If I were actually doing any teaching of math students, I'd simply try to find out:
i) Do they 'get' it (the 'it' refers, of course, to Newton's method for square roots; Duveen's development; or whatever).
I do (to some extent; not entirely) accept Robert Hansen's [RH's] critique of Peter Duveen's reasons for doing this.
ii) But the heart of the matter is simply this:
DO THE STUDENTS FIND THE MATH INTERESTING?
If yes, excellent - you're in very good shape! Proceed!
If not, the teacher needs to find out out just *HOW* to make it interesting.
Skills and fluency in using specific methods and tools will follow automatically - IF [and that's a very big "IF", indeed!] students find it (i.e., math) basically interesting.
I'd guess a teacher should seek to ensure that at least 60% of his/her students find the math interesting. (To expect 100% students to find it interesting is probably too ambitious. But that could lead to another useful 'Mission': how to make math interesting to the 'other 40%'? This one would probably require some 'special handling' - and would be MUCH more complex).
If at least 60% of students DO NOT find the math class interesting, some 'foundational work' is needed - on "how to make the class interesting".
That work may involve redeveloping available methods further (as Duveen has done);
It may involve something else entirely: like, for instance, *demonstrating* that math CAN be interesting, fascinating, etc, etc.
[As stated, I'm NOT a math teacher - but I did solve the underlying problem for my 13-year old grand-daughter Mimi, who had come to me with the complaint "Oh, math is SO UTTERLY boring!"
[Well, I convinced Mimi that all the 'boring' stuff of math could lead to a whole lot of fascinating things like this. She really got turned on by that, took the cube we'd constructed to school, managed to turn on a few of her student peers - then later she started a 'math club'. She's 16 now, will be appearing in a couple of months' time for her IOS exams (the first level of high school in the Indian Open School system).
[I had at that time developed a few models that helped to show me just how to proceed in the case, specifically wrt the Mission: "To demonstrate to Mimi that math need not be boring at all". These models helped in great measure to guide me, and later Mimi as well. If I recall rightly, I had attached some of those models to some messages at Math-teach, to illustrate my point. (I also recall that RH - of course - got the horse by the wrong end (by the tail instead of by the reins, and misunderstood entirely [as he continues doing to this very day]).
However, my then 13-year old grand-daughter Mimi did not misunderstand; she later made some small models for herself that, continuingly developed, very successfully guided her through the rest of her school career - in math and in many other things.
Most importantly (as far as I was concerned), she NEVER EVER found math boring again! (Which was the real point of the whole exercise)].
Information about the tools that enable us to model, in exactly the kind of detail that's needed for any specific issue is available in the attachments herewith.