The hot topic of the week for Econ bloggers is the use of math in econ. This topic and the direction it is going is captivating- at least for a crowd of one. The "normal" sides are not the same. Bryan Caplan and Noah Smith each had complaints about math in econ. Robert Murphy, Daniel Kuehn, and Jonathan Finegold reply with praise on math for enhancing economic thinking, a specific math. They do not discuss the math appearing in most AER articles. I have nothing worthy of adding to that specific question (my views closely resemble Robert Murphy's).
However, as a student, I believe I have something to add about the way math is introduced in grad programs. Since most undergraduate economics programs barely delve into math, besides maybe a top-level econometrics course, the first introduction for formal students is the August before their graduate studies. That is when students take their "Math Camp". The standard format is a few week crash course in what the department believes is the necessary math to get through a program.
The instructor of my own math camp started off by saying something like "economics can be described verbally, geometrically, or algebraically depending on whatever works best" and came up with some examples for each. It is hard to argue with such a statement. However, this is at the start of math course entirely removed from economics besides token uses of labor or profit as variables. It is a math (here math means calculus) camp, not a rhetoric course so one can guess where the stress out of the three lay. In the pure math that is done in this camp, algebra is almost universally the easiest format. Once the real world is abstracted away, math can wow and impress. Most schools have the same format (see here and here). We physicists turned economists can forget econ for the time and slide into our comfort zone.
But econ is not applied math. All of the proponents of math above talk about math as a tool within the toolkit of economics. The math has to follow the economics. However, the institutions of graduate economics encourage a different use of math, though not explicitly.
Pure math is the mindset that all economists start off their studies with and the one many never shake. If math is just a tool for expressing economic ideas as the proponents claim, at least openly, then this structure of a math course seems wrong. It starts off with the math then the econ. With that design, is it any wonder that the economics can be molded to fit the math? Maybe a crash course in econ/price theory could proceed the math. I remember Deirdre McCloskey suggesting such an idea. If she did not, I'll take credit. Maybe this could lower the problem that Robert Murphy saw at NYU-
Until I saw it with my own eyes, I would not have believed how much the people in top-ranked economics programs are great at math, but bad at basic economics of the kind that I learned by reading op eds from Walter Williams and Thomas Sowell growing up.
P.S. Maybe I am missing the big picture here as a student, but I can only report my experiences and impressions up to this point.