In the last blog we looked at how ‘zero’ is the border that separates negative and positive numbers on a number line and that adding positive values is seen as a journey along the line to the right, the destination being the solution. The nature of the addition was seen as an aggregation, that is, combining all the values resulting in a new, greater value. In this blog we will start to look at how addition is seen as augmentation, in other words how the result of an addition is seen as making one value bigger by adding a second. Finally we will consider how the mechanics of augmented addition makes sense of subtraction.
In the last blog we considered the addition of 3 + 2. We will now look at the same problem and see how the imagery differs when seeing the addition as an augmentation.
In short an aggregation is best described as a collection of the things being added and then counted to get the result and an augmentation is is the result as a new number, of the addition of two or more numbers. To continue this idea of addition of positive numbers as placing one of them at the ‘zero’ starting position if you will, then building up to a solution by adding the next number at the end of the first number. When this is understood in conceptual terms we can begin to look at how subtraction might work under the same rule set. For example what if we wanted to understand what was happening on a number line when we subtract 2 from 5. Well we need to see 5 as the object of the subtraction. It is the number that is having the operation done to it so to speak. We could conceivably see the question as how far away from 0, 2 is from a starting position of 5. In this case we could line the numbers up in the augmented addition model but instead of from a starring point of 0, we line them from the finishing point of 5.
The result is what is left of the 5 that we started with and it can be seen that it is indeed 3 away from zero. The model for this can be illustrated as follows and see how there is a direct link to formal column subtraction.
To recap, the model shows that when adding a positive number to a positive number the result is found to the right of zero and can also be interpreted as how far away from zero the sum of the two numbers are. When subtracting a smaller positive number from a larger one, the problem can be interpreted as how far away from zero the number being subtracted is, from the starting point being the larger number. The visual effect is the reverse of adding two positive numbers. Instead of moving to a position on the right from a starting position on the left (0) we start from the right and move to the left to a finishing position (which could be 0 if the subtraction is of two identical whole numbers, or to the right of 0 if the subtracted number is smaller than the number from which it is being subtracted.)
In terms of formal written processes then, column addition is a model of augmentation, whereas formal column subtraction is a model of ‘finding the difference’ of two numbers.
In the next blog we will look at the mechanics of adding a negative number to a positive followed by subtraction a negative from a positive number.