When I was in high school, the common knowledge was that a stereotypically elite school wasn't much better than any other school, especially at the undergraduate level, so I shouldn't worry too much about where I go to school. My experience at Stanford has convinced me that there is a large benefit to going to a stereotypically elite school, but I think that a motivated student can thrive at any school.
The benefit doesn't come from having funding for travel abroad, community centers, or resume critique sessions (though those things don't hurt). Trying to find all of the opportunities like that probably doesn't make much sense -- your university also gave you access to a hammer, but you probably aren't going out in search of all the nails you can find. Instead, you should start with the assumption that you are at college so that you can learn a lot, be happy, and make the world a better place. Then, you can look for the best ways to do those things.
As such, I'll outline the things that I got at Stanford that I don't think I would have gotten as easily at other schools, and I'll also discuss some ways that motivated students at any school can get the benefits of an elite education on their own.
My classes, both in computer science and in other disciplines, were hard. That's because the teachers knew that they could set the expectations high and we would perform accordingly. In the operating systems course at Stanford, for instance, you program a computer operating system. Stanford is on the quarter system, so classes are ten weeks long (versus semester systems where classes might be 17 weeks). I compared our OS class to some OS classes at semester schools, and even though we had much less time, we still programmed more than many other schools. This isn't just computer science. I took a design school class, Design for Extreme Affordability, where we made a product with the intent of it being marketed to the bottom of the pyramid. In some weeks, I spent 30 hours on that class in addition to all of the other things I was doing. Going to a stereotypically elite school will probably make it easy for you to work really hard. That's not to say that the quality of instruction is unimportant -- I really enjoyed almost all of my teachers at Stanford -- but I think that the work load is probably a bigger difference than teaching abilities.
If your school doesn't work you to the limits of your stamina and you want to work harder, you have a lot of options. Some students take tons and tons of classes so that they can finish faster or get a wider breadth of experience. Some students do research projects or internships during the school year. Nowadays, you could take classes for free online in addition to your regular classes.
The key limiting factor is your time. At Stanford, many classes had an expectation of a high workload, so any student who took those classes had hard work as a default. If your school doesn't have a heavy workload as the default, you need to take it upon yourself to find things to work on.
My teachers were Nobel laureates, bestselling authors, nonprofit CEOs, and great educators. I attended talks by half of the US Supreme Court, plenty of other politicians, founders of companies, and some of my favorite musicians. My peers were scaling the companies that they started in high school and were volunteering with nonprofits all over the world.
The hardest part about doing something revolutionary like leading a social movement, starting a big company or nonprofit, or developing a technology that changes the world is getting the confidence to believe that such a thing is possible. It's hard to think of "important" people as people just like you. At Stanford, I was surrounded by enough people who changed the world that I realized that most of them started out just as nervous, unconfident, and unskillful as anyone else and mostly changed the world with a little bit of luck and a lot of hard work. Stanford teaches students to take for granted that they will achieve the impossible.
The key limiting factor is your aspirations. At Stanford, it's easy to dream big and to pretend that your dreams are reality, and then they might actually become reality. If your school doesn't make it seem obvious that you will change the world, you need to find ways to humanize leaders and to expect greatness from yourself. Listen to big talks (TED talks do a pretty good job of this) and become part of a community of changemakers (eg, the Ashoka U program). Don't just start a nonprofit with the goal of helping a hundred people in your local community; start an organization with the goal of dramatically improving the livelihood of a million people.
A related issue is the way you think about money. I was raised to be frugal, I am still frugal to this day, and I think that frugality is a virtue. I didn't ever have an allowance as a kid, and my dad would make me think about (and often consider against) small purchases like eating out or buying a video game. As such, I don't spend very much money on meaningless things (so I have plenty of money to donate to nonprofits, now). The conception of what amounts of money are big and small is different from the conception of frugality.
