, 6x5 was related to repeat addition of 5 for six times. 6x5=5+5+5+5+5+5=30.

b. Students used squares as a starting point to find the answer for multiplication. For 7x8, they would start at 7x7=49, which is a square of seven. Then they added 7 onto 49 to obtain 56. On the other hands, students related multiplication to subtraction also. If they were asked to find 6x7, they would subtract 7 from 49 to obtain 42.

c. Students broke down multiplication into simpler forms. For 6x7 they broke it down into Zx7,2x7 and 2x7, which is equal to 14. Then they added up 14,14, and 14 to obtain 42.

d. Students use certain multiplication, which they remembered very well as a starting point to find answer. For example, to find the answer for 13x2, they started with 2x12, which is equal to 24. Then they added 2 onto 24 to obtain 26.

The findings revealed the strengths and weaknesses of the students in using mathematical knowledge and mathematical operations. A few creative strategies used by the students to work out the problems were identified. However, these findings showed that some students were confused in certain concepts when solving mathematical problems.

From the verbal protocol-coding scheme obtained, eight conceptual operators and six knowledge states are identified as shown in Figure 8.

Data Examination Data Exploration Data Explanation Solution Generation Mathematical Operation Evaluation Retrieval Summarisation

Knowledge States

Number Operation Counting Grouping Row and column Multiplication jump

Figure 8: Conceptual Operators itrid Knowledge States for solving the mathematical problems

A mental model is developed based on the conceptual and knowledge states identified. However, it should be noted that the result of this study is just indicative and not representative of how the students solve mathematical problem.

This result is useful and can he used to enhance the learning of mathematics in primary schools. Nevertheless, there are certain weaknesses and liniitations in this study as stated below.

a. The verbal protocol recorded was based on the verbalisation of a group of three students at a time. Frequently, all three of them could be talking at the same time and each one has hislher own way of reasoning. This brought distraction to the flow of reasoning and posed difficulty transcribing the data.

b. The students were very young and all of them were about the age of eight to nine, they were very shy, and were self-conscious about the presence of the tape recorder. Thus, there was a tendency for them to whisper and the researcher had to keep reminding them to go on talking aloud.

c. All the students used local Malay dialect in their conversation. This posed difficulty to the researcher, as she does not speak the dialect, while transcribing the data. The researcher had to consult people who speak the dialect on the. meaning and the spelling of the words used.

d. Due to their young age, it was quite difficult to make the students talk freely about what they thought in their mind. The researcher had to probe in order to help the students to proceed in their "think-aloud" process. The questions used by the researcher consisted mainly of"how", "why", "what" ... etc.

5.2 Conclusions The findings of this study can be used to enhance the teaching and learning process in the classroom especially for mathematics. This study came out with knowledge constructed by Primary Two students connecting multiplication to addition, a verbal protocol-coding scheme, and mental model of the students while solving the mathematical problems. Teachers can make use of this knowledge to further understand the cognitive processes that took place in the mind of the students while solving the mathematical problems. This verbal protocol data helps to reveal where and how students made mistakes. Thus, teachers can encourage students to verhalise their thinking to help to achieve a better understanding between both parties. Ostler et al. (1998) state that teachers can incorporate problem solving, communication, reasoning, and connections to improve their mathematics instruction. This definitely will enhance the teaching and learning process in the schools.

5.3 Recommendations for the Study A few recommendations are given for further research o'r for future planning purposes

a. The data collection can be further improved and will be more complete if both audio and visual rccordings were used. This is because young students tended to nod and shakc their heads instead of saying yes or no. Another reason was that since this study is ahout solving mathematical problems, it involved using of finger counting which could not he detected by audio rccording.

h. The mental model developed from this study is a mcntal model made up of ;ill the conceptual operators and knowlctlgc states used by the subjects. Further research can be done lo investigate mental model of individual suhjects especially on subjects who gave wrong answers. The mental model obtaincd will then show the tnisconception of the suhject and thus can be used to help te~~chcrs to understand better the difficulty and problems faced by students i n lcarning mnthcmatics.

c. Research can be conducted to compare the ability of boys and girls in solving mathematical problems. Normally, girls are perceived to be weaker in mathematics. Thus, verbal protocol may he able to help to find out the reason.

d. Further research can be done with a single subject solving the problems instead of three subjects at a time. Then it is possible to study the actual flow of the reasoning of the suhject.

e. Research could also be carried out on Form 4 and Form 5 students to find out how they solve problems in additional mathematics. From the experience of the researcher as addition mathematics teacher, students usually face great difficulty and have to struggle very hard to do well in the subject. Thus, it will be useful for both teachers and students to know thc effective ways and strategies to study additional mathematics.

E T o the education policy maker, they could introduce findings of this study to trainees in the training colleges. The constructivist approach should he emphasised in the teaching of mathematics as learning of mathematics involves a lot of reasoning.

g. For the mathematics teachers who are at present teaching in schools, the findings of the study can be made known to them through in-house training or short courses. The findings of this study can enhance the teaching-learning process especially in mathematics as it can bring to the attention of the teachers how, where and why the students make mistakes.

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Knowledge Constructed

(Tujuh tarnbah empat.)

Knowledge

Addition

Number

Multiplication

Number

Counting Number Addition

Addition Counting

Transcript:Group 1 (Kita akan mula dengan aktiviti ini, sama-sama buat & bincang. Tengok aktiviti pertarna, semua ini soalan apa?) Tamball (Tambah, sudah belajar tambah?) Sudah (Mana, bila?) Hm.. .darjah satu (Sampai nombor berapa?) Seratus.. . (Adakah adik guna jari untuk buat tambah?) Ada pakai jari jua. (Kalau tidak pakai jari, adik pakai apa?) Otak (Pandai. Ok yang nombor dua ini soalan apa?) Darab - (Darab, sudah adik belajar?) Sudah (Sampai mana belajar?) Dua belas (Dua belas, boleh hafal?) Hm.. .tak hafal.. . he.. . he.. . he (Ok kalau tengok anak tangga ini, kamu perlu cuba jawab setiap soalan di sini. Kamu boleh mula sekarang, sesiapa pun boleh buat. Macam mana dapat dua belas? Cakap.. .cakap, . . .kuat sikit) Sembilan, sepuluh, sebelas. dua belas.

(Sembilan tambah empat sama dengan tiga belas.. . Macam mana dapat tiga belas?) Seouluh, sebelas, dua belas, tiga belas.. .lapan tambah kosone laoan.

Operation

Retrieval: Recognise

Retrieval: Recall

Retrieval: Recognise

Retrieval: Recall

Mathematical operation: Count

Mathematical operation: Count Data expioration: Examine: Compare

Number

(Sembiian dulu, kemudian baru tiga.. . kalau tiga tambah sembilan, .macam mana buat? Ambil sembilan dulukah atau ambil tiga dulu?) Tl-Q

Transcript:Group 1 Sebelas ...

(Sembilan tambah tiga.. .sama tak tidak sembilan tambah tiga dengan tiga tambah sembilan?) Sama (Kalau sembilan tambah tiga, macam mana buat?). Sepuluh. sebelas. dua belas.. .

(Tiga darab sembilan, macam mana kira? Sebut.. .sebut.. sebut., kena cakap) Hm.. .tuiuh darab tiga.. .tipa.. .enam.. .sembilan.. .dua belas.. . Hm.. .m belas. iauan belas. dua uuluh satu, hm.. .dua uuluh emuat . . .sikit lagi, dua puluh tuiuh

Operation Mathematical operation: Add

Mathematical operation: Count & add

(Oh, pandai, siapa ajar macam itu?) Ci kgu (Oh, cikgu ajar.. .hm.. .tujuh darab dua?) Emuat belas (Macam mana kamu tahu empat belas?) Baru lepas belajar (Ok, empat kali empat.. .enam belas. Macam mana dapat?) Sepuluh (Sepuluh, enam darab tujuh? Sebut apa yang kamu fikirkan. Macam mana enam darab tujuh? ) Hm.. .dua.. .ties.. . empat.. . lima.. . enam.. . tuiuh.. .empat puluh dua

(Boleh tunjuk macam mana dapat empat puluh dua?) Hnl.. . tuiuh darab tuiuh.. .hm.. .hm .... emuat vuluh sembilan tolakkan tuiuh.. .

Data exploration: Examine: Identify

Mathematical operation: Add

Retrieval: Recall

Mathematical operation: Count

Retrieval: Recall Mathematical operation: S~~htrsct

Addition Counting Number

I Number

Counting Number Addition

Multiplication

Counting Number

Multiplication Subtraction

~ranscript:~roup 1 I Operation I Knowledge 1 Knowledge Constructed (Oh, pandai, kamu ingat tujuh darab tujuh, oh.. .mengapa ingat tujuh darab tujuh-dan bukan tujuh darab enam. Jadi guna tolak, tujuh darab tujuh empat puluh sembilan, tujuh darab enam.. .) Tolak tuiuh.. .

(Kalau tujuh darab lima? Kuat sikit) Ah.. .tujuh darab lima.. . Lima, seuuluh. lima belas. dua uuluih. dua uuluh lima. tiea puluh. tiga uuluh lima.. .

Mathematical operation: Subtract

Mathematical operation: Add

Subtraction

(Kalau saya kata ah. ..tujuh darab lapan?j Hm ... (Tadi tujuh darab tujuh.. .) Empat puluh sembilan (Sekarang kalau tujuh darab lapan?) Hm.. .lima uuluh enam (Macam mana dapat?) Tambahkan

(Sekarang kita masuk aktiviti dua, aktiviti dua ini dipanggil lompatan darab. Tengoh soalan pertama.. . tiga darab dua jawapannya.. .) Enam (Jadi kalau kita nak tunjukkan macam mana dapat enam dengan lukisan ini, dua bererti lompat sebanyak dua dari kosong ke dua dan tiga darab dua bererti ia lompat tiga kali.. .sekali lompat 2, lompatan kedua dua juga, jadi sekarang jumlah sudah.. . Enarn.. .eh.. .empat (Empat dan lompatan lagi sekali.. .) Enam (Enam, jadi jumlahnya ialah enam dan kita boleh tulis ayat matematik sebagai dua ...) Dua tarnbah dua tambah dua. enam

Addition

Mathematical operation: Add

Retrieval: Recall

Data examination: Read

Addition

Relate 7x5 to 5 - 3 1 0 - 3 1 5 + 2 0 - + 2 5 -+30+35 (5+5+5+5+5+5+5)

Relate 7x8 by adding 7 to 7x7=49 to get 49+7=56

Multiplication

(Macam mana kamu tahu?) Ini sebab kira ini senang.. .senang banding.. (Berapa kali lompat?) Hm.. .&. . .

Transcript:Group 1 (Itu sama dengan tiga darab dua. Cikgu mahu kamu tengok soalan ini, tunjuk pada nombor garis ini berapa banyak lompatannya d m tulis jawapan di sini, kemudian tulis ayat matematik, bincang. Macam mana dengan lima darab tiga? . . .Lompat berapa jauh? Cakap kuat.) Ke lima belas

(Sekali lompat berapa jauhnya?) Lima belas.. .hm. .. (Sekali lompat?) Hm.. . tiga kali.. . (Sekali lompat berapa nombor ia lompat?) Lima kali -

(Limakan? Sekali lompat lima, dua kali dah jadi . . .) S e ~ u l u h

Operation

(Tambah lagi sekali?) Lima belas

(Ok bagus dan sekarang ayat matematik.) Lima darab tiga sama dengan lima belas (Kalau nak tunjuk lompatan, macam mana?Macam saya buat di sini, ia lompat dua)

Knowledge

Data exploration: Examine: Identify Mathematical operation: Multiplication jump

Knowledge Constructed

Data exploration: Examine: Identify Mathematical operation: Multiplication jump

Mathematical operation: Multiplication jump

Mathematical operation: Multiplication jump

Multiplication jump

Multiplication jump

Multiplication jump

Multiplication jump

Relate 5x3 to 5 jumps 3 steps in each jump.

Transcript:Group 1 Betul (Macam mana awak? Cuba dua darab tujuh.. .lompatan berapa?) Dua -

(Duakah? Mengapa bukan tiga?) Sebab ini ada dua.

(Oh, itu dua, dua kali.. .) Tiga dah. ..empat eh.. . Seouluh, dua belas, emoat belas

(Cukup?) Cukup (Cakap.. .cakap.. . satu) Satu, dua, tiea. empat, lima, enam. tuiuh

(Ok jawapan berapa?) Empat belas (Bolehkah adik tulis? Dua darab tujuh. Sebut.. .sebut.. .sama dengan.. .) Empat belas (Ok, dua.. .) Dua tambah dua tambah dua tambah dua tambah dua tambah dua. Satu, dua, tiga, empat, lima, enam, . . .sigek lagi, tambah dua . . .empat belas jawapan.

(Jawapan empat belas. Empat darab empat, lompat berapa jauh?) Emvat, lavan, sepuluh. Eh.. .sebelas.. .dua belas.. .enam belas

(Oh, pandai, jawapan berapa?) Enam belas (Tulis ayat) Empat darab empat sama dengan enam belas.. .

Data exploration: Examine: Identify

Data explanation: Explain

Knowledge Constructed Operation

Mathematical operation: Count

Knowledge

Mathematical operation: Count Evaluation: Confirm

Mathematical operation: Add & count

Mathematical operation: Multiplication jump

Multiplication jump

Multipljcation jump

Counting Number

Counting Number

Addition Counting Number

Multiplication jump

Relate 2x7 to 7 jumps with 2 steps in each jump

Relate 2x7 to 2+2+2+2+2+ 2+2= 14

Relate 4x4 to 4 + 8 + 12 + 16 (4+4+4+4) Relate 4x4 to 4 jumps with 4 steps in each jump

(Jawapan) Jawapan lima belas, hm.. .tiga darab lima sama dengan lima belas.Tiea tambah tiga tambah tiga tambah tiga tambah tiga sama denean lima belas (Oh, very good, soalan terakhir, sepuluh darab satu) Satu, eh ... dua, ... eh salah, lomuatan seuuluh

Transcript:Group 1 Emuat tambah emuat tambah emuat, sigek lagi, empat

(Cikgu mahu kamu semak empat ini.) Empat. lapan. dua belas. enam belas.. .lebih.. .enam belas betul (Siapa buat tiga darab lima? Oh.. .adik nak buat.. .sebut.. .sebut) Tiga. sembilan, dua belas. lima belas.. .

(Sampai.. . sampai.. .) Sepuluh (Pandai.ok now) [l]Seuuluh darab satu sama dengan seuuluh, [2] satu tambah satu tambah satu tambah satu tambah satu tambah satu tambah satu tambah satu tambah satu tambah satu, ...[ 3]satu, dua. tiea. emDat, lima, enam, tuiuh, lauan, sembilan, seuuluh.. . sama dengan sepuluh

(Ok, kalau cikgu bagi satu darab sepuluh, macam mana kamu buat. .., lukis di sini juga.) Hm.. . (Sama tak tidak?) Tak -

(Sekarang aktiviti ketiga melukis juga. Dalam aktiviti ini kamu dikehendaki kelompokkan. Bilangan ahli dalam kelompok itu mesti sama. Kita tengok tujuh darab dua.) Tujuh darab dua.. . (Macam mana?)

Knowledge Constructed

Relate 3x5 to

Operation Mathematical operation: Add

Evaluation: Confirm

Mathematical operation:

Knowledge Addition

Multiplication Multiplication jump

Mathematical operation: Add

[lIRetrieval: Recall Mathematical operation: [,]Add & [3]count

jump I 3 3 6 + 9 + 1 2 3 1 5

Mathematical operation: Multiplication jump

Data exploration: Examine: Compare

Addition

Multiplication jump

Addition Number

Multiplication

Relate 3x5 to 3+3+3+3+3=15

Relate 10x1 to I multiplication jump of 10 steps but with methematical

- -

Transcript:Group 1 Tujuh dua.. . (Macam mana, tolong dia.. .cakap, cakap.) Satu. dua, tiga, emuat, lima, enam. tuiuh. lauan, sembilan, sepuluh, sebelas, dua belas. tiea belas, emuat belas. lima belas.. .udah...betul.. .betulkah ia?

(Siap? Tapi itu satu saja? Satu kelompok sajakah?) Satu kelompok (Apakah maksudnya tujuh darab dua, adakah cikgu sebut tujuh kumpulan dua?) Ada (Apa maksudnya tujuh kumpulan dua?) Satu, dua, tiga, emuat, lima, enam, tuiuh ... tuiuh. Satu, dua, tica. empat. lima, enam. tuiuh.

(Jadi ada berapa kelompok?) Dua -

(Setiap kelompok berapa? Berapa biiangan?) Tui&

(Jadi jawapan berapa? Macam mana dapat jawapan?) Empat belas (Ok, next one, dua darab tujuh.) Hm ... dua. Satu. dua ... satu. dua ... satu dua ... satu, dua. tiga, emvat, lima, enam, tuiuh.. .hitunglah.. .empat belas dua, dua. emuat, enam. lavan, se~uluh, dua belas, emuat belas

(Sembilan darab empat.) Satu, dua, tipa. empat. lima. enam. tuiuh ... eh ... satu. dua, tiea, emuat, lima, enam, tuiuh, lauan. sembilan.. .semuakah? Empat, habis semua. ..hitung .. .dah. Dua, tiga, empat, lima, lapan belas, hm .... Satu. dua, tiea, emvat. lima ... tiga puluh enam (Macam mana dapat tiga puluh enam?) Hm.. .tambahkan

Mathematical operation: Count & group

Mathematical operation: Count & group

Data exploration: Examine: Ldentify

Data exploration: Examine: [dentify

Mathematical operation: Zount & group Evaluation: Confirm

Mathematical operation: Count & group

Counting Number Grouping

Counting Number Grouping

Number

Number

Counting Number Grouping Addition

Counting Number Grouping

Relate 7x2 to 1 group with 14 items

Relate 7x2 to 7 groups of 2 items in each group

Relate 2x7 to 7 groups of 2 items in each group 2+2+2+2+2+2+2 (2,4,6,8,10,12,14)

Transcript:Group 1 Operation Knowledge Knowledge Constructed (Oh .. .ini tambah ini berapa semua?) Sembilan Mathematical operation: Addition Relate 9x4 to 9+9=18,

Add 18+9=27,27+9=36 (Tambah lagi) Lapan belas (Tambah lagi) Hm ... dua puluh tujuh. (Tambah lagi satu?) Tiga puluh enam (Ok very good. Next one. Eight times five. Adik, sebut...) Lapan ... tok darab ... tak ada .... (Bentuk itu bolehjadi apa-apa, ok boleh lukis di mana-mana ... sebut, sebut. .. )

Satu. dua. tiga. empat. lima. enam. tujuh. lapan ... satu. dua. tiga. empat. lima, Mathematical operation: Counting Relate 8x5 to 5 groups with enam, tujuh, lapan ... Count & group Number 8 items in each group

I Grouping (Ok semua berapa?) Satu. dua. tiga, ... Mathematical operation: Counting

Count Number (Kuat sikit) Sebelas, dua belas. tiga bel as, empat belas, lima bel as, enam belas, tujuh belas. Mathematical operation: Counting lapan belas, sembilan belas, dua puluh, dua puluh satu, dua puluh dua, dua Count Number puluh tiga, dua puluh empat, dua puluh lima, dua puluh enam, dua puluh tujuh, dua puluh lapan, dua puluh sembilan, tiga Quluh, tiga Quluh satu, tiga puluh dua, tiga puluh tiga, tiga Quluh empat, tiga Quluh lima, tiga puluh enam, tiga puluh tujuh, tiga puluh lapan, tiga puluh sembilan, empat 12uluh. (Empat puluh. Ok next one. Tiga darab tiga.) Satu ... boleh bawahjua (Oh, very fast. Enam darab empat. .. ) Satu. dua, tiga, empat. .. Mathematical operation: Counting

Count Number

(Empat. . .Iepas itu ... cakap, cakap)

62

Transcript:Group 1 Empat, lapan. dua belas, lima belas, eh.. .enam belas.. .betul, dua puluh. dua puluh empat. Satu. dua. tiga. emuat. lima. enam, tuiuh. lauan. sembila. seuuluh. sebelas. dua belas. tiga belas, emuat belas. lima belas, enam belas, tuiuh belas, lapan belas, sembilan belas, dua uuluh. dua uuluh satu. dua puluh dua. dua puluh tiga. dua uuluh emuat. (Ok. kalau kita ingin tulis macam tadi itu, dua tambah dua tambah dua.. . macam mana di sini? Dapatkah kita tulis ayat matematik macam itu?) Dapat, dapat. (Ok macam mana dengan tiga ini, ayat matematik nya?) Hm.. .[l]tiga darab tiga sama dengan sembilan. [2]Tiga tambah tiga tambah tiga sama denean sembilan.

(Macam mana dengan enam darab enpat?. . .Enam) Darab.. .. (sebut) Tambah empat, darab empat sama dengan dua puluh empat (Lepas itu?) Enam tambah enam tambah enam tambah enam tambah.. .udah.. .enam.. .udah ini.. . empat . . .sama dengan dua uuluh emuat.. . (Sebut, sebut) IllLima tambah lima tambah lima tambah lima tambah lima tambah lima tambah lima tambah lima sama dengan empat uuluh ...[ 2lsembilan darab empat sama dengan ti pa puluih enam.. .eh.. .[l]~embilan tambah sembilan tambah sembilan tambah sembilan sama denean tiea ~ u l u h enam.. .tiga puluh dua. Tuiuh kali dua . . .i 1 isama dengan dua tambah dua tambah dua tambah dua tambah dua tambah dua tambah dua sama dengan emuat belas. (Macam mana dengan ini?) Tuiuh darab dua sama dengan emuat belas (Lepas itu?) Tuiuh tambah tuiuh sama dengan emuat belas.

(Ok. ini aktiviti nombor empat, kita tengok enam darab dua sama deng an...)

I Operation - 1 ~ n o w l e d ~ e I Knowledge Constructed

-

( Mathematical operation: I Addition ( Relate 6x4 to 6 groups of 4

[IIRetrieval: Recall [2]Mathematical operation: I ~ d d i i o n I .. I

Multiplication

Mathematical operation: Addition Relate 6x4 to 6+6+6+6=24 / Add Multiplication

Relate 3x3 to 3+3+3=9

Relate 8x5 to 5+5+5+5+5+ 5+5+5=40 Relate 9x4 to 9+9+9+9=36 Relate 7x2 to 2+2+2+2+2+

Mathematical operation: []]Add & [2]multiply

I Mathematical operation: I Addition / Relate 2x7 to 7+7=14

Addition

I Data examination: Read I Number

Transcript:Group 1 ( Operation (Jadi kalau kita kira ini, ini enam darab dua.. .jadi satu baris ini ada berapa bintang?) Enam.. . enam

(Kita Ada dua.. .baris, jadi macam mana kita dapat dua belas?) Hm.. . tambah (Ok. sekarang enam darab tiga? Macam mana?) Hm.. . tak ingat (Darab berapa?) Lapan belas (Macam mana dapat lapan belas?)

Data exploration: Examine: Identify

m.. . tuiuh darab enam . . .empat puluh dua.

Tambahkan Enam

(Soalan nombor dua, jika tujuh darab lima ialah sama dengan tiga puluh lima, berapakah tujuh darab enam? Jadi kamu diberi segitiga di sini, setiap bans ada.. .

I (Macam mana dapat empat puluh dua?) Tambah ... macam tadi [refer to activity one] (Macam mana? Terangkan.. .) Empat puluh sembilan tolak tujuh (Oh ya, pandai.ok. dua darab lima sama dengan sepuluh, dua darab enam?) Hm ... (Berapa? Cakap.. .) Enam.. .eh.. . enam belas.. .. (Enam belas?) Eh.. .kan enam belas.. .dua, emoat, enam. laoan. sebelas.. .dua belas.

Mathematical operation: Add

Retrieval: Recall Solution generation: Relate & inverse

Mathematical operation: Add & count

( (Tujuh darab dua sama dengan empat belas, tujuh darab tiga?)

Knowledge

Number

Addition Number

Multiplication

Addition Counting Number

Transcript:Group 1 Emuat belas.. . empat belas, enam . . .lima belas.. .enam belas.. . tuiuh belas.. .lavan belas.. .dua puluh.. ..dua uuluh satu

(Dua puluh satu. Empat darab lima dua puluh, enam darab enam tiga puluh enam.. .) Tiga puluh enam. tiea uuluh tuiuh, t i ~ a uuluh lapan. tiza wuluh sembilan. empat puluh. emuat puluh satu, emvat uuluh dua

(Now, kita tengok aktiviti lima, tengok contoh ini, lapan.. .lapan bahagi dalam empat petak, kita dapat dua tambah dua tarnbah dua tambah dua sama dengan dua darab empat. Maksud lapan ini boieh dianggap sebagai dua tambah dua tambah dua tambah dua. Macam mana dengan dua belas?) Macam tadi. (MuIa-mula bilang petak, syaratnya dia nombor yang kamu pilih mesti serupa.) Satu, dua. tiga, emuat, lima, enam. tuiuh. lavan. sembila, se~uluh. sebelas. dua belas

(Jadi berapa dalam petak itu?) Satu (Very good ... macam mana kamu tulis?) Ikut itu bah.. .dua belas darab satu.. .hm.. .dua belas.. .eh, belum.. . (Bolehkah tulis satu darab dua belas?) Boleh (Ada beza?) Tak ada beza

(Samakah? Sama. Ok, ini berapa petak?) Enam -

(Jadi setiapnya.. .) Dua (Macam mana kamu tahu dua?)

Operation Mathematical operation: Add & count

Mathematical operation: Add & count

Solution generation: Relate

Mathematical operation: Count Summarisation: Summarise

Data exploration: Examine: Compare

Data examination: Read Data exploration: Examine: Identify

Knowledge Addition Counting Number

Addition Counting Number

Number Counting

12x1 & 1x1 2

Number

Knowledge Constructed Relate 7x3 to 7x2 =14, then add 7 onto 14 to obtain 21

Relate 6x7 to 6x6 by adding 7 to 36

Relate 12 to 1+1+1+1+1+1+1+1+1+1+1 +I=12x1

(Enarn tambah enam dua belas.) Tok dua aja.. .hm.. .satu dua.. . (Sama dengan.. .) Dua darab dengan dua belas sama dengan.. .hm.. .hm.. . (Terangkan . . . ) Hm.. ..dua puluh empat (Dua puluh empat, adakah kita dua puluh empat di sana?) Dua puluh empat .:.eh.. .dua puluh dua, dua belas, dua, empat, enam. lapan, sepuluh, dua belas.. .ah ha.. .

(Ok next one, berapa petak semua?) Tiea.. .dua darab senang.. . -

Knowledge Constructed Transcript:Group 1 Sebab ada enam ~ e t a k , enam tambah enam dua belas

(Macam mana kamu tahu tiga itu?) Tahu, satu dua tiea

Mathematical operation: Count & add

Operation Solution generation: Relate

Data exploration: Examine: Identify

Solution generation: Relate

Knowledge Number of boxes

I Addition Counting Number

(Oh, macam itu.. .ok.. .sama dengan.) Dua darab enam.. . (Ok now you see it. Very good, seterusnya, tiga darab empat. ) Dua belas (Dua belas juga. Seterusnya . . .empat . . .sama dengan.. .cakap.. .cakap.. .) Empat darab tipa sama denean dua belas

(Ok seterusnya dua petak sahaja) Sekali lagi, satu saja.. .lima.. .lima tambah seouluh.. .idah.. ..enam tambah enam.. . .dua belas.. ..enam tambah.. . .enam tambah dua sama dengan dua belas

(Ok soalan terakhir, dua belas sama dengan . . .satu petak saja)

Number

Solution generation: Relate

Retrieval: Recall

Mathematical operation: Multiply

Solution generation: Trial & error Mathematical operation: Add

Number

Number

Multiplication

Multiplication

Addition

Relate 12 to 2+2+2+2+2+2

Answer to 1'' & 2"d part is 1 & 2 respectievely, so student deduce that answer to part 3 is 3

Transcript:Group 1 Adoi. .. Satu, dua, tiga, empat, enam.. .dapat dah?

(Cakap, macam mana? Kalau dia sekali saja. ..sekali saja) Sama dengan dua belas. ..dua belas (Ok now, cikgu mahu kamu bandingkan dua ini, ini dua belas, sama dengan satu petak dua belas, dua belas darab satu. Ini dua belas petak pun dua belas darab satu.. .) Satu darab dua belas.. .enam darab dua dua belas.. . tok kau, tok aku.. .. (Nak tukar?) Tak (Sekarang kita tengok dua puluh empat, enzm petak.. .) Dua belas.. .sini, dua, dua. dua, dua.. .dua. emuat, enam, lauan. seuuluh.. .eh.. . dua belas darab emuat.. . . . .

(Atau kamu nak guna bongkah kayu ini untuk tolong kamu)

Tambah dua tambah dua tambah dua tambah dua tambah dua tambah dua tambah dua tambah dua tambah dua tambah dua tambah dua tambah dua, satu dua, satu dua ... two, four, six, eight, ten, twelve, fourteen, sama dengan.. .two, four, empat eh ... two. four, six, eight, ten, twelve. fourteen, sixteen. eighteen, twenty. twenty two. twenty four (Sekarang kamu boleh guna kayu ini, atau kamu nak terusnya, dua, dua, dua.. .) Dua belas (Tapi ini ialah dua puluh empat.. . Satu dua tiga.. .macam mana? Dua juga.. . dua tambah dua tambah dua berapa sernua?) Lapan, dua tarnbah dua ernpat, lima, enam, tujuh.. .ernpat, lima, enam, tujuh, lapan.. . (Lapan saja) Tigalah.. .emuat.. .satu. dua, tiga, ernuat, emuat. emuat.. .enam.. .

Knowledge Constructed

Relate 12 to 12 (1 box)

Relate 24 to 2+2+2+2+2+2 +2+2+2+2+2+2 (2,4,6,8,10,12,14,16,18,20, 22,24)

Operation Solution generation: Relate

Solution generation: Trial & error Mathematical operation: Add

Mathematical operation: Add & count

Solution generation: Trial & error

Knowledge Counting Number

Addition

Addition

Transcript:Group 1 (Macam mana kamu dapat enam? Adik macam mana dapat enam?) Tok dua, tok enam (Enam, enam, enam, enam.. .) Hm.. . betullah.. .dua belas darab dua.. .dua puluh empat.. .enam darab dua.. .

(Macam mana, boleh? Tak boleh? ) Enam (Enam.. .ok kamu ada dua puluh empat. Nak masuk dalam petak.. .nak masuk dalam enam kotak, macam mana kamu letak?) Enam. tiga.. .

(Letak dulu, cuba dengan ini.. .nak masuk dalam enam kotak. Misalnya ada enarn kotak, macam mana? Masuk dalam enam petak, satu, dua, tiga, empat, lima, enam.. .lepas itu?) Tiga. . . .enam,enam.enam

(Belum lagi, belum habis) Satu. dua, tiga, e m ~ a t , lima, enam. tuiuh. l a ~ a n

(Kita mahu enam kotak. Macam mana kita agih? Nampak? Ok sekarang guna kayu ini.. .kamu kena agihkan, ada enam petakkan? Kita agih ke dalarn enam petak.. Tengok setiap petak ada berapa?) Enam - (Enam. Pasti?) Ya (Ok kamu cuba.. .cuba masuk satu, satu, satu.. .jadi sekarang setiap petak ada berapa?) Satu, dua, empat, enam, lapan, berapa? Sepuluh, dua belas (Jadi sekarang setiap petak ada berapa?) Dua -

(Tapi belum habis masuk lagi, macam mana?)

Operation

Solution generation: Trigger

Solution generation: Trial & error

Solution generation: Trial & error

Mathematical operation: Count

Data exploration: Examine: Identify

Data exploration: Examine: Identify

Knowledge

Multiplication

Number Counting

Number

Number

Knowledge Constructed

Relate 24 to 12x2

~ r a n s & ~ t : ~ r o u p 1 Empat belas, enam belas, lapan belas, dua puluh, dua puluh dua, dua puluh empat.. .tiga (Ok belum habis, ada lagi, agih lagi.. .ok apa yang kamu nampak sekarang?) Empat

(Ah, enam betulkah?) Salah. (Isilah, macam mana, empat petak?) Empat . . .empat petak (Kita letak empat petak, agih lagi.. .biar dia empat.. .) Empat. (Ini sudah empat.. .tambah lagi.. .apa yang kamu lihat sekarang? Sama tak tidak?) Tak, tak sama (Setiap petak samakah?) Tak (Ini enam.) Enam, enam, enam.. . (Semuanya ada berapa petak?) Enam, enam, empat.. . (Enam kayu dalam empat petak.. .) Enam.. .dua puluh empat (Seterusnya, tiga petak sahaja.) Agih dulu. .. bongkah ... lap an... lapan darab dua belas sama dengan dua puluh empat. ..satu, dua. tipa, emuat. lima, enam, tuiuh.. .sepuluh, satu, dua, tiga, empat, . ..sebelas. ..lima, enam, tuiuh. lauan, sembilan, dua belas, seuuluh. dua u.. .ini dua.. .sama dengan itu, dua emuat.. .dua. emoat. enam, lauan.. . enam, emuat ,... tipa ... tipa. satu. dua, tiga. empat, lima, enam, tuiuh, lauan ... tiga ... betul ... satu, dua, tiga, empat ... eh ... dua empat, enam, lapan, sepuluh, dua belas.. .dua belas.. . Dua belas darab dua. dua belas darab dua dua empat, dua belas boleh iua. dua darab dua belas betul, dua belas, dua ia boleh juga.. .

