Mathematics is rich in visual representations. Such visual representations are the means by which mathematical patterns "are recorded and analyzed." With respect to "vocabulary" and "symbols," numerous educators have focused on issues inherent in the language of mathematics that influence students' success with mathematics communication.…

Learning in educational settings most often emphasizes declarative and procedural knowledge. Studies of expertise, however, point to other, equally important components of learning, especially improvements produced by experience in the extraction of information: "Perceptual learning." Here we describe research that combines principles of…