### Place Value - Reading and Writing Numbers

Reading and writing numbers might seem intuitive and easy to us as adults, but there is a lot to understand before it becomes second nature to a young child. Think about how you would explain to a child why, in the number 0303, we do not need the first 0 but do need the second one. How would you help a child to really understand that the first 3 has a very different meaning to the second 3? Could you explain why 0.3 and 0.30 are actually the same number? And why can't you always just add a zero when multiplying by 10? The key to a child truly understanding all of these concepts is in allowing them to discover and understand place value and the fact that it is the position of each digit within a number that is important.

(To see a full list of available posts click here.)

In a previous post, I highlighted how important it is to really understand place value and the way our number system works. The post explains what place value is, as well as discussing common misconceptions and practical activities that help children develop an initial awareness of place value.

This later post focuses on reading and writing numbers now that children have had practical experience and developed some comprehension of place value. It includes task ideas and resources that can help develop these skills, particularly focusing on the relative value of each digit within a number. We will also briefly discuss the requirement of zero as a place holder and why this is a much more advanced concept and should be avoided in initial teaching and learning.

Our number system makes life easy!

A key part of our number system (that is often overlooked) is the fact that we only have 10 digits (0, 1, 2, 3, 4, 5, 6, 7, 8, 9). This means that we only need to learn and be able to recognise 10 symbols, making life a lot easier. Rather than having each digit value assigned on a 1-1 basis (so 99 representing 9+9=18) we use different values for the position. This allows us to count to much larger values with less digits and removes the need for calculations. As we can count up to 9 with a single digit, we use the next digit (to the left) to represent how many times we have counted to 10. This means that the second digit from the right will always have a value of tens. We are then able to count to 99 (9 tens and 9 units) so naturally, the next digit can be used to count how many times we have counted to 100. Thus the third column from the right will have a value of hundreds. Knowing this means that reading a number is easy, as all we need to know is 10 symbols and the relative value of each digit. To work out the relative value of each digit, it will simply be ten times the value of the digit to the right of it.

This all means that adding and subtracting numbers also becomes easier as you can add (or subtract) numbers position by position. For example, 234 + 152 can be done by adding 1+2 for the first digit of the answer, 3+5 for the second digit and 4+2 for the third digit (making sure that the corresponding digits are in line with each other). Obviously we will need to learn about 'carries' and borrows' if we are dealing with numbers that add to more than 9 or subtract to less than 0, but it certainly makes life easier than if we didn't have a position based number system.

To help realise why all this makes our lives easier I am going to briefly discuss Roman Numerals, which is not a position based number system.

If you know anything about Roman Numerals, you will know that there are different symbols for 1, 5, 10, 50, 100, 500, 1000... and you can place smaller numbers before larger numbers to adjust the value. It can get quite complicated and just reading a number can require a lot of understanding and some calculations. For example, let's work out the following number

MCMLIV

We first need to understand the value of each symbol. M=1000, C=100, L=50, V=5 and I=1. Now we work from left to right.
• The first digit (M) tells us we have 1000, so nice and easy so far.
• The next digit appears to be a C, which is 100, However it precedes a digit of larger value (M=1000) so we need to know that these digits go together and actually mean C(100) less than M(1000) so 900. So we now have 1000+900=1900.
• The next digit is L(50) and does not precede a larger digit so we have 50, totaling 1950 so far.
• The next digit I(1) precedes a large digit V(5) so this again tells us that these go together and we have I(1) less than V(5) so 4. We now have a total of 1954.
I don't know about you, but this seems like a lot more effort than reading a number that is based on position. The only advantage that Roman Numerals has over our number system is that there are no zeroes required as place holders.

Try adding or subtracting using Roman Numerals and there are no nice shortcut methods of lining up columns and carrying numbers - you have to be able to mentally add them and then know the symbols for the answer.

So a number system being position based simplifies things quite a lot and allows you to break numbers up easily for calculations or rounding. Our number system could have been a different base - different cultures do (or have) used other bases, such as 2, 5 or 60. If, for example, our number system had been base 8, we would be used to working and envisaging numbers in base 8 and it would be just as easy as our current base 10 number system. The numbers would read 1, 2, 3, 4, 5, 6, 7, 10, 11, 12, 13, 14, 15, 16, 17, 20, 21… and so on. We would still be able to count in tens and hundreds, just each 10 would actually only contain a quantity of eight and each 100 would only contain a quantity of sixty-four. The ease with which we can work in base 10 relative to working in other bases is just due to it being what we are used to. It is widely believed that our number system is base 10 purely due to the fact that we have 10 fingers, so it is easy for us to count 10 digits.

