What are some strategies for using automatic grading technologies to get students to express their reasoning?
I think you've hit the limit on what automatic grading technology can achieve at this current point in time. Generally, this technology is only going to allow you to automatically evaluate questions with pre-defined answers. This doesn't work for theory or structuring. It probably isn't going to, either.

However, you can use it to evaluate concepts. I'm not exactly sure how your class works, but here are some example concept questions.


  Given [Situation X], what methodology would you use to solve [Problem Y]?


The answer would be from multiple choices.

You also combine this with over method comprehension. Let's do true/false.


  [Method X] is the process of..... (True/False).


From there, you can then start going through some actual calculation questions. If structured correctly, you can deduce the information you're trying to achieve.


If the student does well on the conceptual questions and bad on the math questions, they likely made some arithmetic errors.
If the reverse is true, the student probably just memorized the formulas.


These situations aren't always going to be true, though. They will never be as accurate as a human evaluating a student's process. But the estimates may still prove useful.