Rebecca has some interesting ideas. I walk 2 miles four or five times a week. I do it for exercise, not for the thoughts that may or may not arise. My first impression is that sitting silently may do the same as what Rebecca says walking does.
Being mindful is not necessarily thinking in a cognitive sense. Awareness precedes thinking and I have found that excessive thinking by persons such as René Descartes have led Western society to want to be overly certain. Richard Rohr, a Franciscan priest, has stated that wanting certainty is our original sin. I tend to agree with Rohr. Excessive thinking may cause one to be certain and that is a major problem. As the poet Keats said, one should make up one's mind about nothing. Rebecca probably would agree but I am not sure based on what she said above. Thanks for the opportunity to respond. Warm and kind regards to everyone.