Number Theory: From Primary School; Through Secondary School; And Beyond.

The purpose of this essay is to examine how a student's understanding of number systems evolves from primary school, through secondary school and beyond. I recently observed a lesson on the concept of negative numbers. The teacher demonstrated the concept by using a temperature scale. The concept of negative numbers is a progression from the positive, or natural numbers that a pupil may first encounter prior to primary school. Orton (1996) explains how children encounter a variety of numbers before they begin school. Parents unconsciously introduce new ideas to their children in everyday situations. Orton and Frobisher explain that "nursery rhymes contain a wealth of number references, as well as other mathematical concepts" (1996, pg.92). They also argue that children attach meaning to numbers through their frequent use in different contexts (Orton and Frobisher, 1996). Thus pupils reach primary school with their own personal awareness and understanding of numbers. As pupils embark on primary school the National Curriculum requires pupils to learn the numbers up to a hundred (Department for Education, 2012). Pupils are also given the definitions of even and odd numbers. Orton and Frobisher (1996) suggest that pupils are comfortable when dealing with the natural numbers. As they advance, the concepts of negative numbers, fractions and decimals are introduced. The National Curriculum suggests pupils should consider fractions as parts of whole numbers (Department for Education, 2012). Teachers are encouraged to use fractions in the context of dissection of chocolate bars (Department for Education, 2012). The National Curriculum also suggests that decimals should be introduced using money as the context (Department for Education, 2012). Thus pupils should leave primary school with at least a contextual understanding of fractions and decimals. Orton (1996) states that pupils struggle with fractions and integers because they are not easily listed and to avoid conceptual issues pupils should experience various contexts. As student's progress to secondary school, pupils are encouraged to think of fractions as the ratio of two integers or the division of two integers (Department for Education, 2012). These concepts lead to the derivation of rational numbers and the converse leads to the derivation of irrational numbers (Orton and Frobisher, 1996). Research has shown that pupils have difficulties with understanding the concept of rational and irrational numbers because they tend to memorise the concept rather than understand it (Behr and Post, 1992). Pupils who study A-level further mathematics are introduced to the concept of imaginary numbers, initially via the study of solutions to quadratic equations (AQA, 2013). The real numbers and imaginary numbers form the complex numbers (AQA, 2013). Students who progress to read mathematics at university study the complex numbers in detail and the concept of infinity. Infinity is introduced in diverging sequences (Mond, 2010). Hannula (2006) describes the issues undergraduates suffer with the concept of infinity especially when distinguishing infinite decimals. To conclude, pupils' awareness of numbers broadens as they progress through education. Each new concept can be related to previous concepts. Initially students learn the concept of the natural numbers. As they progress through education they study the concept of the integers, rational numbers, irrational numbers, complex numbers and more. Students that read mathematics at university will have a better appreciation for the number system. However some of the less tangible concepts are still misunderstood.