Solution generation: Trial Number & error, relate 8: inverse Counting Mathematical operation: Multiplication Count & multiply Summarisation: Summarise

Knowledge Constructed

Relate 12 to 6+6

Relate 12 to 3x4

Relate 12 to 3x4, 7+5

Relate 12 to 2+2+2+2+2+2

Knowledge

Row & column

Row & column

Row & column

Row & column

Multiplication Addition

Row & column

Addition Row & column

Transcript:Group 1 (Ok, now, last one, cikgu bagi dua belas bongkah kayu ini, kamu bina bangunan. Bina satu bangunan. Satu bongkah ialah satu bilik. Ok.. . syaratnya setiap tingkat mesti sama banyak bilik. Dua belas satu bangunan, semua sekali.. .Tak, ia mesti sama banayk setiap baris.. .Ini tak sama, ini dua, ini tiga, tak boleh.. . buat sekali, buat bersama) (Macam mana dengan bangunan ini? Ada berapa buah bilik semua?) Satu. dua, tiea. empat, lima. enam. tuiuh, l a ~ a n , sembilan. se~uluh, sebelas. dua belas (Ok dua belas, berapa tingkat?) Enam

(Satu, dua, tiga, empat, lima, enam, satu tingkat berapa bilik?) Dua (Dua, jadi kita boleh tulis sebagai.. .enam, macam mana kita dapat dua belas?) Enam darab enam (Enam darab enam?) Enam tambah enam.. . (Ataupun.. .) Enam darab dua.. . (Ada lagi?) Ada enam darab tipa.. .he.. . he.. .he.. ..tipa darab tiea.. .hm.. ..tiga, tiga.. .tiga . . .tuiuh.. .tuiuh.. .ties darab emoat.. .tujuh.. . tuiuh tambah denean lima (Di sini saja, tengok ini saja, tadi kamu kata enam darab dua. Ok kita dapat dua belas, enam tambah enam, ini enam, tambah enam itu, dua belas.. . apa lagi? Kalau kita tengok setiap tingkat ini, berapa tingkat ini?) Dua -

(Dual Empat (Tambah.. .) Dua tambah dua tarnbah dua tambah dua tambah dua tambah dua.. .

(Dapat berapa?) Dua belas

Operation

Data exploration: Examine: Identify

Data exploration: Examine: Identify

Solution generation: Relate

Solution generation: Relate

Solution generation: Trial & error & inverse

Data exploration: Examine: Identify

Mathematical operation: Add

Knowledge Constructed

Relate 12 to 4x3,3+3+3+3, 4 4 4

Relate 12 to 4 - 4 4

Relate 12 to 4 - 4 4

Relate 12 to 4 - 4 4

Knowledge

Row & column

Row & column

Row & column

Row & column

Row & column

Row & column

Row & column

Row & column

Transcript:Group 1 (Ok dua tambah dua, berapa kali?) Empat (Sekali, dua, tiga, empat, lima, enam ... itu sama dengan dua kali enam atau dua darab enam. Buat lagi, lain dari bangunan ini.) Tiga, tiga.. . biru pun ada.. .ah.. . tiga, tiga (Berapa semuanya?) Dua belas

(Dua belas juga, macam mana?) Empat darab tiga

(Ataupun) Tiea tambah tiga tambah tipa tambah tiga (Ataupun.. .) Hm.. .emoat tambah empat tambah empat tambah empat.. . (Very good, sekarang biru.. .lain lagi.. .) Empat, empat.. ..ini (Ok dah? Macam mana dengan ini?) Empat tambah empat tambah empat (Atau.. .) Tiga tambah tiga tambah tiga tambah tiga (Ataupun.. .) Dua tambah dua tambah dua tambah dua tambah dua tambah dua, dua darab.. . (Ini.. .) Empat darab dua.. . Emcat tambah empar tambah empat (Ya, empat tambah empat tambah empat, berapa kali?)

(Ataupun.. .) Hm.. . (Lagi ... ok ... buat lagi, bangunan ... Macam mana dengan ini?)

Operation

Data exploration: Examine : Identify

Solution generation: Relate .

Solution generation: Relate

Solution generation: Relate

Solution generation: Relate

Solution generation: Relate

Solution generation: Relate

Data exploration: Examine: Identify

Transcript:Group 1 Lima tambah lima

(Lima?) Eh.. .enam,enam, enam tambah enam

-

(Ataupun.. .) Enam darab dua (Macam mana kita bina satu bangunan yang paling tinggi.. .tinggi lagi.. .) Satu, satu. ..tinggi, tinggi, tinggi.. .paling tinggi. Satu, dua. tipa, emDat, lima, enam, tuiuh. lavan, sembilan. sepuluh. sebelas, dua belas (Ok. macam mana dengan ini?) Dua belas, dua belas darab satu. dua belas itu, satu tambah satu tambah satu tambah satu tambah satu tambah satu tambah satu tambah satu tambah satu tambah satu tambah satu tambah satu (Habis, macam manabuat satu lagi paling panjang?) (Ok now, macam mana dengan ini?) Hm ... satu. dua. tipa. e m ~ a t . lima, enam, tuiuh, l a ~ a n , sembilan, sepuluh, sebelas. dua belas, dua belas tambah satu sama dengan dua puluh empat.

(Macam mana?) Hm.. . he.. . he.. . he.. . (Ini ialah dua belas darab satu atau satu tambah satu tambah satu tambah satu tambah satu tambah satu tambah satu tambah satu tambah satu tambah satu tambah satu tambah satu. Ini dua tambah dua tambah dua tambah dua tambah dua tambah dua, kalau kita tulis ayat matematik, macam mana?) Dua tambah dua, satu tambah satu tambah satu, hm.. .enam darab dua.. .

(Enam darab dua, ok.. .hm.. .ini berapa tingkat ini?) Satu -

(Satu tingkat berapa buah bilik?) Dua belas

Operation Solution generation: Trial & error Mathematical operaton: Add

Solutiion generation: Relate

Mathematical operation: Count

Mathematical operation: Add Solution generation: Relate

Mathematica! operation: Count

Data exploration: Examine: Identify

Data exploration: Examine: Identify

Data exploration: Examine: Identify

Knowledge Addition

Row & column

Counting Number

Addition Row & coium?

Number Addition

Row & column

Row & column

Row & column

Knowledge Constructed

Relate 12 to 4+4+4

Relate 12 to 12x1,1+1+1+1 +1+1+1+1+~1+1+1+1

Relate 12 to 6x2

Rumah panjang (Ha.. .sure, mmah panjang, betul.. . ~ m a h panjang benvarna hijau.. .Boleh tulis satu ayat matematik? Macam mana?) Hm.. . sik tahu. . . enam darab empat (Baik, sekarang luta ada satu, dua, tiga, empat, lima, enam, enam bangunan, mana mempunyai paling banyak bilik?) (Ini, yang tinggi ini? Yang tinggi ini? Kita tahu ini paling tinggi, ini paling panjang and ini paling banyak tingkat. fadi betulkah yang kamu pilih itu mempunyai paling ban yak bilik?) Salah, ini banyak (Ini banyak, berapa semuanya?) Lima, enam.. .dua belas (Semuanya dua belas)

Operation Transcript:Group 1 (Kita boleh anggap dia sebagai.. .)

Knowledge Knowledge Constructed

Transcript: Group 2 Operation Knowledge Knowledge Constructed (Cikgu akan bagi kamu enam aktiviti, semuanya matematik. Kamu tiga orang perlu bekerja sarna dan bincang. Kalau boleh, selalu cakap supaya dapat dirakam, boleh? Tengok aktiviti pertama, aktiviti pertama ini berkaitan dengan apa?) Tambah Retrieval: Recognise Multiplication (Sudah belajar?) Dah (Di mana? Bila?) Dah lama, darjah satu (Macam mana dengan semasa di tadika, ada belajar ini?) Ada Ini soalan-soalan berkaitan dengan .. , Darab Retrieval: Recognise Multiplication

I (Jadi sudah sampai mana belajar darab?) Hm ... sifir lapan. sembilan Retrieval: Recall Multiplication

I (Boleh cuba sebut?) Boleh (Ok, macam mana?) Sembilan? Satu darab sembilan, sembilan. Dua darab sembilan, laQan belas. Retrieval: Recall Multiplication Tiga darab sembilan, dua puluh tujuh. Empat darab sembilan, tiga puluh enam. Lima darab sembilan ... emQat Quluh lima. Enam darab sembilan ... Iima puluh emQat...tujuh darab sembilan ... LaQan, emQat. .. enam Quluh ... tiga, laQan darab sembilan ... tujuh Quluh dua, sembilan darab sembilan .. .laQan Quluh satu, seQuluh darab sembilan ... sembilan Quluh, sebelas darab sembilan sembilan Quluh sembilan, dua bel as darab sembilan ... seratus laQan (Very good, ok sekarang kita mula. Siapa pun boleh buat. Tulis jawapan. Teruskan, kalau boleh, sebutlah, satu tambah satu ... ) Satu tambah satu, dua Mathematical operation: Addition

Add Tujuh tambah emQat, sebelas. Tiga tambah sembilan, sebelas. Lapan tambah Mathematical operation: Addition kosong lapan. Enam tambah lima sebelas. SeQuluh tambah dua ... eh ... tambah Add empat, dua belas ... Sembilan tambah emQat, tiga belas. Empat tambah enam, seQuluh. Tiga tambah lima, lapan. (Ah ... macam mana buat tiga tambah lima ... ada pakai jari?)

74

Transcript: Group 2 Tak (Terns dalam sini? Otak? Macam mana tiga tambah sembilan?) Sebelas, tiga tambah sembilan, sebelas (Sebelas, macam maria?) Ktra dalam otak (Pilib nombor mana dulu?) Sembilan.

(Kemudian?) Tiea I - I (Kemudian tambah tiga.) Ya (Ada gunakan jari tak? Tidak? Tak terns sembilan, sepuluh, sebelas, dud belas. Macam itu kira? Ataupun sembilan tambah tiga, terns dua belas? Ada sebut sembilan, sepuluh, sebelas, dua belas? Tak ada. Sembilan tambah tiga?) Sebelas, cikgu boleh tak tukarsoalan? (Boleh ... kalau nak semak boleh, sekarang buat darab, satu darab satu) Satu darab satu satu (Ok lima darab enam? Macam mana kamu buat? Macam mana dapat tiga puluh?) Kira lima daiab lima dua ouluh lima, lima darab enam. tiea ouluh

(Oh, dua puluh lima campur lima lagi tapi macam mana kamu tahu lima darab lima dua puluh lima?) Senane. Selalu hafal

(Selalu hafal, jadi kamu hafal dari satu darab lima, dua darab lima, tiga darab lima macam itu sampai lima darab lima?) Hm ... (Tiga darab sembil an... Kamu boleh tolong dia, sama bincang, Dua belas? Macam mana?)

Operation

Data exploration: Examine & locate

Data exploration: Examine & locate

Retrieval: Recall

Mathematical operation: Multiply & add

Retrieval: Recall Data exploration: Examine: Compare

Knowledge

Number

Number

Multiplication

Multiplication Addition

Muliiplication

Knowledge Constructed

Relate 5x6 to 5x5=25, then add 5 to 25 to get 30

Transcript: Group 2 (Sebut ... nak semak lagi? Dah? Ok kita masuk ke aktiviti dua. Tengok aktiviti ini. ia berkenaan denzan iom~atan darab. Ini ialah satu lom~atan. Tiga d a a b

Knowledge I

~ ~ . ~ . -~ ~ - dua bereni lompat sebanyak dua dari kosong sampai dua. Itc kira satu lompatan. Tiga kali jadi dia iompat tiga kali. ok? Ia lompat sekaii dua, kali ke dua lompat, empat, dua tambah dua sama dengan empat, lompat lagi sebanyak dua ... ) Enam (Dua tambab empat tadi ... ) Enam

I Operation I

(Kita boleh tulis agat matematik sebagai dua tambab dua tambah dua sama dengan enam, ok jadi cikgu mahu kamu buat sama. Ok lima darab tiga) Limadarab tiea ... dua belas ... lima belas. Tulis darab lima nak? (Kamu bincang, kuat sikit, bila cakap suara besar.1 Betul, w... Empat kali ... oklah .. .tipa. .. tiza ... tiga. ..rice.. tiga..

(Kuat sikit, kuat sikit ... Lepas itu sampai mana?) Sampai sebelas (Sambung lagi. Sampai ... ) S a m p ~ dua beias. ..dua belas.. . (Cakap, cakap ... sampai. .. lepas itu?) Lima belas ... tiga darab ... lima darab tiga lima belas.. . tiga, enam ... (Kuat sikit, tak dengar) Lime darab tiea, lima belas ... tulis lima ... lima tambah lima tambah lima ... lima tambah lima tambah lima, lima belas (Macam mana dengan lompatan itu?) Darab ... dua.. . Tujuh. .. empat ... enam ... empat, enam, lapan, ... sepuluh, dua belas, lima belas. .. empat.. . dua, dua tambah dua.. . (Temskan, kamu orang tidak cakaplah, senyap sahaja, jangan bisik.)

Data exploration: Examine: Identify

Retrieval: Recall

Mathematical operation: Multiplication jump

Multiplication

Multiplication jump

Knowledge Constructed 1

Relate 5x3 to 5 jumps with 3 steps in each jump (3+3+3+3+3)

Related 5x3 to 5+5+5 in the mathematical sentence

Transcript: Group 2 Operation Knowledge Knowledee Constructed i EmQat darab emQat. enam belas ... emQat. emgat. em12at, emgat. Emgat. lagan Retrieval: Recall Multiplication Relate 4x4 to 4 jumps with I ... pergi ke ... dua belas ... dua belas ... enam belas ... dah. Empat, empat .. sama Mathematical operation: jump 4 steps in each jump dengan emgat tambah emQat tambah emQat tambah. enam beTas. Multiplication jump & add Addition (4~8~12~16)

Data exploration: Examine: Multiplication 4+4+4+4~16

Identify (Ok tiga darab lima) Lima belas ... lime belas .. tiga ... lima? Enam pergi ke lima ... gerl2:i ke lima ... Data exploration: Examine: Multiplication Relate 3x5 to 3 jumps with pergi ke seQuluh ... Qen!i ke lima belas. Sitoklah, pergi sepuluh. Tok pergi ke Identify jump 5 steps in each jump sepuluh ... tiga darab lima sarna dengan tiga tambah tiga tambah til2:a tambah Mathematical operation: I Addition 15~IO~15 tiga tambah-tiga.lima belas MuJ-tiplication jump & add Multiplication Mathematical sentence

3+3+3+3+3~!5

(Sepu!uh darab satu, sepu!uh) Kosong pergj ke sepuluh ... pergi ke sepuluh ... Tok sebelas, betullah sepuluh ... Mathematical operation: Multiplication Relate lOxl to 1 jump with sepuluh darab satu sarna sepuluh, sarna dengan sepuluh tambah kosong ... tujuh, Multiplication jump, add & jump 10 steps in the jump. sepuluh ... Satu sepuluh tulis sepuluh juga ... Bukan multiply Addition (O~ 10)

Evaluate: Confinn Multiplication Reiate IOx!~IO+O (Apa yang kamu fikirkan? Kamu tengok soalan inl, lima darab tiga dan ini tiga darab lima, kita dapati jawapannya) i

Lima belas (Lima belas, apa yang cikgu tengok kamu tulis lima, lima, lima dan ini tiga, tiga, riga, tiga, tiga, macam mana banding dua ini? Bandingkan mereka, apa beza?) ltu lima, ini tiga .... habislah (Macam mana lompatan?) Ini sampai lima. sepuluh, lima belas ... sini ... tok salah tok, tok salah ... tiga, Data explanation: Explain Multiplication enam ... Dari riga pergi .. .iru salah, ini tiga, sembilan, tok salah pergi ke jump enam ... sembilan ... pergi dua belas ... pergj lima belas .. (Very good. Sekarang anda dikehendaki kelompokkan titik-titik di sini, yang penting titik dalam kelompok itu mesti sarna ban yak. Kalau karnu ada dua kelompok, satunya ada empat titik, satu lagi mesti ada empat titik juga? Tengok soaian, tujuh darab dua. Kamu kelompokkan dengan lukis pada titik-titik ini. Boleh lukis di mana-mana saja ... Tak faharn? Ah ... pemah cikgu mengajar tujuh darab dua?) Pemah

78

Transcript: Group 2 Qj>eration Knowle<lge Knowledge Constructed (Ah ... macarn mana kamu buat? Dalam satu kumpulan ada berapa?) Oh ... satu, dua, tiga. em:Qat. lima. enam. tujuh. la:Qan ... dua kali. Satu, dua mesti Mathematical operation: Multiplication Relate 7x2 to 7 groups with bulat dua MUltiplication jump jump 2 items in each group (Mana-mana pun boleh, asalkan bilangan titik sarna) I Empat, lima, enam, tujuh ... bulatkan tujuh ... betul I (Ok ini satu kelompok dah ... ) Satu. dua. tiga. empat. lima. enam. miuh, bulat tujuh samalah ... satu, Mathematical operation: Counting dua ... samalah .. Count & group Grouping

Number

I (Berapa kelornpok semuanya?) Tujuh Data exploration: Examine: Multiplication Relate 7x2 to 2 groups of7

Identify jump items (Empat belas. Macam mana dapat empat belas?) Sebab tuiuh, tujuh Mathematical operation: Addition Relate 7x2 to 7+7

Add Number (Dua darab tujuh?) TUjuh ... lukis ... empat belas (Macarn mana dengan sembilan dengan ernpat?) Tok empat ... kira ernpat ... Satu. dua. tiga, emgat, lima. enam. tujuh. lagan. Mathematical operation: Number Relate 9x4 to 4 groups of 9

I sembilan ... empat kali ... empat kali bulat. Sekali ... betullah tok ... agek ... agek ... Count, group & add items (Jawapan?) Sembilan sembilan ... tiga guluh enam (Ok lapan darab lima ... ) Lima kali, dua igek atau pun dua kali ... hitung, satu. dua. tiga. empat. .. sigek ... Mathematical operation: Grouping Relate 8x5 to 5 groups with

Group Number 8 items in each group (Ok, jawapan cmpat puluh. Tiga darab tiga.) Sekali enam, sembi Ian ... Data exploration: Examine: Multiplication Relate 3x3 to 3 groups with

Identify jump 3 items in each group Mathematical operation: Count & group

(Enam darab dua sarna dengan dua bel as. Sini ada dua barisan bin tang, setiap barisan enam bintang. Kaiau saya tak hafal, macam mana dapat dua belas dengan "tengok" bintang ini saia.)

79

Transcript: Group 2 Operation Knowled2e Knowlede:e Constructed Eoam tambah enam Mathemetical operation: Addition Relate 12 to 6+6

Add (Macam mana dengan enam darab tiga?) Lapan bel as. (Macam mana dapat lapan belas?) Ah ... dia tambah (Tambah lagi. .. tambah apa? Tambah berapa banyak?) Dua beJas tambah enam ... Mathematical operation: Addition Relate 6x3 to 12 by addding

Add Multiplication 6to12toget18 (Ok kalau oak lukis bintang, berapa kamu lukis untuk menunjukkan enam darab tiga?) I Lukis enam Data exploration: Exarr.ine:

Identify (Adakah kamu selalu buat macam ito?) Ada ... di rumah (Sekarang tujuh darab lima ialah tiga puluh lima, tujuh darab en am?) Empat puIuh dua. (Macam mana dapat empat puIuh dua?) Tiga puIuh lima tambah tujuh ... Mathematical operation: Addition Relate 7x6 to 7x5=35 by

I Add Multiplication adding 7 to 35 to get 42 (Kamu dapa! berapa untuk tujuh darab enam?) Hm ... tujuh darab enam. empat puluh dua ... (Empat puluh duajuga, macam mana dapa!?) Tambah satu lagi (Tambah satu? Tarnbah satu apa? Kalau tiga puluh lima tambah satu, tiga puluh enam.) Tujuh darab lima, tiga Quluh lima tujuh darab enam. emQat l2u1uh dua Mathematical operation: Addition Relate 7x6 to 7x5=35 by

Add & multiply Multiplication adding 7 to 35 to get 42 (Karnu tambah bcrapa banyak?) Tujuh Data exploration: Examine:

Identify (Tambah tujuh kepada ... kepada apa? Macam mana dapat crupat puluh dua?) Dia tambah. tLRa Duluh lima tambah tuiuh ... Data explanation: Explain

80

Transcript: Group 2 / Operation 1 Knowledge / Knowiedze Constructed (Sekarang iawab sodan-sodm di bawah ini, dua darab lima, sepuluh; dua 1 1 I -. darab enam?) Dua belas. (Tujuh darab dua, empat belas; tujuh darab tiga?) Dua ~ u l u h satu. Emuat. emuat. enam belas. emuat darab lima.. .dua ouluh. enam, enam tiea uuluh enam.. emDat puluh dua (Semua guna tambahkah?) Hm ... saya ingat (Hafal?) Tak, tuiuh darab enam sini emuat dua. sini enam darab ruiuh. terbalikkan. iadi enam darab tuiuh sama tuiuh darab enam (Ini aktiviti kelima, contoh ini, lapan ini boleh dikira sebagai dua darab empat bererti dua tambah dua tambah dua tambah dua, empat kali. Empat kali dua. Sekaranz macam mana dua beias? Semua ini dua belas. taDi uetaknva tidak - . . sama banyak, yang penama ini ada berapa?) Satu. dua. tisa. empat. lima. enam. tuiuh, lauan. sembilan. se~uluh. sebelas. dua belas ~ e t a k (Ini ada enam, empat, tiga, dua, satu, apa yang penting ialah nombor di dalam petak-petak ini mesti sama. Misalnya, dua, dua, dua, dlia semuanya sempa. Ok kamu boleh mula. Cakap.)

satu.. .dua belas (Setenisnya, berapa petak?) Enam petak. Dua, dua. dua. dua.. . (Macam mana kamu dapat dua, dua, dua, dua?) Kira tiea. sembilm, empat belas tak cukup. Dua. empat. lauan . . .dua, emoat. enam, laoan, seuuluh. dua belas ... dua (Ok) Dua. empat ... emoat ... Emuat. emuat ... tiua, tiea

(Sebui) Tiga, tiga ... tiga, tiga, tiga ... tiga, enam, sembilan. tiea. dua belas darab tiga, liga puluh enam.

Retrieval: Recall

Retrieval: Recall

Solurior: generation: Relate & inverse

Mathematical operation: Count

Solution generation: Relate Summarisation: Summarise

Sohtion generation: Re!ate

Data explanation: Explain

Solution generation: Trial & error

Mathematical operation: Add

Multiplication

Multiplication

Multiplication

Number Counting

I Addition Multiplication

Addition

Addition

Addition

Relate 12 to 1+1+1+1+I+l +I+l+l+l+l+l=12xI

Relate I2 to 2+2+2+2+2+2

Relate 12 lo 3+3+3+3

Transcript: Group 2 (Tiga petak lagi) Empat ... dua belas darab empat sama dengan empat puluh lapan. (Dua petak?) Lima ... enam. enam ... duo belas darab enam sama dengan ... enam puluh

(Satu petak saja) Dua belas darab dua belas sama dengan seratus empat puluh empat (Seterusnya, dua puluh empat. Macam mana dua puluh empat?) Tiea. tiea ... emoat. empat.. .satu. dua. tiea. emoat. lima. enam ... Enzm, enam, i u o h : : ~ ern~Jl;ih. d ~ n . e.3pit. i ~ p i r 2rn321. :3mr J U ? hcl?,. en?n S c l ~ i 3u:i r u i ~ t ti2 rulun c n , ~ . J J ~ puiuh cnip3t Jayat Ju? helj,. i u ~ p.lluh empat ... empat. empat. empat ... Dua darab empat berapa? Dua belas darab empat.. .Dua, empat berapa? (Ok seterusnya, empat petak) Lima. lima.. .laDan.. .enam, enam.. .ah, dua belas. dua belas ...

Solution generation: Trial & error Mathematical operation: Add

1

Solution generation: Trial & error Mathematical operation: Count &add

(Kuat sikit cakap) Tuiuh. enam ... daoat. Sik daoat ... tuiuh. tuiuh. dua belas ... tuiuh. tuiuh. tuiuh ... Solution generation: Trial tuiuh. tuiuh. emnat belas. dua ~ u l u h satu. Sik boleh tujuh ... enam ... Tujuh, & error empat, empat, lapan ... tujuh, lapan ... hm ... tujuh. lapan ... tiga igek itu. Tujuh, Solution generation: Relate tiga, dua belas. ..enam. enam. enam. enam, dua belas, i a ~ a n belas. dua uuluh -... Betullah empat, empat ... lapan. Dua puluh empat darab enam ... enam ... dua puluh empat ... lapan, empat darab enam. Tok empat. . ..empat, empat. .. lapan (Kita isi petak dulu, lapan, lapan, lapan, teruskan) E m ~ a t , emoat. empat ... sik boleh. Emuat ... dua belas ... Solution generation:Trial &

error (Guna bongkah kayu ini. Sekarang kita agih ia, dua petak, bahagi habis dan tengok berapa daiani petak, dua petak ... mesti sama.) Empat, empat, Iima ... lima ... empat (Semua dua puluh empat, belum habis. Agih lagi.)

Addition

Addition Multioiication

Relate 12 to 6+6

Relate 24 to 4+4+4+44+4

Relate 24 to 6+6+6+6

Transcript: Group 2 Operation Knowledge Knowled~e Constructed Tujuh, tujuh .. .lapan, lapan enam puluh empat, sembilan, sernbilan lapan puluh satu ... Dua puluh satu, dua puluh .... (Berapa?) Satu, dua. tiga. emQat.lima. enam. tujuh.1al2:an. sernbilan. seQuluh, sebelas. Mathematical operation: Number dua belas ... Dua belas darab dua belas sarna dengan seratus empat puIuh empat. Count Counting (Jadi satu petak ada berapa?) Dua belas. Dua, empat. .. tigalah ... tujuh, tujuh, tujuh, dua puluh satu Data exploration: Examine: Number

Identify Counting Solution generation: Trial & error

(Kamu boleh buat sekali, agih ke dalam petak. Berapa?) Lapan petak ... Dua. dua Solution generation: Trial Number

& error (Mesti berapa kumpulan itu ... berapa kumpulan? Petak berapa?) Empat, empat, empat (Itu berapa petak?) Lapan (lni, berapa petak?) Enam. Dua. dua, dua, dua, dua, tiga. em.Qat, lima, enam, tuiuh. laQan ... satu, Solution generation: Trial Number Relate 24 to 2+2+2+2+2+2 dua, tiga ... satu, dua, tiga, empat...bolehlah, tigalah, tujuh, tiga. Dua, tiga, & error Addition +2+2+2+2+2+2 empat, lima ... dua, dualahnya, dua. emQat. enam. lagan. seQuluh. dua beJas, Mathematical operation: emQat belas, enam belas. dua Quluh dua Quluh dua, dua QuIuh emQat. .. dua, Count & add dua, dua (Ah ... cikgu tertarik dengan yang kamu tulis. Di sini, misalnya dua tambah dua tambah dua tambah dua tambah dua tambah dua sebenamya dua belaskan? Tapi kamu tulis dua belas darab dua. Dua puIuh empat Dua belas menj adi dua puIuh empat, bolehkah? Sarna tak dua belas dengan dua puluh empat?) Tak Data exploration: Examine: Number

Compare (Tak sarna, tapi tengok contoh ini, lapan, dua darab empat berapa?) Lapan (Sini, dua belas menjadi dua puluh empat. Apa yang tidak sesuai? Sebenamya dua kali, dua kali, tiga kali, empat kali, lima kali, enam kali. Macam mana sebenamya?)

83

Tipa, tiqa ... dua belas. Dua ouluh empat ... laoan darab tiea sama denzan dua

Transcript: Group 2 Oh ... tahulah. (Dua belas darab enam. Apa yang bemiang di sini?) Lapan.. jawapannya empat darab dua (Dua belas) Enam darab dua. .. (Enam darab dua masih ... ) Dua belas (Macam mana ini?)

k a w ini. Kamu bina banmnan. Satu ini kira satu bilik dan macam ini dikira ~~~

dua tingkat. Satu tingkat terdiri daripada dua bilik dan ini lagi satu tingkat kamu guna dua belas ini sekali~us. Susunan boleh tinggi, boleh panjang, suka hati. Tapi, ia mesti sama, setiap tingkat mesti sama banyak bilik.) (Ok macam mana dengan ini, semua ada berapa buah bilik?)

Operation

Mathematicd operation: Multiply

I

Knowledge

Data exploration: examine:

(Macam mana kita boieh daoat dua belas itu?i / Identify

biiik. lima bilik. enam. tuiuh. l a ~ a n . Data expianation: Explain

Kita kata ini satu tinzkat. Ada beraoa

Multiplication

Number

. . . buah biiik?)

Data exploration: Examine:

(Sini ada tiga tingkat. Macam mana kita dapat kaitkan tiga dengan empat ini? Kita kira satu, dua, tiga, empat, lima sampai dua belas. Sekarang ini ada empat bilik dan setinggi tiga tingkat, macam mana kita kaitkan? Macam mana kita dapar dua belas? Macam mana kita hubungkan nombor tiga dengan nombor empat. lni tiga, ini empat supaya kita dapat dua belas)

Identify

. . dapat dua belas, kita tengok sini, ad= tiga ta ibah tiga tambah tiga tambah tiga, kita pun dapat dua belas. Jadi kita anggap bangunan ini sebagai tiga tambah tiga tambah tiga tambah tiga, empat kali atau empat tambah empat tambah empat, tapi tiga kali. Ataupun kita kata empat darab atau empat kali tiga. Jadi ia pun dua belas.) (Buat la@, tapi kali ini lain dari pertama) Dua. dua. dua.. tiea. tiea, tiea ...panj ang ... tiga, tiga aja

(Sekarang satu tingkat ada berapa biiik?)

(Ada berapa tingkat semua?) &@

(Semua ini berapa bitik?)

(Kita boleh kata sebagai tiga Iamb ah... ) Tiea tambah tiea tambah tiea tambah tica (Ataupun tiga kali ...I Empat (Atau empat kali tiga ... macam manadengan ini?) Empat tambah empat tambah empat (Tiga kali empat ... kita boleh sebut sebagai tiga darab dengan empat. Kita daoat.. .beraoa bilik?) Dua belas (Buat lain lagi. h'ak buat tinggi, lebar, panjang, tinggi semua boleh.) (Ok ini panjang, saN tingkat ada berapa buah bilik?) Lima (Lima?) Enam

Solution generation: Trial & error

Data exploration: Examine: Identify

Data exploration: Examine: Identify

Data exploration: Examine: Identify

Solution generation: Relate

Solution generation: Relate

Data exp1oration:Examine: Identify

Row &column

Row &column

Row &column

Row & column

Row & column

Relate 12 to 3+3+3+3

Relate 12 to 4+4+4

Transcript: Group 2 Dua

(Kita boleh anggap sebagai.. .enam.. .) Enam tambah enam

(Ataupun enam.. .) Enam darab dua

(Ataupun dua.. .) Dua tambah dua tambah dua tambah dua tambah dua tambah dua

(Berapa kali?) Ah.. .enam kali (Ok ini bangunan yang panjang. Buat lain lagi. Bangunan ke-empat ada berapa tingkat semua?) Tuiuh tingkat.. .