Hopefully this has helped you to realise that our number system being position based makes our lives so much easier. However, to get the full benefit of this, we need to understand that it is position based and what exactly that means - hence we really need to help kids understand place value.

More than just writing numbers

One of the fundamental ideas of place value and our number system is that the position of each digit within a number matters. If children understand this then reading and writing numbers will be much easier. They will also be able to quickly get to grips with why we need zeros as place holders, what numbers after the decimal point mean, how rounding and estimation works and how to do numerical operations. In order to help children to grasp this concept whilst attempting to write numbers, try the following tasks.

Before attempting any of the tasks below, children should be confident recognising and writing the numerals 0 to 9. If they still need more practice with this, please see my post 'Early Years Counting - techniques to help kids really understand numbers'. It would also help if they already have some understanding of place value and how the number system works.

Count a large number of objects
This might sound boring but if you have a child that collects something (and has a lot of them), ask them how many they have. They will want to know and will want to tell you but may be overwhelmed by the task, especially if they have hundreds. However, talk to them about how you could break it down and discuss how you could group the objects to make counting them easier.

Discuss skip counting and encourage the child to think about how it is easy to "count up in" 10s. Then ask them to start putting the objects into piles of 10. Once they have more than 10 piles, talk about how it is getting difficult to keep track again and encourage them to think about how they could organise the piles into rows of 10.

Talk to the children about how many are in each pile and how many are in each row. They should relatively easily be able to see that there are 100 in each row, so if they have 3 complete rows they know they have 300.

A simple task like this gets children thinking in the same way as our number system, without having to teach them anything!

Games to emphasise that you can only have a single digit in each column
Here at Kiducation UK we have invented the 'single-digit monsters' to help kids really understand the idea of place value. The concept is simple - the monsters can only count up to 9 as they can only deal with single digits. If you try to give them more than 9, they start to get angry.  Check out our introductory video to see how this works. It really provides children with a visual and memorable understanding of how the number system works. This understanding can easily be transferred on to further mathematical concepts, such as column addition and why we carry, or column subtraction and why we borrow.

You can play a game based on this with children by getting them to act as the monsters. Each child can only remember a single digit number to keep count and they can pretend to get angry if they are asked to remember more than 9. Once you get to 10 and the first monster starts to get angry, you.will need another monster to help. You can discuss how you could get this second monster to also count to 9 but then you could only count up to 18 in total. However, if the second monster counts how many times that first monster has got angry (counted to 10) then you can count all the way to 99 with just two monsters.

Use Dienes blocks with sets of 9
Provide the child with a set of Dienes blocks containing 9 ones, 9 tens and up to 9 hundreds (and a couple of thousand blocks if you want larger numbers).

Using the set of Dienes blocks, verbally ask the child to represent the number 14, for example. Start with small numbers and encourage them to count up from 0, remembering to exchange 9 unit blocks for one strip of 10 when they need to. As we have only provided them with 9 unit blocks, they will have no other choice, which is similar to our number system as we have no single digit to represent 10. In our example of 14, the only way the child can represent it is with one ten and four units (as they only have 9 units). Ask the child to explain what they have and talk about which are more significant (larger) - the ones or the tens. Ask the child to arrange the block in an easy to understand way, starting with the least significant (smallest) blocks on the right. This method will lead the child towards thinking of the leftmost number as being the largest and will start to develop their understanding of the position being important. It will also help them develop their awareness of the relative size of each of the numbers.

After representing a few different numbers, you can provide the child with a large table in which to place their blocks, again encouraging them to put the smallest blocks in the most right hand column. Once the child is confident ask them to record the number of items in each column using a digit from 1 to 9 (I suggest avoiding zero until they are confident with the concept, as this is a whole lesson of its own). Then have a discussion with the child about how they have written the number in numerals. Start from right to left and discuss the meaning and relative value of each digit, emphasising that the position of the number tells you what it represents and link it back to their physical object (for example, in the number 364 the 6 represents 6 of your strips of 10).

