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Developing an Educational Systems Theory to Improve Student Learning – i

Proffitt Grant Proposal:

Developing an Educational Systems Theory to Improve Student Learning

and the Quality of Life

Principal Investigator: Theodore W. Frick

Associate Professor and Web Director School of Education

Indiana University Bloomington

Consultant: Kenneth R. Thompson Head Researcher

Raven58 Technologies 2096 Elmore Avenue

Columbus, Ohio 43224-5019

November 12, 2004

Developing an Educational Systems Theory to Improve Student Learning – ii

Abstract

The No Child Left Behind Act of 2001 is likely to spur systemic change efforts so that

significant improvements in student academic achievement can occur. Without an adequate

educational systems theory, however, we will continue to reform education largely by trial-

and-error. It is no wonder that educational practitioners often distrust, resist and undermine

the efforts of educational reformers. The stakes are very high. The consequences of mistakes

can be devastating – particularly when changing a whole system of education such as a K-12

school district as some researchers recommend.

I propose to develop an educational systems theory (EST) for making scientifically

based predictions of the outcomes of education systems change efforts. The proposed theory

will extend that originally developed by Maccia and Maccia (1966), which was based on a

theory model called SIGGS. Thompson (2004) has further extended the Maccia and Maccia

educational theory into a comprehensive behavioral theory, Axiomatic-General Systems

Behavioral Theory. A-GSBT will have many applications, including direct analysis of any

discrete educational or learning system.

I believe that it is necessary to develop an underlying educational systems theory

(EST) in a manner that it can be tested through logical and empirical validation. The main

goal of educational reform is to improve student academic achievement, and if successful,

this would be expected to subsequently improve economic conditions and the quality of life.

An adequate EST will predict which changes are likely to result in improved student learning

achievement versus those that are not, and EST will predict which kinds of learning

achievement are likely to improve the quality of life. Thus, EST can guide decision making

instead of guesswork.

Developing an Educational Systems Theory to Improve Student Learning – iii

Validating EST at this initial stage will demonstrate to potential funding agencies the

usefulness of a theory-based approach to educational decision making and systemic change

efforts. EST validation will further the development of SimEd, a technology designed to help

analyze and project educational system outcomes and expected to be a multi-year, multi-

million dollar effort, spanning 3 to 5 years.

A PDF version of this proposal is available at:

http://education.indiana.edu/~frick/proposals/est2004.pdf .

Developing an Educational Systems Theory to Improve Student Learning – 1

Developing an Educational Systems Theory to Improve Student Learning and the Quality of Life

Purpose and Rationale

No Child Left Behind, passed by the U.S. Congress and signed by President Bush on

January 8, 2002, is likely to impact many K-12 schools throughout the United States (NCLB

Act, 2001). NCLB requires schools to assess annually student achievement at numerous

grade levels. Based on average test scores, schools will be identified as succeeding or

failing. Schools that repeatedly fail to meet current state standards for student achievement

will be held accountable. Parents will have the opportunity to send their children to different

schools, if their present school is not succeeding. However, if as anticipated, many schools

are branded as failures, where will parents send their children? There will not be enough

room in the successful schools, let alone other logistical problems such as transportation.

Failing schools will have real incentive to change. For such change to be successful, it

cannot be piecemeal. It must be systemic change (Duffy, Rogerson & Blick, 2000; Caine &

Caine, 1997; Senge, Cambron-McCabe, Lucas, Smith, Dutton & Kleiner, 2000).

The Need for Understanding Systemic Change

In the decades following the publication of A Nation at Risk in 1983, considerable

effort has been undertaken to improve public schooling. Reform efforts have been typically

referred to as site-based management, school restructuring, and educational systems design.

Researchers such as Banathy (1991), Reigeluth (1992), Frick (1991), Jenlink, Reigeluth, Carr

& Nelson (1996), Caine & Caine (1997), Duffy, et al. (2000) and Senge, et al. (2000) have

argued for systemic change in education. 'Systemic change' contrasts with numerous

piecemeal reform efforts that have largely failed in twentieth century schooling.


However, the rhetoric of systemic change is not likely in itself to make any real

difference in schooling. Such rhetoric has been around for some time. Understanding of

educational systems change is needed for intelligent action.

What If?

