Available Chapters

As the chapters are written, we are making them available for download. Please note that this is a first draft and the final version of the Teacher Guide may differ slightly from what is proposed here.

Assessment

The underlying philosophy of the NumberSense Mathematics Programme is that we want children to experience mathematics as a sense-making, problem-solving activity. Furthermore, the programme is designed to develop knowledge with understanding, that children can apply in unfamiliar situations, all the while being able to reason about what they are doing.

Assessment of children using the NumberSense Mathematics Programme should be aligned with these aims. This should be reflected in the use of appropriate assessment tasks and in the nature of the items (questions) in the tasks.

Recognising that giving no education is better than giving it at the wrong time (Piaget, 1953), an important role of assessment is for teachers to gather information. This information should then be used to make teaching decisions that improve learning. While assessment also serves the role of measuring and reporting on children’s achievement at the end of a unit of work to inform decisions about promotion etc., the key role of assessment in the NumberSense Programme is for teachers and children to get feedback on learning.

To read further, download the Assessment chapter of the Teacher Guide (first draft).

As the chapters are written, we are making them available for download. Please note that this is a first draft and the final version of the Teacher Guide may differ slightly from what is proposed here.

Differentiation

Learning opportunities not presented within a child’s zone of proximal development (Vygotsky, 1978) cannot be effective. Piaget (1953) stated this more forcefully as “giving no education is better than giving it at the wrong time.” In the NumberSense Mathematics Programme, we recognise that mathematics classes are not homogeneous with respect to the mathematical proficiency levels of the children. We reject outright the characterisation of children as weak and strong. We prefer to think of children as being on a continuum. On the one end of the continuum, are those children who have not yet achieved the level of mathematical proficiency we expect of children at their age. On the other end, children whose mathematical proficiency already exceeds the level we expect of children at their age. We hold the position that those who have not
yet achieved the level of mathematical proficiency we expect of children at their age, can reduce the gap between where they are and where we expect them to be.

The purpose of this discussion paper is to address some of the classroom realities that are emerging as teachers use the NumberSense Programme to provide differentiated learning opportunities. At the same time the purpose is also to address the implications of the recently introduced Comprehensive NumberSense Workbooks (2024) which address the complete (or full) school curriculum for mathematics.

To read further, download the Differentiation chapter of the Teacher Guide (first draft).

As the chapters are written, we are making them available for download. Please note that this is a first draft and the final version of the Teacher Guide may differ slightly from what is proposed here.

Patterns, Functions and Algebra

Mathematics is, among other things, the study of patterns. Patterns allow us to describe a changing world. Patterns and structures are fundamental to mathematics and it is for this reason that patterning is critical to the development of mathematical proficiency. Patterns are relationships between variables with some form of regularity and rules that describe them. From a young age we need to nurture children’s ability to recognise and describe variation, and to make predictions about:

• what will happen under certain conditions; and
• what conditions are needed for specific outcomes to occur.

There is evidence that children’s ability to recognise, describe, and extend patterns is predictive of their future success in mathematics. It is for this reason that the study of patterns and patterning is a critical component of mathematics in the early years. The challenge of school mathematics is not so much to ‘teach patterns and patterning’ as a topic but, to support children to notice patterns, to describe them, and to make predictions based on those patterns. In addition, children need to be supported to develop the vocabulary with which to describe patterns.

To read further, download the Patterns chapter of the Teacher Guide (first draft).

As the chapters are written, we are making them available for download. Please note that this is a first draft and the final version of the Teacher Guide may differ slightly from what is proposed here.

