Introducing mathematical concepts in a class can cause lots of moans and grumbling, but only at first. This is because there are tools and methods available for teachers in order to make the students interested in learning maths. Math is absolutely necessary for students to learn because they use it so often every day of their lives. By empowering children to learn math; calculating numbers can become a natural part of their day instead of something to fear or to avoid. Interestingly children have various personal strategies when it comes to the basic concepts of mathematics. Mack (2011) gives an example of how children offered a whole range of different ways of expressing the information symbolized by the number 237. Some of the children were able to see groups of numbers - seven 1s, three 10s, and two 100s- while others were only able to express the very basic addition equation - 200 + 30 + 7 = 237.
Children have different learning styles so a teacher needs to find strategies that will engage all the students in learning. In order to engage and empower young students in the number systems of ancient civilizations has been used not only to catch and hold their interest but also to engage them in developing thinking skills. Ancient civilizations used symbols to keep track of daily activities that need counting. For example, how many sheep in a herd or the number of ceramic vessels filled was done with tally marks. Tally marks developed progressively until the system used today which is Hindu-Arabic. Teaching the concept of ancient number systems has been shown to be a successful way for children to become interested and comfortable with modern day maths.
Development of Hindu-Arabic
Ifrah (2000) has described the very beginning of number systems which began approximately 5000 years ago. It was based on the additive principle which is the “the rule according to which the value of a numerical representation is obtained by adding up the
values of all the figures contained” (Ifrah, 2000, p. 325). The first peoples to use maths counted with the digits on their hands and feet, their fingers and toes. When larger numbers were needed people used their knuckles and other parts of the body. People also made notches on wood or pieces of ceramic to keep track of changes in the moon throughout each month. Then number systems then developed into the use of pictograms and other symbols. The early Egyptian hieroglyphics became unwieldy and almost impossible to handle the larger the numbers became. So some cultures including the Romans, Greeks and Etruscans each had number systems in a base 10 system which helped make the systems more efficient. Changing to base 10 helped but large numbers were still difficult to handle so added separate symbols for 5, 50, 500, 5,000, etc. were added to the number systems. For example, the Roman Numeral ‘V’ was added for five, ‘L’ for fifty, ‘C’ for 100, ‘D’ for 500 and so on. But that still did not solve the problem of having to copy the same symbol over and over again for large numbers.
According to Ifrah (2000) the Assyro-Babylonians and Aramaeans added the “multiplicative principle” for numbers larger than 100. For numbers less than one hundred they continued to use only the additive principle. Unfortunately these developments did not end the need for repetitive copying of symbols. For example, to write 7,009 the symbols were [(1+1+1+1+1+1+1) x 1,000] + (1+1+1+1+1+1+1+1+1). (Ifrah, 2000, p. 331) The problem was that symbols did not have any interaction with any other symbol. So, in order to solve the problem mathematicians in India started developing a number system that recognized the importance of the position of numbers; what we call the place value. Approximately 2000 years ago the Indians developed the number system which is the basis for what we call the Hindu system.
The Hindu number system was used by travelling Arab traders plus Arabs made improvements in the systems. It is still a more practical number system than others because the natural numbers are from 1 to 10, (just like our fingers), the all important zero is available to use to fill empty place values, and the position of a digit (a number) determines its place value. The numbers used almost universally today are based on the ancient Hindu-Arabic number system. Zero is valuable as a place holder. (Haylock, 2010, p. 75) Haylock (2010) defines place value as “the principle underpinning the Hindu-Arabic system of numeration in which the position of a digit in a numeral determines its value; for example, ‘6’ can represent six, sixty, six hundred, six tenths, six hundredths, and so on, depending on where it is written in the numeral” (p. 82).
Table 1 below displays the ordinal numbers of the modern Hindu-Arabic system; the ordinal numbers are the words naming the numbers 1 through 9. The middle row in the table shows how the numerals were written in ancient times. Notice that 7, 8, and 9 are the same as we use today and the number for 1 is similar to the modern numeral for 1.
Table 1. Ancient Hindu-Arabic
Hindu-Arabic Compared with other Ancient Number Systems
Keeping a tally in order to keep track of items was used in ancient times. We know this from the archaeological records. Although calculators have become common for working math problems there is nothing (except maybe fingers and toes) that can keep a tally as well as ancient tallying systems. For that reason people around the world still use the tally number system. A tally is made by writing a mark, such as a short straight line, for each thing being counted. After four lines have been marked, the fifth thing being counted is designated by drawing a straight line through the first four. By creating visually distinguishable subset of 5 the total becomes faster to count.
Table 2 compares some of the ancient number systems with Hindu-Arabic. The systems listed used base 10, base 20, base 12 and base 60 to try to make the number systems -easier and more efficient to use. Finally base 10 was kept for the Hindu-Arabic number system that we use now.
