1. Programme for International Student Assessment (PISA)
2015
BAHAGIAN PEMBANGUNAN KURIKULUM
KEMENTERIAN PENDIDIKAN MALAYSIA
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2. Kandungan Taklimat (1 hari)
Masa
Perkara
7.30 – 8.50
Pendaftaran
9.00 – 10.00
Latar Belakang & Kerangka Kerja PISA
10.00 – 10.15
Rehat
10.15 – 12.30
Perbincangan tentang bahan & kelas intervensi
Pembelajaran secara Hands-on
12.30 – 2.00
2.00 – 4.30
Perbengkelan & Pembentangan/Gallery Walk
4.30 – 5.00
Perbincangan dengan JPN
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3. Kandungan Taklimat (2 hari)
Masa
Perkara
HARI PERTAMA
12.00 – 3.00
Pendaftaran
3.00 – 4.30
Latar Belakang & Kerangka Kerja PISA
4.30 – 8.30
Rehat
8.30 – 10.00
Perbincangan tentang bahan & kelas intervensi
Pembelajaran secara Hands-on
HARI KEDUA
8.00 – 10.00
Perbengkelan & Pembentangan/Gallery Walk
10.30 – 12.30
Perbincangan dengan JPN
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4. APAKAH PISA?
Programme for International Students Assessment (PISA) merupakan penyelidikan pendidikan antarabangsa terbesar di dunia
Dianjurkan oleh Organisation for Economic Cooperation and Development (OECD) yang berpengkalan di Paris. Bermula pada tahun 2000 dan dijalankan setiap 3 tahun. PISA 2015 merupakan pusingan PISA yang ke-6
Melibatkan lebih 70 buah negara di seluruh dunia. Mengkaji kesediaan murid untuk menempuh kehidupan alam dewasa
Mengukur literasi murid dalam sains, matematik, bacaan dan penyelesaian masalah. Mengumpul maklumat dalam konteks amalan pendidikan di negara peserta
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5. 3 tahun sekali 2000 2003 2006 2009 2012 2015 Umur 15 tahun +
Programme for International Student Assessment
2000
2003
2006
2009
2012
2015
3 tahun sekali
Umur 15 tahun +

6. KEPENTINGAN MENYERTAI PISA
Menunjukkan tahap kesediaan murid Malaysia untuk menempuh alam dewasa
Membolehkan penggubal dasar mendapat maklumat tepat mengenai sistem pendidikan bagi mengenal pasti ruang untuk penambahbaikan sistem pendidikan
Membuat perbandingan prestasi dan persekitaran pembelajaran murid antara negara
Menetapkan matlamat polisi seperti yang dicapai oleh negara lain dan belajar daripada polisi dan amalan di negara lain
Salah satu daripada matlamat utama penggubal polisi adalah untuk membolehkan rakyat mendapat manfaat daripada ekonomi dunia yang bersifat global. Untuk itu, fokus yang lebih diberikan kepada penambahbaikan polisi pendidikan bagi memastikan perkhidmatan pendidikan yang berkualiti, peluang pendidikan yang saksama dan insentif diberi kepada kecekapan dalam proses pendidikan. Polisi sedemikian bergantung kepada maklumat yang tepat tentang sejauh mana sistem pendidikan menyediakan murid untuk menjalani kehidupan pada masa hadapan. Kebanyakan negara memantau pembelajaran murid dan prestasi sekolah. Walau bagaimanapun, dalam ekonomi global ini, kayu pengukur kejayaan sesuatu sistem persekolahan tidak lagi boleh diukur berdasarkan kepada standard kebangsaan sahaja tetapi mestilah dibandingkan dengan negara-negara lain pada peringkat antarabangsa.
UNIT PENYELIDIKAN ANTARABANGSA (PISA) BPPDP
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7. Negara Peserta
UNIT PENYELIDIKAN ANTARABANGSA (PISA) BPPDP

