• Developing the Fraction Concept (illustrative trajectory)

In this section we illustrate the developmental trajectory for fractions. This illustration is indicative of the progression with some milestones illustrated along the way.

The NumberSense Mathematics Programme learning trajectory designed for the development of the fraction concept assumes that there are a number of interrelated concepts within the overarching concept fractions. For the primary school years these are:

• The existence of a part of a whole – the fraction
• Naming of fractions & fraction notation (social knowledge)
• Comparing fractions
• Awareness of the equivalence of fractions
• Operations with and involving fractions Fraction Trajectory Illustration
• Developing Pre-Algebraic Thinking (illustrative trajectory)

In this section we illustrate the developmental trajectory for algebraic thinking. This illustration is indicative of the progression with some milestones illustrated along the way.

The NumberSense Mathematics Programme learning trajectory designed for the development of the algebraic thinking assumes that mathematical situations can be described in at least five different ways as illustrated below. Furthermore, the learning trajectory for the development of the algebraic thinking, assumes that the development of robust algebraic thinking relies on an awareness of the interrelatedness of these different representations. That is, different patterns (direct proportion; linear; and non-linear) have different characteristics in each representation while at the same time the manifestation of a given characteristic in one representation can be used to explain the manifestation of that characteristic in the other representations, and the ability to switch between the different representations contributes to confidence with first algebraic thinking and then formal algebra. Algebraic Thinking Trajectory Illustration