On Dec 4, 2012, at 10:58 AM, Joe Niederberger <niederberger@comcast.net> wrote:

Simply working through the logic of some time-reversal scenario or other way of illustrating the sign rule, may in fact stand in the way of deeper understanding, because the person may think that somehow "proves" the matter.
It good, but its not enough by itself.

Can you describe more of what you mean by deeper understanding? If the teacher stands in front of the class and works through the logic or describes some analogy, and the students nod in agreement, then I don't think that is proof of any understanding (on the students' part). But if the students are able to tackle some pretty clever problems involving these elements, and even extend the elements past what the teacher is able to deliver in a limited amount of time, then I don't think you can get any deeper than that.

Taking "negative numbers" as an example, what would represent "deeper" understanding. I know one thing I have now that I didn't have when I was 13, a much more "experienced" understanding. But I do know that there was an aha instant before which, my understanding of negative numbers was not fully ripe and after which it was fully ripe. And when I say fully ripe I mean after that point in time, there is nothing you could describe to me involving negative numbers that I wouldn't understand. In other words, I got it. How do you get deeper than that, other than experience?

Bob Hansen