The issue of standardized tests has been controversial because nowadays, there are many complaints related to standardized tests. Most of these complaints say that standardized tests are not fully discovering students’ knowledge potential, which seems to hurt the students when they fail. Some argue that standardized tests aren’t being used enough.They argue that standardized tests should not only be used to place students into classes, but also should require standardized tests between levels of classes.

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They argue that standardized tests should not only be used to place students into classes, but also be taken as students transition between class levels. (as students pass from one class level to another).

(otherwise it seems that "tests ... require ... tests", IMHO )

Nowadays -- можно вобще выбросить

Most of these complaints say that --> most of these complaints state that

are not fully discovering students’ knowledge potential -- грамматически все верно, но по сути по-моему не очень. Тесты не призваны раскрывать потенциал, для этого есть другие методы. Можно заменить на standardized testing discourages from discovering...

ну и CopperKettle + 1

Выражение "standardized tests" употреблено шесть (!) раз в этом маленьком отрывке. Замените на что-то другое: such tests, testing, they, etc