How much understanding is enough to truly master a course?

[TechieDork]

How much understanding is enough to truly master a course?

I'm a 2nd year physics student. Got a so-so GPA.But sometimes I feel like getting an A in a course isn't really enough.

For instance , I got an A in Calculus II but today I don't even remember how to derive a general form of power series or give an insightful explanations and derivation to someone "on the fly". I mean truly excelled something enough so you can roll it out smoothly just like the experts on stackexchange do.

How much is enough?

This may sound like a subjective or opinion based question. But I don't see any place on the internet that would give me more insightful inputs.

 

[fresh_42]

How much understanding is enough to truly master a course?

Define master! If I take the word literally, the answer is: As much as it needs to teach it.

 

[TechieDork]

Define master! If I take the word literally, the answer is: As much as it needs to teach it.

Yes , "master" is subjective term. So it's loosely defined.

I would like to apply Bloom's theory of learning to this case, My own definition of mastery.

Level 1 : Understand the materials given in class, understanding the underlying relationships.
Level 2 : Understand it enough to solve the problem sets given by your instructor.
Level 3 : Understand it enough to teach someone and communicate it effectively.
Level 3.5 : Understand it enough to derive and explain a whole topic from the first principles insightfully and may be able to build a simulation on software.
Level 4.0 : Understand it enough to apply it to a novel situation , something so complex that require multiple steps of thinking. (Aka unseen challenging problem sets) (just like the experts on stackexchange or PF)
Level 4.5 : Understand research papers in that field enough to give informed corrections.
Level 5.0 : Understand it enough so you can create something new , publishing a paper and writing an original textbook on that topic.

What do you think of this scale? Is it legit?

 

[TechieDork]

In the level 3.5 , My professor has explained damped Anharmonic insightfully to me and my friend on the fly in the office hour right after he woke up from a nap!
He didn't even need to look up on the notes.

 

[Orodruin]

But sometimes I feel like getting an A in a course isn't really enough.

Grades are generally not a good measure of how well you know a course. Depending on how the grades are allocated, it may very well be possible to panic-study for the exam, get the highest mark, and then forget everything the next day because the panic-study main focus was passing the exam, not retaining long-term understanding of the subject.

In most undergraduate courses, you should try to learn in such way that you without much trouble can pick up the exam years later, get the basics, and only with reference material could solve it. If you do not do this in the basic math courses of, e.g., a physics education, then it will come back to bite you as you will encounter physics where understanding the math will be necessary.

 

[TechieDork]

Grades are generally not a good measure of how well you know a course. Depending on how the grades are allocated, it may very well be possible to panic-study for the exam, get the highest mark, and then forget everything the next day because the panic-study main focus was passing the exam, not retaining long-term understanding of the subject.

In most undergraduate courses, you should try to learn in such way that you without much trouble can pick up the exam years later, get the basics, and only with reference material could solve it. If you do not do this in the basic math courses of, e.g., a physics education, then it will come back to bite you as you will encounter physics where understanding the math will be necessary.

Spot on , Orodruin
There was a girl who got a B+ in the same class where I got an A.
I realized that she did really learn the material way more than me.
She got a good study habit ,took good notes, listening actively ,spreaded the reviewing sessions throughout a month and teach it to friends.
I studied the final with panic just five days before the exam. Because I has goofed off the entire month worth of material. I couldn't even explain it simply to others.
I have learned that this isn't a good habit. It indicates my lack of work ethics. It'll backfire somedays when I have to face some weed out classes.

I have planned to review the key topics from lower-division courses ,intensively this school break, before I move up to the upper-division ones.

 

[Orodruin]

The big problem is that it typically works during the first year of a program because the material builds on high-school knowledge. Having taught many second year courses, I can tell you that the main reason for people failing those courses is not retaining knowledge from the year-one courses. If you have the opportunity to review earlier courses you do not control as well as you think you should, it is a very good investment of that time.

 

[Stephen Tashi]

I mean truly excelled something enough so you can roll it out smoothly just like the experts on stackexchange do.

The experts on stackexchange don't necessarily write their answers "on the fly". If you teach a subject or think about it every day, you'll keep its facts in memory. If you don't think about a subject often, you will forget many things about it, even if you completely understood them.

A big factor in whether you remember something is the amount of distress or elation you experienced in understanding it. I'd say that most students forget the epsilon-delta definition of limit after taking Calculus 101. Those few that suffer because they are determined to understand it precisely or experience an "ah-ha" revelation are likely to remember the details. Those that find themselves teaching students will be embarrassed if they cannot explain topics and this puts emotion in the picture.

