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One thing I never realized until I started teaching is why we calculate the sum, then divide by the number of numbers to find mean. We're actually finding how much each person would get if the items were distributed equally. So if there were 3 apples, 5 apples, and 10 apples, if we combined them we would have 18 apples. Dividing by 3 gives 6 apples per person if the apples were shared equally. This also shows why we use median rather than mean for items like income where there can be outliers that skew the results when the items are shared equally.This reminds me of Ron Aharoni saying that when he taught arithmetic he realized there were subtleties to elementary mathematics that he hadn't thought about.
Must find that passage & post!
gasstationwithoutpumps on mean, median, & mode
3 comments:
Singapore Math introduces averages as anonymous suggested. In Textbook 5B (Primary Mathematics, US Edition): "These bags do not have the same number of oranges. If the oranges are rearranged so that the bags have the same number of oranges, how many oranges will there be in each bag? 4 + 9 + 5 = 18. There are 18 oranges altogether. 18/3 = 6. There will be 6 oranges in each bag."
On the same page, there is a cartoon of boy thinking: "The average of 4, 9 and 5 is 6."
By the way, here is a nice illustration of a data set that doesn't have a Gaussian distribution (it's bimodal), so where the median or mean wouldn't tell you much about your likely outcome. It is for first year lawyer salaries, and you can find it at
http://www.elsblog.org/the_empirical_legal_studi/2007/09/distribution-of.html
In general, an average doesn't say anything about an individual outcome, but here the average doesn't say much about the outcomes in general.
Is this the Aharoni passage you were thinking of:
A large part of what I learned wasn't new facts, but something completely different: subtleties. It was like looking at a piece of cloth - from afar it seems smooth and uniform, but up close you discover that it is made of fine, interwoven threads. What I believed to be one piece turned out to consist of a delicate texture of ideas. More importantly, I realized that to be a good teacher one must be familiar with the fine elements and the order by which they are interwoven.
-Ron Aharoni, Arithmetic for Parents - Introduction (p. 3)
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