This semester I find myself again in the enviable position of teaching statistics to psychology graduate students. My cohort is over age 30 and has not studied math for more years than we can count. So how do we teach them? I can tell you what not to do!
First, don't assume they know anything, even what an average is! Assume no knowledge until proven otherwise! We don't currently have a placement exam, though I have recommended one now, since the assumption that undergraduate statistics has been retained has proven false. I am finding myself teaching basic ideas, dividing cake, flipping coins, telling them that statistics has to do with proportionality.
Second, assume no interest, either! My second surprise was to learn how uninterested my students were in statistics. I suppose I should have known better since anyone with any interest would know something and these students largely knew nothing. To know nothing about statistics, as common as it is in modern life requires an active effort to avoid learning. I should have suspected this. We all have stories to explain our behavior and avoiding statistics is no exception. These stories included "I don't do numbers"; "Knowing statistics has nothing to do with being a good psychologist"; "I can't do math"; "My brain can't comprehend math"; and "I don't have time for this", among other good summary lines.
Math education in North America is seriously flawed and biased against women. We know this. Malcolm Gladwell explored math education in Asia and discovered that most of the advantage that Asian students have in understanding math over their North American counterparts comes from their going to school around the entire year and not taking a summer vacation in which they forget what was learned the preceding year. Apparently Asian students didn't have to stop school to help their family with the planting, growing, harvesting, butchering, and other farm chores. Math education has changed, however, even if summer vacation is still observed. We now teach math visually and kinesthetically. We use Lego - to model probability distributions. We cut pieces of pie to teach children about fraction and percent. And, we try to make it interesting.
To be most successful, science education has moved to problem-based learning. Except in the most conservative bastions of pre-med student screening in which courses are designed to fail more students than pass, we've abandoned rote memorization as a technique. One, there's too much to memorize. Two, no one remembers what they memorize after the test. Studies have shown that lectures using power point and other visual aids result in 15% the retention of knowledge that occurs when students work together in small interactive groups to solve a problem. Therefore, statistics is being approached as learning how to solve problems together, interactively.
The problem solving approach more closely mirrors how science and math are really done and how they arose. The neat linear textbook with tight principles and theorems that people of my age encountered in high school geometry and algebra is an artifact. It's a story made up years later to explain what happened in a way in which it clearly never happened. Statistics, for example, was born in the gambling dens of France. Noblemen were losing their shirts (and estates) at the gaming tables. They came to mathematicians like Pascal and de Moivre to solve their problem. A famous initiating problem was, "what are the odds of rolling 4 sixes in a row with a die?". No one had thought about this before, so experimentation was required. The mathematicians rolled dice and collected data. They didn't actually go into a state of deep meditation and receive the answers from another dimension (though that's been known to happen in science and math as in the solution to the problem of the structure of the benzene ring or a recent development in the theory of black holes). They collected data and examined their results for patterns. This is what we humans do very well. They counted the number of times that each combination of dots on the dice appeared. Pascal invented a triangular table for predicting the number of times any number would occur given successively increasing numbers of rolls of the dice. Of course, it's called Pascal's table. In that table the expected number of times that a "3" for instance would appear in 20 rolls would be the sum of the two numbers above and adjacent (to the right and to the left) of the desired number. Wow, who knew that would happen! Then they could really inform the noblemen what their odds were of success. And, of course, we all know the answer -- don't gamble; odds always favor the house. This is one statistical result that almost everyone in North America has heard; though not many follow its advice. And, actually, there's another way to interpret the results, which I follow. If the odds of winning the lottery are quite small, then buying enough tickets to make a difference would be prohibitively expensive. Therefore, buy one ticket and ask the spirits (Forces, God, Ancestors, Lady Luck, etc.) to rig the game and help you win. This is my approach each week. It hasn't worked yet, either, but then, neither have I lost much money. The Gallup polling organization uses a similar approach. Instead of increasing the number of people they poll over 4,000, they work at reducing the bias and the error from how they select the people which they poll -- a much less expensive strategy.
Statistics, then, was discovered as a way to answer practical questions. At the Guinness Brewery, for example, a statistician named Gossett invented a way to reduce the number of beers that had to be sampled (drank!) to do quality control. Apparently, the makers of Guinness were so incensed that anyone would suggest that any of their beers were not perfect, that they demanded that Gossett publish his results under a pseudonym. He chose the name Student -- hence, Student's t-test. Using the t-test and it's t-table, Guinness could waste fewer beers on their employees and still achieve an acceptable degree of quality control.
The problem, I discovered with my students, is that they wished certainty. They wanted to know exactly how things worked including the basic principles for going from a to b to c before they attempted to solve any problems. I suspect this is a function of age. My younger acquaintances handle problems very differently. If given new software, for example, my son tries everything to see what it does. He'd never consider reading a manual. He just plays until he feels like he knows what it does. He doesn't have the belief that many people my age have -- that we will somehow screw up things. He comes from the generation that simply knows that pushing the "reset" button will solve everything and we just start over again. My generation is not so sure of that.