Markov processes in learning theory

Consideration is given mathematical problems arising in two learning theories—one developed by Bush and Mosteller, the other developed by Estes. The theory of Bush and Mosteller leads to a class of Markov processes which have been studied in considerable detail (see [1] and [7]). The Estes model can be treated as a Markov chain, i.e., a Markov process with a finite number of states. For an important class of special cases, it is shown that the Bush-Mosteller model is, in a sense, a limiting form of the Estes model. The limiting probability distributions are derived for the cases treated in both models.This research was supported by the National Science Foundation through a grant given to the Dartmouth Mathematics Project.     

Symmetric limiting distributions of response probabilities

Necessary and sufficient conditions for the symmetry of the limiting distributions of response probabilities are derived for the Bush and Mosteller two experimenter-controlled events learning model [Bush and Mosteller, 1955].     

A sequence of limiting distributions of response probabilities

Explicit solutions are obtained for a sequence of limiting distributions of response probabilities for the two experimenter-controlled events learning model of Bush and Mosteller [2]. A generalization to thes experimenter-controlled events model is found.     

Elemental representations of stimuli in associative learning

This article reviews evidence and theories concerning the nature of stimulus representations in Pavlovian conditioning. It focuses on the elemental approach developed in stimulus sampling theory (R. C. Atkinson & W. K. Estes, 1963; R. R. Bush & F. Mosteller, 1951b) and extended by I. P. L. McLaren and N. J. Mackintosh (2000, 2002) and contrasts this with models that invoke notions of configural representations that uniquely code for different patterns of stimulus inputs (e.g., J. M. Pearce, 1987, 1994; R. A. Rescorla & A. R. Wagner, 1972; A. R. Wagner & S. E. Brandon, 2001). The article then presents a new elemental model that emphasizes interactions between stimulus elements. This model is shown to explain a range of behavioral findings, including those (e.g., negative patterning and biconditional discriminations) traditionally thought to be beyond the explanatory capabilities of elemental models. Moreover, the model offers a ready explanation for recent findings reported by R. A. Rescorla (2000, 2001, 2002b) concerning the way that stimuli with different conditioning histories acquire associative strength when conditioned in compound.     

Proverbs with Solomon: A critical revision of the pre-critical commentary tradition in the light of a biblical intertextual study

The historical criticism of the Book of Proverbs has substituted the pre-Enlightenment view that Solomon was the real author with the finding that Israel's post-exilic sages added the name and prestige of the wisest of kings to their work. However the pre-Enlightenment commentators of Proverbs recognised that the name Solomon is integral to the text of Proverbs. This article recognises this textual datum and reads Prov 1–9 from an unusual angle today, namely as if Solomon were the author and principal speaker. In such a reading of Prov 1–9 the portrait of Solomon emerging from within the text is sharpened by a comparison with the depiction of Solomon in 1 Kgs 1–11.
The article seeks to make the point that the character of Solomon so construed in a textual comparison of Prov 1–9 and 1 Kgs 1–11 is but one of a number of diverse portraits made of Solomon in Proverbs throughout the centuries of commentary. In the first part of the article we study three early commentaries, those of Bede, of Hugo Grotius, and of Augustin Calmet. We show that the portrait of Solomon discovered in our own reading of Prov 1–9 in the second part of the article takes its place alongside numerous others. However we today are able to be more faithful to the biblical text than the early commentators because we need not harmonise Proverbs with 1 Kings, as the early commentators felt obliged to do. A contemporary intertextual study of Prov 1–9 can recoup a valid aspect of the ancient literary tradition, that of making Solomon integral to Proverbs. The Solomon we have discovered in Prov 1–9 is the one that spoke through Proverbs at its first publication. In Prov 1–9 the royal attribute is set aside. Solomon is represented as an Israelite teaching parent.     

