Title

Mathematical Models in Education and Training

Description

Present-day mathematical models of educational systems can provide useful answers to limited but important quantitative questions concerning budgeting, resource allocation, and enrollment planning. Such models are designed, at the conceptual level, by determining the major features of the system, outlining their interrelationships with a flow chart, and choosing the variables to be used. If, in addition, specific assumptions embodying the educational "physics" of the model or empirical relationships are included, the model can then be solved; that is, each dependent variable can be stated as a function of the independent variables, and the consequences of the assumptions can be determined. The model structures that result can be characterized by their scope and complexity, by the degree of aggregation of the variables employed, by the model inputs and outputs, and by the purpose for which the model is to be used.

A number of representative existing models of various types are discussed. Input-output models can be a convenient way to examine large amounts of data on enrollments and student flows, but these models are limited in that a current cross-sectional analysis is generally used to predict the future time series of the variables. input-output models may find wide application for analyzing systems with relatively static structures, however, such as training institutions for specific purposes.

Manpower planning models seem to be less useful than many other models. Because these models do not provide explicit allocations of educational resources and because they do not describe actual student flows, they are perhaps too simplified for the problem they attempt to solve. Optimization models have the advantage of making explicit the basic choices of a resource-allocation problem, when the desired benefits can be quantitatively described. Since such models yield priorities and plans as output, they are more likely to stimulate discussion at the policy level. Simulation models will be of considerable assistance in management and short-term planning for educational systems, but they run the risk of foundering in a wealth of detail. It may well be that optimization and simulation models can serve in complimentary ways in educational planning, one operating on the policy level and the other on the detailed operational level. Finally, where specific educational mechanisms can be identified, relatively simple models can be extremely effective. The usefulness of simple models for flows of students and teachers can be extended further by including cost factors and other simple economic variables, but without attempting to model all aspects of an educational system.

More research is needed to increase our understanding of the dynamics of educational systems. the appropriate mathematical basis for the research suggested here would be very simple; stochastic models for probabilistic problems and simple difference and differential equations for deterministic problems, coupled with optimization or simulation techniques where appropriate, should be adequate for most modeling of educational systems in the near future. These techniques and the mathematical structure of educational-system models are discussed in the appendix. A selected bibliography is also included.

The author is a Consultant to The Rand Corporation

Subject

Education -- Management Planning And Control

Programmed Instruction -- Mathematical Models

Optimization

Simulation

Models (Simulations)

Game Theory

Matrices (Mathematics)

Creator

Hammond, Allen

Publisher

Santa Monica, CA : The RAND Corporation

Date

1970

Format

vii, 31 pages ; 28 cm.

Type

report

Identifier

AD0712347

AD0712347

Date Issued

1970-09

Corporate Author

The RAND Corporation

Report Number

RM-6357-PR

AD Number

AD712347

Contract

F44620-67-C-0045

NTRL Accession Number

AD712347

Distribution Conflict

No

Access Rights

THIS DOCUMENT HAS BEEN APPROVED FOR PUBLIC RELEASE AND SALE; ITS DISTRIBUTION IS UNLIMITED.

Index Abstract

Contrails and DTIC truncated

Photo Quality

Not Needed

DTIC Record Exists

Yes

Report Availability

Full text available by request

Abstract

Present-day mathematical models of educational systems can provide useful answers to limited but important quantitative questions concerning budgeting, resource allocation, and enrollment planning. Such models are designed, at the conceptual level, by determining the major features of the system, outlining their interrelationships with a flow chart, and choosing the variables to be used. If, in addition, specific assumptions embodying the educational "physics" of the model or empirical relationships are included, the model can then be solved; that is, each dependent variable can be stated as a function of the independent variables, and the consequences of the assumptions can be determined. The model structures that result can be characterized by their scope and complexity, by the degree of aggregation of the variables employed, by the model inputs and outputs, and by the purpose for which the model is to be used.

A number of representative existing models of various types are discussed. Input-output models can be a convenient way to examine large amounts of data on enrollments and student flows, but these models are limited in that a current cross-sectional analysis is generally used to predict the future time series of the variables. input-output models may find wide application for analyzing systems with relatively static structures, however, such as training institutions for specific purposes.

Manpower planning models seem to be less useful than many other models. Because these models do not provide explicit allocations of educational resources and because they do not describe actual student flows, they are perhaps too simplified for the problem they attempt to solve. Optimization models have the advantage of making explicit the basic choices of a resource-allocation problem, when the desired benefits can be quantitatively described. Since such models yield priorities and plans as output, they are more likely to stimulate discussion at the policy level. Simulation models will be of considerable assistance in management and short-term planning for educational systems, but they run the risk of foundering in a wealth of detail. It may well be that optimization and simulation models can serve in complimentary ways in educational planning, one operating on the policy level and the other on the detailed operational level. Finally, where specific educational mechanisms can be identified, relatively simple models can be extremely effective. The usefulness of simple models for flows of students and teachers can be extended further by including cost factors and other simple economic variables, but without attempting to model all aspects of an educational system.

More research is needed to increase our understanding of the dynamics of educational systems. the appropriate mathematical basis for the research suggested here would be very simple; stochastic models for probabilistic problems and simple difference and differential equations for deterministic problems, coupled with optimization or simulation techniques where appropriate, should be adequate for most modeling of educational systems in the near future. These techniques and the mathematical structure of educational-system models are discussed in the appendix. A selected bibliography is also included.

The author is a Consultant to The Rand Corporation