Q&A: Does Self-Studying Advanced Math Create Bad Habits?

by Justin Skycak on

Sure, accelerating via self-study not as optimal as accelerating within teacher-managed courses, but it's way better than not accelerating at all.

Cross-posted from here.


Is there any data or research on the effects of access to advanced material early in kid’s education?

Gifted students will access more advanced material earlier than their peers and I would applaud the kids reading almost anything they want. Yet, I have the feeling that sometimes this is just too early and reading advanced books without certain level of mathematical maturity may cause more harm than good.

I’ve witnessed negative effects of such behavior multiple times, such as overconfidence and closed-mindedness (as a result of over-reliance on already-learned material), missing fundamentals and bad intuitions, and sometimes even an inability to form one’s own theories/interpretations.


Whether a student is ready for advanced mathematics depends solely on whether they have mastered the prerequisites.

  • If a student has not mastered the prerequisites, then of course the issues you're concerned about will indeed manifest.
  • But if a student has mastered prerequisites, then it is fully appropriate for them to continue learning advanced math early, and not appropriate to stunt their development by holding them back.

There’s a lot of supporting evidence in the literature on academic acceleration. For example:

  • Bernstein, Lubinski, & Benbow (2021): "[There is] extensive empirical literature showing positive effects of acceleration on academic achievement (Kulik & Kulik, 1984, 1992; Lubinski, 2016; Rogers, 2004; Steenbergen-Hu et al., 2016) and creativity (Park et al., 2013; Wai et al., 2010). … Presenting students with an educational curriculum at the depth and pace with which they assimilate new knowledge is beneficial. Other studies have shown that academic acceleration tends to enhance professional and creative achievements before age 50 (Park et al., 2013; Wai et al., 2010)."
  • Wai (2015): "...[F]or many decades there has been a large body of empirical work supporting educational acceleration for talented youths (Colangelo & Davis, 2003; Lubinski & Benbow, 2000; VanTassel-Baska, 1998). ... The educational implications of these studies are quite clear. They collectively show that the various forms of educational acceleration have a positive impact. The key is appropriate developmental placement (Lubinski & Benbow, 2000) both academically and socially. … Educational acceleration is essentially appropriate pacing and placement that ensures advanced students are engaged in learning for life. Every student deserves to learn something new each day (Stanley, 2000). The evidence clearly supports allowing students who desire to be accelerated to do so, and does not support holding them back. ... [T]he long-term studies reviewed here show that adults who had been accelerated in school achieved greater educational and occupational success and were satisfied with their choices and the impact of those choices in other areas of their lives."

Granted, studies on academic acceleration are typically following students who are accelerating their study in teacher-managed courses, whereas the heart of your question is about students who are accelerating through their own self-study while simultaneously being enrolled in lower-level teacher-managed courses.

In that case, a student’s self-study is obviously less likely to be successful than study within a teacher-managed course (for instance, the student may have issues assessing their own knowledge, leading them to think they’ve mastered prerequisites when they actually haven’t).

But what’s the alternative? If some kid is ready to be accelerated, but the educational system they’re enrolled in doesn’t do acceleration, what option do they have aside from self-studying the accelerated material?

Sure, accelerating via self-study not as optimal as accelerating within teacher-managed courses, but it’s way better than not accelerating at all. I mean, this is basically equivalent to asking the following:

If a kid has passion and talent in some domain, but their environment is not optimally supportive of their developing talent in that domain, should they 1) take initiative to pursue this domain to the best of their ability, or 2) just give up and let their environment dictate their future?

I’d be willing to bet my life on option 1 being the correct approach. In fact, I did bet my life on option 1 being the correct approach. I was one of these kids.

Personally, I can’t say I was fully immune from all the concerns that you list, but I can say with 100% confidence that self-studying advanced material had a massive positive impact on my life. It equipped me with skills I never would have otherwise developed, and opened up all sorts of opportunities for me that I would never have dreamed of. Basically, it kicked off a “virtuous cycle” in my life. Plus, it was also an emotional outlet for me and I’m sure I would have released all that teenage angst in far less productive ways if not for the ability to disappear into my math books.