*In 1995/1996 I trained as a primary school teacher. My specialist subject was mathematics. I became particularly interested in the way children develop the ability to “think mathematically”. One of assignments involved writing an essay on this subject, and subsequently presenting my ideas to my fellow students. Here is the essay that I wrote at that time. Warning – this is a long post:*

*As always, comments on this article are most welcome.*

## Mathematical Thinking in the Early Years

### INTRODUCTION

For some people mathematics is a source of joy as an object of elegance and mystery and as a tool of profound utility. For many others, however, mathematics is none of these things. School mathematics has been an area of poor performance for many children. Even those who are fairly successful may still find the subject both unpleasant and un-useful. A review of pertinent material (Open University, 1982, page 8 ) indicated that:

- Levels of attainment in mathematics has never been high,
- Children forget a large part of what they learn in mathematics,
- Children have difficulty in applying what they have learned to new contexts.

Sadly, negative attitudes to mathematics often persist into adult life, leading to a situation where there is, “a public perception of mathematics as precise, logical, cold, austere and (for all but a ‘clever’ minority) unattainable”. (Anghileri, 1995, page 55).

It has been suggested that one reason for these failures is that much mathematics teaching has focused children’s attention upon surface features of mathematical methods, rather than the principles which underlie these methods. This leads to ignorance of why particular procedures give the desired results. Under these conditions it is hardly surprising that mathematics becomes difficult to apply to new situations, difficult to remember, and virtually useless outside the immediate concern of surviving mathematics classes.

The solution would appear to be a switch to a style of teaching which emphasis the thinking behind what is visible, and which focuses on process rather than products.

### MATHEMATICAL THINKING

“Mathematical thought can have no success where it cannot generalise.”

(Charles Peirce, cited Mason, 1990, page 24).

#### Generalisation

A fundamental quality of all human thinking is our ability to see patterns of similarity and difference. As a result of this ability it is possible to move from concrete experience to generalities. “Generality…is a natural part of everyone’s experience. There are so many impressions which fill our senses that we need some way to categorise and organise them”, (Mason, 1990, page 16). Indeed, it would be impossible to function without forming generalities.

Mathematics begins with the expression of these generalities. “To participate in mathematical thinking, to appreciate the strengths and limitations of mathematics, it is essential to express perceived patterns and generalities, so that they can be considered, challenged, and where appropriate, modified. Conjectures need to be tested on other people; you need to try to convince others that generalities which you can see are perceivable by them. This is how mathematical thinking develops” (Mason, 1990, page 5). The result of expressing a generality is awareness of it, and it is that awareness which gives mathematics its power.

#### Classification

An important form of generalisation is classification. Classification of actions, objects, situations or whatever, can only take place where “certain aspects are stressed, and consequently others are implicitly ignored. (Mason, 1990, page 16).

In his discussion of classification, Mason (1990, page 16) pointed out that the way in which we classify is not just a result of features external to ourselves, but depends to a large extent upon what is relevant to the classifier. For example, “When you choose a pen to write with, your main concern is that it has ink in it. When you choose a pen as a present, you attend to quite different features. Two pens, which before were ‘no different’ or ‘equivalent’, are now quite different because one is acceptable as a present and the other is not. Much of mathematics can be thought of as making precise in the context of some question, what is meant by the phrases ‘is the same as’, and ‘is equal to’”.

NB. Mason went on to observe that “inappropriate stressing and ignoring often lie at the root of later confusion”.

#### Patterns

It has been said that “pattern spotting is at the heart of mathematical thinking” (Mason, 1990, 10). Children can be encouraged to seek out all kinds of patterns – in pictures, picture sequences, music, stories, their own movement etc. Word games and rhymes are also a rich source of pattern, and “children are usually good at detecting verbal patterns” (Mason, 1990, page 10).

Some patterns that primary school children are likely to meet are specific to mathematics. These are:

- Repeating patterns, eg. A B C A B C A B C …
- Sequences – a non-repeated series generated by a rule.
- Structural patterns – patterns in the relationships between numbers, symbols, etc.

It is only important to encourage pattern seeking, but also to “encourage children to become more articulate about patterns that they find, and to use them in their thinking” (Mason, 1990, page 9).

Action is one form of expression a pattern or rule, and the ability to follow a rule often develops before other forms of expression. It is often easier, for example, for children to place objects according to a rule than to say what the rule actually is.

#### Sequences

Mason (1990, page 25-27) stated that, “At the heart of recognition of sequences is recognition of repetition or development in time, which can be detected through behaviour such as copying, continuing and devising repeating patterns…For a list to be a sequence there must be more than a random collection of objects, words or events. There must be something which is seen to connect them.

