Understanding summary statistics and graphical techniques are building blocks to comprehending concepts beyond basic statistics. It's known that motivated students perform better in school. Using examples that students find engaging allows them to understand the concepts at a deeper level.

Journals provide a history of each student's thinking over time and allow this history to be easy to review. Journaling in mathematics has been found to be a valuable tool both for students and for teachers. Students benefit from journaling because it advances their mathematical understanding and ability to communicate in mathematics; teachers…

Pre-assessment has a bad reputation, write Hockett and Doubet--but that's largely because it's often misused solely to compare pre- and post-results. Used this way, pre-assessment serves as little more than a thief of instruction and a discouraging exercise for students. The authors describe how teachers can design thoughtful…

Great science teachers recognise the importance of providing children with practical, real-life experiences to develop their understanding of, and enthusiasm for, this truly inspirational subject. However, Gareth Metcalfe believes that the process by which children understand and remember their scientific experiences can be enriched. This article…

When teaching mathematics with technology, the author does so in the belief that technology enables students to experience mathematical ideas in a way that might not otherwise be possible. However, teachers must be careful: Sometimes technology does not produce results in the way that they or their students expect. Rather than allowing unexpected…

This article describes the development of the Precalculus Concept Assessment (PCA) instrument, a 25-item multiple-choice exam. The reasoning abilities and understandings central to precalculus and foundational for beginning calculus were identified and characterized in a series of research studies and are articulated in the PCA Taxonomy. These…

The power of mathematical inscriptions, such as graphs, is often attributed to the fact that they summarize a lot of information independent of their contextual particulars. There is evidence, however, that even quintessential experts and scientists have difficulties interpreting graphs when they are unfamiliar with the entities represented and…

This article describes the use of limits in topics that are usually covered in a high school advanced placement chemistry course or a first-year college chemistry course. Such an approach supplements the interpretation of the graph of an equation since it is usually easier to evaluate the limit of a function than to generate its graph. In…

Provides examples in which graphs are used in the statements of problems or in their solutions as a means of testing understanding of mathematical concepts. Examples (appropriate for a beginning course in calculus and analytic geometry) include slopes of lines and curves, quadratic formula, properties of the definite integral, and others. (JN)

Hierarchical graphs represent relationships between objects (like computer file systems, family trees etc.). Graph nodes represent the objects and interconnecting lines represent the relationships. In two experiments we investigated what concepts are necessary for understanding hierarchical graphs, what misconceptions evolve when some of the…

In this paper, we discuss issues in planning and conducting research into mathematics learning. We emphasize two central themes: (1) the learners' mathematics (especially the issues and ideas, in given problem situations, that learners choose to think about and to present; and (2) the kinds of knowledge that learners may be building (including…

This article describes an approach to teaching linear equations using multiple representations in a highly motivating context. Connections among representations are made explicit to enhance understanding and empower students. (Contains 4 figures.)