A Developing Approach to Studying Students’ Learning through Their Mathematical Activity

We discuss an emerging program of research on a particular aspect of mathematics learning, students’ learning through their own mathematical activity as they engage in particular mathematical tasks. Prior research in mathematics education has characterized learning trajectories of students by specifying a series of conceptual steps through which students pass in the context of particular instructional approaches or learning environments. Generally missing from the literature is research that examines the process by which students progress from one of these conceptual steps to a subsequent one. We provide a conceptualization of a program of research designed to elucidate students’ learning processes and describe an emerging methodology for this work. We present data and analysis from an initial teaching experiment that illustrates the methodology and demonstrates the learning that can be fostered using the approach, the data that can be generated, and the analyses that can be done. The approach involves the use of a carefully designed sequence of mathematical tasks intended to promote particular activity that is expected to result in a new concept. Through analysis of students’ activity in the context of the task sequence, accounts of students’ learning processes are developed. Ultimately a large set of such accounts would allow for a cross-account analysis aimed at articulating mechanisms of learning.     

Task analysis in curriculum design: a hierarchically sequenced introductory mathematics curriculum

A method of systematic task analysis is applied to the problem of designing a sequence of learning objectives that will provide an optimal match for the child's natural sequence of acquisition of mathematical skills and concepts. The authors begin by proposing an operational definition of the number concept in the form of a set of behaviors which, taken together, permit the inference that the child has an abstract concept of "number". These are the "objectives" of the curriculum. Each behavior in the defining set is then subjected to an analysis that identifies hypothesized components of skilled performance and prerequisites for learning these components. On the basis of these analyses, specific sequences of learning objectives are proposed. The proposed sequences are hypothesized to be those that will best facilitate learning, by maximizing transfer from earlier to later objectives. Relevant literature on early learning and cognitive development is considered in conjunction with the analyses and the resulting sequences. The paper concludes with a discussion of the ways in which the curriculum can be implemented and studied in schools. Examples of data on individual children are presented, and the use of such data for improving the curriculum itself, as well as for examining the effects of other treatment variables, is considered.     

Distributing mathematical practice of third and seventh graders: Applicability of the spacing effect in the classroom

We examined the effect of distributed practice on the mathematical performance of third and seventh graders (N = 213) in school. Students first received an introduction to a mathematical topic, derived from their curriculum. Thereafter, they practiced in one of two conditions. In the massed condition, they worked on three practice sets in 1 day. In the distributed condition, they worked on one practice set per day for 3 consecutive days. Bayesian analyses of the performance in two follow‐up tests 1 and 6 weeks after the last practice set revealed a positive effect of distributed practice as compared with massed practice in Grade 7. In Grade 3, a positive effect of distributed practice was supported by the data only in the test 1 week after the last practice set. The results suggest that distributed practice is a powerful learning tool for both elementary and secondary school students in the classroom.     

初中生数学学习策略的可控心理影响机制

运用问卷调查和访谈的方法,探讨了主要可控心理因素对初中生数学学习策略的影响机制。结果表明,所纳入的可控心理因素具有良好的代表性;可控心理因素对初中生数学学习策略的影响机制主要体现为数学学习动机、策略意识、积极归因和困难应对感对数学学习策略有显著的直接效应。研究提示,数学学习策略的教学应重视可控心理因素的作用并处理好各因素之间的关系。     

Stimulus set size and statistical coverage of the grammar in artificial grammar learning

Adults and children acquire knowledge of the structure of their environment on the basis of repeated exposure to samples of structured stimuli. In the study of inductive learning, a straightforward issue is how much sample information is needed to learn the structure. The present study distinguishes between two measures for the amount of information in the sample: set size and the extent to which the set of exemplars statistically covers the underlying structure. In an artificial grammar learning experiment, learning was affected by the sample’s statistical coverage of the grammar, but not by its mere size. Our result suggests an alternative explanation of the set size effects on learning found in previous studies (McAndrews & Moscovitch, 1985; Meulemans & Van der Linden, 1997), because, as we argue, set size was confounded with statistical coverage in these studies. nt]mis|This research was supported by a grant from the Netherlands Organization for Scientific Research. We thank Jarry Porsius for his help with the data analyses.     