Use an abacus Using an abacus is a great tool for making the transition from objects such a Dienes blocks. Dienes blocks help children to visually understand the relative size of each digit as the strips of 10 are the same as 10 units stacked together. However, children need to be able to understand that when we read and write numbers, it is purely the position of each digit that tells us what it is representing. For example, in the number 25, children need to understand that the 2 actually represents more than the 5.
When using an abacus, we use the second row to count how many times we have counted to 10 on the first row, thus one ball on the second row represents 10 on the first row even though the balls are the same size. This will really help children to understand that each digit is worth ten times more than the digit to its right even though they are written in the same way.

Use arrow cards
Once you are sure that the children have an appreciation for the relative value of each digit within a number, and that they understand that the position of the digit within the number is important, you can move them on to arrow cards to help them understand how to write and interpret written numbers.

You first need to make sure that children are aware that they must always align the arrows and demonstrate how to make the number up. The act of assembling the numbers will help children to make the link between the size of the number and the position it ends up in within the larger number. For example, the 300 has the same value as the 3 in the made up number. It is also a great task for helping children to appreciate the parts of a number, e.g. 364 being made up of 300 and 60 and 4. Arrow cards can also be used when moving on to numbers that contain zeros.

Understanding zeros as place holders

Understanding that we need to have zeros as place holders can be difficult to explain and even more difficult to grasp for young children. In fact, it can be such a tricky concept with numerous possible misconceptions that it deserves a whole post of its own (coming soon). However, I will briefly explain it here.

Children need to understand that we start with a ones column and then build on that, moving to the left each time we need larger numbers. We only move as many digits to the left as we need but we need to keep all of the digits to the right. For example, in the number 102, we have had to move to the left twice to account for the one hundred. However, we need something in the tens position to show that the one is two places to the left. As we have no tens, we need a zero. However, for the number 12, we do not need anything in the front (012) as we have only had to move to the left once (for the tens) so do not need a hundreds column at all.

If children can understand this they will more easily be able to understand how the numbers work after the decimal point. They will be able to understand why the zero in 1.03 has significance whilst the zero in 1.30 does not.

If you try introducing zeros too soon to a young child who is just starting to learn about the number system, it can be more detrimental than helpful. They will start getting confused and adding or removing zeros when they shouldn't, meaning that they start focusing on the digits again rather than the positions. However, if the child has had a lot of practical experience investigating the number system and how place value works it will be much easier. I would therefore strongly advise leaving out numbers that contain zeros until the child has a very strong grasp of place value.

A Case Study
A five year old boy has learnt how to count, can count in 2s, 5s, 10s and 100s and can even go higher than 1000. He also managed to figure out for himself how to go beyond 1900 (to 2000) when counting in 100s. When discussing the relative size of numbers, he demonstrated a good comprehension of what is bigger or smaller, and how some numbers are much bigger than others. However, when asked to write the number 512 this is what he wrote:

'50012'

This answer is quite revealing. It demonstrates that he understands that 500 is bigger than tens and suggests that he has learnt that hundreds are written as the required digit followed by two zeros. He therefore believes that he needs to write the full 500 before writing any other part of the number.

However, he is capable of writing double digit numbers correctly - when asked to write 54, he wrote 54 rather than 504. This suggests that he, like many children, defaults to writing numbers purely from repetition and recognition rather than understanding of place value. He is currently aware of all double digit numbers and can correctly read and write most that were asked for up to 99. However, he currently has no awareness that a third digit represent hundreds, even when it is not followed by two zeros. For example, when asked to read the number 243, his reply was

"Twenty four and three"

This confirms that he has learnt double digit numbers just from becoming familiar with them, but he is not understanding place value or the relative size of each digit. It also explains why he would occasionally get his digits the wrong way round and write 32 when meaning 23 - he is trying to recall from memory but does not understand that the position of each digit is significant.

This is just one case study, and at five years old the child is only just beginning the learning process - he would not be expected to have a true grasp of place value at this stage. However, I have seen examples of this lack of understanding of place value time and time again, and even in much older learners. For these children, many areas of maths are therefore much more difficult, including the use of written methods for addition/subtraction, and simple rounding and estimating. However, if we were to help all young children really understand our number system and make sure that they learn numbers through understanding place value rather than recognition then their mathematical futures would be much brighter.

Children will learn how to read and write numbers through repetition and recognition even with no other teaching. However, they need to be able to do more than read/write the numbers - they need to understand why the numbers are written the way they are and how place value actually works. This is something that is so often overlooked or misunderstood by learners. Learning the intricacies of how our number system works will make future maths much easier, but it must start at a young age when children are first learning to read and write numbers.