All posts by Jonah Sinick

Perfectionism and readjustment of expectations

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By Jonah Sinick

Cross-posted from the Davidson Institute Gifted Issues Discussion Forum and Quora

Something that many gifted children struggle with is perfectionism, especially in relation to academics. See for example Perfectionism and the Gifted Child at Hoagies’ Gifted Education Page. There are a number of possible reasons for this, but one can be that their sense of identity becomes heavily dependent intertwined very well (achieving near perfect performance), and this can lead them to feel that if they don’t do very well, then they’re worthless.

Sometimes the standard that they judge themselves against is a relative one: e.g. being ranked #1 in a high school graduating class. Sometimes the standard that they judge themselves against is an absolute one: e.g. getting a perfect score on a math test.

It seems to me that it helps to realize that in the long run, one’s performance will almost certainly fall far short of perfection, regardless of who one is.

The academic environment that a child is exposed to is usually artificial in a sense. For example, exams are designed so that all of the questions are doable with a reasonable amount of study, such that it can be possible for gifted children to get perfect sores. One could construct much harder exams, such that even a highly gifted child would score only 50% or lower.
Even if being the best student in a K-12 class is within the realm of the possible, one will almost inevitably end up in a context in which one isn’t exceptional. It’s common for valedictorians to go on to Ivy League schools and to be shocked to find that they’re only average within that population. There are 4 million American high school students of a given age. Even if one is the best student in a class of 400, the a priori probability of being the best student in the country is 1 in 10,000.

Even if one is the best student of a given age in the country (for example, in mathematical research ability), if one is ambitious, one will still fail routinely. The Riemann Hypothesis in mathematics has been unsolved since it was first 1859. Five+ generations of mathematicians have grappled with it without success. Even if one is the among the best mathematicians in the world, and works on it full time, one probably has a < 10% chance of solving it.

Being exposed to this perspective can initially be jarring. As a matter of reality, one almost certainly isn’t going to be the best in the world at something hard, and one almost certainly isn’t going to do something regarded as amazing. Almost all of those very smart people who dream of proving the Riemann Hypothesis will be disappointed. But once one accepts this, one can learn that it’s possible to have a happy life anyway. The sooner one comes around to this view, the better. It can be very liberating.

Procrastinating because of uncertainty

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By Jonah Sinick

Cross-posted from Less Wrong and Quora

Procrastination accompanied by guilt comes from an internal conflict about whether one should do the activity. Sometimes the conflict comes from partly wanting to cater to one’s present self (by engaging in more gratifying short-term activities) and partly wanting to cater to one’s future self, or to others (by doing something that’s less rewarding in the short term but that will pay for others, or pay off in the long run).

But this is often not the only element present when one procrastinates and is guilty. Often another element present is uncertainty as to whether the activity is what one should be doing, even when considering the indirect consequences. This can be subconscious: one might consciously think “I know I should be doing X, but I just can’t motivate myself to do it” while simultaneously believing on some level that one shouldn’t be doing X (even when considering indirect consequences). The conscious self isn’t always right in these situations – sometimes rather than trying to overcome procrastination, one should instead abandon the activity that one is procrastinating, for example, when the activity isneither interesting nor important. The subconscious self isn’t always right either: sometimes it’s operating based in false information or insufficient reflection.

If one can recognize and resolve the uncertainty, this can increase one’s motivation to do the work if it’s the right thing to be doing, and help one decide not to do the work if it’s not the right thing to be doing. So determining whether there’s uncertainty and trying to resolve it can have high value.

It’s not always possible to resolve the uncertainty. When this is the case, recognizing that there is uncertainty may not be helpful. Unfortunately, uncertainty can be demotivating even when completing the task is expected value maximizing. The question of how to stay motivated in the face of uncertainty is an important one that I don’t know the answer to in general.

Below, I give some examples of beliefs that can coexist with “I know that I should do the work” that give rise to uncertainty, together with commentary. Some of the beliefs described overlap in character, or can be present simultaneously.

 

A belief that it’s more effective to do the work later on

Sometimes there are higher priority things to do (even if one should do the work later on). Sometimes one is in an unusually poor state to do the work (for example, if one is sleep-deprived and this is not a regular condition). In such cases, procrastination can be rational.

