Dynamic Modeling of a Learning Process

N. Eftekhar
D.R. Strong
Mechanical & Industrial Engineering Department, University of Manitoba, Winnipeg, Manitoba, Canada


The primary aim of the current work is to build a simple model of a student learning process and simulate it to examine the behavior of the system. This effort is a comprehensive combination of the Education metrics and Engineering computer programs where major variables of a learning process are investigated and modeled based on a system approach. The main focus is on the study of relationship between structure and behavior of the system.

INTRODUCTION

THE PROCESS of learning in colleges and universities is a topic of current interest and debate among experts in Education. Many educational approaches exist regarding the probable outcome of any course of action that could influence or reinforce the learning process. This work is an ambitious undertaking that implements a unique effort on the combination of education metrics and computer simulation. The work is based on using a package of a simple control engineering concept, a model of an educational process, and a computer simulation. The main purpose is to develop a model by which one can gain a better insight into the possible dynamic behavior of a learning process. The ultimate intention is to provide an Engineering framework for studying and improving the situation. To have a foundation on which to build the present model, this paper draws on the works of Wilbert J.McKeachie, Cameron Fincher, Barbara Schneider Fuhrmann and Anthony F. Grasha.

The methodology used herein is known as "system analysis.” The application of this methodology have enabled the authors to gain a basic understanding of how structure, time delays and policies were interrelated to influence the amount of knowledge learned by an average undergraduate student in a university or college course. The approach includes two phases: qualitative and quantitative system dynamics. The qualitative phase of the method is based on (1) developing an understanding of system concept and the theory of feedback process by mapping the resource flows and interrelations in a simplified learning system, and (2) developing insight into the dynamic behavior of the system by converting the map to a simulation model. Although comprehensive simulation is not advocated by the method at this stage, it is possible from a study of the feedback loop structure of the diagram, to estimate their likely general direction of behavior (say growth or decline). The second phase of the method is that of quantitative computer simulation modeling using the purpose built software. This involves deriving with system actors the shape of relationships between all variables within the diagram, the calibration of parameters and the construction of simulation equations and experiments. Although numbers are attached to variables during this phase, it should be stressed that the method is not aimed at accurate prediction or solutions. It is more concerned with the shape of change over time. In this phase, emphasis is on (1) investigating the dynamic behavior of the system to obtain a better understanding of how a learning process works, and (2) applying the concept and methodology by trying proposed policy changes to the model to maintain sustainable improvement in performance of the system (re-engineering) while considering the feasibility of implementing these changes in its actual situation. Worth mentioning is that the later step will be using a validated model as a guiding device by testing its prediction against an actual student behavior.

The model of a learning process system is constructed using iThink/Stella II modeling language. This software bought from High Performance Systems Inc. is a continuous-simulation based language. The software is one of the appropriate multi-level, hierarchical environment for constructing and interacting with complex models. The language is built around a progression of structures. At its lowest level are stock levels, flow rates, and other primary building blocks of structure. Supporting structures are the next step in the progression. While vary in size and complexity, they are built up from various combinations of stocks, flows, and other building blocks. Feedback loops are the subsequent step in the progression. Feedback loops are the relationships that link the two generic blocks (stock levels and flow rates) in various ways. In so doing, they enable supporting structures to exhibit interesting dynamic behaviors. The final step is the top layer of the model that wraps up the top-down approach to model development.

In summary, using the above methodology may allow one to observe and identify problematic behavior in the learning process of a student over time. This method creates a valid diagrammatic representation of the existing system behavior capable of reproducing by computer simulation and of facilitating the design of improved system behavior (re-engineering).

DIFFERENT APPROACHES ON LEARNING

To build a model of learning process, first, we have to decide which class or combination of classes of thought on learning are more appropriate and applicable to our work. The goal is to hire an effective approach that provides a much more defensible structure for our model. For this purpose, before proceeding, it is appropriate to focus on some of the results of recent study and research into learning methods.

A Quick Look at Current Views on Learning

In concise, the current approaches on learning can be classified as those based on: (1) Personal Viewpoints, (2) Quantitative Approach, and (3) Learning Theories. [1,2]

(1) Personal Viewpoints represent those ideas that experts in the field of teaching and learning and anyone else with an opinion have about the learning issues. It seems that there are as many definitions as there are people trying to define it. Such definitions range from concerns about student learning to lists of ideal traits that instructors must possess. This approach provides us with a rich and broad set of perspectives on the nature of teaching-learning processes. The demerit of this approach is the variety of ideas with no consensus that lead people to defend their point of view rather than accept alternative views.

