Adding and Subtracting Integers
One of the reasons we get these migraines over integers is that, at least up to the point that we as students were actually introduced to operations with negative numbers, we had been taught (correctly) that addition is an operation that describes combination and subtraction describes extraction. We know, for instance, that adding values is like combining collections of objects, and subtracting values is like removing a collection of objects from another collection.
Then we get to integer math, at which point we are asked, judging by present-day treatments in textbooks, to understand the idea that we should be able to, for example, add a "negative collection" to another "negative collection." Or we must throw away and disregard as incredulous all that "collection" talk.
Mathematics is always described as a beautifully and rigorously universal subject in every detail--when an idea is laid down and proven in mathematics, it applies everywhere and always. But, to my mind, this is not the way mathematics works, and people should really stop spreading this "universal" rumor.
When you add or subtract with integers, you are NOT combining collections or extracting from collections; you are moving in certain directions.
As an example, below is a (poorly drawn) submarine at a depth of -5 whatevers. This doesn't mean that depth can be negative (how could it?); it simply means that the submarine's position, in relation to a number line, can be described as -5.
If we add 3, we are adding buoyancy. Because 3 is positive, we are adding positives. Thus, the sub goes up.
If the sub starts at -5, and we subtract 3, however, we are "getting rid of" positives. When we get rid of positives, the sub must sink:
Finally, let's find -5 - (-3). Here we are "getting rid of negatives." When we get rid of negatives, the sub must go up.
Then we get to integer math, at which point we are asked, judging by present-day treatments in textbooks, to understand the idea that we should be able to, for example, add a "negative collection" to another "negative collection." Or we must throw away and disregard as incredulous all that "collection" talk.
Mathematics is always described as a beautifully and rigorously universal subject in every detail--when an idea is laid down and proven in mathematics, it applies everywhere and always. But, to my mind, this is not the way mathematics works, and people should really stop spreading this "universal" rumor.
When you add or subtract with integers, you are NOT combining collections or extracting from collections; you are moving in certain directions.
As an example, below is a (poorly drawn) submarine at a depth of -5 whatevers. This doesn't mean that depth can be negative (how could it?); it simply means that the submarine's position, in relation to a number line, can be described as -5.
If we add 3, we are adding buoyancy. Because 3 is positive, we are adding positives. Thus, the sub goes up.
If the sub starts at -5, and we subtract 3, however, we are "getting rid of" positives. When we get rid of positives, the sub must sink:
Finally, let's find -5 - (-3). Here we are "getting rid of negatives." When we get rid of negatives, the sub must go up.
Labels: mathematics