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57. Students use representations
– whether text-based, symbolic, graphical, numerical, geometric
or in
programming code
– to organise and communicate their mathematical thinking.
Representations enable us to present mathematical ideas in a succinct way which, in turn, lead to
efficient algorithms. Representations are also a core element of mathematical modelling, allowing
students to abstract a simplified or idealised formulation of a real world problem. Such structures
are also important for interpreting and defining the behaviour of computational devices.
58.
Having an appreciation of abstraction and symbolic representation supports reasoning in the
real-world applications of mathematics envisaged by this framework by allowing students to move
from the specific details of a situation to the more general features and to describe these in an
efficient way.
Seeing mathematical structures and their regularities
59. When elementary students see: 5 + (3 + 8) some see a string of symbols indicating a
computation to be performed in a certain order according to the rules of order of operations; others
see a number added to the sum of two other numbers. The latter group are seeing structure; and
because of that they don’t need to be told about the order of operation, because
if you want to add
a number to a sum you first have to compute the sum.
60. Seeing structure continues to be important as students move to higher grades. A student who
sees
f(
𝑥) = 5 + (𝑥 − 3)2 as saying that
f(
x) is the sum of 5 and a square which is zero when x = 3
understands that the minimum of
f is 5. This lays the foundation for functional thinking discussed in
the next section.
61. 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. Such structures are also important for interpreting and
defining the
behaviour of computational devices. Being able to see structure is an important conceptual aid to
procedural knowledge.
62. The examples above illustrate how seeing structure in abstract mathematical objects is a way
of replacing parsing rules, which can be performed by a computer, with conceptual images of those
objects that make their properties clear. An object held in the mind in
such a way is subject to
reasoning at a level that is higher than simple symbolic manipulation.
63. 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. An
irregular shape can be approximated by simpler shapes whose area is known. A geometric pattern
can be understood by hypothesising translational, rotational, or reflectional transformations and
symmetry and abstractly extending the pattern into all of space. Statistical analysis is often a
matter of imposing a structure on a set of data, for example by assuming it comes from a normal
distribution or supposing that one variable is
a linear function of another, but measured with
normally distributed error.
64.
Being able to see mathematical structures supports reasoning in the real-world applications of
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