“Connections are experienced by people, and do not lie in the objects themselves….meaning and connection are something that we bring to sequences as part of our attempt to organise and make sense of the world. What [we] stress, quite unconsciously and due to [our] past experiences and propensities, focuses [our] attention in particular directions.

“One difference between school and play is that in lessons, children’s attention is drawn explicitly to the presence of patterned sequences…Mathematical thinking can be fostered and developed by taking opportunities to children to think about what they are doing when copying, etc.”

The rule is more important than instances of the rule. “It is important to see and state a rule, for all sequences can be extended in more than one way, and children are especially good at seeing novel ways…a rule…should account for all the terms shown, and be capable of being extended indefinitely (Mason, 1990, page 36). Moreover, “children need to realise that they can never be sure what will come next in a sequence unless they know the rule for the sequence…the only way to be really sure how a pattern will continue is to know what the maker of the pattern is thinking! In other words, you need to know the ‘rule’”, and that, “a guess at a rule is completely wrong if just one example can be found which contradicts [it].” ( Anghileri, 1995, page 132-133). Consequently, then, “children are best served when directed to the source of the sequence, rather than just the sequence itself” (Mason, 1990, page 30).

The importance of sequence and pattern work is that it provides children with opportunities to express why things happen the way that they do. “Explaining why a sequence grows the way it does, or why some numbers behave the way they do, is well on the way to a formal proof”. (Anghileri, 1995, page 133).

#### Structural Patterns

Structural patterns arise as a result of the relationships between numbers (or symbols, or patterns or whatever). They are patterns of mathematical relationships, such as the relationships between whole numbers through addition, subtraction, etc.

These relationships are often discussed in the context of algebra, and for many that is where problems start.

#### Algebra

Algebra may usefully be defined as “a language for communicating and exploring mathematical relationships and methods of proving such relationships” (Anghileri, 1995, page 124).

It is with algebra, however, that many people get bogged down. As Mason (1990, page 32) has said, “Algebra is a watershed in many people’s experience of mathematics, the point at which they give up”. It is likely that the reason for this is that many people don’t really understand what algebra is about. “For many people, the word ‘algebra’ conjures up a forest of letters, numbers and ‘=’ signs, a memory of equations, and a collections of rules with mysterious origins for doing things to equations, such as solving them. It is most unfortunate that this impression is widespread, because it is similar to having the impression that cooking is about opening cans and pushing buttons on a microwave, or that exercise is about sweating and having sore muscles. The manipulation of symbols is only a small part of what algebra is really about, the traces that are left behind after mathematical thinking has taken place” (Mason, 1990, page 5).

The fact is that Algebra is something in which people engage naturally. “Algebra is an attitude, a fundamental power of the mind. Not of mathematics only” (Caleb Gattegno, cited Mason, page 5). Algebra is to do with expressing relationships between relationships. As such, algebra is essential, an extension of the basic ability to perceive generality in the first place. Indeed, Mason (1990, page 32) has stated that “algebra is not so much a topic as a way of thinking, a way of responding to and dealing with generality”.

As a natural power of the mind, it follows that mathematics is accessible to all, given that it is presented appropriately.

Mason (1990) states that there are four stages in the development of algebraic competence. The first two stages are important to KS1, particularly Stage I.

##### Stage I – “Say What You See”

Algebraic thinking begins with detecting patterns and expressing them. “There is no mechanical method, no rule; but on the other hand, pattern spotting is what human beings are good at doing” (Mason, 1990, page 35). Different people may see different things, or may express the same thing in a different way. “What each person sees is what they see…there may be many different ways of seeing things…discovering that other people see things differently…as you develop mathematical confidence, you begin to see more mathematical possibilities” (Mason, 1990, page 33). It is important to keep an open mind, therefore. “Flexibility is of most value, so that you can hear what others are saying” (Mason, 1990, page38). Ultimately, these different ways of seeing lead to expressing things succinctly and usefully.

Expressing the pattern may begin with copying or saying out loud what is seen. Gradually, children will learn to formulate and record that pattern in words, and as confidence increases, more and more succinctly until they are using traditional symbols.

##### Stage II – “Multiple Expressions and Brackets”

As a result of Stage I, children will have encountered the same pattern in different ways. This raises the possibility of manipulating expressions directly to see if they are indeed saying the same thing.

“There are two aspects to manipulation: the use of brackets to write down the way things are grouped, and the ways in which different expressions can be transformed from one to another, which really amounts to how brackets can be removed and inserted without changing the value of an expression…the manipulative aspect of algebra arises from deciding that there ought to be some rules for inserting and removing brackets so that you can get from one expression to another” (Mason, 1990, page 40,44).