At Stanford, I was exposed to a lot of people who were wealthier than me, and I was also exposed to a lot of companies, government agencies, and nonprofits. Startups might get hundreds of thousands of dollars in angel investment and millions of dollars in venture capital, and most startups don't succeed. Many nonprofits have hundreds of millions of dollars in their budgets, and they raise that money through grants and individual fundraising. There are a lot of mediocre companies and nonprofits with hundreds of thousands of dollars, which is to say that you shouldn't think of that as a big number.
There are also a lot of students with great ideas to change the world who feel held back because they need a few hundred or a few thousand dollars and they think that that is a big number. That is bad. I think that, unless you grew up wealthy, your sense of big and small numbers is probably not conducive to you changing the world. You should still be frugal, but if you have a good idea, you want to put a lot of time and energy into working on it, and you only need a few thousand dollars, you shouldn't think of that as a barrier.
The key limiting factor is your belief about the resources available to you. You have access to more money than you think, especially with things like university grants and crowdfunding.
Social science research differentiates between strong ties and weak ties. Your network consists of both. Emotionally, your strong ties are useful to give you personal advice. Financially, your weak ties are useful in getting a job, and your strong ties are usful in bankrolling your startup (or political campaign or whatever). Networking gets a bad rap because people think it means meeting people for the sole sake of getting a job or for selfish reasons. I, apparently, am good at networking, but I think of people as an end in themselves, not as a means to an end. I enjoy meeting cool people who do cool things, learning more about their life and their work, and helping them out. I enjoy connecting cool people with other cool people so that they can both enjoy each other's company and help each other out in their jobs. Running Code the Change and being a Stanford student made it easy to meet lots of people doing cool things.
The key limiting factor is your confidence in reaching out to people. As a nice, eager college student, pretty much everyone at every company, nonprofit, and government agency will be happy to talk to you. Cold call them. Cold email them. Use your alumni network. Get to know a professor. You don't even need a real reason to talk with any of them ("Hi -- I don't have any questions about your class, but I have really been enjoying it, and I wanted to get to know you better and learn about how you got involved with ______"). It is nice to have an excuse to break the ice -- for me, Code the Change serves that purpose well, and if you want, you can start up a Code the Change chapter at your school, but there are plenty of other reasons that you could have to talk with someone. But, really, just go into it with the attitude of wanting to meet people and learn about them and keep doing it. At work, I made a personal goal of meeting one new person each week, and I have been mostly sticking to that so far.
In high school, I was much more abstract than I am now, and now I'm fairly pragmatic. That is to say, in high school I was philosophically oriented (eg, I thought about philosophical issues), whereas now I'm value oriented (eg, I try to live my life in accordance with my ethical philosophy). Before, I thought about research, whereas now I think about taking concrete action to improve people's livelihoods.
This is less an effect of Stanford being an elite institution and more an effect of it being an entrepreneurial institution. I saw people starting social movements, nonprofits, and companies, so I began to think about those as methods of social change. Thinking abstractly is a good thing, but you need to stay grounded and make sure that you are putting your ideas into practice.
The key factor is cognitive consonance rather than cognitive dissonance. Cognitive dissonance means that your beliefs don't line up with your actions (eg, hypocrisy). A lot of people don't have a clear ethical philosophy, so they spend much of their lives in cognitive dissonance, unable to really justify their actions. That's a problem. One of the reasons I write my thoughts publicly is because I think that if I can't publicly defend my thoughts or my actions, then I might be thinking or doing the wrong things. You need to find a way to be introspective, and you need to find a way to pull your thoughts and actions into harmony.
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Then again, if you've made it this far, you may realize that my vision of pragmatism is saving the world by pretending that you have more money for your nonprofit than you actually do, overworking yourself, talking to strangers, and as a result, doing the impossible. I guess that's the benefit of an elite education -- you can say things like that and actually take yourself seriously!
Note: this was cross-posted to Quora (a post by someone at Stanford and two posts asking a similar question in general).