Mathematical Learning Theory Research Paper

1. The Earliest Precursor of Mathematical Learning Theory

The first small step toward mathematical learning theory occurred in the course of a massive experiment on the learning and retention of lists of artificial words (‘nonsense syllables’) conducted by the founder of the experimental study of memory, H. Ebbinghaus, during the period 1879–80. The learning of each list to a criterion was followed after an interval by a memory test. In plots of amount remembered (denoted b) vs. time between learning and testing (t), b declined over time in a reasonably orderly fashion but with some fluctuations. Ebbinghaus noted that, in view of the fluctuations, the empirical values of b could not reveal a law of retention, but he hypothesized that a smooth curve made to pass through the choppy empirical graph might reveal the underlying trend. In fact, a curve representing the function

 b =100k/k(logt)ᶜ (1)

where k and c are constants and logt is the logarithm of the time between learning and testing, proved to yield an excellent account of the retention data.

This finding must have been satisfying, but Ebbinghaus did not overemphasize what he had accomplished, ‘Of course this statement and the formula upon which it rests have no other value than that of a shorthand statement of the above results … Whether they possess a more general significance I cannot at the present time say’ (Ebbinghaus 1885/1964, p. 78).

2. The Mathematical Description of Learning Functions

2.1 Fitting the Learning Curve

Discovering how to demonstrate some general significance for mathematical functions fitted to empirical learning and retention curves was to be the central task of the next half century, during which the pace of relevant work gradually accelerated. A review by H. Gulliksen (1934) identified two studies in the decade 1901–10, followed by four in 1911–20, and 11 in 1921–30. in J. McGeoch’s influential text on the psychology of human learning, published in 1942, an extensive chapter on learning curves was conspicuously placed immediately following the opening chapter on concepts and methods.

The early successors of Ebbinghaus followed his example in choosing equations purely for their ability to describe the forms of empirical curves. More interesting were some instances in which investigators started with analogies between learning and other processes. For example, the first of Ebbinghaus’s successors in this strain of research, A. Schukarew adopted an exponential function that had appeared in chemistry as a descriptor of monomolecular chemical reactions. This effort led to no immediate sequel, but the approach was revived many years later with more fruitful results (Audley and Jonckhere 1956, Estes 1950, Thurstone 1930).

2.2 Seeking Rational Foundations for Equations of the Learning Curve

An important step toward providing a rational foundation for mathematical learning functions appeared in a study reported by L. L. Thurstone (1930). Thurstone’s advance in methodology was to go beyond defining parameters of learning functions by fiat and, rather, to derive their properties from an underlying process model. He conceived performance to be made up of a population of elementary acts, some of which led to success and some to failure in a task and assumed that acquisition of a skill is the outcome of a random process in which acts leading to error are discarded from the population and successful acts strengthened. Further, Thurstone went beyond the fitting of his model to mean learning curves and showed how it could be used to interpret situations in which the empirical curves exhibited systematic irregularities.

For example, research of W. Kohler and R. M. Yerkes during the 1920s on problem solving in the great apes had found that curves for acquisition of the ability to solve a problem typically started with a sometimes lengthy plateau (period of no improvement) at a chance level of responding followed by an abrupt transition to perfect performance, suggesting that the animal had suddenly acquired insight concerning problem solution. Whether this interpretation was justified was a live issue for some years thereafter. Analyzing the issue in terms of his model, Thurstone concluded that abrupt transitions should occur when a rate parameter in his learning function had a sufficiently high value or a limit parameter (assumed to be related to task complexity) had a sufficiently low value. Thus the occurrence of abrupt transitions in learning curves would be predicted to occur under quite different conditions as a consequence of the same basic learning process and need not be taken as evidence for a qualitatively different process of insight. Gulliksen (1934) continued the effort to rationalize learning functions with a forward-looking approach that emphasized the importance of treating data from individual learners, rather than only group averages, and the value of going beyond judgments of goodness of fit of equations to data and using estimates of parameters as bases for inferences about underlying processes.

