Imagine that Jim asks me for the number that, if you were to square it, would give you 5. And imagine that I would tell Jim: ” I know the answer to that! Why, it’s the square root of 5!” Which of these would you think I was doing: (a) bluffing, (b) giving Jim a correct answer, (c) both, or (d) neither?
In some ways, I may be doing both. I’m giving Jim a correct answer, but it may not be a useful answer. In a sense, I have merely restated Jim’s question. For “the square root of 5” is mostly just a shorthand name for the number that, if you were to square it, would give you 5. It is bluff, but accepted bluff. I may have satisfied Jim without giving him anything particularly useful, and lived another day. If Jim comes back and asks: “Yes, but how much is the square root of 5?” I’m no worse off than I was to begin with.
Some of the way that mathematics intimidates students might be lessened if you consider various mathematical notations to be bluff, accepted bluff. When we don’t know the answer to a question, or we don’t want to bother finding a good approximate value – well, what if we introduced a notation that gives a name to the answer, lets us pretend for the moment that we have tamed the monster, or at least distracted it for the moment.
Kids are intimidated by fractions? Kids don’t know what to make of 3/7? Well, we might let them in on a little secret – that we don’t know either. Maybe the main difference between kids and us is that we’ve gotten used to answering the question “How much is 3 divided by 7?” with “I know the answer to that! Why, it is 3/7!” The beauty of fractions is that you can get away with the bluff.
We have successfully deferred the computation of 3 divided by 7 till a later day.