8. Sampel Kajian PISA 2015 Bahagian Pembangunan Kurikulum
9660 org murid: 97% dari Tingkatan 4 & 3% dari Ting. 3
Murid berumur 15+ di semua jenis sekolah di Malaysia
Populasi:
Semua jenis sekolah di Malaysia yang mempunyai murid berumur 15+ tahun
Kerangka Pensampelan:
230 buah sekolah dipilih secara rawak dengan menggunakan kaedah Stratified Multi Stage Cluster Sampling
Pemilihan Sampel Sekolah:
42 murid berumur 15+ dipilih secara rawak dengan menggunakan aplikasi KeyQuest
Pemilihan Sampel Murid:
Trends in Mathematics and Science Studies (TIMSS) ialah kajian perbandingan antarabangsa anjuran International Association for the Evaluation of Educational Achievement (IEA). Kajian ini menghasilkan maklumat tentang input, proses dan output pendidikan yang boleh digunakan untuk penambahbaikan dasar dan peningkatan pengajaran dan pembelajaran matematik dan sains dalam kalangan negara peserta.
Satu pusingan kajian ini mengambil masa empat tahun. Sehingga kini lima kajian telah dijalankan iaitu pada tahun 1995, 1999, 2003, 2007 dan Malaysia menyertai kajian ini pada tahun 1999, 2003, 2007 dan Reka bentuk kajian digubal sedemikian rupa untuk membolehkan trend pencapaian murid antara negara dan dalam negara dikaji.
TIMSS melibatkan murid Tahun 4 (Gred 4) dan Tingkatan 2 (Gred 8). Penyertaan Malaysia hanya melibatkan murid Tingkatan 2. Malaysia dan negara peserta hemisfera selatan mentadbir kajian ini setahun lebih awal.
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9. Instrumen PISA 2015 UJIAN KOGNITIF
Gabungan item-item aneka pilihan dan soalan terbuka
Item-item disusun dalam kumpulan berdasarkan teks yang menggambarkan situasi dalam kehidupan sebenar (real - life situation)
UJIAN PENYELESAIAN MASALAH SECARA BERKOLABORATIF
Murid bekerjasama dengan beberapa agen (komputer) untuk menyelesaikan masalah
Menguji kesediaan/kebolehan murid bekerjasama dengan orang lain
SOAL SELIDIK MURID
Latar belakang murid, persepsi dan sikap murid terhadap sains, matematik dan kemahiran membaca
Tahap motivasi dan amalan strategi pembelajaran murid
SOAL SELIDIK SEKOLAH DAN GURU
Mengumpul maklumat berkenaan sekolah
Mengumpul maklumat berkenaan latar belakang dan strategi pengajaran guru
Instrumen kajian terdiri daripada:
Ujian Bertulis (2 jam)
Gabungan item aneka pilihan dan soalan terbuka. Item disusun dalam kumpulan berdasarkan teks yang menggambarkan situasi dalam kehidupan sebenar.
Soal Selidik Murid (35 minit)
Latar belakang, persepsi dan sikap murid terhadap sains, matematik dan bacaan.
Tahap motivasi dan teknik pembelajaran murid.
Soal Selidik Sekolah
- Mengumpul maklumat berkenaan sekolah
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10. KITARAN FOKUS PISA

11. KEDUDUKAN MALAYSIA DALAM PISA 2009

12. Kedudukan Dalam PISA 2012 Matematik
Shanghai-China - 613
Singapore - 573
Hong Kong-China - 561
Chinese Taipei - 560
Korea - 554
Macao-China - 538
Japan - 536
Liechtenstein - 535
Switzerland - 531
Netherlands – 523
Estonia - 521
Finland - 519
Canada – 518
Poland - 518
Belgium - 515
Germany- 514
Viet Nam – 511
Austria - 506
Australia - 504
Ireland – 501
Slovenia – 501
Denmark – 500
New Zealand – 500
Czech Republic – 499
France - 495
U Kingdom- 494
Iceland - 493
Latvia - 491
Luxembourg - 490
Norway - 489
Portugal - 487
Italy - 485
Spain - 484
Russian Fed. – 482
Slovak Republic – 482
United States – 481
Lithuania - 479
Sweden - 478
Hungary - 477
Croatia – 471
Israel – 466
Greece - 453
Serbia – 449
Turkey - 448
Romania – 445
Cyprus – 440
Bulgaria – 439
UAE – 434
Kazakhstan – 432
Thailand – 427
Chile – 423
MALAYSIA - 421
Mexico – 413
Montenegro – 410
Uruguay – 409
Costa Rica – 407
Albania – 394
Brazil – 391
Argentina – 388
Tunisia – 388
Jordan - 386
Colombia – 376
Qatar – 376
Indonesia – 375
Peru – 368
OECD
Ave - 494
Kedudukan Malaysia dalam Literasi Bacaan adalah yang ke 55 daripada 74 negara/wilayah.
International
Ave - 456

13. DEFINISI LITERASI MATEMATIK
For the purposes of PISA 2015, mathematical literacy
is defined as follows:
Mathematical literacy is an individual’s capacity to formulate, employ, and interpret mathematics in a variety of contexts. It includes reasoning mathematically and using mathematical concepts, procedures, facts and tools to describe, explain and predict phenomena. It assists individuals to recognise the role that mathematics plays in the world and to make the well-founded judgments and decisions needed by constructive, engaged and reflective citizens.
Mementingkan aplikasi matematik dalam kehidupan sebenar.
Mengukur keupayaan murid menggunakan konsep dan kemahiran yang dipelajari di bilik darjah untuk menyelesaikan masalah kehidupan seharian
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14. DEFINISI LITERASI MATEMATIK
Kapasiti seseorang individu untuk mengenal pasti dan memahami peranan matematik dalam membuat keputusan yang berasas dan dapat memenuhi keperluan kehidupan individu sebagai warga yang konstruktif, prihatin dan reflektif.
Mementingkan aplikasi matematik dalam kehidpan sebenar.
Mengukur keupayaan murid menggunakan konsep dan kemahiran yang dipelajari di bilik darjah untuk menyelesaikan masalah kehidupan seharian
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15. A Model of Mathematical Literacy in Practice
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16. PISA FRAMEWORK CONTENT PROCESSES CONTEXT
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17. PISA FRAMEWORK Change and Relationship Space and Shape Quantity
These four categories characterise the range of mathematical content that is central to the discipline and illustrate the broad areas of content used in the test items for PISA 2015:
Change and Relationship
Space and Shape
Quantity
Uncertainty and Data
The four content categories serve as the foundation for identifying this range of content, yet there is not a one-to-one mapping of content topics to these categories.
For example, proportional reasoning comes into play in such varied contexts as making measurement conversions, analysing linear relationships, calculating probabilities and examining the lengths of sides in similar shapes. The following content is intended to reflect the centrality of many of these concepts to all four content categories and reinforce the coherence of mathematics as a discipline. It intends to be illustrative of the content topics included in PISA 2015, rather than an exhaustive listing:
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18.  Functions: the concept of function, emphasising but not limited to linear functions, their properties, and a variety of descriptions and representations of them. Commonly used representations are verbal, symbolic, tabular and graphical.
 Algebraic expressions: verbal interpretation of and manipulation with algebraic expressions, involving numbers, symbols, arithmetic operations, powers and simple roots.
 Equations and inequalities: linear and related equations and inequalities, simple second-degree equations, and analytic and non-analytic solution methods
 Co-ordinate systems: representation and description of data, position and relationships.
 Relationships within and among geometrical objects in two and three dimensions: Static relationships such as algebraic connections among elements of figures (e.g. the Pythagorean theorem as defining the relationship between the lengths of the sides of a right triangle), relative position, similarity and congruence, and dynamic relationships involving transformation and motion of objects, as well as correspondences between two- and three-dimensional objects.

