It is widely acknowledged that communicating mathematical thinking is complex, difficult, and often messy. In this article, the authors explore how mathematical thinking is communicated, providing examples of strategies for teachers and students to use in the mathematics classroom.

As a group of middle school mathematics teachers and teacher educators, the authors have focused their attention on using innovative practices coupled with high cognitive demand tasks to engage students, pique their interest in mathematics, and promote conceptual understanding. This article explores their three-act task design. In act 1, students…

A renewed emphasis was placed on ratio and proportional reasoning in the middle grades in the Common Core State Standards for Mathematics (CCSSM). The expectation for students includes the ability to not only compute and then compare and interpret the results of computations in context but also interpret ratios and proportions as they are…

As a college biology instructor, I often see graphs in lab reports that do not meet my expectations. I also observe that many college students do not always adequately differentiate between good and poor (or misleading) graphs. The activity described in this paper is the result of my work with students to improve their graphing literacy. The…

When students understand content deeply, they recognize main concepts and understand the relationships among ideas. But the typical processes students use to study are generally not designed to generate conceptual understandings. Instead, they tend to become passive learners. In this article, the authors present three commonly used classroom…

In this article, the author considers a student exercise that involves determining the exact and numerical solutions of a particular differential equation. He shows how a typical student solution is at variance with a numerical solution, suggesting that the numerical solution is incorrect. However, further investigation shows that this numerical…

This article describes a lesson unit in which students constructed tables and graphs to represent the growth of different trees. Students then compared the graphs to develop an understanding of slope.

The description about the changed instruction to encourage student exploration of the graphical and then the algebraic representations of functions is presented, which enables the students to understand how the graph, equation, and table of a function are related. The activity addresses both the Learning Principle and the Connection standard and…

This paper describes the possible effects of feedback on learning (associations) using a connectionist tool, the delta rule. Feedback in instruction can be described in terms of the interaction of stimulus inputs and response outputs, an associationist perspective. Here the delta rule is applied to each instance that an input and an output likely…

Describes a series of microcomputer-based laboratory exercises that are designed to assist undergraduate students in acquiring basic psychological concepts in memory retention. Mnemonics used in the exercises are explained, and computer-generated tabular and graphic displays designed for use in classroom demonstrations are described. (Author/LRW)

This paper reports on efforts to design a computer-based instructional unit for 7th grade students that would not only address state standards but would also provide some groundwork for the students' subsequent formal study of algebra. SimCalc Mathworlds software was used to explore the same motion from different perspectives. The overall…

Maintains that preparing graphs is an essential part of data analysis and representation. Asserts that, although psychologists use graphs extensively, most textbooks provide limited techniques for designing graphs. Reviews research on graphical techniques and includes recommended articles on the topic. (CFR)

Explores several instructional strategies and approaches for teaching charts, tables, graphs, and diagrams to social studies students. Discusses the various qualities of these visual aids and identifies specific strategies for each. Includes activities designed to teach students how to construct these representations. (MJP)

Describes an instructional unit using graphic organizers to teach about federalism in a U.S. government course. Includes models of charts and lists of appropriate student responses. (CFR)

Described is the V-scope which is a Micro-Based Laboratory system that provides direct and immediate displays of real motion as spatial images (stroboscopic snapshots), as time graphs of any combination of position, velocity, and acceleration, or as vector diagrams of velocity and acceleration. Illustrations of system components and of typical…