Really though, unless you have a strong understanding of both calculus and statistics you'll never be a "data scientist", you'll just be a library jockey.
Yes, yes, it's not like you started from something. Were you born data scientist? Just try to accept that there may be people with almost full skill set who didn't know where to apply it? I'm pure mathematician by education and I really didn't know where to apply my skills, having followed ML, NLP, Big data, PGM courses I have much better understanding now.
Now even if I weren't a mathematician it doesn't mean that following Coursera courses for a couple years and doing a lot of work at home wouldn't get me somewhere. You don't have to switch jobs as soon as you finish ML course but you can certainly practice your skills at home.
I'm not saying the Coursera classes are bad---just that there's a chasm between being able to implement or derive Naïve Bayes and being able to do meaningful work or study in applied statistics.
The way I see it you also learned it somewhere sometime ago. Coursera offers more and more courses some of which are quite in-depth. Motivated person may be capable to finish university degree in couple years. Coupled with the fact that this person may be gainfully employed in similar area, it will not be that hard to imagine him or her to eventually switch to a position that requires all the old and new knowledge. IMHO, examples that were given in the original article are in line with what I described.
Eh, maybe. I have a strong background in mathematics (my focus was Game Theory), and certainly understanding the underlying math in ML models is very useful, tools are coming out on a regular basis that make it simple for good developers and DBA-types to apply some ML modeling to their data.
The "science" part of the term involves testing and iterating, as the scientific method would imply. Not necessarily the knowledge of esoteric mathematics.