Ancient number systems compared with Hindu-Arabic
Ancient Egyptian B hieroglyphics was an additive system as represented on stone engravings. Egyptian B is additive because the powers of the base number (10 or 12 depending on the base used) can be repeated as many times as necessary. Bennett and Nelson (2006) explained that “b = 10 so the powers of the base are 1, 10, 102, 103, etc.” (p.43). Human beings have ten fingers so it is not surprising that base 10 has been commonly used as a base for counting systems. The ancient Egyptian B numbers are written from the right to the left with the largest number at the furthest right and the smallest number at the furthest left. This is the opposite from the way we write the Hindu-Arabic numbers today. The Roman Numerals used what we identify as letters to symbolize the number of items being counted. The Romans tried using base 10 and also base 12 to make writing large numbers less cumbersome. Roman Numerals were developed in approximately 500 BCE. One of the strategies developed was to draw a horizontal line over number to mean ‘multiply.’ The Romans also introduced a subtractive element. For example, the Roman Numeral IX means subtract one from ten so ‘IX’ stands for nine items.
In modern Hindu-Arabic the ability to calculate was greatly enhanced by using place values so that the position of a unique number designates its place where it can interact with the rest of the numbers. In the number 423 we immediately see that there are 4 one hundreds, 2 tens and 3 ones. If we want to add, multiply or subtract 423 with another number the 423 always stand for the same amount because of their position. Their position designates their place value. Fewer numbers are needed because the place value designation was adopted.
Table 3 offers a quick look at how different symbols were used in some ancient number systems compared to the numbers (symbols) we use today. The example in Table 3 displays the modern symbols used to write twenty three, 23. The Romans, Egyptians, and Greeks all used five symbols for the number 23, two identical symbols for twenty (2 tens) and three vertical straight lines for the three ones. The Mayans used a drawing of a shell to
Number Twenty Three
symbolize twenty items and three straight lines for the three ones. Some ancient number systems used horizontal lines to symbolize numbers; an example from China is shown in Table 3; two crossed t-shapes stand for 2 tens and three vertical lines that get smaller from bottom to top stand for the 3 ones.
Teaching Math to Empower
How many children in the class have a birthday in the same month, how many sports fans could travel to the sports tournaments if only two buses are available, recycling how many aluminium cans are needed to earn $150? Problem solving is an unavoidable part of everyone’s daily life. Using ancient symbols to teach math in once a week, hourly sessions, has been found to “help(ed) students grow in their knowledge of place-value ideas, (understand) equivalent representations for numbers, and (improve) mental computational fluency” (Mack, 2011, p. 102). “Fluency refers to having efficient, accurate, and generalizable methods (algorithms) for computing that are based on well-understand properties and number relationships” (NCTM, 2000, p.144). Table 4 lists the teaching goals for teaching ‘Every Day Maths.’
Every Day Maths
Haylock and Manning (2010) have examples of math exercises that can easily be adapted to ancient number systems. Numbers and place values can be taught starting with teaching sets and subsets of numbers. A number line is the set of all countable numbers so smaller groups of countable numbers are subsets of the number line. Number lines can be made in other number systems or learning place value can also be done. For example children can be asked to write 3 + (5 x 10) + (6 x 102) + (7 x 106) or 5 x 107 in Roman Numerals or ancient Egyptian. (Haylock & Manning, 2010, p. 1) Place values can be taught using the example of a number system that does not use place values in order for students to understand the difference between using place value and not using place values. Haylock and Manning (2010) suggest that “to appreciate efficiency and elegance of the Hindu-Arabic place-value number system we use, it is instructive to compare it with number systems that were not based on the principle of place value” (p. 3). The system of Roman Numerals is a good example to use as a contrast to the Hindu-Arabic system.
Bennett and Nelson (2006) have explained that “In the middle grades it is essential that students become comfortable in relating symbolic expressions containing variables to a verbal, tabular, and graphical representations of numerical and quantitative relationships” (p. 43). This is another opportunity to use ancient number systems for teaching. The NCTC (2000) defines fluency as “having efficient, accurate and generalizable methods (algorithms) for computing that are based on well-understood properties and number relationships” (p. 144). Table 5 explains the goals desired from teaching computational fluency.
Table 5. Computational Fluency
Computational Fluency
Deep understanding of computational methods
Able to use more than one method to problem solve
Able to develop strategies on their own (naturally)
Algorithms (process of problem solving, series of calculations)
Able to chose method for context and purpose appropriate
computations: (a) mental computation, (b) estimations,
(c) calculators, (d) paper and pencil
Ref. adapted from NCTC , 2000
Conclusions
This paper has explored the history of the development of the Hindu-Arabic number system which we use today. A comparison has been made to other systems such as the Roman Numerals to demonstrate that learning about ancient number systems makes sense for students. Using ancient number systems to teach mathematics has been shown to engage children in thinking about problem solving. Comparing ancient number systems can be used to teach concepts such as equivalent relationships between symbols that look different. Using every-day math works well when children have been taught the root meaning of numbers such as the importance of place value in understanding quantities and how numbers interact with each other. The ideal goal is to give students a mental computational fluency they can use every day and throughout their lives.
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