At the crossroads, teaching the “Math-y” stuff to the self proclaimed non-math-y.

Hugo, my Colonial Literature of the Americas professor once introduced me to the class (I was a senior in a freshman class that I hadn’t come around to taking) as:

“This is Pablo, you will find that as a Literature student he is a great Mathematician, and I’m sure that in Mathematics they say he is a great Literature scholar”

It is indeed true that I very often have found myself at the intersection of disciplines, first as a dual Literature and Mathematics student in college (which happens to be the reason my schedule was too crammed to take Colonial literature until I was a senior and well known by the professor in question);  later as a graduate student in Linguistics with an expertise in semantics and then in computational linguistics, becoming the odd one now in the middle between the Computer Science and Linguistics departments.

Because of this confluence of disciplines, I have often been called to teach the courses that lie in the fringes, the ones the students feel they are not good at, or, to put it differently, the ones they always feel they “didn’t sign up for” when deciding a field of study. In humanities this means teaching the “math-y” subjects. Mathy in a broad sense of course, since I count my time T.A-ing for the latin and linguistics classes  as the start of this trend. In general, I count here as math-y, courses that required to learn a different formalism to the usual ones in the field, formal languages and strict formal rules, like grammar or formal logic.

If you want to find the math-y subjects in a humanities department it’s easy, just look for the ones no one wants to take, in Linguistics it’s Syntax or Semantics and nowadays Programing (often disguised under a title like Methods in Computational Linguistics as to lull the students into a sense of security). Students taking these classes get exposed to new formalisms, have to handle formulae and derivation processes, new codes that seem inaccessible and often inscrutable or arbitrary, the most common reaction to this is panic.

Teaching a subject that produces this kind of reaction is a mixed experience, on the one side, the frustration of your students can easily transfer to you, they will constantly say that they are no good for this, they will see their efforts as fruitless and because of this stop trying, they will often not mind having bad grades and even having to re-take the class.  More than once have I heard: “I failed Syntax but most people fail it once right?” or “All I care is to get a passing grade and forget semantics, after all it is not my area.” When you see your students stop caring about actually learning your subject it is easy to stop caring about actually teaching it.

On the other hand, this unpopularity makes it all the more rewarding when students finally “get it”, not only in the accomplishment you feel but in the accomplishment you see them feel. These courses very often feature an “aha!” moment, when the student suddenly realizes they can wrap their head around the formalism and use it to their advantage. When students perform well in a task that they once deemed impossible their happiness is contagious too.    

There is often this idea that you somehow have to suffer through the first stages of these processes to come out the other end tempered, that the moment of enlightenment will come after enough tears have been shed (a very judaeo-christian approach if I may say so). I have even seen instructors tell their programming students that, in their first semester, programming often brought them to tears too; as if this was some sort of gauntlet that has to be overcome through tears and blood. This feeling is often reinforced by older students who have already suffered through the test. This is, in my opinion and experience, the wrong approach, the students can be eased into these formalism in ways that are more gentle and effective, it does require however a lot of patience and time but this will save effort and time in the long run. If your class is seen as a gauntlet, don’t take pride in it, work to change this perception.

  In teaching these subjects, I have come to realize that the reactions of your students must be tampered from day one, any moment spent by them brooding about their inadequateness will mean extra work later, when you have to undo that feeling of powerlessness. Empowering the students starts with understanding that they come already with their own formalisms and you can piggyback on them. The students must come to perceive that the “new formulation” is nothing but a reformulation of the old ones. They already think in ways that may be translated into this new field.

Think for instance of teaching formal logic to linguistics students, the traditional way to do this starts introducing formulae and truth tables as a new tool that must be learned by heart. However, propositional logic follows rules very similar to those of natural language, there is a syntax to be followed and you can ease the students with examples from natural language. I, for instance, always talk about the necessity for verbs in natural languages when speaking about the necessity of a relation sing in mathematical formulae, an equation is no more than a sentence and when a student understands this they relate the new formalism to existing structures thus lifting the feeling of newness and inadequacy.

This approach has, of course, to be refined for every class and even for every background or student in your class, which I realize might be a tall order and will take a lot of time, especially at the beginning of the semester. All the time spent in introducing basic notions so that they articulate with students’ previous expertise will however be rewarded eventually. Avoiding any complaining will be the first boon; in making the class feel more tailored to your student’s backgrounds you are eliminating a lot of the objections and that feeling that your class does not really belong in the field. I guarantee that this will lead your class into a more efficient learning process that will make the late semester, when the more difficult material is introduced, way more manageable.

In my teaching experience, it has become evident that most problems with formal languages originate from an incomplete understanding of the concepts that underlie them. Even engineering students will often mislabel any mathematical expression as an equation or fail to provide accurate definitions of every symbol that they use. In math-y courses for humanities this gets even worse, there is a propensity to use lax language and jump to a formal representation only as a formalism, a set of symbols that you don’t truly understand but have valiantly learn to operate on. Even students that show no difficulties on the surface are prone to this, very algorithmically minded students will often process semantic derivations or sets of equations without having an inkling of an idea about what it is that happens between line and line of formalism or how to put their final answer in words.

One of the best approaches to mend this structural problem can be (you probably guessed it) writing. Asking your students to explain how a problem gets formalized, what the result of the derivation means or what the definitions of different symbols are can be a huge help to bridge their understanding. You will never see a math or semantics paper that is just streams of equations (granted a few exceptions, but these are often the bad papers). Why then do math or semantics homework so often take this exact form? Don’t jump to the formalism and the algorithm, have the students explain what the symbols mean, ease them into being comfortable with the transitions, have them translate back and forth. In my time teaching semantics I often implemented this by doing the exercises on the board and having the students explain the reasoning. I now realize that I needed to go one step further, people hand wave when they speak and space out when others speak, low stake writing might have been the key. If the students realize that formal language is just an abbreviation of something that might as well be written in full sentences, not only will they get it, but they will also come to cherish it (after all, writing equations is so much easier than writing text).

I believe this hand in hand approach, between making formalism friendlier by taking the time to relate it to the student’s already existing frameworks and using WAC methods to cement a solid understanding of this formalism, although time consuming, proves way better at introducing “math-y stuff” to all students, but is exceptionally suited  to all the ones that would have previously declared themselves naturally incompetent for math (not to use some of the more expressive language that I have heard over the years to describe this “disability”).

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