Home Again

For the last three weeks I've been leading AP Calculus Suimmer Institutes in Virginia, Alabama and Massachusetts. I'm finally home and have been in the office this week. I hope that now I'll have a little more time to write.

The AP Calculus EDG has been going on and on about two things: the em-dash and the square root of 9. Both discussion have been long (as in lots of E-mails), somewhat repetitive, but nonetheless informative.

The em-dash, —, to use its typographical name, is the symbol used to represent (1) subtraction, (2) the opposite of a number or expression, and (3), when used with a numbers as opposed to an expression, to indicate that the number is negative as in —3. It is having three uses that makes it confusing to beginning and not-so-beginning students. There was a time when it was suggested that the em-dash be used only for subtraction and a raised en-dash be used for the other two uses. Makes a little sense since the opposite of 3 is negative three; but then the opposite of x might really be positive. And then calculators do have a "subtraction" key and an "opposite" key.

This can be confusing especially since the three uses are closely related to each other. The best suggestion was not more symbols, but simply to honestly face the problem and teach the kids the three meanings, how to tell the difference, and how to use the proper one.

I'm afraid I may have started the other thread when I related that a teacher asked me "What is the square root of 9?" I had to ask him if he meant or a solution of the equation . This to led to many replies with all sorts of opinions many as to whether was the same as the square root of 9. Then the function came under discussion: is this the "square root function" or the "radical function."

Again the moral is that we have to be careful how we explain things to students and be very precise in how we (or our textbook) defines things.

One additional thing that came from this discussion was the suggestion that if the square root "function" is the inverse of the squaring function then its range must be restricted to make it a function. This is probably the first time that students come across this situation and it is a great place to stop and explain why this restriction is necessary and why it is allowed. Because the next time this comes up is when you are defining the inverse trigonometric functions and restricting their ranges.

So maybe we all have too much time on our hands this summer - no that can't be right. But the discussions were interesting and point to places where teachers need to be very clear about what's going on.