Mathematical Understanding and Multiple Representations

What convinces (or doesn't convince) you in the authors' argument?

I would have to say that the authors’ arguments were quite convincing to me personally, not only because of the points made and studies cited, but also because I am predisposed to agree based on my personal experience and observation. The main conclusions—that students must practice using multiple representations, that representation is a fundamentally social activity, and that instruction must use varying techniques—are in general agreement with what I have personally realized in my time as a student and my limited time as a teacher. Furthermore, the way that the authors used not only examples, but also real case studies, to support their points was effective in convincing me of their position. For example, Tchoshanov’s pilot study where three groups of students were taught trigonometry with controlled levels of representation (one group with just an analytic approach, one with just a visual approach, and one with both) was very insightful and convincing. Not surprisingly, the group that received instruction through a combination of different representations were more comfortable using these different representations to solve problems, and consequently scored better than their peers in the other groups. 

What kinds of mathematical representations are included and excluded in this article?

I think the authors made a very exhaustive list of both internal and external representations. As for external representations (which will be easier to define and classify), I think they covered all of numerals, algebraic equations, graphs, tables, diagrams, and charts. The only possible forms that I could think of adding to this list would be games and stories. Although this might be a bit of a stretch, many students might easily learn and internalize probability concepts through games in which they already have some intuition. For example, I might introduce the idea of conditional probability through a memory card game. I also think that stories can help students retain more arbitrary definitions by creating a schema that connects them. These are not academically founded opinions per se, but just some thoughts I have.