(2) Quantitative approach assesses the extent to which students possess various characteristics that demonstrate the level of their effective learning. In this view, students receive typical scores that rate them as below average, average, or above average on particular attributes. This approach appears to be objective, although such appearances are deceptive. Because, underlying the numbers are many different points of view about the important characteristics that students should possess. The demerit of this approach is that fixating on the characteristics that we measure prevents recognizing the learning situations vary. Merit of having so-called objective data on learning from this approach is offset by these problems.

(3) Learning Theories often suggest classroom procedures and teacher behaviors that facilitate students' learning along certain lines. These Theories are classified as behaviorist, humanistic, and cognitive views on learning.

The behaviorist view suggests that our behaviors are controlled by stimuli in our environments. Accordingly, anyone can be taught to become anything-by the proper manipulation of environmental stimuli. A modified version of this view introduces what is called a technology of operant conditioning. This technology stresses the need to shape behaviors in small steps and to reward each small success a learner has. It also emphasizes that organisms learn at different rates and that some custom designing of learning environments is necessary to accommodate such variations. Teaching machines, personalized systems of instruction and computer-assisted instruction have evolved based on behavioral principles. In general, the behaviorist approach to learning suggests that good learning is demonstrated when the instructor can write objectives relevant to the course content, specify classroom procedures and student behaviors needed to teach and learn such objectives, and demonstrate that students have achieved the objective afterward.

The humanistic view recognizes that learning is something that students must do for themselves. Teachers must not merely transmit, but must involve and engage students in the activities of discovery and meaning making. This emphasis on student needs and the study of oneself as part of the study of humanity is sometimes also called student-centered education. Teachers are encouraged to guide and direct less and to facilitate or act as a catalyst for students to initiate and take responsibility for their own learning. It is an attempt to personalize education. It represents a reaction against the excesses of the technological emphasis in education during the late 1950s.

The cognitive view provides a conceptual base for understanding the results of the earlier studies in the area of learning and teaching. From a cognitive view, effective learning is demonstrated when teachers use classroom procedures that are compatible with a student's cognitive characteristics, can organize and present information to promote problem solving and original thinking on issues, and demonstrate that students are able to become more productive thinkers and problem solvers. Cognitive theory sees a learner at center stage. The teacher, on the other hand, becomes a facilitator of learning, rather than one who delivers information. This perspective on learning contrasts sharply with views that imply that learners get the point as long as the instructor provides an appropriate stimulus. Cognitive approach says that the learner plays a critical role in determining what he or she gets out of instruction.

Bloom's Taxonomy

The least weakpoint of theories of learning is their inability to agree on an acceptable typology for learning. None offer a taxonomy that make sense in modeling a learning process. The only taxonomy with some promise is from the 1950s. It is called Bloom's Taxonomy.[3,4] It divides the learning skills and intellectual abilities into six major levels identified as: (1) Acquisition of knowledge or information (memorizing), (2) Comprehension or understanding the message, (3) Application of the material comprehended, (4) Analysis the meaning of a comprehension, (5) Synthesis of parts for a whole to constitute a new pattern/structure, and (6) Evaluation (make quick opinion or judgment) about the value of an idea, solution, material, or method.

As shown in Fig.1 (Bloom's Taxonomy), the first two levels are known as lower-order abilities and the others as higher-orders. Higher-order learning skills are more difficult to master than lower-orders. They are required to ever-greater extents as students progress to upper level courses. Although it is possible to conceive of these major levels in several different arrangements, the present classification represents something of the hierarchical order of the different levels of abilities. The skills or abilities in one level are likely to make use of and be built on the behaviors found in the preceding levels. For instance; since creative learning occurs at the synthesis level, it requires the preceding four levels of learning as a basis upon which to build.

Probably the most common learning skill or intellectual ability in any learning system is the acquisition of knowledge or information. That is, it is desired that as the result of completing an educational unit, the student will be changed with respect to the amount and kind of knowledge he or she possesses. Frequently acquisition of knowledge is taken as the primary educational objective in a learning process. In almost every course it is an important or basic one.

Our Chosen Approach

One reasonable way to overcome the problems with above views is not to make a priori assumption about what approach is more appropriate for building the subject model. Rather, to take the view that there are many forms of a learning process [2].It is more appropriate to design a model based on what we have learned from our personal learning experience and what we have grasped from the existing viewpoints and theories (that seems to fit well with our learning values and practices). Thus, our chosen approach emerges from a kind of eclectic viewpoint. It includes a combined look at our personal learning experience and those parts in our learning environments that match well with the theories mentioned earlier. So doing, our first attempt is to start with the higher layer of our model of interest.

Globally, any model of a learning process can be conceptualized in a simple control engineering approach as shown in Fig. 2. (Structure of a General Engineering Control Process.)