I've heard the, perhaps inaccurate, tale that "in ancient times", when a tutor told a student a crucial fact, he would smack the student on the head so the student would remember the fact through it's association with a painful incident. To be fluent in technical topics, you need to feel some emotion about them or deal with them every day.

I agree that there is a correlation between how precisely or superficially students understand material and how much of it they remember months later. However, I think the primary cause of this is not the depth of understanding they attained. I think the cause is due to the emotional experience (or lack of it) that is associated with the material and whether they are interested in related technical topics enough so they think about them every day.

 

[PeroK]

How much understanding is enough to truly master a course?

You've mastered a course if you could subsequently teach it.

 

[TechieDork]

The experts on stackexchange don't necessarily write their answers "on the fly". If you teach a subject or think about it every day, you'll keep its facts in memory. If you don't think about a subject often, you will forget many things about it, even if you completely understood them.

A big factor in whether you remember something is the amount of distress or elation you experienced in understanding it. I'd say that most students forget the epsilon-delta definition of limit after taking Calculus 101. Those few that suffer because they are determined to understand it precisely or experience an "ah-ha" revelation are likely to remember the details. Those that find themselves teaching students will be embarrassed if they cannot explain topics and this puts emotion in the picture.

I've heard the, perhaps inaccurate, tale that "in ancient times", when a tutor told a student a crucial fact, he would smack the student on the head so the student would remember the fact through it's association with a painful incident. To be fluent in technical topics, you need to feel some emotion about them or deal with them every day.

I agree that there is a correlation between how precisely or superficially students understand material and how much of it they remember months later. However, I think the primary cause of this is not the depth of understanding they attained. I think the cause is due to the emotional experience (or lack of it) that is associated with the material and whether they are interested in related technical topics enough so they think about them every day.

Agreed , From my personal experience , I only retain information best if I find that topic interesting or enjoyable enough , if the topic was so dry and boring , I forgot everything after I walked out of the exam room and don't even bother try to understand it again.

 

[Dr. Courtney]

How much understanding is enough to truly master a course?

I'm a 2nd year physics student. Got a so-so GPA.But sometimes I feel like getting an A in a course isn't really enough.

Majoring in physics is a big time squeeze. For most students, mastering the material enough for an A is "enough" the first pass through a course. You'll see most of the material from first year physics again in downstream courses.

If you prep for the GRE, you'll see a lot of undergrad material again during this process. If you go to graduate school, you'll see most material again in your graduate courses and preparing for your general exams. Each time through in a later course you'll go deeper while also reviewing the simpler material from earlier coursework. Preparing for the PGRE and general exams is a review at the same level as recent coursework.

The cyclical nature of the process over years tends to bring a level of mastery that is very unlikely in a single semester. Most students would do well to trust the process as well as the individual professors. Trust that earning an A in each course is "enough" mastery for now.

Having learning quickly evaporate after a semester ends is somewhat different as that involves more the difference between short-term and long-term memory. A more truly comprehensive review prior to the final exam in each course can be helpful here, as can review of a semester's material over the break between semesters. The issue is not a lack of mastery, but a failure to put sufficient mastery into the long term memory. I worry less about this than not achieving mastery in the first place. My experience is that knocking the rust off is much easier than building skills, abilities, and knowledge that were never their in the first place yet are needed for later courses.

 

[TechieDork]

Do anyone here continue studying on the subjects that you got a not-very-good grades in?

(Say, You got a B- in Statistical Mechanics but continue studying on to master the subject anyway , even after the final exam)

What was your experience? And how did it benefits you?

 

[Dr. Courtney]

Do anyone here continue studying on the subjects that you got a not-very-good grades in?

(Say, You got a B- in Statistical Mechanics but continue studying on to master the subject anyway , even after the final exam)

What was your experience? And how did it benefits you?

My worst grade as an undergrad was in Stat Mech, but since I had already taken the PGRE and was very busy and research focused, I waited and re-took the undergraduate course in grad school rather than attempt independent study.

In fact, since I went from a top 100 undergrad school to a top 5 grad school, I spent my whole first year in grad school re-taking undergrad courses in Stat Mech, Mechanics, E&M, and Quantum Mechanics. Circling back and re-taking undergrad courses was excellent preparation both for the general exams and for the graduate courses in those areas. It provided a level of mastery that I probably could not have achieved through self-study.

