20762_Problem_solving_Year_5_Number_and_place_value_2

FOREWORD
Each teacher page concludes with two further aspects
critical to successful teaching of problem-solving. A
section on likely difficulties points to reasoning and
content inadequacies that experience has shown may
well impede students’ success. In this way, teachers
can be on the look out for difficulties and be prepared
to guide students past these potential pitfalls. The
final section suggests extensions to the problems to
enable teachers to provide several related experiences
with problems of these kinds in order to build a rich
array of experiences with particular solution methods;
for example, the numbers, shapes or measurements
in the original problems might change but leave the
means to a solution essentially the same, or the
context may change while the numbers, shapes or
measurements remain the same. Then numbers,
shapes or measurements and the context could be
changed to see how the students handle situations
that appear different but are essentially the same
as those already met and solved. Other suggestions
ask students to make and pose their own problems,
investigate and present background to the problems
or topics to the class, or consider solutions at a more
general level (possibly involving verbal descriptions
and eventually pictorial or symbolic arguments).
In this way, not only are students’ ways of thinking
extended but the problems written on one page are
used to produce several more problems that utilise
the same approach.
Mathematics and language
The difficulty of the mathematics gradually increases
over the series, largely in line with what is taught
at the various year levels, although problem-solving
both challenges at the point of the mathematics
that is being learned as well as provides insights
and motivation for what might be learned next. For
example, the computation required gradually builds
from additive thinking, using addition and subtraction
separately and together, to multiplicative thinking,
where multiplication and division are connected
conceptions. More complex interactions of these
operations build up over the series as the operations
are used to both come to terms with problems’
meanings and to achieve solutions. Similarly, twodimensional
geometry is used at first but extended
to more complex uses over the range of problems,
then joined by interaction with three-dimensional
ideas. Measurement, including chance and data, also
extends over the series from length to perimeter, and
from area to surface area and volume, drawing on
the relationships among these concepts to organise
solutions as well as giving an understanding of the
metric system. Time concepts range from interpreting
timetables using 12-hour and 24-hour clocks while
investigations related to mass rely on both the concept
itself and practical measurements.
The language in which the problems are expressed is
relatively straightforward, although this too increases
in complexity and length of expression across the books
in terms of both the context in which the problems
are set and the mathematical content that is required.
It will always be a challenge for some students
to ‘unpack’ the meaning from a worded problem,
particularly as problems’ context, information and
meanings expand. This ability is fundamental to the
nature of mathematical problem-solving and needs to
be built up with time and experiences rather than be
iv
Problem-solving in mathematics www.ricpublications.com.au R.I.C. Publications ®