What if we had a scientifically based theory that could explain and predict the

behavior of educational systems? Not a learning theory, not a pedagogical theory, not an

instructional method, not a leadership theory, not a classroom management theory, not a

curriculum theory – but an educational systems theory, a theory to describe, explain and

predict whole educational systems and their transactions with societies in which they are

embedded. What if we had a theory where the concepts and relationships were defined in

precise terms that suggest clear operational ways in which they can be observed and

measured? Education does not have the equivalent of a General Theory of Relativity, as in

the discipline of physics. Educators seldom agree on definitions of terminology. We do not

have well established and clearly defined terms such as mass, energy, light, force,

acceleration, velocity, time, gravity, etc. as in physics. In short, we lack a scientific

educational systems theory.

A Start: General Systems Theory and SIGGS

Ludwig von Bertalanffy (1972), often cited as the father of general systems theory,

first introduced GST in the 1930’s, which gained recognition in scientific circles in the

1950’s and 1960’s. As Ackoff (1980) has summarized, we have moved from a machine age

– characterized by analysis, reductionism, mechanism, and the Industrial Revolution – to a

systems age. Systems thinking has become a new way of looking at the world, according to


well-known authors such as Ackoff & Emery (1972), Bar-Yam (2003), Banathy (1991), Lin

(1987, 1999), Mesarovic & Takahara (1975), Senge (1990) and Wolfram (2002).

In the 1960’s Maccia and Maccia made the first and only serious attempt to develop

and formalize an educational systems theory. Their EST was derived from the SIGGS theory

model. SIGGS, in turn, was retroduced from theories in other disciplines: set theory,

information theory, di-graph theory and general systems theory. Their theory consists of 201

hypotheses about school systems, which is now available on the Web at:

http://www.indiana.edu/~tedfrick/siggs.html . For example, one of the hypotheses is: If

centralization in an educational system increases, then active dependence decreases.

Centralization is concentration of channels within a system. Active dependence is

components that have channels from them. One only need look at the many instances of this

pattern in the past 50 years. In the school consolidation movement during the middle part of

the 20th century, many American school systems increased in size and became highly

centralized with respect to administrative decision making. Individual teachers, students,

their parents, and community members now have relatively little impact on administrative

decision making (active dependence has decreased). Affect relations from these individuals

to the superintendent and school board have decreased over time, while centralization has

increased. Parents, students and teachers have few channels to influence administrators.

The complexity of the Maccia and Maccia (1966) educational theory made it difficult

to test empirically. As Wolfram (2002) has noted, “…general systems theory was concerned

mainly with studying large networks of elements – often idealizing human organizations.

But a complete lack of the kinds of methods [that Wolfram now uses] … made it almost

impossible for any definite conclusions to emerge.” (p. 15)


A Further Step: Analysis of Patterns in Time

When I experienced the severe limitations of quantitative methods based on general

linear models in the 1970s (e.g., regression analysis), I sought an alternative research

methodology. Based on probability theory and parts of information theory in the SIGGS

theory model, I developed an observation and measurement methodology, Analysis of

Patterns in Time (Frick, 1983; 1990). APT has shown promise in computer-adaptive

mastery testing (Frick, 1990; 1992) and several other kinds of research studies done by my

doctoral students (e.g., An, 2003; Plew, 1989; Wang, 1996; Yin, 1998).

In the 1990s, I began to conceive of an educational systems simulation tool, SimEd,

that was based on SIGGS educational theory and APT – e.g., the doctoral seminar I taught on

Understanding Systemic Change in Education in 1995 ( see:

http://education.indiana.edu/ist/courses/r695fric.html ).

9/11 Spurs Axiomatic-General Systems Behavioral Theory

Kenneth Thompson, who originally worked on the SIGGS theory model as a graduate

student in the 1960s, was motivated by the events of September 11, 2001 to try to do

something to combat terrorism. In attempting to analyze terrorist networks, he found on the

Web some of my work with SIGGS and APT. These ideas helped him make a breakthrough

in his development of what he now calls Axiomatic-General Systems Behavioral Theory

(see: http://www.raven58technologies.com/index.html ).

