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There are some positive feedback loops in school that cause gaps in ability between students in a subject to widen. There are also some negative feedback loops (e.g., intervention), but the net effect is still the gap widening. Therefore, the system's behavior is chaotic (small differences in students' abilities eventually lead to big differences). If this is true, it means that some variation between students' successes is extremely difficult to predict.
Three examples of these positive feedback loops:
Suppose that Student A has less knowledge in a particular subject and is therefore performing worse than Student B in that subject. Then, it is likelier than not that:
(ergo B studies more than A)
I have low confidence in this model, but I could not come up with a simple, testable prediction that the model makes.
The author of Bring Up Genius would agree with you about the positive feedback loop. If your children get better than average before they join the school, they will keep getting rewards, which will increase their motivation, etc.
Now the question is how much of "children getting better than average before joining school" is about nature or nurture. If your child has good genes, it is definitely worth it to make the difference visible. If your child is average, your options are limited. But still, kids can spend enormous amounts of time talking about dinosaurs or pokemons; if you succeed to redirect some of that energy into something academically relevant (e.g. by teaching them to read the names of the dinosaurs, then some short texts about them), it may help.
The act of studying the subject becomes more aligned with B's self-concept than with A's self-concept.
(ergo B studies more than A)
It's not just "more" or "less", it's often studying different things. The child failing at math will study the textbook, and will hate it. The math prodigy will read some interesting books on math instead. Which again increases the gap.
That said, it sometimes also happens that the smart children stop studying things that are not interesting for them. Why study something, if you are smart enough that you can guess the answer or in the worst case just read the textbook the night before the exam? Sometimes these smart kids get in trouble later when the strategy they successfully used at previous school suddenly stops working when they get to high school or university, when suddenly they are surrounded by people just as smart as them, except that many of those people also have good study habits. I have seen talented people drop out, because they couldn't switch to the "in this environment, I am not so special anymore, and I need to start working hard" mode fast enough.
If this were true, your expect grades/scotes to not be a uniform gaussian. You'd expect the middle to be slightly lower than expected.
Also, you'd expect crossover events (below-average to above-average or vise versa) to be less common than chance would predict.
You need some mathematical model. The bell curve emerges as a sum of many things. For example you have many genes that can contribute to higher intelligence, so it's a question of how many times the coin of fate had landed the right way, and the result is approximately the sum of the contributions.
Now if we assume that school just adds some knowledge to children -- even if we assume that each child gets a random amount of knowledge, but it's a random amount independent on the starting value -- the result is still a bell curve.
If we had a model assuming that school e.g. doubles each child's knowledge, that would increase the gaps, but it would still be a bell curve (only twice as wide).
However, if we assume that each child gets a random multiplier by school, let's say, everyone's starting knowledge is multiplied by a random number between 5 and 20, then the result is no longer a bell curve.
Basically, bell curve × bell curve ≠ bell curve, but instead (assuming all numbers are positive) it is asymmetric, with longer right side. Imagine {1, 2} multiplied by {1, 2}, you get {1, 2, 2, 4}, with the 4 far away from the center.
If depends a lot on which model you choose.