Wednesday, December 22, 2004

No Wonder They Teach Like That

Last night I was trying to sketch out a rough calendar for my class for pre-service elementary teachers this spring. The master syllabus packs in way more material than my students can learn well, so I'm trying to identify sections that can be skipped. Selfishly, I've decided to leave out the topic on normal distributions and standard deviation. The book's presentation is so awful that it would take a lot of time (time that I won't have) to reframe the material in a reasonable way -- this would be especially challenging as I've never taken a real statistics class.

Here's how the book introduces the standard deviation:
Two other commonly used measures of dispersion are the variance and the standard deviation. These measures are based on how far the scores are from the mean. To find out how far each value differs from the mean, we subtract each value in the data from the mean to obtain the deviation. Some of these deviations may be positive, and others may be negative. [...] Squaring the deviations makes them all positive. The mean of the squared deviations is the variance. [...] [W]e take the square root of the variance and obtain the standard deviation.

The steps involved in calculating the variance v and the standard deviation s of n numbers are as follows:
  1. Find the mean of the numbers.
  2. Subtract the mean from each number
  3. Square each difference found in sep 2.
  4. Find the sum of the squares in step 3
  5. Divide by n to obtain the variance v
  6. Find the square root of v to obtain the standard deviation, s.
REMARK: In some textbooks, this formula involves division by n-1 instead of by n. Division by n-1 is more useful for advanced work in statistics.
Yes, the standard deviation is a mysterious incantation where you feed in the data and get out a number whose meaning is just some vague "measure of dispersion." It is presented as a solution in search of a problem -- the book continues on to calculate (by hand!) the standard deviation of populations that each contain six data points. Normally I'm not a big fan of the calculator/gratuitous use of technology in the math classroom, but for statistics topics I think it's a valuable resource. I'd rather have my students put effort into figuring out what the standard deviation means and what it is used for. How it's calculated is just not a priority.

I think there's a two-fold value in teaching these pre-service teachers about statistics. The first, obviously, is that a certain amount of basic statistics is part of my state's standards. But with all the standardized testing that is being done in the schools and all of the assessments of teaching effectiveness that are made based on the students' scores, teachers should have the skills necessary to double-check the conclusions drawn about the amount of learning that occurs in their classrooms. But describing the standard deviation solely in terms of its computational algorithm is not helpful.