The above control process structure is easily applicable to each student's unique learning behavior. Inputs are the resources, physical or non-physical, internal or external, and used or un-used by the student. They drive the learning process and through the process create the outputs. Outputs are the different outcomes of the learning. A feedback loop represents many feedback information that exist in the learning system. It depicts the continuing effort that persistently is seeking an adjustment for the trend of the process. Through feed-back signal, an output is assessed against a reference policy, say student target knowledge, and a corrective measure, say his or her higher or lower effort for study hours, is taken accordingly. The resulting behavior of the system for each student may overlap with those of other students, or may not.

However, to facilitate this approach, we have to identify what components each part of the system possesses. This, subsequently, demands us to focus on the lower layer of the model; e.g., the model construction layer that betrays details about the learning process itself and the major components in the process. This automatically, urges us, to have a look on the definition of some of the basic elements.

LEARNING AND LEARNING CONDITIONS

Technically speaking, although learning and teaching are complementary processes, in any attempt to model an educational system, one should give the heavier part to the "learning process side.” The reason, of course, is uncovered when one recognizes that the quality of an educational system has to be measured mainly by its resources for learning. Having this in mind, let us first figure out what is "learning” and under what conditions learning processes occur.

Learning may be defined as a function of students interacting with instruction method and subject matter variables. The variables influencing a learning process are almost numberless. Because their interactions change from day to day, one needs to move from pretest-posttest measures to studies of ongoing processes, from single-variable studies to individual students interacting in groups, and from studies of outcomes of learning to studies of what goes on in the thoughts, feelings, and desires of students [5]. Despite many efforts to define learning in educational terms, one comprehensive effort gives a definition for learning as follows [3]:

Learning is a process of acquiring and integrating through a systemized process of instruction of organized experience varying forms of knowledge, skill, and understanding that the learner may use or apply in later situations and under conditions different from those of instruction.

This definition specify that: (1) some learning result from instruction and some from other forms of organized experience; (2) the process of learning include cognitive, behavioral, and experimental dimensions or components; (3) learning should be seen in relation to its future uses or applications and its transfer to situations and conditions; and (4) learning and teaching are dual processes that must be treated systematically if they are to make educational sense. One concise definition of learning also is given as follows:

Learning is a change in human disposition or capability, which persists over time, and which is not simply ascribable to processes of growth.

From both above definitions one may infer that learning is taken place when a learner observes a change in his or her performance during a period of time. This, apparently, conveys a notion that learning, in principle, may possess two major sets of components: those that are very sensitive to any change in time and those that are not. The first set of components may be considered as those that take different value at any time. The second set of components are those that can be assumed do not vary in short terms and have their effects on the process in long terms. Some traits of a learner can be assumed in the first set of the components while the others in the second set.

However, the major components within a learning process are the main conditions under which the learning situation occurs. These can be identified as [3]:

Summing up what was mentioned in the last section with what concluded above, the major featured components in a learning process system can be classified as shown in Table 1 (Major Components in a Learning Process). Note that all of these components, more or less, are the main actors in a learning process system. Those items in the first column have relative tendency to play a role in the input side of the system. Likewise, the second and third columns have relative tendencies to play roles in process and output sides respectively.

MODELING SOFT VARIABLES

Modeling of a learning process requires involvement with those variables that are internal to the human beings. Variables like motivation and knowledge or quality of instruction are not things that can be computed. They do not get numeric or precise value.

They do not reflect any "hard” bottom line. However there should be some mechanism for capturing and using them in a model. This notion obliges us to recognize, in the first place, an important distinction that exists between measurement and quantification [6].

Measurement means "assessing the magnitude of." The result of the assessment is often expressed numerically. Apparently, all physical quantities or "hard” variables have their pre-defined units-of-measures. By contrast, quantification means "assigning a numerical index to.” While assigning a quantitative index usually is a pre-condition to measuring something, the two activities are not one and the same. We can quantify anything. However, there are some things that it will never be possible to measure. Fortunately, it is not necessary to measure these variables in order to be able to incorporate them in our simulation model. For instance, to quantify student motivation we can assume 0 represent the complete absence of motivation and 100 represent as much motivation as is possible for a student to have. A similar quantitative index would work equally well for student aptitude and abilities, and quality of instruction. Likewise, to quantify learning rate we can assume 0 represent the complete absence of effort to learn and 100 represent as much knowledge as is possible for a student to acquire per a given period of time. Doing so will cause us to think in a rigorous manner about the relationship each variable bears to other variables in the system. The more we can quantify, the better our model resembles to the real one. In addition, we will be able to simulate the variables to examine their role in the dynamics of a learning process. This is what we are really aiming for.

DESCRIPTION OF SYSTEM STRUCTURE

The analysis of the dynamic behavior of a learning process is undertaken using what is termed a "system approach.” This approach [7] calls for the consideration of a "complex” set of relationships as a system. "Complexity” refers to a high-order, multiple-loop, nonlinear feed-back structure. All social systems belong to this class. Educational systems, and specifically a learning-teaching process that is a complicated set of interrelationships and activities has all the characteristics of a complex system. Typically, in these systems, any decision leads to a course of action that changes the state of the system and gives rise to new information on which future decisions are based.

