In schools today, tests and curricula are primarily suited to the types of students who excel in mathematics and verbal reasoning. The missing factor in testing and education policy is the measurement, selection, and talent development of students with strengths in spatial reasoning. Spatial reasoning is defined as "the ability to generate,…

We consider here a number of ideas for the classroom or lecture theatre associated with the mensuration of solids. In particular, the volumes of various tetrahedra are obtained in an indirect manner (by way of prisms and square-based pyramids). This activity develops problem-solving skills, spatial visualization and a from-first-principles…

Linear algebra students are typically introduced to the problem of how to convert from one coordinate system to another in a very abstract way. Often, two bases for a given vector space are provided, and students are taught how to determine a transition matrix to be used for changing coordinates. If students are successful in memorizing this…

Measurement is an important component of K-grade 12 mathematics curricula. The concepts of area and perimeter of polygons are first introduced in third grade and serve as the basis for teaching in the upper grades. Without a strong understanding of measurement, students will struggle to meaningfully grasp three-dimensional measurement concepts…

The Lorentz length contraction for a rod in uniform motion is derived performing two measurements at arbitrary times. Provided that the velocity of the rod is known, this derivation does not require the simultaneous measurement of two events. It thus avoids uncomfortable superluminal relationships. Furthermore, since the observer's simultaneous…

Most teachers feel confident espousing the benefits of using picture books in English lessons, talking about the importance of using the illustrations to enhance the text, engaging students and fostering a love and appreciation of literature. How many teachers passionately advocate the use of these same picture books in mathematics lessons? This…

Invented by English mathematician and pedagogist, Caleb Gattegno (1911-1988), the geoboard was designed as a manipulative tool for teaching primary geometry in schools (Williams, 1999). Traditionally made out of plywood and nails, geoboards today are usually made out of plastic and come in a variety of different sizes and colours. Rubber bands are…

"Focus in Grade 1: Teaching with Curriculum Focal Points" describes and illustrates learning paths for the mathematical concepts and skills of each grade 1 Focal Point as presented in Curriculum Focal Points for Prekindergarten through Grade 8 Mathematics. It includes representational supports for teaching and learning that can facilitate…

"Back to the Basics: Birmingham, Alabama" is the fourth in a series of workshops that focus on teaching foundational map reading and spatial differentiation skills. It is the second published exercise from the Back to the Basics series developed by the Wetland Education through Maps and Aerial Photography (WETMAAP) Program (see…

Several times a week I walk by a metal chair that is fastened to a flat concrete slab at an outdoor bus stop here in Boulder. One day I noticed on the concrete a nice shadow image of the woven metal seat of the chair (Fig. 1). The seat and back of the chair are formed from 3.8-cm wide strips of metal spaced 3.8 cm apart. The seat is about 39 cm…

Learning about mathematics begins in the early years. Through their everyday activities, young children build important foundational knowledge and understanding of mathematics, especially when adults provide specific opportunities to learn and practice concepts and skills. Young children develop early on an informal understanding of important…

Rhymes are often useful teaching devices for small children, and can even become a creative resource for teaching mathematics. The students and their teachers can change many of them to better fit their environment and/or extend the rhyme so that higher numbers are involved. A number of rhymes and folk tales lend themselves to more detailed…

Teachers often encounter difficulty in teaching estimation of measurement to young children. The intention when teaching the typical "jelly bean jar" classroom activity is to help children develop estimation skills, but most children cannot conceptualise the difference between guessing and estimating. The fact is that many students view…

Good early mathematics is broader and deeper than early practice on "school skills." High-quality mathematics should be a joy--not a pressure. It can emerge from children's play, their curiosity, and their natural ability to think. This article describes the areas of mathematics that young children can learn, and encourages elementary teachers to…

Presents a distance and time map exercise designed as a crime scenario. Explains that students must determine whether one individual could have committed a series of crimes in a given time period. Includes graph production that gives practice in interpreting and expressing aspects of spatial information. (DK)