In my view, speed plays a role, though I can't say that it should play such a central role.
I think that in a lot of disciplines, you have to become fluent at manipulations, and at seeing and thinking in higher level patterns. Being able to think your way through a more complex problem would seem to benefit from, if not require, seeing multiple steps ahead in a progression. At least this is my perception.
My experience in school math was that it wasn't enough to satisfy myself that I knew how to solve a problem. I then had to work my way through a whole bunch of similar problems until I could perform the manipulations quickly. This is also how I managed to commit the definitions, axioms, and theorems to memory. If I didn't do that stuff, then I got my arse handed to me on the exam. I gave my kids pretty much the same advice.
I think that in a lot of disciplines, you have to become fluent at manipulations, and at seeing and thinking in higher level patterns. Being able to think your way through a more complex problem would seem to benefit from, if not require, seeing multiple steps ahead in a progression. At least this is my perception.
My experience in school math was that it wasn't enough to satisfy myself that I knew how to solve a problem. I then had to work my way through a whole bunch of similar problems until I could perform the manipulations quickly. This is also how I managed to commit the definitions, axioms, and theorems to memory. If I didn't do that stuff, then I got my arse handed to me on the exam. I gave my kids pretty much the same advice.