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 this generalizes to more numbers. If I give you a set like <62, 69, 72, 58, 64, 69, 61, 57> you can give me a rough estimate of the average without doing any addition. You have a lot of statistical expertise, but I think anyone could do it. Or at least anyone could do it if I asked about the center. If I graphed the numbers out, it would probably be even easier. So I guess I am talking about some very primitive ability to find the center, somehow without adding. We call it average, fine, but that might disconnect average from the student's idea of center, and then when average is defined by a formula and the formula involves adding the numbers, they might think that the more numbers there are, the higher the average. Donald Burrill wrote: > Hello again, Bob. Thought this vignette (a true story!) might provide > some mild interest... > > When I was in the fourth grade, attending a three-room school with 8 > grades (grades 4, 5, and 6 were in one room), the teacher gave us an > assignment to keep busy while she was doing something else -- whether > necessary desk work for that marking period, or work with another grade, > I do not now recall and wouldn't have been paying much attention anyway. > She asked us to calculate our average math scores. > > Well, as you know, merely adding up a bunch of numbers and dividing by > their cardinality is boring, even at age 9; so to relieve the tedium I > put all the scores in order (there were maybe a dozen to a score of > them), and starting with the lowest, which was around 70% or so, I > averaged the bottom two; then averaged this with the next one up; and > continued in this way till I'd finished. (Of course, by this time the > teacher was wondering what was taking me so long!) I was mildly > surprised to find that the result I'd calculated was 99% -- I'd expected > the average to be high, but not THAT high. Not surprisingly, when she > found out what I'd done, the teacher made me recalculate the RIGHT way. > At the time it was something of a revelation, though not then a very > complete one: after all, an average was an average, wasn't it, and I'd > had an intuitive notion that the value one got oughtn't to depend much > on the procedure one used (not that I'd have used that kind of language > then). > > -- Don. > ------------------------------------------------------------ > Donald F. Burrill [EMAIL PROTECTED] > 56 Sebbins Pond Drive, Bedford, NH 03110 (603) 626-0816 > . > . > ================================================================= > Instructions for joining and leaving this list, remarks about the > problem of INAPPROPRIATE MESSAGES, and archives are available at: > . http://jse.stat.ncsu.edu/ . > ================================================================= . . ================================================================= Instructions for joining and leaving this list, remarks about the problem of INAPPROPRIATE MESSAGES, and archives are available at: . http://jse.stat.ncsu.edu/ . =================================================================
