soft question – How hard do mathematicians have to work to learn?

So this is a soft question, but I’m looking for a collection of specific instances or stories that provide a reasonable breadth or representative picture. The target audience for this question is those with PhD-level knowledge of math. That doesn’t mean you have to have a PhD to respond, but you should ideally possess the rough equivalent of PhD expertise in some mathematical content area even if self-taught.

How hard did you struggle with learning math? Let’s break it down into content categories:

  1. Subject matter up to before calculus, say typical topics before university, arithmetic, algebra, introductory trig, statistics. What is typically taught in secondary/high school or lower.
  2. For calculus and other typical lower-level undergraduate university/college coursework.
  3. For typical upper-level undergraduate courses/content.
  4. Lower-level graduate, e.g. core courses or general electives.
  5. Upper-level graduate, research level, and beyond.

Question: How common is it for PhD-level mathematicians to have struggled and to what degree of struggle:

  • in high school or lower?
  • in lower-level undergraduate classes?
  • in upper-level undergraduate classes?
  • in lower-level/core graduate classes?
  • in upper-level, research level, and beyond content?

Maybe this could be gotten at by a question like: How many hours do you spend obtaining understanding of a typical research paper that is not within your field of expertise? When you want to learn a new concept, does it always come easy, or does it sometimes come with a lot of struggle, how many hours/days?

I’ll argue that this is a useful question to ask as many undergraduates struggle with upper level content, for example, and might get discouraged thinking that it should be easy for them. I discuss this with my students and openly talk about what I struggled with and what I found easier to learn. I imagine my experience is fairly typical for PhD-level mathematicians, like I’m somewhere in the middle in terms of how hard I had to work. But I don’t know this for sure.

Maybe the way to answer the question is to give a rough estimate for how many hours outside of class you needed to work (or would have needed to work) to achieve A/B equivalent scores in the content.

As an example: Math was essentially easy for me before college. I mean, there were times in high school where I played with equations trying to figure things out, but I at most rarely struggled to do what was required of an assignment. Calculus and trigonometry in college was a little more difficult, especially towards the later topics, and it took a bit of practice/going to office hours. I truly struggled with some content of upper level math. Sometimes I was lazy about it, but sometimes I worked for many hours to learn it (that generally paid off too). Also some (say roughly half) upper level content came fairly easily. Graduate school was definitely much more difficult where learning definitely required many hours outside of class working. I still find high level research mathematics extremely difficult (~15 years post-PhD). It definitely comes faster/easier now conceptually, but much of it is still just way beyond me, and I have to work very hard. Before college, I would have said I was the best student at my school (having little to no struggle essentially). During university, I was probably somewhere in the upper half (struggling less than most). In graduate school, I was roughly in the middle (struggling similarly to the average student, but about half of them learning with less effort than me). I feel that is a fairly honest assessment (to within a rough degree of accuracy).

What I’m looking for is similar assessments from people roughly at the PhD-level of mathematics knowledge, anyone from professional researchers who don’t teach at all and have lots of grant funding, to those in teaching positions who do little to now professional research (I’m in the latter category). And even maybe recreational practitioners who just study on their own but have no formal credentials. The key is being honest with the assessment. I’d like to normalize both struggling and having difficulty, but also excelling and having an easy time at it. I think having a broad sampling of experiences would be very useful information for students in particular.

I know this is a soft question, but I’d like to argue that it isn’t opinion-based. It is asking for specific information, and presumably mathematicians are careful enough to give reasonably accurate answers. I didn’t know of another place to post such a question because it requires access to the specific audience that happens to be present at this site. It didn’t seem appropriate for math-meta either, nor for math educators forum because I don’t just want to target those who teach.

Also, if there has already been some academic research into this, please share references/info!