From the perspective of connections, new ideas are seen as extensions of previously learned mathematics. Students learn to use what they already know to address new situations. Elementary school students link their knowledge of the subtraction of whole numbers to the subtraction of decimals or fractions. Middle-grades students recognize and connect multiple representations of the same mathematical idea, such as the ratio that represents rate of change and the tilt or slope of a line. High school students connect ideas in algebra and geometry.
Some activities can be especially productive for featuring mathematical connections.
For instance, the relationship between the diameter and the circumference of a circle can be studied empirically by collecting a variety of circular objects and measuring their circumferences and diameters. The problem involves ideas from measurement, data analysis, geometry, algebra, and number. As students progress through their school mathematics experience, their ability to see the same mathematical structure in seemingly different settings should increase. High school students are ready to look for connections among the many mathematical ideas they are encountering. As students develop a view of mathematics as a connected and integrated whole, they will have less of a tendency to view mathematical skills and concepts separately.