Warfield, Systems, G.S. Chandy, and Mathematics Teaching

Above is a picture of John N. Warfield, "the author of two U.S. patents on electronic equipment, and the inventor of Interpretive Structural Modeling, Interactive Management, and General Design Science." Professor Warfield's work has had a significant influence on the contributions of G.S. Chandy to the math-teach@mathforum.org list. In connection with my recent blog entries about "important open questions in mathematics pedagogy," ("Is Mathematics Teaching A Closed Book?"), Mr. Chandy has tried to offer an approach to discussing these and related points of contention on that most contentious of lists that is grounded in the work of Professor Warfield.

I do not profess to know Warfield's work, but I found what Mr. Chandy posted to be sufficiently heuristic to warrant posting what he wrote in its entirety in hopes of generating interest among readers of this blog, as well as those on various lists to which I contribute who many not have ventured into math-teach (probably with good reason).

G.S. Chandy's Post

The 'elements' below are generated from MPG's posting on Aug 22, 2008 8:43 PM at this very thread. (Many of the elements listed below have been generated by GSC - they should be properly articulated by real teachers of K-12 & other school math):

Possible Mission Statement:

M: "To put into place a highly effective way of teaching math (for teachers); a highly effective way of learning math (for students)"

First trigger question:
"What, in your opinion, are some THINGS TO DO to help accomplish Mission M?"
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Some possible elements, generated in response to above trigger question - these elements largely derived from MPG's posting above-noted:
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1. To ensure that the ideas of those who actually teach K-12 are taken into the action planning for future teaching of K-12

2. To learn how to go beyond the lecture-driven, teacher-centered math classes

3. To ensure active student participation in the math class

4. To stimulate the student's genuine interest in math

5. To generate real interest from students

6. To ensure that students become actively engaged in mathematical thinking and math discourse

7. To provide (point to) ways demonstrating HOW students could become truly active in math

8. To ensure that 'grades' are not treated as jokes (by students or by teachers)

9. To use an effective grading system through which both students and teachers could benefit

10. To convince students that it is possible for each of them to become productive and creative in math

11. To convince students that it is inevitable that some would be more productive, others maybe less productive - but that all can be truly productive

12. To help each student 'fill in the gaps' in his/her understanding of math

13. To convince students of the real benefits that a sound understanding of math could bring to them

14. To convince students to put forth enough real effort to become productive and creative in math

15. To convince students of the value of math that goes beyond just 'book-learning' or 'exam-learning'

16. To ensure that students are fairly and effectively tested at various levels

17. To ensure that math-teaching should go beyond vapid 'multiple choice' questions

18. To enable and encourage students to take a real cract at truly challenging math questions (suited to their current level)

19. To ensure that all students always get needed further chances to improve their standings in math

20. To prevent students from attempting to cheat on math - demonstrate that it is really futile to try and cheat in math

21. .... (Etc).

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There will obviously be a great many such 'elements'. I suggest that generated such elements should be a continuing task, on which teachers and students could very beneficially spend a few minutes each day.

As we generate such elements, the most useful thing to do with them is to find out just how they may "CONTRIBUTE TO" each other (and to M).

This is done by asking questions like:

"Does, in your opinion,

Element X

CONTRIBUTE TO

Element Y?"
(The strength of the contribution relationship should initially be taken as small, i.e., we initially create the model predicated on the relationship "MAY contribute to". Later, as we become more certain of our understanding of the elements (and of the system in which those elements are active), we could strengthen the relationship to "SHOULD contribute to", and still later to "CONTRIBUTES TO".

Asking and answering those questions about these elements will give participants the opportunity to enhance their understanding of the whole system in which the 'teaching and learning of math' operate. They will also become clear in their minds about each separate element in itself, in its specific context.

(Almost) needless to state, each class would have its own listing, and the lists (and the models generated from them) would see considerable variation. Also, it is entirely possible that there would be considerable disagreement about the contributions or otherwise of various elements. Most of such disagreements can be worked out to the mutual satisfaction of all involved - and it is often possible to work out models representing a satisfactory consensus. Sometimes we cannot come to agreement - I suggest in such cases that the persons involved should create and follow their own models.

