So: Reading from the Syncopation section (pp. 32-44) of Progressive Steps to Syncopation, ignore the stems-down part, play the stems-up part with your hands— alternating sticking, natural sticking, improvised sticking, or both hands in unison— and wherever there's an 8th note of space between hand notes, add a bass drum note. Where there's a longer space between hand notes, play as written— don't add anything. Swing the 8th notes, if you're playing this as a jazz method, and you're in that tempo range— around quarter note 100-250.
The first two lines from p. 37 in Syncopation (p. 38 in the new edition):
Would be played like this:
There are rare exceptions where we'll add a bass drum to the longer spaces. Further on on that same pages, there are a few isolated single notes. Where there are single notes with long spaces before and after them, add a bass drum after them. So the first two measures of line 5 of p.37/38:
Would be played:
Same thing when there are two notes isolated. On the last two measures of line 6 from the same page:
In the book that first quarter note is the end of a longer run, so we don't add bass drum after it with this method. Of course this is all a means to an end— small reading errors don't matter, and small errors applying the interpretation really don't matter. Instead of sweating small stuff, go for continuity, and a good feel, sound, and dynamic shape.