### Overview

“What it means to do mathematics?” and “How children learn mathematics?” are questions that guided the development of the NumberSense Workbook Series. In this section we will first discuss what it means for young children to do mathematics, and then we will discuss how children learn mathematics.

### What it means to do mathematics

A teacher’s understanding of what it means to do mathematics profoundly influences how she will teach mathematics. If a teacher’s understanding of doing mathematics involves memorising facts, rules, formulae and procedures to determine the answers to questions then she will teach in a particular way. By contrast, if a teacher regards doing mathematics as a sense-making problem-solving activity then her teaching approach will be quite different.

In our view doing mathematics is much more than “getting lots of answers right”. In the book Adding It Up (Kilpatrick, Swafford & Findell, 2001) the authors describe doing mathematics as being mathematically proficient. The authors introduce mathematical proficiency as follows:

“Our analyses of the mathematics to be learned, our reading of the research in cognitive psychology and mathematics education, our experience as learners and teachers of mathematics, and our judgment as to the mathematical knowledge, understanding, and skill people need today have led us to adopt a composite, comprehensive view of successful mathematics learning. Recognizing that no term captures completely all aspects of expertise, competence, knowledge, and facility in mathematics, we have chosen mathematical proficiency to capture what we think it means for anyone to learn mathematics successfully.”

(Adding it Up, Kilpatrick, Swafford & Findell, 2001, page 5)

The authors go on to describe *mathematical proficiency* (i.e. being able to do mathematics) as having five interrelated dimensions – what they call strands.

The strands of mathematical proficiency are:

**Conceptual understanding**(understanding) – comprehension of mathematical concepts, operations, and relations**Procedural fluency**(computing or calculating) – skill in carrying out procedures flexibly, accurately, efficiently, and appropriately**Strategic competence**(applying) – ability to formulate, represent, and solve mathematical problems**Adaptive reasoning**(reasoning) – capacity for logical thought, reflection, explanation, and justification**Productive disposition**(engaging) – habitual inclination to see mathematics as sensible, useful, and worthwhile, coupled with a belief in diligence and one’s own efficacy

The five strands are interwoven and interdependent. We want children to develop a wide range of calculation **strategies** (procedures) that they can **apply** with confidence and **understanding**, and are able to **reason** about what they have done. Finally, we also want children to develop a disposition towards doing mathematics in which they recognise that they need to **engage** in doing mathematics in order to learn mathematics.

### Mathematical proficiency and the NumberSense Workbook Series

In the NumberSense Workbook Series this view of what it means to do mathematics is translated into the activities in a range of different ways:

- In the workbooks we do not prescribe calculation strategies, but rather introduce children to a range of different strategies with the expectation that they will develop the ability to apply these strategies fluently (with confidence) and flexibly (selecting calculation strategies or problem solving approaches that are appropriate to the situation at hand).
- Children are presented with unfamiliar situations in which they are expected to apply what they already know to makes sense of the situation and solve the problem.
- Children are frequently asked either explicitly (for example, “What did you notice?”) or implicitly (for example, in extending patterns) to reflect on what they have done, forcing them to reason and develop their understanding of what they are doing. The role of the teacher in this regard cannot be overstressed – the expectation is that teachers will instead of checking answers only, review activities by asking questions such as: “What did you notice?” “How did you do that?” “How was this activity similar to or different from previous activities?” and so on.
- There is a lot of repetition. Children revisit the same concepts again and again although in increasing number ranges and in different representations. This is done first to provide practice and second to allow children to increase their confidence.

In developing the NumberSense Workbook Series we have adopted a complex understanding of what it means to do mathematics, an understanding that doing mathematics means much more than the production of correct answers. We regard mathematics as a sense-making problem solving activity.

See the resources section for how to obtain a copy of Adding it up.

### How children learn mathematics

In developing the NumberSense Workbook Series our understanding of how children learn mathematics was informed by a range of factors. These include:

- Stages of number development
- The role of problems
- Different kinds of knowledge
- Mathematics as an integrated/interrelated way of thinking

In the following sections we will describe each of the factors that shaped our understanding of how children learn mathematics. Each of these factors contributed to the design of the NumberSense Workbook Series.

### 1. Stages of number development

As children develop their sense of number there are clearly identifiable developmental stages/milestones:

- counting all
- counting on, and
- breaking down and building up numbers.

When we observe children at work with numbers – in particular, solving problems with numbers – we can tell at what stage of number development they are at.

- If we ask a child to calculate 3 + 5 and we observe that she “makes the 3” and “makes the 5” (using fingers or objects) before she combines the objects and counts all of them to determine that 3 + 5 = 8, then we say that this child is at the
*counting all*stage. - If we observe the child becoming more efficient by “making” only one of the numbers (using fingers or objects) and then counting these objects on from the other number 5: 6, 7, 8… to conclude that: 5 + 3 = 8, then we say that this child is at the
*counting on*stage. - If we observe a child, manipulating numbers to make the calculation easier, for example by saying that 8 + 7 = 8 + 2 + 5 = 10 + 5 = 15, we say that she has reached the
*breaking down and building up*stage. What the child has done is to break up one of the numbers: 7 into 2 and 5, which allows her to “complete the 10” by adding the 2 to the 8 and then adding the remaining 5 to the 10 to get 15. We refer to this stage (more formally) as the*decomposing, rearranging and recomposing*stage.

