This is truly the most frequent of FAQs. So here are my answers.
- To parents: start young -- but keep it low stakes.
However, don't go overboard with this. Elementary and middle school kids shouldn't be doing work for hours every day in the summer. Your goal is not to teach your kids as much as you possibly can; they'll have plenty of time for intensive learning later. Your real goal here is to inspire a love of learning that will last them throughout their lives. Make it fun, and cater to your kids' interests as much as you can.
Schoolwork is not the only important thing: encourage your kids to pursue whatever creative/exploratory things they want. Try to keep lots of books in the house, or make regular trips to a library. Reading lots of books will improve your kids' vocabularies and writing abilities. I also think it's good for kids to do martial arts, sports, or another kind of athletic activity. Tae kwon do helped me learn self-discipline and perseverance. You can also suggest to your kids that they sign up for the science fair, or robotics team, or quiz bowl, etc, or write stories, or make art projects, or do theatre, etc. That said, it's important that academic and creative activities should be fun, low-pressure, and at least somewhat self-directed -- if a kid tries some activity and consistently dislikes it, then it's not for them. Kids will be much more motivated and passionate about their activities if they're doing things they actually like, rather than things they're being forced to do.1
- A strong academic record is basically a must.
Regarding SATs specifically, colleges publish the 25th and 75th percentile scores of their admits. This is a decent way to figure out if your academic record is approximately in range for the school. If you beat 75th percentile that's good, and it doesn't really matter by how much. If you're below 25th percentile (and you've already done a reasonable amount of SAT prep), then you should pay extra attention to the next bullet point.
- Do what you can to stand out, at a state or even national level.
- Make good use of your summers.
MIT lists a number of STEM-focused summer programs here, and the AMS has a longer list of programs in math specifically.
- Don't procrastinate on your college essays.
- Make use of your counselor rec letter.
- Don't add extra materials to your application, unless you've got a good reason.
If it's not really, actually going to make a big difference to your application, leave it out. You don't want your admissions officer to start grumbling about how you are making them read even more stuff than they already have to read. This goes for extra rec letters too -- if you've got three teachers who could write you strong letters, just pick your top 2 (and tell the third to email your guidance counselor!)
- Take care of your mental health and get enough sleep.
A final aside: None of my three closest friends from childhood went to highly selective colleges. One went to a large public school to train as a pharmacist; she's also contemplating going into real estate, and I'm sure she's going to make more money than I ever do. One is applying to psych master's programs so she can become a social worker, and I'm sure she's going to make a bigger difference in the lives of vulnerable people than I ever will. And the third went to community college, is now married with a toddler and another little one on the way, and has spent the past few years living their dreams of traveling all over the world. You should only try to get into a top college if that's genuinely part of what you want your life to look like -- happiness and fulfillment in life are not contingent on it.
1 As an example, here's my own experience. In elementary and middle school, I didn't really learn any advanced math or science. School was easy, and my creative/academic energy was mostly directed elsewhere. I think it was very good for me to have lots of free time and no internet, TV, XBox, Wii, etc -- my brother and I were limited to 1 hour of internet time per day. So I filled my time with reading books, writing stories, learning miscellaneous bits of physics and chemistry from my parents, working on art and photography projects, taking tae kwon do classes, playing with Scratch (a programming language for kids), and inventing silly games with my friends. I played saxophone for a few years, but I really didn't like practicing, so my parents let me quit. I was in stage crew for two years. I went to MathCounts once, in 8th grade, with no preparation. I didn't do well at all, but I didn't care because it was just for fun.
I think I would have benefited from getting to learn more challenging math in 7th-8th grade, e.g. by working through some AoPS books, or taking AoPS classes online slightly earlier than I actually did (the summer after 9th grade). I think I also would have enjoyed doing more science fair projects -- I didn't do any until 8th grade. However, I don't think I would have benefited from a serious focus on math contests at that age; I think the competitive aspect would have made me unhappy.
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