Communicating mathematics online

How does one quickly and easily communicate upper-level mathematics entirely online?

I never interact with my students in a physical classroom. All communication happens via email, virtual meeting room (Adobe Connect), Skype, and/or phone. All of the course materials are hosted on Moodle, but I do not feel that having my students use multiple choice (or free response limited to a few characters) for all assessments is sufficient to gauge their understanding and it's certainly not sufficient to engage them in a mathematical dialogue aimed at sharpening their exploration and written communication of mathematics.  

All of the graded assessments in the AP & College math courses I supervise and instruct are free response so I can clearly see a student's thought process and offer feedback as needed. There is a great deal of material per student per course and the logistics of communicating all of this marvelous math can become overwhelming. 

Fortunately, there are two main options for students and instructors which work very well, both are illustrated below. 

1. Write work by hand, use a free app called CamScanner to send to instructor. 

• Pros
• Quick and easy -- write, scan, email. 
• Automatically color-balances & arranges images into albums (for sending an entire test at once)
• Saves to pdf, jpg, tiff, or other formats, each page ~ 200 kb 
• Very high quality image output
• Cons
• Requires an Android or iOS device, so accessibility may be an issue.
• Can be difficult or slow to edit without special software (Foxit reader is free and has nice PDF editing features) 

Example output:

CamScanner (Android | iOS ) (click to enlarge)

Detail of above (click to enlarge)

2. Use free & open-source LaTeX to typeset work. 

• Pros
• Forces students to articulate their reasoning clearly (crossing out, drawing arrows, and writing all over the page simply won't transfer.)
• Output is beautiful
• Graphics can be integrated very easily
• File size is very tiny (5 kb + images)
• Easy to edit
• Typing math becomes as natural as typing English
• Professional documents can be typed for any course, not just math!
• Learn it now instead of later!
• Cons
• Heavy up-front time investment to learn the syntax. I have a continually updated page devoted to getting students started featuring: videos, templates, and additional resources. 
• Installation of a LaTeX distribution can be tricky, but fortunately there is http://www.sharelatex.com
• Errors can be difficult to spot at first and frustrating to correct without help from an experienced user. 

Example Output: 

LaTeX (click to enlarge) download source here.

Detail of above. (click to enlarge)