Sometimes one rationalizes procrastination with the justification that there are higher priority things to do in the near term, even when it’s not true. Sometimes thinking that one will be in a better state to do the work later on is wishful thinking. So this belief may or may not be good reason to abandon the activity.

A belief that one can’t do the work

Sometimes the belief is well-grounded, for example, for most people who are working on solving a famous unsolved mathematical problem or working creating a tech startup. It tends not to be true for people who are trying to do things that many others have done successfully before. Sometimes the belief can arise from it not immediately being clear how to do the work, even though one could figure out how to do the work if one thought about it. For example, if one is having trouble learning to code, one can ask friends for help, or use Google to find answers to questions.

A belief that one is poorly suited to the work

Even if one can do the work, one might procrastinate it because one has the sense that even if one does it, it won’t move one forward.

I know a number of former engineering majors who found it very hard to motivate themselves to work on their first year math, science and engineering classes because they struggled to learn the material, decided that engineering wasn’t for them, and felt liberated upon coming to this conclusion, feeling much better doing work that they’re better at and enjoy more.

Their motivational problems may have been a valuable signal to their conscious selves that they should be doing something else, and their decision to drop engineering may be rational: they could have been picking up on not being good enough at engineering (or find it enjoyable enough) to be able to get good engineering jobs relative to the other jobs that they could get.

They may have underestimated their ability to improve (c.f. How my math skills improved dramatically). They may have been misinformed about the extent to which engineering jobs are similar to learning the material in the required courses. So their procrastination may not have been a reliable signal that they should abandon the path that they were on.

A belief that the work is inefficient or unimportant

All else being equal, we flinch away from work that’s inefficient or unimportant. So procrastination can be a signal of this belief.

Sometimes there’s a better way to accomplish the goal that a task is supposed to accomplish. For example, it might be possible to write a computer program that automatically carries out a tedious task, whether it be computational, information-gathering or sorting. Also, carrying out the task may not help achieve the goal at all.

On the flip side:

  • Even if it’s possible for a goal to be achieved more efficiently in the abstract, one may not have the resources to accomplish it more efficiently.
  • In the modern economy, many important jobs are several steps removed from the tangible results, so that one can get a subjective sense of not getting anything important done even when one is.
  • Even if work is of no intrinsic importance, it may still be important to do it so as to meet credentialing requirements (e.g. in the context of school).

So here too, it may or may not be rational to act on this belief.

“Children’s stuff” gets less independent reviews and coverage relative to its userbase size

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By Vipul Naik

Cross-posted from Quora

As part of research that my collaborator Jonah Sinick and I have been doing for Cognito Mentoring, we’ve repeatedly noticed that products aimed at children rarely get high-quality independent reviews. This isn’t just bad in and of itself; it also means that these products can’t get Wikipedia pages of their own because they don’t pass Wikipedia’s notability test.
Why might that be? Possible explanations:

  • The children who use the products themselves aren’t old enough or mature enough to write first-person reviews.
  • Publications are targeted at adults, so children’s stuff isn’t that interesting to them.
  • Any other explanations?

Here are some of the resources we looked at (many are listed on our Online mathematics learning resources page; others are listed elsewhere on Cognito)

It’s also noteworthy that almost none of the best books aimed at young people have Wikipedia pages, although it’s common for Wikipedia to have pages on books aimed at adults. For instance, Arthur Engel’s Problem-Solving Strategies is a widely used book for contest mathematics, but neither the book nor the author makes it to Wikipedia.
Even resources that do receive some press coverage generally receive very little. For instance, the Wikipedia page about College Confidential scarcely does justice to College Confidential’s stature as a go-to resource for information about college admissions.
Thoughts?

Rely on self-learning, not school

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By Jonah Sinick
I learned most of what I know on my own, outside of school. This is also true of the most intellectually impressive people who I know.School is rarely well optimized for learning, especially for gifted people. This is what one should expect:

  • The curriculum is a hodge-podge of subjects grouped together by historical accident, that wasn’t developed with a view toward teaching the most useful skills.
  • The materials used in public schools are often determined by fashion (c.f. the “ “math wars”) by political figures who are motivated by ideological agendas and ignore evidence that contradicts their views.
  • Public schools narrowly optimize for improving standardized test scores (to the exclusion of learning) because their reputations and funding are dependent on getting good scores.
  • Public schools’ teacher’s unions prevent poor teachers from being fired.
  • Teaching is a relatively low status job that doesn’t attract many intellectually talented people. (This is true both at public schools and at private schools.)
  • Gifted children are a small minority and not a major focus of schools.

