Teaching addition to children using the concepts of carrying and borrowing is easy if you remember to do the following:
~ Concretise the experience
When teaching addition to children, the first lesson objective is to make sure that the children know what numerals stand for. They must understand that the numeral 0 (zero) means nothing, that 1 (one) means a singular item, that 2 (two) means 1 more than 1 item and so on by demonstrating that in a concrete manner.
Involve your children in counting items they see in the real world. When you go shopping, get them to count the number of fruits, cartons of milk, boxes of chocolate, pieces of fresh shrimp and so on, together with you. They will soon realize that 1,2,3 is not just a song that Michael Jackson or the Muppets sang but that each of those numerals mean a concrete amount. Move on to counting more than 10 only if your children can demonstrate confidently the numerals 1 to 10.
~ Use items are easy to manipulate
Use items that are easy to manipulate and lasts a long time. These include popsicle sticks and bread tabs. I prefer bread tabs to that are colored and similar in size to popsicle sticks. When dealing with amounts more than 10, bread tabs are easier to arrange and you can use a different color to connotate different amounts. Get a few friends involved in helping you collect them and you would have enough to last a life time in a few months.
I would not use items that may connotate another amount for a beginning Mathematics learner. For example, while coins are attractive, in counting a ten-cent coin as a unit, because the numerals 10 may appear when the coin is flipped to the tail, children may think that 10 means all the 1, 2, 3 and so on that they are saying out as they count.
~ Get children involved rather than just demonstrate to them
Not many children learn well by just watching and listening to you telling them what you have to do. Let them be the ones moving the items and saying out the numerals as they do so. Be prepared to get some children to work through many examples before they actually grasp the concept of addition.
~ Be very structured
A little change in your approach may confuse a young child, especially when he is learning disabled. For example, in adding two numbers, a child may be unduly stressed when you use yellow and blue tabs instead of the usual white and green tabs. Make slight variations only when the children have mastered one part proficiently enough to demonstrate it to you independently and without being given hints along the way.
~ Get children to understand key terms taught
In teaching the topic of addition and subtraction to children, the terms carrying and borrowing are frequently used. What is carrying and borrowing in addition? In teaching these terms to children, we have to be mindful of what these terms may mean to a child. For example, carrying may conjure the image of carrying items for the teachers and may have no relation to addition processes to the child. Likewise, the term borrowing makes no sense to a child who has everything and sees no need to borrow anything.
When we neglect the language aspect of learning Mathematics, the children lose out in understanding the Mathematics they are learning. It is not true that language is not involved in the learning of Mathematics, as many have the misconception. Thus terms such as carrying and borrowing must be explained and illustrated during teaching and children must have a hand in demonstrating their understanding of the concept.
When we teach addition and subtraction, while it is easy to teach children how to carry out the operations of carrying and borrowing using stories, manipulatives and audio-visual aids, we must bear in mind the real meaning behind the terms.
What we refer to as ‘carrying’ is also referred to as ‘renaming’. What is known as ‘borrowing’ is also known as ‘regrouping ‘.
When we see a two-digit number, say 11, although both digits are the same, the ‘1’ to the left has a different meaning from the ‘1’ to the right and is 10 times the value of the other.
Prior to teaching addition and subtraction, first introduce the concept of tens and ones using groups of 10 sticks as Tens and individual sticks as Ones.
Use popsicle sticks combined with the concept of place values in a number story.
Let us refer to the number 15. Numbers live in houses. The number 15 is made of two digits, 1 and 5 which live in the same House and so they are written along the same horizontal line. The digit 1 uses the Tens Room while the digit 5 uses the Ones Room.
Tens Ones 1 5
The Ones Room can only hold the numbers 1 to 9. Once you have the number 10, it must move together to the Tens Room and be renamed as 1 Ten. But we write only the digit 1 under Tens because now the Tens Room is a bigger room. Each digit there stands for Tens and not Ones.
Example: 15 + 27
Say that 15 and 27 want to be friends and live together.
Write the words tens and ones on a sheet of paper, large enough to arrange popsicle sticks or bread tabs under each. Note in the example below that (IIIIIIIIII) means a bundle of ten popsicle sticks.
(IIIIIIIIII) IIIII 15
5 and 7 gives 12 which is regrouped as 10 in a bundle (1 ten) and 2 individual sticks (2 ones). Move the 5 and 7 individual sticks together and band the sticks together to show 1 bundle and 2 individual sticks.
Now, the 2 ones can still stay in the Ones Room and it is written as 2 in the Ones Room below the line drawn under 27.
But the 1 ten must move to the bigger Tens Room to the left together with 1 ten in 15 and 2 tens in 27. Write 1′ between the Tens and 1 to remind you 1 ten has just moved from Ones Room. Together the numbers 1, 1 and 2 in the Tens Room make 4 tens which is written as 4 in the Tens Room below the line drawn under 27. Move the 1 bundle from 15 and 2 bundle from 27 as well as the 1 ten together to show 4 bundles.
So, 15 + 27 = 42 Repeat the story with other sets of 2 numbers.
Example: 42 + 15
Say that 15 quarreled with some members of 42 and wanted to leave the House of 42.
4 2 – 1 5
Show the 4 bundles of tens and 2 individual sticks. Since 5 cannot go away from 2, we regroup 1 ten into 10 ones. That makes 3 tens and 12 ones.
We have no more 4 tens but we now have 3 tens, so cancel the number 4 and write the number 3 above. We have 12 individual sticks now instead of 2, so cancel the number 2 and write the number 12 above the canceled 2.
Do not teach children to just write the digit 1 next to the number 2, right in between 4 and 2. The reason is that this digit 1 does not have the same meaning as the digit 1 written between Tens and 1 in the addition story above. It also does not represent a value 1 one but a value 1 ten.
Next, move 5 ones from 12 ones and 1 ten from 3 tens. The remaining bundles and individual sticks will show 2 tens and 7 ones.
So, 42 – 15 = 27
The concepts of tens and ones, renaming and regrouping are very important to the four operations of numbers, namely addition, subtraction, multiplication and division.
By teaching children to represent the regrouping of numbers, as in the subtraction story above, children become clearer that the digit 1′ represents a different quantity below Tens and below Ones. They will be less confused when they learn the Division Algorithm which involves renaming and regrouping as well.
Why do children get more confused and grades fall as they grow older? It is not because they grow more stupid or that the Mathematics is beyond their understanding. It is because they do not understand the basic concepts taught to them at a younger age. They could ‘do’ Mathematics by memorizing the steps of an addition or subtraction algorithm. As they grow older, they need understanding and thinking skills to be able to solve problems which cannot be memorized.
It is thus important for teachers to think through teaching processes carefully, rather than use short cuts which help children do Mathematics without a clear grasp of the concepts to be learnt.