USING NUMBER LINE PROCEDURES AND PEER TUTORING TO IMPROVE THE MATHEMATICS COMPUTATION OF LOW-PERFORMING FIRST GRADERS

Recommendations for mathematics instruction frequently include the use of manipulatives as a critical component. There are few experimental analyses of teaching strategies involving the use of manipulatives (e.g., the number line). This investigation used a multiple baseline design across three groups of students to examine the effectiveness of an experimental procedure for improving low-performing children's skills in solving missing addend arithmetic problems using the number line. To address concerns about inadequate time for instruction and insufficient practice in most mathematics instruction, trained peer tutors implemented the procedure. The results suggest that student performance improved when trained tutors taught the students number line procedures and gave them feedback on accuracy. Further, social validation data indicate that the students, their tutors, and their classroom teachers liked the procedures.     

Matching under nonindependent variable-ratio schedules of drug reinforcement.

Response-contingent deliveries of oral pentobarbital maintained responding of 3 rhesus monkeys during daily 3-hr sessions. Deliveries of pentobarbital were arranged under nonindependent concurrent variable-ratio variable-ratio schedules. Responses to either schedule counted toward completion of both variable-ratio schedule requirements. This schedule is similar in some respects to conventional concurrent variable-interval variable-interval schedules, in which passage of time counts toward completion of the interval value on both schedules. Restricted nonindependent concurrent variable-ratio variable-ratio schedules were also studied. On that schedule, when a drug delivery was assigned to one spout, it had to be collected before responses on the opposite spout again counted toward completion of the schedule requirements. Relative reinforcer magnitude was varied by changing the drug concentration on one schedule while keeping the drug concentration constant on the other variable-ratio schedule. Under both types of concurrent variable-ratio schedules, the relative rate of responding corresponded to the relative drug intake. Unlike earlier studies of concurrent variable-interval variable-interval intravenous cocaine reinforcement, preference was proportionate to concentration, and exclusive preferences did not develop. The relationship between relative rate of responding and relative drug intake was well described by the generalized matching law.     

一般社会情境中风险认知的实验研究

风险认知是一个新的研究领域,随着经济的发展应愈来愈受到重视。本研究在一般社会情境下,讨论了对46个风险因素的风险程度及风险特征的知觉;个性特征与风险认知的关系;群体的风险认知类型特征,以及各样本间的比较分析。研究结论提供了公众风险认知结构的初步轮廓。     

Effects of social context, reinforcer probability, and reinforcer magnitude on humans' choices to compete or not to compete.

In the first two experiments, subjects' choices to earn points (exchangeable for money) either by competing with a fictitious opponent or by not competing were studied. Buskist, Barry, Morgan, and Rossi's (1984) competitive fixed-interval schedule was modified to include a second response option, a noncompetitive fixed-interval schedule. After choosing to enter either option, the opportunity for reinforcers became available after the fixed-interval's duration had elapsed. Under the no-competition condition, points were always available after the interval had elapsed. Under the competition condition, points were available based on a predetermined probability of delivery. Experiments 1 and 2 examined how reinforcer probabilities and reinforcer magnitudes affected subjects' choices to compete. Several general conclusions can be made about the results: (a) Strong preferences to compete were observed at high and moderate reinforcer probabilities; (b) competing was observed even at very low reinforcer probabilities; (c) response rates were always higher in the competition component than in the no-competition component; and (d) response rates and choices to compete were insensitive to reinforcer-magnitude manipulations. In Experiment 3, the social context of this choice schedule was removed to determine whether the high levels of competing observed in the first two experiments were due to a response preference engendered by the social context provided by the experimenters through instructions. In contrast to the first two experiments, these subjects preferred the 60-s fixed-interval schedule (formerly the no-competition option), indicating that the instructions themselves were responsible for the preference to compete. This choice paradigm may be useful to future researchers interested in the effects of other independent variables (e.g., drugs, social context, instructions) on competitive behavior.     

Confirmation of linear system theory prediction: Changes in Herrnstein's k as a function of changes in reinforcer magnitude

Eight human subjects pressed a lever on a range of variable-interval schedules for 0.25¢ to 35.0¢ per reinforcement. Herrnstein's hyperbola described seven of the eight subjects' response-rate data well. For all subjects, the y-asymptote of the hyperbola increased with increasing reinforcer magnitude and its reciprocal was a linear function of the reciprocal of reinforcer magnitude. These results confirm predictions made by linear system theory; they contradict formal properties of Herrnstein's account and of six other mathematical accounts of single-alternative responding.     

