Tangent lines are often first introduced to students in geometry during the study of circles. The topic may be repeatedly reintroduced to students in different contexts throughout their schooling, and often each reintroduction is accompanied by a new, nonequivalent definition of tangent lines. In calculus, tangent lines are again reintroduced to…

Fragile understanding is where new learning begins. Students' understanding of new concepts is often shaky at first, when they have only had limited experiences with or single viewpoints on an idea. This is not inherently bad. Despite teachers' best efforts, students' tenuous grasp of mathematics concepts often falters with time or when presented…

Recommended learning goals for students in introductory statistics courses include the ability to recognize and explain the key role of randomness in designing studies and in drawing conclusions from those studies involving generalizations to a population or causal claims (GAISE College Report ASA Revision Committee, 2016). The purpose of this…

In abstraction, or conceptual behavior, people discriminate features or properties of their surroundings. This permits people to respond selectively and precisely to specialized features of their environment, which has had many benefits, including steady advances in science and technology. Within psychology, J. R. Kantor and B. F. Skinner…

Although discussions about inductive reasoning can be traced back thousands of years (Fitelson 2011), the implementation of the Standards for Mathematical Practice (SMP) within the Common Core State Standards (CCSSI 2010) is generating renewed attention to how students learn mathematics. The third SMP, "Construct viable arguments and critique…

Discusses Vygotsky's theories about concept formation, his distinctions between everyday and theoretical concepts, and how empirical generalizations can lead to misconceptions. Examines the implications of these theories for mathematics instruction and its relationship to the current mathematics reform. (34 references) (MDH)

Identifies an induced incorrect generalization as a generative cause for the formation of misconception. Analyzes students' protocol of an exploratory learning experiment about the laws of reflection of light. Suggests recommendations for textbook, teaching, and practical work. (Author/YP)

The 9th International Conference on Educational Data Mining (EDM 2016) is held under the auspices of the International Educational Data Mining Society at the Sheraton Raleigh Hotel, in downtown Raleigh, North Carolina, in the USA. The conference, held June 29-July 2, 2016, follows the eight previous editions (Madrid 2015, London 2014, Memphis…

Investigates the conceptual-change process in grade 10 students during a computer-supported physics unit. Finds that many students vacillated between alternative and scientific conceptions from one context to another during instruction, and those few students who did exhibit context-independent and stable conceptual change appeared able to accept…

Discussed are the recognition and the relevance, in terms of teacher/student awareness and subsequent effective teacher treatment techniques, of cognitive disequilibrium as experienced by the student, when such disequilibrium is provoked by mathematical contradictions and/or counterexamples, within the classroom paradigm of constructivism. (JJK)

This study tested the effectiveness of having fifth-grade students generate their own examples of selected concepts on energy. Hypotheses were that subjects would perform better on sections of a test related to definitions and recognition of exemplars of the concepts for which they found their own examples than on sections related to comparible…