These are things that are immutable, so prospects for substantially improving your child’s education through advocacy aren’t great.

When you learn on your own, you have the freedom to learn

  • The most important subjects
  • From the best materials
  • At your own pace

For gifted children who don’t have unusually good teachers and are self-motivated, school can’t compete with this. There are prospects for dramatically improving education for gifted children – they just don’t come from school.

The magnitude of the benefits hinge on finding the most important subjects to learn and the best resources to learn from. It’s not obvious what these are. Cognito Mentoring has been researching these things (which is one reason why I compiled a list of links to math resource forum threads) for example).

What do you think? Have I missed important points?

Is math privileged for gifted children?

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By Jonah Sinick

Cross-posted from the Davidson Institute Gifted Issues Discussion Forum and Quora

In Underconfidence in gifted girls I suggested psychology, philosophy, economics and evolutionary biology as candidate subjects for gifted children to learn. I’d add history of science. Thomas Percy wrote:

As a practicing researcher in one of the areas you mentioned in point a, I actually don’t think a child should focus too much in them. I think social science is still relatively subjective and requires experience even a PG child would not necessarily have an easier time acquiring beyond their age. Time is better spent in mastering a more foundational subject. Math and etymology are both fine use of time.I think that learning math is very important for gifted children, as I argued in Gifted children could learn math much earlier. See also Cognito Mentoring’s page on mathematics learning benefits.I’d like to address the question of whether gifted children don’t have enough life experience to be ready for the other subjects that I mention, relative to their readiness for math. I’ve worked primarily with gifted young people of ages 10 and higher, so my remarks are primarily of relevance to that age group, though they may be relevant to exceptionally gifted children who are younger than that as well.

  • Study of Exceptional Talent has found that many more children qualify based on the math section of the SAT than on the verbal section of the SAT. This suggests that gifted children can, on average, excel more in math than in subjects that require verbal reasoning. (On a recent thread it was suggested that the modern SAT’s verbal section isn’t a good measure of verbal reasoning, but many more people qualified for Study of Exceptional Talent before 1995 as well.) It’s been hypothesized that this is because high performance in math can come either from strong verbal reasoning or from strong abstract pattern recognition (of the type that the Raven’s matrices test measures).
  • The case for learning the other subjects that I mention is stronger for verbally gifted children than for gifted children whose strengths are nonverbal.
  • Because math is a subject that’s taught in K-8 school whereas the other subjects that I mentioned aren’t, one would expect gifted children to learn more math independently of whether they’re more developmentally ready for it. It can be argued that the reason that math is taught in schools when the other subjects aren’t is because children are more developmentally ready for math. But there are other possible explanations for this, such as the practical importance of arithmetic. In any case, one would have causality in both directions even if it were true.
  • Similarly, the fact that there are more math enrichment activities (largely in the form of contests) available for gifted children makes them more likely to excel in math than in the other subjects. My understanding is that math contest culture originated at least in part from the Cold War, when the Soviet Union worked to train children in preparation for quantitative occupations in research and development to feed into the Soviet Union’s military power.
  • It may be that life experience enables one to understand economics more deeply. But it’s equally true that learning economics early could prepare one to learn more from one’s early life experiences, on account of seeing relevant economic concepts in them.
  • I think that for children, improving reading and writing skills is more important than learning the subjects that I mentioned. But one can pick up reading and writing skills through them.

Nontechnical, nonfiction books aimed at adults that have few prerequisites such as:

may be well-suited to gifted children with broad curiosity who are reading at the adult level.

How my math skills improved dramatically

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By Jonah Sinick

Cross-posted from Less Wrong and from the Davidson Institute Gifted Issues Forum

When I was a freshman in high school, I was a mediocre math student: I earned a D in second semester geometry and had to repeat the course. By the time I was a senior in high school, I was one of the strongest few math students in my class of ~600 students at an academic magnet high school. I went on to earn a PhD in math. Most people wouldn’t have guessed that I could have improved so much, and the shift that occurred was very surreal to me. It’s all the more striking in that the bulk of the shift occurred in a single year. I thought I’d share what strategies facilitated the change.