For example, 3 + 2 = 5 and 2 + 3 = 5 both say the same thing in different ways, and there are rules for getting from one expression to the other. How do we know that they will always give the same answer? They say the same thing!

Mason sees this kind of thing as “typical mathematical movement…we began by counting and by manipulating numbers. Then we encountered generalities, which we expressed. Expressions can then become objects to manipulate. They can also become the elements of further generalising, and it is in this way that mathematics builds both its complexity and its usefulness” (Mason, 1990, page 45).

#### Stages III and IV – Generalised Arithmetic and Acknowledging Ignorance

Generalised Arithmetic involves becoming explicitly aware of the rules for manipulating numbers, dealing with brackets, and so on. “Algebra is probably most often thought of as generalised arithmetic – you do to the letters what you usually do to the numbers. But this only makes sense if you already know what it is that you do to the numbers! The roots of algebra as generalised arithmetic lie in children having their attention drawn to how they are doing computations rather than just to what the answer is” (Mason, 1990, page 46).

This is relevant at KS1 in that it is possible to invite children to attend to addition as a process applied, rather than simply provide a particular answer, so focusing attention on the generality underlying particular computations. This may be achieved by providing opportunities to express how calculations are done rather than always focusing on numerical answers. (Mason, 1990, page 53).

Acknowledging Ignorance refers to using a symbol (often “x”) to stand for an as-yet-unknown quantity, and expressing what is known about it – leading to the more x = 2x + 3 type of algebra which is more familiar.

### THE DEVELOPMENT OF MATHEMATICAL THINKING

NB. The ‘constructivist’ paradigm provides a framework for the following discussion.

#### Stages of Thinking

Researchers have been able to identify three stages through which children’s thinking develops. (Anghileri, 1995, page 41).

- Enactive – real experiences and thinking by doing.
- Iconic – using signs, diagrams, or objects to represent mathematical relationships.
- Symbolic – using symbols which are “disembeded” from the task.

This developmental sequence has implications for mathematics teaching. Firstly, it highlights the importance of involving children in practical tasks and in action. This is particularly significant when encountering new ideas, and so is vital throughout KS1. Starting points should be real-world, concrete problems which offer opportunities for direct manipulation of objects. As Atkinson (1992, page 12-13) says, all mathematical activity at KS1 should be “rooted in action”.

Secondly, children should move on to using structural apparatus and materials which will promote meaningful, useful images.

Thirdly, children can invent their own symbols to express and support their mathematical thinking. The advantage of using these symbols is that “children’s own symbols hold enormous meaning for them. For example, invented symbols for price labels in the class shop, introduced and interpreted by a child, can be accepted and understood by the class for several weeks”. (Atkinson, 1992, page 40, italics mine). Additionally, “children’s own inventive notations are likely to be far more appropriate in these early stages than the conventional symbolism of arithmetic” (Hughes, 1986, page 170).

Finally, only after a progression through the stages outlined above, children will move on to formal symbols. It is important not to move on to formal work too early.

#### Implications for Learning

The reasoning so far developed has further implications for teaching mathematics, and indeed other subjects as well. Teaching needs to begin with concrete examples from which generalisations can be made. As these are understood, they should gradually be applied to new situations and problems.

#### Children’s Natural Abilities and Prior Leaning

Children come into the classroom with natural abilities in mathematics. They are already able to see patterns, to generalise, to classify. Children have a natural curiosity, and a “strong urge to master the environment” (Floyd, 1981, page 71). Bird (1991, page 7) observed “reception children sustaining an activity by themselves, continually developing and controlling it”. Moreover, she also provided detailed descriptions of very young children engaging in a very wide range of activities, all of which demonstrated skills that are essential to the development of mathematical thinking.

Not only do children come to the classroom with natural abilities that underpin mathematical thinking, they also have a wealth of mathematical knowledge that they have accumulated in their home environment. This understanding is particularly solid, in that it was acquired quite naturally, in circumstances that made sense to the child.

As in other areas of learning, is is important that new understanding is build upon what is already known. Concepts which are not developed in this way cannot possibly have their roots in understanding. This has been one of the main areas of difficulty for children taught mechanical pencil and paper methods – child may appear to have an understanding because they can perform tricks with symbols, but the underlying principles are lost. Clearly, then, Mathematical Thinking should firmly rooted in a children’s natural ability to generalise, and its development will be most secure and most rapid if it is founded upon skills and knowledge that a child has already acquired – i.e. on ‘home learning’.

