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…

In 2 studies (Ns = 55 and 54), the authors examined a basic form of conceptual understanding of rational number arithmetic, the direction of effect of decimal arithmetic operations, at a level of detail useful for informing instruction. Middle school students were presented tasks examining knowledge of the direction of effects (e.g., "True or…

Learners often insufficiently monitor their comprehension, which results in overconfident judgments of learning and underachievement. In the 3 present experiments, we investigated whether insufficient comprehension monitoring is due in part to the fact that learners are not sufficiently aware of the benefit of comprehension monitoring and thus…

The main purposes of this study were to test the effects of teaching at-risk 4th graders to provide explanations for their mathematics work and examine whether those effects occur by compensating for limitations in cognitive processes. We randomly assigned 212 children to 3 conditions: a control group and 2 variants of a multicomponent fraction…

Previous research on children's conceptual and procedural understanding of fractions, and other arithmetic skills, has led to contradictory conclusions. Some research suggests that children learn conceptual knowledge before procedural knowledge, some suggests that they learn procedural knowledge before conceptual knowledge, and other research…

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)…

New methods for representing the structural properties of domain-specific knowledge were studied. Network representations of data from 40 college students' ratings of the relatedness of domain concepts were derived using the Pathfinder scaling algorithm, and similarity between students' and instructor's networks was then assessed using a…

The role of analogical mapping in the formation of conceptual misrepresentations was studied by analyzing target-domain inferences generated by 193 third graders learning from an analogy. Explicit knowledge of the instructional goal decreased the number of conflicting inferences. Implications for learning are discussed. (SLD)

In the continuation of an earlier study, 69 high school students who had completed an epistemological beliefs questionnaire in 1992 completed the questionnaire again as seniors in 1995. Beliefs in fixed ability to learn, simple knowledge, quick learning, and certain knowledge changed as they grew older. (SLD)

The development of the associative structure of physics concepts as a result of the content presented to the learner and the relationship between associative structure and academic achievement was investigated using a treatment and a control group. Assessment of associative structure may allow insight into a learner's progress which complements…

Five boys and five girls each in grades one through six (n=60 students) asked controversial questions in interviews, recognized the lack of social consensus on the controversial topics and made subtle distinctions between controversial and noncontroversial topics. Implications for discussion of controversial issues in the classroom are discussed.…