I became motivated to learn more

I took a course in chemistry my sophomore year, and loved it so much that I thought that I would pursue a career in the physical sciences. I knew that understanding math is essential for a career in the physical sciences, and so I became determined to learn it well. I immersed myself in math: At the start of my junior year I started learning calculus on my own. I didn’t have the “official” prerequisites for calculus, for example, I didn’t know trigonometry. But I didn’t need to learn trigonometry to get started: I just skipped over the parts of calculus books involving trigonometric functions. Because I was behind a semester, I didn’t have the “official” prerequisite for analytic geometry during my junior year, but I gained permission to sit in on a course (not for official academic credit) while taking trigonometry at the same time. I also took a course in honors physics that used a lot of algebra, and gave some hints of the relationship between physics and calculus.

I learned these subjects better simultaneously than I would have had I learned them sequentially. A lot of times students don’t spend enough time learning math per day to imprint the material in their long-term memories. They end up forgetting the techniques that they learn in short order, and have to relearn them repeatedly as a result. Learning them thoroughly the first time around would save them a lot of time later on. Because there was substantial overlap in the algebraic techniques utilized in the different subjects I was studying, my exposure to them per day was higher, so that when I learned them, they stuck in my long-term memory.

I learned from multiple expositions

This is related to the above point, but is worth highlighting on its own: I read textbooks on the subjects that I was studying aside from the assigned textbooks. Often a given textbook won’t explain all of the topics as well as possible, and when one has difficulty understanding a given textbook’s exposition of a topic, one can find a better one if one consults other references.

I learned basic techniques in the context of interesting problems

I distinctly remember hearing about how it was possible to find the graph of a rotated conic section from its defining equation. I found it amazing that it was possible to do this. Similarly, I found some of the applications of calculus to be amazing. This amazement motivated me to learn how to implement the various techniques needed, and they became more memorable when placed in the context of larger problems.

I found a friend who was also learning math in a serious way

It was really helpful to have someone who was both deeply involved and responsive, who I could consult when I got stuck, and with whom I could work through problems. This was helpful both from a motivational point of view (learning with someone else can be more fun than learning in isolation) and also from the point of view of having easier access to knowledge.

Underconfidence in gifted girls

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By Jonah Sinick
The 1996 study Self-Efficacy Beliefs and Mathematical Problem-Solving of Gifted Students found thatAlthough most students were overconfident about their capabilities, gifted students had more accurate self-perceptions and gifted girls were biased toward underconfidence.

Facebook chief operating officer Sheryl Sandberg discusses high potential women being underconfident in her book Lean In.

I’ve had many gifted female students and classmates/colleagues who have struggled with intellectual insecurity.

Parents sometimes ask me if I have any suggestions for what they might do to help.

For those of you who have daughters, does this sound familiar? If so, are there resources / strategies that you’ve found helpful for improving their self-confidence?

 

Gifted children could learn math much earlier

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By Jonah Sinick

Cross-posted from the Davidson Gifted Issues Discussion Forum

(As background context, I’m a mathematician, and I’ve taught gifted children math over a span of ~10 years.)Something that I’ve noticed lately is a widespread implicit acceptance of the norm for gifted children to learn math at grade level, or just 1 year above grade level. My experience has been that even moderately gifted children (IQ ~130) can learn algebra in at age ~11, and that highly gifted children (IQ ~145) can learn algebra at age ~8. Moreover, I think that there are strong arguments in favor of this.Developmental capacity

  • It’s not uncommon for moderately gifted children to be 2+ years ahead in reading and for highly gifted children to be 5+ years ahead in reading, so one might expect them to have mathematical potential that’s 2+ or 5+ years ahead of grade level (respectively).
  • IQ was for a time believed to be “mental age divided by chronological age” multiplied by 100. This notion has (rightly) fallen out of favor, but it’s sufficiently close to the truth for people to have believed it. Under this assumption, a 10 year old with IQ 130 has mental age 13 and a 10 year old with IQ 145 has mental age 14.5, and these 10 year olds are correspondingly cognitively ready for curricula aimed at people of their mental age.
  • I know of people of IQ ~160 who have learned calculus at age 7: this suggests that in some respects IQ understates “mental age.”

Desirability

Some people have suggested that it’s better for gifted children to learn a broad range of things rather than accelerating, because if they accelerate then they’ll be out of sync with their peers. I think that this is true in some contexts. But I don’t think that the benefits of being better in sync with one’s peers outweigh the benefits of accelerating through the K-12 math curriculum specifically.

