Mathematics Curriculum Framework - November 2000
This curriculum framework envisions all students in the Commonwealth achieving mathematical competence through a strong mathematics program that emphasizes problem solving, communicating, reasoning and proof, making connections, and using representations. Acquiring such competence depends in large part on a clear, comprehensive, coherent, and developmentally appropriate set of standards to guide curriculum expectations.
Problem solving is both a means of developing students' knowledge of mathematics and a critical outcome of a good mathematics education. As such, it is an essential component of the curriculum. A mathematical problem, as distinct from an exercise, requires the solver to search for a method for solving the problem rather than following a set procedure. Mathematical problem solving, therefore, requires an understanding of relevant concepts, procedures, and strategies. To become good problem solvers, students need many opportunities to formulate questions, model problem situations in a variety of ways, generalize mathematical relationships, and solve problems in both mathematical and everyday contexts.
The ability to express mathematical ideas coherently to different audiences is an important skill in a technological society. Students develop this skill and deepen their understanding of mathematics when they use accurate mathematical language to talk and write about what they are doing. They clarify mathematical ideas as they discuss them with peers, and reflect on strategies and solutions. By talking and writing about mathematics, students learn how to make convincing arguments and to represent mathematical ideas verbally, pictorially, and symbolically.
Reasoning and Proof
From the early grades on, students develop their reasoning skills by making and testing mathematical conjectures, drawing logical conclusions, and justifying their thinking in developmentally appropriate ways. As they advance through the grades, students' arguments become more sophisticated and they are able to construct formal proofs. By doing so, students learn what mathematical reasoning entails.
Mathematics is not a collection of separate strands or standards. Rather, it is an integrated field of study. Students develop a perspective of the mathematics field as an integrated whole by understanding connections within and outside of the discipline. It is important for teachers to demonstrate the significance and relevance of the subject by encouraging students to explore the connections that exist within mathematics, with other disciplines, and between mathematics and students' own experiences.
Mathematics involves using various types of representations for mathematical objects and actions, including numbers, shapes, operations, and relations. These representations can be numerals or diagrams, algebraic expressions or graphs, or matrices that model a method for solving a system of equations. Students must learn to use a repertoire of mathematical representations. When they can do so, they have a set of tools that significantly expands their capacity to think mathematically.