The Directorate of Education was the Tang body in charge of the official schools, and under it were six schools: the National, Grand and Four Gates schools, which taught the Confucian classics, and the Law, Calligraphy and Mathematics schools, which taught law, writing and arithmetic respectively.
In the early years of Zhenguan under Emperor Taizong, around 627, the National University was made the independent Directorate of Education, and the six schools above all came under it.
There was also the Hongwen Academy, which was outside the six schools and took only members of the imperial family and high nobility.
The name "one academy and six schools" refers to this one academy together with the six schools.
Tang Six Schools is the class in the chapter on Education on the official schools of the Tang. It covers the early Zhenguan years of Emperor Taizong, when the National University became an independent Directorate of Education overseeing six schools, the National, Grand, Four Gates, Law, Calligraphy and Mathematics schools, while the Hongwen Academy was set up separately for members of the imperial family and high nobility.