  • Grade school math is key to understanding the natural sciences, statistics and economics. Remaining at grade level in math substantially delays a gifted child’s ability to understand these things.
  • Learning math well builds general reasoning ability, which has benefits across domains.
  • Many gifted children find math especially enjoyable once they become deeply involved in it.
  • Being far ahead in math can build confidence on account of being an unambiguous signal of intellectual ability.

In The Calculus Trap Richard Rusczyk at Art of Problem Solving argued that rushing through the standard curriculum is not the best answer for mathematically talented young people, suggesting that students should instead focus on learning how to solve complex problems. I agree with him that learning how to solve complex problems is more important than acceleration through the standard curriculum. But the two things aren’t mutually exclusive: gifted children can both learn how to solve complex problems and accelerate through the standard curriculum.

Learning precalculus and calculus was a transformative experience: it allowed me to understand physics, it gave me a thrill, and it made me better understand myself on account of tapping into my latent mathematical ability. It was when my intellectual development really accelerated. I was 16 at the time. I wish somebody had encouraged me to start earlier. A sizable minority of the most intellectually impressive people who I know I know had similar experiences.

There are large potential returns to gifted children learning more math earlier on.

 

Managing your time spent learning

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By Jonah Sinick

Cross-posted from Less Wrong and Quora. Related to the information wiki page managing your time.

This article is written for people who are looking for advice on prioritizing activities, in particular, what to spend time learning.

In thinking about how to budget your time, it’s helpful to explicitly prioritize the activities that you engage in in terms of their relative importance, and distinguish between what’s important and what you find interesting. Sometimes we exaggerate the usefulness of interesting but only slightly useful activities in their minds, on account of wanting to believe that time spent on them is productive. If you think about how useful an activity is and, how interesting the activity is separately, you’re less likely to do this. It’s helpful to consider the following four categories of activities:

  • Important and interesting: Do, and take your time. Get it right!
  • Important and not interesting: Do as much as necessary, and maybe a bit more; look into ways of overcoming procrastination. Also consider ways to make them more interesting.
  • Not important and interesting: Do only if you feel like it, don’t try to press yourself, and consider substituting with activities that are interesting and important.
  • Not important and not interesting: Avoid.

More below

Interesting and important

When you find an academic subject interesting, and when it’s important (e.g. for your future job, as a prerequisite to courses that you’ll take in the future or otherwise related to your future goals), you should delve deeply into it. Gaining deep understanding takes time, and you shouldn’t feel as though you’re working inefficiently if you find yourself spending a disproportionate amount of time on it.

Interesting but not important

Intellectually curious people often have intellectual interests that don’t advance their career goals. Such interests can absorb a lot of time and hinder one’s professional success.

The question of how to balance these interests with one’s career goals is a very personal one.

In general, if there are two activities that are of comparable interest to you, but of unequal importance, you should choose the more important one.

If you find that your time is uncomfortably crowded with things that are interesting but not important, you should look for instances where you’re exerting willpower on them, and cut back on those, reserving the time that you spend doing things that you find interesting to activities that require relatively little energy, to conserve energy for doing things that it’s more difficult to get yourself to do.

Important but not interesting

Sometimes you have to do things that are uninteresting to achieve your goals. If you have trouble motivating yourself to do these things, you might benefit from our recommendations for overcoming procrastination (forthcoming). Also, consider ways that you might find these specific activities more interesting, by checking out targeted learning recommendations for those activities.

Not important and not interesting

These activities should be avoided. This point might seem obvious, but despite this, people often do engage in activities that are neither important nor interesting. This most often happens when:

  • One hasn’t carefully considered the question of whether the activity is important. For example, one might uncritically internalize the view that it’s important to learn a language because learning a language said to keep the mind sharp, without considering that there might be other more interesting or important activities that keep the mind sharp to an equal or greater extent, and therefore try to learn a language, even if one doesn’t find it interesting. There are benefits to learning a language that one can’t get from other activities: the point here is just that keeping one’s mind sharp specifically isn’t a good reason to learn a language rather than do other things.
  • The activity seems interesting at first, and one sets a goal connected with it, but then the activity turns out not to be interesting, and one feels an obligation to continue on account of having already set the goal. For example, one might hear good things about a long novel and set a goal of reading it, find that one doesn’t enjoy it, and feel pressure to plow through to the end.

