Mathematical Learning Model Research Articles

Ingestion of environmentally contaminated Lake Ontario salmon by laboratory rats increases avoidance of unpredictable aversive nonreward and mild electric shock.

To determine what behavioral changes are caused by consumption of Lake Ontario salmon, a 30% diet of Lake Ontario or control Pacific Ocean salmon was fed to rats for 20 days. In Experiments 1 and 2 (preference-for-predictability E-maze test), rats fed Lake Ontario salmon developed a preference for predictable food rewards more quickly than did the control rats. In Experiments 3 (passive avoidance) and 4 (conditioned suppression), rats fed Lake Ontario salmon suppressed responding to food far more after the introduction of mild electric shocks than did control rats. All results supported the hypothesis that ingestion of Lake Ontario salmon, contaminated with polychlorinated biphenyls, mercury, lead, etc., increases the reactivity of rats to aversive events. The results were successfully simulated by DMOD, a mathematical model of learning, using the assumption that rats fed Lake Ontario salmon find unpredictable nonreward and mild shock more aversive.

L. L. Thurstone: A new assessment

Most psychologists have heard of Thurstone, and know him as a pioneer of mathematical psychology and factor analysis. But his appearances in history books are rare and peripheral, as a man of technical prowess rather than intellectual achievement, and at best he is seen as a link in the tradition culminating in signal detection and mathematical learning theory. Nineteen eighty‐seven is the centenary of his birth, so it is a timely moment to correct this picture. Thurstone was influenced by G. H. Mead, and his work is best placed within the pragmatist movement, not as moving towards the mechanistic psychology that now dominates psychology. Viewed in this way, his true status and the reasons for his relative neglect become apparent. This is illustrated by a detailed consideration of his mathematical learning models.

Applications of mathematical learning models to student performance on general chemistry: Microcomputer drill and practice programs

AbstractFour drill and practice programs in general chemistry (Butler & Hough, 1982) were altered to keep records of response, scores, and response times as students used the programs. Mathematical learning models were applied to the resulting score and time data. When data for groups of students using the same programs were analyzed using the polynomial fit model of Kerlinger and Pedhazur (1973), significant results were found and in most cases the linear fit provided the only significant explanation for the variance in the data. The performance of individual students on selected programs was also analyzed using a linear model, a simple exponential saturation model, and a more complex model which incorporated ability and motivation factors into the exponential saturation model (Aldridge, 1983). Most of the students improved their performance linearly with time and none exhibited patterns characteristic of the Aldridge model. In some cases, the simple exponential saturation model provided an explanation of the data as probable as that of the linear model. The implications of the findings for science teaching are discussed.

Superstitious Behavior in a Robot: A Mathematical Model and Computer Simulation Based on Adams' Theory of Motor Learning

A mathematical model and computer simulation of motor learning is presented. The model is based on Adams' theory of motor learning and incorporates detection of errors and correction which is fed back to modify input.

Examining a mathematical model of mastery learning in a classroom setting

AbstractThe purpose of mathematical models in any discipline is to describe accurately the relationships among significant variables of a system. The use of mathematical models is widespread in the sciences, but has rarely found its way into educational research. In developing a mathematical model for mastery learning, empirical research has shown that prior learning, motivation, and time on task are part of the significant variables that work together in some way in a determination of student achievement. A mathematical model that shows the relationship among these variables has been developed. To test the appropriateness of this mathematical model, carefully designed and controlled experiments must be conducted to collect numerical data on the significant variables. Using the model, the accuracy of its predictions can be compared with actual results. This is the technique used in testing mathematical models in the sciences and should be applicable to mathematical models of learning.

Observing response acquisition: Preference for unpredictable appetitive rewards obtained under conditions predicted by DMOD.

In five E-maze experiments, rats were given a choice between receiving reward and nonreward in a situation where stimuli were correlated with reward outcome (predictable situation) versus one where the stimuli were uncorrelated with reward outcome (unpredictable situation). Preference for the unpredictable situation occurred under the following conditions: (a) small (one 37-mg pellet), immediate rewards; (b) small, delayed (15 s) rewards, if the cues correlated with reward outcome were absent during the delay interval; (c) large (15 pellets), immediate rewards if a difficult discrimination was required; and (d) if the stimulus predicting nonreward was present at the choice point. Preference for the predictable situation was strongest if reinforcement was delayed and large or the stimulus predicting reward was present at the choice point. A weaker preference for the predictable situation occurred if reinforcement was immediate and large and a simple discrimination was required or if reinforcement was large and delayed and the cues that correlated with reward outcome were absent during the delay interval. The results support the predictions of DMOD (Daly modification of the Rescorla-Wagner model), a mathematical model of appetitive learning (Daly & Daly, 1982).

