Mathematics of Social Justice

Tuesday, September 12, 2006

Update in the 3rd week

Well, I'm in the third week of my writing course, and I have lots of work looming ahead, so this seems like a propitious moment to share my experiences to date. The course meets in two sections of 16 & 17 students in 75 minute periods on Tuesdays and Thursdays. The introductory module is an introduction to quantitative literacy (QL), for which the students are reading Mathematics and Democracy and What the Numbers Say and writing a personal assessment of their own quantitative literacy. The readings seem to be appreciated by the students, but it seems to be a minority of the class that is actually doing them. Despite this, the opportunity to share personal QL experiences has kept classroom discussions lively.

The time I spend teaching the students to write is substantial, and perhaps the big surprise to me in actually teacing the course. We are using the St. Martins Handbook as a writing text, and the students seem to need a great deal of assistance in writing.

I am creating a module, like the one for Fair Division, chronicling the assignments and the discussion questions I have worked up for class. I hope to post it when the module is finished next week, when the students hand in their personal QL assessment.

Wednesday, August 02, 2006

Course Module on Fair Division

The result of the Fair Division Working Group's summer efforts is a module for the First-Year Seminar being taught at Lafayette in the fall. It is available as a pdf file through this link.

Thursday, July 06, 2006

Annotated Bibliography -- How I did it

It's been pointed out that folks might be interested in what I did to create the annotated bibliography. First of all, I used Google Scholar's forward citation linking to find 176 sources that cited Brams and Taylor's seminal book "Fair Division: From Cake-Cutting to Dispute Resolution." Then I went through that list and found the few that made connections to issues of social justice. Lafayette has hooks into Google Scholar that allow one to automatically search our collection for a source, and order it via ILL if we don't have it. Most articles recent enough to cite "Fair Division" (published in 1996) have abstracts available online, and many have full text available, as well. In any case it didn't take me more than a few hours to decide which of these 176 I thought were salient to our work. Lafayette also subscribes to RefWorks, which is an online citation management package that imports information (although not perfectly) from Google Scholar with a single click. So I have a RefWorks copy of my (unannotated) bibliography, if anyone is interested.

To create the annotated version, I chose to use LaTeX/BibTeX, including the handy application BibDesk (for Mac; can anyone suggest a good Windows analogue?). RefWorks exports (once again, not perfectly) to BibTeX format, so I was easily able to create an unannotated version of the bibliography. I found the skeleton for an annotated bibliography in LaTeX out in cyberspace; the key ingredient is the modified BibTeX style file that uses the annote field in a BibTeX record to create annotations in the LaTeX output. There is one small annoyance to this solution: in multi-paragraph annotations, a blank line is ignored; one must insert a "\par" command to cause a new paragraph.

Altogether this took about a day of my time. I was able to do so much that quickly mostly because of the ease of integration of Google Scholar, RefWorks, and BibTeX. The links here will make the job go more quickly, but in fairness, there aren't many other groups with topics narrow enough that the sources citing a single book will yield a reasonable survey. The blog post for 4 July includes a link to the workshop website page where there are links to the resulting *.pdf and *.bib files, but here are direct links from blog to files.

Tuesday, July 04, 2006

Fair Division Bibliography Posted

I have finally gotten a draft bibliography on Fair Division together. It exists primarily as a BibTeX file, but non-LaTeX'ers will appreciate that it is easy to make a hard copy, which is posted as a pdf on the workshop web site. The link is on the Sample Curriculum page. Most of the work went into the bibliography, not the link. It's just a sentence or two below the list of topics covered near the top of the page.

Although the bibliography doesn't make this clear, there aren't many sources that make a solid connection between fair division, as expounded by Brams and Taylor, and social justice. Brams in particular has gone to great effort to attempt to cement a connection between these two, for instance in his 2004 article and his joint work with Denoon (Google puts his name in all lower-case letters). Right now, the most promising articles by other authors are by Gersbach (2004) and Haugestad (2003). The other articles cited are all either more mathematical or deal with fair division only tangentially.

I still haven't read all these materials; when I do I'll be able to make final judgements about what to include in my course, and hence in the module I'm preparing.

Monday, July 03, 2006

A Possible Exercise

While cruising the web, I found the site for gapminder, a Swedish organization whose vision is "Making sense of the world by having fun with statistics!" The 2005 presentation on human development trends seems to fit that vision admirably. This is an excellent presentation that was given only slightly modified at the 2006 TED conference. The TED version includes some helpful suggestions for classroom activities to accompany the presenations, as well as a very helpful audio narrative. This is a great tool for understanding international development.

I was inspired by this to create the following exercise. I am not sure that I will use it in my class, as it leads in directions I don't intend to go. Nonetheless, it is thought-provoking, and raises clear issues of social justice. The problem from my point of view is that those issues are not ones I intend to cover in my course. Here is the exercise:


Below are groups of countries, grouped by their 2003 child mortality rate. All the countries listed have child mortality of 1% or less, meaning that in all these countries, 99% (or more) of newborns live to 5 years old. In each group, the countries are listed from poorest to wealthiest, as measured by GDP per capita.

Try to list these groups from lowest child mortality rate to highest, and put the United States in its group.