Application of system analysis to a learning process requires the definition of the structure of interacting functions. The definition of the structure must identify not only the separate functions but also their methods of interconnection. According to the Theory of Systems Structure, the four conceptual hierarchies are the closed boundary, components of the system especially stock level and flow rate variables, feedback loops, and policy structure.

The closed boundary defines the higher layer of the model. In fact, it is the control system of our interest as discussed earlier. In this study, the boundary encloses a single system for a single student learning process. Interaction between this system and other sub-systems in a learning environment is simplified at this stage.. The model structure developed is basically includes a main center-part for a learning process and some arbitrarily supporting infra-structures inside the defined boundary. Parts of the infra-structure represent sub-models and interact with the center-part.

The next hierarchy of system structure are the components of the system. There are four basic components or building blocks in the system: the stocks, the flows, the converters, and the connectors. Stock levels and flow rates relate to the accumulations and activities within the system. Stocks can be referred to as system state variables. They are integrations or accumulations of system flows that represent measurements of the state of the system at any given point in time. Flows are the instantaneous rates of flows that represent the means by which the system is controlled and represent activity points in the system. Converters are auxiliary functions converting system states to system activities. They represent the decision processes in the system. Finally, the connectors are links that connect the components forming arc that influence the flows that regulate the system. [6]

Feedback loops represent the structural setting within which all decisions are made. It is any structure of two or more causally related components that close back on themselves. Thus, the feedback loops provide a format for identifying flows of information and the relevant variables. They articulate system circularity giving rise to cause and effect. For example information about student achievement can provide an input to decisions concerning degree of student comfort, which in turn, controls the demand of student's effort. Any system which has a purpose has an internal structure of feedback loops through which the system is controlled. Entire feedback loops, as well as the individual relationships within a loop, are described as either positive or negative. When any variable in a positive loop changes, the resulting interactions cause that variable to change further in the same direction. The positive loop, in other words, characteristically produces self-reinforcing change (unrestrained growth). By contrast, when any variable in a negative loop is changed, then the loop causes that variable to readjust in the opposite direction. The negative loop produces self-regulating change (controlling and restorative behavior).

Implicit in rate equations, therefore, are the actions and policies which reflect the administration of the learning process. For example, the learning rate equation which controls the amount learned by a student reflects the policy of the student regarding the standards required for acquiring knowledge. Thus, the last hierarchy in a general systems structure can be defined as policy structure. Decisions are made for a purpose which, in turn, implies a goal which the decision process is trying to achieve. Policy structure is mainly reflected in the definition of the rate variables although there exist other policy variables with their determining impact on the behavior of the system as well.

DESCRIPTION OF THE BASE MODEL

The system flow diagram for the Base Model of a learning process is as shown in Fig 3 (System Flow Diagram for a Learning Process). The diagram has been constructed by iThink/Stella II simulation language [8]. The definition of each variable has been given in the List of Equations (Appendix). Note that the Stella equations created from Fig.3 in the List of Equations are two types. The first type are stock level equations (which are generated by the software directly from the diagram) and their associated initial conditions. The remainder are converter equations that are generated by the modeler. In this section we will describe the role of each variable and show the detailed nature of the interactions within the entire system.

Referring to the diagram, the main non-conserved system is demonstrated at the center of the diagram by the knowledge stock of the Amount-Learned. The cloud on the left hand side of the flow of Learning depicts the boundary of the model. It represents an infinite source for Learning flow, as shown. (For the purpose of our model, we do not care about what is in the cloud.)

In the present model, Learning is treated as student effort-based. Learning is generated by a student working a certain number of hours per day, and giving rise to a certain quantity of output. On the other hand, Learning flow is sending knowledge from the "student” and placing it in the stock of the Amount-Learned. The flow of learning by a student is defined as the product of his or her Learning-Drivers, Study-Hours, and Productivity. Learning-Drivers consist of five primary drivers that are necessary for each student in a learning process. They are: Information-Preparation, which is a conversion of his or her Total Amount Learned (conversion from hour-based dimension to the percentage-based dimension), Student-Abilities-&-Aptitude; Course-Structure-&-Content; Nature-and-Quality-of-Instruction; and finally Motivation. All of these drivers have their own components defined in separate sub-models (not worked out here for the sake of model simplicity). Worth mentioning is that Course-Structure-&-Content and Nature-&-Quality-of-Instruction represent the role of administrators and faculty in the model respectively.

The stocks of Study-Hours and Motivation allow these parts of the system to have initial values. These stocks change in value according to the amount they receive or lose, (since theirs bi-directional flows can get both positive and negative values). Both stocks with their related bi-flows are parts of the non-conserved system, as defined earlier.

