Wachusett Regional School District
Wachusett Mountain

District Frameworks - Mathematics

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Standard I

Mathematics as Problem Solving

Problem solving may be defined as applying knowledge, skills, and experiences in efforts to resolve a dilemma or situation that is new or perplexing. "Problem solving should be the central focus of the mathematics curriculum. As such, it is a primary goal of all mathematics instruction and an integral part of all mathematical activity. Problem solving is not a distinct topic, but a process that should permeate the entire program and provide the context in which concepts and skills can be learned." (NCTM 1989, p. 23)

Problem solving as discussed here means something quite different from typical word problems. The following two problems illustrate that most problems that are encountered in real life are quite complex. There are many factors affecting the situations, there are many ways to view and approach the problems, there are many strategies for solving the problem and there can be more than one solution.

  • How do I choose the best car for my family? Should I buy now or wait another year and risk substantial repair bills? What are the most important factors for me to consider in selecting a car? What weight should I give to each factor? 
  • I have a fairly secure, steady job with a stable company, but there seems to be little chance for advancement. Should I invest in myself and return to school? Full-time or part-time? Should I apply for a position in a fast-growing but unproven company? How do I calculate the estimated costs and probable benefits?

To become good problem solvers, students need to have many opportunities to create and solve problems in both mathematical and real world contexts. Students need to pose questions, define problems, and consider different strategies and solutions. They need to realize that there are many ways of looking at and solving a problem, and that some ways are more effective and/or efficient than others in particular circumstances. In addition, they need to develop self confidence as competent problem solvers.

In grades K-4 the study of mathematics shall emphasize problem solving so that students will: 

  • Use problem-solving approaches to investigate and understand mathematical content; 
  • Formulate problems from everyday and mathematical situations; 
  • Develop and apply strategies to solve a wide variety of problems; 
  • Verify and interpret results with respect to the original problem; 
  • Acquire confidence in using mathematics meaningfully.

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

  • Use problem-solving approaches to investigate and understand mathematical content; 
  • Formulate problems from situations within and outside mathematics; 
  • Develop and apply a variety of strategies to solve problems, with emphasis on multi-step and non-routine problems; 
  • Verify and interpret results with respect to the original problem situation; 
  • Acquire confidence in using mathematics meaningfully.

In grades 9-12, the mathematics curriculum shall include the refinement and extension of methods of mathematical problem solving so that all students will:

  • Use, with increasing confidence, problem-solving approaches to investigate and understand mathematical content; 
  • Apply integrated mathematical problem-solving strategies to solve problems from within and outside mathematics; 
  • Develop and apply a variety of strategies to solve problems, with emphasis on multi-step and non-routine problems; 
  • Apply the process of mathematical modeling to real-world problem situations.

ADULT BASIC EDUCATION

In the adult basic education classroom, curriculum design shall include approaches which allow the learner to:

  • Explore and employ multiple strategies for solving problems; 
  • Determine, collect, and analyze appropriate data with respect to the original problem or in new problem solving situations; 
  • Have access to and the ability to use appropriate problem solving tools including the use of calculator, computers, and measurement instruments; 
  • Generalize problem solving strategies to a wide range of adult-oriented, real-world situations.