PROBLEM SOLVING

What is the role of the teacher?

“Through problem solving, students can experience the power and utility of mathematics. Problem solving is central to inquiry and application and should be interwoven throughout the mathematics curriculum to provide a context for learning and applying mathematical ideas.”

NCTM 2000, p. 256

Instructional programs from prekindergarten through grade 12 should enable all students to-

•build new mathematical knowledge through problem solving;•solve problems that arise in mathematics and in other contexts;•apply and adapt a variety of appropriate strategies to solve problems;•monitor and reflect on the process of mathematical problem solving.

Teachers play an important role in developing students' problem-solving dispositions.

1. They must choose problems that engage

students. 2. They need to create an environment

that encourages students to explore, take

risks, share failures and successes, and question one another.

In such supportive environments, students develop the confidence they need to explore problems and the ability to make adjustments in their problem-solving strategies.

• Three Question Types– Procedural

– Conceptual

– Application

• Procedural questions require students to:– Select and apply correct

operations or procedures– Modify procedures when needed– Read and interpret graphs, charts,

and tables– Round, estimate, and order

numbers– Use formulas

• Sample Procedural Test Question

A company’s shipping department is receiving a shipment of 3,144 printers that were packed in boxes of 12 printers each. How many boxes should the department receive?

• Conceptual questions require students to:– Recognize basic mathematical concepts– Identify and apply concepts and

principles of mathematics– Compare, contrast, and integrate

concepts and principles– Interpret and apply signs, symbols, and

mathematical terms– Demonstrate understanding of

relationships among numbers, concepts, and principles

• Sample Conceptual Test QuestionA salesperson earns a weekly salary of $225 plus $3 for every pair of shoes she sells. If she earns a total of $336 in one week, in which of the following equations does n represent the number of shoes she sold that week?(1) 3n + 225 = 336(2) 3n + 225 + 3 = 336(3) n + 225 = 336(4) 3n = 336(5) 3n + 3 = 336

• Application/Modeling/Problem Solving questions require students to:– Identify the type of problem represented– Decide whether there is sufficient

information– Select only pertinent information– Apply the appropriate problem-solving

strategy– Adapt strategies or procedures– Determine whether an answer is

reasonable

• Sample Application/Modeling/Problem Solving Test QuestionJane, who works at Marine Engineering, can make electronic widgets at the rate of 27 per hour. She begins her day at 9:30 a.m. and takes a 45 minute lunch break at 12:00 noon. At what time will Jane have made 135 electronic widgets? (1) 1:45 p.m.(2) 2:15 p.m.(3) 2:30 p.m.(4) 3:15 p.m.(5) 5:15 p.m.

What is Problem Solving?

According to Michael E. Martinez There is no formula for problem solving How people solve problems varies Mistakes are inevitable Problem solvers need to be aware of the

total process Flexibility is essential Error and uncertainty should be expected Uncertainty should be embraced at least

temporarily

An Effective Problem Solver

Reads the problem carefully Defines the type of answer that is required Identifies key words Accesses background knowledge regarding

a similar situation Eliminates extraneous information Uses a graphic organizer Sets up the problem correctly Uses mental math and estimation Checks the answer for reasonableness

Improving Problem-Solving Skills

Solve problems out loud Explain your thinking process Allow students to explain their thinking

process Use the language of math and require

students to do so as well Model strategy selection Make time for discussion of strategies Build time for communication Ask open-ended questions Create lessons that actively engage learners

Jennifer Cromley, Learning to Think, Learning to Learn