(Tujuh tingkat?) Enam

(Ok enam tingkat. Ada berapa buah bilik semuanya?) Dua belas

(Macam mana dapat dua belas?) Hm.. .dua tambah.. .dua.. .enam tambah enam. enam darab dua. dua darab enam (Lagi satu) Dua darab enam (Sudah disebut. Kamu tadi kata itu, tambah apa pada awal? Kita sudah sebut enam darab dua dan juga dua darab enam.. .enam tambah enam.. . apa lagi yang tinggal?) Enam.. .dua tambah dua tambah dua tambah dua tambah dua tambah dua (Ya, dua tambah dua tambah dua tambah dua tambah dua tambah dua sama dengan enam kali dua)

Operation Data exploration:Examine: Identify

Solution generation: Relate

Solution genration: Relate

Solution generation: Relate

Data exp1oration:Examine: Identify

Data exp1oration:Examine: Identify

Data exp1oration:Examine: Identify

Solution generation: Relate Summarisation: Summarise

Knowledge Row & column

Row & column Addition

Row & c o l u m ~ Multiplication

Row & colitmn Addition

Row & column

Row & column

Row & column

Row & column

Knowledge Constructed 1

Relate 12 to 6+6 ! Relate 12 to 6x2

I I

Relate 12 to 2+2+2+2+2+2

Relate 12 to 2+2+3+2+2+2, 6x2,2~6,6+6

I

Data exploration:Examine:Identify

(Ada lagi yang lebih tin& da-i ini?)(Ok berapa semuanya?)Dua

(Satu, dua, tiga, empat, lima, enam; tujuh, iapan, sembilul, sepuluh, sebelas,dua b&s. Macam mana kita sebut kali ini?)Enam tambah enam...satu rambah sag . ..ah...dua belas darab satu. satutambah sate tambah saul tambah sate tambah sat” tambah sat” tambah sat”tambah satu tambah sate tambah satu tambah sat” tambah safu(Jumlahnya...dua b&s Ada lain?)Dua darab enam.. .Menan Kuala Lumpu:(Bust yang p&w)Panjang...satu, dua...buat panjangkah?(Sekarang berapa tingkat?)g&l

(Berapa bush biiik?)Dua

(Macam mana lata sebut d&m ayat matematik?)Hm. .dua belas .ah.. .dua b&s darab satu.. .ah.. .satu tambah satu tambahsatu tambah sat”.. .(Satu tambah satu tambab satu tambab sate juga? l n i p u n sat” tambah satujHm...dua tambah dua tambah dua tambah dua tambah dua tambah dua...tigarambah rizza tambah tiea tambah tiea, enam tambah enam.(Tadi kita guna dua tambah dua sebab ini dua bilik, dua bilik, dua bilik. Ini sat”tingkat saja sekahgus...)&

(Kita ada enam bangunan, mana sate mempunyai paling banyak bilik?)Tok, tok(Yang hijau? Yang merah? K&u kita kin, berapa bulb bilik semuanya?)Dua b&s(Ok yane hijau iru berapa buah bilik?)

Solution generation: Relate Ram 8r column Rela te 12 to 6+6, l+l+l+lSummtisation: Summa&e +1+i+1+1+1+1+1+1,12xl,

Solurion generation: Relate Row &column / Relate 12 IO 2x6

Data exploration:Examine: Row &columnIdentify

Data exploration:Examine: Row & columnIdentify

Solution generation: Relate Rou, & column Rela te 12 to l+I+i+l+I+lSummarisation: Sumnmise +1+1+1+1+1+1,12x1

Solution generation: Relate Row&column Relate 12 to 2+2+2+2+2+2,3+3+3+3,6+6

Data exploration:Examine: Ron 8: column

Transcript: Group 2 Operation Knowledee Knowledee Constructed Dua belas (Yang putih?) Dua belas, sernua dua belas ...

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dua. Ini banyak petak, berapa semuanya?) Satu. dua. tiea. emuat. lima, enam, tuiuh. lapar.. Lapan petak.

I (Dua belas tambah dua belas sama dengan dua puluh empat Dua belas darab

(Macam mana kira itu kalau lapan petak?) T&g (Betulkah? Bukan empat? Macam mana kira, ada cuba dua?) Cuba tiga terus (Yang terakhir ini, berapa petak ini?) Satu. dua. tiea. emuat, lima, enam, tuiuh, lauan. sembilan, seuuluh, sebelas, dua belas. Dua belas uetak. Hm.. .dua. dua.. .

I

(Satu tak bolehkah? Kalau satu berapa petak saya perlu?) Dua puluh empat (Kenapa? Satu darab dua puluh empat. Dua belas itu berapa kali?) Dua darab dua belas sama denean dua puluh emuat

(Ok dua darab dua belas. Aktiviti terakhir. C i k p bagi bongkah kayu. Kamu akan bina bangunan dengan kayu ini. Setiap tingkat mesti sama banyak bilik. Kalau kita mula dengan enam, tingkat dua pun mesti enam. Kalau kita buat lima ia mesti lima, ok? Satu bongkah ialah satu bilik.) (Kira tengok bangunan pertama ini, bangunan ada berapa tingkat?) Ti ea - (Setiap tingkat itu ada berapa buah bilik?) Empat

(Ada berapa buah bilik semuanya dalam bangunan ini?) Dua belas

(Macam mana adik dapat dua belas? Kuat cakap.) Satu darab empat, empat darab dua. i a ~ a n : emuat darab tipa. dua belas

Mathematical operation: Count

Solution generation: Relate

Mathematical operation: Count Solution generation: Trigger

Solution generation: Relate Summarisation: Summarise

Data exploration: Examine: Identify

Data exploration: Examine: Identify

Data exploration: Examine: Identify

Mathematical operation: Multiply

Knowledge

Number Counting

( Transcript: Group 3 1 Operation

Number Counting

1

Multiplication

Number Row & column

Number Row & column

Number

Multiplication

Knowledge ~onstructed-

Relate 24 to 2+2+2+2+2+2 +2+2+2+2+2+2

Relate 24 to 2x1 2=24

Relate 12 to 4x3

Transcript: Group 3 Op~ration Knowledge Knowledge Constructed I

Boam darab dua Solution generation: Relate Row & column Relate 12 to 6x2 Multiplication

(Kalau oak guna tambah macam mana?) Eoam tambah enam Solution generation: Relate Row & column Relate 12 to 6+6

Addition (Ataupun? Apa Jagi?) Dua tambah dua tambah dua tambah dua tambah dua tambah dua Solution generation: Relate Row & column Relate 12 to 2+2+2+2+2+2

Addition (Buat lain lagi, cuba buat bangunan yang tinggi lagi dan ini) (Ada berapa tingkat tinggi bangunan ini? Jumlah bilik?) Dua bel as Data exploration: Examine: Number

Identify (Sekali kita boleh kaitkan enam dengan dua belas. Macam mana kira kaitkan?) Dua darab enam Solution generation: Relate Row & column Relate 12 to·2x6

Multiplication (Selain dari itu?) Boam darab dua. enam tambah enam. dua tambah dua tambah dua tambah dua Solution generation: Relate Row& column Relate] 2 to 6x2, 6+6, tambah dua tambah dua & inverse Multiplication 2+2+2+2+2+2

Addition (Ok kita boleh simpulkan ada banyak pilihan untuk dapat dua belas. Kita boleh guna dua, enam, dan sebagainya tapi ini bukan bangunan yang paling tinggi yang adik boleh bina. Buat lagi satu lebih tinggi dari inL) (Berapa bilik semua? Dua belas juga. Sekarang macarn mana ayat matematiknya? Satu tingkat satu biIik dan ia ada dua belas tingkat.) Saru darab dua belas Solution generation: Relate Row & column Relate 12 to Ixl2

Multiplication (Ada cara lain? Kalau kita nak guna tarnbah macam mana? Satu .. ) Satu tambah satu tarnbah satu tambah satu tambah satu tarnbah satu tambah Relate 12 to 1+1+1+1+1+1 satu tambah satu tambah satu tambah satu tambah satu tambah satu +1+1+1+1+1+1 (Berapa kaii?) Dua belas kali Data examination: Number

Examine: Identify (Ada lagi cara lain?) Sebelas tambah satu ! Solution generation: Relate Number Relate 12 to 11+1

98

Transcript: Group 3 Operation Knowledge Knowledee Constructed (Tapi sebelas dengan satu tidak sarna. tengok tadi, kita buat dua, dua, dua ... nombomya sarna.) (Sekarang bangunan yang lebih panjang dari yang ada di sini. Boleh buat lagi yang lebih panjang dari itu. Lagi? Dah?) (Sekarang berapa tingkat bangunan ini?) Satu Data exploration: Examine: Number

Identify Row & column (Tapi biliknya ada ... ) Dua belas Data exploration: Examine: Number

Identify Row & column (Ayat matematik? Macam mana? Kuat sikit, eoam tambah enam, da,.'; mana enam itu? Ada enam? Sekaligus berapa? Dua belaskan? Dua belas sekah saja, lOta boleh tulis sebagai saru tingkat dengan dua belas bilik semuanya. Kalau pakai dua belas, macam mana kita tunjuk dua bel as? Bangunan tadi boleh dianggap sebagai dua belas darab satu, satu tambah satu tambah satu ... atau satu darab dua belas. Sekarang satu tingkat, tapi ada dua belas buah bilik. macam mana ayat matematik?) Satu tambah satu ... Mathematical operation: Addition Relate 12 to 1+1+]+1+1+1

Add Row & column +1+1+1+1+1+1 (Sarna ataupun dua belas. Tadi dua belas, satu tambah satu tambah satu. Sekarang dua belas sekali jadi?) Dua belas, satu darab dua belas ... yang tadi dua belas darab satu ... sebab satu tambah satu ... (Antara bangunan-bangunan ini mana satu bangunan mempunyai paling banyak bilik?) (Yang biro? Kita ada enam bangunan ... yang mana satu paling banyak bilik? Cuba kira berapa banyak bilik?) Dua belas, dua belas, dua belas, dua belas (Sarna sa;a, tidak ada yang paling banyak bilitL

99

Transcript: (;roup4 Operation Knowledge Knowledge Constructed (Cikgu akan bagi adik enam aktiviti matematik. Kamu boleh bincang Apa yang adik cakap akan dirakam. Cakap kuat. Tak boleh cakap perIahan. Jangan takut salah? Ok tengok aktiviti ini, ada duajenis saalan. Apa yang adik nampak: dalam aktiviti pertama ini?) Tambah Retrieval: Recognise Addition (Ok, kalau adik pandai buat, setiap anak tangga ini ada satu saalan, adik kena jawab, lepas itu oaik satu tangga lagi dan akhimya boleh oaik kereta ... macam mana dengan yang dibawah ioi?) Darab Retrieval: Recognise Multiplication (Sudah belajar darab?) Dah (Dah sampai sifir berapa?) Sembilan Retrieval: Recall Number

Multiplication (Dah pandai? HafaP) Dah (Sembilan darab empat berapa?) Tiga puluh enam Retrieval: Recall Multiplication (Macam mana dapat? Cepa! sangat. Macam mana kamu dapa! tiga puluh enam? Tak tahu? Sembilan darab tujuh?) Sembilan darab tujuh ... enam puluh tiga Retrieval: Recall Multiplication (Macam mana dapat enam puluh tiga?) Otak (Tak pakai jari? Ada pakai campur-campur atau tambah-tambah? Tak ada! Ada sebut bermula daripada sembilan satu sembilan, sembi Ian dua lapan belas? Ada mula dari sana? Tak.? Terus ke sembilan darab tujuh? Sembilan darab lima berapa?) Empat puluh lima Retrieval: Recall Multiplication (Oh ... macam mana dapat?) Otak (Ada hi tung dari sembilan satu, sembilan dua ... tak? Macam mana adik mengira tadi, cikgu nampak adik buat macam ini, apa yang adik fikir? Sembilan darab enam.) Hm ... Hma puluh empat Retrieval: Recall Multiplication

100

Transcript: Group 4 Operation Knowled.e Knowledge Constructed Tujuh tambah empat ... sebelas. Tiga tambah sembilan ... dua belas. Lapan Mathematical operation: Add Addition tambah sifar, lapan. Sembilan tambah empat, tiga belas. EmQat tambah enam ... sepuluh. Tiga tambah lima, lapan. (Macam mana? Otak? Nombor mana dalam otak? Tak ada simpan mana satu nambor dalam oral<, tak ada? (Kita mula dengan damb. Ok satu darab satu ... satu. Lima darab enam ... tiga Retrieval: Recall Multiplication puluh. Tiga darab sembilan ... dua puluh tujuh. Lapan darab kosong, kosong. Tujuh darab dua ... empat belas ... empat darab empat. Enam belas ... Lapan darab iima ... berapa? Ernpat puluh ... satu darab sepuluh ... cakap) Satu darab sepuluh sepuluh Retrieval: Recall Multiplication (Enam darab tujuh ... ) Empat Quluh dua Retrieval: Recall Multiplication (Ok macam mana dapat satu darab sepuluh ini? Ada guna tambah-tambah tadi? i\.tfacam mana dengan tiga darab sembi Ian? Terus hafal? Aktiviti kedua, lompatan darab, pernah tengok ini?) Pemah (Kita nak tunjuk tiga darab dua atau tiga darab dua. Satu lompatan ini menunjukkan dua. Jadi ada riga kali maka ia lompat tiga kali. Sekali lompat dua, lompatan kedua dah sampai empat. Dua tambah dua dab empat. Kemudian lompatan,ketiga, sejauh dua lagi. Dua tambah empat tadi jadi enam. Ayat matematiknya tiga darab dua boleh tulis sebagai dua tambah dua tambah dua dan ia sarna dengan enam. Jadi cikgu mahu adik buat seropa untuk semua ini. Lima darab riga. Macam mana lima darab tiga? Cakap. cakap kuat.) Lima belas Retrieval: Recall Multiplication (Macam mana lompatan?) Lompat sampai lima belas (Lompat teros sampai lima belas? Lima kali tiga ... ) Lima belas (Tolong dia. Mengapa lukis dua, dua, dua itu? Lompat dua? Bincang. Macam mana? Lompat berapajauh? Lima? Mengapa lima? Cakap saja. Cakap kuat-

I kuat.) --- --

102

I Transcript: Group 4 1 Operation I Knowledge I Knowledge Constructed I Lima ...p erg , .

(Lompat berapa kali?) Lima ...&... (Ok jawapannya berapa?) Lima belas (Tulis lima belas. Macam mana dengan ayat matematik? Lima darab tiga sama dennan Lima tambah lima tambah lima ... Sekarang pergi ke dua darab tuiuh. - -. - Macam mana dengan lompatannya berapa kali?) Dua.. .tuiuh kali ...

(Sampai? Kuat sikit cakap. Lepas itu samp ai... ) Tujuh (Cakap kuat tak payah bisik, jawapan berapa? Cikgu nak dengar kamu cakap kuat-kuat ... dua tambah dua .. .sebut. ..) Dua tambah dua emoat. emuat tambah dua enam. enam tambah dua lapan. laoan tambah dua seouluh. seouluh tambah dua. dua belas. dua beias tambah duaemuat belas ... Dua emuat. enam. lauan. seouluh. dua belas. emuat beias. (Ok empat darab empat.. .berapa jauh ia lompat?) Empat.. .emuat

(Berapa kali ia lompat? Empat sampai ke iapan. ..dua belas.. .enam belas. Ayat matematik sama dengan empat darab empat sama dengan ... ) Emoat tambah emoat. .. (sebut) Empat tambah emuat tambah emoat tambah emoat sama dengan enam belas

(Tiga darab lima sekarang.. .)

ath he ma tical operation: Multiplication jump

Data exploration: Examine: Identify Mathematical operation: Multiplication jump

Evaluation: Confirm Mathematical operation: Add &count

Data exploration: Examine: Identify Mathematical operation: Multiplication jump

Mathematical operation: Add

Relate 2 x 7 to 2+2=4, 4+2=6,6+2=8,8+2=10,10+ 2=12,12+2=14

Relate 4 x 4 to 4+44+4 = 16

/ T r a ~ c r i p t : Group 4 / Operation j Knowledge I (Yane mana patut lompat, vane mana? Cikrm mahu kamu bandinekan. Rasa / I

nak Gkarkah'atau dua-duahiterima? ~ u a - & a betul. Luks~an mi betui untuk yang lni atau betul untuk yang itu, nlacam mana7 Nak tukar? Tak nak tukar, 1

i setuiu semua?) 1 Setuju ... (Ok kita tengok aktiviti tiga. Adik dikehendaki rnembulatkan atau

I kelompokkan, buat kumpulan untuk menunjukkan tujuh darab dua. Adik kena kelompokkan fitik-titik. ~ a n g penting setiap kumpul& itu titiknya lnesti sarna bilanean. Kamu boleh lukis di mana-mana asalkan ia mesri sama banvak iaitu

I . . - jlka tiga dalam satu kelompok, kelompok l a g pun mesti tiga. Takboleh satu kumpulan satu, satu kumpulan lain dua. satu kumpulan lain tiga. Kamu tengok soalan tujuh darab dua, boleh bincang dan Iukis.) Dun. emoat, enam, lapan, sembilan. sepuluh. dua belas. emoat belas.

/ (Kuaf sikit, iadi iawapan berapa?) - - . Empat belas (Macam manadaoat emoat belas itu? Ha ... senwm saia. cakaolah. Meneaoa -. I adik tak kumpulk&tiga.titik atau pun empat tiiik, mengapa dua titik saja?) Sebab sifir dua. (Oh ... sifir dua, dan mengapa tuiuh kumpulan?) - . .

I Untuk dapat tahu jawapan (Jadi macam mana kamu dapat empat belas ini?) Kira (Macam mana kira?) Dua. emoat, enam. laoan, sembilan. seauluh. dua belas. empat belas

(Adik pakai tambahkah? Dua tambah dua tambah dua tambah dua tambah dua tambah dua tambah dua sampai habis. Sekarang macam mana dengan dua

/ darab mjuh? Macam mana nak lukis?) Tujuh M i ... salah... [Berapa titik satu kumpulan?)

Data exploration: Examine: Identify Mathematical operation: Group &count

Counting Grouping Number

Counting Number

Relate 7 x 2 to 7 groups with 2 items in each group.

I Relate 7 x 2 to 2+2+2+2+

(Cukupkah? Jumiah berapa?) Elnuat belas juga

Transcript: Group 4 / Operation / Knowledge I Knowledge Constructed

(Ok mengapa adik tidak lukis macam ini? Tak boleh? Salah? Ok macam mana dengan sembilan darab empat?) Empat, lapan, sepuluh ... empat, lapan, dua belas ... enam belas ... empat. puluh .. .tipa ouluh ... tiea. emoat. lima. enam. tuiuh. lapan, sembilan. seouluh, sebelas. duo belas. tiea belas, empat be la , lima belas. enam belas. tuiuh belas, Iauan belas. sembilan belas. dua ouluh, dua ouluh satu. dua uuluh dua. dua puluh tipa. dua ~ u l u h emoat. dua ouluh lima. dua uuluh enam. dua ouluh tuiuh. dua ouluh lapan. dua ouiuh sembilan.. . (Ok lapan darab lima) Satu, dua. tiea. empat, lima ...

Data exploration: Examine: Identify

(Berapa titik?) . . L:ma -

(Lepas itu?) Seouluh. lima belas, dua ouluh. dua wluh lima. tina ieek ... tiea puluh...tiza puluh Lima ... emuat ouluh. Satu. dua. tiza. emuat, lima. enam, tuiuh. lapan..

Grouping Number

(Tiga darab tiga ... cakap kuat-kuat, berapa titik satu keiompok?) Tiza -

Relate 2 x 7 to 2 groups with 7 items in each

(Ada berapa kumpulan?)

hlathematical operation: Group 1 Mathematical operation : Count

Counting Number

Data exploration: Examine: Identify Mathematical operation: Group, count & add

Data exploration: Examine: Number Identify Counting

Counting, grouping Number Addition

Data exploration: Examine: Identify Mathematicai operation: Group &count

Mathematical operation: I counting Group &count Grouping

Number

Counting Grouping Number

group. I

Data exploration: Examine: Identify

Data exploration: Examine: Identify

Relate 9 x 4 to 4+4+4+4+4 +4+4+4+4 = 36 (4,8,12,16,20,24,28,32,36)

Number Counting

Number Counting

Relate 8x5 to 5+5+5+5+5 +5+5+5=40. (5,10,15,20,25.30,35,40)

Transcript: Group 4 Operation Knowledge Knowledee Constructed (Sekali lagi, enam darab eDam tiga puluh eoam. Enam darab tujuh?) Empat puluh dua Retrieval: Recall Multiplication (Empat puluh dua juga Ooi pandailah, cepat sangat. lni ialah aktiviti kelirna, adik tengok soalan ini, ada lapan batang pensil. Kita oak agih masuk ke dalam empat kotak pensii dan nampaknya kita boleh simpan dua batang pensil dalam setiap kotak. Kita boleh tulis empat darab dua, Sekarang kita nak buat sarna untuk dua belas. Kita oak agih ke dalam petak. Hitung berapa petak dalarn soalan pertarna Satu. dua, tiga. empat. lima. eoam. tujuh. laQan. sembilan, seQuluh. sebelas. Data exploration: Examine: Counting dua belas. Idenitfy Number

Mathematical operation: Count

(Nornbor dalarn setiap petak ini mesti serupa. Satu? Macam mana tahu satu? Cakap kuat sikit.) Hitun£ satu ... satu. dua. tiga. emQat. lima, enam. tujuh. laQan. sembilan. Data exploration: Examine: Number Relate 12 to sepuluh seblas, dua belas. Satu darab dua belas Jdenitfy Addition 1+1+1+1+]+1

Mathematical operation: +1+1+1+1+1+1=1 x 12 CQunt Summarisation: Summarise

(Ok setersunya. berapa petak itu?) Satu. dua. tiga. emQat. lima. enam Data exploration: Examine: Number

Idenitfy Counting Mathematical operation: Count

(Enam, jadi berapa nombor dalam itu?) Dua, ... enam, la12an, se12u1uh, dua belas ... enam darab dua Mathematical operation: Addition Relate 12 to

Count & add Multiplication 2+2+2+2+2+2=6x2 Solution generation: Relate & Coutning trigger Number Summarisation: Summarise

(Ok seterusnya empat petak)

108

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Transcript: Group 4 / Operation / Knowledge I Knowledxe Constructed (Ada berapa tingkat semuanya?) Dua belas (Tak, tingkat,ini dikira satu tingkat) Enam

(Satu tingkat ada berapa buah bilik?i &a

(Dua bilik jadi macam mana kaitkan enam dengan dua kepada dua belas? Adik beritahu tadi semua ini ada sejumlah dua belas bilik. Jadi macam mana kamu hubungkan dua ini dengan enarn menjadi dua belas?) Dua darab enam (Apa lagi?) Enam darab dua (Selain dari itu kalau kita guna tambah, macam mana?) Enam tambah enam

(Ada lagi satu.) Emoat darab tiea (Mana dia empat? dia mesti samakan? Ini dua, dua, dua, dua, dua, dual Satu darab dua belas (Satu darab dua belas. tak sesuai di sini. Kamu teneok ini dua bilik. Tadi adik - kata enam tambah enam, kita pun boleh kata dua tambah dua tambah dua tambah dua tambah dua tambah dua. Kita boleh lihat sebenamya dua darab enam boleh dikira sebagai enam tambah enam atau dua tambah dua tambah dua tambah dua tambah dua tambah dua. Ok ini bangunan pertama. Buat lagi bangunan yang lain dari ini.) (Ini berapa buah bilik semuanya?)

(Satu tingkat berapa bilik?)

Data exploration: Examine: Identify

Data exploration: Examine: Identify

Solution generation: Relate

Solution generation: Inverse

Mathematical operation: Add

Solution generation: Relate

Solution generation: Relate

Data exploration: Examine: Identify

Row & column Number Counting

Multiplication Inverse 2x6 to get 6x2

Row & column Number Counting

Row & column

Addition Relate 12 to 6+6 Row & column

Relate 12 to 2x6

Row & column Relate 12 to 4x3

Row & column Relate 12 to 1x12

Transcript: Group 4 Operation Knowled2e Knowledo-e Constructed Tiga Data exploration: Examine: Counting

Identify Number Row & column

(Berapa tingkat?) Empat Data exploration: Examine: Counting

Identify Number Row & column

(Kita boleh kata ia sebagai ... ) EmQat darab tig-3 atauQun tiga darab emQat ataupun empat tambah .... lapan Solution generation: Relate & Row & column Relate 12 to 3x4or4x3, t3mbah empat. inverse 4+8 (Lapan tambah empat tak sesuai, kita mahu nombor yang sarna) Empat. empat. empat Data exploration: examine: Addition Relate 12 to 4+4+4

Identify Row & column Mathematical operation: Add

(Kita boleh kata empat t3mbah empat tambah empat. Kalau kita tengok macam ini apa dia?) Tiga tambah tiga tambah ti23 tambah riga Data exploration: examine: Addition Relate 12 to 3+3+3+3

Identify Row & column Mathematical operation: Add

(Very good. Ok macam mana dengan ini? Kamu beritahu cikgu macam mana dapat dua belas?) Tiga darab emgat. Emgat darab tiga, emgat tambah emQat tambah emgat. tiga Data exploration: examine: Addition Relate 12 to 3x4, 4x3, tambah tiga tambah tiga tambah tiga Identify Row & column 4+4+4,3+3+3+3

Mathematical operation: Add Multiplication & multiplY Solution generation: Relate & inverse

(Buat bangunan yang tinggi. Lebih tinggi daripada ini lagi ... ) Lima, enam, tujuh, lagan, sembilan, seQuluh, sebelas, dua belas ... empat, Mathematical operation: Number enam ... sigek lagi Count Counting (Oh ... berapa tingkat?) Dua belas Data exploration: examine: Row & column

Identify (Macam mana a~t matematik?)

112

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Transcript: Group 5 Operation Knowledge Knowledge Constructed (Cikgu panggil adik ke sini untuk buat aktiviti matematik. Cikgu mahu adik kuat cakap. Sebab cikgu nak rakam semua ini, jangan bisik-bisik nanti tak dapat dirakam. Ok adik tengok aktiviti pertama, ia melibatkan apa?) Tambah Retrieval: Recognise Mathematical

operation (Bayangkan adik di sini, nak naik kereta, adik mesti cubajawab semua soalan di sini barulah naik kereta. Macam mana dengan nombor dua ini?) Darab Retrieval: Recognise Mathematical

operation (Oh, darab. Jadi dah belajar sifir?) Dah (Dah sampai nombor berapa?) Sava samoai lima ... sava dua belas ... Retrieval: Recall Multipl~cation

(Oh. dua belas, pandail) Saya lapan bel as (Kenapa ketawa?) Cikgu tak ajar lapan belas (Oh, cikgu tak ajar lapan belas. Jangan malu, adik pandai sifir berapa? Empat pandai?) Tak, lima dengan sepuJuh dengan dua Retrieval: Recall (Dh ... lima dengan sepuluh dengan dua, mengapa lima dengan sepuluh dengan dua?) Senang (Oh ... senang, macam mana sebut siflr sepuluh? Kuat) Satu darab sepuluh, sepuluh; dua darab se12u1uh, dua puluh; tiga darab se12u1uh, Retrieval: Recall Multiplication tiga Quluh; empat darab seQuluh, emQat Quluh; lima darab seguluh, lima puluh .. (Oh, sampai lima puluh saja? Ok adik sesiapa pun boleh buat, baleh tolong dan bincang. Ok boleh mula sekarang. Tapi cikgu mahu adik cakap, cakap dan bincang.) Tujuh darab empaL ..

(Darab?) Eh, darab lagi. Tujuh tambah empat, sebelas Mathematical operation: Addition

Add (Macam mana dapat sebelas itu?)

114

Transcript: Group 5 .ie:nbil~n ranthah emu31 lien bclar e n m t,m-sn emuat. ,ec~!l'-h T!%- dardh lima, tiea tambah lima, lapan. (Boleh terangkan bagaimana dapat tiga tamhah sembilan tadi? Macam mana adik buat?) Fikir dalam otak (Macam mana adik fikirkan?) Sembilan tambah tipa, tok ... tambah satu sudah se~uluh. tiza dah iadi dua.. .dua. dua belas.

(Oh, macam itu. Sembilan dulu, tambah satu, sepuluh.kemudian tiga tinggal dua. Campur dua lagi. Macam mana ah ... lima dengan enam?) Lima tambah enam sama denean sebelas

/ (Ada pakai iaii?) Tak aha - (Sudah pandai sangat. Macam mana dengan darab ini, satu darab satu ... ) Satu. Tiea uuluh ... (Sehut. sebut. Lima darab enam) -~~ . ~ ~

T ~ z a dx3h sembtian dua p u l ~ h rx~uu Lapin d;uh kwonr. ko,c:~x 03mh J ia . cnlp:$t o c ! . ~ ~ F.mrlt d?,.~!, i.:t:>u, ~ 1 . 1 ~ hcl.:. L..?J? ~ n r 3 l . A cmr.tt rulu:i Sat3 d?:.~h se~u iuh , seplll2h. En.;n darih ri!i.:k empst . ;u& h l t u n ~ d u : ~ . t!na k:dt tu~.ih. t le:~ k3l1 tutu11 r?nlh3h .empat rcl;,. Jus uuluh lapan: . .lambah emoat belas enam. .. .emuat puluh dua ... (Ooi, very good. Jadi macam mana kamu dapat empat puluh dua tadi?) Senang ... tuiuh. tuiuh, dua kali ... iadi emuat belas ... tiea kali tambah ... (Tiga kali tambah berapa?) Empat belas tambah ... emoat belas ... empat belas emvat culuh dua

(Oh, kamu guna apa tadi? Kamu sebut enam kali ... macam mana buat?) Enam, dua belas, lauan belas. dua puluh emoat. dua vuluh iauan.. . tiea culuh enam. ..emuat uuluh ... emuat culuh dua

Operation Mathematical oueration: Add

Data exploration: Examine: Identify & iocate Mathematical operation: Add

Mathematical operation: Add

Retrieval: Recall

Retreival: Recall Mathematical operation: Add

Mathematical operation: Add

Retrieval: Recall Mathematical operation: Add

Addition

Addition

I Multiplication

Multiplication Addition

Multiplication Addition

Knowledge Constructed

To do 3+9, take 1 from 3 to add to 9. 1+9=10, then 10+2 (remainder)=i2

Break down 6x7 to 3 x 7 ~ 2 1 & 3x7=21 then 21+21=42.

Break down 6x7 to 2x7=14, 2x7=14,2~7=14 then 14+14+14=42

Relate 6x7 to 6 + 12 + 18-+24+30+36+ A?

Transcript: Group 5 (Macam mana dengan adik, enam darab tujuh berapa?) Empat puluh dua (Ialah. Macam mana dapat empat puluh dua, kawan anda ada cara sendiri. Macam mana dengan anda? Sarnakah? Macam mana anda tahu empat puluh dua?) Entah (Ok macam mana adik dapat empat darab empat, enam belas?) Empat tambah emuat tambah emuat tambah emuat

(Oh, pakai jarikah? Empat tambah lagi empat, empat lagi, empat lagi jadi enam belas, siapa ajar?) Mak (Kita pergi ke aktiviti dua. Ini ialah lompatan darab.?iga darab dua, kita boleh sebut tiga kali dua. Tiga kali dua, tiga kali lompat. Mula dari nombor kosong hingga dua, lompat sekali lagi, sekali ia pun lompat dua langkah, jumlah sekarang ialah empat.. .]ompat lagi, dua lagi.. . dua tambah lagi yang empat tadi, enam. Jadi ayat matematik tiga darab dua kita boleh tulis sebagai dua tarnbah dua tambah dua sama dengan enam, faham? Adik lukis, tulis jawapan dan tulis ayat matematik.) (Mengapa lompat lima?) Lima darab tiga.. .salah. tiga, tiga

(Lompat berapa banyak?) Lomuat tipa.. .pergi ke enam, sembilan.. . .dua belas.. .lima belas

(Ayat matematik?) Lima darab tiea, sama denean tiga tambah t i ~ a . . .sama dengan enam.. .sama dengan enam.. . lima igek, sama dengan lima belas (Sama dengan lima belas. Macam mana dua darab tujuh?) Satu, dua, tiea, ern~at. lima. enam, betullah tuiuh.. .emuat belas Dua darab tujuh sama dengan dua tambah dua . . .tujuh igek, dua, sama dengan . ..Betullah dua tambah dua tambah dua tuiuh igek. Dua. dua, tiea, emoat, lima.. .enam.. . tuiuh.