On the first point, it’s important to critically reflect on whether the activities that one is involved in are important. On the second point, all else being equal, fulfilling commitments is good, because it’s good for one’s self-esteem, but one should still consider whether the cost of fulfilling the commitment is worth it, and also try not to set ambitious goals when the value of fulfilling them is questionable.

College Admissions for Homeschoolers

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By Jonah Sinick

Cross-posted from Quora and the Gifted Homeschoolers Forum

If you’re considering home-schooling for high school, you’re likely wondering how it will affect your college prospects.

Public Colleges

It can be difficult or impossible to get into public colleges as a homeschooler, owing to bureaucratic requirements. For example, UC Davis writes:

The courses of homeschools and unapproved high schools are not accepted by the University of California and cannot be used to establish minimum UC admission requirements. If you are a homeschooled student or attended a California high school without a UC-approved course list, you must establish your academic record through test scores or as a community college transfer student.

While it may be possible to qualify for the UC system via test scores, it’s unclear what one’s prospects are for getting into a campus of your choice. See University of California and homeschoolers at The Well-Trained Mind.

Other public colleges are more receptive to admitting homeschoolers. For example, University of Illinois writes:

We encourage home schooled students to apply to the University…We are very interested in having talented, well-qualified applicants from a variety of settings. Home schoolers would provide a diversity of academic experiences to the campus.

If you’re considering home schooling, be sure to check out what the situation is at the public colleges that you anticipate applying to, in particular, those in your home state.

Private Colleges

Elite private colleges accept home schoolers.  The elite private college students who were home schooled appears to be smaller than the fraction of high schoolers in the general population who are home schooled. About ~3% of students are home schooled nationwide. By way of contrast:

  • Princeton reports that only 0.5% of Princeton students were homeschooled.
  • A University of Chicago student on College Confidentialreported that 13 students in his or her grade were homeschooled. University of Chicago’s class size is about 1,400, so about 1% of the students were home schooled.
  • MIT reports that less than 1% of MIT students were homeschooled.

Some possible reasons for the discrepancy are:

  1. The fraction of homeschoolers who apply to elite colleges may be significantly smaller than the fraction of members of the general population who apply to elite colleges. For example, MIT reports that less than 1% of the applicant pool consists of homeschoolers.
  2. It could be more difficult for homeschoolers to get into elite colleges on average.

On the second point, even if it is more difficult on average, that doesn’t mean that it would be more difficult for you personally. With suitable preparation for the admissions process along the lines described below, homeschoolers could have equal or better odds for getting in (though the situation is ambiguous).

Something that pushes in favor of homeschooling for admissions prospects is that if you homeschool, you’ll have more flexibility in regards to how you arrange your coursework (for example, you can pick which textbooks to use), and if you use this flexibility well, your chances of excelling could increase.

Some points to keep in mind, based on a reading of webpages of elite colleges about applying as a homeschooler:

  • Standardized test scores are weighted more heavily for homeschoolers. Some colleges encourage homeschoolers to take more than the minimum requirement of 2 SAT subject tests, and some refer to AP scores as a way for students to demonstrate their achievement. If you’re unusually capable of getting high standardized test scores, the case for homeschooling is strengthened.
  • Taking college courses at local colleges or summer programs seems to help establish a homeschooler’s academic record. It also gives a homeschooler the chance to solicit recommendations from professors who can vouch for his or her performance.
  • If you homeschool, it’s important to document your academic program.
  • Colleges expect that homeschoolers study the standard academic subjects (math, English, social studies, science and languages): if you homeschool, you shouldn’t design an overly idiosyncratic program that doesn’t include these things.
  • Some colleges want evidence that homeschoolers can integrate well with other students, presumably in the form of extracurricular activities that have a social component.
  • If you homeschool and can give a compelling reason for why you’ve done so in your college applications, this will strengthen your case for admissions.

Process

For our research, we looked at pages published by: Yale, MIT,Princeton, Columbia, University of Chicago, Caltech, New York University and Homeschool Success, as well as College Confidential’s forum with relevant threads. See in particularHomeschool students’ admission rate to Harvard/Princeton/Yale and How do homeschooled students attend Ivy leagues?.

Resources

You might find the following resources helpful for learning more about college admissions for homeschoolers: