A high-school third-year or undergraduate first-semester general chemistry laboratory experiment introducing simple-cubic, face-centered cubic, body-centered cubic, and hexagonal closest packing unit cells is presented. Latex balls and acrylic resin plates are employed to make each atomic arrangement. The volume of the vacant space in each cell is…

When children learn to count, they map newly acquired symbolic representations of number onto preexisting nonsymbolic representations. The nature and timing of this mapping is currently unclear. Some researchers have suggested this mapping process helps children understand the cardinal principle of counting, while other evidence suggests that this…

The equilibrium between phases is a key concept from the introductory physical chemistry curriculum. Phase diagrams display which phase is the most stable at a given temperature and pressure. If more than one phase has the lowest Gibbs energy, then those phases are in equilibrium under those conditions. An activity designed to demonstrate the…

The purpose of this article is to identify the factors which affect students' creative thinking in problem solving. The research which was performed was quasi-experimental. It used one experimental group and two control groups from three second-grade high school classes. They received either traditional, active or heuristic problem-solving…

How many squares can be found on a typical 5 × 5 geoboard? Teachers who are unfamiliar with this question may wish to stop here and reflect a bit. The question can lead to wonderful student discourse: How can someone tell if something is a square? Should squares be counted that are the same size (but in a different location) or just different…

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 "large" and…

Representations that create informative visual displays are powerful tools for communicating mathematical concepts. The National Council of Teachers of Mathematics encourages the use of manipulatives (NCTM 2000). Manipulative materials are often used to present initial representations of basic numerical principles to young children, and it is…

The transition to distance learning in the spring of 2020 caused by COVID-19 was particularly challenging for Montessori educators and students because key elements of the Method were not directly transferable to this new and hastily designed format. Hands-on learning with Montessori materials and learning in a community, as well as careful…

This book represents a crop of wide-ranging research conducted by renown scholars in sub-Sahara Africa revolving around mathematics teaching and professional development programs for mathematics teachers. The research-based proposals and actual how-to-conduct professional development initiatives that enhance effective mathematics instruction are…

Educators often use concrete objects to help children understand mathematics concepts. However, findings on the effectiveness of concrete objects are mixed. The present study examined how two factors--perceptual richness and established knowledge of the objects--combine to influence children's counting performance. In two experiments, preschoolers…

Regular hexaflexagons mysteriously change faces as you pinch flex them. This paper describes a different flex, the V-flex, which allows the hexahexaflexagon (with only 9 faces under the pinch flex) to have 3420 faces. The article goes on to explain the classification of triangle orientations in a hexaflexagon and gives an example of the…

Students use concrete manipulatives to form an imperative affiliation between conceptual and procedural knowledge (Balka, 1993). Hence, it is necessary to design specific mathematics manipulatives that focus on different mathematical concepts. Preservice teachers need to know how to make and use manipulatives that stimulate students' thinking as…

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.…

Multiple Intelligence Theory suggests that individuals perceive knowledge in eight different ways. This article reports on a study that explored the role of manipulatives in the teaching and learning of trigonometric ratios in grade 10. The approach attempts in addressing three domains of the Multiple Intelligence Theory (linguistic/verbal…

This article describes a series of two linear measurement activities designed to integrate the related content and processes of Common Core State Standards for Mathematics (CCSSM) into classroom practice using nonstandard units. In the article, nonstandard units refer to same-size, repeatable units (e.g., paper clips, feet) for a specified…