@Marsigit: Only a few short comments, as it is late in my time zone, and I am going to bed.

First, when I wrote about the separation between mathematics and
mathematics education, I did not mean they should be separated in
general; I meant that your utterances were confusing the two. You should
not talk about mathematics when you really mean mathematics education.

Secondly, you are contradicting yourself in saying that you are against
introducing logic but are for helping the students to think.

Thirdly, one cannot wait until students come up with the mathematics
which they will need. I presume you are familiar with the NP vs PP
problems. Students can discuss all they like, but it is unlikely that
they will re-invent the calculus. At some point, the teacher must
introduce material.

Finally, as I have been trained both as a mathematician and as a
mathematics educator in two separate programs (and different countries,
different decades, etc.), I can be a one-man experiment to see if pure
mathematics helps mathematics education. Outcome: yes.

@
David: My very short response that I am interested with your succinctly
description of Ernest works. Further, I will say that I am witnessing
the pure mathematicians are endangering the younger learner of
mathematics. The instantly outcome: possibly YES; but the long-term
architecturing creativities: clearly NO.

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