PISA 2021 Mathematics Framework

Explore the main sections below, click on the interactive framework components, or download the full PISA 2021 Mathematics Framework Draft in PDF format.

The PISA 2021 mathematics framework defines the theoretical underpinnings of the PISA mathematics assessment based on the fundamental concept of mathematical literacy, relating mathematical reasoning and three processes of the problem-solving (mathematical modelling) cycle. The framework describes how mathematical content knowledge is organised into four content categories. It also describes four categories of contexts in which students will face mathematical challenges.

The PISA assessment measures how effectively countries are preparing students to use mathematics in every aspect of their personal, civic and professional lives, as part of their constructive, engaged and reflective 21st Century citizenship.

Mathematical literacy is an individual’s capacity to reason mathematically and to formulate, employ and interpret mathematics to solve problems in a variety of real-world contexts. It includes concepts, procedures, facts and tools to describe, explain and predict phenomena. It helps individuals know the role that mathematics plays in the world and make the well-founded judgments and decisions needed by constructive, engaged and reflective 21st Century citizens.

PISA 2021 aims to consider mathematics in a rapidly changing world driven by new technologies and trends in which citizens are creative and engaged, making non-routine judgments for themselves and the society in which they live. This brings into focus the ability to reason mathematically, which has always been a part of the PISA framework. This technology change is also creating the need for students to understand those computational thinking concepts that are part of mathematical literacy. Finally, the framework recognises that improved computer-based assessment is available to most students within PISA.

The ability to reason logically and present arguments in honest and convincing ways is a skill that is becoming increasingly important in today’s world. Mathematics is a science about well-defined objects and notions that can be analysed and transformed in different ways using “mathematical reasoning” to obtain certain and timeless conclusions.

In mathematics, students learn that, with proper reasoning and assumptions, they can arrive at results that they can fully trust to be true in a wide variety of real-life contexts. It is also important that these conclusions are impartial, without any need for validation by an external authority.

At least six key understandings provide structure and support to mathematical reasoning. These key understandings include

- understanding quantity, number systems and their algebraic properties;
- appreciating the power of abstraction and symbolic representation;
- seeing mathematical structures and their regularities;
- recognising functional relationships between quantities;
- using mathematical modelling as a lens onto the real world (e.g. those arising in the physical, biological, social, economic and behavioural sciences); and
- understanding variation as the heart of statistics.

Use the arrows below to review the key understandings in-depth:

The word *formulate* in the mathematical literacy definition refers to the ability of individuals to recognise and identify opportunities to use mathematics and then provide mathematical structure to a problem presented in some contextualised form. In the process of formulating situations mathematically, individuals determine where they can extract the essential mathematics to analyse, set up and solve the problem. They translate from a real-world setting to the domain of mathematics and provide the realworld problem with mathematical structure, representations and specificity. They reason about and make sense of constraints and assumptions in the problem. Specifically, this process of formulating situations mathematically includes activities such as the following:

- selecting an appropriate model from a list; **
- identifying the mathematical aspects of a problem situated in a real-life context and identifying the significant variables;
- recognising mathematical structure (including regularities, relationships and patterns) in problems or situations;
- simplifying a situation or problem in order to make it amenable to mathematical analysis;
- identifying constraints and assumptions behind any mathematical modelling and simplifications gleaned from the context;
- representing a situation mathematically, using appropriate variables, symbols, diagrams and standard models;
- representing a problem in a different way, including organising it according to mathematical concepts and making appropriate assumptions;
- understanding and explaining the relationships between the context-specific language of a problem and the symbolic and formal language needed to represent it mathematically;
- translating a problem into mathematical language or a representation;
- recognising aspects of a problem that correspond with known problems or mathematical concepts, facts or procedures;
- using technology (such as a spreadsheet or the list facility on a graphing calculator) to portray a mathematical relationship inherent in a contextualised problem; and
- creating an ordered series of (step-by-step) instructions for solving problems.

** This activity is included in the list to foreground the need for the test-item developers to include items that are accessible to students at the lower end of the performance scale.