Operation

Mathematical operation: Add

Data exploration:Examine: Identify

Mathematical operation: Multiplication jump

Mathematical operation: Add

Data exp1oration:Examine Identify Mathematical operation: Count Evaluation: Confirm

Knowledge

Addition

Multiplication jump

Multiplication Addition

Addition

Multiplication jump

. Addition

Knowledge Constructed

Relate 4x4 to 4 4 4 4

Relate 5x3 to 5 jumps with 3 steps in each jump

Relate 5x3 to 3 + 6 + 9 + 1 2 3 1 5

Relate 5x3 to 3+3+3+3+3

Relate 2x7 to 2+2+2+2+2+ 2+2

[ Transcript: Group 5 I Operation I Knowledge 1 Knowledge Constructed

(Ok muka surat dua. Ini empat darab empat, berapa lompat dia?) Eh, pergi ke tiga.. .lapan.. .dua.. . tiga.. .empat. Cun-cun enam belas.. .empat darab empat.. .sama dengan satu, dua, tiga.. .empat dia. Betulkah.. .eh sekali aja.. .emuat darab empat semua empat. ..semua empat.. .empat igek aja.. .sama enam belas (Tiga darab lima) Langkah tiga.. .lima. sama lima itu.. . terbalikah.. .betullah.. .hin.~ea se~uluh, lima belas.. .kamu tulis tok

(Tiga darab lima sama dengan apa?) Lima belas. ..tambah lima, lima, lima . . .lima belas

(Sepuluh darab satu) Senang ... satu aia ... dua. tiga, emuat, lima. enam, tuiuh, lapan, sembilan, se~u luh . . .sepuluh darab satu.. .sepuluh darab satu. Satu tambah satu tambah satu tambah satu tambah satu tambah satu.. .satu, dua. tiga. empat. lima, enam. tuiuh. lapan, sembilan, sepuluh sama dengan sepuluh betullah. Satu, dua, tiga, emuat. lima, enam. tuiuh, lapan, sembilan. seuuluh.

(Lima darab tiga, tiga darab lima sama jawapankan, tapi lompatannya?) Ini sampai lima belas .... tok sampai lima belas (Apa yang beza?) Tok langkah tiga. tok langkah lima ...

(Oh, sebab ini lima di depan dan ini lima di belakangkah? Seterusnya, tengok gambarajah yang ada titik-titik.) Main hitung (Ya, main hitung. Adik tolong kelompokkan. Kelompok itu mesti sama banyak titik kalau ada dua titik, semua dua titik. Adik boleh lukis di mana-mana, tujuh darab dua berapa?) Empat belas

Mathematical operation: Multiplication jump

Data exploration: Examine: Identify Solution genaration: Trial & error, inverse

Data exploration: Examine: Identify

Data exp1oration:Examine: Identify Evaluation: Confirm Mathematical operation: Multiplication jump

Data exploration: Examine: Compare

Retrieval: Recall

Multiplication jump

Multiplication jump

Addition

Multiplication jump Counting Number

Multiplication

Relate 4x4 to to 4 jumps with 4 steps in each jump

Relate 3x5 to 3 jumps with 5 steps in each jump 5 + 1 0 + 1 5

Relate 3x5 to 5+5+5=15

Relate 10x1 to 10 jumps with 1 step in each jump. (1+1+1+1+1+1+1+1+1+1)

1 Knowledge Constructed I

Relate 4x9 to 3x9 by adding 1 9 to 3x9=27 to get 36

Knowledge Multiplication

Addition

Grouping Counting Number

Coutning Number Grouping

Coutning Number Grouping

Coutning Number Grouping

7x2 & 2x7

Transcript: Group 5 Emuat belas juga ... sembil an... dua puluh tujuh. Tok dua puluh empat. Tok tiga sembilan, dua puluh tujuh.. .empat darab sembilan tiga puluh enam (Ooi, . . .cepat. Macam mana kamu dapat tiga puluh enam?)

I Tadi dua ~ u l u h tuiuh, tambah sembilan saia.

(Kelompokkan.) Dua. dua. dua.. . .

(Berapa banyak?) Tiga agek ... cukup (Ini dua darab tujuh, kelompoidcan.. .Tujuh) Satu, dua, tiga. emuat. lima, enam, tuiuh

(Sembilan darab empat) Sembilan juga ... Satu, dua, tiga. emuat, lima, enam, ... sembilan. .. Satu, kelompok dua, kelompok tiga.. .

(Semuanya ada berapa titik?) Sembilan (Berapa kelompok?) Tiga.. .empat.. .dah. (Lapan darab lima) Lima igek. lauan igek.. .ah, lapan.. .satu, lapan.. .empat igek. Semua sembilanbah . . .lima.. .tiga, empat, lima.. .toklah lima.. .satu, dua, tiga, empat . ..satu, dua, tiga, empat.. .

(Berapa?) Empat (Adik tengok sini, ini tujuh darab dua dan ini dua darab tujuh, apa bezanya?) Dua beza. Ini ada dua. ini ada tuiuh.

Relate 7x2 to 7 groups with 2 items each group

Operation Retrieval: Recall

Mathematical operation: Add

Data exploration: Examine; Identify

hlathematical operation: Count & group

Data exploration: Examine; Identify Mathematical operation: Count & group

Data exploration: Examine; Identify Mathematical operation: Count & group

Data exploration: Examine:

Relate 2x7 to 2 groups with 7 items each.grocp

Relate 9x4 to 9 groups with 2 items each group

Relate 8x5 to 5 groups with 8 items in each group

Knowledge Constructed

Relate 6x2 to 6 groups with 2 items in each group

Knowledge

7x2 & 2x7

Multiplication

Multiplication

Multiplication

Multiplication

Multiplication

Grouping Coutning Number

Transcript: Group 5 (B etul?) Betul (Cikgu ajar macam ini buat, ada belajar dalam kelas?) Ada, ada, tok sama.. .tujuh, tujuh, dua, dua sama nombor.

(Mana yang adik suka? Tujuh darab duakah atau dua darab tujuh? Mana senang?) Hm.. .dua darab tuiuh.. .eh.. .samalah, empat belas. Itu senang.

(Macam mana tiga darab tiga?) Sembilan (Empat darab empat?) Emgat darab empat. enam belas (Lima darab lima?) Dua puluh lima (Sukakah yang nombor sama? Sembilan darab sembilan.. .berapa?) Lapan ~ u l u h satu (Senang ingat?) Darab darab sembilanbah (Ok ini lagi satu aktiviti tapi senang saja. Adik tengok, enam darab dua. Adik tahu jawapannyakan? Dua belaskan, kalau belum pandai lagi sifir, apa adik boleh buat?) Hitung (Ya, hitung macam mana?) Hitung dari bulat-bulat. (Macam mana buat?) Buat telurbah.. .isi dulu (Ah, . . .macam mana buat telur?) Dua ipek isi dalam.. .enam.. .enam ieek bulat besar.. .dua, tipa, emgat, lima, enam.. .dua igek dalam, satu. dua, tiea. empat. lima, enam.. .

(Ini yang cikgu ajar? Yang mula-mula sekali?) Hm ... (Kemudian barn belajar sifir? Bukan belajar sifir dulu?)

Operation

Data exploration: Examine: Compare

Data exploration: Examine: Compare

Retrieval: Recall

Retrieval: Recall

Retrieval: Recall

Retrieval: Recall

Data explanation: Explain Mathematical operation: Count & group

tujuh darab enam.) Tiga belas.. .salah.. . e h . . (la mesti lebih ...I Emoat nuluh dua (Empat darab lima.) Tok empat puluh dua (Very fast. Sekarang tengok aktiviti ini, adik dikehendaki isi ruang-ruang . ..) Macam ini ya, ruang-man"

Transcript: Group 5 Bukan (Jadi sama dengan inilah, ada satu kumpulan dua bintang, dua bintang, dua bintang. Kalau enam darab tiga?) Tiga ... eh, laoan belas ... hitung iagi, satu la@, dua, dua, dua, tiga ... (Kalau enam darab satu?) Isi satu saia dalam

(Enam darab tiga, lapan belas. Sekarang dengan tujuh darab Lima.. .lepas itu

(Macam mka?) - -

Sik tahu ... tambahbah ... dua belas, kosong ah ... dua belas tambah kosong ... Oh! dua belas darab kosong, kosong. Dapatlah dua belas darab kosong, kosong.. . dua belas, sahl. Dua belas darab satu ... dua belas darab enam ... dua belas darab satu ... dua belas darab tiga, dua belas kali. Aku tahu, aku tahu, tok dua belas tok, dua belas darab satu. Eh, dua belas darab satu. dua belas. Tokkan dua belas darab satu ... dua belas tambah, dua puluh empatbah, betullah . d u a belas darab dua belas, seratus empat empat.. Dah , cikgu. (Dah? Sudah setuiu?) . .

I E:dua ouiuh emoat ... )

Operation

Retrieval: Recall

Solution generation: Deduce

. . 1)d.a. cni?..t. enaln. I3p:tn. 5 e r ~ l u h . dua bc.lx\ en s!k c d : ) ~ ! . 5 3 i ~ , du3, t:g:l t n l p n l , cmp: . cn.ubi. c m p ~ t . Iip:tn uJ.h rok Ju* p ~ l u o . i d . 11n1.l. 11,. t i

enam belas.. .enam belas. emoat, dua ~ u l u h . Dua puluh, lima, dua puluh lima, betullah, beful ... dah tadi, satu. dua. tiea. emoat, lima, enam Tok enam darab m. Dua puluh empat lagi. Empat, enam, lapan, sepuluh.. .Dua, empat, iapan ... eh, enam, lapan.. .dua puluh, dua puluh empat. Cunlah. Tok hujung tliacam itu

Knowledge

Multiplication

Grouping

Retrieval: Recall

Knowledee Constructed

Relate 6x1 to 6 groups with 1 item in each group

Mathematical operation: Multiply & add Solution generation: Trial & error Evaluation: Confirm

Solution generation: Trial - & error Evaluation: Disconfirm Mathematical operation: Add & count Summarisation: Summarise

Multiplication

Multiplication Addition

Addition Coutning Number

Relate 12 to 12+0=12 Relate 12 to 12x1=12

Relate 24 to 4+4+4+4+4+4 =6x4

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Transcript: Group 5 Operation Knowledo-e Knowledge Constructed Dua dua. dua. dua. dua. dua. Tiga ... hm ... satu. satu. satu. satu. satu. satu. satu, Solution generation: Relate Row & column Relate 12 to 2+2+2+2+2+2, hi tung satu. dua. tiga. empat. lima. enam. Darabkan en am lag; dengan sebelah Mathematical operation: Addition 1+1+1+1+1+1+1+1+1+1+1 jadi dua bel as Add & count Multiplication +1

Counting Number

(Darab enam lagi?) Hm. (Darab enam lagi? Enam darab enam berapa?) Dua belas ... eoam darab enam, dua belas. Eoam tambah eo am. dua belas. Solution generation: Relate Addition Relate 12 to 6+6

Mathematical operation: Row & column Add

(Kalau kita nak guna darab macam mana kita sebut?) Enam darab dua. Solution generation: Relate Row & column Relate 12 to 6x2

Multiplication (Ataupun?) Enam kali dua (Ataupun?) Dua kali enam (Semua itu kita dapat ... ) Dua belas (Iaitu kita boleh guna dua tambah dua tambah dua tam bah dua tambah dua tambah dua ataupun enam tambah enam atau dua darab enam. Sekarang buat lagi. Ia mesti lain dari yang tadi) Apa agek? Bangunan apa gek? Bangunan ABC. .. ia riga gek, sik cukup. (Sekarang ada berapa buah bilik semua?) Dua belas Data exploration: Examine: Number

Identify (Dua belas juga. Macam mana adik dapat dua belas itu?) Tiga, tiga, riga .. dua belas ... tiga darab empat ... sama dengan dua belas Mathematical operation: Row & column Relate 12 to 3+3+3+3, 3x4,

Add & multiply Addition Summarisation: Summarise Multiplication

(Atau?)

122

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I Ju; 13rrhsh d.!.: tamb::k :-> t m ? > h Jua t:rmnln =.,i , .AeJ & rcu![:p:) I 3!,:1t~p11::tIt~n :-?-?-:-:-?

Transcript: Group 5 Dua belas ... tiga ... enam agek. Lima. lima udah ... lapan, iima, satu tingg al.... satu. dua, tioa. emoat ... dua damb tuiuh ... empat ... betul.. .satu. dua. tiza. emoat. lima, enam darab enam iru

(Berapa dah, semua?) Dua belas

(Macam mana d a ~ a t dua belas?)

(Ini samakah? Bina yang tinggi l a a , lebih tinggi daripada ini) Buat satu.. .rumbang.. .adoh bengkok. ..pakai lif, dah, dah.. .Goyanpya (Macam mana kita kait dengan dua belas ini?) Satu. satu. satu. satu. satu. satu. satu. satu, satu. satu. satu.satu

(Berapa kali satu?) Dua belas

Operation Solution generation: Trial & enor Mathematical operation: Count

Data exploration: Examine: Identify

(Ataupun) Dua belas darab sat"

(Ataupun) Satu darab dua belas enam. enam ... enam tambah enam

Knowledge

Number

(Buat lain)

- Knowledre Constructed

Tixa. tiea ... tira. tipa ... dah emnat udah, lima belum. lima tidak daoat. lima. seouluh, lima belas. dua ouluh ... dua belas ... empat, lain empat. lapan ...

(Macam mana?) Tiga darab tiga darab tiga, emoat iambah emDat tambah emDat.. .tiga darab tiga darab tiga darab tiga ... emDat darab tira, tiga darab tiga

~ ~

Solution generation: Inverse & relate

Solution generation: Relate

Data exploration: Examine: Identify

Solution generation: Relate

Solution generation: Relate

Solution generation: Trial & error Evaluation: Disconfirm

Solution generation: Relate

~ d d ~ n o n

Row & column Addition

Number

Row & column Multiplication

Row & column Multiplication

Row & column Addition

Row & column Addition Multiplicaiion

Relate 12 to !+1+1+1+!+1 + l + l + l + l + l + l

Relate 12 to 12x1

Relate 12 to 1x12, 616

Relate 12 to 3+3+3+3

Relate 12 to 4 4 + 4

Transcript: Group 6 Operation Knowledge Knowledee Constructed (Cakap kuat-kuat, salah betul tak penting. Cakap saja apa yang adik fJkir. Kamu akan buat matematik. Cakap dengan kuat supaya dapat dirakam. Kamu boleh bincang, buat bersama-sama) Hm ... (Tengok soalan-soalan ini, apa dia?) Tambah Retrieval: Recognise Addition (Sudah belajar?) Dah (yang ini?) Darab Retrieval: Recognise Multiplication (Karou akan jawab satu demi satu kalau nak: naik kereta ini. Dapat selesaikan yang berikut barn boleh naikjet. Ok sekarang boleh mula. Sebut.) Satu tambah satu. dua; tujuh tambah emgat. sebelas: tiga tambah sembilan, Mathematical operation: Addition dua belas: .. .laQan tambah kosone:. laI2an; lima tambah enam. sebelas: se12u1uh Add tambah dua. dua belas; sembilan tambah emQat.. .. tiga belas: eml2at tambah enam. sepuluh: ... lapan (Macam mana kamu buat lima tambah enam? Tadi jawapan sebelas, macam mana kamu buat.) Senang cikgu, senang cikgu. (Jadi macam mana sebelas itu?) Tok lima, lima sepuluh. lima. enam, sebelas Data exploration: Examine: Addition Relate 5+6 to 5+5::;:10, so

Identify 5+6=11 Solution generation: Deduce

(Lima, empat?) Sembilan (Adakah kamu buat dengan jari dalam otak itu?) Tidak (Macam mana dengan tiga dengan sembilan?) Oh, sembiJan tambah satu seQuluh; seQuiuh tambah dua, dua belas. Data exploration: Examine: Addition To add 3 to 9. Take I from

Identify 3 to make 9 a 10. Then 10 + Mathematical operation: the remainder 2 to get 12 Add

(Macam mana dengan anda?)

126

Transcript: Group 6 Operation Knowledge Knowledge Constructed Sarna (Kita pergi ke aktiviti dua. Sebut, sebut.) ,

,

Satu darab satu ... satu. lima darab enam. tie:a guluh ... ti2a darab sembilan .. Retrieval: Recall Multiplication (Sebelum kamu sebut dua puluh tujuh, apa yang kamu sebut tadi yang saya tak dengar? Macam mana kamu buat?) Hitung (Macam mana hitung? Lukis?) Macam tiga dulu, ada sembilan, tambah la2:i sembilan tambah la2:i sembilan, Mathematical operation: Addition Relate 3x9 to 9+9+9=27. sembi Ian tambah sembilan. laI!an bel as. tambah sembi Ian la2i dua Quluh tujuh Add (9+9=18.18+9=27) (Sembilan tambah sembilan, lapan belas,lepas itu?) Tambah sembi lan, dua puIuh tujuh (Tapi sebelum itu, kamu buat lapan belas, sembilan belas, dua puluh, dua puluh satu macm itu kira sampai dua puluh tujuh?) Tak Lagan darab kosong. koson£!:. Tujuh darab dua. emQat belas .. emQat darab Retrieval: Recall Multiplication enmat. eDam belas (Hafalkah? Macam mana buat kalau empat darab empat?) Sarna (Yalah, tapi saya nak tengok cara kamu ... ) I

Empat darab dua, lapan. Tambah lagi empat. .. tambah lagi dua .. (Seorang, seorang cakap,nanti kamu terangkan, kamu dulu) Ok empat darab empat cikgu? EmQat tambah emQat jadi lagan, tambah lai!i jadi Mathematical operation: Addition Relate 4x4 to 4+4=8; 8+4;:; dua belas. tambah la2:i jadi enam belas. Add 12;12+4=16

(4+4+4+4) (Macam mana dengan kamu?) Emllat darab dua,laQan ... tambah la};l:an , enam belas,laIlan, laQan, enam belas. Mathematical operation: Addition Break down 4x4 to 4x2=8

Add & multiply Multiplication (2 times); 8+8=16 (Lapan, lapan, enam belas. Macam mana anda? Akan lukis lagi? Ok lapan darab lima.) Lapan darab lima, empat puluh Retrieval: Recall (Cepat, macam mana kira? Terus? Macam mana cikgu ajar dalam kelas?) Kira cikgu Kamu buat sangat cepat, macam mana kamu dapat lapan darab lima?) Ingat cikgu. Tambah-tambah

127

Transcript: Group 6 (Mula dengan apa? Satu d m b satu? Kalau lapan darab lima pakaj sifir mana?) Sifir lima. Bahaa boleh. Empat puluh darab lima ... eh ... empat puluh bahagi

I lima, lapan (Ok satu darab sepuluh ... ) Satu darab seuuluh. seouluh (Temskan.. .) Enam darab tujuh, empat puluh (Empat puluh?) Eh. .. eh ... salah. Enam darab tujuh, empat puluh lapan. (Empat puluh lapan, macam mana adik? Setuju?) Salah ya, aku tahu ... Salah ... empat ~ u l u h dua (Sekarang ingatkah? Macam mana kamu tiba-liba tahu empat puluh dua? Tadi kamu tulis empat puluh lapankan? Sekarang empat puluh dua, betulkah?) Empat puluh lapan, ... eh, tuiuh darab tuiuh. emoat puluh sembilan ... emDat puluh dualah. Takkan tambah tuiuh azek?

(ladi macam mana dari empat puluh sembilan itu boleh jadi empat puluh dua? Apa yang kamu buat di sama?) Tambah cikgu, tambah lagi cikgu (Empat puluh dua tambah tujuh, empat puluh sembilan?) Ah.. .betullah (Sekarang kamu tengok aktiviti ini. Ini dikira sebagai lompatan darab. Tiga darab dua atau tiga kali dua beieni sekali dia lompat sejauh dua. Sekali lompat dua, dua kali lompat, dua tambah dua. Tiga kali lompat, dua tambah dua tambah dua. ..jadi enam. Ayat matematik tiga darab dua sama dengan dua tambah dua tambah dua sama dengan enam. Ok cikgu mahu kamu lukis ... lepas itu jawab dan tulis ayat maternatik. Cakap-cakap. ..) Lima darab tiea, lima belas (Macam mana dengan lompatan itu?) Seuuluh tambah lima ... ha.. .ha.. .

(Bincanglah, kalau tidak pasti bincang dulu. Nombornya mesti sama, tengok ini, dua, dua, dua . . nombornya mesti sama.)

Transcript: Group 6 Operation Knowledge Knowledge Constructed Buat tiga iQ:ek ... salah.lima, lima. lima ... betullah. Lima, lima, lima betullah. Solution generation: Trial Addition Relate 5x3 to 5+5=10, Lima tambah lima. seQuluh: seQuluh tambah lima. lima belas. Dapat Japan & error 10+5=15 tambah tujuh ... Evaluation: disconfirm & Relate 15 ta 8+7

confirm Mathematical operation: Add

(Kamu beIum lukis Iagi ... kuat sikit cakap, tak dengar.) Lan2kah samRai lima. langkah samgai seguluh. langkah samRai lima belas Mathematical operation: Multiplication Relate 5x3 to 3 jumps with

Multiplication jump jump 5 steps in each jump (5--710~ 15)

(Ok teruskan, dua darab tujuh) EIDQat belas Retrieval: Recall Multiplication (Mengapa tak mahu buat ini duIu? Mungkin buat ini dulu lagi senang. Lukis dulu tak mahu? Tulis dulu. Sebutlah.) Tarnbah dua tambah dua tarnbah dua tambah dua, .. dua tambah dua tambah dua Mathematical operation: Addition Relate 2x7 to 2+2+2+2+2+ dah enam. Tambah dua, dah tujuh. Eh .. .lapan. Sepuluh, sepuluh, sepuluh .. Add 2+2 (Apa diary) Lima tambah lima tambah emgat jadinva emQat bel as. Mathematical operation: Addition Relate 14 to 5+5+4

Add (Ah ... itu nombor tak sarna!) Tujuh tambah tujuh .. sini Mathematical operation: Addition Relate 14 to 7+7

Add (Cakap kuat) Tok pergi lima, enam, tujuh ... oh ... tujuh ... empat belas. Habis. (Mengapa pergi ke tujuh dan empat belas? Macam mana dapat tahu?) Tujuh tambah tujuh. empat belas ... Data explanation: Explain Addition (Sambung lagi. Sehut ... ) EmQat darab emQat. enam belas ... Retrieval: Recall MulripJkatjon (Ok lampat berapa langkah?) Effigat. lagan. dua belas. enam belas ... jadilah lagi. Empat darab empat, en am Mathematical operation: Multiplication Relate 4x4 to 4 jumps with belas ... enam belas. Emgat tambah emRat tambah emgat tambah emgat. Empat Multiplication jump jump 4 steps in each jump. igek, tiga, empat betullah. (4--78~12~16)

(4+4+4+4) (Sekarang tiga darab lima)

129

Transcript: Group 6 I Operation 1 Knowledge I Knowledge Constructed (Ok sebut1ah.i I 1 I . Tulun ddrdb dua empar hd3i 101, >diu 1,:lompok bemp*ull Dua -

(Sebutlah, lepas itu?) Tuiuh kali ... tuiuh kali

(Semuanya berapa titik?) Dua. dua ... emuat ... emoat belas titiknva. (Berapa jumiah titik itu?) Dua.. . betullah.. .emDat belas betullah

(Sekarang dua darab tujuh.) Empat belas jua. Samajua, tujuh, mjuh itu lain (Sebut, sebut ... ) Dua darab tuiuh ... empat belas (Bila kamu hitung itu; cakap juga.) Satu. dua, tiea. emDat, lima. enam, tuiuh ... tiea belas. emoat belas.

(Ok sembilan darab empat) Senang, senang ... emoat, emuat, emuat ... Satu, dua, tiga, empat ...- darab emuat. tiea uuluh enam ... tiga dah. ..empat, lima agek.. ., enam.. .dua agek.. tiga.. .dah

(Kuat sikit)

Retrieval: Recall

Data exploration: Examine: Identify Mathematical operation: Group

Multiplication

Grouping Number

Evaluation: Confirm Number Relate 7x2 to 2+2+2+2+2+ Mathematical operation: Addition 1 2+2

Relate 7x2 to 7 groups with 2 items each ~ o u p

Data exploration: Examine: Identify Mathematical operation: Group

Add & count

Retrieval: Recall

Grouping Number

Mathematical operation: Count & group

Data exploration: Examine: Identify Mathematical operation: Count & group Retrieval: Recall

Multiplication

Counting Grouping Number

Grouping Coutning Number

Relate 2x7 to 2 groups with 7 items in each group

Relate 9x4 to 9 groups with 4 items each group

(Sebut, sebut, tak dengar) Tok dua puluh ... hitung, hitung ... emoat. lapan. dua Dela.5. enam belas. lapan belas, dua uuluh. dua ~ u l u h empat ... dua puluh lapan ... lira ouluh dua ... tiea

Transcripl: Group 6 1 Operation 1 Knowledge I Knowledge Constructed

pulilh enam (Ok lapan darab lima. Berapa kumpulan ini?) Lap an... empat di dalam ... ~ a t u , dua. tiea. emoat. lima. enam. tuiuh. laoan. sembilan. seouluh. sebelas, dua belas. tiea belas. empat belas, lima belas. enam belas. tuiuh belas, laoan beias. sembilan beias. dua ouluh. dua puluh satu. dua puluh dua. dua ouluh tiea. dua ouluh empat, dua ouluh tuiuh. dua ouluh lapan, dua ouluh sembilan. tioa ouluh. tiea ouluh satu. tiva Puluh dua. tiea puluh tiea, tisa uuluh emoat. tiea uuluh lima. tiea ouluh enam. tiea ouluh tuiuh. tiea ~ u l u h lapan. tiea ouluh sembilan. emuat ouluh. Ooi, satu, dua. tiea. emoat. lima, enam. tuiuh. laoan. Betullah ... empat puluh, betullah (Teruskan, tiga darab tiga) Tioa darab tiea. sembiian, salah. Tok ada tigabah. Tok ada tiga igek bulatnya ... bulatlah tiga igek (Lepas itu. Bulat lagi tiga igek, cukuplah. Enam darab empat. Bulatkan berapa?)

Counting Number

Emoat. lapan. ... eh ... salah, tok lapan ... tok lap an...

(Sebut, sebut) Enam darab emoat. dua ouluh emoat. Tambah lagi einpat iDah)

Evaluation: Disconfirm Mathematical operation:

~ e t u i , betul (Lagi. ..aktiviti empat jika enam darab dua kita dapat dua belas. Macam maoa kita dapat dua belas?) Ah ... tambah, tambah iKita boleh tambah, kalau Lima darab tiga? Macam mana?)

Count

Mathematical operation: Add &count Evaluation: Confirm

Data exploration: Examine: Identify Mathematical operation: Count Evaluation: Confirm

Retrieval: Recall

Data exploration; Examine: Identify Mathematical operation: Group

Retrieval: Recall

Solution generation: Relate

Addition Number Counting

Counting Number Grouping

Multiplication

Grouping

Multiplication

Addition

Relate 8x5 to 8 groups with 5 items in each group

Relate 3x3 to 3 groups with 3 items in each group

Relate 6x4 to 6 goups with 4 items in each group

m z E x m .- 5 s :n X Y O -- 3 3 - " M 2 g 3 '. m 2 .z - Y x '2 s 7 = 5 v z a

m E ... 4

, e m " Y w a m - m o 2 M O a Z k i s E ~ : ~ e $ $ j - 2 c p ? m o , - z g 5 % g;; 5 % 6 m 2 g m aaa 2 w Z Z Z Y 5 5 2;:

-

Transcript: Group 6 LaDan. enam belas. dua ~ u l u h empat. ..betullah. Dua belas darab dua. dua puluh emoat ... Satu, dua, tiga, empat, lima, enam ... hafal aja. Empat, empat, lap an... lapan, dua belas. Dua tambah enam. Tuiislah. Tok iapan belas, tambah enam.. .dua puluh empat. Enam duiu, hitung, saya b ra , dua belas, dua belas, enam agek. ..bukan euam. Empat, empat, empat, empat banyak ... lapan, empat, empat, empat, empat.. .Empat, empat, iapan, enam belas. ..dua puluh. ..dua puluh empat. Dua, dua, dua.. .betullah (Setuju?) Dua, dua, dua, dua, dua, dua ... dah. Enam, lapan, dua, dua, dua, dua, dua, dua ... kira dua, dua, dua. ..tiga, tiga. ..tiga. ..tiga betul Tiea. tira. tiea. tisa. tiea. tiza. tiea, tiea, lap an... dua belas darab tiga. Beres cikgu. Dua darab dua belas, dua ~ u l u h emDat. Betullah.

(Sekarang kamu bina bangunan. Kamu pakai kalu ini untuk buat bangunan satu bongkah ialah satu bilik. Bilangan bilik untuk setiap tingkat mesti sama, tak boleh ada tiga ada satu) (Ada berapa buah bilik semuanya?)

(Macam mana dapat dua belas?) Enam, enam, dua belas

(Apa lagi? Ini satu jenis bangunan, kamu kata ada dua belas buah bilik, kamu kata enan tambah enam mana enam?) Atas enam. bawah enam. Enam tambah enam. dua belas. Dua tambah dua emoat, emoat tambah dua enam. enam tambah dua laoan. lapan tambah dua se~uluh. seuuluh tambah dua dua belas.

(Oh ... dua tambah dua tambah dua tambah dua tambah dua tambah dua, enam kali. lagi?) Emoat tambah emDat tambah empat

I (Kalau pakai darab macam mana?)

/ Operation / Knowledge / Knowledge Constructed / Mathematical operation: / Addition / Relate 24 to 8+8+8

Data Exp1oration:Examine: ldentiiy

Add & count Summarisation: Summarise

Mathematical operation: Add

Data exploration: Examine: Identify & locate Mathematical operation: Add

Counting Number

Mathematical operation: Add

(8,16,24) Relate 24 to 12+12=12x2

Relate 24 to 3+3+3+3+?+3 +3+3 relate 24 to 2x12=24

Relate 12 to 6+6

Relate 12 to 2+2+2+2+2+2 (2+2=4,4+2=6,6+2=8,8+2= 10,10+2=12)

Relate 12 to 4 4 4

Knowledge Constructed Relate 12 to 4x3

Relate 12 to 6+6

Relate 12 to 6+6

Relate 12 to 3+3+3+3

Relate 12 to 4+4+4

Relate 12 to 6+6

Relate 12 to 2+2+2+2+2+2

Knowledge Multiplication

Number

Row & column

Number

Row & column Addition

Row & column Addition

Row & column Addition

Row & column Addition

Transcript: Group 6 Eh.. .emuat darab tiea

(Di mana empat, di mana tiga?) Darab tiga, dua belas.. .tok darab tiga (Buat lagi satu, mesti lain dari ini Apa cikgu kata tadi. Bilik mesti sama banyak di setiap tingkat.. .Ini tak sama. Bina semula.) (Berapa buah bilik semua itu?) Dua bels

(Dua belas juga. Macam mana kaitan dengan dua belas?) Sama juga. Enam. enam igek. (Buatlah lain sikit. Buat yang tinggi. Tak sama.) Tok empat, tok empat, tok empat (Tiga bangunan sudah. Buat dengan semua dua belas bongkah kayu. Mesti guna dua belas. Buat bangunan lain dari dua bangunan ini. Berapa bilik?) Dua belas

(Macam mana kita kaitkan dengan dua belas?) Tiea, tipa, tiga, tiea

(Lagi?) Empat tambah emuat tambah emuat

(Selain dari itu?) Enam, enam

(Mana ada enam?) Dua tambah dua tambah dua tambah dua tambah dua tambah dua

(Lain lagi, guna darab.) Empat, empat, empat.. .enam, enam (Pakai darab) Tiga, tiga, tiga

Operation Mathematical operation: Multiply

Data Exp1oration:Exarnine: Identify

Solution generation: Relate

Data Exp1oration:Examine: Identify

Mathematical operation: Add Solution generation: Relate

Mathematical operation: Add

Mathematical operation: Add

Mathematical operation: Add

Knowledge Constructed

Relate 12 to 4x3

Relate 12 to 6+6

Relate 12 to 2x6

Relate 12 to 1 +1+1+1+1+1 +1+1+1+1+1+1

Knowledge

Row & column Multiplication

Row & column

Number Counting

Addition

Multiplication

Row & column Addition

Transcript: Group 6 (Macam mana dengan darab?) Tiga, tiga, tiga, tiga (Itu tambah) Emuat darab tiea

(Empat dari mana?) Tok emuat cikrm, ada tiea kumuulan

(Empat darab tiga, tiga darab empat boleh?) Dua belas bahagi empat, tiga (Boleh buat yang tinggi lagikah?) Tok sama, buat tinggi agek. Dah. (Buat lain dari ini lagi. Ada lagi lebih tinggi.) Sambung, sambung (Buat yang panjang, dua belas juga) Buat ~ m a h panjang ini. Dah habis. (Samakan ini? Cikgu kata ia mesti sama setiap tingkat. Teruskan.) Menara Kuala Lumpur (Ini ada berapa buah bilik?) Satu, dua, tipa. emuat. lima, enam, tujuh, lauan, sembilan, seuuluh, sebelas, dua belas. (Macam mana kaitkan dengan dua belas?) Ada lif tak? Enam tambah enam.