FEEDBACK MECHANISMS

Several feedback mechanisms are included in the model. Two of these loops have a major effect on the resulting behavior of the system. The first mechanism is acts along the connector running from Amount-Learned to student Achievement. This linkage closes a feedback loop in which as Amount-Learned increases, the student will achieve more and have a higher test result (Assessment). This leads to more Comfort and subsequently, less Demand for further (instantaneous) learning. A lower Demand will decrease the student's Running-Target that has a direct influence on the Study-Hours. In fact, the connector running from Amount-Learned to Frac-Chg-in-Study-Hrs reinforces this feedback loop so that as Amount-Learned falls below Running-Target, additional hours are added, and as excess Amount-Learned builds up, hours are cut back. The adjustment of hours is one of the feedback mechanisms included in the model for responding to changes in Demand. However, a reduction in Running-Target will send a negative signal to Change-in-Study-Hrs and causes a subsequent reduction in Study-Hours. This, in turn, leads to a decrease in Learning flow and finally in the Amount-Learned. Thus, we started with an increase in the Amount Learned but feedback to Amount Learned makes it decrease. This phenomenon is the characteristic of a negative feedback loop that tries to restore the process and maintain the goal of the system.

The second mechanism acts in parallel to the first one, keeping the same track but diverts from Comfort to Motivation. Thus, an increase in Comfort reinforces the Indicated-Motivation of the student. The result is a higher Chg-in-Motivation that gives rise to more Motivation. A higher Motivation, subsequently, increases the Impact-of-Motivation on the student Productivity. This, in turn leads to an increase in the flow of Learning and a subsequent increase in the Amount-Learned. Summing up, we started with an increase in the Amount-Learned and the feedback to Amount-Learned makes it increase faster. This phenomenon is typical of a positive feedback loop that tries to give an unrestrained growth to the Amount-Learned by the student. In summary, we observe, the overall behavior of the model is the result of the interaction between these two feedback loops. A negative feedback loop that acts in the Achievement side and a positive feedback loop that acts in the Motivation side.

Two other feedback loops are worth mentioning. One acts as a second negative feedback loop on the Achievement side of the model, while the other acts as a second positive feedback loop on the Motivation side. Note that an increase in Amount-Learned by the student, concurrently, increases his or her Total-Amount-Learned. This, in turn, decreases the Demand (bottom of diagram) for more learning. The result is a reinforcement in the negative impact on Demand, that is already established by the Comfort. In the second loop, increase in Motivation has a positive effect on the student Learning-Drivers. This, strengthens the rate of Learning and the Amount-Learned respectively.

In general, both negative feedback loops act on the Achievement side and due to their controlling effect, seek a goal maintaining pattern. They try to restore the Amount-Learned by the student to an overall target that is imposed by existing limited time for study (length of the academic term). In the meantime, both positive feedback loops act along the Motivation side exhibit a typical pattern of behavior of growth. Consequently, the overall behavior of the learning process is the combination of these two general patterns of behaviors.

BEHAVIOR OF THE SYSTEM

Referring to Fig.4 (Diagram of Base Run for a Learning Process System), and List of Equations, it can be seen that the simulation starts with an initial knowledge stock of the Amount-Learned of 0 hours, a Prior-Knowledge of 0 about the subject matter, an initial Study-Hours of 0.5 and an initial Motivation of 10 percent. The other initialization values are as defined and assumed in the List of Equations (Appendix). The time horizon for the model is assumed to be as 120 nominal days, that is; the length of a regular academic term.

As the learning proceeds, more knowledge moves to the stock of Amount-Learned. The maximum value of both the flow of Learning and Amount-Learned happen at the end of the simulation period. The maximum Motivation of the assumed student at any one time is approximately 83 percent and occurs on day 98. The maximum Assessment is about 90 out of 100 and occurs on day 92. This causes a direct effect on the student Demand. That is; due to the attained high Assessment, the resulting comfort tends to flatten the Demand (of course, with a couple of days time-lag). The result is a subsequent collapse in the student testing result (Assessment). In the result, the final Assessment remains constant at about 72 out of 100 for the last days of the term.

The behavior of any other variable can be simulated and tracked on the similar graphs. Worth mentioning, is the behavior of the assumed student at the second half of the term. The weak result of the Assessment at midterm gives rise to his or her high Demand for more studying and learning. This, in turn, speeds up both the flow of learning and the accumulation of Amount-Learned. (Note that the slope of the curves that are much greater in the second half in comparison to the first half of the term.). All the assumed initial values for the different variables in the model are based on the data available for an average student. They can be changed and set differently according to the students individual traits.

What is it that is learned? How we treat it in the present model?

