Electrifying diagrams for learning: principles for complex representational systems

Six characteristics of effective representational systems for conceptual learning in complex domains have been identified. Such representations should: (1) integrate levels of abstraction; (2) combine globally homogeneous with locally heterogeneous representation of concepts; (3) integrate alternative perspectives of the domain; (4) support malleable manipulation of expressions; (5) possess compact procedures; and (6) have uniform procedures. The characteristics were discovered by analysing and evaluating a novel diagrammatic representation that has been invented to support students' comprehension of electricity—AVOW diagrams (Amps, Volts, Ohms, Watts). A task analysis is presented that demonstrates that problem solving using a conventional algebraic approach demands more effort than AVOW diagrams. In an experiment comparing two groups of learners using the alternative approaches, the group using AVOW diagrams learned more than the group using equations and were better able to solve complex transfer problems and questions involving multiple constraints. Analysis of verbal protocols and work scratchings showed that the AVOW diagram group, in contrast to the equations group, acquired a coherently organised network of concepts, learnt effective problem solving procedures, and experienced more positive learning events. The six principles of effective representations were proposed on the basis of these findings. AVOW diagrams are Law Encoding Diagrams, a general class of representations that have been shown to support learning in other scientific domains.     

New methods for solving the Rescorla-Wagner model

Two new methods for generating predictions from the Rescorla-Wagner model are presented. One is to solve the simultaneous equations of the model as differential equations, and the other is to solve them as difference equations, by using recent computer software. The model has been described in terms of simultaneous difference equations, and its predictions have traditionally been derived by performing an iterative computer simulation. But the limit can be taken, and the model can be considered in terms of simultaneous differential equations. Computer software, such as Mathematica and Maple, can solve simultaneous differential or difference equations. The author shows how to use Mathematica for some experimental paradigms, from simple acquisition to more complex cue competition paradigms, and also explains the rule for constructing input. The relative merits of these methods and of simulation are discussed.     

Are diagrams always helpful tools? Developmental and individual differences in the effect of presentation format on student problem solving

Background. High school and college students demonstrate a verbal, or textual, advantage whereby beginning algebra problems in story format are easier to solve than matched equations ( Koedinger & Nathan, 2004 ). Adding diagrams to the stories may further facilitate solution ( Hembree, 1992 ; Koedinger & Terao, 2002 ). However, diagrams may not be universally beneficial ( Ainsworth, 2006 ; Larkin & Simon, 1987 ). Aims. To identify developmental and individual differences in the use of diagrams, story, and equation representations in problem solving. When do diagrams begin to aid problem‐solving performance? Does the verbal advantage replicate for younger students? Sample. Three hundred and seventy‐three students (121 sixth, 117 seventh, 135 eighth grade) from an ethnically diverse middle school in the American Midwest participated in Experiment 1. In Experiment 2, 84 sixth graders who had participated in Experiment 1 were followed up in seventh and eighth grades. Method. In both experiments, students solved algebra problems in three matched presentation formats (equation, story, story + diagram). Results. The textual advantage was replicated for all groups. While diagrams enhance performance of older and higher ability students, younger and lower‐ability students do not benefit, and may even be hindered by a diagram's presence. Conclusions. The textual advantage is in place by sixth grade. Diagrams are not inherently helpful aids to student understanding and should be used cautiously in the middle school years, as students are developing competency for diagram comprehension during this time.     

Aristotelian diagrams for semantic and syntactic consequence

Several authors have recently studied Aristotelian diagrams for various metatheoretical notions from logic, such as tautology, satisfiability, and the Aristotelian relations themselves. However, all these metalogical Aristotelian diagrams focus on the semantic (model-theoretical) perspective on logical consequence, thus ignoring the complementary, and equally important, syntactic (proof-theoretical) perspective. In this paper, I propose an explanation for this discrepancy, by arguing that the metalogical square of opposition for semantic consequence exhibits a natural analogy to the well-known square of opposition for the categorical statements from syllogistics, but that this analogy breaks down once we move from semantic to syntactic consequence. I then show that despite this difficulty, one can indeed construct metalogical Aristotelian diagrams from a syntactic perspective, which have their own, equally elegant characterization in terms of the categorical statements. Finally, I construct several metalogical Aristotelian diagrams that incorporate both semantic and syntactic consequence (and their interaction), and study how they are influenced by the underlying logical system’s soundness and/or completeness. All of this provides further support for the methodological/heuristic perspective on Aristotelian diagrams, which holds that the main use of these diagrams lies in facilitating analogies and comparisons between prima facie unrelated domains of investigation.

