Explore PISA 2012 Mathematics and Problem Solving Test Questions

The OECD's Programme for International Student Assessment (PISA) evaluates education systems worldwide by testing 15-year-olds in key subjects. The focus of PISA 2012 was mathematics. Some countries chose to assess problem-solving too. To understand more about the PISA 2012 mathematics and problem-solving tests, click below to answer sample questions, explore the concepts and skills being tested and learn what 15-year-olds students at different proficiency levels can do.

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Test Levels

The PISA test contains questions

representing 6 levels of proficiency.

Learn more about the levels for each subject.

representing 6 levels of proficiency.

Learn more about the levels for each subject.

Question Categories

The PISA test contains question which provide different contexts and test different skills. Click on the subject below to learn more about the subject categories and skills being tested.

- FAQ: OECD PISA
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Problem solving test levels

Level 6

At Level 6, students can develop complete, coherent mental models of diverse problem scenarios, enabling them to solve complex problems efficiently. They can explore a scenario in a highly strategic manner to understand all information pertaining to the problem. The information may be presented in different formats, requiring interpretation and integration of related parts. When confronted with very complex devices, such as home appliances that work in an unusual or unexpected manner, they quickly learn how to control the devices to achieve a goal in an optimal way. Level 6 problem-solvers can set up general hypotheses about a system and thoroughly test them. They can follow a premise through to a logical conclusion or recognise when there is not enough information available to reach one. In order to reach a solution, these highly proficient problem-solvers can create complex, flexible, multi-step plans that they continually monitor during execution. Where necessary, they modify their strategies, taking all constraints into account, both explicit and implicit.

Level 5

At Level 5, students can systematically explore a complex problem scenario to gain an understanding of how relevant information is structured. When faced with unfamiliar, moderately complex devices, such as vending machines or home appliances, they respond quickly to feedback in order to control the device. In order to reach a solution, Level 5 problem-solvers think ahead to find the best strategy that addresses all the given constraints. They can immediately adjust their plans or backtrack when they detect unexpected difficulties or when they make mistakes that take them off course.

Level 4

At Level 4, students can explore a moderately complex problem scenario in a focused way. They grasp the links among the components of the scenario that are required to solve the problem. They can control moderately complex digital devices, such as unfamiliar vending machines or home appliances, but they don't always do so efficiently. These students can plan a few steps ahead and monitor the progress of their plans. They are usually able to adjust these plans or reformulate a goal in light of feedback. They can systematically try out different possibilities and check whether multiple conditions have been satisfied. They can form an hypothesis about why a system is malfunctioning, and describe how to test it.

Level 3

At Level 3, students can handle information presented in several different formats. They can explore a problem scenario and infer simple relationships among its components. They can control simple digital devices, but have trouble with more complex devices. Problem-solvers at Level 3 can fully deal with one condition, for example, by generating several solutions and checking to see whether these satisfy the condition. When there are multiple conditions or inter-related features, they can hold one variable constant to see the effect of change on the other variables. They can devise and execute tests to confirm or refute a given hypothesis. They understand the need to plan ahead and monitor progress, and are able to try a different option if necessary.

Level 2

At Level 2, students can explore an unfamiliar problem scenario and understand a small part of it. They try, but only partially succeed, to understand and control digital devices with unfamiliar controls, such as home appliances and vending machines. Level 2 problem-solvers can test a simple hypothesis that is given to them and can solve a problem that has a single, specific constraint. They can plan and carry out one step at a time to achieve a sub-goal, and have some capacity to monitor overall progress towards a solution.

Level 1

At Level 1, students can explore a problem scenario only in a limited way, but tend to do so only when they have encountered very similar situations before. Based on their observations of familiar scenarios, these students are able only to partially describe the behaviour of a simple, everyday device. In general, students at Level 1 can solve straightforward problems provided there is only a simple condition to be satisfied and there are only one or two steps to be performed to reach the goal. Level 1 students tend not to be able to plan ahead or set sub-goals.

Mathematics test Levels

Level 6

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 this 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. Student 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.

Level 5

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.

Level 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.

Level 3

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 reporting their interpretations, results and reasoning.

Level 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.

Level 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.