Does a student learn specific responses and or patterns of behavior or does he or she learn to connect or associate responses to differential situations? In a learning process does a student acquire habit and skill or cognitive structure, in the form of expectancies, schemata, or images?[3] These are the typical questions usually raised in Educational terms about what that is learned in a learning process. But as far as an answer to these questions concerned, the present model seeks an entirely different but sensible response. The answer can be explained, based on a logic that governs a learning process. This logic is based on the dimensional compatibility of the featured elements in a learning process. Basically, what is learned by a student can be considered as an accumulation of his or her productive hours of learning per unit time. Mathematically, this means the algebraic sum of the flow of learning of any subject during an assumed period of time, say 120 days or the length of a semester. Flow of learning, in the meantime, is the product of student Productivity, student Learning-Drivers, and Study-Hours used in the process of learning. These statements can simply be expressed by the following equations:

Flow of Learning=[Student Learning-Drivers] *[Student Productivity]*[Study-Hours]

Amount Learned = [Flow of Learning]

Thus, the flow of learning is continuous and places continuously knowledge into Amount-Learned stock. Student Learning-Drivers is the arithmetic mean of five Learning-Drivers as defined in the model. Eventually, the Amount-Learned by the student will be added to his or her Previous-Knowledge and will result in the Total-Amount-Learned.

Motivation

Many experts in the area of teaching and learning believe that in a learning process, greater value should be placed on the student motivation. This is what the author has taken care of in building this model. The stock of Motivation in the present model is, in fact, the sum of student intrinsic and extrinsic motivation at any time. The initial value for Motivation is assumed to be what its sub-model dictates. This provision will work effectively when Motivation is defined as a sub-model in later stages of this work. In the present model, Motivation is defined as a stock level that takes an arbitrary initial value as a starting point. More elaboration on the behavior of Motivation will be given in the next section.

ASSUMPTIONS

As mentioned earlier, this model is the first attempt in a chain of models that will evolve from this study. The structure of the present model provides a principal backbone for future models that necessarily will have more complicated components and linkages. However, while the present model includes all the major actors of a learning process, it is founded on a few assumptions to maintain its simplicity at this stage.

First, three Learning-Drivers namely, Student-Abilities-&-Aptitude, Nature-&-Quality-of-Instruction, and Course-Structure-&-Content have been introduced as single variables. Apparently, each of these Drivers is a complex variable that demands to be defined in a separate sub-model with its characteristics' constituents. Moreover, some components of these variables have reciprocatory inter-relationships on each other.

Second, all of these Drivers are assumed to remain constant during the time of the study (one academic term). This assumption is not so far from reality. Considering their negligible change in short-term, these Drivers actually can not vary much to any extent during a four month period. Anyhow, as mentioned above, this assumption will be set aside later when each of these Drivers are defined as sub-models.

Third, Motivation, despite its complexity, also has been represented by a single entity. This major Learning-Driver, similarly, should be demonstrated in its own sub-model with the effects of its internal and external components. To reduce the weight of this inadequacy in present model, Motivation is defined as a function and represented by a stock level that changes due time. Indicated-Motivation reflects the impact of student Comfort which is an internal effect. Adjustment, on the other hand, reflects the external effect of other Learning Drivers. Such assumption on Motivation, fortunately, holds enough room for future expansion around it.

Fourth, interactions between Learning-Drivers themselves have not been shown in this model. For instance, the effect of Course-Structure-&-Content on student Motivation or vice versa has not been worked out. These linkages will be defined later by using the results of statistical analysis in this area. By the way, the effect of these interactions in short term (length of one term) is again minor and hence can be neglected for present study.

Fifth, "knowledge” is the sole content of the Learning flow and the stock of Amount-Learned. Worth mentioning that Bloom's Taxonomy has been used as a fundamental concept for modeling the content flow of Learning and the Amount-Learned. This assumption bounds the nature of the learning objective to the lowest level in the cognitive domain; that is, acquiring knowledge or sole memorizing. Clearly, this assumption is not much inconsistent with the type of knowledge that a first year college or university student does acquire. However, this reservation will be set aside in the later models when higher intellectual abilities and learning skills will be shown as well.

Sixth and last, some other miscellaneous assumptions that have been made include but not limited to:

EXPERIMENTAL RUNS

Two experiments were carried out with the simulation model to gauge the effects of Prior-Knowledge and Nature-&-Quality-of-Instruction as policy variables on the behavior of the system. Both of these variables have been chosen intentionally. Prior Knowledge represents one of the student trait variables in the model while Nature-&-Quality-of-Instruction is as the sole representative for the faculty trait variables.

Experiment # 1: A sensitivity analysis was made of the student Prior-Knowledge for 0, 15 and 30 units to gauge its effect on the flow rate of his or her learning (Fig.5 - Sensitivity of Flow of Learning to Changes in the Student Prior-Knowledge).