Of course, there may be people participating who are, so to speak, more interested in empty argumentation rather than productive resolution of disagreements and issues - just "trolls" - we have a few such in every set of participants. We need to learn how to handle these effectively: not easy at all to do, but it is possible to ensure that trolls do not disturb the flow too much. If we are all actively involved in generated useful elemetns, working on creating productive models of our perceptions about relationships of those elements, then the trolls will disturb working partipants much less. James Wysocki (and others) have suggested practical ways in which such trolls could be dealt with.
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Below my signature, I provide a list of some other useful 'trigger questions' about Mission M.

GSC
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Useful trigger questions (all are in relation to the Mission M identified above):

Creating the OPMS

Step 1: Identify a desirable Mission M:

MISSION: “______________________________________________”

(In case you have a problem, just put it into ‘Mission’ format. Various examples of ‘Missions’ are available in various examples illustrated).

Step 2: THINGS TO DO (to accomplish Mission):

Generated from “1st Fundamental Trigger Question”:

“What, in your opinion, are the THINGS TO DO to accomplish the above Mission?”

Step 3: Action Planning:

We shall construct an Interpretive Structural Model (ISM) with the responses to above 1st Trigger Question. This ISM (which is technically known as an ‘Intent Structure’) will develop into the ongoing Action Plan for the Mission.

The participants in the Mission should spend about 20-30 minutes each day to develop models relating to progress in their individual parts of this Mission (as seen in their individual One Page Management Systems that should start developing as the organization works towards the global Mission). As a whole, the organization should meet from time to time, as found convenient, to model progress towards the global Mission.



Step 4: Identifying DIFFICULTIES, BARRIERS and THREATS – and overcoming them

These are generated from the “2nd Fundamental Trigger Question”:

“What, in your opinion, are the BARRIERS, DIFFICULTIES & THREATS that may hinder or prevent accomplishment of the chosen Mission?”

Generally, responses to the above trigger question may be:

a) converted into appropriate THINGS TO DO, which would then be integrated into the ongoing Action Plan
b) inserted into a Field Representation – the OPMS software will then help users create Dimension Titles for the Field Representation and also to link up various elements in the Field with appropriate relationships. Various other actions are to be done with Field Representations, which are described in our Workbook.

c) These BARRIERS, DIFFICULTIES & THREATS are also most usefully inserted into a ‘Problematique’ (the governing relationship of which is “aggravates”). It takes a while to learn to develop and use a problematique effectively, but it will be well worth the effort, for the reasons noted in our ‘Basic Presentation’, a copy of which is provided for reference.

The above exercises would help us identify and implement, on a continuing basis, means to overcome BARRIERS, DIFFICULTIES and THREATS discovered during our progress towards our Mission.

Step 5: Identifying STRENGTHS (available/required)

Generated from “3rd Fundamental Trigger Question”:

“What, in your opinion, are the STRENGTHS (available/required) that could help accomplishment of the chosen Mission?”

Responses to the above Trigger Question, if sizable in number, may be inserted into a Field Representation.

The STRENGTHS identified as “Required, not available” would be translated into appropriate THINGS TO DO format and integrated into the ongoing Action Plan to enable users find ways to develop the required
STRENGTHS.

Step 6: Identifying WEAKNESSES – and overcoming them

Generated from “4th Fundamental Trigger Question”:

“What, in your opinion, are the WEAKNESSES that may hinder or prevent accomplishment of the chosen Mission?”

The means of handling WEAKNESSES are exactly the same as we have for handling BARRIERS, etc. That is, responses to the above trigger question would be:

a) converted into appropriate THINGS TO DO, which would then be integrated into the ongoing Action Plan
b) inserted into a Field Representation – the OPMS software will then help users create Dimension Titles for the Field Representation and also to link up various elements in the Field with appropriate relationships. Various other actions are to be done with Field Representations, which are described in our Workbook.

c) These WEAKNESSES are also most usefully inserted into a ‘Problematique’ (the governing relationship of which is “aggravates”). It takes a while to learn to use a problematique effectively, but it will be well worth the effort, for reasons cited in our ‘Basic Presentation’, a copy of which has been provided with our Workbook.