19.  Measurement: Quantification of features of and among shapes and objects, such as angle measures, distance, length, perimeter, circumference, area and volume.
 Numbers and units: Concepts, representations of numbers and number systems, including properties of integer and rational numbers, relevant aspects of irrational numbers, as well as quantities and units referring to phenomena such as time, money, weight, temperature, distance, area and volume, and derived quantities and their numerical description.
 Arithmetic operations: the nature and properties of these operations and related notational conventions.
 Percents, ratios and proportions: numerical description of relative magnitude and the application of proportions and proportional reasoning to solve problems.

20.  Counting principles: Simple combinations and permutations.
 Estimation: Purpose-driven approximation of quantities and numerical expressions, including significant digits and rounding.
 Data collection, representation and interpretation: nature, genesis (raw data) and collection of various types of data, and the different ways to represent and interpret them.
 Data variability and its description: Concepts such as variability, distribution and central tendency of data sets, and ways to describe and interpret these in quantitative terms.
 Samples and sampling: Concepts of sampling and sampling from data populations, including simple inferences based on properties of samples.
 Chance and probability: notion of random events, random variation and its representation, chance and frequency of events, and basic aspects of the concept of probability.

21. PERCENTAGE OF SCORE POINTS
Approximate distribution of score points by content category for PISA 2015
CONTENT CATEGORY
PERCENTAGE OF SCORE POINTS
Change and Relationships
≈25%
Space and Shape
Quantity
Uncertainty and Data
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22. PISA FRAMEWORK CONTENT PROCESSES CONTEXT
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23. PROCESSES Bahagian Pembangunan Kurikulum Peneraju Pendidikan Negara
Formulating
Employing
Interpreting
In particular, the verbs ‘formulate,’ ‘employ,’ and ‘interpret’ point to the three processes in which students as active problem solvers will engage.
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24. PROCESSES
Formulating
Employing
Interpreting
Indicates how effectively students are able to recognise and identify opportunities to use mathematics in problem situations and then provide the necessary mathematical structure needed to formulate that contextualised problem into a mathematical form.
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25. PROCESSES
Formulating
Employing
Interpreting
Indicates how well students are able to perform computations and manipulations and apply the concepts and facts that they know to arrive at a mathematical solution to a problem formulated mathematically.
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26. PROCESSES
Indicates how effectively students are able to reflect upon mathematical solutions or conclusions, interpret them in the context of a real-world problem, and determine whether the results or conclusions are reasonable.
Formulating
Employing
Interpreting
Students’ facility at applying mathematics to problems and situations is dependent on skills inherent in all three of these processes, and an understanding of their effectiveness in each category can help inform both policy-level discussions and decisions being made closer to the classroom level.
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27. PERCENTAGE OF SCORE POINTS
Approximate distribution of score points by process category for PISA 2015
PROCESS CATEGORY
PERCENTAGE OF SCORE POINTS
Formulating situation mathematically
≈25%
Employing mathematical concepts, facts, procedures
≈50%
Interpreting, applying and evaluating mathematical outcomes
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28. PISA FRAMEWORK CONTENT PROCESSES CONTEXT
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29. Contexts Personal Societal Occupational Scientific PISA FRAMEWORK
An important aspect of mathematical literacy is that mathematics is engaged in solving a problem set in a context.
Personal
Societal
Occupational
Scientific
For the PISA survey, it is important that a wide variety of contexts are used. This offers the possibility of connecting with the broadest possible range of individual interests and with the range of situations in which individuals operate in the 21st century.
For purposes of the PISA 2015 mathematics framework, four context categories have been defined and are used to classify assessment items developed for the PISA survey:

30. PERCENTAGE OF SCORE POINTS
Approximate distribution of score points by context category for PISA 2015
CONTEXT CATEGORY
PERCENTAGE OF SCORE POINTS
Personal
≈25%
Occupational
Societal
Scientific

