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73. Statistics is essentially about accounting for or modelling variation as measured by the
variance or in the case of multiple variables the covariance matrix. This provides a probabilistic
environment in which to understand various phenomena as well as to make critical decisions.
Statistics is in many ways a search for patterns in a highly variable context: trying to find the single
defining “truth” in the midst of a great deal of random noise. “Truth” is set in quotes as it is not the
nature of truth that mathematics can deliver but an estimate of truth set in a
probabilistic context,
accompanied by an estimate of the error contained in the process. Ultimately, the decision maker
is left with the dilemma of never knowing for certain what the truth is. The estimate that has been
developed is, at best a range of possible values
– the better the process, for example, the larger
the sample of data, the narrower the range of possible values, although a range cannot be
avoided. Some aspects of this have been present in previous PISA cycles, the growing
significance contributes to the increased stress in this framework.
74.
Understanding variation as a central feature of statistics supports reasoning about the real-
world applications of mathematics envisaged in this framework in that students are encouraged to
engage with data based arguments with awareness of the limitations of the conclusions that can be
drawn.
Problem solving
75. The definition of mathematical literacy refers to an individual’s capacity to formulate, employ,
and interpret (and evaluate) mathematics. These three words, formulate, employ and interpret,
provide a useful and meaningful structure for organising the mathematical processes that describe
what individuals do to connect the context of a problem with the mathematics and solve the
problem. Items in the 2021 PISA mathematics test will be assigned to either mathematical
reasoning or one of three mathematical processes:
Formulating situations mathematically;
Employing mathematical concepts, facts, procedures and reasoning; and
Interpreting, applying and evaluating mathematical outcomes.
76. It is important for both policy makers and those engaged more closely in
the day-to-day
education of students to know how effectively students are able to engage in each of these
elements of the problem solving model/cycle.
Formulating 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 in a
mathematical form.
Employing refers to 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.
Interpreting (and
evaluating) relates to how effectively
students are able to reflect upon mathematical solutions or conclusions, interpret them in the
context of the real-world problem and determine whether the result(s) or conclusion(s) are
reasonable and/or useful. Students’ facility at applying mathematics to
problems and situations is
dependent on skills inherent in all three of these stages, and an understanding of students’
effectiveness in each category can help inform both policy-level discussions and decisions being
made closer to the classroom level.
77. Moreover, encouraging students to experience mathematical problem solving processes
through computational thinking tools and practices encourage students to practice prediction,
reflection and debugging skills (Brennan and Resnick, 2012
[24]
).
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