The College Classroom

Fortunately, I was able to find an article (http://link.springer.com/article/10.1007/s10649-011-9307-4) that focuses on the class I will be teaching this summer: linear algebra. I have come to realize that linear algebra is a very interesting and challenging class to teach for several reasons. One, it is a class “in between” two types of math classes: computational (like the calculus sequence) and proof-based, rigorous upper division math classes, which will likely only be experienced by math majors. Some schools teach linear algebra as a purely computational subject the first time around, while other use it as an introduction to formal math logic. Most, like at this university, treat it as an in-between class. In particular, it is hard to justify to students who do not intend to take more advanced classes why they need to understand definitions and proofs when the calculus sequence did not emphasize this (which some contend is a problem with the undergraduate curriculum in the United States). At the same time, with this approach, many math majors will not learn linear algebra at the level that they will need in later classes, and in many cases (such as my own), will need to teach themselves advanced linear algebra later on.

But everyone agrees that linear algebra is one of the most important math classes for those who intend to pursue theoretical or applied mathematics, and it is one of the most important classes for students who major in any kind of technical field. Therefore, it is a necessity that we try to help the students not only perform computations but understand the concepts so that they will be able to utilize the techniques from linear algebra later in their career. So while this type of class may fall short in many ways, it is the least we must do.

There are several key concepts from linear algebra that students experience trouble with: linear transformation, basis, linearly independent, etc. This article focuses on the notion of a subspace and how students take what they visualize or already know about subspaces (such as the examples in 2 and 3 dimensions, a line or a plane) and how they reconcile this with the formal definition. The article calls the latter the students’ “concept image.” An advanced mathematics student will have a concept image of subspace that agrees with the formal mathematical definition and be able to explain why what they visualize, or their own stock examples, satisfy and also demonstrate the definition. This is the goal of such an “in-between” mathematics course: to produce students who can take their mathematical intuition, which is honed in calculus, and reconcile it with abstract, formal mathematics.

The paper explains results from a set of interviews with students who have just taken such a linear algebra course. The researchers picked 8 students, 4 male and 4 female, which had gotten grades from A-C, as representatives. The students described their concept image of subspace. Then they were given the formal definition and asked how it related to their concept image. Finally, the students were given a problem to solve (“Do these vectors form a subspace?”) to see how students utilized both the concept image and the formal definition. Some of the students had a geometric/visual concept image of subspace (that is, they relied on visualization to understand its properties), and the paper argues that such students who do not also have an algebraic concept image will be led astray when solving linear algebra problems. This is consistent with my own experience teaching linear algebra and multivariable calculus—the students who rely on visualization find that they cannot solve problems consistently. Then there are students who never use visualization and who only rely on the rules given in the definition (this is how I was as a linear algebra student). The advanced students have both understandings, and they complement each other and are both utilized when solving problems.

One issue I did have is that, while they changed the names of the students, the pseudonyms were gendered, and all of the students who demonstrated “advanced” thinking were male, while there was one female student who clearly had the most trouble with the questions. I think that 8 is a very small sample size and it’s hard to draw any conclusions, but I would be interested to know if they thought that there was a gender difference in terms of advanced understanding and if so, how would they explain that. It would be interesting to repeat such a project with a larger sample of students.

The paper was helpful to me in that it verified what I already thought about students’ misconceptions in linear algebra. Students come in thinking of vector as a visual object, a line with an arrow, which is consistent with what they have been taught in calculus and physics, and teachers must build on this concept in order to introduce the notion of an abstract vector space. I hope that the actual material presented in the paper will help me construct effective peer instruction questions.

I found an article by David Concepcion called “Reading Philosophy with Background Knowledge and Metacognition” in Teaching Philosophy 27:4 (2004).

It gave a nice discussion of how one might teach students how to read articles in current analytic philosophy. He pointed out that this sort of task is necessary in order to get students to actually engage in philosophy rather than assuming they are reading the text to pillage a bunch of facts out of it. He incorporates this sort of instruction into the class by adopting the following methodology: (i) students read, summarize, and evaluate a short passage in class. (ii) they then describe the process they engaged in while doing this. (iii) they then read a “how to read philosophy” handout. (iv) they then re-read the passage using the skills outlined in the handout. (v) they compare this latest result with their previous one and note what they have learned. (vi) students look at each others self-assesments. (vii) students turn in a written summary of the text. (viii) questions from the “how to read handout” is included on the final exam to make sure that the students remember the skills they have learned.

Overall, Concepion brought out some interesting pedagogical points but I worry that his methodology won’t work for all kinds of philosophical texts. In order to work, he needs (i) to distinguish what he calls problem-based philosophy from historical philosophy and (ii) assume that both could adopt a similar method. I’m not sure whether either of these assumptions are justified. Can we really get clarity on contemporary philosophical problems without attending to the way they have been historically and socially mediated? Can we really find a methodology that can apply to reading all different kinds of historical philosophers e.g. Hume, Leibniz, Kant, Hegel, Schelling, Hoelderlin, Nietzsche, Heidegger, Deleuze, etc.?

Here’s the link to the free version of the paper I found:

http://secure.pdcnet.org/8525737F00588478/file/C125737F0061DCC6C125756D0060B335/$FILE/teachphil_2004_0027_0004_0055_0072.pdf

Unfortunately I had a great amount of trouble finding an appropriate article as all of my search terms yielded results about teaching *language* rather than teaching Linguistics (I’m just not sure if this type of work is out there as I am in a pretty ‘fringe’ field. In the end I decided on an article about how the formal linguistic knowledge that linguists possess can be incorporated into a classroom setting in order to teach first languages to children: “Some aspects of the impact of Linguistics on the teaching of English in disadvantaged children” (http://www.jstor.org/stable/41386369), though unfortunately the article doesn’t really contain information about how to learn my field but rather how my field can influence the learning of natural languages.

The author points that two that there are two primary groups that enter language classes: proficient speakers of the native language, and those learning it as a second language, but notes that this leaves out an important third group: those who speak a non-standard variety of the language who need additional/modified types of instruction.  The author then goes on to discuss, at each linguistic level the types of issues that these students have. In terms of phonology issues such as neutralizing the final sound in /with/ from a /th/ to an /f/, resulting in /wif/ and advocates the use of this type of linguistic information to inform training on standard pronunciations. In terms of grammatical features, she notes that although many students can pronounce the /s/ sound in /verse/ they will omit the /s/ sound in /printer’s/, even though one is a simple phoneme and one is a morpheme expressing possession. These linguistic distinctions must be expressed to the student, as well as the importance of ‘linking’ verbs, and the correct way to form verbal/tense aspect combinations. The author advocates making these distinctions clear by clearly stating/describing the components of a complex VP such as “have been cooking” by explaining the function of each auxiliary verb as well as the /-ing/ morpheme. This runs counter to the lower level approaches that dominate language teaching, informed by linguistically sophisticated terminology that rather than a statement to student such as “no thats wrong, its “have been cooking” not “have be cook” or “have been cook.”

Ultimately the author advocates an increase in this type of approach, early exposure to formal texts, and a move away from the piecemeal correction approach that is so often adopted (as well have having the teacher slowly enunciate words for the students as they will never actually hear the form in natural speech). Though this does not quite get at the issue of how students learn the concept of linguistics, I believe it is interesting in that it shows how adopting formal linguistic concepts can result in better language outcomes than traditional teaching practices.