The word *employ* in the mathematical literacy definition refers to the ability of individuals to apply mathematical concepts, facts, procedures and reasoning to solve mathematically formulated problems to obtain mathematical conclusions. In the process of employing mathematical concepts, facts, procedures and reasoning to solve problems, individuals perform the mathematical procedures needed to derive results and find a mathematical solution. They work on a model of the problem situation, establish regularities, identify connections between mathematical entities and create mathematical arguments. Specifically, this process of employing mathematical concepts, facts, procedures and reasoning includes activities such as

- performing a simple calculation; **
- drawing a simple conclusion; **
- selecting an appropriate strategy from a list; **
- devising and implementing strategies for finding mathematical solutions;
- using mathematical tools, including technology, to help find exact or approximate solutions;
- applying mathematical facts, rules, algorithms and structures when finding solutions;
- manipulating numbers, graphical and statistical data and information, algebraic expressions and equations, and geometric representations;
- making mathematical diagrams, graphs and constructions and extracting mathematical information from them;
- using and switching between different representations in the process of finding solutions;
- making generalisations based on the results of applying mathematical procedures to find solutions;
- reflecting on mathematical arguments, and explaining and justifying mathematical results; and
- evaluating the significance of observed (or proposed) patterns and regularities in data.

** These activities are included in the list to foreground the need for the test-item developers to include items that are accessible to students at the lower end of the performance scale.

The word *interpret* (and evaluate) used in the mathematical literacy definition focuses on the ability of individuals to reflect upon mathematical solutions, results or conclusions and interpret them in the context of the real-life problem that initiated the process. This involves translating mathematical solutions or reasoning back into the context of the problem and determining whether the results are reasonable and make sense in the context of the problem.

Specifically, this process of interpreting, applying and evaluating mathematical outcomes includes activities such as the following:

- interpreting information presented in graphical form and/or diagrams; **
- evaluating a mathematical outcome in terms of the context; **
- interpreting a mathematical result back into the real-world context;
- evaluating the reasonableness of a mathematical solution in the context of a real-world problem;
- understanding how the real world impacts the outcomes and calculations of a mathematical procedure or model in order to make contextual judgments about how the results should be adjusted or applied;
- explaining why a mathematical result or conclusion does or does not make sense given the context of a problem;
- understanding the extent and limits of mathematical concepts and mathematical solutions;
- critiquing and identifying the limits of the model used to solve a problem; and
- using mathematical thinking and computational thinking to make predictions, to provide evidence for arguments, and to test and compare proposed solutions.

** This activity is included in the list to foreground the need for the test-item developers to include items that are accessible to students at the lower end of the performance scale.

An understanding of mathematical content – and the ability to apply that knowledge to solving meaningful contextualised problems – is important for citizens in the modern world. That is, to reason mathematically and to solve problems and interpret situations in personal, occupational, societal and scientific contexts, individuals need to draw upon certain mathematical knowledge and understanding.

The following content categories used in PISA since 2012 are again used in PISA 2021 to reflect the mathematical phenomena that underlie broad classes of problems, the general structure of mathematics and the major strands of typical school curricula:

Four topics have been identified for special emphasis in the PISA 2021 assessment. These topics are not new to the mathematics content categories. Instead, these are topics that deserve special emphasis:

- growth phenomena (change and relationships),
- geometric approximation (space and shape),
- computer simulations (quantity), and
- conditional decision making (uncertainty and data).

The notion of quantity may be the most pervasive and essential mathematical aspect of engaging with and functioning in our world. It incorporates the quantification of attributes of objects, relationships, situations and entities in the world; understanding various representations of those quantifications; and judging interpretations and arguments based on quantity. To engage with the quantification of the world involves understanding measurements, counts, magnitudes, units, indicators, relative size, and numerical trends and patterns.

Quantification is a primary method for describing and measuring a vast set of attributes of aspects of the world. It allows for the modelling of situations, for the examination of change and relationships, for the description and manipulation of space and shape, for organising and interpreting data, and for the measurement and assessment of uncertainty.

Both mathematics and statistics involve problems that are not so easily addressed because the required mathematics is complex or involves a large number of factors all operating in the same system. Increasingly in today’s world, such problems are being approached using computer simulations driven by algorithmic mathematics.

Identifying computer simulations as a focal point of the quantity content category signals that, in the context of the computer-based assessment of mathematics, there is a broad category of complex problems. For example, students can use computer simulations to analyse budgeting and planning as part of the test item.