(Apa lagi?) Dua darab.. . enam

(Sudah) Dua darab dua, empat, dua, tiga, empat, lima, enam (Berapa buah bilik ini, macam mana kita kaitkan dia?) Satu, satu. satu. satu.. .

(Berapa kali?)

Operation

Solution generation: Relate Mathematical operation: Multiply

Data exploration: Examine: Locate & identify

Mathematical operation: Count

Mathematical operation: Add

Mathematical operation: Multiply

Solution generation: Relate

Transcript: Group 6 Dua belas

(Ini?) Dua. dua. dua, dua. dua. dua .... Tambah

(Sudah) (Kalau adik ingin mengadakan parti harijadi di sebuah bangunan yang mempunyai paling banyak bilik, bangunan yang mana adik akan pilih?) Ini.. .ini.. .tok saya.. . I (Berapa bilik itu? Adakah bangunan yang mempunyai paling banyak bilik?) Sama dua belas. Semua sama.

Operation Data Exploration: Examine: Identify

Mathematical operation: Add

Knowledge Number

Addition

Knowledge Constructed

Relate 12 to 2+2+2+2+2+2

Transcript: Group 7 (Cikgu panggil kamu tiga orang sini untuk bagi kamu buat matematik. Penting semasa kamu buat apa yang kamu fikir perlu disebut. Misalnya, hitung dua campur dua adik sebut dua campur dua, empat. Jawapan betul salah tidak penting, yang penting adik sebut. Cikgu akan rakam apa yang adik cakap. Kalau adik tak cakap, tak dapat dirakam. Kita tengok aktiviti pertama ini. Sama-sama buat. Sesiapa pun boleh buat, boleh tolong dan bincang. Apa yang adik nampak di sini?) Sifir dan tambah

(Ok sifir dan tambah. Adik nak naik kereta ini, mestilah cuba selesaikan semua soalan? Tapi penting, semasa adik buat adik kena cakap.) Satu tambah satu sama dengan dua

(Ok macam mana adik buat satu tambah satu? Semasa satu tambah satu, mula dengan jari, bilakah kali pertama belajar tambah?) Sekolah (Ok yang penting, adik cakap, cakap dengan suara besar.) Tuiuh tambah empat sama dengan sebelas

(Macam mana adik dapat sebelas?) Saya pakai tangan (Macam mana pakai tangan?) Saya pakai tujuh tambah empat, tuiuh ka hati, emDat ka tanean. Saya kira tuiuh, lapan sembilan, seuuluh, sebelas.

(Siapa ajar itu?) Cikgu, mak ajar juga. (Ok seterusnya.. .) Tiga tambah sembilan.. .dua belas

(Macam mana dapat dua belas? Bagaimana hitung?) Tiga dalam hati.. .

(Kuat sikit.. .)

Operation

Retrieval: Recognise

Mathematical operation: Add

Mathematical operation: Add

Data exploration: Locate Mathematical operation: Add & count

Mathematical operation: Add

Data exploration: Examine: Locate

Knowledge

Addition Multiplication

Addition

Addition

Addition Coutning Number

Addition

Number

Knowledge Constructed

Knowledge Constructed Transcript: Group 7 Sembilan.. . hitung (Oh, tiga dalam hati, lepas itu?) Tiqa, emuat, lima. enam, tuiuh. lauan, sembilan. sepuluh, sebelas. dua belas

(Ok.. .teruskan.) Ah.. .tapan tambah kosong, lauan.

(Lima tambah enam, sebelas ... dalam hati, macam tadi juga?) Sepuluh tambah dua, dua belas

(Simpan nombor mana di hati?) Se~u luh simuan di hati, dua di tangan.. .

(Macam mana lima dengan enam tadi?) Ah, enam.. . enam dalam hati. lima di tanean

(Bukan lima dalam hatikah? Mana-mana pun boleh? Tadi adik pakai tujuh dengan empatkan, adik simpan mana satu dalam hati?)

(Tujuh dalam hati, macam mana di sini, simpan mana satu dalam hati tadi?) k m

(Ok mengapa pilih tujuh bukan empat?) Tak tahu (Bolehkan simpan empat dalam hati?) Boleh (Mengapa adik pilih tujuh?) Kerana ia lebih besar

(Oh, I see. Ini sepuluh dengan dua, simpan sepuluh dalam hati, tambah lagi dua ok sembilan dengan empat) Sembilan dengan emoat, tiga belas

Operation

Mathematical operation: Count

Mathematical operation: Add

Mathematical operation: Add

Data exploration: Examine: Identify & locate

Data exploration: Examine: Identify B locate

Data exploration: Examine: Identify

Data exploration: Examine: Identify

Data exploration: Examine: Compare

Mathematical operation: Add

Knowledge

Number Counting

Addition

Addition

Number

Number

Number

Number

Number

Addition

Knowledge Knowled e Constructed

hati? Lain? Ok empat dengan enam) Empat tambah enam sama dengan seuuluh

(Macam mana?) Enam dalam hati. emuat di tangan

(Semua ok dah. Semua dah sampai sini bolehlah naik kereta. Sekarang naik kapalterbang ok tengok soalan ini, semuanya.. .) Sifir . - (Darab.. .ok mula.. .sebut.. .) Satu darab satu sama dnegan satu (Lepas itu?) Lima darab enam sama tiga puluh (Ok macam mana tahu tiga puluh?) Hafal sifir (Macam mana dengan anda, hafal juga? Adik boleh hafal sampai sifir berapa?) Sembilan tapi kadang-kadang lupa (Mana yang paling senang?) Lima

(Mengapa lima paling senang?) Sebab dia lima, lima sepuluh, lima be!as, dua puluh, dua uuluh lima.. . (Pakai jari?) Tak, dia tambah tambah, tambah, tambah.. . (Dia tambah, tambah, tambah. Ok . .. lima itu pakai lima tambah lima, cepat, jadi macam mana lima darab enam tadi?) Eh. ..lima darab enam, ah ... lima tambah ... eh ... lima tambah lima enam kali. Lima darab enam tiga puluh. Lima tambah lima seuuluh, tambah lima lagi lima belas, tambah lima lagi dua puluh, tambah lagi dua puluh lima, tambah lima lagi tiga puluh. (Oh, good. Siapa ajar ini?) Mak (Oh, mak yang ajar, cikgu pun ajar sama dalam kelas? Tiga darab sembilan.)

Mathematical operation: Add

Data exploration: Examine: Identify- & locate

Retrieval: Recall

Retrieval: Recall

Retrieaval: Recall

Retrieval: Recall

Data exploration: Examine: Compare

Data explanation: Explain

Mathematical operation: Add

Addition

Number

Multiplication

Multiplication

Multiplication

Multiplication

Multiplication

Addition

Addition Multiplication

Relate 5x6 to repeat addition of 5 for 6 times 5+5+5+5+5+5

(Oh, empat belas macam itu. Lepas itu, tambah lagi. ..) Ah.. . tambah la@ kalau saya tak ingat (Kalau besar lagi? Tujuh darab lapan, oh. ..banyak ... macam mana?) Ya.. .kira.. .sampai.. .ah habis jika saya tak hafal. (Oh, mula daripada satu darab tujuh, lepas itu tarnbah tujuh, tambah tujuh, tambah tujuh.. .ok lapan darab kosong) Kosong (Kosong. Tujuh darab dua?) Tuiuh darab dua emoat belas (Empat belas, macam mana dapat empat belas?) Ah.. .ah.. .ah.. .satu, dua, Oiga, empat, lima, enam, tujuh.. . (Oh, dua kali.. . tambah lagi dua) Tambah lagi dua iadi empat, tambah iadi enam, tambah lagi iadi laoan, tambah dua iadi se~uluh. tambah lagi iadi.. .ah.. .dua belas, tambah laei emoat belas

Transcript: Group 7 Tiea darab sembilan sama dennan dua puluh tuiuh (Macam mana dapat dua puluh tujuh?) Saya hafal sifir ...ah, dua hingga enam (Dua hingga enam, hafal sifir. ..kalau nombor besar lagi macam mana?) Hafa1 sikit saja. (Ok adik pandai sarnpai sifir berapa?) m, kadang-kadang lupalah (Kadang-kadang lupa. Kalau lupa macam mana?) Saya ingat ah ... misalnya.. .ah.. .ah.. . tuiuh.. . ah.. .ah.. .ah ... darab dua, saya lupa. Sya ingat tujuh tadi. Tuiuh darab satu sava ineat. tuiuh. laoan. sembilan, seouiuh. sebelas. dua belas. tiea belas.. .emoat belas

Retrieval: Recall

Retrieval: Recall

Operation Retrieval: Recall

Retrieval: Recall

Mathematical operation: Add, multiply & count

Mathematical operation: Add

(Oh, I see, I see. Ok macam mana dengan empat darab empat?) Ah.. .ah.. .saya ingat sifir empatlah (Sifir empat?) H afal (Hafal? Adakah adik mula dengan satu darab empat, dua darab dengan empat, lnula dari sana, empai tambah empat. ..)

Multiplication I I

1 Knowledge Multiplication

Multiplication Relate 7x2 to 7x1=1, then Addition count and add with 7 to get Counting Number

i 1 4

Knowledge Constructed

Multiplication I I

Addition Relate 7x2 to 2+2+2+2+2+ 2+2 [2+2=4,4+2=6,6+2=8,8+2= 10,10+2=12,12+2=14)

Tuiuh tambah tuiuh empat belas. tambah tuiuh.. .tambah laei tuiuh

(Kuat, kuat siht.. .lepas itu?)

Knowledge Constructed

Relate 8x5 to 5+5+5+5+5+5+5+5

Relate 6x7 to 12 then count and add till 42

Knowledge

Multiplication

Addition

Multiplication

Number Addition Counting

Transcript: Group 7 Tak. Ah ... kadang-kadang kalau saya tak ingat sifir, kalau saya ingat saya.. . saya terus keluar. (Oh.. .ok lapan darab lima) Empat puluh (Macam mana dapat empat puluh?) Ah.. .lima tambah lima tambah lima

(Oh.. .lima tambah lima tambah lima tambah lima.. .lapan kali. Ok satu darab sepuluh) Sepuluh (Ok enam darab tujuh? Kuat cakap. ..) Dua belas. tiea belas, empat belas, lima belas. enam belas, tuiuh belas, lapan belas, sembilan belas, dua puluh, dua puluh satu. dua uuluh dua. dua puluh tiea. dua ~ u l u h empat, dua ouluh lima. dua uuluh enam. dua uuluh tuiuh. dua puluh lapan, dua puluh sembilan, tiga puluh. tiga puluh satu, tiea puluh dua, tiea puluh tipa, tiea uuluh empat (Tiga puluh empat. Cukup dah?) Empat dua (Ok mengapa empat dua?) Tambah (Macam mana? Kuat sikit.. .enam.. .)

Dua puluh satu (Oh.. .Dua puluh satu, lepas itu? Kuat.. .kuat.. .) 1

Operation

Retrieval: Recall

Mathematical operation: Add

Retrieval: Recall

Mathematical operation: Add & count

Mathematical operation: Add

Dua ~ u l u h tuiuh (Lepas itu?) Tiea empat. tambah lapi tuiuh (Oh ... macam itu, sila tulis jawapan, empat puluh dua) (Sekarang adik tengok aktiviti dua. Cuba adik fahamkan apa yang diberitahu oleh gambarajah ini.)

Addition Relate 6x7 to repeat addition of 7 for 6 times (7+7=14,14+7=21,21+7=28 ,28+7=35,35+7=42)

Transcript: Group 7 ( Operation I Knowledge Oh .. .tompatan darab. I Data examination: Read I Multiplication

I Data exploration: Examine: I jump

(Lompatan darab, pemahkah adik belajar?) Pemah (Ok jadi.. .apa yang ditunjukkan? Adik tengok conioh pertama ini, apa yang adik faharnkan? Cakap.) &&I

(Apa dia? Ya. .. cakap apa saja yang adik fikirkan.) la lompat lomoat.

(Ok ia lompat lompat.) Jawa~an enam

(Ya, jawapan enam, macam mana dapat enam itu?) Ah.. . tambah satu satu.. .dua tambah dua empat tambah dua enam (Macam mana dengan ayat matematik ini?) Oh.. .dua tambah dua tambah dua, enam darab.. .ayat matematik darab tiga darab dua, enam (Darab tiga darab dua enam juga. Ok jadi . ..apa kaitannya tiga darab dua dengan dua, dua, dua ini? Tambah dua tambah dua tambah dua?) Oh.. .ini, satu, dua, tiga, sama tiga.. .itu.. ..eh.. .soalan nombor dua, dua sini, jawapan senang (Senang.. . jadi.. .eh.. . apa kaitannya tiga darab dua dengan dua tambah dua tambah dua?) E m ... (Berapa kali dua di sini?)

1

1 (Apa kaitan dengan tiga darab dua?) T ~ g a darab dua.. .

Identify & extract

Data exploration: Examine: Mvltiplication Identify

Data exploration: Examine: Multiplication Identify Sr extract jump

Data exploration: Examine: Identify

Data explanation: Explain Addition

Addition

Data exploration: Examine: Number identify

Knowledge Constructed '

lompatan. Lepas itu, ah.. .adik tulis ayat matematik. Ok lima darab tiga.) Aku dulu.. .aku dulu.. .aku dulu.. . (Lima darab tiga macam mana? Mula dari.. . kosong, berapa jauh ia lompat?) Ljma

Transcript: Group 7 (Boleh adik cuba? Tak dapat, ok tak apa. Jadi cikgu mahu adik buat yang ini. Ini ialah lima darab tiga, cikgu mahu adik lukis juga anak panah menunjukkan

(Lepas itu? Sampai ke.. .)

Operation

S e ~ u l u h (Sepuluh, lepas itu ke.. .) Lima belas (Ok lima darab tiga ialah lima belas, macam mana dengan ayat matemtik dia?) Ah ... lima tambah iima tambah lima. lima kali tiga, lima tambah lima tambah lima sama denzan lima belas

(Macam mana dengan dua kali tujuh? Cakap.. .cakap) Dari kosonn sampai oergi ke tuiuh, dari tuiuh lompat ke emaat belas

(Ok, empat belas sama dengan dua darab tujuh.. . Sebut.. .sebut.. .) Dua darab tujuh sama dengan ah.. .tujuh.. .Tujuh tambah tujuh sama dengan empat belas.. . (Ok very good. Sambung lagi. Empat darab empat) Enam belas

Data exploration: Examine: Identify Mathematical operation: Multiplication jump

Mathematical operation: Add & multiplication jump

Mathematical operation: Add L2 multiplication jump

Retrieval: Recall

Knowledge

(Oh.. . .enam belas. Satu lompatannya ... Dari sifar ke empat . . .) Lapan, dua belas.. .enam belas

(Ok sebut.. .) Empat darab empat sama dengan enam belas.. .ah.. .emoat tambah emaat tambah emapt tambah empat enam belas.

Multiplication jump

Mathematical operation: Add

Data exploration: Examine: Identify

Addition Multiplication jump

Addition Multiplication jump

Multiplication

Addition

Multiplication jump

Knowledge Constructed

Relate 5x3 to 3 jumps of 5 1 steps in each jump

Relate 5x3 to 5+5+5=15 (0+5+10+15)

Relate 2x7 to 2 jumps of 7 steps in each jump 7+7=14 (0+7+14)

Relate 4x4 to 4 jumps of 4 steps in each jump ( 0 + 4 + 8 + 1 2 + 16)

Relate 4x4 to 4+4+4+4 -

Transcript: Group 7 ( Operation I Knowledge 1 Knowledge Constructed (Very good. Sekarang tiga darab lima, tadi lima darab tiga sekarang tiga darab 1 lima, berapa ia lompat?) Tiea, enam. sembilan. dua belas, lima belas.. .

(Ok ayat matematik. ..) Tica darab lima sama denpan lima belas. lima tambah lima tambah lima sama denoan lima belas. (Ok.. . adik nak bandingkah tidak?) Banding ... tok salah, tok tiga, lima kali ... Lima kali, bukan, bukan, buk an... Padam. Tiga, dua, tiga, empa:, lima ... tok, lima belas, sitok ... sepuluh (Dah? Sepuluh darab satu) Sepuluh (Macam mana lompatannya?) Se~uluh

(Oh.. . terus ke sepuluh? Sepuluh darab satu) Sepuluh.. . tambah.. .eh.. .se~uluh tambah kosono,. . .eh.. .betullah

(Adakah kita tambah kosong di sini? Tak ada. Ayat matematik yang adik tulis itu nombor lompatannya semua sama, kalau tiga, tiga, tiga, tiga, tiga, tujuh, tujuh, kemudian empat, lima .. .Sepuluh dengan kosong, adik setujukah? Kalau ok, kita temskan.) Sebab.. . tak ada, ingat nak tambah. Kalau tambah sepuluh nanti dua puluh.. . (Macam mana kita . . .perlukah kita tulis kosong? Macam mana? Sebut.. . tak tahu? Tak tahu atau tak perlu?) Rasa sik perlu (Ok kita tengok aktiviti ketiga, ada banyak petak dan banyak titik. Apa yang adik perlu buat ialah kelompokkan titik-titik. Titik dalam setaip kelompok mesti sama banyak. Kalau kelompok ada lima titik jadi semua mesti lima. Sebut ... ) Tuiuh darab dua sama dengan emoat belas. (Ok kelompoknya macam niana? Lukiskan, bila kira sebu:, sebut.. . h a t . . . j

Mathematical operation: Multiplication jump

Mathematical operation: Add

Retrieval: Recall

Mathematical operation: Multiplication jump

Mathematical operation: Add

Retrieval: Recall

Multiplication jump

Addition

Multiplication

Multiplication jump

Addition

Multiplication

Relate 3x5 to 5 jumps of 3 steps in each jump ( 3 - + 6 + 9 - + 1 2 + 1 5 ) 3+3+3+3+3

Relate 3x5 to 5+5+5 (mathematical sentence)

Relate 10x1 to a jump of 10 steps

Relate 10x1 to 10+0

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Transcript: Group 7 i Operation Knowledoe Knowledge Constructed (Lepas itu? Lukis ... ) Sembilan kali empat ... satu, dua, tiga, empat ... sembilan lagi empat, empat lagi, Data exploration: Examine: Number Relate 9x4 to 9 groups with satu. dua. ti2:a. empat, satu. dua. tiga. empaL .empat. lima. enam. tujuh.lapan, Identify Counting 4 items in each group sembi Ian. Betullah ya ... eh ... tidak Mathematical operation: Grouping

Count & group (Lapan kali lima.) Lapan darab lima empat puluh Retrieval: Recall Multiplication (Lukis) Lapan kali lima, betullah lima butang. Lima puluh empat, betul, satu. dua. tiga. Data exploration: Examine: Number Relate 8x5 to 8 groups with empat. hma, satu. dua. tiga. empaLlima ... satu, dua. tiga, empat. lima ... tiga Identify Counting 5 items in each group ... satu. dua. tie:a, empat.lima ... satu. dU3. tiga. empat. lima ... empat. lima. Mathematical operation: I Grouping satu. dua. riga. empat.lima, enam ... satu. dua, tie:a. empat.lirna ... Satu. dua. Count & group

I tiga. empaLlirna. cukup dah. (Ok tiga darab tiga ... ) Sembilan Retrieval: RecaH I Multiplication (Macam mana lukis? Berapa?) Betullah, betuUah ... jangan takut.. .satu. dua. tiga ... satu. dua. riga ... satu. dua. Data exploration: Examine: Grouping Reiate 3x3 to 3 groups with

~ Identify 3 items in each group (Ok, enam darab empat) Dua puluh empat, enam ... enam ... enam .. satu, dua, tiQ:a. empaL. bukan enam Data exploration: Examine: Number Relate 6x4 to 6 groups with igek ... satu, dua, riga, empat. .. satu, dua tiga, empat ... satu. dua, tis::a, empat.. Identify Counting 4 items in each group sam. dua, tiga, empat. lima enam Mathematical opera[ion: Grouping

Count & group (Very good) (Tenogk sini, ini senat'Jg, enam darab dua ialah dua belas, macam mana buat enam darab dua 1) Enam darab dua, .. enam tambah enam ... dua bel as. Mathematical operation: Addition I Relate 6x2 to 6+6

Add (Enam tambah enam, dua belas. Ok kita boleh lukis bin tang, sebaris ini ada enam bintang, tambah, sekarang enam darab tiga, macam mana?) Lapan belas Retrieval: Recall Multiplication (Macam mana tahu lapan belas?) Saya dab hafal sifir (Oh ... hafal sifir. Apa kaitannya enam darab til!3 dengan enam darab dua?)

-------- ~----. --- -_ ..

148

Transcript: Group 7 Oh.. .dua belas belakang itu.. . darab satu.. .darab satu.. .dua belas darab dua.. . dua belas darab tiea.. .dua belas darab emoat.. .dua belas darab enam.. .& belas darab dua belas . . .betullah, dua belas. Darab satu.. .oh.. . t a u , tahu.. .sik tahulah.. . (Macam mana dengan yang baki ini? Dua belas darab dua berapa?) Dua belas (Dua belas darab satu dua belas) Tok salah, salah.. . takkan semua dua belas, dua belas satu betul tok betul.. . Toknya.. .enam kali dua. emuat kali tiea. tiga kali emuat.. .salah terbalik.. . emDat kali tiea. ..empat kali tiga.. .tiga, empat, lima, enam.. .enam dah.

(Macam mana sekarang soalannya dua puluh empat?) Kau dulu ... eh ... enam (Kamu boleh mula dengan mana-mana satu.. .) Enam, enarn, enam. enam.. .tok ia satu, satu.. .dua, tipa. emuat. lima, enam, tuiuh, lapan, seuuluh. sebelas, dua belas.. .dua belas igek.. .ha.. .dua.. .dua, empat, enam, lapan. dua belas, empat belas. enam belas, lauan belas. dua puluh, dua dua, dua ~ u l u h emuat.. .dua puluh empat darab.. . Dua belas tambah dua belas.

Knowledge Constructed

Relate 12 to 6 ~ 2 . 3 ~ 4 & 4x3

Relate 24 to 2+2+2+2+2+2 +2+2+2+2+2+2 & 12+12

Operation Solution generation: Trial & error

Solution generation: Trial & error, relate & inverse Evaluation: Confirm

Evaluation: Confirm Mathematical operation: Count Solution generation; Trigger

Knowledge

Multiplication

Counting Number

Knowledge Constructed Knowledge

Mathematics

Addition & multiplication

Addition

Addition Counting Number

Number

Number

Addition

Transcript: Group 8 (Adik perlu kuat cakap. Salah, betul tak penting. Cikgu akan rakamkan semua yang kita cakap. Apa yang adik nampak?) Matematik (Ada jenis apa?) Tambah dan kali.. .m (Kalau adik nak naik kereta ini adik mesti cuba semua soalan yang ada di sini. Kalau nak naik kapal terbang mesti buat semua soalan juga. Semua yang adik fikirkan adik cuba sebut. Kalau adik tak sebut, cikgu tidak dapat tahu apa yang berlaku dalam otak dan tidak dapat dirakam. Mula, sesiapa pun boleh buat. .

Boleh bincang. Yang penting sebut dengan kuat ok? Kita mula satu tambah satu.. .) Satu tambah satu.. . (Tulis, lepas itu?) Tujuh tambah emoat, seuuluh . . . (Tujuh tambah empat, sepuluh? Macam mana dapat sepuluh? Tujuh.. .ok macam mana?) Laoan, sembilan, seouluh. sebelas.. .

(Macam mana dapat sepuluh tadi? Sila adik tunjukkan macam mana dapat sepuluh tadi?) Terfikir (Terfikir. Pakai jari? Macam mana buat? Kamu fikir mana dulu, tujuh dulukah atau empat dulu?) nJ&l

(Lepas itu?) EmDat (Terus tambah? Ada kira dengan kayu manciskah dalam otak sini? Ada? Tapi silap. Ok tiga tambah sembilan.. .) Dua belas (Macam mana dapat dua belas? Hafal? Jangan malu cakap saja.) Seperti rantai.. .

Operation

Retrieval: Recognise

Retrieval: Recognise

Mathematical operation: Add

Mathematical operation: Add & count

Data exploration: Examine: Locate & compare

Data exploration: Examine: Locate & compare Mathematical operation: Add

Transcript: GrouJl8 OJ!eration Knowledge Knowledge Constructed (Kamu fikir mana dulu?) I Sembilan Data exploration: Examine: Number

Compare (Lepas itu?) Tiga (Tadi adik ada sebut rantai, apa rantai? Tolong lukis. Dua, sebut, sebut kuat...) Dua, tiga. emQat. lima. enam, tujuh. laQan. sembilan, itu tambah tiga, lukis satu Data exploration: Examine: Counting rantai lagi, campurkan ... Identify Number

Mathematical operation: Count

(Kamu mula dengan sembilankah atau mula dengan tiga?) Sembilan Data exploration: Examine: Number

Identify Addition Data exploration: Examine: Compare

(Lepas itu?) Tiga (Sambungkah lepas sebut sembilan? Tambah'l Terus menjadi dua belas atau

I kamu kira sepuluh, sebelas, dua belas Terus jadi dua belas (Ok, lapan tambah kosong ... ) Ah ... lapan Mathematical operation: Addition

Add (Teruskah sebut. .. ) Lima tambah enam, sebelas. Mathematical operation: Addition

Add (Macam mana dapat sebelas? Cikgu minat mac am mana kamu buat. Kamu letak en am, Jepas itu tambah lagi lima? Tak?) * Ada enamkan, ini ada emQat, lebih satu lagi sebelas. Data explanation: Explain Addition To obtain 5+6, take 4 from

5 & add onto 6 to make a 10, then add remainding J to get 11

(Terus kepada sebelas?)

152

(Macam mana adik buat tambah?) Sembilan, sepuluh, sebelas, dua belas. ti ga belas

Transcript: Group 8 Sebelas.. .sepuluh tambah dua, dua belas, sembilan tambah emuat.. .tiga belas

(Teruskan.. .) Emuat tambah enam. seuuluh

(Perlu guna jari? Mana dulu?) Enam duiu. baru emuat.

Operation Mathematical operation: Add

(Mengapa tak pilih empat dulu?) Susah.. . (Kalau tiga tambah sembilan, mana pilih dulu?) Sembilan

Knowledge Addition

(Empat tambah enam?) m... (Mengapa?) Lebih senang (Apa bezanya, mana lebih besar?) m (Ini, tiga dengan sembilan?) Sembilan

(Maksud adik kalau ada dua nombor, adik pilih yang besar? Ok.. .tuliskan) Tiga tambah lima.. .laoan (Kita tengok soalan di bawah.. .) Kali - (Kalau boleh, tolong juga sebutkan macam mana dapatnya) Satu darab satu sifar.. . Lima darab enarn.. .lima darab enam, tiga uuluh (Macam mana dapat tiga puluh?)

Mathematical operation: Count & add

Mathematical operation: Add

Data expioration: Examine: Compare

Data exploration: Examine: Compare

Data exploration: Examine: Compare

Addition Counting Number

Addition

Number

Number

Number

Data exploration: Examine: Identify

Mathematical operation: Add I Addition

Retrieval: Recognise Multiplication

Retrieval: Recall Multiplication I

Transcript: Group 8 Fikir (Macam mana fikir? Itu yang cikgu rninat) Kami dah hafal (Dah hafal dah? Tak payah kira? Adakah adik mula dari satu darab enam, dua darab enam, tiga darab enam.. .sampai ke lima darab enam?) Tak, mula dengan enam dulu.. . (Mula dengan enam dulu, enam darab enam berapa?) Enam dzrab enam. tioa puluh enam (Lepas itu?) Enam darab tuiuh.. .emDat uuluh dua (Enam darab lima?) Enam darab lima, tioa ~ u l u h (Tadi, adik mula dengan enam darab enam, tiga puluh enam: enam darab tujuh, empat puluh dua. Macam mana dapat empat puluh dua?) Tiga uuluh enam tambah lacri enam

(Ok tolong tulis jawapan. Sebut.. .cakap.. .) Tiea darab sembilan, dua uuluh tuiuh (Macam mana dapat dua puluh tujuh?) Fikir (Macam mana adik fikir, terangkan apa saja dalam otak.) Macam benda-benda lain yang boleh terpilih. Macam.. .macam kalkulatorkan, kita pakai kalkulator sebab saya selalu main kalkulator dekat "office" mak saya, saya kira-kira tengok buku kali, tengok bukukan.. .lepas tulis dalam otak, bila saya balik rumah saya hafallah. (Tadi tiga darab sembilan kamu hafal saja. Kamu terus ke tiga darab sembilankah atau lain? Jadi tiga darab sembilan, dua puluh tujuh. Tolong tulis.) Lavan darab k o s o n ~ . . . kosonq (Tujuh darab dua.. .) Tuiuh darab dua. ..emvat belas (Macam mana dapat empat belas?) Mula dengan dua belas

(Macam mana dapat dua belas?)

Operation

Retrieval: Recall

Retrieval: Recall

Retrieval: Recall

Solution generation: Relate Mathematical operation: Add

Retrieval: Recall

Retrieval: Recall

Retrieval: Recall

Mathematical operation: Add

Knowledge

Multiplication

Multiplication

Multiplication

Multiplication Addition

Multiplication

Multiplication

Multiplication

Addition Number

Knowledge Constructed

Relate 6x7 to 6x6=36 by adding 6 to 36

Relate 7x2 to 10+2+2

Transcript: Group 8 ( Operation I Knowledge I Knowledge Constructed Kerana tambah.. .ah.. .dua belas dari seuuluh.. . (Lepas itu?) Tambah (Empat darab empat) Enam belas (Lapan darab lima) Emuat ~ u l u h (Macam mana dapat empat puluh? Macam mana buat sifir lima?) Hafal. (Hafal terns? Senangkah sifir lima? Jadi macam mana kamu dapat empat puluh?) Hm.. .sebab tambah. tambah. tambah.. .lima, sepuluh, lima belas: dua puluh. dua uuluh lima, tiga uuluh. tiga ~ u l u h lima. emuat uuluh.

(Berapa kali?) Ah.. .lapan (Ok tuliskan, satu darab sepuluh?) Seuuluh (Enam darab tujuh?) Empat puluh dua (Ok aktiviti dua. Cikgu mahu adik tenpok ini dan cuba fahamkan apa yang - - diberitahu oleh contoh ini? Cakap, cakap dengan kuat.. Tiea. tiga, tiga. tiga

(Macam mana lukis itu? Soalan ialah lima kd i tiga) Tiga tambah tiga tambah tina tambah tiga tambah tiga

(Berapa kali?) Ah ...m (Berapa semua itu?) Ah. . . h a belas. Ok, kalau kita nak lukis lompatannya? Macam mana?)

Retrieval: Recall

Retrieval: Recall

Data exploration: Examine: Identify Mathematical operation: Add

Retrieval: Recall

Retrieval: Recall

Data exploration: Examine: Extract

Data exploration: Examine: Identify

Data exploration: Examine: Identify

Multiplication

Multiplication

Addition

Multiplication

Multiplication

Multiplication jump

Multiplication jump

Relate 8x5 to 5+5+5+5+5+ 5+5+5 (5,10,15,20,25,30,35,40)

Relate 5x3 to 3+3+3+3+3 ( 3 + 6 + 9 + 1 2 + 1 5 )

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i (Kita tengok seterusnya. ..macam mana dengan ini?) Sila kelomuokkan, dua, dua, dua, dua darab tujuh. ..

Transcript: Group 8 Hm ... dari sifar ke dua. ke empat.. .enam. ke lapan. ke sepuluh ...

, '~ . ~ ~ ~ - . Sila kelomookkan titik-titik untuk menuniukkan ... (Kelompo &an... titik dalam kelompok itu mesri sama banyak Kita tengok tujuh darab dua.. .) Empat belas (Ya, empat belas ... lepas itu, lukis kelompok, kelompok itu boleh lukis di mana-mana. Asalkan bilangan titiknya sama banyak. Macam mana?) Dua tambah dua tambah dua tambah dua tambah dua tambah dua tambah dua

Operation Mathematical operation: Multiplication jump

(Berapa semua itu?) Satu. dua. tiea. emoat, lima.enam, tuiuh ... tuiuh

(Setiap kelompok ada berapa titik?) 1 &

Knowledge Multiplication jump

(Seterusnya, dua darab tujuh) ffitak...kitak... (Tak apa, siapa pun boleh buat. Dua darab tujuh. ..empat belas juga?) Dua, dua, dua.. .sama juga.. . (Sebut. ..berapa dah? Temskan sembilan darab empat) Sembilan darab empat tiga puluh enam.. .empat...

Knowledge Constructed Relate 10x1 to 5 jumps with 2 steps in each jump ( 0 - + 2 - + 4 + 6 + 8 +

1 (Kelompoknya berapa?)

