# 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.

Fundamentally, the NumberSense Mathematics Programme provides structured daily practice of mathematical concepts. These concepts are developed gradually over time and in an interrelated manner. This explains why there are no chapters or sections, no worked examples, and why children 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. A context where differentiated teaching is valued and where children are respected as sense-making, problem-solving individuals capable of and interested in seeing patterns 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 the mathematics that is being revealed through the activities that they complete in the workbooks.

When children develop knowledge with understanding, that they can apply in unfamiliar situations, while being able to explain (reason about) what they are doing, they experience mathematics as a sense-making, problem-solving activity. The development of the NumberSense Mathematics Programme has been guided by clear developmental trajectories for all of the mathematical concepts. These developmental trajectories draw on mathematics education research about how children learn mathematics with a particular focus on developing mathematical reasoning skills. By building connections and exploring similarities and differences between activities and tasks, the NumberSense materials support children to develop a robust structural understanding of mathematics.

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

:__Developmental Trajectories__The NumberSense Mathematics Programme is based on the understanding that the development of mathematical concepts progress along a developmental trajectory. This trajectory typically begins with contexts that reveal the mathematical concept. It then moves on to more advanced contexts that force the child to use increasingly sophisticated strategies. Thereafter, the child should be 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), and finally, reification (where the idea becomes an object independent of the context that introduced it).

We cannot present children with mathematics as ready-made objects, just as we cannot expect infants to speak fluently from the outset. Furthermore, another feature of developmental trajectories is that, in the early stages, when the concept is context-based or concrete, it is quite transparent. In contrast, 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, the child is then unable to apply, with understanding, the concept in even slightly unfamiliar situations, and certainly not with the reasoning that we would expect them to be able to do so.

:__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.

- Who can use the NumberSense Mathematics Programme?
The richly-illustrated and engaging workbooks are used in a wide range of different ways: from schools who use the NumberSense Mathematics Programme as their mathematics programme, to schools that use the NumberSense Mathematics Programme as an integral part of both their extension and remediation programmes, corporates and volunteer organisations that provide the NumberSense Mathematics Programme to children and schools as part of their corporate social investment projects, to parents who homeschool their children. As the NumberSense Mathematics Programme is being used more widely, the body of both anecdotal and more rigorously researched evidence in support of the efficacy of the series is growing.

In all applications, the programme allows for differentiated learning support, independent engagement by children and the opportunity to experience mathematics as a meaningful, sense-making activity. Teachers and parents play an important role in the success of the programme. Not only is there a need to ensure that children are working in the most appropriate workbook, but teachers also need to monitor children’s progress for the identification of problems, misunderstanding and necessary interventions. Furthermore, it is critical that teachers and parents discuss the activities with the children: ask children to explain their answers, describe any patterns they may have observed and invite them to ask questions. This contributes to their development of a strong sense of number.