It is expected that all children should reach the *breaking down and building up* stage within age appropriate number ranges.

In the early grades, we support children’s development of number sense and progression through the stages of number development through three distinct but interrelated activities:

- Counting,
- Manipulating numbers, and
- Solving problems.

The amount of time that is spent on each of these activities will change over time as illustrated in the figure.

### Stages of number development and the NumberSense Workbook Series

In the NumberSense Workbook Series we are mindful that children’s sense of number progresses through the three stages. The activities in the workbooks are sensitive to the stage at which the child is expected to be while simultaneously encouraging the child to move along the developmental trajectory. That said, by the time that children reach Workbook 6 we expect children to be *breaking down and building up* numbers within age appropriate number ranges and by means of an increasingly larger range of strategies. These different strategies are more explicitly developed in workbooks 13 to 24.

Each page of the first twelve workbooks typically addresses the three key activities – counting, manipulating numbers and solving problems in an integrated way. The amount of counting gradually decreases in workbooks 9 through 14.

For more on counting, click here.

For more on manipulating numbers, click here.

For more on solving problems, click here.

### 2. The role of problems

Mathematics is a tool for solving problems, that much is self-evident. Problems, however, also provide a way of supporting the learning of mathematics.

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. They just don’t know that they have “divided” the sweets among themselves.

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, he/she 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 so-called 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/develop. This approach is not limited to the basic operations of addition, subtraction, multiplication and division, this approach applies throughout school 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.

Note that the problem centred approach is by no means a “hands off” approach. It involves a teacher actively thinking about what problems will best introduce what they want a child to learn. It also involves a teacher thinking carefully about the questions they ask in order to provoke the child to see the structure of the problem (the mathematical concept).

### Problems and the NumberSense Workbook Series

In the NumberSense Workbook Series 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 solving equations: workbooks 13 to 24

For more about the problem centred approach watch these videos or click here.

### 3. Different types of knowledge

Piaget distinguished between three different types of knowledge – physical knowledge, social knowledge and conceptual (logico-mathematical) knowledge. When thinking about how children learn mathematics is it important to be mindful of these different knowledge types and how they are related.

Before discussing how the NumberSense Workbook Series takes account of these different knowledge types, each one type is described in some detail.

#### Physical knowledge

This is the kind of knowledge which children acquire through interaction with the physical world, for example through observing and handling objects.

Physical knowledge is derived from concrete experiences – touching, using, playing with and acting on concrete/physical material. Children need a lot of concrete experiences in the numeracy/mathematics classroom to develop their physical knowledge of number by counting concrete apparatus. It is through counting physical objects that children develop a sense of the size of numbers: 50 takes longer and more actions to count than 5 does, but 250 takes a lot more. Five counters can be held in one hand; 50 in two hands; while 250 require a container – there are too many for our hands. Five counters look different from two counters.

Counting physical objects like counters is called rational counting – the counters are physically handled and moved from one place to another. The children observe the pile of counters grow as they count them.

The implication of physical knowledge for the mathematics classroom is very simply that there must be both concrete apparatus (counters, shapes such as building blocks and other construction materials, and measuring apparatus) and the opportunity for children to work/play with the apparatus. It is the teacher’s responsibility to provide the materials and the time for children to use them.

Interestingly, when it comes to learning geometric thinking “play” or “physical knowledge” is vital to the formation of geometric understanding at any age. See an article about this by Pierre van Hiele by clicking here.

#### Social knowledge

The number five is an unproblematic concept for an adult who has known and used the word for many years. An adult can imagine five items and can even calculate with five without having to recreate the number using physical counters or representations in their minds. However, if we take a moment to reflect and think about this then we realise that the word five has no intrinsic properties that hint at the number of items it represents. So it is with people’s names, place names, days of the week and months of the year. The words we use to describe these are all “names” that we have assigned – and because the people in our community (society) all associate the same thing with the same name (word) we are able to communicate with each other. In order to know these names (social knowledge) we need to be told the names and to remember them.

We refer to knowledge that must be both told and remembered as social knowledge.

Conventions are another example of social knowledge. In some countries people drive on the left side of the road while in others they drive on the right side of the road. There is no “correct” side of the road to drive on. However, if all of the people in a country don’t agree to drive on a particular side then there would be chaos. A good example of a mathematical convention is the so-called order of operations convention that guides us in determining the value of the statement 5 + 2 × 3. It is possible to imagine at least two different values for the statement:

- 21 which is the result of first adding the 5 and the 2 and multiplying the answer by three (i.e. working from left to right)
- 11 which is the result of first calculating the value of 2 × 3 and then adding the 5

As with the sides of the road example there is no inherent reason for either solution being correct. However, because we want everybody who calculates the value of the statement 5 + 2 × 3 to get the same value we have introduced a convention called the order of operations convention.