In science, technology and everyday life, uncertainty is a given. Uncertainty is therefore a phenomenon at the heart of the mathematical analysis of many problem situations, and the theory of probability and statistics as well as techniques of data representation and description have been established to deal with it. The uncertainty and data content category includes recognising the place of variation in processes, having a sense of the quantification of that variation, acknowledging uncertainty and error in measurement, and knowing about chance. It also includes forming, interpreting and evaluating conclusions drawn in situations where uncertainty is central. Quantification is a primary method for describing and measuring a vast set of attributes of aspects of the world.

Identifying conditional decision-making as a focal point of the uncertainty and data content category signals that students should be expected to appreciate how the assumptions made in setting up a model affect the conclusions that can be drawn and that different assumptions/relationships may well result in a different conclusion.

The natural and designed worlds display a multitude of temporary and permanent relationships among objects and circumstances, where changes occur within systems of interrelated objects or in circumstances where the elements influence one another. In many cases, these changes occur over time. In other cases, changes in one object or quantity are related to changes in another. Some of these situations involve discrete change; others involve continuous change. Some relationships are of a permanent, or invariant, nature. Being more literate about change and 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.

Understanding the dangers of flu pandemics and bacterial outbreaks, as well as the threat of climate change, demands that people not only think in terms of linear relationships but recognise that such phenomena need non-linear models reflecting a very rapid growth. Linear relationships are common and easy to recognise and understand, but to assume linearity can sometimes be dangerous.

Identifying growth phenomena as a focal point of the change and relationships content category does not signal an expectation that participating students should have studied the exponential function, and certainly the items will not require knowledge of the exponential function. Instead, the expectation is that there will be items that expect students to recognise (a) that not all growth is linear and (b) that nonlinear growth has profound implications on how we understand certain situations.

Space and shape encompass 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, and 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.

Today’s world is full of shapes that do not follow typical patterns of evenness or symmetry. Because simple formulas do not deal with irregularity, it has become more difficult to understand what we see and to find the area or volume of the resulting structures.

Identifying geometric approximations as a focal point of the space and shape content category signals the need for students to be able use their understanding of traditional space and shape phenomena in a range of a typical situations.

An important aspect of mathematical literacy is that mathematics is used to solve a problem set in a context. The context is the aspect of an individual’s world in which the problems are placed. The choice of appropriate mathematical strategies and representations is often dependent on the context in which a problem arises. For PISA, it is important that a wide variety of contexts are used.

Problems classified in the personal context category focus on activities of one’s self, one’s family or one’s peer group. Personal contexts include (but are not limited to) those involving food preparation, shopping, games, personal health, personal transportation, sports, travel, personal scheduling and personal finance.

Problems classified in the occupational context category are centred on the world of work. Items categorised as occupational may involve (but are not limited to) such things as measuring, costing and ordering materials for building, payroll/accounting, quality control, scheduling/inventory, design/architecture, and job-related decision-making. Occupational contexts may relate to any level of the workforce, from unskilled work to the highest levels of professional work, although items in the PISA survey must be accessible to 15-year-old students.

Problems classified in the societal context category focus on one’s community (whether local, national or global). They may involve (but are not limited to) such things as 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.

Problems classified in the scientific category relate to the application of mathematics to the natural world and issues and topics related to science and technology. Particular contexts might include (but are not limited to) such areas as weather or climate, ecology, medicine, space science, genetics, measurement and the world of mathematics itself. Items that are intra-mathematical, where all the elements involved belong in the world of mathematics, fall within the scientific context.

There is increased interest worldwide in what are called 21st Century skills and their possible inclusion in educational systems. The OECD has put out a publication that focuses on such skills and has sponsored a research project entitled *The Future of Education and Skills: Education 2030*. Some 25 countries are involved in this cross-national study of curriculum including the incorporation of such skills. The project has as its central focus what the curriculum might look like in the future, focusing initially on mathematics. Some of the key 21st Century skills are:

- critical thinking;
- creativity;
- research and inquiry;
- self-direction, initiative and persistence;
- information use;
- systems thinking;
- communication; and
- reflection.

Although test-item developers recognise these 21st Century skills, the mathematics items in PISA 2021 are not specifically developed according to these skills.

Below are some example exercises from the PISA 2021 Mathematics assessment. Each button below opens an overlay that shows an example experience from the application.