31. Computer-based Assessment
In PISA 2015, for the first time, the computer will be the main mode of delivery for all tests and questionnaires.
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32. Why Computer-based Assessment?
First, computers are now so commonly used in the workplace and in everyday life that a level of competency in mathematical literacy in the 21st century includes usage of computers (Hoyles et al., 2002). Computers now touch the lives of individuals around the world as they engage in their personal, societal, occupational and scientific endeavours. They offer tools for—among other things— computation, representation, visualisation, modification, exploration and experimentation on, of and with a large variety of mathematical objects, phenomena and processes. The definition of mathematical literacy for PISA 2015 recognises the important role of computer-based tools by noting that mathematically literate individuals are expected to use these in their endeavours to describe, explain, and predict phenomena. in this definition, the word “tool” refers to calculators and computers, as well as to other physical objects such as rulers and protractors used for measuring and construction.
First, computers are now so commonly used in the workplace and in everyday life that a level of competency in mathematical literacy in the 21st century includes usage of computers (Hoyles et al., 2002). Computers now touch the lives of individuals around the world as they engage in their personal, societal, occupational and scientific endeavours. They offer tools for—among other things— computation, representation, visualisation, modification, exploration and experimentation on, of and with a large variety of mathematical objects, phenomena and processes. The definition of mathematical literacy for PISA 2015 recognises the important role of computer-based tools by noting that mathematically literate individuals are expected to use these in their endeavours to describe, explain, and predict phenomena. in this definition, the word “tool” refers to calculators and computers, as well as to other physical objects such as rulers and protractors used for measuring and construction. A second consideration is that the computer provides a range of opportunities for designers to write test items that are more interactive, authentic and engaging (Stacey and Wiliam, 2013). These opportunities include the ability to design new item formats (e.g. drag-and-drop), to present students with real-world data (such as a large, sortable dataset), or to use colour and graphics to make the assessment more engaging

33. Why Computer-based Assessment?
A second consideration is that the computer provides a range of opportunities for designers to write test items that are more interactive, authentic and engaging (Stacey and Wiliam, 2013). These opportunities include the ability to design new item formats (e.g. drag-and-drop), to present students with real-world data (such as a large, sortable dataset), or to use colour and graphics to make the assessment more engaging
First, computers are now so commonly used in the workplace and in everyday life that a level of competency in mathematical literacy in the 21st century includes usage of computers (Hoyles et al., 2002). Computers now touch the lives of individuals around the world as they engage in their personal, societal, occupational and scientific endeavours. They offer tools for—among other things— computation, representation, visualisation, modification, exploration and experimentation on, of and with a large variety of mathematical objects, phenomena and processes. The definition of mathematical literacy for PISA 2015 recognises the important role of computer-based tools by noting that mathematically literate individuals are expected to use these in their endeavours to describe, explain, and predict phenomena. in this definition, the word “tool” refers to calculators and computers, as well as to other physical objects such as rulers and protractors used for measuring and construction. A second consideration is that the computer provides a range of opportunities for designers to write test items that are more interactive, authentic and engaging (Stacey and Wiliam, 2013). These opportunities include the ability to design new item formats (e.g. drag-and-drop), to present students with real-world data (such as a large, sortable dataset), or to use colour and graphics to make the assessment more engaging

34. The suite of tools available to students is also expected to include a basic scientific calculator. Operators to be included are addition, subtraction, multiplication and division, as well as square root, pi, parentheses, exponent,
square, fraction , inverse and the
calculator will be programmed to respect the standard order of operations.
PISA Editor tool
Allow students to enter both texts and numbers.
Students can enter a fraction, square root, or exponent. Additional symbols such as pi and greater/less than signs are available, as are operators such as multiplication and division signs.

35. The released PISA item Litter calls most heavily on students’ capacity for interpreting, applying, and evaluating mathematical outcomes. The focus of this item is on evaluating the effectiveness of the mathematical outcome—in this case an imagined or sketched bar graph—in portraying the data presented in the item on the decomposition time of several types of litter. The item involves reasoning about the data presented, thinking mathematically about the relationship between the data and their presentation, and evaluating the result. The problem solver must and provide a reason why a bar graph is unsuitable for displaying the provided data.
Computer-based

36. Structure of Items Bahagian Pembangunan Kurikulum
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37. Multiple-choice Questions (MCQ) Open-constructive
Types of Items
Multiple-choice Questions (MCQ)
Open-constructive
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38. Structure of the Items
PISA items in this form comprise a piece of stimulus, one or more questions related to that stimulus (with each question being referred to as an ‘item’) and, for each question, a set of guidelines that define the possible student response options and a proposed scoring scheme based on the defined response codes (the ‘coding guide’).
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39. Stimulus HEIGHT Scoring scheme Question PISA ITEMS 1
There are 25 girls in a class. The average height of the girls is 130 cm
M421Q01 –
Question 1: HEIGHT
Explain how the average height is calculated.
Scoring scheme
Question

40. PISA ITEMS 1
HEIGHT
There are 25 girls in a class. The average height of the girls is 130 cm
M421Q01 –
Question 1: HEIGHT
Explain how the average height is calculated.
Answer the question
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41. PISA ITEMS 1
Full Credit
Code 1: Explanations that include: Sum the individual heights and divide by 25.
You add together every girl’s height and divide by the number of girls.
Take all the girls’ heights, add them up, and divide by the amount of girls, in this case 25.
The sum of all heights in the same unit divided by the number of girls.
No Credit
Code 0: Other responses.
Code 9: Missing.