The implication of social knowledge for the mathematics classroom is that there is certain knowledge that teachers have to tell (teach). That is, they have to introduce children to the vocabulary and conventions of mathematics. The way in which we write the number symbols is also a socially agreed on convention/habit and it is the job of the teacher to introduce children to these conventions.

The words “addition, subtraction, division and multiplication” and the symbols that we use to denote them are all also examples of social knowledge.

The challenge for the mathematics classroom is to distinguish between knowledge that must be told (social knowledge) and knowledge that children can construct for themselves (see conceptual knowledge).

#### Conceptual (logico-mathematical) knowledge

When children reflect on activities and begin to see patterns, relationships, regularities and irregularities within and between the numbers and the operations, they are constructing what is known as logico-mathematical knowledge. Logico-mathematical knowledge is internal knowledge and is constructed by each individual for themselves.

The teacher’s role in the development of children’s logico-mathematical knowledge is two-fold. On the one hand the teacher is responsible for creating activities and situations (problems) that will reveal the underlying structures of numbers, operations, and relationships. On the other hand the teacher needs to actively encourage children to reflect on what they are doing and what they are thinking. The teacher should be helping them to express these ideas in words so that they can explain their actions to others, discuss their respective methods, and even argue about the validity of each.

Since the teacher is unable to teach logico-mathematical knowledge through direct instruction, one of the most important tasks of teaching is to design situations from which children can construct/develop their logico-mathematical knowledge. That is, the teacher needs to ask the question when designing a lesson/task: “What do I want children to learn from this situation/problem/activity?” Having established what the teacher wants children to learn she then needs to shape the situation/problem/activity in a way that will provoke children to “see” the patterns and structures. Then, both during and on completion of the activity, the teacher needs to facilitate reflection on the activity by the child –this reflection, more than anything else, will provoke the development of logico-mathematical knowledge.

### Knowledge types and the NumberSense Workbook Series

In the NumberSense Workbook Series we are very mindful of the different knowledge types and the relationship between them:

- Physical knowledge: In the early workbooks there are a lot of counting activities involving physical objects – the physical counting activities are critical for children to develop a sense of the muchness of number. When children physically draw solutions to problems (an early problem solving strategy) they using their physical understanding of the problem situation to solve the problem.
- Social knowledge: Throughout the workbooks social knowledge (vocabulary and conventions) is introduced in a deliberate way – often with a box.

For example:

Workbook 10, page 39:

Workbook 16, page 13:

- Conceptual (logico-mathematical) knowledge: The overriding approach adopted in the NumberSense Workbook Series is to present children with situations that they engage with. These situations are designed to reveal the underlying mathematical structure and through the reflection questions in the workbooks and the teacher’s all important role of facilitating reflection (see earlier discussion) children develop a deeper conceptual understanding of the mathematics they are learning.

### 4. Mathematics as an integrated/interrelated way of thinking

It is tempting to think that children learn to do mathematics in neat little compartments (curriculum topics). Such an approach is evident in programmes where children first learn to add and then to subtract, next to multiply and finally to divide – as if there is no relationship between these operations. These operations are however far more integrated than such an approach would suggest.

- The problem 5 – __ = 2 can be solved in at least three different ways: we can count down from 5 to 2 counting the steps; we can count up from 2 to 5 counting the steps; and we can subtract 2 from 5. The point is that the problem is not an addition or a subtraction problem, but rather a problem that can be solved using either addition or subtraction.
- Division can be thought of as repeated subtraction or repeated addition and multiplication is simply repeated addition.

In the same way that the basic operations are interrelated (as illustrated above) so most of mathematics is interrelated, for example:

- The number 24 can be thought of in a range of different ways:
- 20 + 4
- 30 – 6
- 6 × 4 and hence 3 × 2 × 4 etc.
- 48 ÷ 2
- and so on

- Fractions (at least at first) are the result of sharing (division).
- Tables, graphs, formula and equations are all different ways of representing relationships and any relationship can presented in all of these different ways, with each representation revealing different properties of the relationship.

Cognitive psychologist and neuroscientists describe the thinking associated with an awareness of the interrelated nature of mathematical concepts as constellatory thinking. They go on to make the point that in order to really understand mathematics it is important to be aware of the interrelated nature of mathematical concepts. This awareness can only be developed through experiencing mathematics as interrelated.

Of course, it is tempting to break mathematics up into discreet topic or concepts. This is useful for developing curricula and work schedules for teaching etc. However, the evidence is that children learn mathematics more effectively if they experience mathematics as an integrated whole.

### Mathematics, integrated thinking and the NumberSense Workbook Series

In developing the NumberSense Workbook Series we were guided by the content (curricula) that we wanted children to be exposed to, developmental learning trajectories for each of the concepts in the content list, and the interrelated nature of the content and concepts. All that to say that there is an enormous amount of planning that underpins the design of the NumberSense Workbook Series. However, in presenting the learning opportunities to the children, the workbooks are not organised by topics. There are no section headings and the same concept is not introduced and practiced for 5 to 10 pages before progressing to the next concept. This is deliberate. In the workbooks we recognise that mathematics is not made up of isolated concepts and mathematical thinking needs to be “constellatory”—aware of the interrelated nature of the subject.