Introduction to the Early Years SelfStudy Modules
In this module you will:
 reflect upon your own mathematical experiences
 examine the nature of mathematics and why it is important.
At the end of this module you will:
 have examined the effects of your mathematical experiences
 have developed a deeper understanding of what mathematics is and be able to suggest why it is important
 be able to offer a definition of mathematics.
Most people have some recollection of their school years, however long ago it was. Strong emotions help to fix memories, so it is likely that remembered events are either positive or negative, rarely neutral. Some recollections are likely to be about people, often something a key figure said or did. Others may be of an event, large or small.
"I remember being chosen to take part in a trial of some new mathematical materials at primary school. I was 7 or 8 and the materials were Dienes blocks. I don’t know why I was chosen, but I loved exploring place value and different bases. I still feel very comfortable with this area of mathematics and enjoy using a range of models and images in my teaching.
"My first secondary school maths teacher actively encouraged me and I began to consider myself to be ‘good at maths’. When I moved to a new school, everyone was terrified of the maths teacher and when I put my hand up to tell her about a mistake on the board, everyone tried to tell me to put my hand down. But I knew I was right and persevered. We rubbed along pretty well after that and I was never afraid of her, unlike everyone else. I continued to love maths and felt I had a good understanding too. My last school mathematics memory is of my A level teacher. I struggled with calculus at first and I have never forgotten her sarcastic comment, ‘Oh, you got there first for a change!’ That really dented my confidence and I took some years to recover it. It was only when I began an Open University course in mathematics around eight years later that it all came flooding back and I rediscovered the joy of mathematics."
Primary teacher
"I enjoyed maths at school. It came quite easily to me and I gained huge satisfaction from pages of ticks. However, looking back now, I actually feel a certain amount of anger. My experience of mathematics was a very narrow one, characterised by explanation and practice. I don't remember having to think in maths lessons. I got by very nicely simply by having a good memory.
"This was brought home to me in my first year of primary teaching. The time had come to teach my class the decomposition algorithm for subtraction. I suddenly realised that I had no idea why it worked. I had a little mantra that I'd recite in my head: "cross off the zero and write a nine, cross off the number to its left ..." Could I really teach a stepbystep process in this way simply as a recipe? I decided that I couldn't and for the first time ever, I sat down to work out why the algorithm worked and what I was really doing by crossing out a zero and writing a nine.
"That was certainly not the last time that I realised quite how much I had relied on my memory at school. What about my classmates who didn't have good memories? I suspect that maths had been completely inaccessible to them. How I wish that, first and foremost, my teachers had focused on understanding."
Primary mathematics consultant
Pretask activity
Think back to your own experience of mathematics at school. Are your memories positive or negative? Who are the key players in your mathematical memories? How have those experiences impacted on your life and in particular, your feelings about mathematics? Has your attitude to mathematics changed since your school days? If so, in what way?
Record your reflections in your Learning Journal within your Personal Learning Space.
Being Stuck
Being stuck is an honourable and valuable state: it's the best way to learn something! Being stuck is often an indication that a learner is pushing the boundaries of their learning and trying to learn something new. Learners need to have strategies that help them to get unstuck. Can you remember a time when you were stuck when working on some mathematics?
Read the first three paragraphs of the Mathemapedia entry Being Stuck. Use the two questions suggested in paragraph 2 of the article  What do I know? What do I want? – to help you to tackle the Nine pin triangles activity in the NRICH article, I’m Stuck! After you have explored the problem, read the rest of the article.
Did you get stuck? How did that feel? What did you do to become ‘unstuck’? How will you use your experience to help yourself in future?
Record your reflections on being stuck in your Learning Journal within your Personal Learning Space.
Heather Cooke offers the following advice on being stuck on pages 119 – 120 of her book, Success with Mathematics (2002):
“Being 'stuck' is an honourable state and there are a number of strategies that can make this a positive experience rather than simply a negative feeling.
 Acknowledge that you are stuck  relax and recognise that this is a learning opportunity. Different people develop different strategies for dealing with being stuck. Whatever you do, do not panic.
 Next, try to identify exactly why you are stuck. This process is, in effect, identifying what you already know and what you still want. Doing this can sometimes be enough to see a way of building a bridge between know and want...and so become unstuck.
"Here are some possible strategies.
 If the question seems too complicated or too general, try simplifying it in some way. For example, break it down into a subset of smaller problems or rewrite it using simpler numbers or easier words.
 If there does not seem to be enough information, list what else you think you need. (Some problems may deliberately not have enough information included). Sometimes you may find that you do have the information but it was not in quite the form you expected.