Data examination: Read

Data examination: Read i I Retrieval: Recall I

Data exploration: Examine: Identify Mathematical operation: Add &group

Mathematical operation: Count

Retrieval: Recall Data exploration: Examine: Identify Mathematical operation: Group

Murliplication

Grouping Addition

Counting Number

Grouping

Relate 7x2 to 7 goups with 2 items in each group (2+2+2+2+2+2+2)

Relate 9x4 to 9 groups with 4 items in each group

/ Transcript: Group 8 I Operation I Knowledge / Knowledge Constructed I Se;nbilan Empat tambah emuat tambah emoat ... tambah emuat tambah empat tambah empat tambah emDat (Lagi, berapa sudah?) Tuiuh ... lauan ... sembilan

(Kalau adik hitung semua titik itu, ada berapa semua?) Satu laei ... satu. dua. tiea. emDat. iima. enam. tuiuh. lapan. sembilan. seuuluh. .c@elir J.1- hc:i, :ici h c l ~ :mnJ: h s l ? ~ llma hells. c n ~ i . ~ J I U ~ I b::+>. -- larm W13i. crrn.h!lz~ br!:rj. oua c.i!!!: 2-2 n n l ~ h s?rc Jc.; P I : I J ~ J1.9. dua puiuh tiea. dua puluh emuat, dua ~ u l i i h lima. dua puluh enam. dua puluh tuiuh. dua puiuh lapan, dua puluh sembilan, tiza puluh. tiea puiuh satu, tiea puluh dua. tiea ouluh tiea. tiea uuluh empat. tiea ~ u l u h lima. tiea puluh enam.. . (Macam mana dengan lapan darab lima?) Emuat puluh (Empat puluh, ok rnacam mana lukis?) Lima tambah lima tambah iima tambah lima ... tambah iima tambah lima ... tambah lima. Sitok bolehlah

(Berapa dah?) Tujuh ...-

~ u k u p (Ok tiga darab tiga.. .cakap.. .cakap.. .cakap.. .) a (Sembilan, macam mana lukis?) Enam (Sebut, sebut ... ) Dua. dua, dua. dua ... tiea ... tiga, tiga ...

(Enam darab empat)

Matehmatical operation: Add

Mathemarical operation: Count

Mathematical operation: Count

Retrieval: Recall

Data exploration: Examine: Identify Mathematical operation: Add & group

Mathematical operation: Count

Retrieval: Recall

Solution generation: Trial & error

Addition

Counting Number

Counting Number

Mulripiication

Grouping Addition

Counting Number

Multiplication

Relate 9x4 to 4+4+4+4+4+4+4+4+4

Relate 8x5 to 8 groups with 5 items in each g o u p (5+5+5+5+5+5+5+5)

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(Kita tengok bansnan ini, berapa buah bilik?)

Transcripk Group 8 I Operation I Knowledge I Knowledge Constructed 1

(Dua belas juga. Macam mana dapat dua belas im?) Eh ... emoat. empat. emoat ... cnam. enam

(Enam, enam juga?) Tioa, tioa. tiza. tiaa

Row & column Ini enamkan? 1ni dua.. .. enam darab dua ... dua belas

(Lagi?) Dua. dua, dua. dua. dua. dua

Data exploration: Eaxmine: Tdentifv

(Kalau darab?) Enam. ..darab empat ... (Dari mana dapat enam?) Tiga darab dua ... dah...dua, dua, dua ... emoat darab t i ~ a .

/ (Macam mana tahu itu?)

dua, dua, dua. dua .... dua tambah enam, eh ...-

(Tapi macam mana kaitkannya? Kita tengok ini ada dua, ini ada enam . . )

/ ~ z i i n generation: Relate

Data examination: examine: ldentify Solution seneration: Relate

Data examination: examine: Identify

Data examination: examine: Identify Solution generation: Relate

Mathematical operation: Add Solution generation: Relate

Mathematical operation: Add Solution generation: Relate

Data exploration: Eaxmine: ldentify Solution generation: Relate

Data examination: Locate

Data exploration: Eaxmine: ldentify Solution generation: Relate Summarisatiom: Summarise

Row & column

Number

Addirion ROM, & column

Addition Row &column

Addition Row 8- column

Multiplication Row & column

Row & column

Addition Row &column

Relate 12 to 4x3

Relate 12 to 4 4 4 , 6 i 6

Relate 12 to 3+3+3+3

Relate 12 to 2+2+2+2+2+2

Relate 12 to 4x3

Relate 12 to 2+2+2+2+2+2 =2x6

Transcript: Group 9 1 Operation (Cikgu panggil adik sini untuk buat sedikit matematik. Betnl salah tidak pentingT yang penting adik cakap. Adik mesti pandai dalam matematik, sebab itulah diuilih. Cikgu akan rakam semua yang adik cakap. Lebib kuat adik cakap lebih bagus, usah bisik-bisik. Cikgu bagi aktiviti, sesiapa pun boleh buat, boleh buat bersama, bincang bersama, yang penting sebut dengan kuat supaya dauat dirakam. Kadang-kadang cikgu akan tanya ...ah... macam mana buat, &alnya.. . satu tambah satu macam mana buat?) &

(lalah dua. Cikgu minat macam mana dapat dua im.)

Matematik (Apa operasi adik tengok?) HA.. .kereta (Setiap anak tangga ini ada satu soalan, kalau adik pandai buat, sampai sini adik akan naik kereta.) Oh. ..dapat kereta (Oh ... kalau pandai lagi dapat kapal terbang) Oh ... senang. Kitak dulu ... kitak dulu (Mula ... sebut, jangan bisik, lain boleh tolong.) Macam biasa, cikgu? (Ya, macam biasa.) Satu tambah satu. dua

(Tulis, tnlis.. .) Tuiuh tambah emuat, dua be1 as... eh, &&s

(Macam mana dapat sebelas?) Sebah kira (Macam mana?) Ambil tuiuh letak ke mulut. emoat di tanean

Mathematical operation: Add

Data exp!anation: Explain

Retrie\zal: Regconise

Mathematical operation: Add

Mathematical operation: Add

Data exploration: Examine: Identify & locate

Addition

Addition

Mathematics

Addition

Addition

Number

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171

Sama ini. sila tuniuk lompatan darab untuk soaian-soalan berikut. please show ... Lima darab tiga, number line ... Iima belas ... avat matematik ... P

Data examination: Read Data exploration: Examine: Extract

(Terns) Lima darab tiza. lima belas.. . lima belas, lima tambah lima.. . Lima tambah lima tambah lima.. .ah.. .betullah

Transcript: Group 9 (Ya, betul. ..)

(Lompat? Macam mana nak lompat?) Lompat.. .oh, lompatan senang aja. Lompat.. .kosong.. .Kita lompat lima sekali, satu, dua eh.. .[l]satu. dua. tiga. empat. lima ... Salah, lompatan dua. Lompatan dua? Dua lompat, dua lompat macam tokkah? Tuiuh kali.. .Oh macarnnya. Satu, dua, tlga, emuat, lima. enam, tuiuh ... eh ... jawapan em~at w... tambah satu ... ah j a w a ~ a n emDat belas. .. ah betul ... emuat belas betullah, lima belas jawapannya. Betullah.. . dua, salah kau.. .eh.. . betul. Darab tujuh tlea lanekah, tiea lanpkah, tiea lanekah, lagi berapa? Satu, dua, cuba dulu.. .[2]Tiga langkah. lima kali.. .eh, besar. ia empat langkah.. . (Ok, lepas itu? Sebutlah) Empat. empat. tiga. empat, lima. lima.. ..

(Ok) [IjTuiuh darab dua, dua darab tuiuh. senang empat belas ... iukislah. Oh ...[ 2ldua tambah dua tambah dua tambah dua tambah dua.. . Satu, dua, tica, emuat. lima, enam. tuiuh. Dua, empat. enam. lapan, sepuluh, dua belas, empat belas .... dua lompat ... satu, dua, tiga, empat, lima, enam, tuiuh

(Dah? Sambung.. .sebutlah)

Knowledge Constructed Operation

Retrieval: Recall Data exploration: Examine: Identify Mathematical operation: Multiplication jump

Knowledge

[lIMathematical operation: Count [2]Solution generation: Trial & error

Solution generation: Trial and error

[lISolution generation: Inverse [2]Mathematical operation: Add & count

Multiplication Addition Multiplication jump

Multiplication jump Number Counting

Multiplication jump Multiplication Addition Counting Number

Relate 5x3 to 3 jumps with 5 steps in each jump (5+5+5=15)

Relate 2x7 to 7 dumps with 2 steps in each jump

Relate 2x7 to 2+2+2+2+2+ 2+2

Transcript: Group 9 Empat darab emuat.. .enam beias.. .dua. dua.. . dua darab dua darab.. .bolehlah, bolehl ah... lima. enam. tuiuh. lauan, sembilan, seuuluh ... enam belas ...

(Sebtu ... sebut ... ) Emuat tambah emuat sama denean lauan. lauan tambah emuat sama denean dua belas, emuat dua belas sama denoan enam beias.. .Berapa Iompatan nak? Empat.. . tok satu, dua. tiea, emuat.. .dua belas.. .dua belas.. .lapan.. .dua, lima.. .senang buat, jangan takut. Macam tadibah.. . (Sebut.. .sebut) Tiea darab lima lima belas (Ok, lompat lagi, berapa jauh iompat?) Tiea. tiga.. .

(Lepas itu pergi mana? Lepas itu?) Betullah, betullah.. .salah.. .berapa lompatan? Tiga. Oh . . .tiga. enam, sembilan, dua belas, lima belas.. .

(Sepuluh darab satu) Sepuluh (Lepas itu.. .itu saja?) Sepuluh ... eh ... satu tambah sepuluh, satu tambah satu tambah satu tambah satu tambah satu tambah satu tambah satu, tuiuh, agek.. .lauan, sembilan ... satu tambah satu tambah satu tambah satu tambah satu tambah satu tambah satu tambah satu.. .emuat, lima, enam, tuiuh. lauan. tambah satu sembilan.. . lomuatan satu. Senang. Tiga. emuat, lima, enam, tuiuh. lauan.. .lapan, sembilan. seuuluh. (Ok, lukis kelompok dan titik dalam kelompok mesti sama banyak. Tujuh darab dua) Eh.. . senang (Berapa itu? Berapa titik dalam satu kelompok?)

Operation Retrieval: Recall Solution generation: Trial and error Mathematical operation: Count

Mathernnatical operation: Add & count

Retrieval: Recall

Data exploration: Examine: Identify

Mathematical operation: MuItipIication jump

Retrieval: Recall

Mathematical operation: Multiplication jump & count

Knowledge Multiplication counting Number

Addition

Multiplication

Multiplication jump

Multiplication jump

Multiplication

Addition Counting Number

Knowledge Constructed

Relate 4x4 to 4+4=8, 8+4=12, 12+4=I6 (4+4+4+4)

Relate 3x5 to 5 jumps with 3 steps in each jump (3+6+9+12+15)

Relate 10x1 to 1+1+1+1+1 t l + l + l + l + l

Transcript: Group 9 Dua ... satu, dua, tipa, empat, lima. enam. tuiuh ... dua, emuat, enam, lauan. - seuuluh. dua belas, emuat belas.. .

(Ok, pandai, dua darab tujuh) Emuat belas. ..dua, tiga, empat.. .dua. tipa, emuat. lima. enam, tuiuh.. . satu. dua, tiea. emuat. lima, enam, tuiuh.. .empat belas. Betullah.

(Sembilan darab empat) Emuat. emuat.. .eh.. .dua.. .emuat, emuat, emuat.. .sembilan emuat.. ..

(Lepas itu? Sebut, sebut.. .) Sembilan. l a ~ a n belas ... lapan belas, lauan belas tambah sembilan, dua puluh berapa? Dua ~ u l u h tuiuh.. .lagi.. .tambah sigek.. .betul, betul (Dah? Lapan darab lima) Emuat puluh.. .tiea, emuat, lima.. .tuiuh, lauan. lauan.. .enam, tuiuh, lauan

(Ok tiga darab tiga) Sembilan (Sebut.. .ok) Tiea, enam, sembilan

(Ok, enam darab empat.. .lepas itu?) [1][2]Emuat, emuat. empat ...[ 3]emuat, lauan, dua belas.. .dua belas, enam belas.. .dua belas. ..enam belas, dua uuluh, dua uuluh emuat.. . -

(Sebut, sebut. Ok tengok sini) Enam darab tipa.. .lauan belas (Teruskah? Tak perlu inikah?)

Operation Data exploration: Examine: Identify Mathematical operation: Count, add & group

Mathematical operation: Count, group & multiply

Data exploration: Examine: Identify h$athematical operation: Group

Mathematical operation: Add

Retrieval: Recall Mathematical operation: Count & group

Retrieval: Recall

hlathematical operation: Add & group

[l]Data exploration: Examine: ldentify Mathematical operation: [2]Group, [3]count 8: add

Retrieval: Recall

Knowledge Number Counting Addition Grouping

Multiplication Number Grouping

Grouping

Addition

Multiplication Grouping Counting Number

Multiplication

Addition Grouping

Addition Grouping Counting Number

Multiplication

Knowledge Constructed Relate 7x2 to 7 groups ctf 2 items in each group

Relate 2x7 to 2 groups of 7 items in each group

Relate 9x4 to 9+9+9+9

Relate 3x3 to 3 groups of 3 items in each group

Relate 6x4 to 6 groups of 4 items in each group

Transcript: Group 9 Tak (Tak, hafal? Ok) Tuiuh darab enam empat puluh dua (Ya, lagi? Sebut, sebut.. .) Dua darab enarn dua belas. dua uuluh satu. empat darab lima dua puluh.. .enam tujuh emuat dua (Ok tengok sini) Oh. ..dua belas.. .[l]satu, dua. tiga, emuat. lima, enam. tuiuh. lauan, sembilan, seuuluh. sebelas. dua belas, .. .[2]satu, satu. satu. satu. satu. ..dua belas, dua belas. ok semua dua belas,..:.[3]dua, dua, dua.dua, dua, dua ... ooi.. .salah

(Macarn mana tahu ia dua?) Dua dua empat ... eh ...[ lldua, emuat, enam, l a ~ a n , seouluh, dua belas.. .[2]tiga, tiga.. .emuat. lapan. dua belas.. .lima, lima.. .lima, lima seuuluh.. .enam, enam.. .enam. enarn.. .tuiuh. tuiuh.. .satu.. .dua belaslah. Sepuluh. dua belas tok.. .kosone, kosong dua belas (Kosong? Dah? Nak betulkan?) Tak, tak, salah betul tak apa. (Sekarang dua puluh empat) Dua puluh ernpat. Dua.

(Ok, dua puluh empat. Macam mana dua puluh empat?) Sembilan. sembilan tarnbah sembilan, sembilan

(Boleh mula dari mana-mana satu yang kamu rasa lebih senang) [IISatu. dua. tiea ... senang ... dua, tiga, [2]dua. emoat. enam. laoan. seouluh, dua belas ... Oh ... dua belas .. .enam belas. laoan belas, enam, enam. dua belas, tiea belas, ernoat belas. lima belas. tuiuh belas, laoan belas.. . laoan belas, sembilan belas, dua ouluh. dua ~ u l u h satu. dua uuluh dua. dua ouluh tiga, dua puluh emoat.. .mahu samakah? (Ya, mahu sarna nombor itu. Fikirlah) Hm.. . (Mula dengan yang kamu rasa lebih senang)

Operation

Retrieval: Recall

Retrieval: Recall

[lIMathematical operation: Count [2]Solution generation: Trial & error [3]Evaluation: Disconfirm

[IJMathematical operation: Add [2]Solution generation: Trial & error

Solution generation: Trial & error

Solution generation: Trial & error

[l]Solution generation: Trial & error [2lMathernatical operation: Count & add

Knowledge

Multiplication

Multiplication

Counting Number Addition

Counting Number Addition

Addition Counting Number

Knowledge Constructed

Relate 12 to 1+1+1+1+1+1 +1+1+1+1+:+1

Relate 12 to.2+2+2+2+2+2

Relate 24 to 6+6+6+6

Knowledge Constructed Relate 24 to 2+2+2+2+2+2 +2+2+2+2+2+2; Relate 24 to 3+3+3+3+3+ 3+3+3

Relate 24 to 4+4+4+4+4+4 (4+4=8,8+4=12,12+4=16, 16+4=20,204=24)

Transcript: Group 9 Tujuh.. .satu, dua. tiga, emuat. lima, enam, tuiuh, lauan. sembilan, seuuluh, sebelas. dua belas. ..dua, dua, dua. dua. dua, dua. dua, dua, dua.. .dua emuat. k, tiga.. .tiga, enam, sembilan, dua belas. lauan belas, dua uuiluh satu, dua puluh emuat Tiga. tiga, tiga, tiga, tiga, tiga. tiea, tiga .... dua uuluh emuat (Lagi?) Dua puluh empat.. .sik boleh.. .oh.. . (Oh.. . nombor yang sama, jawapan dua puluh empat) Semua dua puluh empat.. .tok enam.. .sitok enambah, Illenam, tuiuh. lauan. sembilan, seuuluh, sebelas, dua belas, tiga belas, emuat belas, lima belas. enam belas. tuiuh belas, lauan belas, sembilan belas, dua uuluh, dua ~ u l u h satu, dua puluh dua, dua uuluh tiea, dua uuluh empat, dua uuluh lima. dua puluh enam.. .ah ha.. .[2]seeuluh, lima belas. dua uuluh. dua puluh lima.. . [3]empat, - empat, lauan.. .lauan emuat dua belas, dua belas emuat enam belas, enam belas emuat dua uuluh. dua puluh tambah emuat yes! Empat, emuat, emuat, emuat, empat. empat.. . (Tinggal tiga lagi.. .) Tuiuh. tuiuh emuat belas. emuat belas tambah tuiuh ... dua uuluh satu ... bukan, lima, lima lima seuuluh, sebleas. dua belas, tiga belas. emuat belas. lima belas. enam belas, tuiuh belas, lauan belas. sembilan belas, dua puluh

(Dah?) Tiga.. .dua belas.. .tok.. .lauan

(Tak susah ... bincang dulu.. .ah, bincang dulu, nanti dapat. Fikirlah, cuba satu, satu, sama-sama bincang.. .) Berapa? Tuiuh. tuiuh emuat belas. emuat belas tambah tuiuh dua uuluh satu.. .sik dauat. Sik boleh.. .

(Sambung) Eh, lauan. lauan enam belas. lapan belas. sembilan belas, dua uuluh, dua uuluh satu, dua vuluh dua. dua puluh tiea. dua puluh emoat, dua uuluh limo. Dua puluh enam.. .tuiuh. tuiuh . . .

Operation Mathematical operation: Count & add

[ l ]Mathematical operation: Count & add [2]Solution generation: Trial & error [3]Evaluation: Confirm

Solution generation: Trial & error Mathematical operation: Count & add

Solution generation: Trial & error

Solution generation: Trial & error Eva1uation:Disconfirm Mathematical operation: Add

Mathematical operation: Count & add

Knowledge Addition

Counting Number Addition

Addition Counting Number

Addition

Addition Number

Transcript: Group 9 (Tujuh, tujuh berapa?) Empat belas ... dua belas, tiga belas, emuat belas. lima belas, enam belas. tuiuh belas, lapan belas. sembilan belas, dua uuluh, dua uuluh satu. dua uuluh dua, dua ~ u l u h tiea. dua puluh emuat.. .ha, ha. ..betul, betul, betullah. Dua, dua empat, satu, satu dua.. .dua puluh empat, betullah, betul. Kitak sudah.. .he, he, he.. . dua puluh tambah empat tambah empat, dua puluh lapan.. . (Tadi, awak cuba tujuhkan? Cukup?) Tak (Ok, cubalah nombor lain) Lapan.. .lapan.. .sik dauat.. .dua puluh enam, . ..[2]lauan, lauan, enam belas ... sembilan belas. dua uuluh, dua uuluh satu. dua ~ u l u h dua. dua uuluh tiya. dua puluh empat. dua ~ u l u h lima, dua uuluh enam.. .dauat.. .dua uuluh emaut ok ya, [jltiea darab lauan beraua? Dua uuluh emuat, va? Tiga darab lauan dua puluh empat nak, ah . . .lapan eh betullah, betullah.. . Lapan, lapan, lapan.. .lauan. lauan enam belas. tuiuh belas. lauan belas. sembilan belas, dua puluh, dua uuluh satu, dua uuluh dua, dua uuluh tiya, dua uuluh emuat. [l]Sembilan, lauan belas.. .lavan belas sembilan dua uuluh tuiuh .. .tiga uuluh lima eh.. .dua uuluh tuiuh. tiga uuluh enam.. . bukan. .. Tolak, [lltolak dua belas, tolak dua, tolak lauan, tolak sembilan.. .lima belaskah? Lima belas, lima belas sama beraua? Tiea uuluh ... boleh tambah lima belas? Banyak lagi soalan, macam mana? (Tinggal satu lagi saja, tinggd satu lagi) Enam, dua belas, dua belas tambah enam, lauan belas, sembilan belas, dua puluh, dua uuluh satu, dua uuluh dua, dua uuluh tiea, dua uuluh emuat. Ye! Dapat! Enam, enam. enam, enam.. .ah.. .ok (Ini dua belas, kamu bina satu bangunan dengan dua belas bongkah kayu . . .) Enam. ..dauat buat empat, emuat ... emuat tok tiga ... tiga ieek. tiga ieek ...

(Setiap tingkat mesti sama banyak bilik, kalau satu bilik, semua satu bilik, tak boleh, ini dua, ini satu.) Oh.. .enam, enam

( (Dah? Cuba lagi.)

I Operation 1 Knowledge ( Knowledge Constructed I

Mathematical operation: Count & add

[2]Mathematical oper~tion: Count & add Solution generation: [3]Relate & [l]Trial B error Summarisation: Summarise

Mathematical operation: Add & count

Solution generation: Trial & error

Solution generation: Trial & error

Addition Counting Number

Addition Counting Number

Addition Counting Number

Relate 24 to 6+6+6+6

Knowledge Constructed Knowledge

Counting Number

Counting Number

Counting Number

Transcript: Group 9 Lapan

(Sudah?) Sudah, bentuk lain.. .sik boleh.. .Satu, dua, tiga, empat, lima, enam, tujuh. Satu, dua, tiga, empat. lima, enam, tuiuh, lapan. sembilan. sepuluh. sebelas. dua belas.. .

(Berapa itu, dua belas?) Ini empat.. .buat tiga, tiga.. .tiga igek.. .dua igek lagi, enam, sembilan, dua belas.. .saEu, dua, tiga.. .dua belas, tiga belas.. .eh, tiga belas. (Buat tinggi) Emuat, empat. emuat ... sama. .. Satu, dua. tiga, emuat, lima, enam, tuiuh. lavan, sembilan, seuuluh, sebelas. dua belas. Ah.. .macam.. .samanya nak... berapa lagi? Tolak empat igek.. .sik dapat.. .macam ia lagi.. . tok lain.. .Satu, dua, tiga, empat, lima, enam, tujuh, lapan, sik dapat, sik dapat. Lima igek barn dapat.. .lima, sepuluh.. .tok lain.. .sama tak? Satu, dua. tiga, embat, lima. enam, sama ooi, ia sama.. . (Ubah, ia mesti semua sekaligus) Cantum.. .sekaligus.. .ubah.. .lain.. .empat.. . (Ok, buat lagi) [IISatu, dua. tiea, emnat. lima. enam, tuiuh. lapan. sembilan, se~u luh , sebelas, dua belas, betullah. Satu, dua, tiga, empat, lima buah ...[ 2]Satu, dua, tiea, empat, lima.. .enam. enam, enam.. . ye belum.. .panjang . . .dua igek lagi samalah, samalah.. .[3]sama emuat dah, enam dah, nak. ..enam, enam, enam. Satu, dua, tiga, empat.. .

(Itu sudah lebih dua belas, sama, empat, empat, empat. ..sama, bolehlah.. .buat sekaligus sambung.. .Ini sama tak?) Tak sama (Itu dah lebih, mahu dua belas.. .sama juga. Lagi.. .buat tinggi lagi.) Buat tinggi.. .tinggi, satu, dua, tiga, emDat.. .empat, empat, boleh, boleh.. .ah tinggi, bolehlah, boleh.. . (Rumah panjang?)

Operation Solution generation: Trial & error

Solution generation: Trial & error Mathematical operation: Count

Data exploration: Examine: Identify Data exploration: Examine: Compare

[l]Mathematical operation: Count [2]Solution generation: Trial & error [3]Data exploration: Examine: Identify

Solution generation: Trial & error

Transcript: Group 9 Macam itu, macam itu.. .rumah panjang. Satu, dua, tiea. emuat, lima, enam. tuiuh, lavan, sembilan. seuuluh.. .satu lagi.. .bengkang bengkok.. . (Ada lagi? Mahu lagi.. . habislah?) Satu. dua. tiea, emuat. lima. enam. ..empat igek lag, sigek

(Berapa semua itu?) Dua belas

(Tak ada lagi, habis?) Ya, semila sama. Satu, dua:. .dua belas (Dah? Ia sama.. .) Sama.. .ha, ha (Ok, sekarang, cikgu mahu kamu tengok bangunan ini, berapa semua ini?) Satu. dua, tiea. emuat, lima, enam. tuiuh. lapan. sembilan. seuuluh, sebelas. dua belas.. . .dua belas (Macam mana dapat dua belas itu?) Dua (Kamu dah hitung, satu, satu, satu, ada cara lainkah? Kalau pakai darab?) Ada, ada, ada, dua darab dua darab dua.. . (Tengok bangunan ini saja.. .) Ernpat darab tiea. empat darab empat.. .Empat darab . . .lapanbah.. .

(Itu saja? Empat darab tiga saja? Ada lain, kalau pakai tambah?) Emuat tambah emuat tambah emuat

(Lain lagi? Lagi?) Bagi boleh, cikgu? (Macam mana bagi?) Dua belas baai emuat tiga.. .

(Macam itu? Bolehkah dapat dua belas?)

Operation Mathematical operation: Count

Mathematical operation: Count

Data exploration: Examine: Identify

Mathematical operation: Count

Solution generation: Relate Mathematical operation: Multiply

Solution generation: Relate Mathematical operation: Add

Mathematical operation: Divide

Knowledge Counting Number

Counting Number

Number

Counting Number

Row & column Multiplication

Row B column Addition

Division

Transcript: Group 10 (Sepuluh tambah dual D U ~ belas (Macam mana sembilan dengan empat?) Sembilan simpan di otak, tambah iari empat ... sembilan. seuuluh. sebelas. dua belas. t i ~ a belas.

(Ok tulis tiga belas, macam mana empat dengan enam?)

(Tiga dengan lima?)

(Sekarang kita tengok darab. Satu darab satu.. .) Satu - (Lima darab enam?) Tlea uuluh (Macam mana dapat tiga puluh?)

Operation

Mathematical operation: Add

Data exploration: Examine: Identify & locate Mathematical operation: Add &count

Mathematical operation: Add

Mathematical operation: Add

Retrieval: Recall

Retrieval: Recall

Knowiedge

Addition

Ah ... (Betullah tiga puluh, tapi macam mana teringat tiga puluh itu, hafal? Perlu p a l l satu darab enam enam, dua d a n b enam dua belas? Tak payah. Macam mana dengan kamu?) Tiea darab sembilan. dua ~ u l u h tuiuh (Macam mana dapat dua puluh tujuh?)

Counting Addition Number

Retrieval: Recall

Addition

Addition

Multiplication

Multiplication

(Tahukah? Macam mana dapat dua puluh tujuh?) Saya ... ah ... riea hitune sembilan kali.

(Lapan darab sifar.) Sifar - (Tuluh darab dua?) Emuat belas (Macam mana dapat? Hafal? Boleh hafal sampai sitir berapa?) Sebelas

Data explanation: Explain

Retrieval: Recall

Retrieval: Recall

Retrieval: Recall

Additon

Multiplication

Multiplication

Multiplication

Relate 3x9 to repeat addition of 3 for 9 times. (3+3+3+3+3+3+3+3+3=2 7)

Transcript: Group 10 I Operation 1 Knowledge / Knowledge Constructed (Ok sebelas darab sebelas berapa?) Ah ... sebelas ... (Berapa sebelas kali sebelas? Ok sebelas darab sepuluh?) Ah ... serams seouluh (Ok tujuh darab dual Empat belas (Macam mana dapat empat belas? Ada pakai jari? Tak perlu, haiai? Kamu hafalkan tujuh darab dua atau dua darab tujuh? Mana senang?) Dua darab tuiuh

(Jadi empat darab empatl Enam belas (Cepatnya. Macam mana?)

( J a d i ) Enam belas (Ok lapan darab Iima) Empat ~ u l u h (Macam mana?] Lima hitung ... lapan hitune lima kali.

(Lapan tambah lapan) LaDan lambah l a ~ a n ... enam belas (Ok l a ~ a n darab lima) ~ m ~ a t o u l u h (Ada pakai lima, sepuluh, lima belas, dua puluh ... tak? Hafal? .... Satu darab sepuluh) Seouluh (Ok akhirnva enam darab tuiuh) Empat oulih dua (Macam mana dapat empat puluh dua?)

Retrieval: Recall

Retrieval: Recall

Data exploration: Examine: Compare

Retrieval: Recall

Data explanation: Explain

Retrieval: Recall I Data exploration: Examine: Identify Mathematical operation: Add

Mathematical operation: Add

Retrieval: Recall

Retrieval: Recall I Retrieval: Recall I

Multiplication

Multiplication

Multiplication

Multiplication

Addition

Multiplication

Addition

Addition

Multiplication

Multiplication

Multiplication

Relate 4x4to repeat addition of 4 for 4 times. (4+4+4+4=16)

Relate 8x5 to repeat addition of 8 for 5 times. (8+8+8+8+8=40)

Transcript: Group 10 I Operation I Knowledge I Knowledge Constructed Tak I I I (Macam mana awak buat?) Tiga.. .per@ ke.. .. (Macam mana buat dua darab tujuh dulu? Kalau dua darab tujuh?) Dua darab ruiuh, emuat belas (Ya, nak lulus macam mana?) Sekali. dua kali. tiza. emoat ... enam. tuiuh. l a ~ a n , sembilan. se~uluh. sebelas. dua belas. tiea belas. emuat belas ...

I (Setuiu? Cakao ... kalau adik buat ini macam mana adik lukis?) . <

Eh ... (Tadi adik buat empat darab empat, enam belas. Macam mana adik lukis?) Satu. dua, tiea. emDat, lima. enam. tuiuh, laoan, sembilan. seouluh. sebelas, dua belas. tiea belas. emoat belas. lima belas. enam belas ... (Ayat matematik? Tulis. Bincanglah jika tak pasti.. .bincang, bincang.) Eh ... (Berapa ini?) Dua darab lauan

(Dua darab lapan.. .) Dua darab lau an... sama denean .. .enam belas .... (Cikgu rasa kamu tahu ... ini lompat berapa jauh?) Dua...

(Duakan dan ini empat darab empat.. .jawapannya.. .) Enam belas. (Tulis enam belas ... ada lagi? Macam mana dengan ayat matematik? Sebut .. .jangan malu ... Ok buat ini, tiga darab lima) Lima belas (Lepas itu? Sebut ... ) Dua . ..lima, enam ... l a ~ a n . sembilan, se~uluh. sebelas, dua belas. tiea belas, emDat belas, lima belas

Retrieval: Recall Multiplication

Mathematical operation: Count

Mathematical operation: Count

Retrieval: Recognise

Retrieval: Recall

Addition Counting Number

Data exploration: Examine: Identify

Relate 2x7 to 2+2+2+2+2+ 2+2

Retrieval: Recall

Retrieval: Recall

Mathematical operation: Count

Counting Number

Number Multiplication

Multiplication

Multiplication jump

Multiplication

Multiplication

Counting Relate 3x5 to / Number I 1+1+1+1+1+ 1+1+1+1+1+1+1+1+1+1=

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Transcript: GrouplO QJ>eration Knowledge Knowledge Constructed Tak (Macam mana dengan ini?) Dua darab enam ... enam darab dua sarna dengan dua belas .. dua puluh ... tie:3 Retrieval: Recall Multiplication Relate 2x6 to 6x2 which is darab tujuh. dua Duluh sam ... empat darab lima. dua ou1uh; enam darab tujuh Solution generation: Inverse given earlier sarna dengan empat puIuh dua (Macam mana ingat empat puluh dua?) Hafal (Aktiviti nombor lima) Sila isi ruang-ruanQ: koson2: berikut. nombor-nombor van!! anda Qilih mesti Data examination: Read sarna untuk ceraian." (Nombor dalam petak mesti sarna, macam mana tahu ini satu?) Eh ... 12etak ini dua belas ... lima. enam. tujuh.la,Qan. sembilan. sel1u1uh, sebelas. Solution generation: Relate Counting Relate 12 to dua belas. Mathematical operation: Number 1+1+1+1+1+1

Count +1+1+1+1+1+1=12xl Summarisation: Summarise

(Ayat matematik?) Dua belas darab dua belas .. (Dua belas darab dua bel as") Dua belas darab satu .. (Ok teruskan ... ) Dua. dua. dua. eml2at. enam, 18.l2an. seQuluh. dua belas ... enam darab dua .. tiga. Mathematical operation: Addition Relate 12 to enam. sembi lan, dua belas ... emQat darab tiga .. darab tiga ... Count & add Multiplication 2+2+2+2+2+2

Solution generation: Relate Counting =6x2 Summarisation: Summarise Number (2->4->6-*8 -> 10

-> 12) Relate 12 to 3+3+3+3=4x3 (3->6->9->12)

(Sebut) Empat empat empat. .tiga darab empat. .. Mathematical operation: Addition Relate 12 to 4+4+4=3x4

Multiply Multiplication Solution generation: Relate Summarisation: Summarise

(Macam mana tahu enam, enam?)

