¶ … Base ten number system and on common misconceptions, which young children might develop when trying to learn about the use of numbers. What is Place Value Notation

Place value notation, also otherwise known as positional notation, is a system of encoding numbers such that it simplifies the arithmetic. It is the basis for understanding arithmetic and is essential to the way we read, write, speak or use whole numbers. Another important use of the place value notation is that it allows us to string together and make sense of a sequence of whole numbers put together in order. Quite simply, the place-value system allows us to make sense of a sequence of numbers which would otherwise have just appeared to be random digits put together.

What is the Base Ten number system?

The Base ten number system uses digits for the numbers zero to nine for all number values no matter how small or large. The number in each position has a value ten times greater than that to its right while zero is used as the placeholder. The position of each digit denotes the quantity which it represents -- tens, hundreds, thousands, ten thousands and so on. Thus, the value of the digit multiplied by the power of its position gives us the quantity which the digit represents (Kennedy, Tipps, & Johnson, 2008).

What is number sense and what does it help us achieve

Number sense refers to a person's understanding of the numerical system including their ability to make numerical judgments and solve complex mathematical problems. It is the way we visualize number patterns and develop an understanding of their positional hierarchy as the value moves from tens towards the thousands. For students of mathematics, it is not only important to learn about the magnitude of every quantity and place in the system, but also the relationship that exists between each place (Sue Willis, Jacob, Powell, Tomazos, & Treacy, 2004).

A heightened place value sense allows students to effectively group numbers together and understand their significance. It also helps in allowing us to easily divide numbers into different parts in order to make calculations. Take the following example into consideration:

257 + 312 = ?

200 + 300 = 500

50 + 10 = 60

7 + 2 = 9

500 + 60 + 9 = 569

A developed number sense can also help students achieve endless calculation possibilities, among which are the following:

Count, read, write, recognize, order and understand number sequences.

Solving problems including addition, subtraction, multiplication and division of numbers.

Link, arrange and convert between decimals and fractions.

Solve complex calculations involving fractions and decimals.

Understand and use the properties of odd and even numbers

Be able to develop mental and written problem solving strategies and use digital technology where appropriate

Essentials ideas that need to be taught to children about place-value

As mentioned earlier, the value of each place increases by a power of ten as we move from right to left. Thus, it is important that children, when learning the number system, are taught that numbers are grouped together into tens, hundreds and thousands so that it becomes really clear that numbers are not to be learnt as a never-ending continuum.

Secondly, in order to help children develop a thorough understanding of the place-value system, we need to realize that it is a slow and gradual process and thus must treat it as such. To begin with, children should be taught to count using small numbers and be able to name these numbers while being able to perform simple additions and subtractions using these small numbers. It is vital that they understand the importance of counting in groups so as to understand that it comes in handy when counting a large number of objects. Then, slowly, they can work their way up by counting in twos, five, hundreds...

(Liedtke, 2010).
Use of physical objects such as rubber bands, pebbles or poker chips can also be a very effective way of teaching children about the numeric system.

Common misconceptions that children might develop when learning the place-value system

When dealing with children, it is important to be prepared that they will land into many misconceptions about the number system and will need time to adjust to it. When making the move from counting small numbers to handling larger, more complex numbers, they often come unstuck.

For example, moving above 10 and entering into the teens presents the first major challenge to children. The children often cannot grasp the concept that '13' is to be pronounced thirteen while '23' if pronounced twenty-three. They often cannot make the link as to why numbers between 10 and 20 have a different pronunciation while all others follow a different format. Often children ask that if 2 and 3 make 23, then why 1 and 3 can't be pronounced as onety-three and so on.

Another obstacle that often presents a major challenge to children is when they attempt counting beyond 100. An example here would be that since 116 is spoken as one hundred and sixteen, then the obvious deduction would be to write it as 100 and 15 hence 10016. It is important for teachers to clarify the concept of ones, tens and hundreds and how an increase in 1 would be simply mean a change in the right-most digit rather than an addition of a digit (Shumway, 2011). Therefore, one hundred and one would be written as 101 rather than 1001. Similarly, one hundred and two would be 102; one hundred and three would be 103 and so on and so forth. Often, children do the opposite of this and increase the value of the ten placed digit when they should be making an increased in the one placed digit. These children would subsequently be writing 101 as 110. In both these cases, the child is displaying a weak grasp of the place-value concept and needs more tutoring on the concept. Teaching these children the concept of having a set of three places -- hundreds, tens and ones (illustrated below) -- shall greatly improve their ability to get a clearer understanding of the number system.

hundreds tens ones hundreds tens ones thousands ones

A working understanding of this grouping together will also help solve the misconceptions that children might develop over the magnitude of a digit placed in a sequence of numbers. Failure to understand the power of a position relative to the digit on either side can cause misconceptions to develop such as mistakenly assuming that the value of 1 is smaller than the 7 in the number 17.

Another misconception, this time dealing with numbers higher than 1000 can also be avoided if the child develops a clear understanding of place-value. Often it is found that children make the mistake of increasing the digits when making calculations such as 2999 + 1 = 29100.

How children can be taught to champion the place-value system

In order to easily and effectively teach children to conquer the place-value system, it is important that they be taught the nature of the relationship between digits, respective to their place in the number sequence. Each number has a multiplicative relationship with the digit to its right. The value of each position increases by a power of ten moving from right to left. The right most number therefore, shall have the lowest value while the left-most shall represent the highest.

There are also many interactive methods which can be used to develop a child's number sense. For instance, giving children a large number of objects such as poker chips, then teaching them to count them in groups will not only improve their understanding of numbers, but also teach them that it is easier to count in groups, e.g. 7 groups of ten, rather than count each object or chip individually. This will also aid their mental math skills and enable them to solve complex problems quickly and…