Seeing negative numbers

When I lived in New York City, I had the opportunity to tutor neighborhood children in subjects such as math and reading. On one occasion, the boy I was tutoring was struggling with adding negative numbers. The problem was that he lacked an intuitive grasp of what the operation meant.

The concept of adding positive numbers is easy to understand. We can take three apples, add two apples and see that we end up with five apples. We can also subtract two apples and end up with three again. But what happens if you subtract ten apples from five apples? Or add negative two apples? This no longer makes intuitive sense.

The solution is to find practical applications where the concept of negative numbers does make sense. I noticed that the floor was constructed of tiles. So I got my student to stand on a tile which we marked as the starting point (this was our number line origin). Taking a step forward to the next tile corresponded to adding the number one. Taking a step backward corresponded to subtracting the number one. It was then possible to step backwards until he was one or more tiles behind the starting point. Unlike with the apples scenario, it was now possible to "see" a negative number and meaningfully add and subtract with them.

Other practical applications of negative numbers include calculating budget deficits (as against surpluses) and measuring temperatures below freezing point. Essentially, negative numbers are applicable whenever the subject matter can be considered in terms of opposite magnitudes.

Interestingly, up until the eighteenth century, negative numbers were often considered absurd or meaningless. But as the above examples show, it is just a matter of applying them appropriately and any absurdity goes away.