     

Computing the real solutions of Fleishman's equations for simulating non-normal data

Fleishman's power method is frequently used to simulate non-normal data with a desired skewness and kurtosis. Fleishman's method requires solving a system of nonlinear equations to find the third-order polynomial weights that transform a standard normal variable into a non-normal variable with desired moments. Most users of the power method seem unaware that Fleishman's equations have multiple solutions for typical combinations of skewness and kurtosis. Furthermore, researchers lack a simple method for exploring the multiple solutions of Fleishman's equations, so most applications only consider a single solution. In this paper, we propose novel methods for finding all real-valued solutions of Fleishman's equations. Additionally, we characterize the solutions in terms of differences in higher order moments. Our theoretical analysis of the power method reveals that there typically exists two solutions of Fleishman's equations that have noteworthy differences in higher order moments. Using simulated examples, we demonstrate that these differences can have remarkable effects on the shape of the non-normal distribution, as well as the sampling distributions of statistics calculated from the data. Some considerations for choosing a solution are discussed, and some recommendations for improved reporting standards are provided.     

The use of diagrams in analogical problem solving

In four experiments, we examined the impact of perceptual properties on the effectiveness of diagrams in analogical problem solving, using variants of convergence diagrams as source analogues for the radiation problem. Static diagrams representing the initial problematic state (one large line directed at a target) and the final state for a convergence solution (multiple converging lines) were not accessed spontaneously but were often used successfully once a hint to consider the diagram had been provided. The inaccessibility of static diagrams was not alleviated by adding additional diagrams to represent intermediate states (Experiment 1), but spontaneous access was improved by augmenting static diagrams with a verbal statement of the convergence principle (Experiment 3). Spontaneous retrieval and noticing were increased markedly by animating displays representing converging forces and thereby encouraging encoding of the lines as indicating motion toward a target (Experiments 3 and 4). However, neither static nor animated diagrams were effective when the arrows were reversed to imply divergence rather than convergence (Experiment 2). The results indicate that when animation encourages the interpretation of a diagram as a helpful source analogue, it can greatly enhance analogical transfer.     

Mathematical models of central pattern generators in locomotion: II. Single limb models for locomotion in the cat

Three mathematical models of central pattern generation for locomotion in the single limb of the cat are presented. In each model, the activities in populations of neurons controlling limb joint flexors and extensors are described by a system of nonlinear differential equations. Each solution of the system for a different set of parameters corresponds to a simulation of some gait of the cat. Model I is based on unit generators for each limb joint muscle group and assumes that flexors inhibit their paired extensors, but not vice-versa. Model IIa assumes that flexors and extensors are mutually inhibitory, but that only the flexors have inherent oscillatory capability. Model IIb assumes flexors and extensors are mutually inhibitory and that both flexors and extensors have oscillatory capability. The properties of each of these models are explored, compared and contrasted, and discussed in relation to the experimental literature. All three models are shown to be capable of generating patterns consistent with various stepping rates of the cat and to show appropriate muscle sequencing and flexor-extensor interactions. Further, all three models exhibit smooth initiation and termination of stepping. However, Model I seems to provide a more parsimonious account of producing changes in stepping rate and is preferred, therefore, over models IIa and IIb.     

The twofold role of diagrams in Euclid’s plane geometry

Proposition I.1 is, by far, the most popular example used to justify the thesis that many of Euclid??s geometric arguments are diagram-based. Many scholars have recently articulated this thesis in different ways and argued for it. My purpose is to reformulate it in a quite general way, by describing what I take to be the twofold role that diagrams play in Euclid??s plane geometry (EPG). Euclid??s arguments are object-dependent. They are about geometric objects. Hence, they cannot be diagram-based unless diagrams are supposed to have an appropriate relation with these objects. I take this relation to be a quite peculiar sort of representation. Its peculiarity depends on the two following claims that I shall argue for: (i) The identity conditions of EPG objects are provided by the identity conditions of the diagrams that represent them; (ii) EPG objects inherit some properties and relations from these diagrams.     

