I think that when teachers make their students research on the internet, they are just being lazy.
Also, most of changes in learning at school is made for money, so whether learning is social or not, the people who write the program for public schools will do what is financially better for them by fooling everyone everyone that it is better. No offense.

I benefited a lot from a competitive environment as a kid in Russia (note that it is quite different from what "social learning" usually means in the USA. We exchanged the ideas freely and had a lot of team work and such but it was always "who'll get it first?" or "who'll do it best?"). It was quite a shock to realize how low I stood compared to others when I was first exposed to it in honest but it was fun to fight my way up and the learning rate was much faster than it would be without such pressure.

It is worth mentioning though that 3 years spent in this competitive environment were preceded by 4 years of essentially solitary learning with very little guidance from anybody (my mother would buy me any math. related book I would ask for but she could hardly help me to choose them and I had essentially nobody to talk about mathematics to).

I like collaborative research a lot (I doubt I've published a single paper without co-authors in the last 4 years) but it is always periods of discussions followed by periods of solitary thoughts followed by periods of discussions followed by etc. It is a good idea to spend some time discussing what to try (both because other people may see how to overcome your difficulties and because they may see fatal flaws in your ideas) but then there is a lot of work to be done on your own before you proceed to the next "public" stage.

That's my view of social vs. individual for whatever it is worth.

As far as the "educational tricks" are concerned, I prefer the model (that was used in some team competitions) that tells each team: you are one unit; here is the problem, nobody cares how exactly you cooperate on it but in 30 minutes the opposing team will choose one of you to present the solution.

Anyway, IMHO there are no universal recipes and general theories in education and thinking in general terms here is counterproductive. You just try this and that and see what works and what doesn't in your hands. You've already made Alcumus and FTW. They are based on completely opposite "social interaction" principles as far as I can tell. See if the people who use them are the same or whether you have a clear split in preferences and ages and what is popular there and what isn't. That'll tell you something. AoPS Wiki, which might become a good learning source and a fun community project, turned primarily into the collection of "2000 AIME I Problems/Problem 3"'s, so that's, probably, not the way to go. Mathlink contest is another interesting thing to look at (I really wish you would revive it) not only as a contest per se but as a model of interaction between more and less experienced mathlinkers (you can try to get some people to create the problems, some to grade it, and some to participate as contestants, though you'll have to put the whole thing together yourself, not to rely on it running on its own steam). Math. jams are also there, and so on and so forth.

What I mean is that you have made enough experiments to start drawing conclusions based on them instead of deliberating too much over how what some other people are doing fits general "education philosophy". There will always be more ideas (both good and bad) out there than you can evaluate, not to say to implement. If this particular one stirs your thoughts too much, just try it at the next math. camp you'll run and see how much you and the students like it. Nothing is good or bad per se according to some "self-evident general principles", "past individual experiences", or "possible benefits and side effects one can imagine". You just try it on small scale and observe the output. If it is favorable, you keep it, if not, you discard it. We were all taught how to make lists of possible pro's and contra's and know perfectly well that either one can be made longer and look more convincing on paper. So why to bother? If there is no grave danger and irrecoverable loss associated with trying something and it is really tempting, why not just to try and see?

Sounds like your experiences are not that unlike mine, despite our very different upbringings. It also sounds like your conclusions are not that different from mine. There isn't a silver bullet in education, and we are best off if we create many delivery systems and many avenues to, and definitions of, success. But I periodically force myself to question my convictions about education, partially because I think it's dangerous to automatically believe everything I think. Team-based learning has always seemed faddish to me, but that could be because my model of learning was so different. The social aspect was important for social reasons, and for status-based reinforcement, I suppose. But the vast majority of my learning was done alone.

I am a bit sad that the wiki project didn't really work out that well. But I guess I shouldn't be that surprised. It doesn't even work that well on Wikipedia, which has a much larger number of contributors than we ever could.

Interesting question about FTW and Alcumus. I'm a bit embarrassed that I hadn't thought to investigate the question earlier. They do have distinct differences in their potential competitive appeal.

IMO, the best way to learn is to work with what you're learning until the subjects changes the way you feel about it.

For example, when I think about 2, the feeling I get is liking it, as if it is the "good guy." On the other hand, 3 gives me a dislike feeling, like the "bad guy," even though I have nothing against it. Then, I associate all even numbers with 2 (they give me a similar feeling) and all odd numbers with 3 (I don't know why I don't use 1). The look and sound give me the "feelings" as well. In a similar way, I associate with , 1.7 with 2.89 (2.89 is a common gas price here, so I see it often).

That is pretty much how I learn everything, so it is impossible for me to learn directly from a teacher or socially. When the teacher uses a song or a mneumonic device to help memorize something, it doesn't help me, and in some cases, makes it harder for me (ex. SOH CAH TOA just doesn't work for me).

Another point I wanted to mention is memorizing vs remembering. IMO, they are quite different. Memorizing can be something like memorizing the the US Presidents in order (just the names, nothing else), while remembering can be remembering the names of your cousins. Memorizing the Presidents just put names in your brain, but cousin's names give you certain feelings and memories about them.


Of course, this is just me, and everyone learns in different ways.

Yeah, I think one of the biggest problems with public education is that it is "factory-based". The standardization system is based on using the minimum resources to send out info to the maximum number of students.

Maybe it's also the nature of college apps. They encourage students to have interest in not so much their area of study but in the scores at the end. Students can study their teachers' notes/homework well enough to get an A but won't get much more than a trained chimpanzee would from a group of biologists.

So when group "learning" activiites are dished out, very little learning takes place. But given an environment in say, a math circle, where say 5 or six students are given a difficult problem to work on, the competitive nature of the problem as well as the presumably high interest of the students encourages at least roughly equal attempts and thinking, which are the basis of learning, even if just 2 or 3 students dominate most of the solution writing.

Speaking from personal experience, I also had good experiences with an interactive competitive environment, and also with collaborative study and research. Most of my mathematical friends also say they work better when working with others, seeing mathematics as a social activity.

That being said, I do know several mathematicians that do not pursue mathematics as a social activity, and in fact prefer working on their own. They form a minority among my circle of friends, though that could be because it is naturally more likely that I will meet the more socially inclined mathematicians. In any case, I believe it is a highly individual preference, and I'm glad that both Alcumus and FTW exist, so as to accommodate both types of students.