Which way is better? — Wheatley

The best way is the one that works best for you. People have different learning styles.

It depends on what the goal is. If the goal is to catch fish as efficiently as possible then A), but if the goal was to take up a new hobby or to relax then B) might make more sense.

Which way is better? — Wheatley

I regularly read the math-oriented subreddits on Reddit. There's a steady stream of people trying to learn math at their own at literally every level, from adults who never learned math as kids trying to pick up what they missed at the grade school level, to former math majors now in other careers trying to self-learn graduate level math.

I think it's a very hard way to go. It's a method that works for geniuses. Galileo was a college dropout, and Newton made his most brilliant discoveries while away from school during London's plague year.

But for most of us mortals, having a teacher is essential. In theory you can pick up a math book and start grinding through the definition/theorem/proof exposition and work hard on the problem sets. In reality, a good teacher makes all the difference. It's very difficult to go it alone.

Copying is better in nearly every way but there is an enjoyment of being free to experiment and enquire and learning something on your own.

1) You can learn from others by copying their instruction — Wheatley

Pros: You won't have to reinvent the wheel.

Cons: You won't discover anything new.

2) You can figure it out yourself. — Wheatley

Pros: You might discover something new.

Cons: You'll probably reinvent the wheel.

It boils down to, as usual, risk. If you feel the downside of reinventing the wheel is offset by the likelihood of a discovery and the fame and money that comes with it, figuring things out for yourself is the right choice, wander off into uncharted territories. Explore.

On the other hand, if you don't wish to waste precious time and resources as would happen if you reinvent the wheel and there's only a snowball's chance in hell that you'll discover anything worthwhile, learn from others and stay within territory that's already mapped. Colonize.

Speaking for myself, I prefer a mashup of the two. Learn from others and once you have what's known about a particular topic under your belt, activate explorer mode and set forth into the unknown. Mind you, this is only true of ideal situations, few and far between I'm afraid.

In theory you can pick up a math book and start grinding through the definition/theorem/proof exposition and work hard on the problem sets. In reality, a good teacher makes all the difference. It's very difficult to go it alone — fishfry

Yes, indeed. Even with a good instructor you can find yourself reading a paragraph over and over until its meaning sinks in. It's no picnic. You should always read a math book with pen and paper at hand. It's not like reading a novel.

I've known literary autodidacts, but one thing they may miss is not having their writings carefully criticized. It's hard to see our own work through another's eyes.

I wrote the following about intuition vs cognition, but it could be applied to learning:

The mode of travel can matter. I used to be jealous of those who, after my long cognitive slog to a place, I find already there, having arrived on the wings of intuition. But then I remember I have found along my way; the truth is often counterintuitive. While others may wonder what took me so long, I’d rather arrive knowing what I don’t know. We may be in agreement; we may be in the same place. But if I must have company, I choose those who arrive by foot.

Which way is better? — Wheatley

We have to learn it ourselves -- whether somebody teaches us or whether we use trial and error. The thing is, whatever we learn, we have to fold it into our repertoire by processing the information. It can't just be "poured in". If it could, then languages, math, or milking a cow could be known by anyone with no effort.

A teacher presents information; maybe makes it interesting; maybe threatens a pop quiz on the next chapter tomorrow. But you have to read it and incorporate it into your structure of knowledge. Trial and error works too, at least for many procedures (if we survive our errors). The results of our T & E efforts also have to be integrated into what we already know how to do.

I suppose, if you don't reinvent the wheel, but build it as taught, then you're both a builder and also in a great position to find flaws. And you'll have saved time, too. Might even have time to build wagons. And carts. :)

The sky is the limit