Characteristics of students who benefit from high-quality mathematics instruction

The present study examined characteristics of students who took advantage of the opportunity to learn Algebra II from a highly skilled teacher. In particular, student information was gathered concerning: (a) the amount of mathematics students knew prior to the class, (b) their motivation for learning mathematics, (c) the number and types of interactions they had with the teacher during class, (d) degree of self-regulation, and (e) general aptitude. A model for achievement was used that suggests both opportunity and propensity are important (i.e., students must be willing and able to take advantage of opportunity). Support for the model was found. Propensity to take advantage of a highly favorable opportunity accounted for almost sixty percent of the variance in achievement. In particular, students who were self-regulated, had strong mathematics backgrounds, and had low levels of frustration benefited more from high-quality instruction.     

The rewarding effects of number and surface area of food in rats

Visual cues have an important role in food preference for both rats and humans. Here, we aim to isolate the effects of numerosity, density, and surface area on food preference and running speed in rats, by using a forced-choice maze paradigm. In Experiment 1, rats preferred and ran faster for a group of multiple smaller pellets rather than a single large pellet, corroborating previous research (Capaldi, Miller, & Alptekin Journal of Experimental Psychology: Animal Behavior Processes, 15(1), 75–80, 1989). Further experiments tested the prevailing hypothesis that multiple food pieces are more reinforcing because they occupy a larger surface area. Experiment 2 controlled for numerosity by utilizing a continuous food: mashed potatoes flattened to cover a larger surface area or rounded into a ball. The rats preferred and ran faster for the flattened potatoes, suggesting surface area plays a role in quantity estimations. Finally, in Experiment 3, rats displayed no preference or difference in running speed between a group of scattered and clustered pellets when number of pellets were kept constant. Taken together, these results suggest that density has an important role in food perception—that is, the rewarding effect of higher numerosity or larger surface area is removed when the food does not fill out the entire space. Alternative explanations and implications for human diet are discussed.     

Using Variables in School Mathematics: Do School Mathematics Curricula Provide Support for Teachers?

This study employed content analysis to examine 3 popular middle-grades mathematics curricula in the USA on the support they provide for teachers to implement concepts associated with variables in school mathematics. The results indicate that each of the 3 curricula provides some type of support for teachers, but in a varied amount and quality. More specifically, whereas the University of Chicago School Mathematics Project (UCSMP) curriculum provides support for teachers on several aspects of using variables in school mathematics, the support found in the Connected Mathematics 2 and the Math Connects curricula focused mainly on one conception of variables—namely, the use of variables as quantity that varies in the Connected Mathematics 2 curriculum and the use of variables as specific unknowns in the Math Connects curriculum. Overall, the UCSMP curriculum provides the most support for teachers, followed by the Connected Mathematics 2 curriculum, with the Math Connects curriculum recording the least support for teachers to enact variable concepts. It is worth pointing out that, although the 3 curricula collectively provide guidance for teachers to implement variable ideas within meaningful real-world contexts, the supports identified in the respective curriculum were not sufficient in addressing all of the areas that are essential for teaching the many concepts associated with variables in school mathematics effectively. Recommendations for curriculum developers and for international researchers with interest in the roles of variables in school mathematics are provided.