Experiment # 2: A sensitivity analysis was made of Nature-&-Quality-of-Instruction for 50, 75, and 100 units to gauge its effect on the Amount-Learned by the student (Fig.6 - Sensitivity of Amount-Learned to Changes in Nature-&-Quality-of-Instruction).

The effect of other policy variables like Motivation, Study Hours, Course-Structure-&-Content, and Student-Abilities-&-Aptitude can be investigated as well. However, for the purpose of this study, the results of the above experiments seem reasonable and sufficient.

RESULTS OF EXPERIMENTS

To examine the results of these two experiments, the data at three points of our interest (Day 60, 90, and 120) were extracted and tabulated as shown in Table 2 (Results of Sensitivity Analysis). The similar result of the Base Run is also included, so then the change in "performance" would be clearer. Flow of Learning and Amount-Learned by the student are taken as the measures of "performance." These look like reasonable choices as everything runs on rate of Learning, and Learning itself is the basis for the Amount-Learned or the knowledge acquired by the student.

The result of Experiment #1 indicate that, the greater the student Prior-Knowledge about the subject, the higher the rate of his or her Learning would be. The difference in the amount of Learning is more evident when the student speeds up his or her effort during the second half of the academic term. Also, comparing the values of Learning at Day 60, 90, and 120, imply another finding. The increasing rate of the student Learning on a specific subject slows down as his or her Prior-Knowledge increases further. Practically, This seems reasonable; because, the influence of Prior-Knowledge on the rate of Learning may have an optimum value for each specific subject.

On the other hand, according to the results of Experiment # 2, any improvement in Nature-&-Quality-of-Instruction has a considerable effect on the Amount-Learned by the student. Interestingly, in Run 2, a 50% increase in Nature-&-Quality-of-Instruction results in a 45% increase on the Amount-Learned by the student at Day 90 and a 33% increase at Day 120. Also, in Run 3, a further 33% increase in Nature-&-Quality-of-Instruction results in an additional 30% increase at Day 90 and about 32% increase at Day 120 on their respective previous level of Amount-Learned.

The two above experiments demonstrate the strength of our approach in predicting changes in behavior of the system due to using different policy actions. The important characteristic of the methodology used here is its power to show the insight of the system. Note that the obtained results are valid only for the particular student under the conditions and limitations defined in the boundary of the system. Apparently, each individual student has his or her particular traits that in similar situations may give or not give rise to identical pattern of behavior.

Influence of a combination of two or more policies on the behavior of the system can be examined as well. For instance, in this model, a student at a Prior-Knowledge of 15 interacting with a Nature-&-Quality-of-Instruction of 75% can be taken as an alternative option. In general, by comparing the resulting behavior of the system under different options, the most appropriate policy or course of action can be identified. This step, eventually, directs us to a re-engineered structure for the system.

CONCLUSION

The modeling effort made on a learning process in this study is a unique combination of Educational metrics and Engineering simulation programs. On one side, the work consists largely of inferences drawn from available Educational experience and viewpoints with an absence of a defensible, universal mechanism. On the other side, it heavily relies on a series of activities drawn from a methodology of system analysis to build a solid Engineering framework for the reinforcement and improvement of the process.

Since modeling is an emerging process, any "model” represents only one of a sequence of models, that provide insight to the situation and form a basis for continued evolution. The model worked on in this study is presented in this spirit. This model is to be viewed as a vehicle that can be used to identify the important variables and necessary interactions for implementing policies and tracing the resulting behavior of a learning process. It is an initial attempt for the other stages of this study that will come later. The present model is useful in developing an understanding of a total system of teaching and learning. The ultimate model, thus, will provide a general model for the development of an "improved system of higher education, a model that will support students, faculty, and administrators in maintaining a quality education in the era of "inevitable changes.”

REFERENCES

  1. Barbara Schneider Fuhrmann and Anthony F. Grasha,"The Past, Present, and Future in College Teaching: Where Does Your Teaching Fit?” from Teaching and Learning in the College Classroom edited by Kenneth A. Feldman and Michael B. Paulsen, ASHE Reader Series, Ginn Press, MA (1994).
  2. Barbra Schneider Fuhrmann & Anthony F. Grasha,"Toward a Definition of Effective Teaching,” from Teaching and Learning in the College Classroom edited by Kenneth A. Feldman and Michael B. Paulsen, ASHE Reader Series, Ginn Press, MA (1994).
  3. Cameron Fincher, "Learning Theory and Research,” from Teaching and Learning in the College Classroom edited by Kenneth A. Feldman and Michael B. Paulsen, ASHE Reader Series, Ginn Press, MA (1994).
  4. "Bloom's Taxonomy: A Forty-year Retrospective,” Ninety-third Yearbook of the National Society for the Study of Education (NSSE), Edited by Lorin W. Anderson and Lauren A. Sosniak, Distributed by The University of Chicago Press, Chicago, IL (1994)
  5. Wilbert J. McKeachie,"Research on College Teaching: The Historical Background,” Journal of Educational Psychology, Vol.82, No.2, 1990, ERIC Document Reproduction Serivce No. ED043789
  6. Barry Richmond et al., Introduction to System Thinking and iThink, High Performance Systems Inc. Hanover NH (1992).
  7. Eric F. Wolstenholme, System Enquiry: A System Dynamics Approach, John Wiley & Sons Ltd. Chichester, England (1990).
  8. Steve Paterson and Barry Richmond, Stella II Technical Documentation, High Performance Systems Inc. Hanover NH (1994).

