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The PISA 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.
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 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. 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. This fundamental and ancient concept of quantity is conceptualised in mathematics by the concept of number systems and the basic algebraic properties that these systems employ.
The overwhelming universality of those systems makes them essential for mathematical literacy. It is also important to understand matters of representation as symbols involving numerals, as points on a number line or as geometric quantities and how to move between them; the ways in which these representations are affected by number systems; and the ways in which algebraic properties of these systems are relevant for operating within the systems.
The fundamental ideas of mathematics have arisen from human experience in the world and the need to provide coherence, order and predictability to that experience. Many mathematical objects model reality, or at least reflect aspects of reality in some way. Abstraction involves deliberately and selectively attending to structural similarities between objects and constructing relationships between those objects based on these similarities. In school mathematics, abstraction forms relationships between concrete objects, symbolic representations, and operations including algorithms and mental models.
Students use representations � whether symbolic, graphical, numerical or geometric � to organise and communicate their mathematical thinking. Representations can condense mathematical meanings and processes into efficient algorithms. Representations are also a core element of mathematical modelling, allowing students to abstract a simplified or idealised formulation of a real-life problem.
Structure is intimately related to symbolic representation. The use of symbols is powerful, but only if they retain meaning for the symboliser, rather than becoming meaningless objects to be rearranged on a page. Seeing structure is a way of finding and remembering the meaning of an abstract representation. Being able to see structure is an important conceptual aid to purely procedural knowledge.
A robust sense of mathematical structure also supports modelling. When the objects under study are not abstract mathematical objects, but rather objects from the real world to be modelled by mathematics, then mathematical structure can guide the modelling. Students can also impose structure on non-mathematical objects in order to make them subject to mathematical analysis.
Relationships between quantities can be expressed with equations, graphs, tables or verbal descriptions. An important step in learning is to extract from these the notion of a function itself, as an abstract object of which these are representations. But reading a graph, coordinating the values on the axes, also has a dynamic or process aspect. And the graph of a function is an important tool for exploring the notion of a rate of change. The graph provides a visual tool for understanding a function as a relationship between co-varying quantities.
Models represent an ideal conceptualisation of a real-life or scientific phenomenon. They are, in that sense, abstractions of reality.
A model may present a conceptualisation that is understood to be an approximation or working hypothesis concerning the object phenomenon or it may be an intentional simplification.
Mathematical models are formulated in mathematical language and use a wide variety of mathematical tools and results e. Therefore, they are used as ways of precisely defining the conceptualisation or theory of a phenomenon, for analysing and evaluating data does the model fit the data?
Models can be operated � that is, made to run over time or with varying inputs, thus producing a simulation. When this is done, it can be possible to make predictions, study consequences, and evaluate the adequacy and accuracy of the models. Living things as well as non-living things vary with respect to many characteristics. As a result of that typically large variation, it is difficult to make generalisations in such a world without characterising in some way to what extent that generalisation holds.
Accounting for variability is one, if not the central, defining element around which the discipline of statistics is based. As a result, they suggest sweeping generalisations that are often misleading, if not wrong, and therefore very dangerous. Bias in the social science sense is usually created by not accounting for the variability in the trait under discussion.
Ultimately, the decision-maker is left with the dilemma of never knowing for certain what the truth is. The estimate in the end is a set of plausible values. 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:.
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. 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:. 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 are again used in PISA 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 assessment.
These topics are not new to the mathematics content categories. Instead, these are topics that deserve special emphasis:. 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.
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. 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.
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 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. 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.
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