This set of Standards reinforces the dual goals that mathematics learning
is both about making sense of mathematical ideas and about acquiring skills
and insights to solve problems. The calculator is an important tool in
reaching these goals in grades 35 (Groves 1994). However,
» calculators do not replace fluency with basic number combinations,
conceptual understanding, or the ability to formulate and use efficient
and accurate methods for computing. Rather, the calculator should support
these goals by enhancing and stimulating learning. As a student works
on problems involving many or complex computations, the calculator is
an efficient computational tool for applying the strategies determined
by the student. The calculator serves as a tool for enabling students
to focus on the problem-solving process. Calculators can also provide
a means for highlighting mathematical patterns and relationships. For
example, using the calculator to skip-count by tenths or hundredths highlights
relationships among decimal numbers. For example, 4 is one-tenth more
than 3.9, or 2.49 is one-hundredth less than 2.5. Students at this age
should begin to develop good decision-making habits about when it is useful
and appropriate to use other computational methods, rather than reach
for a calculator. Teachers should create opportunities for these decisions
as well as make judgments about when and how calculators can be used to
Teachers in grades 35 make decisions every day that influence their students' opportunities to learn and the quality of that learning. The classroom environment they create, the attention to various topics of mathematics, and the tools they and their students use to explore mathematical ideas are all important in helping students in grades 35 gain increased mathematical maturity. In these grades teachers should help students learn to work together as part of building a mathematical community of learners. In such a community, students' ideas are valued and serve as a source of learning, mistakes are seen not as dead ends but rather as potential avenues for learning, and ideas are valued because they are mathematically sound rather than because they are argued strongly or proposed by a particular individual (Hiebert et al. 1997). A classroom environment that would support the learning of mathematics with meaning should have several characteristics: students feel comfortable making and correcting mistakes; rewards are given for sustained effort and progress, not the number of problems completed; and students think through and explain their solutions instead of seeking or trying to recollect the "right" answer or method (Cobb et al. 1988). Creating a » classroom environment that fosters mathematics as sense making requires the careful attention of the teacher. The teacher establishes the model for classroom discussion, making explicit what counts as a convincing mathematical argument. The teacher also lays the groundwork for students to be respectful listeners, valuing and learning from one another's ideas even when they disagree with them.
Because of the increasing mathematical sophistication of the curriculum in grades 35, the development of teachers' expertise is particularly important. Teachers need to understand both the mathematical content for teaching and students' mathematical thinking. However, teachers at this level are usually called on to teach a variety of disciplines in addition to mathematics. Many elementary teacher preparation programs require minimal attention to mathematics content knowledge. Given their primary role in shaping the mathematics learning of their students, teachers in grades 35 often must seek ways to advance their own understanding.
Many different professional development models emphasize the enhancement of teachers' mathematical knowledge. Likewise, schools and districts have developed strategies for strengthening the mathematical expertise in their instructional programs. For example, some elementary schools identify a mathematics teacher-leader (someone who has particular interest and expertise in mathematics) and then support that teacher's continuing development and create a role for him or her to organize professional development events for colleagues. Such activities can include grade-level mathematics study groups, seminars and workshops, and coaching and modeling in the classroom. Other schools use mathematics specialists in the upper elementary grades. These are elementary school teachers with particular interest and expertise in mathematics who assume primary responsibility for teaching mathematics to a group of studentsfor example, all the fourth graders in a school. This strategy allows some teachers to focus on a particular content area rather than to attempt being an expert in all areas.
Ensuring that the mathematics outlined in this chapter is learned by all students in grades 35 requires a commitment of effort by teachers to continue to be mathematical learners. It also implies that districts, schools, and teacher preparation programs will develop strategies to identify current and prospective elementary school teachers for specialized mathematics preparation and assignment. Each of the models outlined heremathematics teacher-leaders and mathematics specialistsshould be explored as ways to develop and enhance students' mathematics education experience. For successful implementation of these Standards, it is essential that the mathematical expertise of teachers be developed, whatever model is used.