Set Theory Meets Immigration
We know about issues surrounding U.S. immigration since these issues have been in the news on a daily basis for the last few months. However, you might ask “What the heck is set theory?”
Set theory was developed by Georg Cantor in the latter part of the 19th century. It is abstract, but lies at the heart of modern mathematics and Computer Science (CS). If you are seeking the mathematical equivalent of the periodic table of elements, look no further than set theory. It is the foundation. Much of what you think throughout the day can be formalized in set theory: how to create categories, comments like “you are with us or against us,” and “he/she is not one of us.” If you want to think computationally, set theory is a great place to start. Even better than coding? Yes — more on that at the end.
My goal is to teach students about mathematically oriented concepts — especially those found in CS— to as many people as I can. To do this effectively, these concepts need to be framed in context. That context is ideally situated within each individual student. The student likes ice skating? Assume that the learning objective to be delivered is “data structures” (a foundational topic in CS). So, allow the student the freedom of exploring data structures within the context of ice skating. This may seem like a difficult task, but it is easier than you might think. I could just draw lots of alphabetic symbols on the whiteboard, but I risk not achieving my teaching goals in the process.
Back to immigration. For the first semester exercise in my Modeling and Simulation class, I told students: you are to use something called “set theory” but use set theory to model a recent NY Times article (linked above in the figure caption) on immigration and DACA. Here are the assumptions: (1) The goal is to learn and explore set theory, and (2) The students are interested in the article’s contents. The use of set theory as a real-world modeling technique is non-traditional, but basically all of mathematics (and computer science) can be seen as modeling. Making connections with the real world is what modeling is all about — representing stuff. To summarize, modeling is the vehicle through which abstract concepts are manifested within real-world contexts.
Here is one team’s approach to capturing the Times article using set theory:
The different colored polygons are set representations. For example, in the set “Support DACA” (colored in red), there are 5 elements, starting with “Dreamers”, the element that most strongly belongs in that set. You may have seen diagrams similar to this in your daily reading. The learning objective in this exercise was not only to understand the immigration issues, but also to learn that, abstractly, these issues can be modeled using set theory (the learning objective). Making connections through abstraction is very useful since the world, and news articles in that world, can be seen as patterns. The identification of patterns is what defines mathematics.
Here is an example by another team of students:
This one is a bit more complicated but the overall structure is the same: sets defined and visualized by enclosing, colored, boundaries. While the first example was mainly a delineation of sets, this picture also defines “relations” and “functions.” The arrows extending from p1, p2, … are functions from the set of dreamers.
Here is what was learned in this exercise:
- Mathematics, and applied mathematics (e.g. computer science), can seen as formal structures for modeling things in the world.
- The target learning objective for students can be situated in familiar spaces — a news article on a current topic.
Questions that arise:
- Do students learn better using this approach?
- Wouldn’t it be better not to assume student interests (i.e., the article), and instead ask them about their values and interests?
- Are these visual set theory representations suitable as a means for a greater purpose — to communicate the immigration crisis to a wide audience beyond the classroom?
- Do we teach core CS concepts, like set theory, or focus instead of teaching students how to program?
One advantage of teaching abstract math concepts, as opposed to relying solely on coding/programming, is that with concepts, one can easily apply them to anything — from news articles, and parks, to art museums. With code, since it is not conceptual, the ability to cross-link to other disciplines is limited. Programming is a good thing, but not the only thing, when it comes to learning how to think computationally.