APPENDIX

List of Equations

Amount_Learned(t) = Amount_Learned(t - dt) + (Learning) * dt

INIT Amount_Learned [1*H]= 0

Learning [1*H/D] = Study_Hours*Learning_Drivers*Productivity

Motivation(t) = Motivation(t - dt) + (Chg_in_Motivation) * dt

INIT Motivation [%] = 10

Chg_in_Motivation = (Indicated_Motivation-Motivation)/Adjustment

Study_Hours(t) = Study_Hours(t - dt) + (Chg_in_Study_Hrs) * dt

INIT Study_Hours [H] = .5

Chg_in_Study_Hrs [H/D] = Study_Hours*Frac_Chg_in_Study_Hrs

Achievement [H]= Amount_Learned*Productivity

Adjustment [1] = IF(Learning_Drivers<=.5) THEN(5) ELSE 6

Comfort [%] = SMTH1(Assessment,Delay_in_Comfort) where SMTH1 is a first order smooth (delay) function.

Course_Structue_&_Content [%] = 50

Delay_in_Comfort [D] = 2

Exam_Cram [!] = (1+STEP(.3,55)-STEP(.3,60))+((STEP(.5,110)-STEP(.5,120))) where 0.3 and 0.5 respectively are 30%

and 50% increase in demand for study at day 55 (lasting 5 days) and day 110 (lasting 10 days).

Learning_Drivers[1]= (Information_Preparation+Nature_&_Qulaity_of_Instruction+Course_Structue_&_Content+Student_Abilities_&

_Aptitude+Motivation)/500

Maximum_Productivity [1] = .8

Nature_&_Qulaity_of_Instruction [%] = 50

Prior_Knowledge [H] = 0

Productivity [1] = Maximum_Productivity*Impact_of_Motivation

Running_Target [H] = (IF(Demand<50)THEN(60)ELSE(Demand))*Exam_Cram

Student_Abilities_&_Aptitude [%] = 50

Total_Amount_Learned [H] = Prior_Knowledge+Amount_Learned

Assessment [%] = GRAPH(Achievement)

(0.00, 21.0), (1.00, 29.0), (2.00, 51.5), (3.00, 71.0), (4.00, 81.0), (5.00, 89.5), (6.00, 90.5), (7.00, 89.5), (8.00, 85.5), (9.00, 72.5), (10.0, 72.5)

Demand [H] = GRAPH((Comfort*Total_Amount_Learned)/100)

(0.00, 30.0), (10.0, 35.0), (20.0, 35.0), (30.0, 40.0), (40.0, 40.0), (50.0, 45.0), (60.0, 50.0), (70.0, 55.0), (80.0, 60.0), (90.0, 65.0), (100, 70.0)

Frac_Chg_in_Study_Hrs [1] = GRAPH(Amount_Learned/Running_Target)

(0.00, 0.00), (0.1, 0.029), (0.2, 0.01), (0.3, 0.04), (0.4, 0.0195), (0.5, 0.055), (0.6, 0.0295), (0.7, 0.069), (0.8, 0.039), (0.9, 0.086), (1, 0.086)

Impact_of_Motivation [!] = GRAPH(Motivation)

(0.00, 0.1), (10.0, 0.4), (20.0, 0.45), (30.0, 0.5), (40.0, 0.55), (50.0, 0.6), (60.0, 0.65), (70.0, 0.7), (80.0, 0.8), (90.0, 0.9), (100, 1.00)

Indicated_Motivation [%] = GRAPH(Comfort)

(0.00, 0.00), (10.0, 1.00), (20.0, 6.50), (30.0, 17.5), (40.0, 25.5), (50.0, 33.5), (60.0, 47.5), (70.0, 68.5), (80.0, 79.5), (90.0, 88.0), (100, 99.5)

Information_Preparation [%] = GRAPH(Total_Amount_Learned)

(0.00, 0.00), (10.0, 10.0), (20.0, 20.0), (30.0, 30.0), (40.0, 40.0), (50.0, 50.0), (60.0, 60.0), (70.0, 70.0), (80.0, 80.0), (90.0, 90.0), (100, 100)