Patterns+and+Narratives


 * Humans are pattern-seeking, story-telling animals. Most are familiar with projective psychological tests like the Rohrshach [[image:rohrshach.jpg align="right"]]test. Individuals are asked to see patterns in a random inkblot like the one on the right. Even though there is no real pattern, each person can "see" something in the blots just like people see shapes in clouds. The pattern we see may say more about the observer than the object.

Since we are "pattern-seeking" animals we tend to see things that may not be there. The same is true of information we get from our social and natural world. For healthcare providers, physical symptoms can be elements of a larger syndrome, or not. For teachers, a student correctly choosing "A" on a multiple choice test may demonstate knowledge of the material or just a really lucky guess.

Humans also like to tell stories or "narratives." Stories gives actions and events context and meaning. Take the following sentence.

Jack killed Jill.

By itself, this simple sentence does not tell us a lot. Immediately, you may think Jack is pretty bad because killing is bad, but we don't know about the context, the motivations, the setting to inform us about making a judgment. For example, perhaps Jill is a terrorist and Jack is a soldier, then perhaps Jack is a hero. Or, Jill is suffering from a terminal disease and Jack is an anguished spouse who, against his own wishes, does what Jill asks. Or, like Othello & Desdemona, Jack killed Jill because he was tricked into believing Jill was unfaithful and "he loved not wisely, but too much." Or, Jack is sent from the future to stop Jill the scientist from inventing a technology that will lead to the end of the human race. As humans, we not only construct narratives in fiction, but also create self-narratives that provide explanations for our own actions.

While being pattern-seeking, story-telling animals can lead to self-deception and self-subversion, it also provides us distinct advantages. Identifying patterns allows us to perceive reality and use language in a way unlike any other species. Telling stories allows us to learn over generations that has made humans today much more advanced than a human 10,000 years ago, while a chimpanzee today is probably just as advanced as a chimpanzee from 10,000 years ago.

This page is about our ability to use visual and mathematical tools to understand patterns in social science and economics and how they support or undermine the narratives we tell about ourselves and our societies. **


 * FUNCTIONS

A //statistic// is a single measurement like your height at age 18 or the average grade in geometry. A //variable// a vector of individual statistics such your height at ages 1, 2, 3,. . . or average grades in geometry across high schools in Westchester and Putnam counties. Key thing: Variables vary, statistics do not. Statistics are photographs, variables are movies.

Functions define the mathematical relationship between two variables, //x// and //y//. For our purposes let's say that //x// is the hours students studied for a test and //y// is their grade on that test. First, we can hypothesize two relationships:

1) Studying more will increase your grade on the test or ** **// y = x // 2) Studying more will decrease your grade on the test or //y = -x//

These two functions are //linear//, meaning that for every in the value of //x,// the value of //y// increases a fixed amount. Examples of linear functions are // y = 3x // or // y = -5x //. Functions can also be //non-linear// meaning that the transformation of one variable into another increases or decreases as the first variable increases. In economics, //non-linear// functions include patterns including // increasing returns to scale, diminishing returns to scale ,// and // diminishing marginal utility //. Increasing returns to scale are usually written as exponential or power functions such as // y = x^ // 2 or // y = 10^2x -1 .// The most famous example of increasing returns to scale is Adam Smith's idea of the //Division of Labor//. When one organizes work in a division of labor the productivity of the each individual worker increases exponentially as more workers are added to the process. Decreasing returns to scale are written as logarithmic functions such as // y = ln x //or //y = 10 log x//

To go back to our grade example, some argue that studying a foreign language helps one learn to read, write, and understand English and vice versa. Personally, I learned more about reading and writing English by studying 3 years of Japanese than I did from my 12 years of elementary and secondary English. If this is true, that means that one hour studying both English and Japanese together will have a positive //non-linear// effect on my English grade. Conversely, studying while listening to your I-Pod Nano may have a negative //non-linear// effect on the relationship between hours studied and your grade. Some students claim -- erroneously -- that listening to music makes them work better, but what they may really is that they can study //longer// while listening to music. This does not mean they are studying more effectively. The slope of the function tells you how effectively / efficiently you are studying -- how much your grade improves for each hour studied -- while the number of hours studied tells you what the value of //x// (hours studied) for an individual student. **

**A good comparison is the economies of China and the USA, the second largest and largest economies in the world respectively. In total, the economies are both large, but China has 4 to 5 times the population of the United States. Obviously, if you have 5 times the number of people, you should be 5 times more work. However, if you looked at the economy in per capita terms, the average Chinese earns about $3000 to $4000 per year, while the average American earns $30,000 to $40,000 per year. Political units come in all shapes and sizes, and therefore, aggregate comparisons sometimes only tell you that a country is bigger.

Functions are not just string of numbers and letters, but can tell an important story if used effectively.**