Standard 1: Mathematics as Problem Solving

In grades 5-8, the mathematics curriculum should include numerous experiences with problem solving as a method of inquiry and application so that students can:

• Use problem solving approaches to investigate and understand mathematical content

Benchmarks 12A (Habits of Mind: Values and Attitudes)
Grades 3-5, page 286
Keep records of their investigations and observations and not change the records later.
 
• Formulate problems from situations within and outside mathematics

Benchmarks 1B (The Nature of Science: Scientific Inquiry)
Grades 3-5, page 11
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.
 
• Develop and apply a variety of strategies to solve problems, with emphasis on multi-step and non-routine problems

Benchmarks 2A (The Nature of Mathematics: Patterns and Relationships)
Grades 6-8, page 28
Usually there is no right way to solve a mathematical problem; different methods have different advantages and disadvantages.

Benchmarks 1B (The Nature of Science: Scientific Inquiry)
Grades 3-5, page 11
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.
 

• Verify and interpret results with respect to the original problem situation

Benchmarks 1B (The Nature of Science: Scientific Inquiry)
Grades 6-8, page 12
If more than one variable changes at the same time in an experiment, the outcome of the experiment may not be clearly attributable to any one of the variables. It may not always be possible to prevent outside variables from influencing the outcome of an investigation (or even to identify all of the variables), but collaboration among investigators can often lead to research design that are able to deal with such situations.

Benchmarks 1B (The Nature of Science: Scientific Inquiry)
Grades 6-8, page 12
What people expect to observe often affects what they actually do observe. Strong beliefs about what should happen in particular circumstances can prevent them from detecting other results. Scientists know about this danger and take steps to try and avoid it when designing investigations and examining data. One safeguard is to have different investigators conduct independent studies of the same question.
 

• Generalize solutions and strategies to new problem situations

Benchmarks 3B (The Nature of Technology: Design and Systems)
Grades 3-5, page 50
The solution to one problem may create other problems.
 

• Acquire confidence in using mathematics meaningfully

Benchmarks 1C (The Nature of Science: The Science Enterprise)
Grades 3-5, page 1
Science is an adventure that people everywhere can take part in, as they have for many centuries.

Benchmarks 1C (The Nature of Science: The Science Enterprise)
Grades 6-8, page 17
Important contributions to the advancement of science, mathematics, and technology have been made by different kinds of people, in different cultures, at different times.