A SYSTEMIC MODEL OF BIOFEEDBACK LEARNING

Mathematical models of biofeedback learning provide empirical laws of these processes and could be used in order to improve experiments. The systemic approach provides a well‐adapted framework, in which learning is studied as a control problem. The model proposed here is a system of recurrence equations involving the successive measurements of the subject's physiological activity collected by the biofeedback device, and also the performance index presented to the subject. A parameter identification method is described and tested on a computer simulated process.

A mathematical model for mastery learning

AbstractPrior research on mastery learning has examined relationships between achievement and time on task, motivation variables, as well as the effects of prior learning. One can show that the results of prior research lead to certain requirements on a model for mastery learning. This article uses those requirements to develop a mathematical model for mastery learning. This model includes measures of a motivational factor and of the specific ability of a learner with respect to the content being learned. The model also contains a variable to reflect prior learning. The model has been developed to include a computer program which accepts student data and from which predictions can be made and tested. Assumptions and limitations of the model are discussed, and some research questions are raised. In most respects the model proposed in this article is susceptible to the same tests and refinements as an empirical law in the physical or life sciences.

A model for learning global properties

This paper reports developments in a mathematical model of cognitive learning. The model describes the processes of formation, testing, and revision of hypotheses held by a learner attempting to understand an environment. The fundamental assumption is that learning proceeds by a feedback cycle, where hypotheses are tested for validity against external reality and reweighted according to the outcome. A particular application of this model to the case of a mechanical “beetle”, mapping a geometrical environment, yields results of interest in artificial intelligence and robot design, including results on the computability of several geometrical predicates.

Three experiences affecting spatial discrimination learning by ambulatory children

Children 1 and 3 years of age were given a two-choice spatial discrimination task. They were placed in front of a barrier and encouraged to walk around it; one route around the barrier was blocked. Three experiments studied the effects of different types of experience with this spatial problem. The first examined trial and error experience, in which the child was allowed to walk the route he or she had chosen. Although most children readily learned to choose the open route, the pattern of choices indicated that 3-year-olds were more likely to achieve a learning criterion after an error than 1-year-olds. A second experiment looked at the effects of showing the children the layout of the problem prior to trial and error experience. The 3-year-olds were again more likely than the 1-year-olds to achieve criterion after an error; furthermore, they were likely to know the correct route on the first trial. The third experiment let the children watch their parents take the correct route. This procedure eliminated age differences in performance. The initial choices of the 1-year-olds indicated that they were likely to know the correct route, and if they did not, they were just as likely to learn from their mistakes as the 3-year-olds. The data were interpreted within a mathematical model of learning. Based upon these analyses, we propose two developmental trends. One involves a growth in sensitivity to the consequences of a choice of route. The other is a progression from social learning to more independent and ideational methods of spatial problem solving.

Developmental Invariance in a Mathematical Model of Associative Learning

Developmental Invariance in a Mathematical Model of Associative Learning

A MODEL OF LEARNING BASED ON FUZZY INFORMATION

Using fuzzy measures and fuzzy integrals, the paper presents a mathematical model of learning which is able to learn through fuzzy information. The characteristics of the model are studied theoretically and in numerical examples, where the model is compared with an ordinary Bayesian learning model. The problem of seeking an extremum of multimodel objective function is given as an example.

A state-space model for automatic instruction

This paper introduces a state-space instructional model, in which instruction is viewed as a process of transforming the student from one state to another. The model is expressed in terms of state-space problem solving as in artificial intelligence research, an area to which computer-assisted instruction has been increasingly turning for solutions to some of its fundamental problems. The model is first described in general terms and it is explained how the usual concerns of computer-assisted instruction (such as response-sensitivity and individualisation) find expression in the model. The use of the model in the design of teaching programs is illustrated by two systems, for paired-associate teaching and concept teaching. The former involves a re-expression of familiar methods based on the use of mathematical learning models; the latter uses more heuristic methods derived from artificial intelligence. Finally, some comparisons with other instructional models are made.

Models of Learning

This review centers on mathematical models of learning and their empirical tests. 2 Trends noted by Greeno & Bjork (77) have continued and accelerated: mathematical learning theory has become 'more cognitive in terminology, learning models have made even greater use of trees or networks to represent process and structure, and material learned in the laboratory has born increasing similarity, if not identity, to real world knowledge. Extension of these trends raises a question whether practice effects or tasks requiring extensive practice during acquisition are a central aspect of learning. Indeed it could be argued that one of the finest learning models avail­ able, that of Norman, Rumelhart, and the LNR groupl (140), discusses learning only peripherally as a side issue to the treatment of memory storage, question answering, and problem solving. Greeno & Bjork's (77) implicit definition of learning as the new storage of infor­ mation in memory or the modification of a structure of knowledge forms a basis for identifying studies to be cited here. The

