Teaching Through Problem Solving

Numeracy

Barbara A. Guillory, M.A.Ed.

National Board Certified Teacher

Three P’s

Purpose: To provide a comprehensive framework for teaching mathematics through problem solving.

Process: Discussions, reflections and small group interactions

Payoff: Engage students thinking about and developing problems of a reflective nature

Important mathematics concepts and procedures can best be taught through problem solving, that is, tasks or problems can and should be posed that engage students in thinking about and developing the important mathematics they need to learn. Van de Walle (2001, p.40)

Why Teach Problem Solving?Problem Solving…

Focuses attention on ideas and sense making.

Develops “mathematical power”.

Helps students develop confidence in doing mathematics

Generates assessment data to make instructional decisions.

Is a lot of FUN!!!

What is a Problem? A problem is any task or activity

where the students have no prescribed or memorized methods or rules for accomplishing the results.

Features of a problem for Learning Mathematics

It must begin where the students are

It must clearly focus on the mathematics students are to learn

It must require explanations for answers and methods

Problem SolvingGroup Dynamics

Individual students

Paris

Groups

Activities

Hands-on materials

Drawings

Paper and pencil

Mental math

Calculators

Japan vs. United States Steps of a typical Japanese

eight-grade mathematics lesson:Teacher poses a complex though-provoking problem. Students struggle with the problem Various students present ideas or solutions to the class. Class discusses the various solution methods.The teacher summarizes the class’ conclusions. Students practice similar problems

Steps of typical American eight-grade mathematics lesson: Teacher instructs student in a concept or skill Teacher solves example problems with classStudents practice on their own while the teacher assists individual students

2 X 21. List two misconceptions of the

teacher’s role in problem solving.

2. List one responsibility of the teacher in creating a problem solving environment.

Developing a Problem Solving Lesson

Before

During

After

Getting Ready

Get students mentally ready to work on the task

Be sure all expectations for products are clear

Students Work

Let go Provide hints

Listen carefully Observe and assess

Classroom Discourse

Accept student solutions with evaluation

Conduct discussions as student justify and evaluate results and methods

Locating Problem Solving Tasks

Your Textbook

Using Traditional Textbooks

Using Reform Programs

Children’s Literature

Other Resources

Teaching Tips and QuestionsFour Suggestions:

Predict! Don’t hope.

Be clear in your own mind about the purpose of the task or activity.

Recognize that there is much more to a problem than the answer.

Do not confuse open-ended problem solving with encouraging creativity.

Problem Solving Steps

• Understand the problem

• Devise a plan

• Carry out the plan

• Reflecting back on the process and outcome

Understanding the Problem • Can you state the problem in your own

words?

• What are you trying to find or do?

• What are the unknowns?

• What information can you obtain from the problem?

• What information is not needed or is missing?

Devising a Plan

• Look for a plan

• Look at a similar problem to determine if the same strategies can be used.

• Make a table

• Make a diagram

• Use guess and check

• Work backward

Carrying out the Plan

• Implement the strategies selected when devising a plan

• Check each step of your plan

• Write all necessary steps to your plan (keeping records)

Reflecting (Looking back)

• Check the results in the original problem

• Look for another method for solving the problem.

• Ask does your answer make sense, is it reasonable?

Building Attitudinal Goals

Build in success

Praise efforts and risk taking

Listen to all students

Provide special successes

“When problem solving is interwoven with learning, students are learning mathematics by doing mathematics!”