Number and time in acquisition,extinction and recovery

We measured rate of acquisition, trials to extinction, cumulative responses in extinction, and the spontaneous recovery of anticipatory hopper poking in a Pavlovian protocol with mouse subjects. We varied by factors of 4 number of sessions, trials per session, intersession interval, and span of training (number of days over which training extended). We find that different variables affect each measure: Rate of acquisition [1/(trials to acquisition)] is faster when there are fewer trials per session. Terminal rate of responding is faster when there are more total training trials. Trials to extinction and amount of responding during extinction are unaffected by these variables. The number of training trials has no effect on recovery in a 4-trial probe session 21 days after extinction. However, recovery is greater when the span of training is greater, regardless of how many sessions there are within that span. Our results and those of others suggest that the numbers and durations and spacings of longer-duration “episodes” in a conditioning protocol (sessions and the spans in days of training and extinction) are important variables and that different variables affect different aspects of subjects' behavior. We discuss the theoretical and clinical implications of these and related findings and conclusions—for theories of conditioning and for neuroscience.     

All Behavior is choice: Revisiting an evolutionary theory's account of behavior on single schedules

The axiomatic principle that all behavior is choice was incorporated into a revised implementation of an evolutionary theory's account of behavior on single schedules. According to this implementation, target responding occurs in the context of background responding and reinforcement. In Phase 1 of the research, the target responding of artificial organisms (AOs) animated by the revised theory was found to be well described by an exponentiated hyperbola, the parameters of which varied as a function of the background reinforcement rate. In Phase 2, the effect of reinforcer magnitude on the target behavior of the AOs was studied. As in Phase 1, the AOs' behavior was well described by an exponentiated hyperbola, the parameters of which varied with both the target reinforcer magnitude and the background reinforcement rate. Evidence from experiments with live organisms was found to be consistent with the Phase-1 predictions of the revised theory. The Phase-2 predictions have not been tested. The revised implementation of the theory can be used to study the effects of superimposing punishment on single-schedule responding, and it may lead to the discovery of a function that relates response rate to both the rate and magnitude of reinforcement on single schedules.     

A comparison of stimulus set size on tact training for children with autism spectrum disorder

Previous studies on skill acquisition have taught targets in stimulus sets composed of different numbers of stimuli. Although the rationale for selection of a stimulus set size is not clear, the number of target stimuli trained within a set is a treatment decision for which there is limited empirical support. The current investigation compared the efficiency of tact training in 4 stimulus set sizes, each of which included 12 stimuli grouped into (a) 4 sets of 3 stimuli, (b) 3 sets of 4 stimuli, (c) 2 sets of 6 stimuli, and (d) 1 set of 12 stimuli. Results of all 4 participants with autism spectrum disorder show tact training with larger (i.e., 6 and 12) stimulus set sizes was more efficient than training with smaller (i.e., 3 and 4) stimulus set sizes.     

A preliminary analysis of self-control with aversive events: the effects of task magnitude and delay on the choices of children with autism

When faced with a choice between two aversive events, a person exhibits self-control by choosing a smaller, more immediate aversive event over a larger, delayed aversive event. Task demands are often aversive to children with autism and other developmental disabilities. The purpose of this study was to evaluate behavioral sensitivity to differences in the amount and delay of tasks as part of a preliminary study on self-control. Participants were 2 children with autism who engaged in problem behavior maintained by escape. Results indicated a lack of self-control with respect to choosing between two aversive tasks and suggested potential strategies for increasing self-control (i.e., choosing a small immediate task over a large delayed task).     

An examination of indexes for determining the number of clusters in binary data sets

The problem of choosing the correct number of clusters is as old as cluster analysis itself. A number of authors have suggested various indexes to facilitate this crucial decision. One of the most extensive comparative studies of indexes was conducted by Milligan and Cooper (1985). The present piece of work pursues the same goal under different conditions. In contrast to Milligan and Cooper's work, the emphasis here is on high-dimensional empirical binary data. Binary artificial data sets are constructed to reflect features typically encountered in real-world data situations in the field of marketing research. The simulation includes 162 binary data sets that are clustered by two different algorithms and lead to recommendations on the number of clusters for each index under consideration. Index results are evaluated and their performance is compared and analyzed.Author names are listed in alphabetical order.This piece of research was supported by the Austrian Science Foundation (FWF) under grant SFB#010 (Adaptive Information Systems and Modeling in Economics and Management Science).The authors would like to thank the anonymous reviewers and especially the associate editor for their helpful comments and suggestions.