42. PISA ITEMS 1 Statement True or False
If there is a girl of height 132 cm in the class, there must be a girl of height 128 cm.
True/False
The majority of the girls must have height 130 cm.
If you rank all of the girls from the shortest to the tallest, then the middle one must have a height equal to 130 cm.
Half of the girls in the class must be below 130 cm, and half of the girls must be above 130 cm.
HEIGHT SCORING 2
Full Credit
Code 1: False, False, False, False.
No Credit
Code 0: Other responses.
Code 9: Missing.

43. PISA ITEMS 1
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44. PISA ITEMS 2
Litter
For a homework assignment on the environment, students collected information on the decomposition time of several types of litter that people throw away:
Type of litter
Decomposition time
Banana peel
1-3 years
Orange peel
Cardboard boxes
0.5 year
Chewing gum
20-25 years
Newspapers
A few days
Polystyrene cups
Over 100 years
Context: Scientific
Content: Uncertainty & Data
Process: interpreting, applying and evaluating mathematical outcomes
This item is set in a scientific context, since it deals with data of a scientific nature (decomposition time). The mathematical content category is Uncertainty and data, since it primarily relates to the interpretation and presentation of data, although Quantity is involved in the implicit demand to appreciate the relative sizes of the time intervals involved. the mathematical process category is interpreting, applying and evaluating mathematical outcomes since the focus is on evaluating the effectiveness of the mathematical outcome (in this case an imagined or sketched bar graph) in portraying the data about the real world contextual elements. The item involves reasoning about the data presented, thinking mathematically about the relationship between the data and their presentation, and evaluating the result. The problem solver must recognise that these data would be difficult to present well in a bar graph for one of two reasons: either because of the wide range of decomposition times for some categories of litter (this range cannot readily be displayed on a standard bar graph), or because of the extreme variation in the time variable across the litter types (so that on a time axis that allows for the longest period, the shortest periods would be invisible). Student responses such as those reproduced in Figure 12 have been awarded credit for this item.
A student thinks of displaying the results in a bar graph.
Give one reason why a bar graph is unsuitable for displaying these data.

45. PISA ITEMS 2 Sample responses for Litter
Response 1: “Because it would be hard to do in a bar graph because there are 1-3, 1-3, 0.5, etc. so it would be hard to do it exactly.”
Response 2: “Because there is a large difference from the highest sum to the lowest therefore it would be hard to be accurate with 100 years and a few days.”
Communication comes in to play with the need to read the text and interpret the table, and is also called on at a higher level with the need to answer with brief written reasoning. The demand to mathematise the situation arises at a low level with the need to identify and extract key mathematical characteristics of a bar graph as each type of litter is considered. The problem solver must interpret a simple tabular representation of data, and must imagine a graphical representation, and linking these two representations is a key demand of the item. The reasoning demands of the problem are at a relatively low level, as is the need for devising strategies. Using symbolic, formal and technical language and operations comes into play with the procedural and factual knowledge required to imagine construction of bar graphs or to make a quick sketch, and particularly with the understanding of scale needed to imagine the vertical axis. Using mathematical tools is likely not needed.
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46. Pizzas
A pizzeria serves two round pizzas of the same thickness in different sizes. The smaller one has a diameter of 30 cm and costs 30 zeds. The larger one has a diameter of 40 cm and costs 40 zeds.
Question 1
Which pizza is better value for money? Show your reasoning.
Pizzas is set in a personal context with which many 15-year-olds would be familiar. The context category is personal since the question posed is which pizza provides the purchaser with the better value for the money. It presents a relatively low reading demand, thereby ensuring the efforts of the
The open constructed-response item Pizzas shown in Figure 8 is simple in form, yet rich in content, and illustrates various elements of the mathematics framework. It was initially used in the first PISA field trial in 1999, then was released for illustrative purposes and has appeared as a sample item in each version of the PISA mathematics framework published since This was one of the most difficult items used in the 1999 field trial item pool, with only 11% correct.
The item draws on several areas of mathematics. It has geometrical elements that would normally be classified as part of the Space and shape content category. The pizzas can be modelled as thin circular cylinders, so the area of a circle is needed. The question also involves the Quantity content category with the implicit need to compare the quantity of pizza to amount of money. However, the key to this problem lies in the conceptualisation of the relationships among properties of the pizzas, and how the relevant properties change from the smaller pizza to the larger one. Because those aspects are at the heart of the problem, this item is categorised as belonging to the Change and relationships content category.
The item belongs to the formulating process category. a key step to solving this problem, indeed the major cognitive demand, is to formulate a mathematical model that encapsulates the concept of value for money. The problem solver must recognise that because pizzas ideally have uniform thickness and the thicknesses are the same, the focus of analysis can be on the area of the circular surface of the pizza instead of volume or mass. The relationship between amount of pizza and amount of money is then captured in the concept of value for money modelled as ‘cost per unit of area.’ variations such as area per unit cost are also possible. Within the mathematical world, value for money can then be calculated directly and compared for the two circles, and is a smaller quantity for the larger circle. The real world interpretation is that the larger pizza represents better value for money.
QUESTION INTENT: Applies understanding of area to solving a value for money comparison
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47. Change and Relationships
Pizzas
A pizzeria serves two round pizzas of the same thickness in different sizes. The smaller one has a diameter of 30 cm and costs 30 zeds. The larger one has a diameter of 40 cm and costs 40 zeds.
Question 1
Which pizza is better value for money? Show your reasoning.
Pizzas is set in a personal context with which many 15-year-olds would be familiar. The context category is personal since the question posed is which pizza provides the purchaser with the better value for the money. It presents a relatively low reading demand, thereby ensuring the efforts of the
The open constructed-response item Pizzas shown in Figure 8 is simple in form, yet rich in content, and illustrates various elements of the mathematics framework. It was initially used in the first PISA field trial in 1999, then was released for illustrative purposes and has appeared as a sample item in each version of the PISA mathematics framework published since This was one of the most difficult items used in the 1999 field trial item pool, with only 11% correct.
The item draws on several areas of mathematics. It has geometrical elements that would normally be classified as part of the Space and shape content category. The pizzas can be modelled as thin circular cylinders, so the area of a circle is needed. The question also involves the Quantity content category with the implicit need to compare the quantity of pizza to amount of money. However, the key to this problem lies in the conceptualisation of the relationships among properties of the pizzas, and how the relevant properties change from the smaller pizza to the larger one. Because those aspects are at the heart of the problem, this item is categorised as belonging to the Change and relationships content category.
The item belongs to the formulating process category. a key step to solving this problem, indeed the major cognitive demand, is to formulate a mathematical model that encapsulates the concept of value for money. The problem solver must recognise that because pizzas ideally have uniform thickness and the thicknesses are the same, the focus of analysis can be on the area of the circular surface of the pizza instead of volume or mass. The relationship between amount of pizza and amount of money is then captured in the concept of value for money modelled as ‘cost per unit of area.’ variations such as area per unit cost are also possible. Within the mathematical world, value for money can then be calculated directly and compared for the two circles, and is a smaller quantity for the larger circle. The real world interpretation is that the larger pizza represents better value for money.
CONTEXT
Personal
PROCESS
Formulating
CONTENT
Change and Relationships