 

[Dr. Courtney]

My worst grade as an undergrad was in Stat Mech, but since I had already taken the PGRE and was very busy and research focused, I waited and re-took the undergraduate course in grad school rather than attempt independent study.

In fact, since I went from a top 100 undergrad school to a top 5 grad school, I spent my whole first year in grad school re-taking undergrad courses in Stat Mech, Mechanics, E&M, and Quantum Mechanics. Circling back and re-taking undergrad courses was excellent preparation both for the general exams and for the graduate courses in those areas. It provided a level of mastery that I probably could not have achieved through self-study.

This approach was very humbling. I was the only grad student or one of few in the undergrad courses, while most 1st year grad students were jumping into graduate coursework or taking the first general exam straight away. I often felt like the dumbest grad student in the department as a Louisiana redneck out of place and far away from home.

But re-taking the undergrad courses was a successful strategy. I earned As in both the undergrad and grad Stat Mech courses, passed both written general exams on the first try, and graduated (PhD) faster than anyone in my experimental research group (5.5 years). The only three Bs on my graduate transcript were in Quantum Mechanics (all three semesters of it: undergrad Quantum Physics II, and grad Quantum Theory I and II.) But Jeffery Goldstone was my prof for the grad Quantum Theory courses, and that was a heck of a ride - the smartest man I've ever met.

As it turned out, most of my theory papers have been in Quantum Mechanics, but more on the computation side than pure theory. It is now the area of physics where I have the most real mastery. But the real mastery came from years of research rather than course work. And the pure theory guys would laugh at my level of mastery, but they have often been interested in borrowing my code.

 

[TechieDork]

My worst grade as an undergrad was in Stat Mech, but since I had already taken the PGRE and was very busy and research focused, I waited and re-took the undergraduate course in grad school rather than attempt independent study.

In fact, since I went from a top 100 undergrad school to a top 5 grad school, I spent my whole first year in grad school re-taking undergrad courses in Stat Mech, Mechanics, E&M, and Quantum Mechanics. Circling back and re-taking undergrad courses was excellent preparation both for the general exams and for the graduate courses in those areas. It provided a level of mastery that I probably could not have achieved through self-study.

I'm planning to intensively review Electromagnetism (my weakness) ,Classical Mechanics and Thermodynamics this break (April to late June) using textbooks , MIT OCW , online science forums and discuss or email with professors if they have enough time.

Not really trying to do something difficult on my own but I feel like studying on my own pace works the best for me. So I'd call this plan a "self-study under expert's advice" . (gonna consult with a professor about this plan before I start studying)

I have heard about the horror of Statistical Mechanics from many seniors and professors. It is said that many professors don't want to teach this class because it's really DIFFICULT. One has to be well-versed in most undergraduate courses to be able to tackle it.

 

[Dr. Courtney]

Most reading and watching video lectures is passive learning. If combined with the active learning of solving problems, it sill be much more effective.

My rule is that work with the pencil moving is three times as effective (in terms of time usage) than work without the pencil moving. In a time squeeze, keep the pencil moving.

 

[TechieDork]

Grades are generally not a good measure of how well you know a course. Depending on how the grades are allocated, it may very well be possible to panic-study for the exam, get the highest mark, and then forget everything the next day because the panic-study main focus was passing the exam, not retaining long-term understanding of the subject.

In most undergraduate courses, you should try to learn in such way that you without much trouble can pick up the exam years later, get the basics, and only with reference material could solve it. If you do not do this in the basic math courses of, e.g., a physics education, then it will come back to bite you as you will encounter physics where understanding the math will be necessary.

There are some students who do everything right.

-Sit in the front row
-Ask questions
-Take neat notes
-Review the material before and after the class
-Do the problem sets
-Teach it to other students
-Hand in the assignments on time and do it by themselves
-Attend the office hours
-Watch online video for more clarification


What could go wrong?
They end up with only passing scores , end up getting lower marks than the students who slack off in the class and study a night before using that golden old exam sheet. Ignoring all other potentially useful informations that won't be on the exam.


This may sounds very discouraging to some students.Some will eventually ditch these habits and start buying into the "one-night-miracle genuis" way of study. Guy who got a 4.00 in my class says he doesn't want to appear studious, He wants to appear effortless while getting the top marks.

But there are hidden benefits to this approach , They are building good habits that will translate to workforce and other area of their life. And showing a professor that you are attentive , hardworking and curious will results in a nice recommendation letter.