We began working together about three years ago. His background in mathematics

and logic, my background in educational research methodologies and APT, and our common

interest in systems theory and experience with SIGGS has led to an unexpected synergy of

ideas. Thompson (2004) intends to apply this theory to military problems – including the


analysis of terrorist networks in order to predict their behavior. I want to apply A-GSBT in

the eventual development of a computer simulation of educational systems (SimEd). The

axioms and theorems of A-GSBT will constitute the rule base for SimEd, which will then be

used to make predictions of educational systems behavior in the simulation.

At present, Thompson has reduced A-GSBT to an initial axiom set (see Appendix A),

and has derived 14 theorems.

Table 1. Sample of Theorems derived from the Initial Axiom Set

T.12. System input increases only if filtration decreases.

T.13. System input decreases only if filtration increases.

T.21. System feedthrough increases only if compatibility increases.

T.29. System openness increases only if efficiency decreases.

The terms of A-GSBT are well-defined using notation from mathematics and predicate

calculus (see Appendix A). Properties of systems such as input, filtration, feedthrough,

compatibility, complexity, adaptability, flexibility, etc. are defined precisely. This is

necessary for proof of EST theorems and for specification of how these properties can be

measured. Logically clear definitions are necessary because commonsense or dictionary

meanings lack the precision necessary to test the validity of the theorems. This kind of

precision then allows us, for example, to measure a property such as compatibility at time 1

and again at time 2, and if the measure of compatibility is greater at time 2, then we have an

empirical instance of ‘compatibility increases’. And if we also measure feedthrough at two

points in time, and observe that ‘feedthrough increases’ then we have support for Theorem

21.


Goals of this Research: Expected Activities and Outcomes

Continued theory development. When developing theory in a formal and rigorous

manner, axioms are assumed – they are given, not proved. From those axioms theorems can

be derived. This derivation process is done by applying rules of logical deduction, and by

retroduction (cf., Peirce, 1896; Steiner, 1988). For example, the 14 theorems in Appendix A

were derived in this manner. See Appendix B for examples of such derivations and

implications for school systems. These theorems are logically consistent with the axioms.

Whether they are valid empirically must be demonstrated further (see below).

If the theorems are shown to be valid, then the axioms can be retained; but if a

theorem is empirically shown to be invalid, then the specific axioms and definitions needed

for proof of that theorem are called into question, and additionally any other theorems that

depend on those axioms and definitions must also be scrutinized. In other words, if empirical

data are inconsistent with the theory, then parts of the theory must be modified (cf., Steiner,

1988).

While we have been working on EST development for some time and have made

good progress, it is far from complete at this time. Thus, we anticipate that most of next year

will be needed to do this (see the Timeline that follows the proposal budget for details).

Empirical validation of theorems. Analysis of Patterns in Time (Frick, 1990) will be

the primary research methodology for validating theorems in EST. Since systems are

dynamic, a methodology which can measure temporal relationships is necessary. Frick

(1983) demonstrated that the linear models approach (e.g., ANOVA, regression, time series

analysis) fails to validate temporal relations which are stochastic. The proof was

mathematical and supported by empirical findings.


To help the reader understand APT methodology in a familiar context, consider the

simple APT score in Table 2:

Table 2. APT Score and Analysis of Weather Events ( ~ means ‘not’)

Time:::::::: 1:00......1:30......1:45.....3:00......11:00 CloudStruct: ~clouds>>>clouds>>>>>>>>>>>>~clouds>>>>>> Precip:::::: ~rain>>>>>>>>>>>>>>>rain>>>>~rain>>>>>>>>

Analysis:

a. From 1 - 1:30: [not clouds and not rain] = true

b. From 1:30 - 1:45: [clouds and not rain] = true

c. From 1:45 - 3:00: [clouds and rain] = true

d. From 3:00 - 11:00: [not clouds and not rain] = true

e. From 1:00 - 11:00: [rain and not clouds] = false

f. APTprob[clouds [rain]] > 0. 'Clouds followed by rain' became true at 1:45 until

3:00. The relative frequency measure for this pattern in the APT score in Table 2

would be 1 out of 1, and the relative duration measure would be 75/600 = 0.125. The

logical implication consistent with analysis of the above APT score is: ‘rain only if

clouds.’