48. An important part of formulation
15 cm
Employing knowledge from space and shape and Quantity.
Formulating a mathematical model to measure value for money
20 cm
The released PISA item Pizzas (see Appendix B) calls most heavily on students’ abilities to formulate a situation mathematically. While it is indeed the case that students are also called upon to perform calculations as they solve the problem and make sense of the results of their calculations by identifying which pizza is the better value for the money, the real cognitive challenge of this item lies in being able to formulate a mathematical model that encapsulates the concept of value for money. The problem solver must recognise that because the pizzas have the same thickness but different diameters, the focus of the analysis can be on the area of the circular surface of the pizza. The relationship between amount of pizza and amount of money is then captured in the concept of value for money, modelled as cost per unit of area.
Interpreting mathematical result in real world terms.

49. Computer-Based Assessment PISA 2015
Single multiple-choice
Full credit dependent on both the selection and the reasoning behind it.
An alternative form of reasoning, which reveals even more clearly the item’s classification in Change and relationships, would be to say (explicitly or implicitly) that the area of a circle increases in proportion to the square of the diameter, so has increased in the ratio of (4/3),2 while the cost has only increased in the proportion of (4/3). Since (4/3)2 is greater than (4/3), the larger pizza is better value.
While the primary demand and the key to solving this problem comes from formulating, placing this item in the formulating situations mathematically process category, aspects of the other two mathematical process are also apparent in this item. The mathematical model, once formulated, must then be employed effectively, with the application of appropriate reasoning along with the use of appropriate mathematical knowledge and area and rate calculations. The result must then be interpreted properly in relation to the original question.
The solution process for Pizzas demands the activation of the fundamental mathematical capabilities to varying degrees. Communication comes in to play at a relatively low level in reading and interpreting the rather straight-forward text of the problem, and is called on at a higher level with the need to present and explain the solution. The need to mathematise the situation is a key demand of the problem, specifically the need to formulate a model that captures value for money. The problem solver must devise a representation of relevant aspects of the problem, including the symbolic representation of the formula for calculating area, and the expression of rates that represent value for money, in order to develop a solution. The reasoning demands (for example, to decide that the thickness can be ignored, and justifying the approach taken and the results obtained) are significant, and the need for devising strategies to control the calculation and modelling processes required is also a notable demand for this problem. Using symbolic, formal and technical language and operations comes into play with the conceptual, factual and procedural knowledge required to process the circle geometry, and the calculations of the rates. Using mathematical tools is evident at a relatively low level if students use a calculator efficiently.
In Figure 9, a sample student response to the Pizzas item is presented, to further illustrate the framework constructs. A response like this would be awarded full credit.
Reasoning

50. QUESTION INTENT: Applies understanding of area to solving a value for money
comparison
Code 2: Gives general reasoning that the surface area of pizza increases more
rapidly than the price of pizza to conclude that the larger pizza is better
value.
• The diameter of the pizzas is the same number as their price, but the amount
of pizza you get is found using diameter2 , so you will get more pizza per zeds
from the larger one
Code 1: Calculates the area and amount per zed for each pizza to conclude that the larger pizza is better value.
• Area of smaller pizza is 0.25 x π x 30 x 30 = 225π; amount per zed is 23.6 cm2
area of larger pizza is 0.25 x π x 40 x 40 = 400π; amount per zed is 31.4 cm2
so larger pizza is better value
Code 8: They are the same value for money. (This incorrect answer is coded
separately, because we would like to keep track of how many students
have this misconception).
Code 0: Other incorrect responses OR a correct answer without correct reasoning.
Code 9: Missing.

