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Suppose you had to teach averages to a teenager. For arguments sake let the question be;

Find the mean of $2, 3, 4, 7$

Of course the simple answer is $\frac{2+3+4+7}{4}=4$ but in my experience the student only reinforces their addition and division skills and is left bereft of any conceptual understanding of mean.

Context and comparison helps;

The length of male mice tails are $2cm,3cm,4cm,7cm$

The mean average female tail is $3.9cm$.

Compare the mice tails.

I would like to know ways in which I can instill conceptual understanding of averages. I'm always left with a feeling students just calculate and don't really understand why they are doing it.

Perhaps worse is range. Practically all students will tell be it's highest take lowest but almost never attach any sensible meaning to it.

How do you teach averages and not the simple calculations?

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    $\begingroup$ Give them examples to solve they would like to solve like give them example of average runs in cricket match, average height in their class (including male and female students). That might be intriguing for them. $\endgroup$ – Sufyan Naeem Mar 21 '15 at 9:46
  • $\begingroup$ It's a good start. I often get them to guess the number of dots on a screen shown for 5 second and ask who's better at guessing boys or girls? I want to improve their robustness in making sensible decisions in new situations. $\endgroup$ – Karl Mar 21 '15 at 10:49
  • $\begingroup$ However it's the way to upgrade their IQ but I will not say that you are wrong but It would be more better to make them lovers of Math first by telling them its number of applications in the Real World..Well it feels good that you are a caring 'n' daring teacher. I think you should spend 4-5 days to make them compatible with average. That is it I can say about that. $\endgroup$ – Sufyan Naeem Mar 21 '15 at 14:46
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    $\begingroup$ I have noticed on many occasions that if you ask a student "What number is exactly halfway between $8$ and $34$?" they will first compute $34-8$ to find the distance between the two numbers, then divide the result by $2$, and then add that to $8$. If you instead ask "What is the average of $8$ and $34$?" they will add the two numbers together and divide by $2$. The fact that these two questions have exactly the same answer always comes as something of a surprise to them. $\endgroup$ – mweiss Mar 22 '15 at 16:22
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    $\begingroup$ I'd suggest making a separate question about ranges. The title of this question, and almost all of the question text, is about averages, and it seems better to keep the two topics separate (but of course you can include links from each to the other). $\endgroup$ – mweiss Mar 22 '15 at 17:59
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BeamMean
$\mbox{}\mbox{}\mbox{}\mbox{}\mbox{}$

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  • $\begingroup$ I like the idea of a balancing point. Is there any programme that'll do this for large data sets? $\endgroup$ – Karl Mar 21 '15 at 15:07
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    $\begingroup$ @Karl: You might attempt this physically, with a beam, fulcrum, and weights. $\endgroup$ – Joseph O'Rourke Mar 21 '15 at 15:10
  • $\begingroup$ Chose this because it has a one point to represent all to it without the calculations. $\endgroup$ – Karl Apr 4 '15 at 7:46
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Another interpretation that can help make the formula meaningful is "equal redistribution". Say a group of 4 friends have (respectively) \$2, \$3, \$4, and \$7. How much money would they each have, if they all had the same amount of money as each other without changing the total?

Or: Tom makes hand-made wooden spoons, which he sells at a local farmer's market. At the most recent market he sold 2 spoons in the first hour, 3 in the second hour, 4 in the third hour and 7 in the fourth (and last) hour. If you imagine all of those spoons being sold at a steady rate over the course of the market, how many would be sold per hour?

In both cases you find the answer by summing the number of "things" (dollars or spoons) and dividing by the number of "places" (people or hours), and end up calculating the mean. The idea that underlies both of these examples is "uniformization" -- you take some distribution of stuff that has an irregular pattern to it, and describe what it would be like if things were equally distributed.

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  • $\begingroup$ (+1)Thanks for the answer uniformity is another spin on mean. $\endgroup$ – Karl Apr 4 '15 at 7:49
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Once upon a time, I designed some versions of the following task to bring about the algebraic nature of 100 number grid. But, @mweiss answer gave me the idea of the possibility of using it for what you've asked for. Thus, it is clear that I've never used it to teach the "average", but I thought it is worth sharing.

Suppose three children are standing on different numbers on a 100 number grid (if you are a school teacher you can indeed make such a suitable-for-standing number grid in schoolyard). The task for them is to move to and meet at the same cell of the number grid without changing the initial sum of the three numbers they were standing on.

It is only work when the average is a natural number. But, it is nice to see that to solve the problem both the smallest number and the biggest number should "move" towards the number that later on will be called mean. As a result, your students may also experience some aspects of "range" as well.

PS. Of course, you can easily use this idea with paper number gird and counters.

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    $\begingroup$ I really like this idea, especially if the rules specify that all students must move simultaneously and in such a way that the total never changes throughout (as it is currently worded, it could be interpreted to mean simply that the ending sum must be the same as the initial sum). Then the students will have the experience of systematically replacing a set of numbers with a new set of numbers that has the same mean but smaller range. $\endgroup$ – mweiss Mar 23 '15 at 16:32
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    $\begingroup$ Great idea for a collaborative task. I think I'll add this task to the scheme of work. Thanks. $\endgroup$ – Karl Mar 23 '15 at 18:26

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