3. The Transition from Learning Curves to Learning Theory

It might have seemed that by the time of the Gulliksen (1934) article, the pieces were all in place for a sharp acceleration of progress toward mathematical learning theory during the second half of the 1930s. In fact, however, the acceleration was delayed for more than 15 years, during which a continuing fixation on the learning curve was accompanied by a dearth of notable new developments.

A major factor contributing to this ‘plateau’ may have been the scantiness of communication between the psychometricians who were working on mathematization of learning curves and learning theorists who might have made use of their results. During the early 1930s, learning theory had abruptly emerged from obscurity and, under the leadership of C. L. Hull, E. R. Guthrie, B. F. Skinner, and E. C. Tolman and their followers, took center stage in the theoretical psychology of the period. With one exception, none of these theorists took quantitative theory as an objective, and some even argued that psychology was not ready for applications of mathematics. The exception was Hull, whose research group carried out a vigorous program of quantitatively oriented studies that set the stage for a dramatic broadening of mathematical approaches to learning phenomena.

Beyond stewarding this program, Hull almost singlehandedly defined the goals and the general character of a genuine mathematical learning theory. In an epoch-making monograph, Hull (1943) set forth a set of axioms for a general theory of conditioning and elementary learning, together with theorems presenting implications for many aspects of learning, including the way in which learned habits combine with motivation to determine performance, and the quantitative form of generalization from original learning situations to new test situations.

Hull’s formal system, the output of a relatively short though intense period of activity in his late years, was programmatic in many aspects and thus was both technically untestable and difficult to work with by anyone who did not share all of his intuitions. Consequently, continuation of Hull’s program for quantification of learning theory was left for a new generation of theorists who came to the field in the 1950s with fresh outlooks and better technical backgrounds for formal theory construction.

3.1 Mathematical Learning Theory in the Midtwentieth Century

With the stage set for accelerated progress by Hull’s heroic efforts, new approaches to mathematical learning theory literally exploded on a number of fronts in the early 1950s. The events of this period have been well portrayed by Bower and Hilgard (1981) and the continuation in the 1960s by Greeno and Bjork (1973), who opened their review with the remark, ‘In the decade or so that constitutes the period of our review, progress toward understanding the processes involved in learning can only be described in terms that sound extravagant’ (p. 81). Concurrently, the range of empirical targets for modeling expanded drastically and the mode of application shifted from fitting mean performance curves to predicting the almost innumerable statistics describing the fine structure of data (Atkinson et al. 1965). Among the models emerging from this wave of new activity, that of Bush and Mosteller (1955) and that of Estes (1950) and his associates (including R. C. Atkinson and P. Suppes) have had enduring roles in the development of mathematical learning theory. However, the two approaches (fully discussed by Coombs et al. 1970) differ in their basic assumptions about learning.

In Bush and Mosteller’s model, an organism’s state of learning in a task at any time was assumed to be characterized by a set of response probabilities; and the effect of the events of a learning trial (e.g., reward or punishment) was expressed by a simple difference equation (‘linear operator’ in their terms) giving the value of response probability at the end of a trial as a function of the value on the preceding trial.

The model of Estes (1950) included difference equations of the same type, but their forms were derived from an underlying theoretical process in which the acquisition of information during learning was represented by changes in the partitioning of an array of abstract ‘memory elements.’

The operator model has been attractive for the ease with which it can be applied to predict almost innumerable statistics of simple learning data, but the multilevel approach has become the norm for models of learning, most notably human concept and category learning, in cognitive psychology of the 1980s and 1990s.

3.2 Concluding Comment

As the twenty-first century starts, the trends that led to the appearance of a diversified array of ‘miniature’ models of different aspects of learning beginning in the 1950s have not given way to any consolidation into a single comprehensive model. However, significant lines of cumulative theoretical development are identifiable. A notable example is the embodiment of ideas drawn from Hull’s theory in the model of conditioning that has been the pace setter since the early 1970s (Rescorla and Wagner 1972) and recently has appeared as a constituent of new ‘adaptive network’ models that are flourishing in contemporary cognitive science.

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