 Tell someone: in trying to explain, you may find that you stress and ignore different parts of the problem and so be able to view it in a new light. Even if there is no one around to help, just saying something out loud to yourself can help considerably  saying it 'in your head' is not as powerful.
 Use the solution if available: you may only need to read a little before you can see what is needed and can continue on your own.
 If you are still stuck, still do not panic: you may need to take a break and do something quite different. Simply freeing your attention can 'unblock' the problem.
 If nothing seems to work, skip over the problem area for the moment. Later studies may help.
"The way to make being stuck a more positive experience is to notice not only what helped to get you going again, but also what led you to getting stuck in the first place. This 'learning from experience' is then available to you for use in future situations.”
What is mathematics?
In the activity above, you considered your experiences of school mathematics. But what is mathematics?
Maths is the study of patterns abstracted from the world around us – so anything we learn in maths has literally thousands of applications, in arts, sciences, finance, health and leisure!
Professor Ruth Lawrence, University of Michigan
Mathematics is not just a collection of skills, it is a way of thinking. It lies at the core of scientific understanding, and of rational and logical argument.
Dr Colin Sparrow, Lecturer in Mathematics, University of Cambridge
Knowing mathematics, really knowing it means understanding it. When we memorise rules for moving symbols around on paper we may be learning something but it is not mathematics. Knowing a subject means getting inside it and seeing how things work, how they are related and why they work like they do.
If they (children) spend most of their time reflecting on how various ideas and procedures are the same or different, on how what they already know relates to the situations they encounter, they are likely to build new relationships. That is they are likely to construct new understandings.
J. Heibert, 1997 – Making Sense:Teaching and Learning Mathematics with Understanding
Mathematicians have only minority audiences, consisting mostly or perhaps entirely of other Mathematicians. The majority have been turned off it in childhood. For these, the music of mathematics will always be altogether silent.
Skemp, 1987
Maths is the truly global language. With it, we convey ideas to each other that words can’t handle – and bypass our spoken Tower of Babel.
Professor Alison Wolf, Head of Mathematical Sciences Group, Institute of Education, University of London.
Activity 1
How would you describe mathematics? What do you consider is vital to knowing about mathematics? What would you describe as mathematical thinking?
Record your reflections in the Learning Journal within your Personal Learning Space.
So what is mathematics?
The National Strategies Guidance paper Mathematics and the primary curriculum offers the following definition of mathematics:
Mathematics is an old and wellestablished subject. It has many rich, historic legacies. Mathematics has been studied by most ancient cultures and its importance in modern society continues. Over time the focus of study has changed in line with the changing needs of society. Who would have predicted that the early number theory of the Greeks would inform the codebreaking work undertaken during the Second World War, or become a tool used on credit card transactions to strengthen security in the financial world?
Mathematics builds from simple definitions and propositions that are based on observation
For example, shapes that can be seen, touched and made in the real world are usually imprecise. Mathematics helps to define them and give them structure. The definition of parallel lines, equal angles and similar and congruent shapes has created a geometry in which to think and explore. Propositions about shapes have led to new observations and proven properties that designers and artists use in construction. The imagined world of precise shapes has influenced the world in which we live. Children enjoy living in imagined worlds and creating stories that take the real world to extremes, rather as mathematics has done down the ages.
Mathematics starts from a desire to explain the real world
For example, the number system used today started because of the need to record quantities of objects or events. Counting the passage of time in days or weeks led to the development of the calendar and the ability to predict the phases of the Moon and changes in tides. This informed our understanding of the movement of planets and, ultimately, led to the ability to put a man on the Moon. The natural inquisitiveness of children and our desire to explain the world around us is the start of this adventure.
Mathematics involves measuring, comparing and classifying objects
For example, local trading led to a need to standardise quantities to replace a system of bartering. The standardisation grew as access to new worlds extended trade. Mathematics provided a tool with which to quantify the standard measures and, from this, abstract the process of measuring that is now applied to quite disparate outcomes. Measures of chance and risk, measures of public opinion and measures of climate change are all developed from the application of mathematical ideas. Within mathematics there are measures of chaos, the infinite and discontinuities: all stimuli to a vivid imagination.
From an early age, children compare and classify items by smell, taste or colour. They begin to start to measure by comparison or by counting the numbers of objects; the playground is alive with the language of comparison, 'more than', and classification, 'same as', and counting games where the outcomes are measured and compared. This naive use of mathematics reflects its development and historic legacy.
