@David - Just time enough now to thank you for the reply and
mention that a venture ("Anthrobotics") built working robotics
software demonstrating that the underpinnings of my model, as applied to AI,
were indeed workable in simulating human cognitive behavior. And I've not been
"rebuffed" (Dozens of Darpa scientists reviewed my software demo,
liked it, and recommended that Darpa contractors looking for help in my areas
contact me), etc., etc.) , just more like "avoided". More later.

Mathematics is known as sign language developed through the
passages of different civilisations,helping the others it has become complicated.All
methods and models become helpless in front of primary / secondary students.
teacher has to develop own models and methods to teach.

@Marty
: sorry for the poor choice of words ("rebuff") on my part, as I
misunderstood your " they are called 'too complex to exist yet'" from
your first post in this thread. I applaud such advances in AI -- which,
together with neurophysiology, will provide more advances in our understanding
of the mind, and hence allow more fruitful pedagogical theories, than all the
speculative philosophy of the mind and all the pedagogical theories heretofore
developed. I'm also glad that DARPA, with all its funding possibilities, is
getting into it. As I understand it, you are the contact person for
Anthrobotics.com.

@prabha rastogi: you are right that presently there are very few pedagogical theories that stand up to the rigours of the classroom. This does not mean, however, that one should abandon the search for a pedagogical theory which is both efficient and practical, while being flexible enough to be adapted to local conditions and personalities. After all, we are still looking for a solution to the Riemann Hypothesis, despite the numerous failures to find the solution so far.

@Margisit: if I understand what you are saying, you wish to give school children simplified mathematics, leaving the more complex mathematics to the universities. The problem is the yet-unsolved problem that, although most people do not need most of the mathematics taught in schools, there will be a small but important percentage of the students who will need the mathematics, and waiting until university is waiting until the brain has passed the development phase most suited to learning the concepts. Since one does not know which children will form this important minority, school systems adopt the strategy of making everyone learns the same mathematics up to a certain point, with some systems trying to make this point as young as possible (such as the fourth year of school) and others waiting until later (e.g., the last years of secondary). Bad solution, of course, but no one has come up with a viable alternative. In any case, models such as Marty's are not meant to be used directly in the classroom, but rather to lead us to a better understanding of the mind, eventually leading to a better pedagogy.

@prabha rastogi: you are right that presently there are very few pedagogical theories that stand up to the rigours of the classroom. This does not mean, however, that one should abandon the search for a pedagogical theory which is both efficient and practical, while being flexible enough to be adapted to local conditions and personalities. After all, we are still looking for a solution to the Riemann Hypothesis, despite the numerous failures to find the solution so far.

@Margisit: if I understand what you are saying, you wish to give school children simplified mathematics, leaving the more complex mathematics to the universities. The problem is the yet-unsolved problem that, although most people do not need most of the mathematics taught in schools, there will be a small but important percentage of the students who will need the mathematics, and waiting until university is waiting until the brain has passed the development phase most suited to learning the concepts. Since one does not know which children will form this important minority, school systems adopt the strategy of making everyone learns the same mathematics up to a certain point, with some systems trying to make this point as young as possible (such as the fourth year of school) and others waiting until later (e.g., the last years of secondary). Bad solution, of course, but no one has come up with a viable alternative. In any case, models such as Marty's are not meant to be used directly in the classroom, but rather to lead us to a better understanding of the mind, eventually leading to a better pedagogy.

@ Praba Rastogi: What you exposed
indicated that we, as adults or teachers need to change in perceiving
mathematics. It is right but it is not only that as you said "Mathematics
is known as sign language developed through the passages of different civilisations,helping
the others it has become complicated". I also say that for younger
learner, according P Ernest, Mathematics is creativity and or even Activity.
Philosophically, I further can say that Mathematics is the students themselves.
I totally agree with you that the teachers should develop their own method of
teaching. More than those, even the teachers need to develop their own learning
resources, textbooks, worksheets, including many kinds of model of
'communication'. The teachers should be simultaneously as a teachers but also
as a researcher of his classroom.

@ David: I totally agree with your last elaboration. From those I found that we need to change the paradigm of teaching i.e. not giving the students a receipt or model but to facilitate them to construct their own math/model. It is not impossible to facilitate different kinds of students competencies if we still implement traditional/orthodoxy way of teaching. One of the role of IT is to facilitate their need to learn.

@Marty: Still I perceive that any kind of model of math/communication produce by adults can only be a trap for the younger.

@ David: I totally agree with your last elaboration. From those I found that we need to change the paradigm of teaching i.e. not giving the students a receipt or model but to facilitate them to construct their own math/model. It is not impossible to facilitate different kinds of students competencies if we still implement traditional/orthodoxy way of teaching. One of the role of IT is to facilitate their need to learn.

@Marty: Still I perceive that any kind of model of math/communication produce by adults can only be a trap for the younger.

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