Context:
There was recently a question on Math.SE: Inferior to Other Younger and Brighter Kids which starts as follows:
I'm a high school student (Junior/Grade 11) and I'm currently studying Spivak's Calculus on Manifolds. I've finished Single/Multi Variable Calculus (the basic ones without analysis/rigor), the first half of Rudin for Real Analysis and Linear Algebra (Lang) through self-study.
And then the OP writes how inferior he feels compared to other younger and brighter kids. Perhaps there are some deeper issues involved, but I don't want to dwell on it, nor discuss the situation of OP publicly.
The issue:
The issue I would like to write about is that more and more often I notice kids studying advanced books. Surely 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.
My experience isn't enough to draw any conclusions, but this is not just a feeling, I've witnessed negative effects of such behavior multiple times. Here are some example motives:
- Over-reliance on the already-learned material. There was an introductory problem given which could be solved using some elementary method. The gifted student thought "this is boring" and solved it faster using an advanced method. Then there was another problem given, but the advanced method couldn't be generalized to this new problem (while the elementary way could) and the gifted student was unable to find a solution until the end of the class.
- Missing fundamentals. Gifted student learned $X$ by himself, but the material depended on the knowledge of $Y$. Yet, the student lacked proper understanding of $Y$ and so build some bad intuitions, which had to be unlearned before progressing further.
- Closed-mindness (compared to other gifted students). This one is hard to describe, some signs are:
- the student thinks there's always a book that explains the area in question;
- the student specializes in solving toy problems and (difficult) homework, but is unable to handle even simple open or research problems;
- the student is unable to form his own theories or own interpretations.
There also others, like burnout, but the causes will be more complicated, let's not diverge too much. Even these three trouble me very deeply (especially the last) and it's like some math-contest rat race rather than curious study of the beautiful area that mathematics is.
To me, rediscovering the wheel (if not too often, human lifespan is still limited) is a necessary part of study, so that when the time comes, the kids will be able to do their own original discoveries. Getting all the best concepts laid out in a book, digested, updated and improved over the years doesn't seem like the best idea. Still, perhaps this view is invalid and unfounded?
Question:
Is there any data or research on the effects of access to advanced material early in kid's education?
Given that gifted students are rare, right now even some anecdotal evidence would be great.
What are your experiences in this matter?
More concrete examples:
@Joseph Van Name asked for examples, so here there are.
Over-reliance on the already-learned material:
- In a geometry class there was a guy that tried to solve every single problem using analytical approach: coordinates, trigonometry, algebra, etc. I admit that he was very good at this, but he didn't learned some techniques we practiced using simpler problems and then wasn't able to solve the more difficult ones which wouldn't yield to his standard method.
- During programming class there was a group who liked suffix trees too much and some students didn't learned some simpler techniques (here I mostly mean variations on the Knuth-Morris-Pratt algorithm) which were necessary later.
- There was another guy who would try to solve all the contest inequalities using Muirhead's inequality. He had a huge problem each time the inequality in question was not similar to any mean (e.g. all the numbers begin equal not causing equality).
Missing fundamentals:
- A person who had learned more advanced probability before university disregarded a few first classes and then assumed that any distribution has a density function. There were later some problems with conditional expected values and their $\sigma$-algebras, but I don't remember it well enough.
- Many times students who learn asymptotic complexity by themselves don't understand some concepts properly and don't pay enough attention to details. Some common misconceptions are
- $n \notin O(n^2)$,
- $n^2 + n \in \omega(n^2)$,
- $f_i \in O(g)$ implies $\sum_{i=1}^{n} f_i \in O(n\cdot g)$.
- Frequently students that learned programming without any supervision or guidance from more experienced people have bad style (and programming contests often reinforce that bad style). It is an issue, because it's hard to unlearn and the drawbacks are not severe enough for the student to want to change it by himself. The effect is that the student penalizes himself for a much longer time (one of the reasons why summer internships or involvement in open-source projects is a good idea).