Students’ images of two-variable functions and their graphs

This paper presents a conceptual analysis for students’ images of graphs and their extension to graphs of two-variable functions. We use the conceptual analysis, based on quantitative and covariational reasoning, to construct a hypothetical learning trajectory (HLT) for how students might generalize their understanding of graphs of one-variable functions to graphs of two-variable functions. To evaluate the viability of this learning trajectory, we use data from two teaching experiments based on tasks intended to support development of the schemes in the HLT. We focus on the schemes that two students developed in these teaching experiments and discuss their relationship to the original HLT. We close by considering the role of covariational reasoning in generalization, consider other ways in which students might come to conceptualize graphs of two-variable functions, and discuss implications for instruction.     

Delivery Method: Does it Matter? A Study of the North Carolina Developmental Mathematics Redesign

In 2009, the North Carolina Community College System (NCCCS) began the process of redesigning developmental mathematics (DMA) courses. The first classes were offered in 2012 with final state implementation in fall 2013. The purpose of this study was to examine the subsequent gateway course success rates achieved by former DMA students in relation to the delivery method (i.e., teacher-centered, student-centered, and computer-centered) used in the redesigned North Carolina DMA courses. Students included in this study took a gateway math course and a developmental math course during the same or previous academic year. Students who participated in student-centered and computer-centered instruction in their DMA course had a statistically significant pass rate in subsequent gateway courses compared with students receiving teacher-centered instruction. Along with the findings, this paper discusses considerations for redesign, including the choice of delivery methods and instructional techniques, and recommendations for further research.     

Transfer of judgments of control to a target stimulus and to novel stimuli through derived relations

Three experiments examined the effect of response?Coutcome contingencies on human ratings of causal efficacy and demonstrated that such ratings transfer to novel situations through derived stimulus relations. Efficacy ratings generally followed the delta probability rule when positive response-outcome contingencies were employed (Experiment 1) and when some outcomes were not contingent on participants?? responses (Experiment 2). Experiment 3 employed a negative response?Coutcome contingency and manipulated performance expectancies in the task. All three groups overestimated their causal efficacy ratings. A learned helplessness effect was observed when the response?Coutcomes were uncontrollable and in the high-expectancy group when participants?? performance in the task was worse than they had expected. In all experiments, ratings transferred to a stimulus presented during the task and often generalized to novel stimuli through derived relations. These results corroborate the view that outcome probability is a determinant of causal efficacy ratings and that schedules can be employed as UCS in procedures that share characteristics of evaluative conditioning procedures.     

The transfer of social exclusion and inclusion functions through derived stimulus relations

Previous studies have found that social exclusion can cause distress to those excluded. One method used to study social exclusion is through a virtual ball-toss game known as Cyberball. In this game, participants may be excluded from or included in the ball-toss game and typically report lower feelings of self-esteem, control, belonging, and meaningful existence following exclusion. Experiments 1 and 2 sought to explore the transfer of feelings of exclusion and inclusion through stimulus equivalence classes. In both experiments, participants were trained to form two three-member equivalence classes (e.g., A1–B1, B1–C1; A2–B2, B2–C2) and were tested with novel stimulus combinations (A1–C1, C1–A1, A2–C2, C2–A2). Thereafter, participants were exposed to the Cyberball exclusion and inclusion games. In these games, one stimulus (C1) from one equivalence class was assigned as the Cyberball inclusion game name, whereas one stimulus (C2) from the other equivalence class was assigned as the Cyberball exclusion game name. In Experiment 2, participants were only exposed to the Cyberball exclusion game. During a subsequent transfer test, participants were asked to rate how included in or excluded from they thought they would be in other online games, corresponding to members of both equivalence classes. Participant reported that they felt they would be excluded from online games if the games were members of the same equivalence class as C2. In contrast, participants reported that they felt they would be included in online games if the games were members of the same equivalence class as C1. Results indicated the transfer of feelings of inclusion (Experiment 1) and feelings of exclusion (Experiments 1 and 2) through equivalence classes.