Wiggins question #7

Question 7

Most teachers assign final grades by using the mathematical mean (the “average”) to determine them. Give at least 2 reasons why the mean may not be the best measure of achievement by explaining what the mean hides.

All measures of center hide variation.

This is what makes them useful, and it is what makes them problematic.

Using the mean makes zeroes a problem in grading. Wildly divergent values (such as a zero in a gradebook) will greatly affect the mean. It is hard to argue that 2 A’s and a zero is the same as consistent D work. Yet this is how the mean plays out.

But going too far down this road will only lead to critiques of the whole system of grading students at all. I find that system to be indefensible and counterproductive. I have made my peace with it, and I try to do as little harm as possible with the responsibility I have to assign grades in my work.

All of which is to say, it is not using the mean that leads to a poor measure of achievement. It is mistaking quantitative measures for accurate ones that leads to a poor measure of achievement.

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Experiments with datasets

Go out and collect a modest-sized, discrete dataset. Name lengths of all of the students in your classroom, say, or the number of people in each of their households.

This bar graph is only tangentially relevant, being more of a case-value plot of four different populations. But it breaks up a texty post. So deal with it.

Now play with that data.

If we add one or more new (hypothetical) cases, can we…

1. Increase both the median and the mean?
2. Decrease both the median and the mean?
3. Increase the mean while decreasing the median?
4. Vice versa?
5. Increase the Mean Absolute Deviation (MAD) while decreasing the mean?
6. Vice versa?
7. Decrease both the MAD and the range?
8. Decrease the MAD while increasing the range?
9. Vice versa?

If we delete one or more actual cases, can we…

1. [same list as before]

Thanks to Susan Friel, Connected Mathematics and tons of other creative folks for getting me started with this. Future elementary teachers to tackle this shortly. I’ll report back.