Einstein appears on media as an effortless absent-minded genuis , but in the reality he did struggle and he devoted 10 years of his life working on the relativity theory.

(Reminder this is opinion-based)

 

[symbolipoint]

TechieDork, back to your post #1,

If you want a number of hours, how does 300 strike you?

 

[Dr. Courtney]

Grades are generally not a good measure of how well you know a course.

The correlation is imperfect, but students who got As in their prerequisite courses tended to do much better in my courses than students who got Cs. Sure, there were some notable counter examples along the way. But these were mostly due to teachers gifting grades to students who did not even demonstrate mastery on tests rather than students who crammed and forgot the stuff later. Math and physics tend to be learned through repeated practice over time rather than cramming.

Can anyone who has actually TAUGHT physics say that students who got Cs in the prereq math or physics course do as well (on average) as students who earned As?

 

[TechieDork]

TechieDork, back to your post #1,

If you want a number of hours, how does 300 strike you?

To be great in something we have to invest a great deal of time.
And 300 can be reduced by using the effective resources and study techniques.
I think there are much more variables to determine the learning outcomes than the number of hours.

 

[Dr. Courtney]

To be great in something we have to invest a great deal of time.
And 300 can be reduced by using the effective resources and study techniques.
I think there are much more variables to determine the learning outcomes than the number of hours.

Says the student who is still unclear how to master a course.

When I hear most students parroting the "work smarter not harder" ideas, they're usually not working very hard in the first place and looking for tricks to succeed in coursework without actually working hard.

Some approaches are more effective than others, but in the end, there is no substitute for hard work.

Accomplishment is the integral of effort over time. Sure, efforts in the wrong things can waste time and be less efficient. But there is no "magic" that bypasses lots of hard work over lots of time for true mastery.

300 hours is about right to truly master a meaty math or physics course. I won't say it is absolutely impossible for someone to do it with less. But I haven't ever seen it done. Most short cuts only fool the gullible students into believing they have achieved mastery when they have not.

 

[Orodruin]

There are some students who do everything right.

-Sit in the front row
-Ask questions
-Take neat notes

In my opinion, your first three examples is not necessarily doing ”everything right”. In particular the note taking is highly subjective and each student has to find what works for them. Personally, I almost never took notes. I tried when I first started university, but it just distracted me from the actual content and I am not the kind of person who learns well that way.

Asking questions, fine, in many cases you should ask if there is something that is unclear. But asking for the sake of asking is bot recommendable, although it projects interest to the teacher.

I also do not think it is possible to throw around numbers like 300 hours at random. People are different and have different talents. It will take different amount of time for different people. Some are naturally gifted and some will struggle even with 600 hours put in. Of course, that is not to say that hard work over time is not necessary, just that how much of it that is needed is individual.

You should also always try to be honest to yourself when you try to gauge how well you know something. Both under and over estimating your ability can be bad.

 

[symbolipoint]

To be great in something we have to invest a great deal of time.
And 300 can be reduced by using the effective resources and study techniques.
I think there are much more variables to determine the learning outcomes than the number of hours.

Yes. You're right. I gave the number 300 hours, as what? Maybe arbitrary. Important is how thoroughly one studies and how much time spent studying, how much repetition, how and where applied the subject or topic or course. How many INSTRUCTION hours would be in most semester course? Maybe 73 hours? Student needs what - maybe 80 hours of STUDY time outside of class-time; so this is an estimated 152 hours. Now imagine that regardless of grade earned, the student , mastery OR not, repeats everything on his own. About 230 hours as an estimate - but the student may spend MORE additional time bringing the hours somewhere between 230 and 300. NOW, what should happen if this student were to be graded on this as if actually going through the course enrolled? Probably he'd get an A grade earned.

 

[symbolipoint]

When I hear most students parroting the "work smarter not harder" ideas, they're usually not working very hard in the first place and looking for tricks to succeed in coursework without actually working hard.

YES, that! Also the student might have been working hard, but still needs much extra studying time. A serious student who works hard for a course the first time, and repeats the course working hard the second time, should expect to learn better the second time than the first time. Student should likely also earn a better grade upon finishing the repeated course. Further, there is no rule saying you must not review something before you officially repeat it.

 

[Vanadium 50]

Can anyone who has actually TAUGHT physics say that students who got Cs in the prereq math or physics course do as well (on average) as students who earned As?

