Unpacking quantitative reasoning

Topics/tags: Grinnell, liberal arts, computational thinking

A group on campus has recently been unpacking [1] one of Grinnell’s six elements of the liberal arts [2] or perhaps an element of a strong education that is closely related to those six elements. They did a nice job and the faculty recently approved their proposal to formalize the list.

I’ve been thinking a bit more about another one of those elements, that of quantitative reasoning. Here’s what we say about that topic.

4. Quantitative reasoning, with emphasis on mathematical models and methods above the secondary-school level, aids in the expression of hypotheses, processes, and theoretical relations. A course in statistics can be helpful for all students, and particularly for those who might work in the social and behavioral sciences. Studies in computer science offer valuable exposure to principles of logic and problem-solving paradigms. [4]

I’ve never liked that paragraph. As far as I can tell, there’s no real connection between the three sentences, other than the tendency of people outside of our departments to link CS, Statistics, and Computer Science. I’m also not sure that mathematical models, hypotheses, and theoretical relations have much to do with the study of quantities. They can, but they need not.

I’ve come to the conclusion that we really need to spend some time unpacking what we expect when we say that students have mastered, or at least encountered, the kinds of skills and concepts intended for this element. In the ideal world, we’d convene a group to study the issue and come up with a clear, well-thought-out, description of the issues [5]. I don’t know when or if such a group will be formed. For the time being, I’ll provide my own perspective [6].

When I hear the term quantitative reasoning used as one of our core liberal arts values, I find myself identifying four different kinds of reasoning that I would hope that all modern liberal arts students [7] posses.

First, there’s the literal meaning of quantitative reasoning; the ability to reason about quantities. From my perspective, that corresponds, in part, to the ability to do the kind of word problems that are the bane of most middle-school students’ existence but are also part of everyday life. If my walls are these sizes and one gallon of paint covers this many square feet, how many gallons of paint do I need? The College-level version of this is mathematical modeling; the ability to model complex systems using quantities and formal relationships between those qualities. But there’s more than modeling; one should also be able to reason about the size of numbers. What does it mean, for example, that someone has a billion dollars or that our national deficit is in the trillions?

Second, there’s the kind of reasoning that the accompanying text describes, which I might term formal reasoning. Can a student work in a formally described system and follow rules for reasoning and deduction within that system? Can a student examine a hypothesis in such a system and demonstrate its correctness using formal mechanisms. While we cover that kind of thinking in our proof-oriented courses in mathematics, students might also get exposure to formal reasoning in a course in Philosophy [8].

Third, there’s the very different kind of reasoning which, for lack of a better term, I’ll call statistical reasoning or reasoning in the face of uncertainty. Since Grinnell aims to graduate individuals who can participate as citizens, they must be able to interpret the data that they receive and ask the kinds of questions that a Statistician knows to ask, such as how the data were gathered or processed. They should understand p values and their use (and non-use). They must also be able to interpret graphs, charts, and the like. I’d say that Grinnell’s Statistics classes do a particularly good job at helping students think through these issues, particularly as students design and carry out their own experiments. But students can learn many of these skills in some courses in the social sciences or the natural sciences.

Finally, modern citizens must have some sense of algorithmic reasoning (or computational thinking as some term it). That is, they should know a bit about how you develop computational models of domains, specify goals, write algorithms that meet those goals, and analyze those algorithms. With the power of computers to change the world, everyone has an obligation to understand a bit about what we can and cannot do with computers.

There are, of course, other reasons to study these four disciplines. Long ago, when I applied to Grinnell, I’m pretty sure that part of my application packet was an essay on how a core aspect of a liberal arts education is the opportunity to think in many different ways [9]. Each of these four ways of reasoning provides a different model of thinking about the world or about problems.

I realize that I’m biased, but I really do think that our students should experience all four kinds of thinking as part of their undergraduate careers. And the four are really different ways of thinking. Someone who can model things mathematically or who can design algorithms can’t necessarily think statistically. Someone who can model the world using statistics may not be able to prove statements correct in a formal system. And so on and so forth.

But I guess we accept the same compromises in other elements of the liberal arts. For example, there may be a large gap between being able to create works and to analyze works; a student who has taken only an art history course is unlikely to be well prepared to paint or sculpt. Similarly, there’s a gap between the artistic disciplines; that art history student is also unlikely to be able to analyze musical or theatrical works. And we accept that a student might only choose one of these modes of expression or understanding [10].

So what should we do about the quantitative reasoning element? As I said, I’d like to see a broader conversation in which we unpack that element a bit. I’d like to see us more explicitly state that these are really four different ways of quantitative reasoning. And since the reasoning is not necessarily about quantities, I’d like a better term. Unfortunately, I can’t come up with one right now. Reasoning in formal systems is the best I can do [11].

But my perspective is not necessarily comprehensive or useful. As I said, I hope this issue is one we’ll discuss more.

[1] By unpacking, I mean something like examining carefully, reflecting upon, and expanding and elucidating relevant details.

[2] https://catalog.grinnell.edu/content.php?catoid=17&navoid=3449#Elements_of_a_Liberal_Education [3].

[3] Given the less-than-sensible design of the catalog, that URL may change. Last year’s was https://catalog.grinnell.edu/content.php?catoid=16&navoid=3312#Elements_of_a_Liberal_Education. If the URL does change, try doing a Web search for Grinnell Elements of a Liberal Education.

[4] Grinnell College (nd). Grinnell College Catalog 2017-18. Education in the Liberal Arts: Elements of a Liberal Education.

[5] I’m serious. While I know that committees are not an ideal vehicle, multiple voices and perspectives do tend to produce better understanding. And I’d generally take the product of a committee over whatever a random administrator comes up with.

[6] I hope that my perspective ends up being better than that of a random administrator. But my worldview is also limited.

[7] And all citizens.

[8] Not just in the courses the specifically deal with abstract systems, such as Symbolic Logic, but more generally in courses in which students must reason within a set of constraints.

[9] And computers certainly think very differently than humans.

[10] Wouldn’t it be cool if every Grinnell student had taken courses in analyzing, understanding, and appreciating visual art, literature, music (in multiple genres), film, theatre, dance, and more? Wouldn’t it be equally cool if every student had the opportunity to learn each of those creative areas in a more formal setting?

[11] Perhaps I should stop making comments about random administrators.

Version 1.0 of 2018-05-07.