> I can learn from a math teacher in an hour what would take me possibly years, or a lifetime for that matter, to discover on my own.
I would be surprised if this is true, unless you can't read. Usually the reverse is true. (Which is not to discount the value of guidance, but real learning happens with you, not anyone else; in an hour (or less), you can find out about a good book or problem to study from a math teacher, which could take years of searching on your own, but the learning doesn't happen by you listening to them talk, especially in math.)
> I would be surprised if this is true, unless you can't read. Usually the reverse is true.
I, and many other students, pre-read our mathematics text before class started.
It was of about 0 usefulness. I'd say maybe 10%-15% comprehension.
Steps to solve a problem were always cut out, and always seemingly the ones that we needed most to see. Example problems were either too complex or too simple. And explanations of complex topics were poorly written all around.
A good professor comes in, sees where we are having problems at. Discusses the topics in ways we can relate to, helps guide us down the path of understanding. Demonstrates techniques on the board at various levels of difficulty, ensuring we can see not only how is an idea applied to a trivial problem, but also how it is applied within a larger framework as well.
I am someone who loves to read, and who learns really well from reading. But when it comes to math I don't think I ever got any comprehension from a mathematics textbook by itself. Typically after a lecture was given, maybe 80% of the book would make sense. Sometimes, on a few chapters, 100%. Eventual understanding of what the book said came only after mastery of the technique, from which I could work backwards to figure out what the book was doing.
Then there is the fact that applying a concept is a huge part of understanding it. Application and practice are what take a shakily understood idea that will be soon forgotten and turn it into an understanding of a new way of thinking that will stick with the student for life.
A good professor can see what areas of understanding their class is struggling with and assign problems appropriately, helping to ensure that students come away from their schooling with life long knowledge.
And of course on top of this there remains the fact that different people have different abilities of reading. One of the smartest people I know is dyslexic, it is faster for him to have someone else read the text to him than for him to read it himself!
Naturally I know people who were able to just read their math textbook and understand everything in it. Kudos to them! For the rest of the students though, who were all actively asking questions and engaging with the professors during classes, it seems that they most certainly gained value out of class lectures!
Indeed, office hours were also popular. The typical rule in study groups I attended was we'd spend 2-3 hours trying to understand a concept together before we marked it down as "ask the professor about it".
That's interesting. For me, in the only somewhat hard math class I took in college, the class meetings were mostly useless. All that I learned came from solving the problems.
For me, solving the problems cemented the concepts. Some problems I couldn't solve, I'd ask another student for help, they'd explain the remainder of the concept that I was failing to grasp, and things would make sense. If they didn't go to prof next day for help.
Basically the professor would show off the technique in various usages, I'd take copious step by step notes, and then try to apply those same techniques to problems given. A really good problem set builds up difficulty and slowly expands upon the initial base techniques.
This is in strict contrast to how CS is taught. Although CS is still largely "throw in pool, hope students figure it out". IMHO I think I learned more in CS from discussions after class with the prof than from most classes, and I also think many other students would agree with me. :)
It's honestly something that i found out entirely through self-experience. I'm a self-taught Perl developer who spent 5 years to reach a proficient level. I managed to get another developer with little Perl experience to nearly my level within 6 months by handing him two excellent books and reading his code and pointing out issues in his algorithms and semantics.
The local maxima i am aware of here are:
1. Outdated books. When i learned Perl i wasted a lot of time on books that are now known widely to be objectively bad, but are still recommended widely. The ones i recommended allowed learning with little friction.
2. Algorithmic and semantic problems that have to be resolved through lengthy investigation sessions long after they're implemented. I spent a LOT of time figuring out what doesn't work well in the long run. He's been able to skip those and spend his time more productively on problems that i haven't encoutered or solved yet.
I'm not trying to toot my horn here and i think i'm not a good teacher. But the advantages he had from me being available were staggering.
> I spent a LOT of time figuring out what doesn't work well in the long run. He's been able to skip those and spend his time more productively on problems that i haven't encoutered or solved yet.
This is one thing teachers are really good for!
I'm sure some students would figure out not to sure GOTO after awhile, but, well, think of all those wasted years! Easier to just explain to them the problems upfront!
> I managed to get another developer with little Perl experience to nearly my level within 6 months by handing him two excellent books and reading his code and pointing out issues in his algorithms and semantics.
Part of this is natural ability as well. I have a friend who much to my dismay is not a software engineer, but who only dabbles.
I was able to explain lambdas and closures to him in a couple of minutes over IM, and he was almost immediately able to see their uses.
Meanwhile, I know experienced software engineers who cannot fathom the purpose of a Lambda, and a few friends I know who are just not intelligent enough (?) to comprehend on a deep level how lambdas and closures work.
Now all this said, the college professor who was supposed to teach us functional programming did such as piss poor job at it that I ended up not understanding basic FP concepts until I started using them in C# and reading about how they are implemented in the CLR. (Part of this is because Eric Lippert is a damn good writer...)
I think the text book we had at the time defined them solely in terms of mathematical constructs. Ugh.
I would be surprised if this is true, unless you can't read. Usually the reverse is true. (Which is not to discount the value of guidance, but real learning happens with you, not anyone else; in an hour (or less), you can find out about a good book or problem to study from a math teacher, which could take years of searching on your own, but the learning doesn't happen by you listening to them talk, especially in math.)