Referring to Liping Ma's and Mark Saul's "Notices to the AMS", it really should be evident that they're both on an entirely wrong track, or 'tack' if we seek a nautical metaphor. In order to develop an effective 'educational system' for any level, we really have to get down to 'fundamentals' in a way neither of them has successfully done in their respective Notices.
I don't have the background or detailed knowledge of the systems under consideration to suggest how the 'whole math educational system' should develop.
That would be a major task for the stakeholders* and the experts in education to work out in detail, by seeking to arrive at a 'reasonable' consensu on the issues involving the systems they are seeking to work with.
(*Stakeholders: Teachers; students; parents; educational experts - if it is a math educational system, then math experts; principals and other educational leaders; administrators of schools; politicians; others interested in education: all of these stakeholders need to 'get together' effectively and put their ideas together to get an effective system going. The kind of 'elementary math education system design' that Liping Ma has proposed and Mark Saul has critiqued would actually develop (actually *evolve*) considerably later, after a fair bit of design work is done at the level I am describing here).
I do have the background and needed knowledge to suggest how an appropriate Mission could be 'developed' for an individual learner and teacher (/guide/ facilitator). I develop some elements and construct a starting model from those elements to demonstrate the process.
Earlier, it has been suggested that the crucial issue for the 'math educational system' is simply to ensure that the learner of math does not come to fear or loathe math through his/her progress through the school system. That (i.e., fear and loathing of math) unfortunately, has been the case in most schools to date, except where there have been exceptional teachers who've *somehow* managed to arouse and maintain the learners' interest in math.
The underlying issue is simple, call it a 'Mission':
"To impart knowledge of school mathematics to math learners". (This is evidently a 'Mission' for the teacher (/guide / facilitator). The learner would have his/her own 'companion Mission'.
We could reasonably suggest the following two elements in a student's model:
Mission: "To learn all my math topics thoroughly enough so as to do well in my math examinations"
1. To get over my 'fear and loathing' of math
2. To understand, effectively, all the topics of my math syllabus.
Then the following 'representation of a system model' may be said to be an adequate picture of the student's 'mental model' about his Mission:
"2 --> 1 --> M", which translates into standard prose as: Prose translation of model: +++++ "To understand, effectively, all the topics of my math syllabus
to get over my 'fear and loathing' of math, which, in turn,
to learn all my math topics thoroughly enough so as to do well in my math examinations". +++++
The above appears to be almost trivially true, and the initial reaction may be: "Why write up something that is so trival?"
However, such 'simple' representations of our mental models can 'lead to' development of astonishingly complex models *representing* some of the thoughts floating around in our minds on the complex issue under consideration. When such models are constructed by an individual, they would represent the individual's 'mental models'. When developed by a group, the models could come to represent the 'consensus mental models' of the group as a whole.
We illustrate the 'initiation' of one such (individual) model, in parts, at the multi-part attachment to this post, which illustrates the model developing from 2,3, 5, 7 and 10 elements.
Back to the *Teacher's Mission*:
There are various conditions that need to be satisfied, things that may need to be done, in order accomplish the Mission 'M':
M: "To impart knowledge of school mathematics to math learners".
Things to be done to accomplish M (10 elements listed below): +++++ 1. To ensure that the learner is not bored out of his/her wits
2. To try to make math interesting for the learner
3. To ensure that the learner does not come to 'loathe or fear' mathematics (Once such a thing happens, in particular at a very young age, it is extremely difficult - perhaps impossible - to overcome such feelings)
4. To ensure that the learner become interested in what he/she can do with math
5. To give the learner the confidence that he/she can master a good bit of math (The learner has in general already learned to speak, read and write, which are really MUCH more complex learning tasks than elementary math. So math should really be no hassle at all, given that the learner is not 'turned off' math by an ineffective/ incompetent 'teaching system')
6. To enable the learner properly to understand his/ her own capabilities for math
7. To help the learner to find needed internal resources to overcome the difficulties that will surely confront him/ her
8. To encourage (NOT PUSH or GOAD!!) the learner to put his/her best resources to the 'Mission' (Because PUSHING/GOADING destroys ENCOURAGEMENT - PUSHING and/or GOADING is the surest way to bring about frustration in an intelligent individual)
9. To help the learner not get frustrated in his/her learning
10. To give the learner to confidence that 'school math' is not beyond his/her abilities.
(Total of 10 elements, to be taken up for modeling to represent initiation of an Action Plan to accomplish the chosen *Teacher's Mission*). +++++ The above list has been developed by GSC, and therefore reflect GSC's own ideas, not the teacher's. For a real model, these 'elements' should develop from the teacher's own ideas.
Some Interpretive Structural Models (ISMs), construjcted from the above list of elements, are illustrated, with proressively greater elements, in the attachments to this post. To begin with the relationship running throgh all of these 'part models' may be taken to be "MAY CONTRIBUTE TO".
A student of 'adequate maturity' would enter into such a process, with appropriate help/guidance from a teacher (/guide/ facilitator).
Where the learner is younger than, say, high school age, appropriate elements would be 'generated' by the teacher (/guide/ facilitator). Any good teacher has been in any case doing this, though probably has not been formally 'writing up' the elements.
As the 'systems modeling' process is not widely known, the models have obviously not been constructed in any formal way.
However, we certainly know that such 'mental models' do appear in the mind of any conscientious teacher, and he/she generally tries to follow them. What we're doing here in the process we're describing is only 'to represent' those mental models in a simple graphical form as a 'structure' that can then serve as a guide and 'ready reference').
Younger students do not need to see either such elements or the model(s) that develop from them. The teacher (guide/facilitator) would be an excellent resource to develop both the element lists and models from them.
Liping Ma and Mark Saul could work together (along with a number of other stakeholders in the education system) on developing *effective models* on issues representing the entire systems they have thus far only superficially discussed in their respective "Notices to the AMS".
That kind of exercise would accomplish something practical for the US educational system they have sought to discuss theoretically in their respective "Notices".