So what do we do during practice? My current understanding

The short answer is that we do two things: 1- We create some space, 2- We look into our mental objects. We shoud first develop some skill about number 1, then having found some skill we can do number 2.
 What dose it mean to create space?
 It means we stop reactivity. It is done by letting go of things in mindfulness meditation and by gratitude and loving kindness practice out of meditation.
 What dose looking into mental objects mean?
 It means we focus on them and investigate the how of them. The natural processes of the mind help us to untangle things gradually.
 So a meditation session has at least two parts, in the first stage we calm down the mind and create some space, then in the second stage we start investigating mental objects, we just look in to them. We do not struggle.
 It is very important not to forget 3 things, we should keep it simple, we shouldn't struggle and we shouldn't be hasty.Right view must also have a role to play here, but it is not so clear for me, so I will write about when I got it!

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One of the most important teachings of mindfulness meditation is that when emotions arise we can, we really can, stop from getting involved.