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of work), or a scientific context (relating to the application of mathematics to the natural and
technological world).
34. Included for the first time in the PISA 2021 framework (and depicted in Figure 2) are selected
21st century skills that mathematical literacy both relies on and develops. 21st century skills are
discussed in greater detail in the next section of this framework. For now, it should be stressed that
while contexts (personal, societal, occupational and scientific) influence the development of test
items, there is no expectation that items will be deliberately developed to
incorporate or address
21st century skills. Instead, the expectation is that by responding to the spirit of the framework and
in line with the definition of mathematical literacy, the 21st century skills that have been identified
will be incorporated in the items.
35. The language of the definition and the representation in Figure 1 and Figure 2 retain and
integrate the notion of mathematical modelling, which has historically been a cornerstone of the
PISA framework for mathematics e.g. (OECD, 2004
[6]
; OECD, 2013
[7]
). The modelling cycle
(formulate, employ, interpret and evaluate) is a central aspect of the PISA conception of
mathematically literate students; however, it is often not necessary to engage in every stage of the
modelling cycle, especially in the context of an assessment (Galbraith, Henn and Niss, 2007
[18]
). It
is often the case that significant parts of the mathematical modelling cycle have been undertaken
by others, and the end user carries out some of the steps of the modelling cycle, but not all of
them. For example, in some cases, mathematical representations, such as graphs or equations,
are given that can be directly manipulated in order to answer some question or to draw some
conclusion. In other cases, students may be using a computer simulation to
explore the impact of
variable change in a system or environment. For this reason, many PISA items involve only parts
of the modelling cycle. In reality, the problem solver may also sometimes oscillate between the
processes, returning to revisit earlier decisions and assumptions. Each of the processes may
present considerable challenges, and several iterations around the whole cycle may be required.
36. In particular, the verbs ‘formulate’, ‘employ’ and ‘interpret’ point to the three processes in which
students as active problem solvers will engage. Formulating situations mathematically involves
applying mathematical reasoning (both deductive and inductive) in identifying opportunities to
apply and
use mathematics
– seeing that mathematics can be applied to understand or resolve a
particular problem or challenge presented. It includes being able to take a situation as presented
and transform it into a form amenable to mathematical treatment, providing mathematical structure
and representations, identifying variables and making simplifying assumptions to help solve the
problem or meet the challenge. Employing mathematics involves applying mathematical reasoning
while
using mathematical concepts, procedures, facts and tools to derive a mathematical solution.
It includes performing calculations, manipulating algebraic expressions and equations or other
mathematical models, analysing information in a mathematical manner from mathematical
diagrams and graphs, developing mathematical descriptions and explanations and using
mathematical tools to solve problems. Interpreting mathematics involves reflecting upon
mathematical solutions or results and interpreting them in the context of a problem or challenge. It
involves applying mathematical reasoning to evaluate mathematical solutions in relation to the
context of the problem and determining whether the results are reasonable
and make sense in the
situation; determining also what to highlight when explaining the solution.
37. Included for the first time in the PISA 2021 framework is an appreciation of the intersection
between mathematical and computational thinking engendering a similar set of perspectives,
thought processes and mental models that learners need to succeed in an increasingly
technological world. A set of constituent practices positioned
under the computational thinking umbrella (namely abstraction, algorithmic thinking, automation,
decomposition and generalisation) are also central to both mathematical reasoning and problem
solving processes. The nature of computational thinking within mathematics is conceptualised as
178
defining and elaborating mathematical knowledge that can be expressed by programming, allowing
students to dynamically model mathematical concepts and relationships. A taxonomy of
computational thinking practices geared specifically towards mathematics and science learning
entails data practices, modelling and
simulation practices, computational problem solving
practices, and systems thinking practices (Weintrop et al., 2016
[14]
). The combination of
mathematical and computational thinking not only becomes essential to effectively support the
development of students’ conceptual understanding of the mathematical domain, but also to
develop their computational thinking concepts and skills, giving learners a more realistic view of
how mathematics is practiced in the professional world and used in the real-world and, in turn,
better prepares them for pursuing careers in related fields (Basu et al., 2016
[19]
; Benton et al.,
2017
[20]
; Pei, Weintrop and Wilensky, 2018
[13]
; Beheshti et al., 2017
[21]
).
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