A one-step method for modelling longitudinal data with differential equations

Differential equation models are frequently used to describe non-linear trajectories of longitudinal data. This study proposes a new approach to estimate the parameters in differential equation models. Instead of estimating derivatives from the observed data first and then fitting a differential equation to the derivatives, our new approach directly fits the analytic solution of a differential equation to the observed data, and therefore simplifies the procedure and avoids bias from derivative estimations. A simulation study indicates that the analytic solutions of differential equations (ASDE) approach obtains unbiased estimates of parameters and their standard errors. Compared with other approaches that estimate derivatives first, ASDE has smaller standard error, larger statistical power and accurate Type I error. Although ASDE obtains biased estimation when the system has sudden phase change, the bias is not serious and a solution is also provided to solve the phase problem. The ASDE method is illustrated and applied to a two-week study on consumers’ shopping behaviour after a sale promotion, and to a set of public data tracking participants’ grammatical facial expression in sign language. R codes for ASDE, recommendations for sample size and starting values are provided. Limitations and several possible expansions of ASDE are also discussed.     

On portfolio management with value at risk and uncertain returns via an artificial neural network scheme

This paper focuses on the computation issue of portfolio optimization with scenario-based Value-at-Risk. The main idea is to replace the portfolio selection models with linear programming problems. According to the convex optimization theory and some concepts of ordinary differential equations, a neural network model for solving linear programming problems is presented. The equilibrium point of the proposed model is proved to be equivalent to the optimal solution of the original problem. It is also shown that the proposed neural network model is stable in the sense of Lyapunov and it is globally convergent to an exact optimal solution of the portfolio selection problem with uncertain returns. Several illustrative examples are provided to show the feasibility and the efficiency of the proposed method in this paper.     

An SEM approach to continuous time modeling of panel data: relating authoritarianism and anomia

Panel studies, in which the same subjects are repeatedly observed at multiple time points, are among the most popular longitudinal designs in psychology. Meanwhile, there exists a wide range of different methods to analyze such data, with autoregressive and cross-lagged models being 2 of the most well known representatives. Unfortunately, in these models time is only considered implicitly, making it difficult to account for unequally spaced measurement occasions or to compare parameter estimates across studies that are based on different time intervals. Stochastic differential equations offer a solution to this problem by relating the discrete time model to its underlying model in continuous time. It is the goal of the present article to introduce this approach to a broader psychological audience. A step-by-step review of the relationship between discrete and continuous time modeling is provided, and we demonstrate how continuous time parameters can be obtained via structural equation modeling. An empirical example on the relationship between authoritarianism and anomia is used to illustrate the approach.     

Evaluating methods for approximating stochastic differential equations

Models of decision making and response time (RT) are often formulated using stochastic differential equations (SDEs). Researchers often investigate these models using a simple Monte Carlo method based on Euler's method for solving ordinary differential equations. The accuracy of Euler's method is investigated and compared to the performance of more complex simulation methods. The more complex methods for solving SDEs yielded no improvement in accuracy over the Euler method. However, the matrix method proposed by Diederich and Busemeyer (2003) yielded significant improvements. The accuracy of all methods depended critically on the size of the approximating time step. The large (∼10 ms) step sizes often used by psychological researchers resulted in large and systematic errors in evaluating RT distributions.     

Testing attitude-behaviour theories using non-experimental data: An examination of some hidden assumptions

This chapter presents a detailed causal analysis of the two leading theories of attitude-behaviour relations, the theory of reasoned action (TRA) and the theory of planned behaviour (TPB). It is noted that the direct path from perceived behavioural control to behaviour in the TPB is causally ambiguous. Focusing on the attitude-intention relationship, timeline and path diagrams are used to illustrate some of the hidden assumptions that underlie the way these theories are usually tested using standard multiple regression applied to nonexperimental data. Randomised experiments are recommended as the best solution to the serious problems arising from omitted causes.     

