The representations of the arithmetic operations include functional relationships

Current theories of mathematical problem solving propose that people select a mathematical operation as the solution to a problem on the basis of a structure mapping between their problem representation and the representation of the mathematical operations. The structure-mapping hypothesis requires that the problem and the mathematical representations contain analogous relations. Past research has demonstrated that the problem representation consists of functional relationships, or principles. The present study tested whether people represent analogous principles for each arithmetic operation (i.e., addition, subtraction, multiplication, and division). For each operation, college (Experiments 1 and 2) and 8th grade (Experiment 2) participants were asked to rate the degree to which a set of completed problems was a good attempt at the operation. The pattern of presented answers either violated one of four principles or did not violate any principles. The distance of the presented answers from the correct answers was independently manipulated. Consistent with the hypothesis that people represent the principles, (1) violations of the principles were rated as poorer attempts at the operation, (2) operations that are learned first (e.g., addition) had more extensive principle representations than did operations learned later (multiplication), and (3) principles that are more frequently in evidence developed more quickly. In Experiment 3, college participants rated the degree to which statements were indicative of each operation. The statements were either consistent or inconsistent with one of two principles. The participants' ratings showed that operations with longer developmental histories had strong principle representations. The implications for a structure-mapping approach to mathematical problem solving are discussed.     

THE EFFECTS OF CHOICE ON SELF‐CONTROL

Three adolescents with traumatic brain injury performed a physical therapy task in the absence of programmed consequences or duration requirements. Next, the experimenter gave the participants the options of a smaller immediate reinforcer with no response requirement or a larger delayed reinforcer with a response requirement. Self‐control training exposed participants to a procedure during which they chose between a smaller immediate reinforcer and a progressively increasing delayed reinforcer whose values varied and were determined by a die roll. The participants chose whether they or the experimenter rolled the die. All participants initially demonstrated low baseline durations of the physical therapy task, chose the smaller immediate reinforcer during the choice baseline, and changed their preference to the larger delayed reinforcer during self‐control training.     

CREATING SINGLE‐SUBJECT DESIGN GRAPHS IN MICROSOFT EXCEL™ 2007

Over 10 years have passed since the publication of Carr and Burkholder's (1998) technical article on how to construct single‐subject graphs using Microsoft Excel. Over the course of the past decade, the Excel program has undergone a series of revisions that make the Carr and Burkholder paper somewhat difficult to follow with newer versions. The present article provides task analyses for constructing various types of commonly used single‐subject design graphs in Microsoft Excel 2007. The task analyses were evaluated using a between‐subjects design that compared the graphing skills of 22 behavior‐analytic graduate students using Excel 2007 and either the Carr and Burkholder or newly developed task analyses. Results indicate that the new task analyses yielded more accurate and faster graph construction than the Carr and Burkholder instructions.     

The dynamics of insight: Mathematical discovery as a phase transition

In recent work in cognitive science, it has been proposed that cognition is a self-organizing, dynamical system. However, capturing the real-time dynamics of cognition has been a formidable challenge. Furthermore, it has been unclear whether dynamics could effectively address the emergence of abstract concepts (e.g., language, mathematics). Here, we provide evidence that a quintessentially cognitive phenomenon—the spontaneous discovery of a mathematical relation—emerges through self-organization. Participants solved a series of gear-system problems while we tracked their eye movements. They initially solved the problems by manually simulating the forces of the gears but then spontaneously discovered a mathematical solution. We show that the discovery of the mathematical relation was predicted by changes in entropy and changes in power-law behavior, two hallmarks of phase transitions. Thus, the present study demonstrates the emergence of higher order cognitive phenomena through the nonlinear dynamics of self-organization.     

A hybrid approach to experimental control

In the present article, I describe a new, hybrid approach to experimental control that combines the best features of general-purpose programs and specific-experiment programs. First, experimental control objects are developed in an object-oriented programming environment. Then, an XML file is used to select and arrange those control objects in order to carry out an experiment. A wide range of experiments can be implemented by specifying the arrangement of the control objects in the XML file, while, at the same time, the set of control objects can be easily extended for more demanding applications. An overview of a system using this approach is provided, along with some examples.     

Make to Relate: Analyzing Narratives of Community Practice

As young people design, build, and problem solve within maker spaces and clubs, they talk about making. We analyze short interviews with young people involved in maker clubs, conducted during public presentations, and hypothesize a progression through frames of participation. Moving from exploration, to exchange, and on to deliberate engagement, these frames reflect changes in the nature of participation, appropriation of community practices, and adoption of making as a tool for identity work and interest development.     

Transfer of Responsibility in Parent–Child Play During the Second Year

The purpose of the present study was to examine changes in parental support for play during the toddler's second year. Thirty toddlers and their parents were observed when the toddlers were 13,14,20 and 21 months of age. Parents' production of facilitating behaviours and exploratory play increased over time, whereas their production of combinatorial, symbolic and sequential play decreased. Correlational and sequential analyses indicated that, at the early visits, parents responded to their toddlers' combinatorial and symbolic play with similar types of play, whereas at the later visits, only symbolic play was followed by comparable actions. Sensitive parents ‘transfer responsibility’ for play actions as they believe their toddlers become more competent. © 1996 John Wiley & Sons, Ltd.     

Establishing derived categorical responding in children with disabilities using the PEAK‐E curriculum

The purpose of the study was to evaluate a procedure to generate derived categorical responding by three children with disabilities and to promote the emergence of untrained intraverbal categorical responses. In the study, three 4‐member equivalence classes including three stimuli (A, B, and C) and a category name (D) for each class were trained using a match‐to‐sample procedure. Test probes were conducted for categorical responding, including both a trained (D‐A) and two derived (D‐B, D‐C) relational responses, as well as the emergence of untrained intraverbal categorical responding (D‐A/B/C) throughout the study. Relational training was effective at promoting the emergence of categorical responding, and two of the three participants demonstrated the emergence of additional intraverbal responding without prior training. The results provide further evidence supporting the practical utility of stimulus equivalence as well as the PEAK‐E curriculum.     

The effect of embedded bonus rounds on slot machine preference

Twenty‐three university students completed a simulated slot machine task involving the concurrent presentation of two slot machines that were varied both in win density and the inclusion of a bonus round feature to evaluate the effect of embedded bonus rounds on participant response allocation. The results suggest that participants allocated a greater percentage of responses to machines with embedded bonus rounds across both dense (Bonus: M = 68.4, SD = 19.2; No Bonus: M = 51.2; 9.6) and lean (Bonus: M = 48.8, SD = 9.6; No Bonus: M = 31.6, SD = 19.2) reinforcement schedules, in which the overall reinforcement rate across all machines was held constant.     

Transfers of stimulus function during roulette wagering

Twenty‐five recreational gamblers were initially asked to place bets on either red or black positions on a roulette board in a simulated casino setting. Each participant was then exposed to a stimulus pairing observing procedure which attempted to develop equivalence classes between one color (black or red) and traditionally positive words (e.g., love, happy, sex) and another color (black or red) and traditionally negative words (e.g., death, cancer, taxes), in the absence of consequence manipulations. Twenty‐one of the twenty‐five participants demonstrated greater response allocation to the color position on the roulette board that participated in a relational network with the positive words. Variations in sequencing of experimental conditions had no impact on poststimulus‐pairing wagers, but did impact tests for equivalence accuracy.