Manipulatives have the potential to be powerful tools in helping children improve their number sense, develop advanced mathematical strategies, and build an understanding of the base ten number system. Physical manipulatives used in classrooms, however, are often not designed to promote efficient strategy use, such as counting on, and typically do…

Place value underpins much of what people do in number. In this article, the author describes some simple tasks that may be used to assess students' understanding of place value. This set of tasks, the Six Tasks of Place Value (SToPV), takes five minutes to administer and can give direct insight into a student's understanding of the number system…

Effective teaching includes enabling learners to make connections within mathematics. It is easy to accord with this statement, but how often is it a reality in the mathematics classroom? This article describes an approach in "connecting equivalent" fractions and whole number operations. The authors illustrate how a teacher can combine a common…

Three tools for teaching symmetry in the context of an upper-level undergraduate or introductory graduate course on the chemical applications of group theory are presented. The first is a collection of objects that have the symmetries of all the low-symmetry and high-symmetry point groups and the point groups with rotational symmetries from 2-fold…

In this article, the author narrates her experience of meeting a Montessori kid for the first time and shares the characteristics she observed in Montessori students. The author was working as director of academic resources in university housing at the University of Wisconsin-Madison and met Jason, a pre-med sophomore who was the resident…

The purpose of this study is to demonstrate how virtual manipulatives, designed to provide multimodal interactions, support richer perceptual experiences that promote conceptual learning. To study this phenomenon, a virtual manipulative called, "Puzzle Blocks," was developed by the researcher. "Puzzle Blocks" introduces the…

This study investigated how the use of virtual manipulatives in integer instruction impacts student achievement for integer addition and subtraction. Of particular interest was the influence of using virtual manipulatives on students' ability to create and translate among representations for integer computation. The research employed a…

Dice are considered one of the oldest gambling devices and thus many mathematicians have been interested in various dice gambling games in the past. Dice have been used to teach probability, and dice rolls can be effectively simulated using technology. The National Council of Teachers of Mathematics (NCTM) recommends that teachers use simulations…

We examined automatic spatial alignment effects evoked by handled objects. Using color as the relevant cue carried by an irrelevant handled object aligned or misaligned with the response hand, responses to color were faster when the handle aligned with the response hand. Alignment effects were observed only when the task was to make a reach and…

The purpose of this embedded quasi-experimental mixed methods research was to use solving simple linear equations as the lens for looking at the effectiveness of concrete and virtual manipulatives as compared to a control group using learning methods without manipulatives. Further, the researcher wanted to investigate unique benefits and drawbacks…

This study aimed to investigate the effect of experimenting with physical manipulatives (PM), virtual manipulatives (VM), and a blended combination of PM and VM on undergraduate students' understanding of concepts in the domain of "Light and Color." A pre-post comparison study design was used for the purposes of this study that involved 70…

An interview with a sixth-grade student illustrates how her number sense and understanding of variability relate to her ability and proclivity to apply a frequentist (statistical) approach to probability tasks. A general suggestion for teaching about mathematics of uncertainty through the gradual strengthening of estimation, as per the historical…

Thin-layer chromatography (TLC) is a widely used method of qualitative analysis in organic synthesis, as it uniquely combines low cost, rapidity, simplicity, versatility, small quantities of sample and low detection limits. The simplest and most economical method for the application of samples onto TLC plates is by hand, using glass capillaries.…

A fundamental component of mathematical learning is spatial visualization. To improve their spatial visualization skills, students should be given opportunities to work with two- and three-dimensional objects. The authors describe two activities that use tetrahedra nested within cubes to help improve students' visualization of two- and…

An embodied approach to reading comprehension suggests that emerging readers must learn to map words and phrases onto their remembered experiences, but this is made difficult by the necessity of focusing attention on decoding. Having children manipulate toys to correspond to what they are reading overcomes this problem, but introduces its own…