A Three‐State Markov Model of Choice Behavior within Spatial Structures

AbstractThis paper derives a simple mathematical model of spatial learning and choice by integrating several psychological theories, whose features accord with the results of empirical investigation in a previous paper [2]. A sample of homogeneous decision‐makers are assumed to make a succession of selections from a given set of initially unknown spatial alternatives (for example, a set of shopping places for a good). The decision‐makers pass through two states of “recognition” and “discrimination” learning, before reaching an equilibrium state. In the equilibrium state, decision‐makers have sufficiently learned the attributes of the alternatives so that satisfactory choices are always made. These assumptions permit the derivation of mathematical expressions to answer two questions: first, what proportion of decision‐makers will be in the equilibrium state after any given time interval, and second, what proportion of decision‐makers will choose any designated spatial alternative after the same time interval. In the second case, the proportion of decision‐makers choosing an alternative is predicted from the individuals' perceptions of the attributes of the alternatives. The paper concludes with an evaluation of the model and suggestions for its testing and further development.

Reliability coefficients as a function of individual differences induced by a learning process assuming identical organisms

In this article we show that a particular mathematical learning model, the Bower-Trabasso (1964) concept identification model taken together with an assumption of independence of replicate measurements, implies the existence of substantial and statistically significant performance differences across individuals. The individual differences in turn imply a sizeable reliability coefficient. These results contradict naive intuition, for this model (like many other mathematical models of learning) assumes that all individuals begin the experiment with identical parameter values for the process under study. Thus at least one such model has the characteristic of implying the generation of individual performance differences among originally identical organisms. Examination of data from an experiment by Cotton shows that the Hoyt reliability coefficient under classical test theory, a lower bound for the (composite) reliability of total scores for a series of trials, increases with the number of trials analyzed and exceeds the corresponding theoretical values implied by the Bower-Trabasso model. An experiment by Levine was also analyzed because its use of blank trials between feedback trials permitted direct calculation of composite reliability (or more properly stated composite consistency). For this experiment, the theoretical development just discussed (Case I) was used together with Restle's hypothesis selection model specialized to include a local consistency assumption, the so-called P 2 model of Gregg and Simon (Case II). Moderate conformity of empirical and theoretical reliabilities was found, with discrepancies between observed and predicted values usually being smaller with Case II. However, the Hoyt reliability coefficient is not a lower bound for composite reliability in Case II, because composite reliability is underestimated when identical stimuli are not used for comparable trials. Despite the Bower-Trabasso assumption of no initial differences, it seems reasonable to attribute the difference between predicted and obtained reliabilities to preexisting individual differences. Implications of the tentative conclusion that individual differences in concept identification performance are attributable to a combination of preexisting differences and differences induced in a current task are discussed briefly.

A mathematical model of learning under schedules of interresponse time reinforcement

A model is proposed for free responding under schedules of interresponse time reinforcement. Each response generates a pattern of stimuli that changes in an orderly way over time. At any time the changing pattern may be sampled, and the state of conditioning of the sampled pattern determines whether or not a response actually occurs. The prediction of this model for the asymptotic distribution of interresponse times is derived. The predictions are tested against data from ratio, interval, and DRL schedules of reinforcement. The fit of the model is sufficiently good to make it worth-while continuing investigation of the model.

Mathematical learning models and neuronal networks

The concept of modifiable synapses in conjunction with a time continuous version of Caianiello's neuronic equation is proposed as a possible basis for mathematical learning models used in experimental psychology. The effects of a large number of memory cells and a large number of modifiable synapses on the time course of the learning system's response probabilities are studied and comparisons with the predictions of mathematical learning models are made. Possible extensions of the formalism are indicated.

Computer-based instruction in spelling

Two mathematical models of paired-associate learning (the One Element Model and the Single Operator Linear Model) were used to develop optimal strategies for presenting speeling words of 42 elementary school children. The study was conducted within a coputer-based drill and practice project using computer terminals with audio capability.For the study two computer programs were developed. One program presented and corrected a daily list of sixteen words for each student. The second program planned lessons for each student from the list of 48 words the student was studying. Words for a lesson were selected by the second program as a function of the mathematical model each word represented and the child's previous performance on each word.An analysis of the correct response made by each child through the study demonstrated that the strategy derived from the One Element Model produced significantly better final performance than the strategy developed from the Single Operator Linear Model.

A neural model for conditioned avoidance learning.

A mathematical model for learning of a conditioned avoidance behavior is presented. An identification of the net excitation of a neural model (Rashevsky, N., 1960.Mathematical Biophysics. Vol. II. New York: Dover Publications, Inc.) with the instantaneous probability of response is introduced and its usefulness in discussing block-trial learning performances in the conditioned avoidance situation is outlined for normal and brain-operated animals, using experimental data collected by the author. Later, the model is applied to consecutive trial learning and connection is made with the approach of H. D. Landahl (1964. “An Avoidance Learning Situation. A Neural Net Model.”Bull. Math. Biophysics,26, 83–89; and 1965, “A Neural Net Model for Escape Learning.”Bull. Math. Biophysics,27, Special Edition, 317–328) wherein lie further data with which the model can be compared.