Robin Rumph

CONSTRUCTING AND DERIVING RECIPROCAL TRIGONOMETRIC RELATIONS: A FUNCTIONAL ANALYTIC APPROACH

Participants were pretrained and tested on mutually entailed trigonometric relations and combinatorially entailed relations as they pertained to positive and negative forms of sine, cosine, secant, and cosecant. Experiment 1 focused on training and testing transformations of these mathematical functions in terms of amplitude and frequency followed by tests of novel relations. Experiment 2 addressed training in accordance with frames of coordination (same as) and frames of opposition (reciprocal of) followed by more tests of novel relations. All assessments of derived and novel formula-to-graph relations, including reciprocal functions with diversified amplitude and frequency transformations, indicated that all 4 participants demonstrated substantial improvement in their ability to identify increasingly complex trigonometric formula-to-graph relations pertaining to same as and reciprocal of to establish mathematically complex repertoires.

A Relational Frame and Artificial Neural Network Approach to Computer-Interactive Mathematics.

Fifteen participants unfamiliar with mathematical operations relative to reflections and vertical and horizontal shifts were exposed to an introductory lecture regarding the fundamentals of the rectangular coordinate system and the relationship between formulas and their graphed analogues. The lecture was followed immediately by computer-assisted instructions and matching-tosample procedures in which participants were e)(posed to computerposted rules regarding the relationship between particular types of formulas and their respective graphs. After participants demonstrated mutual entailment on formula-to-graph and graph-toformula functions, they were assessed for 36 novel relations on complex variations of the original training formulas and graphs. In Experiment 1, 5 of 15 participants demonstrated perfect or near perfect performance on all novel relationships. Experiment 2 was directed at the remaining 10 participants who failed to correctly identify all mathematical relationships assessed in Experiment 1. The error patterns for these 10 participants were classified with the help of an artificial neural network self-organizing map (SOM). Training in Experiment 2 was directed exclusively at the types of errors identified by the SOM. Following remedial training, all participants demonstrated a substantial reduction in errors compared to their performance in Experiment 1. Derived transfer of stimulus control using mathematical relations is discussed.

Behavioral and Physiological Neural Network Analyses: A Common Pathway toward Pattern Recognition and Prediction.

Using 3 diversified datasets, we explored the pattern-recognition ability of the Self-Organizing Map (SOM) artificial neural network as applied to diversified nonlinear data distributions in the areas of behavioral and physiological research. Experiment 1 employed a dataset obtained from the UCI Machine Learning Repository. Data for this study were composed of votes for each U.S. Representative on 16 key items during a particular legislative session. Experiment 2 employed a dataset developed in our human neuroscience laboratory and focused on the effects of sympathetic nervous system arousal on cardiac and inner-ear physiology. Experiment 3 employed the well-known Wisconsin Breast Cancer dataset, which was used to develop a sensitive, automated diagnostic method of distinguishing between malignant and benign cells. We suggest that the SOM is capable of identifying cohesive patterns of nonlinear measurements that would be difficult to identify using traditional linear data reduction procedures and that neural networks will be increasingly valuable in the analysis of a wide range of complex behaviors.

Training and Deriving Precalculus Relations: A Small-Group, Web-Interactive Approach

A small-group, web-interactive approach to teaching precalculus concepts was investigated. Following an online pretest, 3 participants were given a brief (15 min) presentation on the details of reciprocal math relations and how they operate on the coordinate axes. During baseline, participants were tested regarding their ability to construct formulas for a diversified series of graphs. This was followed by online, construction-based, small-group training procedures focusing on the construction of mathematical functions and a test of novel relations. Participants then received group training in accordance with frames of coordination (same as) and frames of opposition (reciprocal of) formula- to-graph relations. Online assessment indicated that participants showed substantial improvement over baseline and pretest performances. This was true even though, during the tests of novel relations, graphs were displayed with scattered data points instead of solid lines on the coordinate axes. Although one participant was unable to complete the second half of the experiment, we were able to train this small group employing approximately the same number of exposures needed for individual training in previous research.

Fixed-Interval Responding During Human Computer-Interactive Problem Solving

Experiment 1 was designed to investigate student patterns of responding during fixed-interval (FI) 30-s reinforcement. During the experiment, students were able to respond to multiplication problems by typing answers on the keyboard. Correct answers/min were calculated by the computer program and automatically recorded on disk. Following the experiment, students were questioned regarding what they believed to be the best way to earn the money while working problems. Outcomes from the first experiment showed that only one of the five students was dominated by a pause-respond pattern of temporal disparity. This student provided a verbal rule that accurately described the contingenCies associated with FI reinforcement. The other four students in this experiment responded at relatively constant rates in the majority of their intervals and provided verbal descriptions of contingencies suggesting that reinforcement became available only after the completion of a number or a changing number of problems. Experiment 2 replicated the preparations of Experiment 1; however, prior to initiated computerinteractive problem solving, the two subjects were shown the accurate rule generated by the subject in Experiment 1 who had exhibited pause-respond performance. Response patterns produced by these subjects showed conspicuous and consistent patterns of pauserespond throughout all intervals of FI 30-s reinforcement. Experiment 3 was conducted to further assess the possibility that scalloping (or some other pattern) might emerge over an extended series of sessions. Outcomes confirmed that performance patterns did not change significantly over sessions. Moreover, the subjects’ verbal description of programmed contingencies conformed to the pattern of responding they produced. Outcomes are discussed in terms of selfgenerated and socially mediated rule-following.