- 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 students 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 (teaching to the developmental level of the child) is valued, students 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 students 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/approaches:
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 student to use increasingly sophisticated (age and number range appropriate) strategies until the student 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 objects (concepts) develop along a well-defined trajectory from procedural (process) to structural (object). 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 idea/concept); to reification – where the idea becomes an object independent of the context that introduced it. There are no short cuts. We cannot present students 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 situation (context) based (“concrete”), the situation/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 a “abstract.” Any attempt to move to the abstract (opaque) stage too quickly (as in presenting students with mathematical objects as ready-made) contributes to the student not understanding what they are doing (although they may appear to be able to mimic “the steps”) and as a result being 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 students to subtract using addition and so on, is a critical to solving algebraic equations. By helping early grade students to become aware of these interrelations we are not only strengthening their ability to perform the operations and do arithmetic, but we are also preparing them with the understanding that they will be critical to the development of their algebraic manipulation skills.
The NumberSense Mathematics Programme understands that students 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 student, 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 students 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.