Tag: math

Social Loafing: Getting Groups to Work Well Together

PsyBlog has an excellent summary of the research on social loafing, the phenomena where people working in a group work less compared to when they work alone. Because we do so much group work, this is sometimes an issue.

The first research on social loafing came from Max Ringelmann way back in 1913 (Ringelmann, 1913). He had people pulling on a rope, and compared the maximum they could have pulled, based on individual test, to how much each person actually pulled. The results were, kind of, sad; with eight people, each one only pulled half as much as their maximum potential strength. A graph of Ringelmann’s data is shown below. If everyone pulled at their maximum the line would have stayed horizontal at 1.

The relative loafing of people working in a group. As the group gets larger, the amount of work per person decreases from its maximum of 1. Data from Ringelmann (1913)

The PsyBlog article points out three reasons why people tend to loaf in groups:

  • We expect others to loaf so we do it, too.
  • We feel more anonymous the larger the group, so we feel less need to put in the effort.
  • We often don’t have a clear idea about how much we need to contribute, so we don’t put in as much as we could.

This can be summed up in Latane’s Social Theory:

If a person is the target of social forces, increasing the number of other persons diminishes the relative social pressure on each person.

— Latane et al., 1979: Many hands make light the work: The causes and consequences of social loafing in the Journal of Personality and Social Psycology. Quote via Keith Rolag’s Website.

How do we deal with this

The key is making sure students are motivated to do the work. We want self-motivated students, but creating the right environment, especially by training students in how to work in a group will help.

  • Make sure students realize the importance of their work; this makes them more motivated.
  • Build group cohesion; team members contribute more if they value the group they’re in.
  • Make sure the group clearly and fairly divides the work. Let everyone be part of the decision making process so students have choices in what to do will help them be more invested in their part of the work.
  • Make sure each group member feels accountable for their share of the work.

A Brief Excursion into Mathematics

Ringelmann’s data falls on a remarkably straight line, so I used Excel to plot a trendline. As my algebra students know, you only need two points to write the equation of a line, however, Excel uses linear regression to get the best-fit line through all the data. Not all the data points will be on the line (sometimes none of them will be on the line) but the sum of the distance from each point to the line is minimized.

Curiously, since the data is pretty close to a straight line, you can extend the line to the x-axis to find out how many people it would take for no-one to be exerting any force at all. Students should be able to determine the equation of the line on their own, but you can get Excel to give you the equation of the trendline. From the plot we see:

y = -0.0732 x + 1.0707

At the x-axis, y = 0, so;

0 = -0.0732 x + 1.0707

solving for x we first subtract the constant, 1.0707 from both sides to get:

0 – 1.0707 = -0.0732 x + 1.0707 – 1.0707

giving:

-1.0707 = -0.0732 x

then divide by -0.0732 to isolate x:

! \frac{-1.0707}{-0.0732} = \frac{-0.0732 x}{-0.0732}

which yields:

x = 14.63

This means that with 15 people, no-one will be pulling on the rope. In fact, according to this equation, they’ll actually start pushing on the rope.

It’s an amazing result, but if you can find flaws with my argument or math, please let me know.

Exercise on Wealth Distribution

Using the actual U.S. wealth distribution data from Norton and Arieli (2011; pdf), I created a little addendum to our exercise on the distribution of wealth.

I started with the definition of wealth. Students tend to think you’re referring to annual income, so I gave the example of someone who does not have a job but owns a house; they have no income but some wealth in the value of the house. Alternately, someone who has $2 million in the bank, but owes $4 million, actually has negative wealth.

Then I drew a little stick figure diagram to represent the population of the United States. With ten figures, paired up, that gives five parts, aka quintiles.

Breaking the population of the U.S. into five parts (quintiles) based on wealth. The least wealthy are to the right and the most wealthy are to the left.

Students were then presented with an empty bar graph and asked, “How much of the U.S.’s wealth is owned by the wealthiest 20% of the population?” Instead of asking in percentages (as are shown in the graph), I asked them to assume that the total wealth in the U.S. is $100 trillion.

Population with empty bar graph.

The first suggestion was $35 trillion, which is shown below. Others offered different amounts, ranging up to $50 trillion. Someone even suggested $15 trillion, which is not possible, since that would mean that the wealthiest 20% have less than 20% of the total wealth of the country.