  • Singapore, Sweden

  • Czech Rep, Slovenia, Spain, Italy, Japan, Iceland, Denmark, Norway

  • Korea, Portugal, Cyprus, Greece, Germany, France, Finland, Austria, Switzerland, Luxembourg

  • Malta, Israel, New Zealand, United Kingdom, Australia, Canada, Ireland

  • Malaysia, Poland, Croatia

  • Cuba, Slovakia, Hungary, United Arab Emirates

  • Chile, Estonia, Kuwait

  • Costa Rica

Answer: The groups are already in order from lowest to highest child mortality. The highest group has child mortality of 0.3%, and each successive group has higher child mortality by 0.1%. These numbers all seem very small, but realize the implications. Costa Rica has double the child mortality of Korea, or any of the other countries in Korea's group. This is not too surprising, given that Costa Rica's GDP per capita of $9,080 is slightly more than half of Korea's $17,000, and Korea is the poorest of the countries in its group.

Sadly, the United States falls in the group with Cuba. Cuba has GDP per capita of $5,400, while the United States has $35,500, close to seven times higher. The only other countries with GDP per capita higher than $30,000 are Ireland, Norway, and Luxembourg, and they all have substantially lower child mortality rates. Norway has half the child mortality rate of the US with roughly the same GDP per capita, and this is accomplished by several other, poorer countries, including the Czech Republic, which has a GDP per capita of $15,500, less than half that of the United States.

It is obvious that most/all of the countries that offer better survival prospects to their infants have national health insurance, indeed, very few countries with wealth comparable to the US do not. (Can anyone name one other than the US? I can't.) But the lurking variables in the situation lead in directions I am not familiar with. What are the leading causes of infant/toddler mortality? What steps are the countries with the best records taking? Why are they successful? What are the arguments against taking them in the US? All of these are fascinating questions that rise naturally from this simple exercise, and I can't answer any of them.

One tack I did briefly investigate: The issue of abortion. I found data on abortion from most of the countries mentioned in the exercise at, and plotted abortion ratios against child mortality. (Abortion ratio is the number of abortions per thousand live births in a specified polity over a specified time period.) There is very little association at all between these two statistics. The association appeared to me to be slightly negative, but with strong outliers that would probably render the correlation positive. In short, there is not much here without considering the myriad other lurking variables (especially the level of legal access to abortion and the cultural/legal status of women more broadly) that affect the prevalence of unwanted pregnancy and the access to abortion as a means to end them. In general, abortions for the purpose of ending pregnancies that are expected to end in the birth of a child with serious health problems seem to be a small fraction of all abortions performed, and so there seems to be little association between abortion ratio and child mortality.

Friday, June 16, 2006

Update from the Fair Division Working Group

Right now, the Fair Division working group is me with a little help from Ron (Thanks, Ron!), so the module we will develop is going to be for my first-year seminar course. That means the mathematical content will be minimal, but there will be some writing expected of the students. I intend to use the book "The Win-Win Solution: Guaranteeing Fair Shares to Everybody" by Steven J. Brams and Alan D. Taylor. The book was published in 1999, but I intend to use the copyright 2000 paperback edition published by W. W. Norton that lists for $14.95, and sells at Amazon for $9.72.

I haven't seen the book yet, but I have an idea of what it contains. The general idea is to divide up a prescribed collection of goods (or bads) amongst interested parties. The canonical example is dividing a cake and assigning the pieces to children. The primary goal for division method is to make the division (and assignment) envy-free, that is, each child believes that their piece of cake is at least as desirable as any other piece. For a uniform cake (and uniform children) this would mean all the pieces are the same size, but for a cake with different flavors that might appeal to different children, it might require sizes that are unequal. This also offers the possibility of _each_ party receiving a part that it perceives as "generous," that is, the child values the assigned piece with more than 1/n of the value of the entire cake, where n is the number of children.

When different parts of the cake are assigned different values by different children, the concept of Pareto-optimality comes into play, and here it is given the name efficiency. That is, we would like to maximize the total value that the parties assign to the pieces they receive. As one might expect, the constraint of envy-freeness might prevent the achievement of the Pareto-optimal solution, but it is often possible to optimize the total value within the constraint of envy-freeness.

A refinement of envy-freeness is an equitable distribution. The idea is to make each party's perceived value assignment equal. The books develops a technique called the adjusted winner process that results in a division between two parties that is envy-free, efficient, and equitable.

I haven't yet read about the technique, but I believe that it will be the central topic of study for the module. this leaves the issue of a writing assignment. Two general approaches occur to me. One is personal, like, "Describe a dispute over division from your own experience. How salient where the issues raised by this book: envy-freeness, equitableness, efficiency? How was your dispute resolved? Describe the fairness or lack of fairness of the resolution from your point of view. Would one of the procedures described in this book have made the resolution more fair? What difficulties would you foresee in implementing one of the book's procedures in resolving a similar dispute in the future?"

That's fairly rough, but it is the start of an assignment I imagine that many students would find worth writing about. To make the link to social justice, the assignment might deal with a social issue, like salaries for hourly workers versus salaries for their managers and executives in a corporation, or a political dispute, like the Groton CT eminent domain case, or the Israeli/Palestinian land dispute. I believe that these would require more research by the students, and would require more exposition. This might be a good thing, depending on the goal of the module. I expect to come up with a couple of examples of this sort of assignment, although I also expect making appropriately motivating assignments will be more difficult in this case.

Saturday, June 03, 2006

RadicalMath Powerpoint Online

Hello everyone.

Just wanted to thank you all again for a very interesting and informative workshop. I look forward to working with all of you in the near future.

The powerpoint presentation I gave, "Integrating Social Justice into a High School Math Class" is now on my website ( if you are interested in downloading it. The site itself will be getting a major overhaul in the next week or two.