To Dr. Marsigit: first, we must separate mathematics from mathematics
education. As I pointed out in my post to Doug, Prof. Ernest's work has
had some good spin-offs, even if indirect, for mathematics education.
With respect to mathematics, his conclusions can be separated into two
categories: the obvious ones about the limitations of mathematics, and
the ones that are fuzzy enough to easily give rise to erroneous
interpretations of the complex relationships between mathematics and
society.

You refer to mathematics education, so I will say no more about
his ideas on mathematics per se. Here, Prof. Ernest's main conclusion
is that student's learning styles will depend on their respective social
backgrounds, and that one must take this into consideration. I have
taught in a large number of different cultures, and in each one I
adapted my teaching style to the student's society's; even in a single
school, my teaching styles for two different groups will usually differ.
But this conclusion is not unique to the style of social constructivism
espoused by the followers of Prof. Ernest.

Therefore Prof. Ernest's
philosophy cannot always take credit for this conclusion. I understand
what you mean when you say that the criteria for proof and truth are
different in different environments. But you need to separate the ways
of learning from mathematics per se. For mathematics per se, these
criteria are not society-dependent, but the way a student to be
convinced of something without formal proof is society-dependent. The
problem is that eventually the two concepts of truth and proof -- one
subjective, one mathematical, need to eventually brought together.
Students should eventually learn that their subjective ideas are not
necessarily correct.

I fully agree with you that one cannot start
immediately in a topic in primary school with this synthesis, especially
in primary school (and you seem to concentrate on primary education),
as one needs to first appeal to their intuition. But even in primary
school one can, little by little, impress upon the students that,
whereas their intuitions are a useful starting point, and whereas one
can develop this intuition quite a ways, one cannot always rely on
intuition to take you where you want to go. Mathematics partly uses
intuition, but also partly uses methods which are no longer intuitive,
nor which can be made to be so.

It is especially important in secondary
to get away from the pure subjective approach. This is also why pure
mathematicians could be useful in organizing a curriculum -- both at
primary and secondary level. An important aspect of mathematics which
could be profitable for all students, regardless of their eventual
professions, would be introducing concepts from logic, to help students
think clearly. But not the formal notation of logic; rather the
concepts. This can start in primary, and continue in secondary.

However,
every time it has been introduced, it has been a fiasco, as most
teachers below university level themselves do not have a clear idea of
these concepts, and have made a mess of it. The solution has, alas, been
to strip school mathematics of the very essence of mathematics, that of
clear analysis. I am not sure what you consider pure mathematics, but
at university level, pure mathematics includes Mathematical Logic, and
some of the more useful and learnable aspects of Mathematical Logic
would be more useful, if put into a form which students could understand
and practice, than most of the formulas which are presently memorized
by students.

@ David:

There are many logical implications for separating mathematics and mathematics education: 1. It will be only the utterances of pure mathematicians about pure mathematics, 2. That mathematics educationist have their separate room for their utterances about both education and mathematics, 3. There will be a kind of demarcation that the pure mathematicians should not talk about primary and secondary education. and also 4. That mathematics education in university should be differentiated with that of primary and secondary schools.

From my above description, I agree with your notion for separating mathematics and mathematics education in the case of pure mathematics room and mathematics education in university. However, I do not agree if it means also for primary and secondary mathematics education room; because, for primary and secondary education, there should be a different nature of mathematics.

Agree or not agree, in my perception, at all level of education (primary, secondary and university) there are existing the ontology of constructivism. Even, for advance mathematicians, they should construct the world of mathematics in each of their math research. At the dimensions of constructivism, it is impossible to separate between your mind and your objects of mind; albeit, it is impossible to separate between your mind and your mathematics. Epistemologicaly, I wish to say that mathematics is ultimately you your self; mathematics is ultimately the students themselves.