The above exercises help us identify and implement, on a continuing basis, means to overcome WEAKNESSES discovered during our progress towards our Mission.

For individuals as well as for organisations, correct identification of BARRIERS & WEAKNESSES (and defining ways to overcome them) is often a very long and painful task – particularly when organisational Barriers and Weaknesses are derived from an individual’s Weaknesses. It could take months – sometimes people NEVER learn to get over their Weaknesses even when they cause disaster. In fact, most (man-made) disasters are caused precisely because people have not learned how to handle their existing Weaknesses!

Step 7: Identifying OPPORTUNITIES available

Generated from “5th Fundamental Trigger Question”:

“What, in your opinion, are the OPPORTUNITIES available
that may help accomplishment of our chosen Mission? –
and what are the THINGS TO DO to avail the
OPPORTUNITIES discerned?”

Responses to above trigger question are inserted into a Field Representation showing OPPORTUNITIES available – and THINGS TO DO to avail the OPPORTUNITIES identified. The THINGS TO DO to avail the OPPORTUNITIES are also integrated into the Action Plan, to enable us to see how we may work towards availing of the OPPORTUNITIES that arise.



Step 8: Identifying EVENTS & MILESTONES

Generated from “6th Fundamental Trigger Question”:

“What, in your opinion, are the EVENTS/MILESTONES
that may occur during our progress towards our
chosen Mission?”

Responses to above trigger question are inserted into PERT/Gantt Charts (as is done in the conventional ‘Project Management’ software, to show the
status of Milestones during progress towards our Mission.

We observe here that the conventional Project Management software deals only with this single dimension of the OPMS (namely, the EVENTS Dimension). Obviously the name ‘Project Management’ software is a serious misnomer for such software – which may accurately be called ‘Event Management’ software.

The OPMS may be seen to fulfill the role of true Project Management, as it enables users to see all dimensions relating to a specified Mission. However, we prefer to regard the OPMS as an ‘aid to problem-solving and decision making’ (which includes ‘Project Management’, and much else besides).


Step 9, et seq: ‘System’ Dimensions of the OPMS

When sufficient numbers of elements have been generated in response to the Six Fundamental Trigger Questions as described above (and, further, those elements have been appropriately inserted into models AND linked up), the users would find it necessary and useful to start working on the ‘System Dimensions’ of the OPMS:

• PLANNING SYSTEM(S)
• INFORMATION SYSTEM(S)
• MARKETING SYSTEM
• PRODUCTION SYSTEM (for manufacturing organizations. For educational institutions, research institutions, etc., the title of this dimension may be modified appropriately).
• PROBLEM SOLVING SYSTEM AND LEARNING SYSTEM: The OPMS itself defines these two systems, and no work is required to be done by users in regard to these two closely coupled systems)
• MONITORING & EVALUATION SYSTEMS
• FINANCE CONTROL SYSTEM
• OTHER(s) (as required)

The OPMS would help users develop all above dimensions.

In general, the models in these ‘system’ dimensions of the OPMS are small, but powerful, ‘meta models’ derived from the larger models appearing above the System Tie Line. It is generally found that these models start developing effectively, in a very natural way after a certain ‘richness of connection’ has been established between the fundamental models above the System Tie Line.

The models within the OPMS are models of human perceptions relating to the chosen Mission. The purpose of creating such models is primarily to show a simple ‘action path’ to each person involved in accomplishment of the Mission. That is, these models show what each person and each group involved should do each day in pursuit of the Mission. The THINGS TO DO identified in the Action Planning structure as ‘FOCUS elements’ are the crucial activities at any point of time – these would be just a few (typically, 3 to 5 each day). These focus elements will naturally change from time to time as we progress towards accomplishment of the Mission.