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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?

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    $\begingroup$ Your profile shows you are a HS student. In what capacity are you helping this other student? I ask because if the student is at your school, you should feel comfortable to ask the department chairman about the process. Keep in mind, it's possible for a student to do well in geometry vs algebra, or the reverse, and wind up in the lower level in the next class. $\endgroup$ – JTP - Apologise to Monica Jun 5 '15 at 12:00
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    $\begingroup$ @nosyarg: No, there is not a standard procedure. Some states may have standardized or partially standardized procedures, but it most of the country, that sort of policy would be set at the level of the school district. (Also, every school I was in had a system which was at best semiformal, and subject to negotiation with the student and parents.) $\endgroup$ – Henry Towsner Jun 5 '15 at 13:57
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    $\begingroup$ I agree with Henry's comment. The issue for you is that the process likely depends on performance the year prior, and might not be indicative of the student's true skills for the year to come. $\endgroup$ – JTP - Apologise to Monica Jun 5 '15 at 14:23
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    $\begingroup$ I recommend that you and this student talk with her parents, and perhaps get them to suggest moving her to a higher level class. $\endgroup$ – Sue VanHattum Jun 5 '15 at 15:40
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    $\begingroup$ Since it's just "the usual", I make only a "comment" that the fact that your friend is female, rather than male, instantly enormously increased the likelihood of her assignment to "not good at math" status... for bad reasons. If she's not ethnically northwestern European, that'd double-down on that bet. With blue-collar parents, immigrants ... again. The reason to mention such stuff is that that's what you'll be pushing back against to try to help. That is, be aware that there are (I'd wager) huge pre-existing biases... that almost surely will not be explicitly mentioned in any discussion. $\endgroup$ – paul garrett Jun 6 '15 at 0:15
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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!

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