David, if you expose your style of teaching, why you seemed not interested to expose the style of learning. For me, the last is more challenging. Again I still found your forcing to use pure mathematics criteria to judge School Mathematics; and I still did not find your interest in School Mathematics. I do not agree with the separation of mathematics and mathematics education if they want to speak about education. However, I agree to expel pure mathematics from primary and secondary teaching; it means that the expelling mathematics will be mathematics is for mathematics, mathematics is an art, mathematics is a King, mathematics is beautiful, mathematics is bla...bla...bla...up to the utterances of pure mathematicians. However, I wish to suggest to pure mathematician for not forcing their self-ego perceptions to the world of young learner. Let the young learners have their efforts to construct/build and get their own perceptions of mathematics.

I do not agree with your argument that because mathematics is partly uses methods which are no longer intuitive, nor which can be made to be so, to make an account and legalize pure mathematicians to organize primary and secondary math curriculum. I agree that in secondary education there should get away from the pure subjective approach; I perceive that is is as the transition stage of learning. It needs much more consideration both from educationist and from mathematicians.

Again I find your immanent "orthodoxy" approach i.e. by INTRODUCING the concepts; you still hard perceive the students as an object of teaching.

I do not agree with your introducing "logic" as a concept; rather I agree to facilitate them to learn it as a way of thinking (method), attitude and activities.

Again, I wish to claim that teaching learning of primary and secondary mathematics is not as simple as pure mathematics think. This is totally not about grounding down every single pure mathematics concepts into the lower level of mathematics education; rather this is about students' constructing of their mathematics.

If I and they have talked much in the same level of ontology and we still have many differences perception; I then have a question about their motivation. I wish to claim that motif is the ground below the truth.

There are many logical implications for separating mathematics and mathematics education: 1. It will be only the utterances of pure mathematicians about pure mathematics, 2. That mathematics educationist have their separate room for their utterances about both education and mathematics, 3. There will be a kind of demarcation that the pure mathematicians should not talk about primary and secondary education. and also 4. That mathematics education in university should be differentiated with that of primary and secondary schools.

From my above description, I agree with your notion for separating mathematics and mathematics education in the case of pure mathematics room and mathematics education in university. However, I do not agree if it means also for primary and secondary mathematics education room; because, for primary and secondary education, there should be a different nature of mathematics.

Agree or not agree, in my perception, at all level of education (primary, secondary and university) there are existing the ontology of constructivism. Even, for advance mathematicians, they should construct the world of mathematics in each of their math research. At the dimensions of constructivism, it is impossible to separate between your mind and your objects of mind; albeit, it is impossible to separate between your mind and your mathematics. Epistemologicaly, I wish to say that mathematics is ultimately you your self; mathematics is ultimately the students themselves.

David, if you expose your style of teaching, why you seemed not interested to expose the style of learning. For me, the last is more challenging. Again I still found your forcing to use pure mathematics criteria to judge School Mathematics; and I still did not find your interest in School Mathematics. I do not agree with the separation of mathematics and mathematics education if they want to speak about education. However, I agree to expel pure mathematics from primary and secondary teaching; it means that the expelling mathematics will be mathematics is for mathematics, mathematics is an art, mathematics is a King, mathematics is beautiful, mathematics is bla...bla...bla...up to the utterances of pure mathematicians. However, I wish to suggest to pure mathematician for not forcing their self-ego perceptions to the world of young learner. Let the young learners have their efforts to construct/build and get their own perceptions of mathematics.

I do not agree with your argument that because mathematics is partly uses methods which are no longer intuitive, nor which can be made to be so, to make an account and legalize pure mathematicians to organize primary and secondary math curriculum. I agree that in secondary education there should get away from the pure subjective approach; I perceive that is is as the transition stage of learning. It needs much more consideration both from educationist and from mathematicians.

Again I find your immanent "orthodoxy" approach i.e. by INTRODUCING the concepts; you still hard perceive the students as an object of teaching.

I do not agree with your introducing "logic" as a concept; rather I agree to facilitate them to learn it as a way of thinking (method), attitude and activities.

Again, I wish to claim that teaching learning of primary and secondary mathematics is not as simple as pure mathematics think. This is totally not about grounding down every single pure mathematics concepts into the lower level of mathematics education; rather this is about students' constructing of their mathematics.

If I and they have talked much in the same level of ontology and we still have many differences perception; I then have a question about their motivation. I wish to claim that motif is the ground below the truth.

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