Knowledge and Understanding 2

This report aims to examine the historical and cultural development of mathematics and how it stands today with the current decimal system and modern curriculum. Exploring the differences between numeracy and mathematics and also the recent introduction of functional skills, I shall also discuss the significance of zero and place value and how links and generalisations can be made between mathematical concepts. Furthermore, this report shall examine how teaching and learning theories influence the way in which we understand maths through the process of problem solving, generalisations, assumptions and proof, and also the common errors and misconceptions that may result from this.

THE HISTORY OF MATHS

From around 2000BC the Babylonian civilisation brought a style of mathematics which succeeded the Sumerian-Akkadian system following their invasion of Mesopotamia. As the oldest example of numeration that used place value systems, the Babylonians had an advanced number system now known as ‘base 60’. This was unlike the ‘base 10’ system that is in widespread use today, although time is still organised in this way. The Babylonians divided the day into 24 hours, each hour into 60 minutes and each minute into 60 seconds (O’Connor and Robertson, 2000). Like our own decimal system today which is a positional system comprised of nine special symbols and a 0 to denote an empty space, the Babylonians only had two symbols to produce their base 60 system. However, although the Babylonians had a fairly positional system, it also had some element of a base 10 within it. This is because the number 39, for example, was built from a unit symbol and a ten symbol (i.e. three tens and a 9). ‘Appendix A’ illustrates the 59 numbers built from these symbols.

Initially the Babylonians had a place value system without a zero feature for over 100 years. This need for the zero started to emerge when difficulties came in distinguishing between numbers such as 61, 601 and 6001. The only means to open to Babylonian scribe was to leave an empty space where a zero would be. However, the disadvantage of this was how was anyone to know if this was a single, double or multiple space, or even if these spaces were at the end or beginning of a number. To overcome this, the Babylonians invented another place marker: the zero. This was represented by a slanting double ‘corner’ sign (often known as a wedge). Although this ‘zero’ sign from around 500BC, functions as a zero, the Babylonians probably had no concept of zero as a number and instead used this symbol as a void (Gullberg, 1997, p56-57).

The Mayan civilization in Central America also developed a place value system in the first millennium. Using a ‘base 20’ system which used a dot for one and a line for five, it is assumed that the reason for this base 20 was because ancient people counted on both their fingers and toes. Furthermore, this civilisation even had a symbol for zero, which was used as a fully functioning number (Place Value System of Numeration, p4). ‘Appendix B’ shows how the Mayans constructed numerals.

Moving onto the Egyptian history of maths, this civilisation used sophisticated mathematical techniques for devising calendars, buildings, administration and accounting. However, their skills were more focused upon solving real world problems, rather than discovering principles of maths. This was unlike the Greeks and the need for mathematical proof (Shuttleworth, 2010). To represent numbers the Egyptians used a numerical grouping system where 1 was represented with an ‘|’. The drawings below illustrate the numerical hieroglyphs used at around 3000BC:

To create the number 276 or 4622, the following combinations would be used:

(O'Connor and Robertson, 2000)

As well as basic number systems the Egyptians also introduced the concept of fractions. If a unit fraction is in the form of 1/n, where n...