Mathematics describes patterns, properties and general concepts
For example, even the symbol for three (3) represents a general concept. It can be a label, peg number 3, a position, third in line, or a quantity, Three Little Pigs. The early identification of numbers that had particular properties led to a detailed study of number theory. Many mathematical hours have been spent trying to derive a formula to predict the prime numbers  but still without success. However, the search for such formulae has provided a language and symbolism in use today. There are odd, prime, square, triangular and perfect numbers. Mathematics, too, has built up a vocabulary to describe patterns and properties in shape, such as 'symmetric' or 'regular'. Children use the terms reflection, rotation and translation to describe pattern, movement and properties of shapes. They acquire general concepts such as quadrilateral, equal to or perimeter through experience, even though the general concept may not have been defined. Children's ability to extract the essential properties and generalise from particular cases is a key skill in mathematics.
Mathematics provides the tools to abstract and work in an imagined world
For example, counting objects, then recording the number and calculating 50% by dividing the recorded number by two moves from a real world activity into the mathematical world. The world that mathematics provides is a representation of the real world, where objects may be represented by symbols, diagrams or statements. To identify the whole numbers in a set as 19 < m < 81 and m  2, or the even numbers from 20 to 80 inclusive, is easier than listing the numbers. Having set out the numbers involved, the representative 'm' can be manipulated and controlled in the mathematical world. Telling children that this particular triangle is isosceles because it has two equal sides conveys the abstraction that all isosceles triangles have two equal sides. When children carry out mental calculations they are working in an abstract world of mathematics. Even using an empty number line still requires imagination. Children visualise a generalised solid shape from a 2D representation and they can describe the properties of the 3D shape from the image. Such visualisation and generalised thinking are essential skills in mathematics.
Mathematics is a creative subject in which ideas can be generated, tested and refined
For example, making observations about the properties shared by a group of objects, inducing general statements from particular cases and making deductions from evidence about the properties of shape are all creative activities. Euclid's geometry created a whole set of propositions and theorems and challenges involving the use of a pair of compasses and a ruler to make particular shapes such as 'squaring the circle'. Later, geometry took new directions, based on assumptions about parallel lines and, in the case of topology, abstracted the geometry in a world where measurement was involved. Like the underground map, the route is clear but no measurements can be taken. These developments resulted from someone asking the question: 'What if…?' Children like to ask 'How…?', 'Why…?' and 'What if…?' questions and to be directed to solutions or be given ideas they can use to find out the answers. Questioning assumptions and conclusions, and testing to see when and why they are valid, is a cornerstone of mathematical thinking and something children enjoy.
This and other guidance papers may well be archived, but an internet search should enable you to locate them if you wish to read the full paper.
Why is mathematics important?
The mathematics developed in the current century will be the basis for the technological and scientific innovations developed in the next one. The thought processes, the ways of looking at things, the habits of mind used by mathematicians, computer scientists and scientists will be mirrored in systems that will influence almost every aspect of daily life. To empower children for life after school, we need to prepare them to be able to use, understand, control and modify a class of technology that doesn’t yet exist. That means we have to help them develop genuinely mathematical ways of thinking.
The following extract from the same guidance paper as above, Mathematics and the primary curriculum, offers these comments on why mathematics is important:
A report into mathematics education, Making Mathematics Count, 2004, by Professor Adrian Smith, offers a number of reasons why mathematics is important. In one of the paragraphs the report outlines the importance of mathematics to modern society:
Mathematics is of central importance to modern society. It provides the vital underpinning of the knowledge economy. It is essential in the physical sciences, technology, business, financial services and many areas of ICT. It is also of growing importance in biology, medicine and many of the social sciences. Mathematics forms the basis of most scientific and industrial research and development. Increasingly, many complex systems and structures in the modern world can only be understood using mathematics and much of the design and control of hightechnology systems depends on mathematical inputs and outputs.
Smith, 2004. Making Mathematics Count
For most of us, our awareness of the role of mathematics in modern society is as a user of the end products rather than from a direct involvement in the process. Children seek the latest electronic goods and adults benefit from the medical advances that result from those processes listed above. But the report also offers a more fundamental set of reasons for children and adults to learn mathematics:
The acquisition of at least basic mathematical skills  commonly referred to as "numeracy" is vital to the life opportunities and achievements of individual citizens. Research shows that problems with basic skills have a continuing adverse effect on people's lives and that problems with numeracy lead to the greatest disadvantages for the individual in the labour market and in terms of general social exclusion. Individuals with limited basic mathematical skills are less likely to be employed and, if they are employed, are less likely to have been promoted or to have received further training.
