This study investigates how students in England and Shanghai understand linear function. Understanding is defined theoretically in terms of five hierarchical levels: Dependent Relationship; Connecting Representations; Property Noticing; Object Analysis; and Inventising. A pilot study instrument presented a set of problems to both cohorts, showing…

The Vygotsky Blocks Test assesses problem-solving styles within a theoretical framework for the development of higher mental processes devised by Vygotsky. Because both the theory and the associated test situate cognitive development within the child's social and linguistic context, they address conceptual issues around the developmental relation…

The learning sciences community has made significant progress in understanding how people think and learn about complex systems. But less is known about how people make sense of the quantitative patterns and mathematical formalisms often used to study these systems. In this article, we make a case for attending to and supporting connections…

To improve mathematics achievement, students' errors should be treated as a source to stimulate their understanding of the conceptual and procedural basis of their errors. The study investigated 20 Chinese and 20 U.S. high school teachers' interpretations and responses to a student's errors in solving a quadratic equation. The teachers' responses…

Although many students struggle with fractions, students with mathematical learning disabilities (MLDs) experience pervasive difficulties because of neurological differences in how they process numerical information. These students make errors that are qualitatively different than their typically achieving and low-achieving peers. This study…

In 4 word-translation experiments, we examined the different representational frameworks theory (Crutch & Warrington, 2005; 2010) that concrete words are represented primarily by category, whereas abstract words are represented by association. In our experiments, Chinese-English bilingual speakers were presented with an auditory Chinese word…

Interpreting and responding to student thinking are central tasks of reform-minded mathematics teaching. This study examined preservice teachers' (PSTs) interpretations of and responses to a student's error(s) involving finding a missing length in similar rectangles through a teaching scenario task. Fifty-seven PSTs' responses were…

Based on a 2-component framework of episodic memory development across the lifespan (Shing & Lindenberger, 2011), we examined the contribution of memory-related binding and monitoring processes to false memory susceptibility in childhood and old age. We administered a repeated continuous recognition task to children (N = 20, 10-12 years),…

In this article we quantitatively and qualitatively analyze a sample of novice programmer compilation log data, exploring whether (or how) low-achieving, average, and high-achieving students vary in their grasp of these introductory concepts. High-achieving students self-reported having the easiest time learning the introductory programming…

This study examines how well second-year nonmajor organic chemistry students are learning to draw, interpret, and understand resonance-related structures. Students were tested seven times throughout an academic year using a set of four tasks that reflected their understanding of what these structures represent and how they relate to each other.…