For many scientists, researchers and students Markov chain Monte Carlo (MCMC) simulation is an important and necessary tool to perform Bayesian analyses. The simulation is often presented as a mathematical algorithm and then translated into an appropriate computer program. However, this can result in overlooking the fundamental and deeper…

Calculus students' difficulties in understanding integration have been extensively studied. Research shows that the difficulty lies with students understanding of the definition of the definite integral as a limit of a Riemann sum and with the idea of accumulation inherent in integration. We have created a set of manipulatives and activities…

This article is a discussion drawing on a learning study with a teacher working with one- and two-year-olds. The focal interest is a scrutiny of the use of manipulative objects, emanating from observations of a teacher's and children's use of, and responses to the use of, different objects. The concept of size and the notions…

The papers in this special issue focus on instructional strategies with manipulatives. Often described as "hands-on learning", these strategies emphasize the use of physical and, more recently, virtual objects to represent target information and concepts. These strategies are frequently suggested as effective techniques for teaching…

Recent literature on learning with instructional manipulatives seems to call for a moderate view on the effects of perceptual and interactive richness of instructional manipulatives on learning. This "moderate view" holds that manipulatives' perceptual and interactive richness may compromise learning in two ways: (1) by imposing a…

A longstanding debate concerns the use of concrete versus abstract instructional materials, particularly in domains such as mathematics and science. Although decades of research have focused on the advantages and disadvantages of concrete and abstract materials considered independently, we argue for an approach that moves beyond this dichotomy and…

The purpose of this project was to investigate the effects of virtual versus physical manipulation using a simulation-based laboratory activity (SBL) and a microcomputer-based laboratory activity (MBL). Both the SBL and the MBL used computers to collect, graph, and analyze data. A major difference was that the MBL allowed the students to…

Pattern blocks are inexpensive wooden, foam, or plastic manipulatives developed in the 1960s to help students build an understanding of shapes, proportions, equivalence, and fractions (EDC 1968). The colorful collection of basic shapes in classic pattern block kits affords opportunities for amazing puzzle-like problem-solving tasks and for…

This is an activity for younger learners in which the concepts of relative frequency and probability are explored, illustrating how a creative, non-transmission teaching approach using bottle tops can enable students to develop their understanding in an engaging fashion and at almost no cost.

We compared the effects of two approaches to decoding on the word reading skills of struggling first-grade students: decontextualized, teacher directed decoding practice with manipulative letters and embedded, incidental decoding practice. Participants were 98 students at risk for reading disability. We randomly assigned participants to three…

This study focuses on three main goals: First, 3-year-olds' spatial assembly skills are probed using interlocking block constructions (N = 102). A detailed scoring scheme provides insight into early spatial processing and offers information beyond a basic accuracy score. Second, the relation of spatial assembly to early mathematical skills…

This study examines the relative instructional efficiency of virtual and physical fraction manipulatives. More specifically, this study uses a randomized experiment to determine if differences in students' knowledge of fractions magnitude exist when students use virtual rather than physical manipulatives to learn basic fraction concepts.…

In this empirical study, we investigate an early childhood education activity in mathematics on the concept of "pattern". The children participating in the study attend an intermediate form of schooling in Sweden called "the preschool class for six-year-olds", intended to facilitate their transition from preschool to school.…

Mathematical learning and teaching are increasingly seen as a multimodal experience involved in cultural and social semiotic registers and means, and as such social-cultural semiotic analysis is expected to shed light on learning and teaching processes occurring in the mathematics classroom. In this research, three social-cultural semiotic…

Curricula for three chemistry camp experiences for blind and visually impaired (BVI) individuals that incorporated single- and multiday activities and experiments accessible to BVI students are described. Feedback on the camps from students, mentors, and instructors indicates that these events allowed BVI students, who in many cases have been…