In the APT score it is never true that we have [rain and not clouds]; it is sometimes

true that we have [rain and clouds] simultaneously occurring; and it is sometimes true that we

have [clouds[rain]] occurring. This last pattern was expressed as 'If clouds, then rain' in

original APT syntax (see Figure 1 and APT query examples in the 1990 AERJ article at:

http://www.indiana.edu/~tedfrick/apt/aerj.pdf). This last pattern is found to be true in the

APT score if first there are clouds and no rain, and next there are clouds and rain. Since a

temporal ‘if…then…’ can easily be confused with a logical implication expressed as

‘if…then…’, APT temporal relations are now expressed with bracket syntax – e.g.,


[clouds[rain]] indicates the temporal ordering of clouds and rain, where clouds are observed

to occur first and then later rain begins while clouds continue to occur.

There is a relationship between logical implication and temporal implication in APT.

The logical implication, 'rain only if clouds', is validated empirically when two temporal

implications as measured by APT are true: (APTprob[clouds [rain]] > 0) and (APTprob[rain

and not clouds] = 0). In simple terms: Clouds are necessary for rain. If we ever observe rain

without clouds in APT, then this invalidates the logical implication.

To put this in context for validating EST, consider now students who might enter an

educational system. Theorem 13 states: System input decreases only if filtration increases.

We could look for temporal patterns of admissions into an educational system. For example,

this past year the freshman student enrollment at IU dropped by approximately 400 students,

compared to the previous year. Thus, student input decreased. Therefore filtration must

have increased, if T13 is valid. So we look for the temporal pattern where it is the case that

filtration had increased prior to the start of the fall semester. As it was discovered, offers of

financial aid to new students were delayed significantly during the previous six months by a

glitch in the new PeopleSoft system. IU had (unintentionally) increased filtration by offering

less financial aid to students who were admitted but could not afford to attend, when

compared to the previous year when PeopleSoft was not used and offers of financial aid

arrived in a more timely manner for students in need. If, on the other hand, we had a

situation where enrollment had decreased, but it was not true that filtration had increased,

then this would not support T13.


This is a relatively simple example, but nonetheless illustrates the methodology that

we will use for testing the theorems. In short, we look for temporal patterns in existing data

on educational systems to support or refute the theorems.

Conclusion

How can EST benefit student academic achievement and the quality of life? As many

government leaders now hope, our education systems can lead to improvement of economic

conditions. The belief is that our universities can create new knowledge that feeds out of the

system into society and leads to new businesses that stimulate economic growth – e.g.

genomics. Another important feedout is students who graduate from our educational

systems, whether from high school or college. Those students leave their educational

systems with knowledge, skills and attitudes which we hope will help them become

productive members of society.

In education, one goal is to maximize compatibility between an educational system

and its negasystem. Compatibility is defined as commonality between system feedin and

feedout. An example from biology, where looking at animals as systems: animals exhale

carbon dioxide (feedout) that is needed by plants for photosynthesis. A by-product of

photosynthesis is oxygen, which animals need to breath (feedin). The environment provides

what the system needs to take in, and the system provides what the environment needs to

take in. If compatibility is high between an educational system and its environment, then this

kind of symbiosis is occurring. One of the reasons there is such concern about our K-12

educational systems in the U.S. (e.g., A Nation at Risk and NCLB) is that many leaders in

business and industry have found our graduates to be ill prepared for the kinds of jobs

available. That is, from their perspective compatibility is too low – too many students who


feedout of the educational system do not have the qualifications needed for successful

participation in the workforce.

One thing EST may reveal is that NCLB could actually decrease compatibility by

forcing schools to measure the wrong things. By narrowing the focus of our K-12 systems to

student achievement in reading, mathematics and writing, students may fail to learn good

decision-making skills, to take initiative, to become good problem solvers and team players,

may develop attitudes of cynicism and disrespect, may lack vision and hope, may have no

sense of aesthetics or moral value, etc. When we measure the feedout patterns in EST, we

look for the commonality between what is provided by our educational systems and what is

wanted and needed by society. And similarly, when we measure the feedin patterns, we look

at what society is providing as resources for an educational system (e.g., money, learning

materials, knowledge), properties of students who enter the system, kinds of teachers who are

selected and hired, etc.

As mentioned in the proposal abstract, the educational systems theory we are

developing will benefit educational decision makers, and it is needed for the rules in a

computer simulation I am developing to help people to learn to think systemically (SimEd).