“Teachers need to become aware, if they are not already, of just what capabilities children bring with them to classrooms. They should be aware that they need not teach children how to do these things. Teachers need, rather, to be come aware of how to present maths to children so that these capabilities can be exercised naturally by children. In short, teachers need to educate themselves in how to assist learners to educate themselves…with respect to mathematics” (Floyd, 1981, page 125).

#### Children’s Own Methods

An alternative to teaching standard pencil and paper methods to children, many mathematics educators believe that it is more fruitful to encourage children to develop their own strategies. (e.g. Anghileri, 1995, page 39).

The main benefit of this approach is that when a child develops a particular strategy, it is self evident that the child must understand why that strategy works. Indeed as Floyd (1981, page 125) reports, “None of the maths now reflected in texts…came into existence in formal ways. The maths inventors doodles, made mistakes galore, agonised over problems for hours, days, weeks, even years, disposed of hordes of paper and chalk, and made haste slowly”. As children develop their own methods, they are engaging in Mathematical Thinking at the highest level, doing real mathematics.

There are other advantages, in that “The motivational aspect of allowing children the freedom to develop their own strategies is significant.”The learning we want is for children to use their own individual thoughts, their own experiences, their own sense of truth, their own time, their own doubts and queries and so on” (Floyd, 1981, page 126).

The power of this approach is reported by Atkinson (1992, page 46), who reports that, “children who are encouraged to use their own methods attain higher levels than children who are taught standard written algorithms first”.

Only later, when children have mastered their own methods, do children need to go on to more formal methods. By this time, they will have a solid understanding of what a particular problem or investigation is all about, and will also have developed a good grasp of many of the mathematical principles involved. “Obviously we want children to move on to new and more powerful strategies, but if these are forced upon children regardless of their own methods they will not only fail to understand the new ones, but will feel ashamed and defensive about their own” (Hughes, 1986, page 177). Encouraging children in their own methods shows that we value children’s own thinking.

In order to foster this kind of learning in the classroom, teachers need to encourage children to feel good about their own methods (Atkinson, 1992, page 46). This is facilitated in an atmosphere where the teacher promotes the ethos are no right or wrong approaches, and that mistakes are not seen as disasters, but as opportunities for learning. The teacher must accept what is done, without necessarily agreeing with it (Floyd, 1981, page 127). Atkinson (1992, page 46) warns that a particular concern may be minimising peer negatives. Ultimately the children themselves must decide whether or not to accept their own methods. The aim is to “Place the responsibility for correction on the learners…assume…that everyone is correct, not in what is said or written, but in the intent behind such outward evidence” (Floyd, 1981, page 127).

#### Collaboration and Communication

It is too much to expect individual children to invent mathematical methods unaided. One way to provide the support that children need is for them to work collaboratively. “Allow children to talk to one another, to check with each other” (Floyd, 1981, page 127). This necessitates children sharing their ideas with one another, a vital activity in itself, as “an essential part of the strategy for developing mathematical thinking is communicating one’s thoughts” (Floyd, 1981, page 68).

The kinds of dialogue which is encouraged is important. “Encourage two-way conversation in pairs, or with everyone, talking about what is not what should be” (Floyd), encouraging awareness of, and reflection on mathematical processes. (Anghileri, 1995, page 39), i.e. how things are done rather than on getting the answer. The most important skill in this context is that children (and teachers) need to be encouraged to listen.

NB. One of the barriers to understanding mathematics has been the language that teachers and children use to communicate mathematical ideas. Unfortunately, an analysis of these difficulties is beyond the scope of this report.

#### A Shared and Relevant Context

Children will best understand the mathematics which they are doing if it is set into a context that they understand. A situation which has meaning shared by a group of children will enable children to talk to each other and the teacher about that situation in language which everyone understands. This is fundamental, as mathematics is as much about expressing the meaning that is to be found in a context.

Other advantages of a shared context include:

- Pertinent features of a problem very according to context, that what is stressed or unstressed changes depending upon the situation. If a situation is not meaningful to a children, they will find it difficult to know what should and should not be stressed.

- A meaningful context may also help a children visualise a situation. This visualisation can give rise to useful mental images. Such images can then be manipulated either mentally or they could form the basis of a child’s written notation.

- A context that is meaningful to children avoids the pitfalls of contrasting common sense and mathematical logic. Some questions just don’t make humans sense to school children – and as a result can be somewhat confusing. “When a child gives a ‘wrong’ answer…it is always worthwhile trying to consider why she gave that answer (Anghileri, 1995, page 57).