191

Transcript: Group 10 m a . . .enam, enam.. .Enam tambah enam, dua belas.. .dua darab enam.. .eh.. . dua darab enam.. .

(Satu lagi) Dua belas.. .

(Sekarang dua puluh empat.. .kalau dua puluh empat, macam mana?) Dua puluh empat, tiaa, enam. sembilan. dua belas.. .lima belas.. .lapan belas.. .

(Sebut, sebut.. .) Empat, lauan, dua belas.. . emuat, lapan. dua belas. enam belas, dua puluh, dua puluh emuat.. .dua puluh lapan.. .emuat. empat, empat, enpat tambah empat tambah empat tambah emuat tambah emuat tambah emuat . . .enam darab empat... (Seterusnya?) [l]Tiga. tiga.. .tiga, enam.. .lima, seuuluh, lima belas, dua uuluh.. .enam.. .dua belas.. . [2]enam. dua belas, . . .lauan belas.. .dua puluh empat.. .enam tambah enam tambah enam tambah enam, empat darab enam.. .Ok hm.. . [l]& emuat belas.. .dua puluh satu.. .lauan,. . . .lapan.. .lauan tambah lauan tambah lauan, tiga darab lauan.. .empat . . .empat.. .tujuh.. .lapan belas.. .dua puluh empat.. . dua puluh empat.. .[l]dua belas, dua belas.. .dua belas, dua belaskah? Dua puluh empat, [2]dua darab dua belas. Tiga belas, empat belas, lima belas, enam belas, tuiuh belas, lapan belas, sembilan belas. dua uuluh, dua puluh satu, dua puluh dua, dua uuluh tiga, dua uuluh empat. m.. .tiga, enam, dua belas. ..dua belas. lima belas. lapan belas, dua puluh satu.. .dua puluh empat. Tiaa tambah tiga tambah tiga tambah tiga tambah tiea tambah tiga tambah tiga tambah tiga tambah.. .

Solution generation: Relate Summarisation: Summarise

Operation I Knowledge Mathematical operation: Add I Addition

Data exploration: Examine: Identify

Knowledge Constructed Relate 12 to 6+6=2x6

Solution generation: Trial & error Mathematical operation: Add

Mathematical operation: Add & multiply Solution generation: Relate Summarisation: Summarise

[lISolution generation: Trial & error [2]Mathematical operation: Add, multiply & count Summarisation: Summarise

Multiplication

Number

Addition

Addition Multiplication

Addition Multiplication Counting Number

Relate 12 to 12

Relate 24 to 4+4+4+4+4+4=6x4 (4,8,12,16,20.24)

Relate 24 to 6+6+6+6=4x6; 8+8+8=3x8; 12+12=2x12; 3+3+3+3+3+3+3+3=8x3

Transcript: Group 10 [l]Satu, dua. ticra, emuat, lima. enam. tuiuh, lauan. sembilan.. .sembilan darab tiga.. .lapan darab tiga. ..satu, dua, tiga, empat, lima, enam, tujuh, lapan.. .[l]satu. dua. tiea, emuat, lima. enam. tuiuh. lauan, sembilan. seuuluh.

Addition I

Operation Mathematical operation: [l]Count 8: [2]Add, [I]Solution generation: Trial

sebelas. dua belas.. . Dua, empat, enam, lauan. seuuluh, dua belas. empat belas, enam belas.. .eh.. .dua, empat, enam, lapan, sepuluh, dua belas.. . Dua, emuat. e s sembilan belas. dua puluh, dua uuluh dua, dua wluh empat.. .Ok [2]& tambah dua tambah dua tambah dua tambah dua tambah dua tambah dua tambah dua tambah dua tambah dua tambah dua tambah dua. Satu, dua. tioa. empat. lima. enam. tuiuh, lauan. sembilan, sepuluh, sebelas, dua belas. dua belas darab dua. (Kita buat aktiviti terakhir. Bina bangunan dengan dua belas bongkah kayu. Setiap satu bongkah mewakili satu bilik. Kalau kamu bina dua bilik, semua mesti dua bilik setiap tingkat. Kalau buat tiga, semua tiga ok?) Dua belas.. .terbalik tiga, tiga, tiga.. . (Yang kamu buat mesti berlainan) (Ok buat lagi lain dari ini.) (Buat panjang boleh?) (Buat lagi satu.) Lima, lauan.. . (Ini tak samakan?Tak ada lagi) (Sekarang cikgu mahu kamu tengok ini, ada berapa buah bilik semuanya?) Dua belas

(h4acam mana kita dapat dua belas itu?) Tipa, enam, dua belas.. .

(Selain dari kira satu, satu, satu. ..macam mana dapat dua belas?) Empat, lauan, dua belas.. .

(Empat, lapan, dua belas.. .ok selain dari itu apa lagi? Kalau pakai darab macam mana dapat?)

Number

& error Summarisation: Summarise

Data exploration: Examine: Identify

Data exploration: Examine: Identify Mathematical operation: Add

Data exploration: Examine: Identify Solution generation: Relate

Row & column

Addition Row & column

Relate 12 to 3+3+3+3 ( 3 - 6 4 9 - + 1 2 )

Relate 12 to 4+4+4 ( 4 - + 8 4 1 2 )

Transcript: Group 10 Emuat darab tioa.. .

(Mana empat mana tiga?) Emuat, tiea

(Selain dari itu? Apa lagi?) Darab (Darab macam mana? Berapa buah bilik semuanya?) Dua belas

(Dua belas juga. Macam mana kita hitung dua belas itu?) Tiga, tiga. tiga. tipa.. .

(Lagi?) Enam tambah enam

(Lagi?) Enam darab dua.. .

(Di mana enam? Di mana dua?) Ini dua.. . (Enam, dua? Yang ini enam, lepas itu dua di mana? Maksdunya ini enam, ini pun enamkah?) Enam.. . . . . tuiuh tambah lima

(Tujuh tambah lima, lepas itu?)

Knowledge Constructed Relate 12 to 4x3

Relate 12 to 3+3+3+3=12

Relate 12 to 6+6

Relate 12 to 6x2

Relate 12 to 7+5

Operation Data exploration: Examine: Identify Solution generation: Relate

Data exploration: Examine: Identify

Data exploration: Examine: Identify

Data exploration: Examine: Identify Solution generation: Relate

Data exploration: Examine: Identify Solution generation: Relate

Data exploration: Examine: Identify Solution generation: Relate

Solution generation: Trial & error Data exploration: Examine: Identify Mathematical operation: Add

Knowledge Row & column Multiplication

Row & column

.Number

Addition Row & column

Addition ROW & column

Addition Row & column

Addition

Transcript: Group 10 Tiga lambah tiga tambah tiea tambah tiga

(Apa lagi? Adakah lagi?) Empat tambah emuat tambah emuat

(Ini?) Enam tambah enam, dua. dua. dua, dua, dua. dua

(Kalau guna darab?) Enam darab dua

(Lagi? Ini satu darab dua belas? Ada lagi?) Tak ada

Knowledge Constructed Relate 12 to 3+3+3+3

Relate 12 to 4+4+4

Relate 12 to 6+6 & 2+2+2+2+2+2

Relate 12 to 6x2

Operation Solution generation: Relate Data exploration: Examine: Identify Mathematical operation: Add

Solution generation: Relate Data exploration: Examine: Identify Mathematical operation: Add

Solution generation: Relate Data exploration: Examine: Identify Mathematical operation: Add

Solution generation: Relate Data exploration: Examine: Identify Mathematical operation: Multiply

Knowledge Addition Row & column

Addition Row & column

Addition Row & column

Multiplication Row & column

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Transcript: Group 11 Tuiuh tambah tiga, tambah satu, sebelas

(Macam mana dengan adik? Macam mana kamu buat tujuh tambah empat?) Kira.. . (Pakai jari juga, tunjuk tujuh, lepas itu?) Tambah empat (Adakah adik hitung tujuh, lapan, sembilan, sepuluh, sebelas?) Tak (Tujuh, lepas itu?) Tujuh tambah empat (Terus ke sebelas? Tidal; hitung lapan, sembilan, sepuluh.. .) Tujuh tambah empat (Ok sekarang tiga tambah sembilan. Sebut.. .sebut.. .) Tiga tambah sembilan, dua belas

(Pakai jari? Macam mana?) Dalam hati (Mula dengan nombor berapa?) Sembilan

(Lepas itu tiga, terus tambah? Ada pakai apa-apa? Tidak ada langsung? Jangan angguk-angguk saja tak dapat dengar. Lapan tambah kosong atau sifar.. .sebutlah) Laoan tambah sifar. laoan

(Lima tambah enam?) Sebelas

(Pakai jarikah? Lima dulu atau enam dulu?) Lima

(Lepas itu?)

Knowledge Addition

Addition

Addition

Addition

Number

Operation Mathematical operation: Add

Mathematical operation: Add

Data exploration: Examine: Identify & locate

Mathematical operation: Add

Mathematical operation: Add

Data exploration: Examine: Compare

Knowledge Constructed Obtain 7+4 by adding 3 to 7 to make 10 then 10+1 (remainder)=l 1

Transcript: Group 11 Lima tambah enam

(Sepuluh tambah dua?) Sepuluh tambah dua. dua belas.

(Sembilan tambah empat.. .) Sembilan tambah emuat, tirra belas

(Macam mana dapat tiga belas?) Sembilan (Sembilan pada jari, tambah lagi empat, sembilan, sepuluh atau terus ke tiga belas? Ada hitung sepuluh?) Tak (Ok ernpat tambah enam. Sebut.. .) Empat tambah enam, se~uluh.

(Tiga tambah lima) Tiga tambah lima, lauan.

(Ok, sekarang buat darab, satu darab satu) Satu - (Satu, lima darab enam?) Lima darab enam, tiga uuluh (Macam mana dapat tiga puluh? Jelaskan, macam mana dapat?) Lima darab enarn, tiga puluh (Tiga darab sembilan) Tiga darab sembilan, dua ~ u l u h tuiuh (Macam mana? Semua yang karnu buat, beritahu cikgu, dua puluh tujuh macam mana dapat?) Sembilan darab tiga (Macam mana dapat? Kamu kena terangkan macam rnana? Hafal sifir? Lapan darab sifar) Lapan darab sifar, sifar (Tujuh darab dua?)

Knowledge Addition

Addition

Addition

Addition

Addition

Multiplication

Multiplication

Multiplication

Multiplication

Multiplication

Operation Mathematical operation: Add

Mathematical operation: Add

Mathematical operation: Add

Mathematical operation: Add

Mathematical operation: Add

Retrieval: Recall

Retrieval: Recall

Retrieval: Recall

Retrieval: Recall

Retrieval: Recall

Knowledge Constructed

-

Transcript: Group 11 Tuiuh darab dua, emuat belas (Empat darab empat.. .sebut.. .sebut) Empat darab e m ~ a t , enam belas (Lapan darab empat?) Lauan darab e m ~ a t , emuat ~ u l u h (Lepas itu, pasti?) Pasti (Hafal? Sepuluh darab satu?) Seouluh (Enam darab tujuh. :.) Enam darab tujuh, emuat uuluh dua. (Sekarang aktiviti dua. Apa yang adik nampak?) Tambah. darab

(Lepas itu?) Nombor (Cuba adik terangkan apa yang adik faham?) Dua tambah dua tambah dua

(Lagi?) Lima darab tipa, lima belas (Lepas itu?) Hm.. . , tuiuh darab dua sama dengan e m ~ a t belas (Belum lagi tunjuk lompatan dan ayat matematik.. .sebut, sebut.. .) Tiga.. .macam tok.. .hm.. . (Sebut, sebut, cakap kuat. Bincang, bincang.. .jangan malu. Cikgu tidak kesah betul salah, apa yang adik nak cakap, cakaplah.. .ah.. .cakap kuat) Lima tambah lima tarnbah lima (Ah.. .lukis) Lukislah.. .lima darab tiga . . .sama dengan . . .lima tambah lima tambah lima sama denrran lima belas (Macam mana dengan itu? Lukis, 1ukiskan.macam mana lukis?)

Knowledge Multiplication

Multiplication

Multiplication

Multiplication

Multiplication

Addition Multiplication

Number

Multiplication

Multiplication

Addition

Operation Retrieval: Recall

Retrieval: Recall

Retrieval: Recall

Retrieval: Recall

Retrieval: Recall

Retrieval: Recognize

Data examination: Read Data exploration: Examine: Extract

Retrieval: Recall

Retrieval: Recall

Data exploration: Examine: Identify

Knowledge Constructed '

Relate 5x3 to 5+5+5 (5+10+15)

D z "a -

Y 3 $ .+ 3 3 L)

5 fj - 8 - :

m.& g E 0 5 5 2 - % a t ? * v:$? - " a : E 2 3 W w o

Transcript: Group 11 Macam mana la& darab dua ... lagi darab, dsrab ... (Kuat sikit cakap) Macamini ... dua, tiea. emoat. lima. enam. tuiuh, Iauan. sembilan. seouluh.

(Kuat sikit cakap) Sigek, dua ... enam ... laoan. sepuluh. dua belas. emoat belas

(Ada nak tukar?) Tujuh (Teruskan,temskan) Ah.. .empat belas (Macam mana ini, jangan bisik ... jangan bisik, kuat cakap) Macam tok.. .tuiuh tambah tuiuh ... eh.. .sama iuza, tujuh, tujuh.. .Tok dua tambah tujuh, dua, tiga, empat, lima, enam, tujuh. Tok, dua. dua. emnat ... satu. dua. tiea. emoat. lima. enam. tuiuh. lao an... seouluh. sebelas. dua belas. tiea belas. emoat belas. ..eh, empat belas ... satu, dua, tiga, empat, lima, enam, rujuh (Setemsnya sembilan darab empat ... ) Sembilan darab emoat ... tiea puluh enam ... sembilan emoat kali. lanya ... sembilan ... satu. dua, tiea. emoat. lima. enam. tuiuh, laoan. sembilan ... & dua. tira. emnat. lima. enam. tuiuh. laoan. sembilan ... satu. dua. tina. emoat, lima, enam, tuiuh, laoan. sembilan ...

(Sekarang lapan darab lima) L a ~ a n darab lima, emoat ouluh (Lukiskan) Lima, laoan ipek . . satu. dua. tiea, emwt. lima ... satu. dua. tiea. empat, lima.. enam. ..tujuh. ..lapan

1 Operation / Knowledge

Mathematical operation: Count & group

Mathematical operation: Count gr group

Data exploration: Examine: Compare Mathematicd operation: Count &group

Retrieval: Recall Data exploration: Examine: Identify Mathematical operation: Couni & group

Retrieval: Recall

Data exploration: Examine: Identify Mathematical operation: Count & erouo

Counting Grouping Number

Counting Grouping Number

Counting Number Grouping

Counting Number Grouping

Multiplication

Counting Number Grouping

Knowled e Constructed n Relate 7x2 to 7+7.2 groups with 7 items in each group Relate 2x7 to 7 goups with 2 items in each group

Relate 9x4 to 4 groups with 9 items each group

Relate 8x5 to 8 groups with 5 items in each group

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Transcript: Group 11 / Operation / Knowledge / Knowledge Constructed

Solution generation: Trial Row & column &error I I

I (Lima darab empat, dua puluh. Ok tak apa. Tegok bangunan ini ... Berapa '1 I 1 1

Relate 12 to 6+6, 10+2 Emuat tambah dua. sepuluh tambah dua, ... enam tambah enam ...

semuanya?) Dua belas ...

(Macam mana dapat dua belas?) Dua. emDat. enam. lapan. laoan darab dua ... eh emuat darab enam. emoat

Solution generation: Trial & error

darab tiza ... (Lagi) Tiza darab emuat

Row & column

(Selain dari itu?) Enam darab dua

(Mana dapat enam?) Eh ... Empat, empat, lap an... (Lagi, kita boleh tengok dia dari segi satu tingkat, satu tingkat) Tiea darab emoat

(Ialah. Sudah itu. Tiga darab empat, empat darab tiga. sudah) Dua helas bahagi enam.. . (Kalau cikgu nak guna tambah. ..) Dua belas ... dua belas.. .ah, empat, lapan, l a ~ a n tamhah emuat.. .

(Nombor tak sama, guna nombor yang sama.) Enam darab enam.. . (Darab enam?) Enam darab dua

Data exploration: Examine: Identify

Solution generation: Trial &error & relate

Solution generation: Inverse

Solution zeneration: Relate

Solution generation: Relate

Mathematical operation: Add

Mathematical operation: Multiply Solution generation: Relate

Number

Row & column Multiplication

Row & column Multiplication

Row &column Multiplication

Row & column Multiplication

Addition

Row &column

Relate 12 to 4x3

Relate 12 t o . 3 ~ 4

Relate 12 to 6x2

Relate 12 to 4x3

Relate 12 to 8+4

Relate 12 to 6x2

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paka

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Transcript: Group 12 Empat ... Satu. dua. tipa. empat; satu. dua. tiea. empat; satu, dua. tiga, emoat: satu. dua, tiea, emuat; satu. dua. tiea, emvat; satu. dua, tipa. emuat. Satu. dua. tipa. empat. lima. enam ... tuiuh. lapan, sembilan.. (Oh, macam itu. Sembilan kelompok, satu kelompok ada ...) Empat (Empat titik, berapa semua itu?) Lapan. sembilan. se~uluh. sebelas. dua belas. tiea belas. emuat belas, lima belas. enan belas. tuiuh belas. lapan belas. sembilan belas. dua uuluh. dua puluh satu. dua puluh dua. dua puluh tiea. dua ~ u l u h empat. dua puluh lima.

I duo ~ c l u h Lnax L a rululi Iuiuh d ~ i D U U ~ i i r s ~ . US D U ! J ~ S C P I ~ : . ~ .I@.,

pulut. tlr.1 r u : w . : : ? lcluh flu,: !id? piiuh t l e s tlra r,u:un s t n p ~ l tic:

I puiuh lima, tiea puluh enam, tiga puluh tujuh, tiga puluh lapan, tiga puluh I r - m h ~ l ~ ! ~ . cmpli ?i$lin. . t trs puluh cnam caru dua. ncn. cnx,?r 1 : ~ : cn3-r.

I ' l l uk . ..!1311 i r r .3: :JT . < P J I u ~ !:b!k Z!ECI. Er1:lil p~1.k (Empat puluhkah itu?) Lebih sigek. Satu. dua. tiea. emuat. lima. enam. tuiuh. lauan. sembilan

I (Ok sekarang lapan darab lima ... sebutlah ... lapan darab lima) Lapan darab Ima, e m ~ a t uuiuh (Ada cara yang baws untuk kira? Hafal sifir limakah? Macam mana hafal sifir

I (Ada pakai iari?) . . Tak (Lapan darab lima, empat puluh. Lepas itu? Sebut) Satu. dua, tiea. emuat. lima: satu, dua, tiea. emuat. lima; satu. dua. tiea, empat. lima; satu. dua, tixa, empat, lima; satu, dua, tiea, empat. lima: satu, dua. tiea. empat, lima; satu. dua, tiea, emuat. lima; satu, dua. tiga, empat, lima, enam, u... Satu, dua. tiea. empat, lima. (Cukup. Ok, tiga darab tiga)

Count & group Number 4 items in each group Grouping

Mathematical operation: Counting Count / Number 1

Retrieval: Recall Multiplication

Multiplicarion operarion: Count

Counring Number

Mathematical operation: Add

Addition Reiatc multiplication of 5 to I repeat addition of 5 . (5-+10-+15+20+ 2 5 j 3 0 - 3 35-+40--3 45 -+ 55 -+ 60)

Mathemat~cal operation: Count & group

Counting Number Grouping

Relate 8x5 to 8 groups with 5 items in each group

Transcript: Group 12 I Operation I Knowledge / Knowledge Constructed (Mesti isi semua petak itu, petak mesti habis isi ... ) 1 I I Dua tambah dua tambah dua tambah dua (Sebut.) Dua. emoat. enam, laoan. seouluh. dua belas, emoat belas. enam belas. lapan belas. dua uuluh. dua ouluh dua, dua puluh emoat

(Bolehkah?) Dua puluh, dua puiuh dua, dua puluh empat (Tapi cikgu mahu apa? Ini berapa?)

(Ini?) Dua puluh empat (Besarlah. Tengok ini, lapan, dua, empat, enam, lap an... adik Cuba lain ... sebab dua puluh empat terlalu besar. Macam mana? Cakap, sebut ... Apa yang adik terfikir? Sesuaikah dua itu? Tak sesuai, habis? Satu? Mengapa Satu?) Cukuo dua belas (Oh, cukup dua belas. Ada berapa buah petak itu?) Dua belas- (Adik macam itu juga Ada adik fikir satu? Ada? ) Satu tarnbah satu tambah satu tambah satu tambah satu tambah satu tambah - -

saru tarnbah salu r:~rnhah saru ramhah saru rambah caru ihn~!,nh u r u s,ma den~an dua bclm darab s ~ t u rarna dencan dud belas (Sckarang rcngok be~rkut. berapa pctak nu?, Dua. emoat, enam, laoan. enam darab dua sama densan dua belas.

(Adik cuba dengan empat petak itu. Macam mana?) Tiea ... tiea, tiga

(Mengapa tahu tiga?)

Mathematical operation: Count & add

Data examination: examine: Identib

Solution generation: Relate

Mathematical operation: Add & multiply Summarisation: Summarise

Mathematical operation: Counting & add

Solution generation: Trigger

Counting Number Addition

Number

Number

Addition mutliplication

Addition Counting Number

Relate 12 to l+ l+ l+ l+ l+ l +I+l+l+l+l+l=12xl=i2

Relate 12 to 2+2+2+2+2+*12 (2- - f4+6+8+10 -3 12)

Transcript: G r o u p 12 I Opera t ion / Knowledge 1 Knowledge Constructed Tiea rambah 1i.a ramball l i ra rambah tira. Tira. enam. sembilan. dua belas. / h'lathemarical operation: 1 Addition / Relate 12 ro 3+3+3+3=4x3

1 Emnat darab tiza sama denran dua helas. I Add & mulripl? / Multiplication / =12 I

(Lenas itu?!

Dua belas darab saiu sama denzar, dua helas (Sekarane macam mana densan dua puluh empa:?) Empat. emoar .. . empal. emna:. lanan. dua helas ... Dua. empar. exam. lapan.

(Sekaranp empa: pelah.) Lima rambah lima tanihali lima iamhah iima ... oh.. .

(Boleh? Berapii lima tambah lima tambah lima tambah lima?) Enam tnmhah enam tarnhah cnam ramhnh enam ... cnam. dua heias. lanan helas. dua puluh empa!

(Sebut) Empat d:mh enam sam:i denyan dua pcluli empa!. Tifa ... ( T i p pcrah. Oii l a p a n ? Cepaun?a.) T i r a damh lapar samn dcriran dua poluli crnnal ...

(Seknrang dua pelah ... sehur . d u a pcrak ... j &&&5

(Schul . . )

Dua helas t;lmhah du:> hcliir snrna dciiil:m dii;! nuluii emrlat

(Sebilt ... srhut . . ) Enam, Iapali. sepuiiili, dua heins ... ( J a n p t ~ bisik. sehur s;iin ap;! ).any adih fihii . . !

Mathematical operarion: Add & mulriply Summansarion: Summarise

hlathemarica! operarion: Add. coun: & multipl! Summansation: Summar:se

Solur,on yeneration: Tna! &error

?4athcmarical operarion: Add & coun!

Malhem.itica1 opcrarion: L'lultipl! Solurio~i gcncretion: Relate

Solution :cncrarion: Relate

Matheniatlcai oiieiation: Multiply 6: add

Addition Multiplication

Addition Mult~plication NumSei Counting

Addirior! Counring Kumber

Murliplication

Motlrpl~cnlion Additloi~

Relate 24 to 4 4 ~ 4 4 - 4 + 4 = 6 x 4

Relate 2 1 to 12+12

empat, e n a n , lapan. sepuluh. dua belrs ... dan . . (Macam mana tahu dua?) Dua. empat. enam. lapan. sepuluh. dua belas. emnar belas. enam helas. l awn belas. dua ouluh, dua ouluh &a. dua puluh empat.

Transcr i P t: Grou P 12 T i ~ a , enam. sembilan ... Tira . enam. sembilan. dua belas. lima belas. lapan belas. dua ouluh satu. dua ouluh dua. dua puluh tiea. dua ouluh emoat. Tigalah. Tiea tambah lira tambah t i ~ a tambah tioa tambah tiza tambah tipa tambah tioa

(Scbur ... sebut . . ) Dua talnbah dim ta~nhal> dua tamhai? dub tamiah dua tamhah dua ralnball dua ramhah dua mrnhah dua ramball du;! ri..:nhall dun tamhah dua sama denoan dua b e l a daiah dua sama denran dua vululi cmnai. (Apa bezanya dengan dua bclas darah dua denfan dua darab dua belas?) Hrr...tak adz bezaTerbaIik.

(Mana yang senan8 dua bclai darah dua aLau dua duiah dua bc!as?) Dua darah du;: belas

tzmhah tisa samadenpaa dua puluh emnat ... Eh ... dua puluh empat. Dua, !

(hlcn;np:i'!> Seha!, duo helai ramha11 dua h c l a dua nuli:li crnoal scnanc. (Kalziu dun bslas darah du;i'.') Tamhali dua tamhah dus tao~hali dua rnoihall dua ... (Ok sekamng clkgr! mahu adik ambil bongkah hayu ini dan guna uii;. bee's hongk;ih bina hangunnn Dua helas sntu bang~!iian. Setiap tingkat mesrl sama b ~ ~ ~ i y n i , hilik. Kalau diio bilik. scmua iner i dua hongkoll. Kaiau t i p . tipa. ok?) (Sudiil dua bclas? Uina Ingi, bua; lain.) Sigch, asek ... Da;!. c~nn;~:, enalii. 1;lvan. sepiiluh. due heI;is dali. Siiti~hah? Oh. t u ~ ~ i h a n ~ l ; ~ l ? . liilia ... Iim a... c u b a . . ~ ! I . p u a . emniit. eilnni. iai>;i!i, sep~lluh. M... Tok ling$ gilak ... saiu. duii. lira. ernpi!:, lima. cnam. tu~ul,. iaoan. senihiliin. senuli~li. schel;is. dii;i heI;ls , A < i k I:!CYI !;q>n Lnl> ' l !

Knowledge Constructed Relate 24 to 3+5+3+3+3+3+3+3

Operation Mathematical operation: Add &count

Knowledge - Addition Cocnting Number

Marhematical operation: Add & count

Data exploralion: Examine: 1 ~ o m p n r c I

Addition. Countlnf Number

Mnrhcmaticai operation: Add Summansation: Sunmarise

I Dara crploration: Examine: 1 hlutliplication Idcotit? I '

Addirion Mu1tiplica:ion

Data esplanarion: Explain / Addilloli

Data explanation: Explain Addifion

Relate 2 1 to repeat addition of 2 io; i 2 times ( 2 + 4 + 6 + 8 + 1 0 -3 1 2 4 1 4 i i 6 i 1 6 + 20 -3 22 i 24)

Solution generation: Tnnl & error htaihematical operation: Coon; br add

Counti:ig Number Additioil

u - r

5 g 2 ; C C c ;.;$ -

g g .- 2 3 .- 2 g g E '3 3 - - - - - - - 3

o - 0 a e -

p u u z .- E E S o c u u 8 C U

2 4 5% $ a .= 9 id 4 .- 2 a 0 - . - 2 ? 2 - 5 5 -5s 3 'c 0

5 3 7 2 r J o 4 E w T I C 2 e : SCL: Q E < ZFL Q e:

Transcript: Group 12 / Operation I Knowledge j Knowledge Constructed Dua tambah dua tambah dua tambah dua tambah dua tambah dua T ~ e a tambah / Solut~on generation Relate I Row & column I Relate 12 to 2+2+2+2+2+2, tiga tambah tiea tambah tiea. Emoar tambah emuat tambah emuat. Enam I Mathematical operation: I Multiplication 1 3+3+3+3,44+4, tsmir3:l :nam . I X a i z Siru t:*rnb;.h ciru 'dmbch 1311: 13rlbah catc t i n h a i .;tu \I.~ltlpl! 6r add A J ~ I I I ~ 1 ~ l - l ~ l ~ l ~ : ~ l - 1 - 1 ~ l T I rarrja'l rat" rdmjnh 53:u r~mj ,h ix 13n;.3? sjt~! t?.nlbah sap3 1 2 7 1 h I i h r I i ~ m m r ? s ~ : l o n . Summsi~re I --

tambah satu sama denean dua belas darab satu I I I (Habis?) Tok udah. Dua darab enam, enam darab dua

(Dari mana dapat enam darab dua?) Ada dua kumuulan. satu ada enam kumoulan dan dua bilik. (Sekarang ini.) Emoat tambah emoat tambah emuat. Tira tambah tiea tambah tiea tambah tiea. Dua tambah dua tambah dua tambah dua tambah dua tambah dua. Satu tambah satu tambah satu tambah sat" tambah satu tambah satu tambah satu tambah satu tambah satu tambah satu tambah satu tambah satu sama denean dua beias darab satu (Lagi?) Emuat darab tiea. tiea darab emuat.. e n a m darab dua

(Di mana enam sekarang? Lain ada?) Satu darab dua belas. dua belas darab satu

Solution generation: Relate & inverse Mathematical operation: Multiply

Data explanation: Explain

Solution generation: Relate Mathematical operation: Multiply & add

Row & column Multiplication

Row & column Multiplication Addition

Relate 12 to 2x6 & 6x2

Relate 12 t0.2+2+2+2+2+2, 3+3+3+3,4+4+4, 1+1+1+1+1+1+l+l+1+1+1 +1=12x1

Solution generation: Relate Row & column Relate 12 to 4x3, 3x4 & & inverse Multiplication 6x2 Mathematical operation: 1 Multiply I Solution generation: Relate Row 8: column Relate 12 to 1x12 & 12x1 &inverse 1 Multiplication 1

Ini (Semua pilih bangunan yang paling tinggi. Berapa buah bilik?) Dua belas (lad1 ada yang paling banyak?) Tak ada, sama saja.

(Ok kalau adik ingin pilih sebuah bangunan yang mempunyai paling banyak bilik, bangunan yang mana satu adik akan pilih?)