If the wealthiest 20% owned 35% of the wealth of the U.S. the graph would look like this.

Once they got the idea, I showed them what the graph would look like in an idealized socialist country, where everyone had the same wealth.

An even (socialist) distribution of wealth.

Finally, I asked my students to fill in what they believed to be the actual case for the U.S. for all five quintiles. The results had to add up to $100 trillion. They gave me their numbers individually before we broke up our meeting, and I entered it in the U.S. distribution of wealth spreadsheet to produce a graph.

After lunch, I showed them the results.

Students' beliefs about the distribution of wealth in the U.S. (S1 through S10 and the average student response), compared to an equal distribution and the actual distribution (bottom).

For dramatic effect, I hid the last two bars at first. We talked over their numbers, then I showed them the equal distribution case (which they’d seen before), and finally the actual distribution.

Actual U.S. distribution of wealth. Data from Holder and Arieli (2011)

The response was salutary; a moment of surprised silence and then whispers. What then followed was a nice, short discussion. I pointed out the pie charts showing the U.S. versus an equal distribution, versus Sweden and asked what they would do, if they were an autocratic monarch, or if they were the president to make the U.S.’s distribution more equal.

How wealth is shared in the U.S. compared to and equal distribution (middle), compared to Sweden. Image adapted from Norton and Aireli (2011).

We talked about the government just taking private property, like the communists did. Then we talked about progressive taxation. We ended by talking about the estate tax, and meritocracy, which we’d touched on in the morning.

I thought the exercise worked very well. Not only did we get into an interesting economic issue, but got some practice with math and interpreting graphs too.

Beating probability

Since we just finished doing a bit of probabilities in math, here’s an article about how one guy figured out how to beat the lottery.

The first lottery Mohan Srivastava decoded was a tic-tac-toe game run by the Ontario Lottery in 2003. He was able to identify winning tickets with 90 percent accuracy.
–Lehrer (2010) in Cracking the Scratch Lottery Code

However, he decided not to just try to get rich of what he’d discovered. It’s an example of using the power of math for good:

“People often assume that I must be some extremely moral person because I didn’t take advantage of the lottery,” [Srivastava] says. “I can assure you that that’s not the case. I’d simply done the math and concluded that beating the game wasn’t worth my time.”

As a side note, my philosophy about the lottery is that it’s basically a tax on the poor:

[H]igh-frequency players tend to be poor and uneducated, which is why critics refer to lotteries as a regressive tax. (In a 2006 survey, 30 percent of people without a high school degree said that playing the lottery was a wealth-building strategy.)
–Lehrer (2010): Cracking the Scratch Lottery Code

Graphing discussion threads

Graphic representation of the Wikipedia discussions about deleting articles. The image links to an interactive version of the graphic at http://notabilia.net .

Swings to the right are arguments for keeping the article, swings to the left are arguments to delete them. Moritz Stefaner and others’ website have created this wonderful graphic of Wikipedia’s discussion threads. They have lots more details and discussions on their website.

U.S. Immigration Data

Raymond Cohn has a great table of immigration data on the Economic History Association website.

This data ties very nicely into the work we’re doing on graphing. The Excel file with the post 1820 data, and another with pre-1790 data, make it easier to work with (note the pre-1970 data comes from the Wikipedia page on the history of immigration; it was the easiest source to find a table of data).

Since each small group of students is responsible for a different wave of immigration, the groups will create bar graphs showing the countries of origin for each wave. They should look like these:

U.S. Immigration from 1820 to 1831. Data from Cohn (2010).

and,

U.S. Immigration from 1900 to 1914. Data from Cohn (2010).

Plotting the time series as a line graph would be another great way to slice the data:

Comparison of U.S. Immigration Rates from Great Britain and Central Europe. Data from Cohn (2010).

Note that the data in the table is as a percentage of total immigration, so the numbers do not compare directly from one time period to the next; however, the proportions still work to show the same patterns.

Graphing calculator pro. – Free app

Until tomorrow (Jan. 5th, 2011), the Graphing Calculator Pro iPod app is free (thanks Josh). It can do a lot, so it might take a minute figure out how to do the graphs and it can be a little fiddly (you can scale both axes separately, for example, which is nice unless you don’t intend to), but I like it. After just a couple of minutes, the student who found it was already playing with slopes.