How would you, as a teacher, decide if a particular topic/ concept should be tested as multiple-choice questions or as free-response questions? This is probably a very broad question. It might be difficult to find one answer across all disciplines. If anyone could share their personal experiences on how they have designed tests and the thought process behind it, I would really appreciate it.
In the absence of other factors, in my opinion, at least in mathematics, a free response is always the more valuable type of question. The educational value of getting shown, where precisely your argument went wrong is much higher than just being told if the final result was right or not. It also allows for a much finer grading, as I can both give partial points for an almost correct reasoning leading to a wrong answer as well as deduct points for incorrect reasoning leading to the correct answer.
That being said, a test question never happens in a vacuum and the problem usually is lack of time. This can occur in two forms, lack of working time to correct the test and lack of time during the test.
Personally, I believe the former is a bad excuse. Unless you have a 200+ people class without any TAs (in which case, clearly something else is wrong), it is always possible to correct a full-text exam in a reasonable amount of time.
This leaves the latter point. Any exam has a limited duration and it might not be possible to cover all topics in this time. My personal response would be to leave out some of the less important ones at random. As long as the topic selection is not announced to the students beforehand, they still have to study all of them and the results will be similar. But this is a point where a good argument can be made for turning part of the test into multiple-choice. In this case, the most important topics should still all be free-responses, but you could add a small multiple-choice section covering the rest. However as these are not the important part of the test and they are there to save time, I would keep them simple. (E.g. picking the right definition or &quot;does A imply B?&quot;) After all, if marking the right answer is not substantially quicker than writing it out, then you are not saving time.
In any case I would always invert the original question. Don't take a concept and decide of it should be multiple choice, but instead decide how many multiple choice questions you have to make and then pick the questions that work best.