Mathematical operation: Multiply

E9 E .3 s m m -- 5 2 E r n E r_ 2 8 m

? 5 2 E .a $ Z "

- $5 " 'r i m i - - C; m g2 m

-2 r m c .- kJ 2 .,M s v - i Y - i - - 5 - ; m -. PC- Y! r m m m

3 3 2 5 r E

Transcript: Group 13 Entah (Kalau adik buat lima darab enam?) Ambil sifir Iima, kira (Y a, macam mana?) Kira jawavan sifir lima. lima. se~uluh, lima belas. dua vuluh. dua vuluh lima, tiga uuluh

(Macam itu, sampai enam kali?) Hm.. .lima darab enam saya ingat (Ingat, hafal? Adakah adik ingat bermula dari satu darab enam, dua darab enam, tiga darab enam.. .sampai lima darab enam?) Ada juga (Cuba tiga darab sembilan.) Ah.. .ambil tiea kum~ulan, dalam setiap kumuulan sembilan

(Lapan darab sifar) Sifar (Tujuh darab dua) Emuat belas (Empat darab empat) Enam belas (Lapan darab lima) Emuat uuluh (Satu darab sepuluh) Seuuluh (Enam darab tujuh) Emvat uuluh dua (Aktiviti dua, adik cuba buat, cuba faham contoh ini.) Dua, emvat, enam. lavan, sevuluh, dua belas. emuat belas, enam belas

Operation

Mathematical operation: Add

Data exploration: Examine: Identify Mathematical operation: Group

Retrieval: Recall

Retrieval: Recall

Retrieval: Recall

Retrieval: Recall

Retrieval: Recall

Retrieval: Recall

Data exploration: Examine: Extract

Knowledge

Addition

Addition Grouping

Multiplication

Multiplication

Multiplication

Multiplication

Multiplication

Multiplication

Knowledge Constructed

Relate 5x6 to repeat addition of 5 for 6 times 5+5+5+5+5+5=30 (5+10+15+20+ 25 +30)

Relate 3x9 to 3 groups with 9 items in each group

Transcript: Group 13 (Apa lagi yang adik narnpak? Cakap.) Lomuatan

(Pemah beiajar?) Pemah (Macam mana dengan ayat matematik itu?) Dua tambah dua tambah dua sama den, man enam (Selain dari itu?) Tiga darab dua (Tiga darab dua, adakah mereka sama?) Sama (Jadi cikgu mahu adik jawab yang seterusnya, lukis, jawab dan tulis ayat matematik juga. Baca soalan) [l]Sila tuniuk lomuatan darab untuk soalan-soalan berikut. [2]Lima darab tiea. lima - belas (Lompatan? Macam mana ia lompat? Sebut) Sifar, tisa, enam, sembilan.. . .dua belas.. .lima belas

(Mengapa? Macam mana adik tahu ia lompat tiga?) Sebab ini ada tiga (Tapi ada lima juga.) Hm.. .lompat lima (Nak lompat tiga atau lompat lima?) Lima (Cakap dengan suara besar) Lima, seuuluh. lima belas. Ayat matematik, lima darab tiga sama dengan lima belas.. .sama dengan lima tambah lima tambah lima sama dengan lima belas

(Dah? Mengapa tak lompat tiga?)

Operation

Retrieval: Recognise

Data examination: Read

Data examination: Read

[l]Data Examination: Read [2]Retrieval: Recall

Data exploration: Examine: Identify Mathematical operation: Multiplication jump

Data explanation: Explain

Data exploration: Examine: Identify Mathematical operation: Multiplication jump

Knowledge

Multiplication jump

Multiplication

Addition Multiplication jump

Number

Addition Multiplication Multiplication jump

Knowledge Constructed

Relate 5x3 to 5+5+5=15 (0 -+ 3 + 9 -+ 12 + 15)

Relate 5x3 to 3 jumps with 5 steps in each jump (5 3 10 + 15)

Transcript: Group 13 I Operation I Knowledge I Knowledge Constructed Entah. Lomuat lima lebih ceuat dari lomuat tiga I Data exploration: Examine: I (Oh.. .I see. Seterusnya, baca soalan) Dua darab tuiuh, empat belas (Lepas itu?) Lomuat tuiuh. tuiuh

(Lompat tujuh, tujuh. Sebut) Tuiuh.. .tuiuh emuat belas

(Ayat matematik?) Dua darab tujuh sama dengan tujuh tambah tujuh sama dengan empat belas (Ok baca soalan) [l]Emuat darab empat. enam belas.. .[2]lomcat empat, emuat.. .uerei ke emuat, lauan. enam belas.. .eh.. .empatnya ia lompat.. .dua belas dulu. dua belas belum.. .enam belas (Lepas itu?) Empat darab empat sama dengan empat tambah empat tambah empat tambah empat sama dengan enam belas (Seterusnya) Saya, saya.. .ties darab lima sama dengan lima belas.. .lomuat lima. enam ... tadi lima darab tiga, tok tiga darab lima. Seuuluh ke lima belas. & matematik tlga darab lima sama dengan lima tambah lima tambah lima sama dengan lima belas. Dah? Sepuluh darab satu sama denvan seuuluh. Satu. dua, tisa. emDat, lima, enam, tuiuh, lauan. sembilan, sepuluh. Seuuluh darab satu sama dengan satu tambah sama lima tambah lima.. .satu tambah satu.. .lima, tambah satu tambah satu tambah satu, lima, tambah satu, enam. tambah satu tambah satu tambah satu tambah satu. se~u luh , sama dengan seuuluh. (Ok kamu bandingkan ini dengan ini) Sama

(Sama soalan?) Ini sama tapi nombor terbalik

Compare

Retrieval: Recall

Data exploration: Examine : Identify

Mathematical operation: Add

[]]Retrieval: Recall [2]Mathematical operation: Multiplication jump

Retrieval: Recall Mathematical operation: Add, count, multiply & multiplication jump

Data exploration: Examine: Compare

Multiplication

Multiplication jump

Addition

Multiplication Multiplication jump

Multiplication Multiplication jump

Multiplication

Relate 2x7 to 2 jumps of 7 steps in each jump

Relate 2x7 to 7+7=14

Relate 4x4 to 4+4+4+4=16 ( 4 + 8 + 12+ 16)

Relate 3x5=5+5+5=15 (5+10+15) Relate 10x1 to 1+1+1+1+1 +1+1+1+1+1=10

Transcript: Group 13 (Ok ini kelompokk an... titik itu mesti sama banyak dalam kelompok-kelompok yang adik lukiskan) [l]Tujuh darab dua. emuat belas.. . [Zldua. empat. enam.. .macam mana? . . .Bulatkah?

(Ya, bulat, kumpulkan) Emuat. lima. enam, tuiuh.. .

(Satu kelompok aja? Sebut.. .) Dua kelom~ok, setiau kelom~ok ada tuiuh

(Dua kelompok? Macam mana adik tahu dua kelompok?) Sebab tuiuh tambah tuiuh, empat belas. (Oh.. .tulis jawapan sini. Berapa titik semuanya yang adik telah kelompokkan?) Empat belas (Sebut.. .sekarang dua darab tujuh.. .) Emuat belas juga

(Macam mana dengan kelornpoknya?) Dua. dua.. .dua. emuat. enam. lapan, se~u luh , dua belas. emuat belas.. .enam belas.. .dab.. .. dua, empat, enam, lapan, sepuluh, dua belas. empat belas

(Jadi adik macam mana tahu ia ini tujuh kelompok?) Tuiuh di d e ~ a n (Cikgu mengajar macam itu?) Hm.. . (Macam mana dengan sembilan darab ernpat.. .sebut-sebut) [IISembilan darab empat.. . t i ~ a puluh enam, [2]ambil sembilan.. .r3lem~at, lima, enam, tuiuh, lauan, sembilan. Ambil laei sembilan. [4]Hingga emuat kelomuok, tiga uuluh enam

Operation

[ 1 ]Retrieval: Recall [ZIMathematical operation: Add & count

Mathematical operation: Count

Data exploration: Examine: Identify

Data explanation: Explain

Retrieval: Recall

Data exploration: Examine: Identify Mathematical operation: Group, count & add

Data explanation: Explain

[IIRetrieval: Recall [2]Data exploration: Examine: Identify Mathematical operation: [3]Count & [4]Group

Knowledge

Multiplication Counting Number Addition

Counting Number

Grouping

Multiplication 7x2

Grouping Addition

Multiplication Grouping Counting Number

Knowledge Constructed

Relate 7x2 to 2 groups of 7 items in each group

Relate 2x7 to 7x2=14

Relate 2x7 to ?+2+2+2+2+2+2=14 (2,4,6,8,!0,12,14)

Relate 9x4 to 4 groups with 9 items in each group

Transcript: Group 13 (Ok lauan darab lima) Nombor lap an...[ lllauan darab iima, emuat uuluh. [2]Satu. dua. tiga. empat, lima. enam, tuiuh, lapan. Satu.. .[3]lauan.. .lauan. samuai lima.. .empat puluh. Tiga darab tiga sama denean sembilan.. .ambil tiga. Ambil tigakah? Betullah . . .enam, sembilan.. .tiga darab tiga sama denean sembilan. Enam darab emuat sama denean dua puluh emuat (Berapa titik itu?) Enam. Setiau kelomuok ada enam kumuulan, tok emuat kelomuok.. . . Enam darab empat sama dengan dua puluh empat

(Baca) Jika enam darab dua sama dengan dua belas berauakah enam darab tipa? Enam darab tiea sama denean lauan belas. (Cepat. Macam mana dapat lapan belas?) Sebab ada enam kumpulan. setiau kumuulan ada tiea

(Ada pakai ini?) Tak (Tujuh darab lima sama dengan tiga puluh lima, berapa tujuh darab enam?) Tuiuh darab enam, empat puluh dua (Empat puluh dua, ada pakai apa-apa?) Tak, dua darab enam sama dengan dua belas, Tuiuh darab tiga sama denzan dua uuluh satu. Empat darab lima sama dengan dua puluh. Enam darab tuiuh sama dengan emDat ~ u l u h dua. (Habis. Oh cepatnya. Sekarang tengok ini) Satu, satu, satu udah. Sila isi mane-manp kosong dengan nombor.. . (Apa yang adik nampak ini?) Petak, tambah dan sama dengan nombor dua belas

(Selain dari itu?) LaDan

( (Macam mana dengan nombor dalam petak itu?)

I Operation I Knowledge

[IIRetrieval: Recall Data exploration: Examine: Identify Mathematical operation: [2]Count & [3]Group

Data explanation: Explain

Data examination: Read Retrieval: Recall

Data explanation: Explain

Retrieval: Recall

I Retrieval: Recall

I Data examination: Read

Multiplication Counting Number Grogping

Multiplication

Grouping

/ Multiplication

Data exploration: Examine: Extract

I Multiplication

Data exploration: Examine: Extract

Number

1 Knowledge Constructed I Relate 8x5 to 5 groups with 8 items in each group Relate 3x3 to 3 groups with 3 items in each group

Relate 6x3 to 6 groups with 3 items in each group

(Dua puluh empat, kita boieh mula dari mana-mana ... sebut, sebut.) Dua puluh empat sama dua puluh empat, [Ildua belas tambah dua belas sama denean dua uuluh emoat. [2]Dua tambah dua emuat. Sam tambah satu dua

(Oh, very good, hebat. Lagi? Adik nak buat mana? Sebut.sebut) Satu. dua. tiea. emoat. lima. enam. tuiuh. Iauan. sembilan. seuuluh. sebelas. dua belas, sama dengan dua puluh empat tambah dua puluh empat. Dua tambah dua - tambah dua tambah dua tambah dua tambah dua Iambah dua tambah dua tambah dua tambah dua tambah dua tambah dua sama denean dua belas darab dua - (Lagi? Sebut ... ) Dua puluh empat, satu. dua. tiea. emoat. lima. enam. tuiuh. lapan. Tiea. tiea tambah tiea tambah tiea ...

Transcript: Group 13 Sebab dua belas darab sifar sama denean sifar. (Oh ... darab sifar sama dengan sifar. Ok sekarang dua puluh empat.) Oh ... hm ...

(Macam mana tahu r i m ? )

Knowledge Operation Data explanation: Explain

denean dua ouluh emoat. Tok tiea darab lauan.. . enam tambah enam

Knowledge Constructed

(Macam mana adik tahu enam?) Sebab enam tambah dua belas. Dua belas tambah enam laoan belas, laoan belas tambah enam dua ~ u l u h emoat.

(Itu yang adik buat tadi dalam otak?) Hm. ..enam tambah enam tambah enam tambah enam jadi empat darab enam

(Sekarang tinggal dua soalan.) Satu. dua. tiea, emoat. lima. enam. enam oetak. Ada empat, emoat tambah emDat tambah emoat tambah emoat tambah emuat tambah emoat sama denean dua puluh emual

[IIMathematical operation: Add [2]Evaiuation: Confirm

Mathematical operation: Count & add Summarisation: Summarise

Data exploration: Examine: Identify Mathematical operation: Count & add

Data explanation: Explain Mathematical operation: Add Summarisation: Summarise

Data explanation: Explain Mathematical operation: Add

Solution generation: Relate Summ%isation: Summarise

Mathematical operation: Count & add

Addition

Addition Counting Number

Addition Counting Number

Addition

Addition Muitiplication

Counting Number

Relate 24 to l2+12

Relate 24 to 2+2+2+2+2+2 +2+2+2+2+2+2=12x2

Relate 24 to 3+3+3+3+3+3+3+3

Relate 3+3+3+3+3+3+3+3 to 3x8

Relate 24 to 6+6+6+6=4x6=24

Relate 24 to 4+4+4+4+4+4=24

(Mana empat?) Ini empat, sini tisa, empat darab tiga

(Selain dari empat darab tiga?) Tiea darab emoat

(Ok tengok ini) Dua. dua. dua. dua. dua. dua

(Dua ia~nbah dua tambah dua tambah dua tambah dua tambah dua. iepas j:u?) Satu. satu, satu, satu. satu. satu. satu. satu satu. satu, satu, satu. Empat. empat.

Enam. enam.

(Kaiau pakai darab?) Empat tambah enipat, enarn darab dua. dua darab enam

I (Ini) Satu tambah satu tambah satu . . tiea tambah tiza..

Solution generation: Relate Mathematical operation: Multiply

Daia exploration: Examine: Identify Data explanation: Explain

Soiution generation: Inverse

Data exploration: Examine: ldentify hlathematical operation: Add

Data exploration: Examine: Identify Mathematical operation: Add

Data exploration: Examine: Identify Mathematical operation: Multiply Soiutioin generation: Reiate 8: Inverse

Row &column

Multipiication

Addition Row & columr

Addi:ion Row 8r column

Multiplication Row 8i column

Data exploration: Exatnine: Addition Identify Mathematical operation: , Add I

Relate 12 to.2+2+2+2+2+2

Relate 12 to l + l + l + l + i + l +1+1+1+1+1+1; 4+4+4 & 6+6

Relate 12 to 6x2 & 2x6

Relate 12 to l + l + l + l + l + l +1+1+1+1+l+l;3+3+3+3

(Macam mana dengan kamu?) Tuiuh dalam otak. emoat iari. tuiuh. laoan. sembilan. seouluh, sebelas

Transcript: Grouq 14 ( Operation I Knowledge / Knowledge Constructed

(Macam mana dengan adik? Tems tujuh, empat, sebelas? Adakah adik simpan dalam otak? Simpan empat dalam otak, tujuh di jari?) Seuuluh di iari ...

Hituna ... tuiuh dalam otak, tuiuh. iaoan. sembilan. sepuluh. sebelas

(Sepuluh, ok, macam mana?) Lima. enam, tuiuh. laoan. sembilan. seouluh. sebelas

(Tiga tambah sembiian.) Sehelas. ..dua belas.. .dua belas

Mathematical operation: Add &count

(Macam mana dapat dua helas?) Dalam otak tiaa ... dalam tanzan sembiian..

Counting Number

(Lepas itu?) Hitune.. .tira. emoat. lima. enam. tuiuh, lapan. sembilan, sepuluh. sebeias, dua belas - (Lapan tambah kosong?)

(Mengapa lapan?) Sebab ada kosons, kosonr tak ada nombor (Lima tambah enam?)

Data exploration: Examine: Identify &locate Mathematical operation: Count & add

Data exploration: Examine: Locate

Mathematical operation: Add &count

Mathematical operation: Add &count

Data exploration: Examine: Identify &locate

Mathematical operation: Count & add

Mathematical operation: Add

Data explanation: Explain

/ Addition

Addition Counting Number

Kumber

Addition Counting Number

Addition Counting Number

Number

Counting Number Addition

Addition

1 Number

Transcript: Group 14

(Sepuluh tambah dua?) Dua belas

(Sembilan tambah empat?) Tipa belas

(Empat tambah enam) SeDuluh

(Tiga tambah lima?) &

(Sekarang kita tengok darab, satu darab satu) & (Lima darab enam?) Tiea nuluh (Macam mana dapat tiga puluh?) Tambah laei ... lima tambah lima senuluh. tambah lima lima belas ... lima belas. dua nuluh. dua nuluh lima. tiea puluh

(Oh, macam itu) Tambah.. ..lima tambah enam.. l ima tambah iima enam kali

(Tiga darab sembiian) Dua nuluh tuiuh (Lapan darab kosong) Kosone (Tujuh darab dua) Emuat belas (Macam mana dapat empat belas?) Tujuh darab dua empat belas . . tuiuh tambah tuiuh dua kali

Operation Mathematical operation: Add

Mathematical operation: Add

Mathematical operation: Add

Mathematical operation: Add

Mathematical operation: Add

Retrieval: Recall

Retrieval: Recall

Data exploration: Exafine: Identify & locate Mathematical operation: Add

Mathematical operation: Add

Retrieval: Recall

Retrieval: Recall

Retrieval: Recal!

Data explanation: Explain

Knowledze 1 Knowledge Constructed Addition I

Addition i

Addition

Addition

Addition

Multiplication

I\lultiplication

Addition Relate 5x6 to 5+5=10, 10+5=l5, 15+5=20. 20+5=25,25+5=30 (5+5+5+5+5+5)

Repeat addition of 5 for 6 times

Multiplication

hlultiplication

Multip!ication

Addition Relate 7x2 to 7+7

. . 1 .d

w m ; -

- w e - D m 2 - s % . E - . - 9 : . .

2 3 ..: .- - - 2 2 D z m : ,c 5 ,: - a . k3-2 .- Z E . 0 D 5 2 ; Z E . o E - ;! g . " 'c a m

g g s z 5

u m m 2 ; : 2 - 5 F m

= . .- E m m a n m 20 E 2 E E . Z - d Y 2 E C 2 8 m , : s g ~5 5

- .- x r L

2 a 5 m 2 ,. 0

2 0

2 5 2 -; '. .-

Z D a n m 2 0

i s m n U i

2 5

2 ~2

(Ok.. .lapan darab lima) Ah ... emuat uuluh (Satu kelompok berapa titik?)

Transcript: Group 14 (Berapa kelompok?)

,... satu, dua. tiea. emoat

(Ok tiga darab tiga ... ) [I]-. ..eh, betul ... [Zltiea.. .tiea. ..enam. ..sembilan,[3] enam emoar dua e r r . 0 3 1 . dui emvdt. ?nm: k,li zii .rr. J8.a 5cl idinp.rn nei;.r d i ~ ?

E r n : I lj?,;n F c l ~ s

Operation

Data exploration: Examine: Identiiy

(Macam mana dapat lapan belas? Apa yang adik buat tadi?) Satu. dua. tiga. emoat. lima. enam. tuiuh. laoan. sembilan

(Berapa?) Dua belas (Ada kaitan tak dengan enam darab tiga?) Tak (Oh ... ok temskan) Emoat dua (Ya, macam mana dapai empat dua?) Kerana ia darab enam (Hafalkah?) Satu, dua, tiea, emDat, lima. enam. tuiuh ... satu, dua. tiea, emuat. lima . . . sebelas, dua belas ... lima. sepuluh. lima belas. dua ouluh. dua ouluh lima, tiea puiuh. tiga puluh lima. .. (Apa tujuan bagi segitiga ini? Ada pakai?) Tak

Mathematical operation: Count

Knowledge

Counting Number

Retrieval: Recall

Knowledee Constructed

Data exploration: Examine: Identify

[IIRetrieval: Recall Mathematical operation: [Z] Group & [3]multiply

Mathematical operation: Count

Retrieval: Recall

Data explanation: Explain

Mathematical operation: Count & add

Multiplication

Grouping

Multiplication Grouping

Counitng Number

Multiplication

Counting Number

Relate 8x5 to 5 groups of 8 items in each group

Relate 3x3 to 3 groups of 3 items in each group Relate 6x4 to 4 groups of 6 items in each group

Relate 7 x 5 to 5+5+5+5+5+ 5+5

Transcript: Group 14 / Operation [IIEmoat, lao an... dua be1 =...empat darab ... [2]dua belas bahaei emoat tiea / Mathematical operation: ... dua belas bahaei emoat tiea ... tiga ... satu empat. Add &divide

Knowledge Addition

(Ok.. .lagi?) En am... enam.. .dua darab enam ... satu ... enam darab du alah... Dapat satu ... padam ... salah...tok betul ... sdah...satu. dua, tiea. empat, lima. enam. tuiuh. laoan. sembilan. seouluh. sebelas, dua belas. .. satul ah... saru. dua. tiea. empat. lima. enam, tuiuh. laoan, sembilan. seouluh, sebelas. dua be1 as... satulah. .. sata .. sa tu... saru .... sa tu... satu ... Sam. .. satu ... Dua. emnat. euam. laoan. ..enam dua .... emoat ... dua ... dua ... betullah tok betullah. ..enam tadi ... enam darab dua ... dzh (Ada lagi ... dua puluh empat) Eh.. .empat.. .emoat.. . em~at . . .emoat.. .

(Macam mana buat ini?) [IIEmoat. laoan, sembilan. seouluh. sebelas. dua belas. tiea belas. emoat belas. lima belas, enam belas. tuiuh belas. laoan belas. ..sembilan belas, dua wluh. dua oulnh emoat. .. betul ... enam darab emoat ....T ok dua puluh emapt lagi ...y e...[2]emoat, lima ... enam ... enamkah ... [I]-, enam, dua belas, lapan belas, sembilan belas. dua buluh, dua ouluh satu. dua ouluh dua. dua puluh tiea, duaouluh emoat. [3]betullab ... enam ... enam ... enam ... empat ... tiga eh ...tig a...[4]Iaoan.. .darab tiza ,... betullah dua belas ... betull ah... [I]- tiea, emoat. lima, enam, tuiuh. lauan ... dua ... [llemoat, enam. lao an... tiea ... eh tiea.. .emuat...betullah tiea ya ... [2]hitune dulu.. .tiea, enam, sembilan. dua b m sembilan. dua be1 as.... l3lbetullah..tiea. enam. sembilan. tiea belas. dua belas, lima belas. emoat belas .... dua emoat.. . laoan darab tiea. tiea darab laoan. .. Satu, dua. tiea, emuat, lima. enam. tuiuh, laoan. sembilan. seouluh. sebelas. dua belas. Dua ... dua ... dua. emoat, enam, lauan, se~uluh. dua belas.. . [ 3 ] m dah.. .dua. ..dua.. . ",-I.:-,,,

Division

Solution generation: Relate & trial &error Mathematicd operation: Count Summarisation: Summarise

Solution generation: Trial & error

Mathematical operation: Count, add & [4]multiply Evaluation: Confirm Solution generation: Trial & error Summarisation: Summarise

Counting Number

Multiplication Counting & number Addition

Addition

Addition Counting Number Multiplication

Evaluate bv 12 1 4 = 3 and

Relate 12 to 1+1+1+1+1+1 +1+1+1+1+1+1 Relate 12 to 2+2+2+2+2+2%- x2

Relate 24 to 4+4+4+4+4+4

Relate 24 to 6+6+6+6 Relate 24 to 8+8+8+8+8+8+8+8 =3x8 Relate 24 to 2+2+2+2+2+ 2+2+2+2+2+2+2

Operation I Knowledge / Knowledge Constructed Solution generation: Trial I Counting I

I & error 1 Number (Kalau buat satu bilik, setiap tingkat satu bilik, kalau dua semua dua ... ) I I

I Emoat.. .emoat.. .ah. ..empat I I 1 (La gi... ) Lima. ..emuat. Empat ... enam ... betullah. .. dah...empat...empat, dah... & I Solution generation: Trial

& error I / (Buat panjang lagi) Lima dulu ... buat enam ... dah.. .sama la gi... Dua belas ... ini dah. ..apa lagi ... enam.. .dua belas.. .sembilan.. .tiea ieek.. .enam. .. ud &...dm belas . . .ini sebelas.. .lagi.. . tinggi.. .tiga.. . tiga dah.. .satu.. .satu dah.. .lima.. .berapa? Dah . . .belum.. .beium.. .lapan.. .empat, lima.. . tiga.. .Empat. ..buat segitiga.. . empat ... empat lagi, empat, lapan ... (Sud ah...tadi adik kata ini dua belas, j ad~ macam mana dapat dua belas?) Enam tambah enam

(Di mana enam?) Atas, bawah. ..

(Lepas itu? Selain daripada itu?) Darab, eh ... darab ... (Darab) Enam darab dua.. . (Enam darab dua, selain daripada itu?) Dua darab enam

(La@?) Bahagi. Dua belas bahagi. Darab tiga. Emoat darab tiea.

Di mana tiga? Tak ada CTengok bangunan ini) Enam darab.. .

Solution generation: Trial & error

Mathematical operation: Add

Data exploration: Examine: Identify

Addition

Row & column

Solution generation: Multiplication Inverse

Solution generation: Relate

Solution generation: Relate Multiplication Row & column

Multiplication Row &column

Relate 12 to 6+6

Relate 12 to 6x2

Relate 12 to 2x6 by inversing 6x2

Relate 12 to 4x3

Transcript: Group 14 / Operation I Knowledge / Knowledge Constructed Ah senang. satu tambah satu tambah satu ... dua belas tambah satu ... satu / Solution generation: Relate 1 Addition i Relate I2 to 1+1+1+1+1+1

/ tambah dua be1z.c i / Rou,6cci i imn / + l + l + l + l + l + l / (Ya. dah?) Eh. belum. dua darab dua darab dua darab dua darab dua darab dua dapat.. . (Apa lagi. i n i ?~ Enam tarnbah enam

(Ini) Sato tarnbah salu lambah satu tamhah satu ramball satu tambah sat" tanbah satu iambah saro tambah satu tambah satu tambah saw tambah satu (Lagi; Ah. enam rambah enain

(Ini )

Dua. emn;ii . . dua. dua. dua. dua. dua. cnam darab cnam

(Eanam tnmhah atau darab?) Eanam lambah enam (Ok ... la& Dus darah e n a m e n a m darab dua

(hlecaai m;ma inill Sudah tad) ... bcluin ... belum ... Tira ramhah tic;] tamhah lira ta~nbah lira ... empsi tamhali empal tambah emrat ... empat darah ti ra. . t l ra darab empat.. (Sudah pnndai.. i n i l Dua iarnbah ... Empat darab empat. dua lambah dua rambah dua ... (Di maw dua . . ) Tak ada . i n i ini ... empiit tambah Cmpdt talnhah cinpai ...

I Solutlon generation: Relate

Solution ;Oeneration: Relate

I I / Solution generation: Relate

I / Solution generation: Relatc

Solution zeneration: Relatc & Invcrse

Solutlon renereuon: Relate I 6 i n i s ;

Solution generation: Relate

Addition Row 6 column

Addition Ro\\ 6 column

Addillon Roii 6 ca!umn

Addlt~on Row 6 column

Multiplication Rou 6 column

Multiplication Row 6 column

Addition Row b. column

Relate 12 to 6+6

Relate 12 to l + l + l + l + l + l + l + i + l + l + l t l

Relate 12 to 616

Relatc I2 to 2+?+2+2+2+2

Relate I ? to 2x6 and 6x2

Relate 12 to 3+3+3+3, 4+4+4.4,3 and 3x4

/ iLap?i Eniitn t:i!nhah cn:im

, (Layi'!)

Solut~on generation: Relate Addition Roo. 6 column

Relate I2 lo 6+6

Appendix U:

Mathematicai Activity

Sila tunjuk lompatan darab untuk soalan-soalan berikut: Please show the multiplication jumps on the number line.

Aktiviti 2lActivity 2

(a) 5 x 3 = I I

3 x 2 = 6 Lompatan darab/

(b) 2 x 7 = 1

Ayat matematik: 3 x 2 = 2 + 2 + 2 = 6

multiplication jump 2 + 2 + 2

mmm

I I

i d - I I

i " I I - N I I I . 3 I I

! 0

Aktivili 3lActivity 3

Sila kelolnpokkan t i t ik- t~~ik berikut untitk menunjukka~~ IPlcase group the clots to show :

(a) 7 x 2 = (b) 2 x 7 =

( c ) 9 x 4 = - (d) 8 x 5 =

Jika 6 darah 2 salna dengall 12, herapakah 6 darah 3? /If 6 timcs 2 is cqual lo 12, what is 6 t i~ncs 3 ?

Jika 7 d a u b 5 sama dengan 35, berapakah 7 darab 6? /If 7 times 5 is equal to 35, what is 7 times 6'?

aanaaan annnana aannann nnannnn nannnnn

Jawab soalan-soalan berikut:/Answer the following questions:

Aktiviti SlActivity 5 Sila isi ruang-ruang kosong berikut, nombor-nombor yang anda pilih mesti sarna untuk ceraian soalan yang sama: (Please fill in the boxes, the number used in each box must be the same for the same portion of the question.)

Example : 8 = + + + = 4 x 2

(d) 12 = + +

(e) 12 =

Aktiviti GIActivity 6

Anda dikehendaki membina bangunan dengan bongkah kayu yang dibelcalkan. Bangunan yang dibina mesti lnempunyai bilangan bilik yang sama dalaln setiap tingkat. Setiap bongkah kayu mewakili sebuah bilik. / You are required to construct building with the wooden blocks provided. The buildings you construct must have the same number of rooms in every floor. Every wooden block represents a single room.

Appendix C:

Letters Relating to the Approval of the Study

BAHAGIAN PERANCANGAN DAN PENYELIDIKAN DASAR PENDlDlKAN (BPPDP) , KEMENTERIAN PENDlDlKAN MALAYSIA PARAS 2,3 .% 5, DLOK ] PUSAT BANDAR DAMANSARA. Telcfi,~ : 03-2583204 50604 KUALA LUMPUR. f&r. 03-2554960

Rui. Kaml : KP(BPPDP) 131 15 J ld .50(538) C; K Tarikh : 18 Ogos 1999 I&. Chien Lee Shing, 475, Taman Hui Sing, 93350 Kuching, SARAWAK.

Tuan,

Maktab-Maktab ~ e r ~ u ~ k c a n , Jabatin-Jabatail Pei~didikan b a n Bahagian-Bnhagian Di Batual~ Kernenterin11 Pendidikan Malaysia

Adalah saya dengan hormatnya diarah memaklumkan bahawn per~uohonan tuan untuk menjalankan kajian bertajuk:

"To Explore Using Cognitive Task Analysis How Students Reason Out Tlie Connection Between Addition And Multiplication By Contructivism Approach"

telah diluluskan.

2. Kelulusan' ini adalah berdasarkan apa yang terkandung di dalam cadangan penyelidiltan yang tuan kemukakan ke Bahagian ini. Kebenaran bagi mengeunakan s a m ~ e l kaiian ~ e r l u di~erolehi dari~ada Ketua BahaxianIPengarah Pendidikan neperi vang berkenaan. Sila kemukakan ke Bahagian ini senaskah laporan kajian tuan setelah ia selesai kelak.

Sekian untuk makluman dan tindakan tuan selanjutnya. Teri~na kasih.

"BERICHIDMAT UNTtTK NEGARA"

Saya yang menurut perintah,

(DR ADlIR BIN MOHD SALLEII) b.p. Pengarah, Bahagian Perancangan dan Penyelidikan Dasar Pendidikan, Kementerian Pendidikan Malaysia.

Pcngarah Pendidiltnn, .Jabatan Pendidikan Negrri Sasawalz.

Ketua, Kurnpulan Terns Pembangunan Osganisasi, Fakulti Sains Kognitif dan Pemb. Manusin, UNIMAS.

Ilrq K~rrrr~ 177JPPERWWK/l

I r k l r 17 September 1999

Cik Chien Lee Shing, Fakulti Sains Kognitif & Pembangunan Manusia, Universiti Malaysia Sarawak, 94300 Kota Samarahan.

Tuan,

Permohonan Kebenaran Untuk Membuat Kajian Di Sekolah-Sekolah Dl Bahagian Kuching.

Surat tuan yang bertarikh 16 September 1999 mengenai perkara di atas adalah dirujuk. Sesalinan surat Kementerian Pendidikan Malaysia 8il. KP(BPPDP)i3/15 Jid50(538) yang bertarikh 18 Cgosi999 adalah berkaitan.

2. Sukacita dimakiumkan bahawa Jabatan ini tiada halangan unluk tuan menjaiankan kajian bertajuk "To Explore Using Cognitive Tasks Analysis How Students Reason Out the Connection Behveen Addition and Multiplication by Constructivism Approach" Sila bejumpa atau menghubungi GUN Besar sekolah berkenaan mengenai kajian itu nanti.

3. Dengan salinan surat ini kami memohon kerjasama daripada semua pihak sekolah yang akan teriibat.

Sekian, harap maklum. Terima kasih

"BERKHIDMAT UNTUK NEGARA"

Saya yang menurut periniah,

Sektor Pengurusan Perkhidmatan Pendidikan, b.p. Pengarah Pendidikan, Sarawak.

s.k. Pegawai Pendidikan BahagianBahagian Kuching.

Guru BesarSRK Merpali Jopang, Kuching

Guru BesarSRK Encik Buyong Adil, Kuching

Guru Besar,SRB St. Thomas, Kuching.

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