David W. Braithwaite

Effects of Variation and Prior Knowledge on Abstract Concept Learning

Learning abstract concepts through concrete examples may promote learning at the cost of inhibiting transfer. The present study investigated one approach to solving this problem: systematically varying superficial features of the examples. Participants learned to solve problems involving a mathematical concept by studying either superficially similar or varied examples. In Experiment 1, less knowledgeable participants learned better from similar examples, while more knowledgeable participants learned better from varied examples. In Experiment 2, prior to learning how to solve the problems, some participants received a pretraining aimed at increasing attention to the structural relations underlying the target concept. These participants, like the more knowledgeable participants in Experiment 1, learned better from varied examples. Thus, the utility of varied examples depends on prior knowledge and, in particular, ability to attend to relevant structure. Increasing this ability can prepare learners to learn more effectively from varied examples.

Fossil Fuels - At What Cost? Government Support for Upstream Oil and Gas Activities in Indonesia

The GSIs first report in the series “Fossil Fuels - At What Cost?” studies the subsidies provided to fossil-fuel producers in Indonesia. The study provides in-depth analysis of Indonesias system of Production Sharing Contracts and other relevant policies to identify and estimate government support for the industry. The study concludes that three subsidies can clearly be identified, totaling US

Flexibility in data interpretation: effects of representational format

1.8 billion in 2008. It estimates that this is a lower-bound figure, as at least seven other potential subsidies were identified that could not be assessed or quantified based on the available information. The researchers recommend that further work could usefully undertake a full assessment of the economic, environmental and social impacts of these subsidies in order to inform a public debate on whether the subsidies should be kept or considered for reform.

A Computational Model of Fraction Arithmetic.

Graphs and tables differentially support performance on specific tasks. For tasks requiring reading off single data points, tables are as good as or better than graphs, while for tasks involving relationships among data points, graphs often yield better performance. However, the degree to which graphs and tables support flexibility across a range of tasks is not well-understood. In two experiments, participants detected main and interaction effects in line graphs and tables of bivariate data. Graphs led to more efficient performance, but also lower flexibility, as indicated by a larger discrepancy in performance across tasks. In particular, detection of main effects of variables represented in the graph legend was facilitated relative to detection of main effects of variables represented in the x-axis. Graphs may be a preferable representational format when the desired task or analytical perspective is known in advance, but may also induce greater interpretive bias than tables, necessitating greater care in their use and design.

An In Vivo Study of Self-Regulated Study Sequencing in Introductory Psychology Courses

Many children fail to master fraction arithmetic even after years of instruction, a failure that hinders their learning of more advanced mathematics as well as their occupational success. To test hypotheses about why children have so many difficulties in this area, we created a computational model of fraction arithmetic learning and presented it with the problems from a widely used textbook series. The simulation generated many phenomena of children’s fraction arithmetic performance through a small number of common learning mechanisms operating on a biased input set. The biases were not unique to this textbook series—they were present in 2 other textbook series as well—nor were the phenomena unique to a particular sample of children—they were present in another sample as well. Among other phenomena, the model predicted the high difficulty of fraction division, variable strategy use by individual children and on individual problems, relative frequencies of different types of strategy errors on different types of problems, and variable effects of denominator equality on the four arithmetic operations. The model also generated nonintuitive predictions regarding the relative difficulties of several types of problems and the potential effectiveness of a novel instructional approach. Perhaps the most general lesson of the findings is that the statistical distribution of problems that learners encounter can influence mathematics learning in powerful and nonintuitive ways.

Children learn spurious associations in their math textbooks: Examples from fraction arithmetic.

Study sequence can have a profound influence on learning. In this study we investigated how students decide to sequence their study in a naturalistic context and whether their choices result in improved learning. In the study reported here, 2061 undergraduate students enrolled in an Introductory Psychology course completed an online homework tutorial on measures of central tendency, a topic relevant to an exam that counted towards their grades. One group of students was enabled to choose their own study sequence during the tutorial (Self-Regulated group), while the other group of students studied the same materials in sequences chosen by other students (Yoked group). Students who chose their sequence of study showed a clear tendency to block their study by concept, and this tendency was positively associated with subsequent exam performance. In the Yoked group, study sequence had no effect on exam performance. These results suggest that despite findings that blocked study is maladaptive when assigned by an experimenter, it may actually be adaptive when chosen by the learner in a naturalistic context.

Corrigendum to ‘‘Non-formal mechanisms in mathematical cognitive development: The case of arithmetic’’ [Cognition 149 (2016) 40–55]

Fraction arithmetic is among the most important and difficult topics children encounter in elementary and middle school mathematics. Braithwaite, Pyke, and Siegler (2017) hypothesized that difficulties learning fraction arithmetic often reflect reliance on associative knowledge—rather than understanding of mathematical concepts and procedures—to guide choices of solution strategies. They further proposed that this associative knowledge reflects distributional characteristics of the fraction arithmetic problems children encounter. To test these hypotheses, we examined textbooks and middle school children in the United States (Experiments 1 and 2) and China (Experiment 3). We asked the children to predict which arithmetic operation would accompany a specified pair of operands, to generate operands to accompany a specified arithmetic operation, and to match operands and operations. In both countries, children’s responses indicated that they associated operand pairs having equal denominators with addition and subtraction, and operand pairs having a whole number and a fraction with multiplication and division. The children’s associations paralleled the textbook input in both countries, which was consistent with the hypothesis that children learned the associations from the practice problems. Differences in the effects of such associative knowledge on U.S. and Chinese children’s fraction arithmetic performance are discussed, as are implications of these differences for educational practice.

Integrating Formal and Grounded Representations in Combinatorics Learning

http://dx.doi.org/10.1016/j.cognition.2016.03.024 0010-0277/ 2016 Elsevier B.V. All rights reserved. DOI of original article: http://dx.doi.org/10.1016/j.cognition.2016.01.004 ⇑ Corresponding author at: Department of Psychology, Carnegie Mellon University, 5000 Forbes Avenue, Pittsburgh, PA 15213, United States. E-mail address: baixiwei@gmail.com (D.W. Braithwaite). Fig. 2. (A) Difficulty ratings and (B) foil error rates for Family 1 problems. Cognition 151 (2016) 113