Earned A's or received A's?

 

[Dr. Courtney]

Earned A's or received A's?

Sharp eye there. Once taught the second half of an algebra-based physics course (E&M) for a few years. We had the first half (mostly mechanics) tightened down pretty well, but lots of students who took the first half at a different nearby school would enroll as transient students and take the second half at the college I taught at.

Over a three year period not a single student who had passed (A, B, or C) our Physics 1 failed to pass our Physics 2 course. In contrast, EVERY student who had passed (A, B, or C) the Physics 1 at the community college in the next county over FAILED our Physics 2 course.

So there's great value in efforts to tighten down the pre-requisites of a course one teaches frequently. If there is a strong trend of students arriving in one's class unprepared in spite of good grades pre-requisite course, one owes it to the student body to begin a discussion and not allow that to continue in one's own institution.

When I was at the Air Force Academy, the institution took it quite seriously if students in pre-requisite courses were not well prepared for downstream courses. Students were tracked downstream, and teachers were assessed not based on student evaluations (which the administration did not even see), but on how one's students performed downstream.

 

[Mondayman]

When I hear most students parroting the "work smarter not harder" ideas, they're usually not working very hard in the first place and looking for tricks to succeed in coursework without actually working hard.

True, but some experienced students are able to learn what works for them and refine their study methods accordingly. Sometimes this results in less hours put in and less struggle with the material. I think the key is consistency and focused, uninterrupted work.

 

[jack action]

I would like to apply Bloom's theory of learning to this case, My own definition of mastery.

[...]

Level 2 : Understand it enough to solve the problem sets given by your instructor.
[...]

Obviously, getting A's can only prove you have reach Level 2.

There are some students who do everything right.

Here, I'm going to agree with @Orodruin and say that 'doing everything right' most likely vary from person to person. I was an 'A' student in university and here's my take on your list:

-Sit in the front row

I did it in high school and it did help because I was goofing around when sitting in the back. But I did not look for a particular place when in university.

-Ask questions

Never. Quite the opposite for me. 'Shut up and listen' was my motto. I did not need to understand while in class, just absorb as much material as possible and deal with it later.

-Take neat notes

Taking notes was important, but not much I do is neat. I have sometimes difficulty reading my own handwritten notes.

-Review the material before and after the class

I mostly only reviewed material before exam. But I was doing it very thoroughly: I could spend hours on a single page until it sinks in.

Sometimes, I wish I would have been told more of what needed to be read before a class. I think it would have help me understand better what some teachers were talking about. For me, the teacher's role is to show how to do it. I hated teachers that just explained the theory and then sent us home with problems to solve on our own. Show me how you do it!

-Do the problem sets

Never. I always hated homework and they never did me any good. Just a waste of time.

In my compressible fluid dynamics class, I only reviewed the material before the final exam (see previous answer). By the time I finished, I was so weary that I didn't care much to do practice problems. I did one problem and got it wrong (but quickly realized why). I did a second one, got it right, and I quit studying there, satisfied with my work. I got an A^- on the exam.

-Teach it to other students

This I did a little and it does help. (It is one of the perk of being on this forum)

-Hand in the assignments on time and do it by themselves

One time, one of my teacher decided to reserve 5% of the total grade to the homework assignments, to encourage (force?) students doing them. I personally decided that 5% wasn't worth it and decided to not do them as usual and accept the loss; contrary to my fellow students who though it was an easy 5% to get.

Once, I was talking with the teacher and he seemed proud of this new method he was using. I told him I still wasn't doing it because homework was not helping me at all and did not feel it was worth 5% to me. I don't know if it was that discussion that changed his mind, but by the end of the semester he told us that he was not going through with it, i.e. homework assignments were worth nothing on the final grade. People were pissed off!

-Attend the office hours

Always.

-Watch online video for more clarification

There was no 'online' in my time, but I did buy a textbook on a class subject once, to help me understand. The teacher was forcing us to buy his notes instead of a regular textbook. Although he was a good teacher for me, it did help me to get another point of view.

-------------------------------------------------​

Most of all, if I'm not interested in something, learning becomes very tedious.

Also, even though I was an 'A' student, I was just like you: It was very difficult to remember what I learned after the exams were passed.