Problem Solving and Investigations

Mathematics is a tool for solving problems. Problems also provide a medium for teaching mathematics. Children can add, subtract, multiply, divide, work with fractions, use formulae and solve equations long before these operations and concepts are formalised and long before they have mathematical vocabulary with which to describe what they are doing. When a mother gives her children some sweets and asks them to share the sweets equally amongst themselves, they can do so without describing what they are doing as dividing and without writing a mathematical expression to describe what they have done. Children come to school with an incredible capacity for solving problems in general. One only needs to watch children at play to realise how inventive and clever they are. When we teach mathematics through problems, we present children with problem situations that they can make sense of and resolve. The organic response that they use is the mathematics we want them to learn more formally. In other words, we use a problem to provoke a response and that response is the mathematics we want to teach/develop. This approach is not limited to the basic operations – this approach applies throughout school mathematics.

To read further, download the Problem Solving chapter of the Teacher Guide (first draft).

As the chapters are written, we are making them available for download. Please note that this is a first draft and the final version of the Teacher Guide may differ slightly from what is proposed here.

Measurement

Measurement has its origin in the desire to compare, who is taller and who is shorter (length); which container holds more and which holds less (capacity/volume); which object is heavier and which is lighter (mass/weight); which object takes up more space and which takes up less space (area and volume) etc. Measurement is one of the most practical topics in the mathematics curriculum. The activities in the NumberSense Mathematics Programme assume a very practical, hands-on approach. Measurement involves assigning a numerical value to an attribute to enable comparison, ordering, and calculation. The attributes that the NumberSense workbooks focus on are:

• length, capacity (volume), mass (weight), area, and time.

There are three key aspects to measuring:

• knowing what attribute is being measured (e.g. what is mass (weight)?).
• being able to describe the attribute that is being measured (e.g. how do you describe how heavy something is?).
• the ability to use the appropriate measuring instrument(s) to measure the attribute (e.g. how do you use a scale?).

The nature of the measuring activities in the NumberSense Mathematics Programme is informed by a developmental trajectory. In broad terms:

• Measurement in Grade R and 1 (Workbooks 00 to 4) focuses on using direct comparison to compare and order.
• Measurement in Grades 2 and 3 (Workbooks 4 to 9) introduces indirect comparison to compare and order.
• In Grade 3 (Workbooks 9 to 12), non-standard units are introduced. The nature of the activities draws attention to the need for selecting appropriate units and the challenges associated with non-standard units.
• From Grade 4 (Workbooks 13 and up), the workbooks begin the transition to standard units and converting between these. By Grade 6 and 7, the workbooks begin to introduce formulae associated with perimeter, area, volume and so on.

Throughout the NumberSense Mathematics Programme, measurement activities are used as a context for exploration, sense-making and problem solving.

To read further, download the Measurement chapter of the Teacher Guide (first draft).

An extract from the Counting, Manipulating Numbers and Solving Problems Handbook (full handbook available here).

Why is Counting Important?
Humans have a natural curiosity and tendency to compare:

• Who has the most? Who has the least?
• Who is tallest? Who is shortest?
• Who is heaviest? Who is lightest?
• Do I have enough? Do I have too much?
• Which route is the longest? Which route is the shortest?

In many situations, we can hold objects next to each other and make direct comparisons. However, when we cannot hold the things we are comparing next to each other, we need a “measuring device”. Measuring devices start out informally: fingers, hands, paces and so on. With time, these measuring devices become both more efficient and standardised. As much as we compare objects, we also need to compare the quantity of a collection of objects.

Counting plays three important roles in the lives of children:

• Counting develops the language of number.
  • Giving meaning to the words often learnt through number songs and rhymes.
• Counting develops a sense of ‘muchness’ – quantity or numerosity:
  • The idea that 5 is a small number, 50 a bigger number and 500 a much bigger number.
  • The last number ‘counted’ tells us the size or measure of the group.
• Counting provides an early tool for calculating and solving problems.

Teacher Activities (detail in the download)

Rote Counting:

• Counting in 1s
• Number songs and rhymes
• Counting in steps

Rational Counting:

• Using number cards
• Counting small piles of counters individually
• Counting out a given number
• Estimating and counting in 1s
• Counting in groups – body parts
• Counting large piles of counters

Written Activities in the NumberSense Workbooks
The NumberSense Workbooks contain various counting activities. This includes (but is not limited to):

• Counting in 1s or groups
• Counting groups
• Counting mixed groups
• Number patterns
• Number lines
• Number worms
• Less and more activities

An extract from the Counting, Manipulating Numbers and Solving Problems Handbook (full handbook available here).

