Many primary school students have difficulties understanding mathematical equivalence with considerably poorer performance in some countries than in others. However, students' formal understanding of equivalence has significant and long-lasting effects, as it predicts arithmetic and algebra achievement throughout school years. Currently, little is…

Does the timing of children's formal understanding of mathematical equivalence matter for algebra readiness? A change-resistance account (McNeil & Alibali, 2005) predicts that it is beneficial for children to construct a formal understanding of mathematical equivalence in the early grades before overly narrow operational patterns become…

A central understanding in mathematics is knowledge of "math equivalence," the relation indicating that 2 quantities are equal and interchangeable. Decades of research have documented elementary-school (ages 7 to 11) children's (mis)understanding of math equivalence, and recent work has developed a construct map and comprehensive…

Worked examples have proven to be effective for knowledge acquisition compared with problem solving, particularly when prior knowledge is low (e.g., Kalyuga, 2007). However, in addition to prior knowledge, executive functions and fluid intelligence might be potential moderators of the effectiveness of worked examples. The present study examines…

Understanding an arithmetic operation implies, at minimum, knowing the direction of effects that the operation produces. However, many children and adults, even those who execute arithmetic procedures correctly, may lack this knowledge on some operations and types of numbers. To test this hypothesis, we presented preservice teachers (Study 1),…

We provide evidence of persistent gender effects for students using advanced adaptive technology while learning mathematics. This technology improves each gender's learning and affective predispositions toward mathematics, but specific features in the software help either female or male students. Gender differences were seen in the students' style…

Providing exploratory activities prior to explicit instruction can facilitate learning. However, the level of guidance provided during the exploration has largely gone unstudied. In this study, we examined the effects of 1 form of guidance, feedback, during exploratory mathematics problem solving for children with varying levels of prior domain…

Researchers in both cognitive science and mathematics education emphasize the importance of comparison for learning and transfer. However, surprisingly little is known about the advantages and disadvantages of what types of things are being compared. In this experimental study, 162 seventh- and eighth-grade students learned to solve equations (a)…

Rapid cognitive diagnosis allows measuring current levels of learner domain-specific knowledge in online learning environments. Such measures are required for individualizing instructional support in real time, as students progress through a learning session. This article describes 2 experiments designed to validate a rapid online diagnostic…

A multitrait-multimethod study collected measures of social-cognitive flexibility, crystallized social knowledge, and academic problem solving from 169 high school seniors. Results support a division of social-cognitive intelligence into declarative and procedural social knowledge (crystallized) and flexible knowledge application as distinct from…

Schematic knowledge was assessed by asking 195 ninth and tenth graders to classify area-of-rectangle problems in terms of whether the text provided insufficient, sufficient, or irrelevant information for solution. The hierarchical ordering of templates for the area of a rectangle is demonstrated, and implications for mathematics instruction are…

Four experiments involving 440 college students in the Philippines and United States examined the effects of symbolic knowledge and problem-information context on translating relational statements into mathematical equations. Results indicate that symbolic knowledge is not always easily accessible in contexts that differ from ordinary…

This study surveyed personal factors influencing reasoning of 61 adult citizens about federal budget deficit solutions. Reasoners were motivated to present a strong argument for their answers and generally did not offer counterargument unless prompted to do so. Education level was not significantly related to quality of argument. (SLD)

Unique associations between three families of mathematical knowledge and individual differences in the mathematics skills of 103 seventh and eighth graders were studied. Conceptual and procedural knowledge adequately explained individual differences in fraction solving and problem set-up accuracy, but only conceptual knowledge adequately expressed…

Interactions between levels of learner knowledge in a domain and levels of instructional guidance were investigated. Inexperienced trade apprentices were presented with either worked examples to study or problems to solve. On subsequent tests, inexperienced trainees benefited most from worked examples. With more experience in the domain, worked…