In this article, the authors share their favorite "Construct It!" activity, which focuses on rate of change and functions. The initial approach to instruction was procedural in nature and focused on making use of formulas. Specifically, after modeling how to find the slope of the line given two points and use it to solve for the…

Research has shown that the types of tasks assigned to students affect their learning. Various authors have described desirable features of mathematical tasks or of the activity they initiate. Others have suggested task taxonomies that might be used in classifying mathematical tasks. Drawing on this literature, we propose a set of task types that…

This document introduces, describes and exemplifies the technical features of some recently implemented automated reasoning tools in the dynamic mathematics software GeoGebra. The new tools are based on symbolic computation algorithms, allowing the automatic and rigorous proving and discovery of theorems on constructed geometric figures. Some…

Empirical arguments rely on examples without necessarily addressing all cases. Students should be skeptical of empirical evidence and should seek more secure arguments for generalizations, such as those that explain why a generalization is true for all cases. Generalizing on the basis of patterns in data is an important mathematical practice;…

Young learners need activities that engage them mentally and physically in processing new information. It is easy to accept such ideas intuitively, but matching the curriculum content to meaningful activities is a challenge. The authors believe that following a learning cycle is a satisfactory way to meet that challenge. A learning cycle is a…

Algebraic thinking in the middle grades involves constructing generalizations, moving beyond the focus of specific calculations in elementary school. This creates exciting opportunities to examine the validity of general arguments. Creating rich mathematical tasks and asking questions related to justification encourages students to examine what…

The Philosophy for Children Program was introduced to a class of deaf fifth graders as an instructional approach for reasoning skills. The program is intended to develop analytic skills required for intellectual functioning (including concept development, generalization, inference making, question formulation, and analogies). The program's major…

A school-based short-term intervention used successfully with impulsive, aggressive, shy, oversensitive, attention-disordered, and language- and learning-disabled elementary school students features exercises and activities such as generating ideas, rehearsing behaviors, and adapting behaviors to help students learn to "enter" play groups and join…

A project was undertaken to develop an experimental program to help young people acquire a generalized strategy for learning after they leave school. Other objectives of the program were to help teachers improve their understanding and use of various learning strategies, to develop an induction course for those purposes (entitled the "How do…