- There is also a motivational aspect to meaningful mathematics. Children are more likely to work hard how many sandwiches are needed for a class picnic than how many sandwiches are eaten by train drivers. (See also, Atkinson, 1992, page 27-28).

Atkinson (1992, page 12-13) points out that mathematics is a powerful tool for interpreting the world. This suggests that it should be relatively straight forward to found much of children’s mathematical experience in real tasks, drawing on everyday situations and the whole curriculum. One area that I feel is neglected is the historical context of mathematics. I feel that children would benefit a great deal from understanding how and why mathematical concepts were developed in the first place.

### ACTIVITIES TO SUPPORT MATHEMATICAL THINKING

Having discussed the nature of mathematical thinking and the context in which it develops, it is useful to consider specific kinds of activities for their opportunities to develop essential skills.

Some kinds of activities lend themselves particularly well to the development of Mathematical Thinking. The key to promoting development in this area is not really the kind of activity, however, but that thinking processes are made explicit (Atkinson, page 12-13).

There are three main areas to consider:

- Harnessing at naturally occurring mathematical activity.
- Problem solving.
- Investigative mathematics.

#### Naturally Occurring Mathematics

To begin with, Mason (1990, page 22) asserts that “Pre-algebra is an integral part of the primary classroom already. By being aware of it, you can encourage children to develop more fully”. He goes on to say (1990, page 24) that ,”to prepare children for algebraic thinking does require less in the way of special pre-algebra tasks and activities and more in the way of recognising opportunities to develop their thinking as it happens”.

The key for Mason is that children should be called upon to “express generality” wherever possible.

Elsewhere, Mason (1990, page 22) states that, “Theme or topic-based work lends itself to many different opportunities for noticing that children are being called upon to express generality, and for drawing their attention to it. Then when they work on specifically mathematical generalising, it will not be seen as something strange and different”.

#### Problem Solving

An oft quote phrase is that “problem-solving is at the heart of all mathematics” (Anghileri, 1995, page 148), and problems certainly provide a great deal of scope for the development of Mathematical Thinking. The kind of problems presented to children is important, however:

- Problems should be rooted in meaningful contexts. Where possible, problems should come from children’s own questions. “An answer has more meaning to someone who has first asked a question”.

- Problems need to be accessible, allowing all members of the group to enter the problem easily, irrespective of their mathematical background.

- Problems should be suited to collaborative work, but should also offer scope for individual expression and the development for children’s own strategies. “[know-how in mathematics is] the ability to solve problems – not merely routine problems but problems requiring some degree of independence, judgement, originality, creativity” (Floyd, 1981, page 117).

Children will need guidance through the various stages of problem solving, and will need support to develop problem solving strategies such as simplification, etc.

#### Investigative Work

Another, closely related way in which more experienced children can explore Mathematical Thinking is through mathematical investigations. As Bird (1991, page 3) observes, “there is growing evidence to demonstrate that top infants, juniors and secondary pupils can work at mathematics in an active way, carrying out their own explorations”.

### CONCLUSION

Floyd, (1981m page 125) states that, “Maths is the birthright of all human beings”. If this is so then it is important that children develop an understanding of the processes of mathematics, rather than a collection of mechanical rules. Mathematical Thinking will only develop in an environment where, “know-how is more important than mere possession of information” (Floyd, 1981, page 11), where children’s own methods are valued, and where adults and children work together to develop their understanding of mathematical ideas. In this way, it is hoped that we can “help all children be joyful at their own maths for general living purposes, spiritually and practically” (Floyd, 1981, page 125).

### REFERENCES

ATKINSON, Sue (1992) “Mathematics with Reason: An Emergent Approach to Primary Maths”, Hodder and Stoughton, London.

ANGHILERI, Julia (1995) “Children’s Mathematical Thinking in the Primary Years”, Cassell, London.

BIRD, Marion H (1991) “Mathematics for Young Children: An Active Thinking Approach”, Routledge, London.

BURTON, Leone (1984) “Thinking Things Through: Problem Solving in Mathematics”, Blackwell, Oxford.

FLOYD, Ann (1981) “Developing Mathematical Thinking”, Open University, Milton Keynes.

HUGHES, Martin (1986) “Children and Number: Difficulties in Learning Mathematics”, Blackwell, Oxford.

MASON, John (1990) “Supporting Mathematics: Algebra”, Open University, Milton Keynes.

OPEN UNIVERSITY (1982a) “EM235 Developing Mathematical Thinking: Introduction”, Open University, Milton Keynes.

WOMAK, David (1988) “Developing Mathematical and Scientific Thinking in Young Children”, Cassell, London.

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