Mathematics Question Categories

Mathematics Content Categories

Space & Shape

Space and Shape encompasses a wide range of phenomena that are encountered everywhere in our visual and physical world: patterns, properties of objects, positions and orientations, representations of objects, decoding and encoding of visual information, navigation and dynamic interaction with real shapes as well as with representations. Geometry serves as an essential foundation for space and shape, but the category extends beyond traditional geometry in content, meaning and method, drawing on elements of other mathematical areas such as spatial visualisation, measurement and algebra.

Change & Relationships

Change & Relationships involves understanding fundamental types of change and recognising when they occur in order to use suitable mathematical models to describe and predict change. Mathematically this means modelling the change and the relationships with appropriate functions and equations, as well as creating, interpreting, and translating among symbolic and graphical representations of relationships.

Quantity

Quantity may be the most pervasive and essential mathematical aspect of engaging with, and functioning in, our world. Engaging with the quantification of the world involves understanding measurements, counts, magnitudes, units, indicators, relative size, and numerical trends and patterns. Aspects of quantitative reasoning – such as number sense, multiple representations of numbers, elegance in computation, mental calculation, estimation and assessment of reasonableness of results – are the essence of mathematical literacy relative to quantity.

Uncertainty & Data

Mathematics Contexts

Personal

Personal questions include those focusing on activities of one’s self, one’s family or one’s peer group e.g. food preparation, shopping, games, personal health, personal transportation, sports, travel, personal scheduling and personal finance.

Occupational

Occupational questions are centred on the world of work e.g. measuring, costing and ordering materials for building, payroll/accounting, quality control, scheduling/inventory, design/architecture and job-related decision making.

Societal

Societal questions focus on one’s community (whether local, national or global) e.g. voting systems, public transport, government, public policies, demographics, advertising, national statistics and economics. Although individuals are involved in all of these things in a personal way, in the societal context category the focus of problems is on the community perspective.

Scientific

Scientific questions relate to the application of mathematics to the natural world and issues and topics related to science and technology e.g. weather or climate, ecology, medicine, space science, genetics, measurement and the world of mathematics itself.

Mathematical Processes

Formulating situations mathematically

Formulating situations mathematically means being able to recognise and identify opportunities to use mathematics and then provide mathematical structure to a problem presented in some contextualised form.

Employing mathematical concepts, facts, procedures and reasoning

Employing mathematical concepts, facts, procedures and reasoning means being able to apply mathematical concepts, facts, procedures, and reasoning to solve mathematically-formulated problems to obtain mathematical conclusions.

Interpreting, applying and evaluating mathematical outcomes

Interpreting, applying and evaluating mathematical outcomes means the ability to reflect upon mathematical solutions, results, or conclusions and interpret them in the context of real-life problems.

Problem Solving Question Categories

Nature of the Problem Solving Situation

Interactive vs Static

The problem is interactive when not all information is disclosed at the outset and some information has to be uncovered by exploring the problem situation. The problem is static when all relevant information for solving the problem is disclosed at the outset.

Problems Contexts

Setting: Technology vs Non-Technology

The problem belongs to the technology category when it involves a technological device. It belongs to the non-technology category when there is no technological device.

Focus: Personal vs Social

The problem belongs to the personal category when the focus is on one’s self, family or close peers. It belongs to the social category when the focus is the community or society in general.

Problems Solving Processes

Exploring and understanding

Exploring and understanding involves building mental representations of each of the pieces of information presented in the problem. This includes exploring the problem situation: observing it, interacting with it, searching for information and finding limitations or obstacles; and understanding given information and information discovered while interacting with the problem situation; demonstrating understanding of relevant concepts.

Representing and formulating

Representing and formulating involves building a coherent mental representation of the problem situation. To do this, relevant information must be selected, mentally organised and integrated with relevant prior knowledge. This may involve: representing the problem by constructing tabular, graphical, symbolic or verbal representations, and shifting between representational formats; and formulating hypotheses by identifying the relevant factors in the problem and their interrelationships; organising and critically evaluating information.

Planning and executing

Planning and executing means planning by setting goals, including clarifying the overall goal, and setting sub-goals, where necessary; and devising a plan or strategy to reach the goal state, including the steps to be undertaken; as well as executing by carrying out a plan.

Monitoring and reflecting

Monitoring and reflecting involves monitoring progress towards the goal at each stage, including checking intermediate and final results, detecting unexpected events, and taking remedial action when required. Reflecting involves reflecting on solutions from different perspectives, critically evaluating assumptions and alternative solutions, identifying the need for additional information or clarification and communicating progress in a suitable manner.

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