This is not an exhaustive list but its aim is to give just a
sense of each strand in the "third-dimension".
Many of these "third-dimension" terms are linked in some way.
For example, "Independence" links with many other aspects,
including "confidence", "resilience" and "familiarity". A learner
who works independently will also be more confident, more resilient
and is more likely to be able to have a go at problems set in
unfamiliar contexts. The reason for separating these terms is to
give space for describing the particular features of each of these
strands. They are not isolated objects, and the aim is certainly
not to tick each off as separate targets to meet. This would not
only be pointless (and probably impossible) but would miss the
opportunity to look more holistically at what is happening when a
problem is investigated. The aim is to give a sense of the
opportunities that need to be available to learners to become
effective problem-solvers and mathematicians, and therefore have an
impact on the role of us as teachers.
Learners work autonomously. They identify the mathematics in a
problem for themselves, they pose their own problems and explore
ideas without prompting. They do not rely on others to make
decisions about what to do and take responsibility for their own
learning. Learners move from imitation to independent application
of mathematical ideas and techniques. They show increasing
confidence in their mathematics and ability to share ideas.
Learners think about what they want to do and anticipate
potential consequences. They can change their lines of enquiry in
the light of experience. They can pull out key features and
identify strengths and weaknesses in approaches. They make
connections with other mathematics and other situations.
Learners will get absorbed in the task. When the going gets
tough, when they get stuck, they will persevere, using their own
ideas whenever they can, but seeking advice from other sources when
necessary. When seeking advice they have some clarity concerning
what their need is and how they believe it will help them move on.
They have a sense of the learning opportunities offered when
something causes them to "struggle".
Learners see themselves as part of a larger learning community
(local and remote), taking an active part in discussions. They
share their ideas and listen to others, offering suggestions and
support. They work collaboratively recognising their own and
others' strengths. They respond constructively to suggestions and
adapt their methods or approaches in the light of interactions.
Learners communicate information about their thinking without
prompting and when presenting ideas they have a sense of
Learners are a source of mathematical ideas. They utilise their
own, and other people's ideas throughout the problem-solving
process. They are able to adapt the ideas of others and utilise
them in mathematical situations. They challenge assumptions and
make unexpected connections.
Learners handle complex mathematical situations using a range of
strategies. For example, they are able to see the mathematics in a
"real-life" situation, or break a complex problem down into
manageable parts. They create models of situations and can explain
how they have simplified the situation. Learners can identify the
variables and the constraints and explain the limitations and
strengths of their own and other people's models.
Learners are able to identify the mathematics in situations that
are not part of their everyday experiences or which they have not
considered in a mathematical context before. They can apply their
mathematics to unfamiliar contexts. They are able to make
connections between different aspects of mathematics, or use a
range of knowledge and skills, in new situations.