I think the real nail in the coffin would be if a young child does not understand a particular concept (say, volume conservation) and it is found that you can teach them this concept before they are supposed to be developmentally ready.
The article doesn't discuss conservation of volume in detail, but it talks about an experiment that's said to be "conceptually similar". And while it's hard to say from the quote, it seems to imply that when children are given feedback on the similar problem, their performance improves (I've bolded that part):
The child is shown two rows of objects, say, pennies. Each row has the same number of pennies and they are aligned, one for one. The child will agree that the rows are the same. Then the experimenter changes one row by pushing the pennies farther apart. Now, the experimenter asks, which row has more? (Pennies might also be added to or subtracted from a line.) Younger children will say that the longer line has more pennies.
When Piaget (1952) developed this task he argued that children go through three stages on their way to successfully solving this problem. Initially they cannot process both the length of the rows and the density of coins in the rows, so they focus on just one of these, usually saying that the longer row has more. The next stage is brief, and is characterized by variable performance: children sometimes use row length and sometimes row density to make their judgment, sometimes they use both but cannot say why they did so, and sometimes they simply say that they are unsure. In the third stage, children have grasped the relevant concepts and consistently perform correctly.
Robert Siegler (1995) showed that children’s performance on this task doesn’t develop that way. Ninety-seven 4- to 6-year-olds who initially could not solve the problem were studied, with each child performing variants of the problem a total of 96 times over eight sessions. After each problem, children were asked to explain why they gave the answer they did, so there was ample opportunity to examine the consistency of the children’s performance and their reasoning. The experimenter found a good deal of inconsistency. Children used a variety of explanations— sophisticated and naïve—throughout, even though they became more accurate with experience (the experimenter provided accuracy feedback, which is a big help to learning). It was not the case that once the child “got it” he consistently used the correct strategy. If the child gave a good explanation for a problem, there was only a 43 percent chance of his advancing the same explanation when later confronted with the identical problem.
I agree that while not exactly 'volume conservation', this addresses the exact same skill.
If the child gave a good explanation for a problem, there was only a 43 percent chance of his advancing the same explanation when later confronted with the identical problem.
Would you interpret this as meaning the children had not acquired the concept, after all? It seems that if the child actually truly understands the concept that moving things around doesn't change their number, then they wouldn't be inconsistent. (Or is the study demonstrating what I found unintuitive, that children can grasp and then forget a concept?)
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.