# The NumberSense Mathematics Programme

- The NumberSense Mathematics Programme
The NumberSense Mathematics Programme is designed to support the development of mathematics as a sense-making, problem-solving activity in which children are expected to develop a body of knowledge that they can apply in unfamiliar situations with understanding and reasoning.

At its most basic, the NumberSense Mathematics Programme provides structured daily practice of mathematical concepts. These concepts are developed gradually over time and in an interrelated way, this explains why there are no chapter or sections, no worked examples, and why students are expected to solve problems using age, grade and number range appropriate strategies.

The programme is most successful when used by skilled teachers in a classroom context where differentiated teaching is valued and where children are respected as sense-making, problem-solving individuals capable of and interested in seeing pattern and structure in the activities that they do. The programme works best when teachers and parents ask questions that force children to reflect on the patterns and structure of mathematics that is revealed through the activities that they have completed in the workbooks.

At a deeper level, the programme is premised on these few key understandings:

**Developmental Trajectories**:The NumberSense Mathematics Programme approaches the development of mathematical concepts from the understanding that concepts develop along a developmental trajectory. This trajectory typically begins with contexts that reveal the mathematical concept, moves to more advanced contexts that force the child to use increasingly sophisticated strategies until the child is finally able to make sense of and use the concept in situations that are independent of the context that initially revealed the concept. To express this in more academic language: mathematical concepts develop along a well-defined trajectory from procedural to structural. This trajectory involves three distinct but interrelated stages: interiorisation (the revelation from contexts of the mathematical idea); condensation (the increasingly confident and more sophisticated use of the mathematical concept); to reification (where the idea becomes an object independent of the context that introduced it). There are no short cuts. We cannot present children with mathematics as ready-made objects, no more than we can expect infants to speak fluently from the get go. There is another feature of developmental trajectories that should be mentioned here. In the early stages of the trajectory, as the concept is still context based or concrete, the concept is quite transparent. At the more sophisticated end of the trajectory, the concept is quite opaque; the situation that gave meaning to the concept early in its development is not visible. At this stage, the concept is often referred to as being abstract. Any attempt to move to the abstract stage too quickly, contributes to the child not understanding what they are doing despite appearing to be able to mimic the procedure. As a result they are unable to apply, with understanding, the concept in even slightly unfamiliar situations and certainly not with reasoning that we would expect them to be able to.

**Interrelatedness of Concepts**:The NumberSense Mathematics Programme recognises that mathematical concepts are highly interrelated. Being aware of the interrelated nature of the four basic operations not only helps young children to subtract using addition etc., it is also critical to solving algebraic equations. By helping children in the early grades become aware of these interrelations, we not only strengthen their ability to perform the operations and do arithmetic, we also prepare them with skills critical to the development of their algebraic manipulation skills.

**Regular Practice**:The NumberSense Mathematics Programme understands that children need regular practice with the same concepts for mathematical concepts to evolve from procedural to structural in nature. It is for this reason that the workbooks are designed to be familiar. Every page consists of activities that are familiar to the child, enabling them to work independently and with confidence, practicing and gaining confidence in working with the concept that is being developed. The routine of the workbooks is important in that it helps children to focus their attention on doing the activities and developing their fluency rather than on working out what they need to do on each page.