Mathematics Standards
(Adopted 8/26/96)

Mathematics Standards Rationale

The four National Council of Teachers of Mathematics (NCTM) standards of problem solving, reasoning, communication and connections are goals interwoven throughout the Arizona mathematics standards. These goals are the reasons people study and use mathematics, and they should permeate everything we do in and outside the classroom.

Whenever possible, mathematical learning should be placed in a broader, problem-solving context and evaluated through performance assessments. In this setting, students discover questions involving numbers or equations from a real-world context which lead to answers that have meaning. Ultimately, all problems should be application problems; more ideally, problems should be presented in the broader context of an investigation or project. This way the students use problem solving, reasoning, communication and connections in every mathematical activity. The spirit of these four goals is a mathematical apprenticeship in which the students solve problems on a daily basis, much as mathematics is used in the real world.

Even the youngest students can use mathematics to solve social science problems, engineering problems and business problems in a meaningful way. As early as possible, students should learn that mathematics is everywhere in the world around them. They should realize that in the real world not all answers are small whole numbers; instead they can be large or small and/or have a fractional part.

As students develop their ability to perceive and conceptualize in problem solving, they should reason about the mathematics they do. Teachers should guide students to ask such questions as: Does the answer make sense? Are there other ways to arrive at the answer? Does the answer bring up more questions? Can I answer those? What other information would I need? It is this kind of reasoning that enriches a mathematical educational experience. If students do answer such significant questions, they then naturally apply mathematics in everyday life. Without this guidance, they remain mathematically deficient.

Teachers should engage students in mathematical discourse at all stages of learning. Mathematics was developed as a means to communicate about quantities, logical relationships and unknowns. To use this language, students should communicate (both orally and in writing) everything they do mathematically. They should explain their mathematical thinking through language, through models, graphically, geometrically, numerically and algebraically. Students should be encouraged to express themselves in as many ways as possible and to learn to translate between one mathematical language and another.

Students should regularly see the mathematical connections within the course of an investigation or project. They must experience mathematics primarily through its connections to other disciplines. For too long we have structured our curricula to reach the few who will use mathematics in isolation rather than the majority who will apply it to their work or study in other fields.

A variety of tools should be available to students as they develop concepts and understandings of mathematics. Graphing calculators and computers should be standard equipment in mathematics classrooms. New technology not only has made calculations and graphing easier, it has changed the very nature of the problems important to mathematics and the methods mathematicians use to investigate them.

As the four essential standards–problem solving, reasoning, communication and connections–and the implementation of technology become functioning parts of our curricula, we can expect all Arizona students will develop the mathematical power to confidently handle the future. They will be able to face the world knowing that they can not only merely compute but also that they can use meaningful mathematics to solve real problems.

The organization of the content in these standards is designed for readability purposes and is not intended to dictate sequence or to define the structure of courses. Topics from all six mathematics standards need to be continuously integrated within the curricula.