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
31
Embed
Developing an Educational Systems Theory to …tedfrick/proposals/est2004.pdfDeveloping an Educational Systems Theory to Improve Student Learning – iii Validating EST at this initial
This document is posted to help you gain knowledge. Please leave a comment to let me know what you think about it! Share it to your friends and learn new things together.
Transcript
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-
Wang, K. (1996). Computerized adaptive testing: A comparison of item response theoretic
approach and expert systems approaches in polychotomous grading. Bloomington,
IN: Ph.D. dissertation.
Welch, R. & Frick, T. (1993). Computerized adaptive mastery tests in instructional settings.
Education Technology Research & Development, 41(3), 47-62.
Wolfram, S. (2002). A new kind of science. Champaign, IL: Wolfram Media, Inc.
Yin, L.C. (1998). Dynamic learning patterns during individualized instruction.
Bloomington, IN: Ph.D. dissertation.
Developing an Educational Systems Theory to Improve Student Learning – 15
*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
Developing an Educational Systems Theory to Improve Student Learning – 16
(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
Developing an Educational Systems Theory to Improve Student Learning – 17
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.
(3) Collect and analyze extant data from educational system #2.
(4) Identify theorems which appear to be invalid across the two systems studied.
Revise EST as needed.
Developing an Educational Systems Theory to Improve Student Learning – 18
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.
(6) Collect and analyze extant data from educational system #3.
(7) Collect and analyze extant data from educational system #4.
(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.
Developing an Educational Systems Theory to Improve Student Learning – 19
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.
Developing an Educational Systems Theory to Improve Student Learning – 20
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.
Developing an Educational Systems Theory to Improve Student Learning – 21
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)
Developing an Educational Systems Theory to Improve Student Learning – 22
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.
Developing an Educational Systems Theory to Improve Student Learning – 23
**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.
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.”
Developing an Educational Systems Theory to Improve Student Learning – 24
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.
Developing an Educational Systems Theory to Improve Student Learning – 25
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.
Developing an Educational Systems Theory to Improve Student Learning – 26
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
Developing an Educational Systems Theory to Improve Student Learning – 27
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.
Developing an Educational Systems Theory to Improve Student Learning – 28
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. ☺