Teaching mathematics

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# 3.1 Formal written methods of addition

Of the four basic arithmetic operations, the addition of whole numbers and associated written methods of addition tend to be reasonably well grasped. It is the other three operations where learners have difficulties.

## Activity 7 Reflecting on two methods for adding three-digit numbers

Timing: Allow 10 minutes

Watch Video 3 on formal written methods and then reflect on the difficulties a learner might experience when using either of these methods.

Skip transcript: Video 3 Demonstrating formal written methods for addition

#### Transcript: Video 3 Demonstrating formal written methods for addition

INSTRUCTOR:
I'm going to work out 426 plus 378. First method I'm going to use will actually partition each number into its hundreds, tens, and units. So 426 is 400 and 20 and 6, and then 378 is 300 and 70, and 8.
And then if I combine the hundreds, that makes 700. If I combine the tens, that gives me 90. And if I combine the units, that's 6 plus 8, is 14. Then I simply add all of those up-- 790, 800, 804. So altogether, that comes to 804.
The other method would be to put 426 and 378 one underneath the other, using the column method. So if I write 426, 378 will go underneath like that so that the units are in the column, the tens are in a column, the hundreds are in a column. Add those up.
6 and 8 makes 14, so I write down the 4 and carry the 1. 2 and 7 makes 9, and then 1 makes 10, so I put down the 0 and carry the 1. 4 and 3 are 7, and 1 makes 8, so that's 804.
The column method is quicker, and may be more efficient. It relies on the digits being placed into columns-- the units in one column, the tens in another, the hundreds in another. And after we've lined up the digits, each of the numbers is added in their colours.
We're effectively treating each number as a string of digits, whereas in the first method, we're actually using the meaning of the 4 in 426 as 400, and the 2 meaning 20, and the 6 meaning six units-- and then similarly for the other number. But it is, obviously, a more chunky method, and not quite so quick to do. But they both come up with the same answer.
End transcript: Video 3 Demonstrating formal written methods for addition
Video 3 Demonstrating formal written methods for addition
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### Discussion

If learners have developed mental calculation skills then these will be built on when moving to written addition methods. If learners have not developed good mental calculation skills, using written methods will be much more difficult.

If you think back to the early years, remember that learners operate addition by counting out the sets of objects and then combining the sets. If they have not progressed from this stage to being able to add up abstract numbers mentally then adding large numbers using a written method will be cumbersome.

A learner who struggles to add 6 + 8 and who uses their fingers or makes marks on paper (and I have seen both these activities used by secondary school pupils) will struggle to add 60 + 80. If they only operate addition by counting sets of objects then they are unlikely to have developed a robust concept of place value and therefore no appreciation that 6 tens + 8 tens gives us 14 tens. This is why it is so important that learners build a firm foundation of understanding in the early years. It also implies that expecting learners to move on to the next stage (e.g. of doing formal written addition before they can mentally add abstract numbers) is not appropriate. As a teacher, you may need to ‘go back to basics’ with learners who are in this position.

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