A lot of the understanding came after, when working on problems that I cared for. For example, I was very excited about thermodynamics because I loved internal combustion engine. I thought I was going to learn about turbochargers in that class. Although I excelled in that class, in the end, I couldn't see how anything I learned related to turbos. When I studied on my own afterward to see how I could apply what I've learned to supercharging, I learned a lot. Not only about supercharging, because I realized that I wasn't really mastering some of the knowledge I acquired in university. It was a tedious process but so much fun, because it was an important subject to me.

IMHO, applied knowledge is much better to understand the general theories than the other way around.

 

[Mondayman]

Never. I always hated homework and they never did me any good. Just a waste of time.

I have to strongly disagree with this. Solving problems is where you do most of your learning. If you can't solve problems, how do you expect to pass exams and whatnot? The value of homework is so much more than the grade percentage it's worth.

It's like claiming to know how to play hockey cause one has watched it so much and knows all the rules, without ever stepping onto the ice to apply their skills.

 

[Orodruin]

I have to strongly disagree with this. Solving problems is where you do most of your learning. If you can't solve problems, how do you expect to pass exams and whatnot? The value of homework is so much more than the grade percentage it's worth.

It's like claiming to know how to play hockey cause one has watched it so much and knows all the rules, without ever stepping onto the ice to apply their skills.

Again, this is highly personal just as the poster you quoted mentioned. The poster was describing what works and doesn’t work for him. This does not mean that it will be the most effective for everyone or even most people. Quite clearly, you can have a general opinion on what methods are typically most effective, but you will have no idea what is best for any particular person.

Personally, I absorbed most of the relevant theory in class, did all homework that gave partial credit, and studied for exams almost exclusively by doing old exams and by helping others. That does not mean this would work for everyone.

 

[Mondayman]

True, I shouldn't jump the gun like that. But I would insist that problem solving is one of the most effective strategies to learn material and make it stick.

 

[symbolipoint]

I have to strongly disagree with this. Solving problems is where you do most of your learning. If you can't solve problems, how do you expect to pass exams and whatnot? The value of homework is so much more than the grade percentage it's worth.

It's like claiming to know how to play hockey cause one has watched it so much and knows all the rules, without ever stepping onto the ice to apply their skills.

LIKE LIKE LIKE LIKE LIKE

 

[symbolipoint]

Homework Assignments just 5% of the course grade? Some professors make it 10% of the grade; some might make it more or less than 10%. Much of the idea is like this: "Show me that you are both studying and learning now, before any tests are given and before the final examination is given".

 

[jack action]

If you can't solve problems, how do you expect to pass exams and whatnot?

When you understand how everything is linked together, you would be surprised how the problems become easy to solve.

My method was to summarize the class notes with as little words/equations as I could with the goal of using no more than one sheet of letter-size paper. Sometimes, I had to write really - really! - small, but I remember summarizing the statics class in less than half a page.

I remember my thermodynamics class where we were ask to know how to draw every P-V & T-s diagram for every cycle we studied. One of my friends had like ten pages of graphs and he was trying to memorize them, which is very hard. At first, I was going to do like him, but that was going to break my 1-sheet rule. I couldn't draw them small enough and still be comprehensible. So I studied them long and hard to find the similarities. Once I've noticed that on similar processes (compression, heating, etc.) behavior were similar for engine cycles (say, P increasing with V) and the reverse for refrigeration cycles, that was all I needed to know. So all I had on my summary sheet was a single line with a bunch of arrows pointing up, down, left and/or right besides a letter identifying the process. The specific details of each cycle were easy to remember to differentiate them.

Once you notice the patterns, it makes a heck of a difference to problem solving.

It's like claiming to know how to play hockey cause one has watched it so much and knows all the rules, without ever stepping onto the ice to apply their skills.

Full-disclosure: I'm not saying that I've never solved problems except in exams.

But you'd be surprise how easily your example can be done sometimes, if one have already played some other sports (like football or speed skating). Skills are transferable and there are a lot of similarities in different fields.

For my part, in a virtual world (i.e. visualizing in your mind), I think it is even easier, once you get it. Not only transferring, say, from fluid mechanics to electricity, but to such different fields as science to accounting or law. I have amazed often people in their field who couldn't believe how I was understanding what they were doing. I can read a set of rules, find the patterns, and zoom into the essentials, which are often done in very similar ways everywhere. After all, they were all created by humans and I am one too.

 

[Orodruin]

But I would insist that problem solving is one of the most effective strategies to learn material and make it stick.

... for most people. The entire point is that it can be counter productive to generalize and tell people ”you have to do this to learn well”. This is particularly true with people who have tested different methods for themselves and found out what works for them.