Teachers should select tasks that help students explore and develop increasingly sophisticated mathematical ideas. They should ask questions that encourage and challenge students to explain new ideas and develop new strategies based on mathematics they already know. For example, asking students to describe two ways they can estimate the cost of twelve notebooks can prompt different strategies. Teachers should encourage students to look for mathematical ideas throughout the school day. For example, geometry can play an important role in art, data should have a prominent role in social studies discussions, and communication and problem solving should be integrated with language arts.
Scientific contexts can be especially productive for exploring and using mathematics.
Mathematics and science have a long history of close ties, and many mathematical notions arose from scientific problems. Teachers should build on everyday experiences to encourage the study of mathematical ideas through systematic, quantitative investigations of phenomena that students can experience directly. These may include applications as varied as studying the relationship between the arm span and height of students, investigating the strength of a particular brand of paper towel, or studying the volume and surface area of different cereal boxes. At times, opportunities for mathematical investigations arise spontaneously in class. For example, after a fifth-grade class spent some time learning about environmental issues, a question arose as to whether the water fountain was an efficient way of getting water to students.