### Standard 1: Mathematics as Problem Solving

In grades 9-12, the mathematics curriculum should include the refinement and extension of methods of problem solving so that students can:
• Use, with increasing confidence, problem solving approaches to investigate and understand mathematical content
• Benchmarks 1C (The Nature of Science: The Science Enterprise)
Important contributions to the advancement of science, mathematics, and technology have been made by different kinds of people, in different cultures, at different times.

• Apply integrated mathematical problem-solving strategies to solve problems from within and outside mathematics
• Benchmarks 1B (The Nature of Science: Scientific Inquiry)
Scientists' explanations about what happens in the world come partly from what they observe, partly from what they think. Sometimes scientists have different explanations for the same set of observations. That usually leads to their making more observations to resolve the differences.

• Recognize and formulate problems from situations within and outside mathematics
• Benchmarks 1B (The Nature of Science: Scientific Inquiry)
Scientific investigations may take many different forms, including observing what things are like or what is happening somewhere, collecting specimens for analysis, and doing experiments. Investigations can focus on physical, biological, and social questions.

• Apply the process of mathematical modeling to real-world problem situations
• Benchmarks 2B (The Nature of Mathematics: Mathematics, Science, and Technology)
Mathematical modeling aids in technological design by simulating how a proposed system would theoretically behave.

Benchmarks 2C (The Nature of Mathematics: Mathematical Inquiry)
Much of the work of mathematicians involves a modeling cycle, which consists of there steps: (1) using abstractions to represent things or ideas, (2) manipulating the abstractions according to some logical rules, and (3) checking how well the results match the original things or ideas. If the match is not considered good enough, a new round of abstraction and manipulation may begin. The actual thinking need not go through these processes in logical order but may shift from one to another in any order.

Benchmarks 4A (The Physical Setting: The Universe)
Mathematical models and computer simulations are used in studying evidence from many sources in order to form a scientific account of the universe.

Benchmarks 8E (The Designed World: Information Processing)
Computer modeling explores the logical consequences of a set of instructions and a set of data. The instructions and data input of a computer model try to represent the real world so the computer can show what would actually happen. In this way, computers assist people in making decisions by simulating the consequences of different possible decisions.

Benchmarks 11B (Common Themes: Models)
The basic idea of mathematical modeling is to find a mathematical relationship that behaves in the same ways as the objects or processes under investigation. A mathematical model may give insight about how something really works or may fit observations very well without any intuitive meaning.

Benchmarks 11B (Common Themes: Models)
Computers have greatly improved the power and use of mathematical models by performing computations that are very long, very complicated, or repetitive. Therefore computers can show the consequences of applying complex rules or of changing he rules. The graphic capabilities of computers make them useful in the design and testing of devices and structures and in the simulation of complicated processes.

Benchmarks 11B (Common Themes: Models)
The usefulness of a model can be tested by comparing its predictions to actual observations in the real world. But a close match does not necessarily mean that the model in the only "true" model or the only one that would work.