Fostering a mathematically thoughtful environment is vital to supporting the development of students' facility with mathematical reasoning. The teacher plays an important role by creating or selecting tasks that are appropriate to the ages and interests of middle-grades students and that call for reasoning to investigate mathematical relationships. Tasks that require the generation and organization of data to make, validate, or refute a conjecture are often appropriate. For example, the examination of patterns associated with figurate numbers discussed above shows how a teacher can use the task both to stimulate students' investigation and to develop facility with mathematical reasoning and argumentation. Suitable tasks can arise in everyday life, although many will arise within mathematics itself.
Teachers also serve as monitors of students' developing facility with reasoning.
In order to use inductive reasoning appropriately, students need to know its limitations as well as its possibilities. Because many elementary and middle-grades tasks rely on inductive reasoning, teachers should be aware that students might develop an incorrect expectation that patterns always generalize in ways that would be expected on the basis of the regularities found in the first few terms. Because the number of segments is large and some students will have difficulty being systematic in representing the segments on their geoboards, the teacher encourages the students to examine simpler cases to develop a systematic way to generate the different segments. The teacher helps the students see that each successive square contains the previous square within it. The students then record the number of segments of different lengths in each square and note the pattern of growth, as shown in figure 6.35.