In your opinion, can the ability to motivate a student something that can be learned or something that just come naturally to the professor? Is it different in the classroom versus online learning?
Good (and adequate) attitude rather than absolute knowledge is a key to motivating students. Ideally, the combination of knowledge, good and personalized attitude, effective communication (communicating knowledge) and charisma is what describes a good teacher -- all of these qualities with the exception of knowledge are a combination of skill/talent. However, despite having a good teacher, students may not be motivated due to various other reasons (personal, financial, mismatch between interests and obligations, change in intent, etc.). I guess an online teacher will have more difficulties or need more time to understand students' motivations and personal learning styles. Overall, teaching is quite an art.
I agree that teaching is an art. The reward for me is when the "light bulb comes on!" This is the moment went the motivation begins and you have to catch it then or you will miss it.
It could be a matter of time as a process rather than a given point of time. Imagine that the light goes on and off repeatedly, it would be wrong to say that you have missed it, you must notice it at some point. Actually, there are too many factors involved, you can't always say where the problem is, how motivation starts and develops, whether it is teachers (in terms of professional and emotional behaviour), students learning on their own terms, other external factors or some combination that made things work.
If a person is not in good emotional and/or physical health, you can't expect that your subject will be their primary obsession. In some cases, learning may spontaneously unlock some ''hidden'' talents and get students out of a miserable state (e.g. music education projects targeting poor kids in California).
Another thing to consider is that there are people who are so self-critical, they might underestimate themselves but be very motivated nonetheless. Would you be able to see that? Or maybe they don't want to move fast. Confucius said that It does not matter how slowly you go as long as you do not stop. Or maybe it is not the right time. For instance, in this article http://bit.ly/1i9sO22 it is argued that at a certain moment we all facing a situation in which we are not prepared to learn (in this case math), not that we will never be able to. A good teacher must also be a good psychologist.
There are people who at a certain point of development will find a reason to be motivated for something they have been ignoring in the past but for others the path of least resistance will continue be the only path forward (or backward...)
I think that the most important role of any teacher is to inspire some respect in his subject, to transfer a message that there is much more to it than what people see in class, the depth, the meaning and the application of what this subject is about. The drawback of investing too much energy in doing so is that if people are not able to see the depth, they will not see it no matter what you're doing.
One of the biggest problems in any education system is that grading is not based on potential, it is based on absolute knowledge and ambitions (i.e. the most ambitious but not necessarily the brightest students being encouraged). I've seen so many smart people failing their exams while at the same time so many stupid and superficial ones passed with excellent grades - it is a disaster, an absolute HR crisis in education.
I think that in the art of teaching, more attention should be paid to hidden patterns and quality vs. quantity: having encouraged one or two students who haven't previously shown much interest to work harder and get the meaning of a problem/subject should matter more than aiming at a decent average grade of the whole class or encouraging the ones who had all the answers but could barely think outside the box.