The general psychophysical differential equation: A comparison of three specifications

The general psychophysical differential equation, dy/dx = W₂(y)/W₁(x), with the solution y = f(x), where x and y are subjective variables and W₁ and W₂ their subjective Weber functions, is (a) compared with a corresponding functional equation, and (b) studied from a stochastic point of view by error calculus, Methods for evaluating and handling divergences are proposed and illustrated for a number of combinations of Weber functions. It is shown that either the differential: and the functional equations have the same solution or the difference between the solutions is negligible compared to empirical scatter. The error calculus gives the same result: either no error at all or a negligible one.     

HOW DIAGRAMS CAN IMPROVE REASONING

Abstract— We report an experimental study on the effects of diagrams on deductive reasoning with double disjunctions, for example:
Raphael is in Tacoma or Julia is in Atlanta, or both. Julia is in Atlanta or Paul is in Philadelphia, or both. What follows?
We confirmed that subjects find it difficult to deduce a valid conclusion, such as
Julia is in Atlanta, or both Raphael is in Tacoma and Paul is in Philadelphia.
In a preliminary study, the formal of the premises was either verbal or diagrammatic, and the diagrams used icons to distinguish between inclusive and exclusive disjunctions. The diagrams had no effect on performance. In the main experiment, the diagrams made the alternative possibilities more explicit. The subjects responded faster (about 35 s) and drew many more valid conclusions (nearly 30%) from the diagrams than from the verbal premises. These results corroborate the theory of mental models and have implications for the role of diagrams in reasoning.     

Between Square and Hexagon in Oresme's Livre du Ciel et du Monde

In logic, Aristotelian diagrams are almost always assumed to be closed under negation, and are thus highly symmetric in nature. In linguistics, by contrast, these diagrams are used to study lexicalization, which is notoriously not closed under negation, thus yielding more asymmetric diagrams. This paper studies the interplay between logical symmetry and linguistic asymmetry in Aristotelian diagrams. I discuss two major symmetric Aristotelian diagrams, viz. the square and the hexagon of opposition, and show how linguistic considerations yield various asymmetric versions of these diagrams. I then discuss a pentagon of opposition, which occupies an uneasy position between the square and the hexagon. Although this pentagon belongs neither to the symmetric realm of logic nor to the asymmetric realm of linguistics, it occurs several times in the literature. The oldest known occurrence can be found in the cosmological work of the 14th-century author Nicole Oresme.     

Modeling Noisy Data with Differential Equations Using Observed and Expected Matrices

Complex intraindividual variability observed in psychology may be well described using differential equations. It is difficult, however, to apply differential equation models in psychological contexts, as time series are frequently short, poorly sampled, and have large proportions of measurement and dynamic error. Furthermore, current methods for differential equation modeling usually consider data that are atypical of many psychological applications. Using embedded and observed data matrices, a statistical approach to differential equation modeling is presented. This approach appears robust to many characteristics common to psychological time series.     

A Newton-Raphson algorithm for maximum likelihood factor analysis

This paper demonstrates the feasibility of using a Newton-Raphson algorithm to solve the likelihood equations which arise in maximum likelihood factor analysis. The algorithm leads to clean easily identifiable convergence and provides a means of verifying that the solution obtained is at least a local maximum of the likelihood function. It is shown that a popular iteration algorithm is numerically unstable under conditions which are encountered in practice and that, as a result, inaccurate solutions have been presented in the literature. The key result is a computationally feasible formula for the second differential of a partially maximized form of the likelihood function. In addition to implementing the Newton-Raphson algorithm, this formula provides a means for estimating the asymptotic variances and covariances of the maximum likelihood estimators. This research was supported by the Air Force Office of Scientific Research, Grant No. AF-AFOSR-4.59-66 and by National Institutes of Health, Grant No. FR-3.     

Mood‐management dynamics: The interrelationship between moods and behaviours

People actively attempt to create and maintain positive moods and to escape from negative moods by engaging in various activities. The principle of homeostasis explains the essence of the mood‐management system: adjustments of individuals' moods and activities help maintain constant their conditions of life. We model the dynamics of mood and mood‐management behaviour through a pair of interdependent, linear differential equations and estimate the equations using mood and behaviour data collected from an adult panel. Because empirically fitting continuous‐time differential equations to intermittent observations is uncommon in the literature, we show how to transform differential equations into equations that can be estimated using simultaneous‐equation regression methods. Our adult panel shows strong homeostasis in mood management with mood episodes of several hours and no evidence of endogenous mood cycles.