183
in 1 hour and 40 minutes? Such problems have a specific answer: to drive 180 km in 1 hour and 40
minutes you must drive at 108 km per hour.
66. At some point students start to consider situations where quantities are variable, that is, where
they can take on a range of values. For example, what is the relation between the distance driven,
d, in kilometres, and time spent driving,
t, in hours, if you drive at a constant speed of 108
km per
hour? Such questions introduce functional relationships. In this case the relationship, expressed by
the equation
d = 108
t, is a
proportional relationship, the fundamental example and perhaps the
most important for general knowledge.
67. 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. The essential elements of the concept are a
domain, from which inputs are selected, a codomain, in
which outputs lie, and a process for
producing outputs from inputs.
68.
Recognising the functional relationships between the variables in the real-world applications of
mathematics envisaged by this framework supports reasoning by allowing students to focus on
how the interdependence of and interaction between the variables impacts on the situation.
Using mathematical modelling as a lens onto the real world
69. Models represent a conceptualisation of phenomena. Models are simplifications of reality that
fore- ground certain features of a phenomenon while approximating or ignoring other features. As
such, ‘‘all models are wrong, but some are useful’’ (Box and Draper, 1987, p. 424
[23]
). The
usefulness of a model comes from its explanatory and/or predictive power (Weintrop et al.,
2016
[14]
). Models 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.g., from
arithmetic, algebra, geometry, etc.). As such, 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?), and for making predictions. Models can be operated
– that is, made to run over time or
with varying inputs, thus producing a simulation. When this is done, it is possible
to make
predictions, study consequences, and evaluate the adequacy and accuracy of the models.
Throughout the modelling process cognisance needs to be taken of the real world parameters that
impact on the model and the solutions developed using the model.
70. Computer-based (or computational) models provide the ability to test hypothesis, generate
data, introduce randomness and so on. Mathematical literacy includes the ability to understand,
evaluate and draw meaning from computational models.
71.
Using models in general and mathematical models in particular supports reasoning about the
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