Thoughts on Skemp article:
For me, the three things that made me stop and think were:(1) The necessity to, more or less, get the right language across when teaching math, because even replacing the character 'x' (multiply) in exchange for 'of' can lead to a mountain of misunderstandings down the road.
(2) Facing the fact that math formulas can be extended in so many different directions, how can we get our students to see that if we give them a formula (like the area of a circle), then they conclude that diameter times 'pi' would work fine, too.
(3) At the end, Skemp digs up a valuable point - should the word "math" be used in schools for purely relational practice, or can we use it for instrumental too? Knowing already there's tons of maths out there, should we be teaching more the path of application, or speak more to theory?
I stand pretty firm with what Skemp writes, because he shows that we can all benefit from looking at something (whatever it is) mathematical from a different angle.