This line of research is likely to go on for the next decade or more, and as the benefits of it

are realized this should increase the likelihood of getting major external funding to help

support it.

Kurt Lewin often said, “There is nothing so practical as good theory.” If we are

going to systemically change education, we need to know what we are doing. A good

educational systems theory can increase the chances that we will succeed – that we increase

student learning achievement and that our graduates will help improve the quality of life.


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*Budget Notes and Justification

The primary consultant is Kenneth Thompson. He is the third author who worked

originally on SIGGS with Elizabeth Steiner Maccia and George Maccia at Ohio State

University in the early 1960s. He is well versed in the SIGGS theory model and educational

theory development in general. I have been working with him on an informal and voluntary

basis for about three years as we have been gradually identifying and solving problems with

the existing educational theory. This work is not complete, and the consulting fees would

help support Ken’s work on extending and refining the educational theory. Kenneth

Thompson lives in Columbus, Ohio, and is a private consultant. He is not associated with

Ohio State University or Indiana University, nor an employee of these or any other

university.

During the second year, a GA will be needed to help collect and analyze empirical

data to validate theorems in A-GSBT. Thompson will be retained, as a consultant, to assist

with interpretation of the findings and to revise the theory as required by the results. In

addition, a relatively small amount is allocated to cover travel expenses to school sites to

collect data during the second year. Finally, a small amount of funds are allocated for costs

of copying and printing reports.

Timeline

Phase 1: Jan. 1, 2005 – June 30, 2005:

Technical Reports Expected:

(1) Logic-Based Model versus Scenario-Based Model

(2) A-GSBT Index of Properties


(3) A-GSBT List of Empirical Axioms

(4) Axiomatic General Systems Behavioral Theory

(5) Basic Properties

a. System Definition

b. General System Definition

(6) Structural Properties

a. Affect Relation Properties

b. Put Properties

c. Feed-Transmission Properties

d. Filtration Properties

e. Interface System Properties

f. Behavior Properties

g. Morphism Properties

(7) Dynamic Properties

a. System Properties

b. State Properties

c. Behavior-Controlling Properties


Phase 2: July 1, 2005 – Dec. 31, 2005:

Technical Reports Expected:

(1) Theorem Construction Logic

(2) List of A-GSBT Theorems and Theorem Schemes

(3) Criteria for Application of A-GSBT to Empirical Systems

(4) Final Report: A-GSBT

Phase 3: Jan. 1, 2006 – June 30, 2006:

(1) Choose K-12 school systems to be studied empirically for validation of EST

theorems. Likely candidates (which require little travel): Monroe County

School System; Brown County School System; Richland-Bean Blossom

System; Bloomington Montessori School; Harmony School; Indianapolis

Metropolitan Career Academies. Contact educational systems to negotiate

agreements for participation of four that vary considerably in size, complexity,

mission and social context. Secure IU Human Subjects Committee approval

as needed.

(2) Collect and analyze extant data from educational system #1.


(4) Identify theorems which appear to be invalid across the two systems studied.

Revise EST as needed.


Phase 4: July 1, 2006 – December 31, 2006:

(5) Continue to identify theorems which appear to be invalid across the two

systems studied. Revise EST as needed.



(8) Identify theorems which appear to be invalid across the last two systems

studied. Revise the educational systems theory accordingly.

(9) Write final report for entire study.

Other Proffitt Grants

Elizabeth Boling (PI), Kennon Smith, Malinda Eccarius, and Ted Frick. Visual

Representations to Support Learning: Effectiveness of Graphical Elements Used to Extend

the Meaning of Instructional Illustrations. (Award amount: $39,864.00, funded from June

2003 - June 2005.) This grant is funded by the Proffitt Foundation. Elizabeth Boling is the

principal investigator. My role is minor, mostly to assist with research methodology issues

such as reliability of measurement and statistical analysis. The project is making good

progress. Data from on interpretation of meanings of graphical elements from different

populations and cultures have been collected and analyzed. Several conference presentations

have been made, a paper submitted for publication, and further populations are currently

being compared.

Other Grant Support and Pending Proposals

At this time, I have no grant support for the research proposed here, and no pending

proposals.


