I am currently working with a student who is on the slowest math track at her school of the three available; however, I cannot understand why. She shows an amazing ability to reason mathematically and she can understand the underlying concepts of the material extremely quickly, even more quickly than some of the students who I have worked with on the fastest math track at her school. I am not sure what her placement in her current math class was based upon, because she seems to be extremely strong in mathematics. In the US, what is the standard procedure for placing students into their earliest math classes?
At my school, in a large urban district in the United States, students are (unfortunately) placed into different math tracks based off a few different factors.
1) Performance on tests (standardized or otherwise) - In this day and age, this is likely the most indicative factor of a student's math placement. A student's ability to take tests, especially the state mandated and facilitated ones, and receive high marks consistently will almost always be placed in the highest track. Unfortunately, these state tests are very influential over a lot of things including teacher rating, school rating, budget appropriations, etc and as such they are deemed a large indicator of students' abilities. If a student consistently "underperforms" on a set of standardized tests they will almost assuredly be placed in the remedial or middle track.
2) Teacher recommendations - I would say that the second most important factor in determining a student's track is the recommendation of past teachers, which is generally trustworthy. However, just because a student is recommended by one teacher does not mean that they will automatically be in whatever track that teacher says. The principal, other teachers, or parents may override a recommendation to either push a student up a level or hold them back one. Additionally, if a class is already at the population limit or only offered once or twice a day, students may not be able to be rostered and be forced to again move levels, either up or down depending on the scheduling conflict.
3) Student input - Last, and at least in my school least, a student's own input is sometimes taken into consideration. Usually this will occur when a parent has unreal expectations for their child and has pushed the admins to move them up a level and the student fights back because they are not comfortable with the more accelerated pace. I have seen this go both ways where the student wins out and also when the parent persists and has the student moved up (this almost always ends disastrously for the student) against their will.
I would love to say that students' placement is determined based off of their capacity for abstraction and critical thinking skills and how well they are able to conceptualize and explain their reasoning but unfortunately, at least in my school and most other schools in my district, this is how it works. If you feel that your friend is in a lower track than they deserve, I would highly recommend having an honest conversation with their teacher, their parents, the principal, rosterchair etc and come with proof of their abilities (written work on a problem or have them come with and demonstrate how they think about a problem). This direct method is likely the only way to get them to where they will thrive best. Best of luck!