51. APARTMENT PURCHASE
This is the plan of the apartment that George’s parents want to purchase from a real estate agency.
Question 1: APARTMENT PURCHASE PM00FQ01 – 019
To estimate the total floor area of the apartment (including the terrace and the walls), you can measure the size of each room, calculate the area of each one and add all the areas together.
However, there is a more efficient method to estimate the total floor area where you only need to measure 4 lengths. Mark on the plan above the four lengths that are needed to estimate the total floor area of the apartment.
APARTMENT PURCHASE SCORING 1
QUESTION INTENT:
Description: Use spatial reasoning to show on a plan (or by some other method) the minimum number of side lengths needed to determine floor area
Mathematical content area: Space and shape
Context: Personal
Process: Formulate

52. Full Credit A = (9.7m x 8.8m) – (2m x 4.4m) A = 76.56m2
Code 1: Has indicated the four dimensions needed to estimate the floor area of the apartment on the plan. There are 9 possible solutions as shown in the diagrams below.
A = (9.7m x 8.8m) – (2m x 4.4m)
A = 76.56m2
[Clearly used only 4 lengths to measure and calculate required area.]
No Credit
Code 0: Other responses.
Code 9: Missing.

53. Stimulus
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54. Question Question 1: SHAPES Which of the figures has the largest area?
M158Q
Question 1: SHAPES
Which of the figures has the largest area?
Explain your reasoning.
SHAPES SCORING 1
QUESTION INTENT: Comparison of areas of irregular shapes
Code 1: Shape B, supported with plausible reasoning.
• It’s the largest area because the others will fit inside it.
Code 8: Shape B, without plausible support.
Code 0: Other responses.
Code 9: Missing.
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55. Example responses Code 1:
• B. It doesn’t have indents in it which decreases the area. A and C have gaps.
• B, because it’s a full circle, and the others are like circles with bits taken out.
• B, because it has no open areas:
Code 8:
• B. because it has the largest surface area
• The circle. It’s pretty obvious.
• B, because it is bigger.
Code 0:
• They are all the same.
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56. M158Q
Question 2: SHAPES Describe a method for estimating the area of figure C.
SHAPES SCORING 2
QUESTION INTENT: To assess students’ strategies for measuring areas of
irregular shapes.
Code 1: Reasonable method:
• Draw a grid of squares over the shape and count the squares that are more
than half filled by the shape.
• Cut the arms off the shape and rearrange the pieces so that they fill a square
then measure the side of the square.
• Build a 3D model based on the shape and fill it with water. Measure the
amount of water used and the depth of the water in the model. Derive the
area from the information.
Code 8: Partial answers:
• The student suggests to find the area of the circle and subtract the area of the
cut out pieces. However, the student does not mention about how to find out
the area of the cut out pieces.
• Add up the area of each individual arm of the shape
Code 0: Other responses.
Code 9: Missing.

57. to which the student describes a METHOD. Example responses Code 1:
NOTE:
The key point for this question is whether the student offers a METHOD for
determining the area. The coding schemes (1, 8, 0) is a hierarchy of the extent
to which the student describes a METHOD.
Example responses
Code 1:
• You could fill the shape with lots of circles, squares and other basic shapes so
there is not a gap. Work out the area of all of the shapes and add together.
• Redraw the shape onto graph paper and count all of the squares it takes up.
• Drawing and counting equal size boxes. Smaller boxes = better accuracy
(Here the student’s description is brief, but we will be lenient about student’s
writing skills and regard the method offered by the student as correct)
• Make it into a 3D model and filling it with exactly 1cm of water and then
measure the volume of water required to fill it up.
wrong)
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58. • Minus the shape from the circle
Code 8:
• Find the area of B then find the areas of the cut out pieces and subtract them
from the main area.
• Minus the shape from the circle
• Add up the area of each individual piece e.g.,
• Use a shape like that and pour a liquid into it.
• Use graph
• Half of the area of shape B
• Figure out how many mm2 are in one little leg things and times it by 8.
Code 0:
• Use a string and measure the perimeter of the shape. Stretch the string out to
a circle and measure the area of the circle using π r2.
(Here the method described by the student is wrong)
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59. Describe a method for estimating the perimeter of figure C.
Question 3: SHAPES M158Q
Describe a method for estimating the perimeter of figure C.
SHAPES SCORING 3
QUESTION INTENT: To assess students’ strategies for measuring perimeters of
irregular shapes
Code 1: Reasonable method:
• Lay a piece of string over the outline of the shape then measure the length of string used.
• Cut the shape up into short, nearly straight pieces and join them together in a line, then measure the length of the line.
• Measure the length of some of the arms to find an average arm length then multiply by 8 (number of arms) X 2.
Code 0: Other responses.
Code 9: Missing.
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60. Question 3: SHAPES M158Q
Describe a method for estimating the perimeter of figure C.
Example responses
Code 1:
• Wool or string!!!
(Here although the answer is brief, the student did offer a METHOD for
measuring the perimeter)
• Cut the side of the shape into sections. Measure each then add them
together. (Here the student did not explicitly say that each section needs to be
approximately straight, but we will give the benefit of the doubt, that is, by
offering the METHOD of cutting the shape into pieces, each piece is assumed
to be easily measurable)
Code 0:
• Measure around the outside.
(Here the student did not suggest any METHOD of measuring. Simply saying
“measure it” is not offering any method of how to go about measuring it)
• Stretch out the shape to make it a circle.
(Here although a method is offered by the student, the method is wrong)