Appendix A

Initial Axiom Set in A-GSBT

1. System input decreases only if fromput decreases.

2. System output increases only if fromput increases.

3. System filtration increases only if adaptability increases.

4. System toput increases and fromput increases only if feedthrough increases.

5. System input is constant and fromput is constant only if output is constant.

6. System toput increases only if centrality decreases.

7. System feedin decreases only if unilateralness decreases.

8. System feedin decreases only if complexity-degeneration increases.

9. System complete-connectivity increases only if feedin increases.

10. System interdependence increases only if feedin increases.

11. System centrality increases only if toput decreases.

12. System complete-connectivity increases or strongness increases only if toput increases.

13. System complete-connectivity increases or strongness increases only if input increases.

14. System filtration decreases only if isomorphism increases.

15. System isomorphism increases only if fromput decreases and feedout decreases.

16. System size increases and complexity-growth is constant only if toput increases.

17. System size increases and complexity-growth is constant only if feedin decreases.


Theorems Derived thus far from the Initial Axiom Set

T.12. System input increases only if filtration decreases.

T.13. System input decreases only if filtration increases.

T.21. System feedthrough increases only if compatibility increases.

T.29. System openness increases only if efficiency decreases.

T.53. System complete-connectivity increases only if flexibility increases.

T.54. System strongness decreases only if wholeness increases.

T.55. System strongness increases only if hierarchical-order decreases.

T.56. System strongness increases only if flexibility increases.

T.57. System unilateralness only if hierarchical-order.

T.179. System size increases and complexity-growth is constant only if vulnerability increases.

T.180. System size increases and complexity-growth is constant only if flexibility decreases.

T.181. System size increases and complexity-growth is constant only if centrality decreases.

T.182. System size is constant and complexity-degeneration increases only if disconnectivity increases.

T.183. System size decreases and complexity-degeneration increases only if disconnectivity decreases.


Sample of Definitions of Properties in A-GSBT

**Active dependence, ADC, =

Df system components that have connections from them.

ADC =

df W _ GO | ∀x∈W ∃y∈S

O ∃i((x,y)∈Ai∈A . x∈

iE)

Active dependence is defined as an object-set; such that, the components are initiating components of an affect relation.

M(ADC) =df M(W) = ν :h: [W _ S

O = {x | ∃y∈S

O ∃i((x,y)∈Ai∈A . x∈

iE)}] q

log2(|W|) ÷ log2(|d(maxHOC)|) = ν

Measure of active dependence is defined as the value ν of initiating components; equivalent to the component-set x of initiating components of an affect-relation, implies the quotient of the base-2 log of the cardinality of the set of initiating components by the base-2 log of the maximum hierarchical order distance, equals ν.

**Adaptable, AS, =

df difference in compatibility under system environmental change.

AS =

df ∆C | ∆S’

Adaptable is defined as a change in compatibility given system environmental change.

M( A

S) =df

| M(C t(1)

) - M(C t(2)

)| = ν

Measure of adaptable is defined as the cardinality of the difference of compatibility at time t(1) and time t(2).

**Centralization, CC, =

df concentration of connections to primary-initiating components.

CC =df W _ S

O | ∀x∈W ∃y∈S

O∃i((x,y)∈Ai∈A . x∈

piE)

Centralization is defined as an object-set; such that, the components are primary-initiating components of an affect relation.

M(CC) =df M(W) = ν :h: [W _ S

O = {x | ∃y∈S

O∃i((x,y)∈Ai∈A . x∈

piE)] q

[log2|W| ÷ d(ΣHOC

pi(E))] × log2 |rEx| = ν

Measure of centralization is defined as the value ν of primary-initiating components; equivalent to the component-set x of primary-initiating components of an affect-relation, implies the product of the quotient of the base-2 log of the cardinality of the primary-initiating component set by the distance of the sum of the hierarchical-order primary-initiating components, equals ν.

**Compatibility, C , =df

is a measure of the commonality between feedin and feedout.

C =df

M(C ) = Ax(fO) + Ax(fI)


Compatibility is defined as a measure; such that it is equal to the quotient of the APT value of feedout by the APT value of feedin. Compatibility can be viewed as the composite function that defines feedthrough where the values are the same as toput, as follows:

C =df

fB =

df σx | σx(x) = (fO ) fN ) fI)(x) = fT(x) = y∈T

P

Compatibility is a system state-transition function; such that it is equal to feedthrough that is equal to toput.

