So what is mathematics?

Mathematics is an old and well-established 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 code-breaking 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 2-D representation and they can describe the properties of the 3-D 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.