61. 6 Proficiency Scale Description PISA 2015 LEVEL DESCRIPTION
At Level 6 students can conceptualise, generalise and utilise information based on their investigations and modelling of complex problem situations. They can link different information sources and representations and flexibly translate among them. Students at this level are capable of advanced mathematical thinking and reasoning. These students can apply their insight and understandings along with a mastery of symbolic and formal mathematical operations and relationships to develop new approaches and strategies for attacking novel situations. Students at this level can formulate and precisely communicate their actions and reflections regarding their findings, interpretations, arguments and the appropriateness of these to the original situations.
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62. 5 4 Proficiency Scale Description PISA 2015 LEVEL DESCRIPTION
At Level 5 students can develop and work with models for complex situations, identifying constraints and specifying assumptions. They can select, compare and evaluate appropriate problem-solving strategies for dealing with complex problems related to these models. Students at this level can work strategically using broad, well-developed thinking and reasoning skills, appropriate linked representations, symbolic and formal characterisations and insight pertaining to these situations. They can reflect on their actions and formulate and communicate their interpretations and reasoning. 4
At Level 4 students can work effectively with explicit models for complex concrete situations that may involve constraints or call for making assumptions. They can select and integrate different representations, including symbolic, linking them directly to aspects of real-world situations. Students at this level can utilise well-developed skills and reason flexibly, with some insight, in these contexts. They can construct and communicate explanations and arguments based on their interpretations, arguments and actions.

63. 3 2 1 Proficiency Scale Description PISA 2015 LEVEL DESCRIPTION
At Level 3 students can execute clearly described procedures, including those that require sequential decisions. They can select and apply simple problem-solving strategies. Students at this level can interpret and use representations based on different information sources and reason directly from them. They can develop short communications when reporting their interpretations, results and reasoning. 2
At Level 2 students can interpret and recognise situations in contexts that require no more than direct inference. They can extract relevant information from a single source and make use of a single representational mode. Students at this level can employ basic algorithms, formulae, procedures, or conventions. They are capable of direct reasoning and making literal interpretations of the results. 1
At Level 1 students can answer questions involving familiar contexts where all relevant information is present and the questions are clearly defined. They are able to identify information and to carry out routine procedures according to direct instructions in explicit situations. They can perform actions that are obvious and follow immediately from the given stimuli.

64. Sample Test Items on OECD Website:

65. FOKUS KELAS INTERVENSI…
Pentafsiran Data
Perimeter
Nisbah
Bentuk 3D
Pola
Nombor
Keserupaan
Teorem Pythagoras
Wang
Inkuiri dan Penaakulan
Kebarangkalian
Kadar
FOKUS KELAS INTERVENSI…
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66. Kelas Intervensi Sekurang-kurangnya 10 kelas
Sebaik-baiknya 12 kelas  ada 12 bahan intervensi
Masa  1 jam
Sasaran  semua murid Ting. 3 berumur 15+ thn
 semua murid Ting. 4
 selepas 15 Feb 2015: murid kajian sebenar
Penggunaan komputer oleh murid amat digalakkan  penggunaan adalah wajib pada 2 kelas terakhir
 murid dilatih menjawab menggunakan komputer
Pembelajaran secara hands-on digalakkan
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67. Kelas Intervensi
Bincang dengan murid bagaimana menggunakan proses formulating, employing dan interpreting apabila mencari penyelesaian sesuatu masalah yang diberi
Aktiviti pembelajaran yang dijalankan dalam taklimat ini perlu dilaksanakan bersama murid
Sebaik-baiknya semua guru Matematik dilibatkan
Peserta taklimat ini perlu menjalankan kursus dalaman tentang kelas intervensi di sekolah masing-masing
Bahan intervensi boleh dimuat turun dari
bit.ly/pisamatematik2015
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69. Perbengkelan Teroka bahan yang disediakan.
Selesaikan masalah dalam bahan kelas intervensi
tunjukkan kepelbagaian strategi penyelesaian sekiranya berkaitan.
Kemukakan cadangan penambahbaikan bahan:
pembetulan untuk sebarang kesilapan dalam bahan, jika ada, dan
Cadangan bahan bantu mengajar untuk aktiviti hands-on, jika berkaitan.
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