**Complexity, X, =df

number of connections.

X =df M(Am∈A) | M:Am → R ∧ M(Am) = |Am|

Complexity is defined as a is a measure of an affect relation; such that the measure is a function defined from the affect-relation set into the Reals, and the measure is equal to the cardinality of the affect-relation set.

Complexity is a measure of the connections in an affect relation.

**Feedin, fI, =df transmission of negasystem toput to system input.

fI =df σ | ∃P(x)∈TP

LC . [∃Am ∀{{x},{x,P(x)}}∈Am∈A (σ: TP % TP

LC → IP )]

Feedin is a system state-transition function; such that there is a P(x) that is an element of the toput system-control qualifier, and there is an affect-relation such that for all elements of the affect-relation, the transition function maps toput to input.

** Filtration, Filtration-by-System, SF, =df

the set of toput system-control qualifiers that preclude feedin of toput.

SF =

df {P(x) | P(x)∈TP

LC . [∃Am ∀{{x},{x,P(x)}}∈Am∈A (σx: TP % TP

LC → TP )]

Filtration, Filtration-by-System, is defined as a set of predicates, P(x); such that, P(x) is an element of the toput system-control qualifier, and there is an affect-relation such that for all elements of the affect-relation, the transition function maps toput onto itself.

**Flexible-connected components, FC, =

Df subgroups of system components that are independently path-

connected between two other components not in the subgroups.

FC =df X = {x| x∈ S

O . ∃y[(x,y)∈cE q (x,y)∈pcE . F(X

i)((x,y))]}; where

F(Xi) = {X

i | X

i _ S

O . i>1 . ∀X

i ∃x∈S

O ∃y∈S

O[(x,y)∈pcE q (x,X

i),(X

i,y)∈pcE]}

Flexible-connected components is defined as a set of components of the object-set; such that, the components are path-connected and are path-connected through two or more subgroups of the object-set.


**Input, IP, =df system components whose value-set of the toput system control-qualifiers is “true.”

IP =df {x| x∈S

O . ∃P(x)∈LC ∃Ai∈A [{{x},{x,P(x)}}∈Ai . P(x) = S | T/IS]}.

Input is the set of system components for which there exists system control-qualifiers of an affect relation of the T/I-put interface system for which the predicate is “true.”

**Size, Z, =df

Number of components.

Z =df M(W⊂ SO) | M : W → R ∧ M(W) = |W|

Size is defined as a is a measure of a subset of the object set; such that the measure is a function defined from the object-set into the Reals, and the measure is equal to the cardinality of the object-set.

Size is a measure of the number of components in an object-set.

**System, S, =df a group with at least one affect relation.

S =df (GO, A) = (SO, Sφ)

System is defined as a set of components and a family of affect relations.

**System affect relation measure, MS(A), =

df a measure that is a function, ƒ, or APT Score, A, defined on

one or more Affect Relation sets, Am, such that a value is determined.

MS(A) =df [∃ƒ(ƒ:Am → R) - ∃A(A:Am → R) | ƒ(Am) = x - A(Am) = y]

System affect relation measure is defined as a value derived from a function or APT score defined on an affect relation set.

**Toput, TP, =df negasystem components that result in a value-set of the system control-qualifiers.

TP =df {x| x∈S’

O . ∃P(x)∈LC ∃Ai∈A [{{x},{x,P(x)}}∈Ai | T/IS]}.

Toput is the set of negasystem components for which there exists system control-qualifiers of an affect relation of the T/I-put interface system. These are the components that will become input if and when the value-set is “true.”


Appendix B

Derived Theorems:

A-GSBT Initial Axiom Set

The following theorems have been derived from the Initial Axiom Set and the Definition-Derived Theorems. The first theorem, T.106.90, is derived from Axioms 106 and 90. It is derived as a result of the transitivity of implication, q, which is defined by Logical Schema 0.

T.106.90. d CCC↑ - SC

↑ q CC↓

Complete connectivity increasing or strongness increasing implies that centrality decreases.

