Robert Frick <[EMAIL PROTECTED]> wrote in news:[EMAIL PROTECTED]:
> Which reminds me of a story. I gave a student two numbers (say, 376 > and 378) and asked him the average. He answered 377, quick as a flash. > Then I asked him how he calculated the average, and he said, quick as > a flash, add the numbers and divide by two. Then I asked him what was > 376 + 378. He couldn't do that without pencil and paper. > > So he wasn't adding the numbers. I thought for a while how he actually > did figure out the average. I thought about the possibility that he > stripped off the front numbers, added 6 + 8, divided by two, and then > restored the front numbers. I don't think he knew math well enough to > do that, or that he could do those calculations as quickly as he could > know the average. Also, I don't think he would have had trouble with > the average of 999 and 1001 (assuming he knew they were two apart). > > My memory is that this worked with several students, and anyway it > works on me -- I can tell you the average of 376 and 378 without doing > any addition, and the answer comes so fast it is not easy for me to > know what I am doing. I think all of us who can do it have simply internalized the identity that (A+B)/2 = A+(B-A)/2, so we really *are* doing addition (and subtraction), just not an addition of the two numbers. We just split the difference and add it to the smaller number or subtract it from the bigger one (it's so internalized that we don't even think in terms of addition; we just think "halfway between the two numbers"). Note that this trick only works as a shortcut for averaging *two* numbers; three or more and the mental bookkeeping involved gets more complicated than just adding all the numbers. And it could lead to confusion in the mathematically naive, who often have a difficult time understanding the difference between the concepts of the mean and the midrange; it's just a coincidence that for two-element sets they're the same thing. Someone who can correctly calculate the mean of two numbers might well, when asked for the mean of three numbers, give the midrange instead. . . ================================================================= Instructions for joining and leaving this list, remarks about the problem of INAPPROPRIATE MESSAGES, and archives are available at: . http://jse.stat.ncsu.edu/ . =================================================================
