Along with the emphasis on tests has come an emphasis on precocity and acceleration in mathematics. It is relatively easy for a bright student to work through the mathematics curriculum far more quickly than the usual pace.
There are several problems associated with precocity. People who skip ahead in the curriculum often have gaps in their background which only show up later. At that point, the person may be too embarrassed to admit the gap and tries to fake understanding. This regularly leads to disastrous results.
Another problem is that precocious students get the idea that the reward is in being 'ahead' of others in the same age group, rather than in the quality of learning and thinking. With a lifetime to learn, this is a shortsighted attitude. By the time they are 25 or 30, they are judged not by precociousness but on the quality of work. It is often a big letdown to precocious students when others who are talented but not so precocious catch up, and they become one among many. The problem is compounded by parents in affluent school districts who often push their children to advance as quickly as possible through the curriculum, before they are really ready.
A third problem associated with precociousness is the social problem. Younger students are often well able to handle mathematics classes intellectually without being able to fit in socially with the group of students taking them. Related to precociousness is the popular tendency to think of mathematics as a race or as an athletic competition. There are widespread high school math leagues: teams from regional high schools meet periodically and are given several problems, with an hour or so to solve them.
There are also state, national and international competitions. These competitions are fun, interesting, and educationally effective for the people who are successful in them. But they also have a downside. The competitions reinforce the notion that either you ‘have good math genes’, or you do not. They put an emphasis on being quick, at the expense of being deep and thoughtful. They emphasize questions which are puzzles with some hidden trick, rather than more realistic problems where a systematic and persistent approach is important.
This discourages many people who are not as quick or as practiced, but might be good at working through problems when they have the time to think through them. Some of the best performers on the contests do become good mathematicians, but there are also many top mathematicians who were not so good on contest math. Quickness is helpful in mathematics, but it is only one of the qualities which is helpful. For people who do not become mathematicians, the skills of contest math are probably even less relevant.
These contests are a bit like spelling bees. There is some connection between good spelling and good writing, but the winner of the state spelling bee does not necessarily have the talent to become a good writer, and some fine writers are not good spellers. If there was a popular confusion between good spelling and good writing, many potential writers would be unnecessarily discouraged.
I think the answer to these problems is to build a system which exploits the breadth of mathematics, by allowing quicker students to work through the material in greater depth and to take excursions into related topics, before racing ahead of their age group.