Cognitive skills in mathematical problem solving in Grade 3

Background. Research on the relationship between cognitive skills and mathematical problem solving is usually conducted on adults or on participants with acquired deficits associated with brain injury (e.g. Cipolotti, 1995 ; Cohen, Dehaene, & Verstichel, 1994 ; McCloskey, 1992 ). Aims. In these studies we wanted to make a contribution to the field of children's mathematical problem solving. The first aim of this study was to investigate whether mathematical problem solving in children is merely determined by semantic elaboration, as hypothesized in some of the models of adult processing (semantic hypothesis). In addition, we aimed to investigate whether there is a continuum from very good to very poor mathematical problem solving among children with mathematical learning disabilities showing immature cognitive skills (maturational lag hypothesis). Sample. The participants were 376 third graders and 107 second graders. Method. The internal structure of the data was analysed with a principal components analysis. In addition, two MANOVA were conducted to compare children with learning disabilities or problems with age‐matched and performance‐matched subjects. Results. Two components, a semantic and a non‐semantic one, were needed to account for an adequate fit of the dataset. In addition, children with mathematical learning disabilities had less‐developed cognitive skills compared with peers without learning disabilities, but they did not differ from younger children on seven of the nine cognitive skills. Conclusions. This study highlighted that children's mathematical problem solving is not determined by one general component. The picture is more complex, since two mathematics components were found. In addition, although our findings point in the direction of the maturational lag hypothesis it may be important to assess the different cognitive skills and especially assess the number system knowledge, since it seems below average in children with mathematical learning disabilities, compared with the knowledge of younger children with comparable skills in mathematics.     

The variance error of the P50-discriminant

The P₅₀-discriminant has been reported elsewhere in connection with its use in predicting whether selective service registrants if inducted would become normal operative soldiers or would commit offenses causing their imprisonment. The standard error of the P₅₀-discriminant is a good measure to use in determining how far to the side of this statistic a particular case falls. The standard error formula itself has also been published elsewhere; but its derivation, as the variance error, is given here.The author gratefully acknowledges the very extensive assistance kindly given to him by Dr. Truman L. Kelley and Dr. Frederick Mosteller. This assistance was given without reference to the utility of the P₅₀-discriminant, upon which matter the author reports elsewhere and for which he takes full responsibility.     

Learning and motivation: Computer simulations of the Rescorla-Wagner,Solomon-Corbit,and Schull models

A suite of programs is described for simulating the Rescorla-Wagner (1972) model of associative learning, the Solomon and Corbit (1974) opponent-process theory, and Schull’s (1979) extension of the latter. The programs are written for the Commodore 64 computer.     

A group-administered lag task as a measure of working memory

The purpose of the present studies was to evaluate the utility of a group-administered version of the n-back, or lag task. Experiments 1 and 2 describe the construction of the task and reveal that the modified lag task (MLT) produces the same performance trends as have been observed in individually administered versions of the lag task; performance decreased significantly as lag conditions increased in difficulty. Experiments 3 and 4 established convergent validity by comparing the MLT to another common working memory task, the operation-word span task, as well as the updated version of this task, the automatic operation span task. The results showed that MLT performance was significantly correlated to scores on both measures. These experiments provide important details about the MLT as a measure of working memory, in a group- or individual-administration setting.     

Stepping Beyond Message Specificity in the Study of Emotion as Mediator and Inter-Emotion Associations Across Attitude Objects: Fahrenheit 9/11, Anger,and Debate Superiority

This experimental research focuses on four distinct, but related theoretical interests regarding the study of emotion and media effects: (a) emotion serving as a potential mediator of the relationship between the consumption of multiple media messages, (b) media messages sparking emotional reactions to attitude objects not specifically referenced or identified in a message, (c) the assessment of inter-emotion relationships across distinct attitude objects, and (d) varied processes of emotion-based influence based on individual-difference attitude object preferences. This work focuses on Michael Moore's Fahrenheit 9/11, its effects on anger felt toward President George W. Bush and Senator John F. Kerry, and subsequent perceptions of candidate debate superiority across Bush versus Kerry supporters. Moore's film was found to influence levels of anger felt toward both Bush and Kerry and these emotions were related to subsequent perceptions of debate superiority. Distinct processes of influence were found for previously identified Bush supporters versus Kerry supporters, with Bush anger serving as a mediator for Bush supporters and Kerry anger serving as a mediator for Kerry supporters. Ramifications of this study's findings are outlined and future lines of research are summarized.     