Calculating Fluently and Efficiently
Numbers and calculations with numbers are at the heart of mathematics. Children need to develop a range of calculating strategies that enable them to calculate flexibly and fluently. Furthermore, it is important that they can perform a wide range of calculations mentally. Mental calculations are central to estimating. It is unlikely that we would expect anybody to spend time calculating 24.382 × 0.248 using paper and pencil in a context where we have calculators. However, it is important that a person has a sense of what the expected answer is. In the case of 24.382 × 0.248 we expect a person to have a sense that 24.382 × 0.248 ≈ 12 × 0,5 = 6 so that when they use their calculator and get 6.046736 as the answer they are not surprised.

• Calculating flexibly means using different calculation strategies for different situations.
• Calculating fluently means confidently using a range of calculation strategies within a number range and for operations appropriate to a child’s developmental state.

Teacher Activities (detail in the download)

• Single-digit arithmetic
• Multiples of 10, 100, 1000
• Completing 10s, 100s, 1000s
• Bridging 10s, 100s, 1000s
• Doubling and Halving
• Multiplication facts

Written Activities in the NumberSense Workbooks
The NumberSense Workbooks contain various manipulating numbers activities. This includes, but is not limited to:

• Equivalent expressions (number sentences)
• Flow diagrams
• Number chains
• Tables
• Bubbles
• Number pyramids
• “Make less and more”
• “Guess my number”

An extract from the Counting, Manipulating Numbers and Solving Problems Handbook (full handbook available here).

The Role of Problems

Mathematics is a tool for solving problems. Problems, however, also provide a way of supporting the learning of mathematics. Problems in the context of a problem-driven approach to learning serve three key purposes:

• They introduce children to the mathematics that we want them to learn.
• They help children to develop efficient computational strategies.
• They help children to experience mathematics as a meaningful, sense-making activity.

Children can add, subtract, multiply and divide long before they know these words. When a mother gives her children some sweets and asks them to share them equally between themselves they can do so.
Living organisms are natural problem solvers. Consider a plant growing in the ground. If the root meets a stone, it grows around the stone. When an animal senses danger, it will run away and hide or change colour or attack the perceived danger. When a young baby is hungry, they will cry to get attention. Children who come to school know how to solve problems, what they do not know are the labels that adults use to describe their natural responses to a problem. This is particularly true in mathematics.

When we present a young child with some toy animals and a pile of counters and ask the child to share the counters equally between the animals they will do so. Try it! You will be amazed. Young children have naturally efficient strategies for sharing the counters between the toy animals. Young children can solve this problem and problems like it long before they can count the counters in the pile, long before they know what it is to divide and long before they can write a number sentence to summarise the problem situation and its solution.

In a problem-driven approach to learning mathematics, we present children with problem situations that they are capable of solving and where the natural solution strategy they use is the mathematics we want them to learn in a more formal way. In other words, we use a problem to provoke a natural response and that response is the mathematics we want to teach.

Teacher Activities (detail in the download)

• Introducing mathematics
• Developing strategies
  • Primitive drawings
  • Sophisticated drawings
  • Primitive number strategies
  • Sophisticated number strategies – low
  • Sophisticated number strategies – high
• Meaningful mathematics
• Classroom management

Written Activities in the NumberSense Workbooks

In the NumberSense Mathematics Programme, we use problem situations to introduce children to the different age-appropriate mathematical concepts. Typically we use problems to introduce children to:

• The basic operations and increasingly efficient number range-appropriate calculation strategies: Workbooks 1 to 12
• Fractions and operations with fractions: Workbooks 7 to 24
• Ratio, rate and proportion: Workbooks 15 to 24
• Patterns, algebra and problem-solving equations: Workbooks 13 to 24