Problem-solving approach
A balanced mathematical programme includes concept learning, developing and maintaining skills,
and learning to tackle applications. These should be taught in such a way that students develop
the ability to think mathematically.
Students learn mathematical thinking most effectively through applying concepts and skills in
interesting and realistic contexts which are personally meaningful to them. Thus, mathematics is
best taught by helping students to solve problems drawn from their own experience.
Real-life problems are not always closed, nor do they necessarily have only one solution. Determining
the best approximation to a solution, and finding the optimum way of solving a problem when several approaches
are possible, are skills frequently required in the workplace. Students need frequent opportunities to
work with open-ended problems. The solutions to problems which are worth solving seldom involve only one item
of mathematical understanding or only one skill. Rather than remembering the single correct method, problem
solving requires students to search the information for clues and to make connections to the various
pieces of mathematics and other knowlege and skills which they have learned. Such problems encourage thinking
rather than mere recall.
Closed problems, which follow a well-known pattern of solution, develop only a limited range of skills.
They encourage memorisation of routine methods rather than consideration and experimentation.
While fluency with basic techniques is very important, such routines only become useful tools when students
can apply them to realistic problems.
The characteristics of good problem-solving techniques include both convergent and divergent approaches.
These include the systematic collection of data or evidence, experimentation (trial and error followed by improvement),
flexibility and creativity, and reflection – that is, thinking about the process that has been followed and
evaluating it critically.
Teachers can create opportunities for students to develop these characteristics by encouraging them to practise
and learn such simple strategies as guessing and checking, drawing a diagram, making lists, looking for patterns,
classifying, substituting, re-arranging, putting observations into words, making predictions, and developing proofs.
Learning to communicate about and through mathematics is part of learning to become a mathematical problem solver
and learning to think mathematically. Critical reflection may be developed by encouraging students to share ideas,
to use their own words to explain their ideas, and to record their thinking in a variety of ways, for example,
through words, symbols, diagrams, and models.
The chance to look for problems as well as to solve them, to create and to produce rather than reproduce what
already exists, is important for all students. Creativity in problem solving is recognised as one of the basic
traits that must be developed if outstanding achievement is to result, and it plays a major role in innovation,
invention, and scientific discovery.
Catering for individual needs
It is a principle of the New Zealand Curriculum Framework that all students should be enabled to achieve
personal standards of excellence and that all students have a right to the opportunity to achieve to the
maximum of their potential. It is axiomatic in this curriculum statement that mathematics is for all students,
regardless of ability, background, gender, or ethnicity.
Students of lower ability need to have the opportunity to experience a range of mathematics which is
appropriate to their age level, interests, and capabilities. Equally, students with exceptional ability in
mathematics must be extended and not simply expected to repeat different permutations of work they have
clearly mastered.
As new experiences cause students to refine their existing knowledge and ideas, so they construct
new knowledge. The extent to which teachers are able to facilitate this process significantly affects
how well students learn. It is important that students are given explicit opportunities to relate
their new learning to knowledge and skills which they have developed in the past. Factors such as
out-of-school experience and language have profound effects on the way students learn mathematics.
In many cases in the past, students have failed to reach their potential because
they have not seen the applicability of mathematics to their lives and because
they were not encouraged to connect new mathematical concepts and skills to
experiences, knowledge, and skills which they already had. This has been particularly
true for many girls, and for many Māori students, for whom the contexts in which
mathematics was presented were irrelevant and inappropriate. These students
have developed deeply entrenched negative attitudes towards mathematics as a
result.
An awareness of these issues has led to improved access for girls to mathematics, but the participation
rate of female students in mathematics continues to be lower than that of male students at senior school
level and beyond. This limits later opportunities for girls and women.
The suggested learning experiences in this document include strategies that utilise the strengths and
interests that girls bring to mathematics. Techniques that help to involve girls actively in the subject
include setting mathematics in relevant social contexts, assigning co-operative learning tasks, and providing
opportunities for extended investigations.
The suggestions also describe experiences which will help girls develop greater confidence in their mathematical
ability. Girls' early success in routine mathematical operations needs to be accompanied by experiences
which will help them develop confidence in the skills that are essential in other areas of mathematics. Girls
need to be encouraged to participate in mathematical activities involving, for example, estimation, construction,
and problems where there are any number of methods and where there is no obvious "right answer".
It is particularly important that mathematical learning experiences for Māori
students acknowledge the background experiences which have led to the formation
of ideas and skills which those students already have. Māori students will
be helped to achieve if teachers acknowledge and value those ideas and experiences.
Traditional time-constrained pencil and paper tests have proved unreliable
indicators of Māori achievement in the past. Among the sample assessment
activities, there are procedures suggested which may be more appropriate for
assessing Māori students. In selecting assessment procedures, teachers
should endeavour to ensure that all of the desired objectives are evaluated
and that the procedures which are selected are culturally appropriate.
The development of more positive attitudes to mathematics and a greater appreciation of its usefulness is
the key to improving participation rates for all students.
Use of Resources
Apparatus
The importance of the use of apparatus to help students form mathematical concepts is well established.
Using apparatus provides a foundation of practical experience on which students can build abstract ideas.
It encourages them to be inventive, helps to develop their confidence, and encourages independence.
Junior school teachers are used to choosing an appropriate range of apparatus to focus students'
thinking on the concept to be developed and modifying the apparatus as the learner's understanding grows.
Teachers know that students are capable of solving quite difficult problems when they are free to use
concrete apparatus to help them think the problems through. Such an approach is equally valid with older
students and should be used wherever possible.
At all levels, students should be introduced to new ideas by having their attention drawn to examples
occurring in their natural environment, and then by modelling them with apparatus. For example, a child's
concept of "four" could be enriched by discussing the number of wheels on a car, legs on a table, or edges
on a piece of paper. The child could then be encouraged to explore the idea further, using materials with
which to make their own models of "four". Similarly, secondary students could be focused on the concept of
"rate of change" by discussing, for example, that younger people grow faster than older people, or by
discussing the slope changes on nearby hills. Students could then model uniform and non-uniform rate situations,
using apparatus such as sand running through an egg-timer or a ball rolling down a smooth slope.
Textbooks
Many textbooks contain material to provide students with practice and enrichment. Increasing numbers
of books contain excellent ideas for problem-solving situations which develop mathematical skills and
understandings. However, teachers must realise that there are dangers in adhering too closely to any
particular textbook. Many texts contain material not included in this curriculum statement, or have
emphases which are different from those advocated for New Zealand. In any event, teachers should continually
re-evaluate the texts they are using in the light of the particular needs of their students.
Technology
This curriculum statement assumes that both calculators and computers will be available and used in the
teaching and learning of mathematics at all levels. Instruction in the correct and appropriate use of
calculators is particularly important.
Calculators, graphics calculators, and computers are learning tools which students can use to
discover and reinforce new ideas. Calculators are powerful tools for helping students to discover
numerical facts and patterns, and helping them to make generalisations about, for example, repeated
operations. Graphics calculators, and computer software such as graphing packages and spreadsheets,
are tools which enable students to concentrate on mathematical ideas rather than on routine mechanical
manipulation, which often intrudes on the real point of particular learning situations. Computer programs,
such as Logo, provide excellent environments for mathematical experimentation and open-ended problem solving.
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