Ultimate choice between two attractive goals: Predictions from a model

A mathematical model for two-choice behavior in situations where both choices are desirable is discussed. According to the model, one or the other choice is ultimately preferred, and a functional equation is given for the fraction of the population ultimately preferring a given choice. The solution depends upon the learning rates and upon the initial probabilities of the choices. Several techniques for approximating the solution of this functional equation are described. One of these leads to an explicit formula that gives good accuracy. This solution can be generalized to the two-armed bandit problem with partial reinforcement in each arm, or the equivalent T-maze problem. Another suggests good ways to program the calculations for a high-speed computer.Support for this research has been received from the National Science Foundation (Grant NSF-G2258), the National Institute of Mental Health (Grant M-2293), and the Laboratory of Social Relations, Harvard University.We wish to acknowledge and express our appreciation for the cooperation and assistance given by Phillip J. Rulon, Albert Beaton, Wai-Ching Ho, and Donald Spearritt, who set up, programmed, and executed numerous calculations connected with the linear equations method of solution, and by Cleo Youtz for extensive calculations at every stage of the work. We also wish to thank Ray Twery and Robert R. Bush for permission to use in Table 3 some of the unpublished results of their calculations. Those calculations were made on the Illiac through the cooperation of the Digital Computer Laboratory of the University of Illinois, Dr. John P. Nash, Director.     

The bimean: A measure of central tendency that accommodates outliers

Observations that depart considerably from the center of a distribution demand special consideration. They may be retained, trimmed, or weighted less than other data. This article provides a BASIC program that implements Mosteller and Tukey’s (1977) technique for weighting observations less as they depart from the middle of a distribution. Influence curves for this “bisquare-weighted mean,” or “bimean,” are displayed and compared with more traditional measures of central tendency.     

World Leaders As Movie Characters? Perceptions of George W. Bush,Tony Blair,Osama bin Laden,and Saddam Hussein

Abstract

This study tested the validity of a theoretical model to explain how viewers perceive real people shown frequently on T V. In particular, we wanted to test whether world leaders are perceived any differently than fictional characters. Participants evaluated media images of George W. Bush, Tony Blair, Osama bin Laden, and Saddam Hussein before and after the second Gulf War. The results demonstrated a good fit of the theoretical model to the data. World leaders were perceived equally realistic and less relevant than Hollywood's protagonists (e.g., Superman, Dracula). Viewers had a negative perception of world leaders. Bush was perceived as negatively as bin Laden and Hussein. With today's hybrid media, it is important to understand viewers’ perceptions of real as compared to fictional characters.     

Richard L. Solomon and learned helplessness

This article reviews the course of development of research on a currently popular explanatory approach to dysfunctional behavior, the learned helplessness analysis. The early history is prominent in this review as it reflects the inspirations of Richard L. Solomon, a scholar who fostered the resurgence of psychologists’ interests in Pavlovian conditioning in the 1950s and 1960s. Current research is characterized as having four separate themes: elaboration of “symptoms,” elucidating the role of fear, explicit modeling, and extensions involving attributional constructs.     

Science,politics, and computers

This address briefly traces post-1940 U.S. science policy from the science policy agenda set forth by Vannevar Bush in 1945 through the recent congressional task force on science policy. Although there are many factors influencing science policy, the focus here is on presidential influence. The effects of U.S. science policy on science are traced by looking at funding patterns in general, as well as at those for behavioral science. The influence of politics on the ability of scientists to freely communicate scientific information is examined. Finally, the role of politics in scientists’ access to computing resources is explored. Presented here is a side of science not often seen by those at the bench.