Published November 2009,December 2009,February 2011.
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 audience.
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.