Tag: formal thinking

Statistical significance

Normal distribution with 95% unshaded. Adapted from Wikimedia Commons.

A discussion of statistical significance is probably a bit above middle school level, but I’m posting a note here because it is a reminder about the importance of statistics. In fact, students will hear about confidence intervals when they hear about the margin of error of polls in the news and the “significant” benefits of new drugs. Indeed, if you think about it, the development of formal thinking skills during adolescence should make it easier for students to see the world from a more probabilistic perspective, noticing the shades of grey that surround issues, rather that the more black and white, deterministic, point of view young idealists tend to have. At any rate, statistics are important in life but, according to a Science Magazine article, many scientists are not using them correctly.

One key error is in understanding the term “statistically significant”. When Ronald A. Fisher came up with the concept he arbitrarily chose 95% as the cutoff to test if an experiment worked. The arbitrariness is one part of the problem, 95% still means there is one chance in twenty that the experiment failed and with all the scientists conducting experiments, that’s a lot of unrecognized failed experiments.

But the big problem is the fact that people conflate statistical significance and actual significance. Just because there is a statistically significant correlation between eating apples and acne, does not mean that it’s actually important. It could be that this result predicts that one person in ten million will get acne from eating apples, but is that enough reason to stop eating apples?

It is a fascinating article that deals with a number of other erroneous uses of statistics, but I’ve just spent more time on this post than I’d planned (it was supposed to be a short note). So I’d be willing to bet that there is a statistically significant correlation between my interest in an issue and the length of the post (and no correlation with the amount of time I intended to spend on the post).

From novices to experts

Socrates teaching (from Wikimedia Commons).

The primary role of an instructor is to transform a novice into an expert within a given subject area. – Cooper (1990)

The above quote comes from a paper on instructional design by Graham Cooper. I don’t quite agree with it entirely since it does not seem to allow for a well rounded view of a student as an individual, or the Socratic ideal, but it does seem applicable to the more strictly academic areas in the middle school curriculum.

In order to figure out what distinguishes experts from novices, cognitive scientists have spent a lot of time observing the two groups. Their key finding has been that as you become an expert on a topic, you construct mental pictures (or schemes) of the shapes of problems, so when you encounter a new problem you can just fit the new problem to the mental pictures you have and see which best fits. It’s a bit like learning rules of thumb that apply to different situations. When a problem comes up, the expert can quickly whip out the right rule of thumb from their mental back pocket while the novice, though equally smart, needs to figure out all the steps with some degree of trial and error.

This is a nice perspective when it comes to teaching something like solving equations, but I think one important distinction of the Montessori philosophy is the belief that adolescents should also be learning flexibility, and be capable of dealing with novel problems. Because adolescence is when students are just becoming able to think abstractly (at least according to Paiget), and abstract thinking needs to be practiced, it is necessary that students encounter novel, challenging problems on a regular basis.

Pangea breakup in reverse (adapted from image in Wikimedia Commons).

A lot of creative and problem solving thinking comes from hashing out new problems. In a globalized world, where technology is capable of dealing with routine tasks, be they constructing a car or solving a series of equations, creative problem solving is becoming a more and more valuable skill. Especially now that “The World is Flat“.