Smith, 2004. Making Mathematics Count
This more pragmatic viewpoint is one to which all primary teachers can subscribe. The primary curriculum is the point at which children's acquisition of these basic mathematical skills starts and, for most children, it is a very successful experience. They leave primary schools well equipped to progress and succeed in the future labour market. But this pragmatic view, while important, does not say anything about how a study of the subject might improve individual children and help them to become more rounded persons. Here is a final quote from the report:
Mathematics provides a powerful universal language and intellectual toolkit for abstraction, generalisation and synthesis. It is the language of science and technology. It enables us to probe the natural universe and to develop new technologies that have helped us control and master our environment, and change societal expectations and standards of living. Mathematical skills are highly valued and sought after. Mathematical training disciplines the mind, develops logical and critical reasoning and develops analytical and problemsolving skills to a high degree.
Smith, 2004. Making Mathematics Count
Here, the justification for mathematics being important is in its ability to develop and support children's thinking, reasoning and problemsolving skills. The skills embedded in mathematics and the discipline of learning and using mathematics provides children with other cognitive skills that they can use across and beyond the school curriculum. The training received through the study of mathematics provides children with skills that are in high demand. The ability to analyse information and to solve problems are key skills embedded in the primary curriculum, within which mathematics has a significant role to play.
In summary, mathematics makes a significant contribution to modern society; the basic skills of mathematics are vital for the life opportunities of our children; and mathematics develops the mind and those highly valued cognitive skills.
Watch Shift Happens on YouTube.
Activity 2
Did you find anything in the above section surprising or challenging? Have you changed or extended your thinking on what maths is?
Revisit your reflections in your Learning Journal within your Personal Learning Space.
Mathematical thinking is important for all members of a modern society as a habit of mind for its use in the workplace, business and finance; and for personal decisionmaking. Mathematics is fundamental to national prosperity in providing tools for understanding science, engineering, technology and economics. It is essential in public decisionmaking and for participation in the knowledge economy.
Mathematics equips pupils with uniquely powerful ways to describe, analyse and change the world. It can stimulate moments of pleasure and wonder for all pupils when they solve a problem for the first time, discover a more elegant solution, or notice hidden connections. Children who are functional in mathematics and financially capable are able to think independently in applied and abstract ways, and can reason, solve problems and assess risk.
Mathematics is a creative discipline. The language of mathematics is international. The subject transcends cultural boundaries and its importance is universally recognised. Mathematics has developed over time as a means of solving problems and also for its own sake.
John Mason is Professor of Maths Education at the Open University. John strongly believes that generalisation is the key to negotiating through the world; seeing the general through the particular, and seeing the particular in the general.
Listen to John Mason reflecting upon The Nature of Mathematics in the audio file included in the Excellence in Maths Leadership materials (length 10.23). As you listen, make some notes to support your reflection on what John has to say.
John talks about threee worlds: the material, virtual and symbolic. He believes in the importance of the power to imagine, and the power to express what we imagine.
Activity 3
Do you ask your children to imagine in mathematics? And then express what they have imagined? Young children usually have vivid imaginations, but may not have used them in a mathematical way before. Consider how you might encourage children to do this. Plan a short activity to carry out with a small group. How do the children respond to the activity?
Record your reflections on John’s comments and Activity 3 in your Learning Journal within your Personal Learning Space.
Digging Deeper
If you would like to find out more about mathematics as a habit of mind, read the Mathemapedia entry Habits of Mind.
Alternatively, download a more detailed article, Habits of Mind: an organizing principle for mathematics curriculum by Al Cuoco, E. Paul Goldenberg and June Mark.
Meet with colleagues online in the Early Years Forum to discuss issues which are of importance to you in the light of your reading.
Record your reflection on this introductory module in your Learning Journal within your Personal Learning Space.
Now that you have completed this module you will:
 have examined the effects of your early mathematical experiences.
 have developed a deeper understanding of what mathematics is and be able to suggest why it is important.
 be able to offer a definition of mathematics.
 be ready to move on to study any of the other early years modules.
References
 Cooke, H. (2002). Success with Mathematics. London, Routledge.
 DfEE (1999) Mathematics, The National Curriculum for England. Norwich, HMSO.
 Hiebert, J. et al. (1997) Making Sense: teaching and learning mathematics with understanding. Portsmouth NH, Heinemann.
 Skemp, R. R. (1987) Psychology of Learning Mathematics. New York, Routledge.
 Smith, A. (2004) Making Mathematics Count: the Report of Professor Adrian Smith's Inquiry Into Post14 Mathematics Education. Norwich, HMSO.
 The National Strategies Guidance paper: Mathematics and the primary curriculum, accessed February 2011.
