Right Mindfulness is part of the Noble Eightfold Path
Maha-satipatthana Sutta: The Great Frames of Reference
"And what is the noble truth of the path of practice leading to the cessation of stress? Just this very noble eightfold path: right view, right resolve, right speech, right action, right livelihood, right effort, right mindfulness, right concentration.
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"And what is right mindfulness? There is the case where a monk remains focused on the body in & of itself — ardent, alert, & mindful — putting aside greed & distress with reference to the world. He remains focused on feelings in & of themselves... the mind in & of itself... mental qualities in & of themselves — ardent, alert, & mindful — putting aside greed & distress with reference to the world. This is called right mindfulness.
I'm going to quote the
right concentration part, as well--for clarification: right mindfulness is needed for right concentration:
"And what is right concentration? There is the case where a monk — quite withdrawn from sensuality, withdrawn from unskillful (mental) qualities — enters & remains in the first jhana: rapture & pleasure born from withdrawal, accompanied by directed thought & evaluation. With the stilling of directed thoughts & evaluations, he enters & remains in the second jhana: rapture & pleasure born of composure, unification of awareness free from directed thought & evaluation — internal assurance. With the fading of rapture, he remains equanimous, mindful, & alert, and senses pleasure with the body. He enters & remains in the third jhana, of which the Noble Ones declare, 'Equanimous & mindful, he has a pleasant abiding.' With the abandoning of pleasure & pain — as with the earlier disappearance of elation & distress — he enters & remains in the fourth jhana: purity of equanimity & mindfulness, neither pleasure nor pain. This is called right concentration.
You don't "drop" the mind:
"This is called the noble truth of the path of practice leading to the cessation of stress.
"In this way he remains focused internally on mental qualities in & of themselves, or externally on mental qualities in & of themselves, or both internally & externally on mental qualities in & of themselves. Or he remains focused on the phenomenon of origination with regard to mental qualities, on the phenomenon of passing away with regard to mental qualities, or on the phenomenon of origination & passing away with regard to mental qualities. Or his mindfulness that 'There are mental qualities' is maintained to the extent of knowledge & remembrance. And he remains independent, unsustained by (not clinging to) anything in the world. This is how a monk remains focused on mental qualities in & of themselves with reference to the four noble truths...