Proof:

1. CCC↑ - SC

↑ q TP

↑ Axiom 106

2. TP

↑ q CC↓ Axiom 90

3. d CCC↑ - SC

↑ q CC↓ Logical Schema 0, Transitivity of q

What this means for a school system is that the central administrative authority of the system is diminished when alliances are established within each school or a principal assumes direct control over the teachers within a school. This type of system may be something to be encouraged, or, if it goes too far, the central administration may need to disrupt the affect relations that have isolated the central authority.


This next theorem, T.13.28, is derived from Theorem 13 and Axiom 28 by the transitivity of implication, q.

T.13.28. d IP

↓ q AS↑

Input decreasing implies that adaptability increases.

1. IP

↓ q SF ↑ Theorem 13

2. SF ↑ q AS↑ Axiom 28

3. d IP

↓ q AS↑ Logical Schema 0, Transitivity of q

What this means for a school system is that as the system receives less input, it must achieve greater adaptability. This happens frequently when bond issues do not pass, or the student population decreases. With a decreasing number of students, certain subsystems, schools, may have to close; that is, the subsystem “dies.” Generalizing this theorem, it asserts that when the input of a system decreases, the system must adapt or die. Hence, the conclusion must be that adaptability increases, since otherwise there will be no system.


This next theorem, T.194.15.33, is derived from three axioms—194, 15 and 33. This proof is more complex than the two preceding theorems, as it entails the use of Modus Ponens, a logical transition rule, as well as applications of the definition of the conjunction operation, ., and the Deduction Theorem. Also, this theorem stresses the importance of “assumptions.” It is important to recognize that this theorem is valid only when the assumptions are “true.”

T.194.15.33. d Z↑ . X +c . O

P↑ q fT↑

Size increases and complexity growth is constant and output increases implies that feedthrough increases.

Proof:

1. Z↑ Assumption

2. X +c Assumption

3. OP

↑ Assumption

4. Z↑ ∧ X +c q T

P↑ Axiom 194

5. Z↑ ∧ X +c Conjunction on 1 & 2

6. TP

↑ Modus Ponens on 5 & 4

7. OP

↑ q FP

↑ Axiom 15

8. FP

↑ Modus Ponens on 3 & 7

9. TP

↑ ∧ FP

↑ Conjunction on 6 & 8

10. TP

↑ ∧ FP

↑ q fT↑ Axiom 33

11. fT↑ Modus Ponens on 9 & 10

12. Z↑ . X +c . O

P↑ d fT↑ 1, 2 & 3 yields 11

13. d Z↑ . X +c . O

P↑ q fT↑ Deduction Theorem


What this means for a school system is that as its student population increases, and the complexity of the student connections within the school increase at a constant rate (i.e., the student-teacher ratio remains constant), and its graduation rate increases, then it will have an increasing output to the community. While this may be simplistically obvious, it often goes unrecognized as many “fixes” are sought for a school system realizing unexpected growth; e.g., from an influx of new students or a school within the system that has to close unexpectedly thus placing a greater burden on the remaining schools. Under these or similar conditions, if the antecedent parameters are not maintained, then a decrease in student output would be expected; e.g., there would be a greater drop-out rate. In the event of unexpected student growth, this theorem alerts the school administration that they must control the one parameter over which they do have control—the growth of the complexity of each school; i.e., the complexity growth must be held constant.

What must be recognized by the school administration is that the increased size of the student population is a given, it has happened. The output is not within their control unless all other factors are maintained. The only parameter over which they have direct control is the complexity growth, which must be held constant. By dumping the increased number of students into classes that already exist will not maintain a constant growth rate. Introducing more “points-of-contact” within each class, possibly by way of teaching assistants, team teachers, or student assistants, can help to maintain a constant growth rate. What this theorem helps school administrators to do is focus on a solution that may not be otherwise obvious, even if the theorem, after being stated, is obvious.


While the cartoon below is intended to be humorous, it nonetheless illustrates an example of compatibility increasing. An educational system consists of student, teacher, content and context subsystems. The affect relation depicted below is between student and context subsystems. Theorem 21 states: System feedthrough increases only if compatibility increases. At time 1, compatibility is low. At time 2 compatibility has increased. It also appears that feedthrough has increased. ☺

developing an educational systems theory to …tedfrick/proposals/est2004.pdfdeveloping an...

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