Is hastily writing down the professor's lecture a good way of learning?
Prior to electronic media generally and after the start of mass, rather than individual instruction, this was the primary way of teaching a lot of things. It can be a good start to learning, actually. However, it isn't enough, and never was. Learning is an active sport, not a spectator sport.
We learn by actively engaging the brain. Copying down notes is somewhat active. Certainly more active than taking a phone picture of the board once it is full, or downloading a lecture. It is even more engaging than (just) reading a lecture.
However, if you really want to learn, or teach someone how to learn, I suggest a few additional steps.
One is to review the notes after a day or so.
A more extreme version is to take the notes by hand (pencil and paper) and then transcribe them and add annotations that are needed, perhaps preparing questions for the instructor or yourself to answer.
One is to extract and write down the most important message in a lecture. Or maybe a couple of them. This can be done at the immediate end of the lecture.
One is to ask questions during the lecture if a step in a proof isn't clear to you. (Yes, instructors make mistakes).
One is to review the most important points of the previous lecture just before the next one.
But the most important aspect is to do some kind of exercises based on the lecture. Perhaps these will be assigned. Perhaps they are available in a text book. Try to get some sort of feedback on your solutions.
Another is to start a discussion group with a few students to review the lecture material and update the notes.
But the real key to this is that you learn by actually rewiring the synapses in your brain and that takes repetition and reinforcement. To assure that you are reinforcing the correct ideas, you need feedback.
Note that watching lectures (say on the internet) without active participation is almost completely ineffective for most people. Especially for long term retention and the ability to put the knowledge to use.

For some of the biology behind learning that suggests reinforcement and feedback see: The Art of Changing the Brain by James E Zull

Note also that, as a professor, I've had to explicitly teach this to students and force some of the activities. For those that learn easily they may not seem to be needed, but eventually things will get hard and having good practice can save you.
For example, I'd end a lecture a couple of minutes early and ask for &quot;the most important lesson of the day&quot;, expecting that the students would have note cards with a few ideas.