Mathematical Problem Solving

MATHEMATICAL PROBLEM SOLVING

• What is it? • Why encourage it?

• How is teaching like a problem-solving endeavor?

WHAT TYPES OF QUESTIONS ARE ASKED IN MATH CLASSROOMS?

recognition/recall: a situation that can be resolved by recalling specific facts from memory

drill/exercise: a situation that involves following a step-by-step procedure or algorithm

simple translation/application: traditional textbook word problems which involve translating the words into a single equation, the format of which has typically been taught directly

complex translation/application: similar to a simple translation, but it involves at least two steps

nonroutine/process problem: solutions require the use of thinking processes; a situation that does not have an apparent solution path available–there is no set format available for finding a solution

applied problem: real-world, or at least realistic, situations where solutions require the use of mathematical skills, facts, concepts, and procedures.

open-ended problem: a situation in which the student is often required to identify interesting questions to pursue and/or to identify...not only is there no apparent solution path available, often there is no direct question to purse

SO . . . WHICH OF THESE TYPES OF QUESTIONS ARE

“PROBLEMS?”

WE FIRST NEED TO ASK: WHAT IS A PROBLEM?

A problem is a difficult question; a matter of inquiry, discussion, or thought; a question that exercises the mind.

. . . . and

IT IS A SITUATION OR TASK FOR WHICH:

the person confronting the task wants or needs to find a solution.

the person has no readily available procedure for finding the solution.

the person makes an attempt to find the solution.

SO . . . NOT ALL CATEGORIES OF QUESTIONS ARE PROBLEMS . . . WHEN IS THE MATH OR TASK “PROBLEMATIC?”

WHAT IT MEANS FOR MATH TO BE “PROBLEMATIC”

“Allowing the subject to be problematic means allowing students to wonder why things are, to inquire, to search for solutions, and to resolve incongruities. It means that both curriculum and instruction should begin with problems, dilemmas, and questions for students.”

“Problem Solving as a Basis for Reform in Curriculum and Instructions: The Case of Mathematics”

Hiebert, et al.

Educational Researcher

May, 1996 (p. 12)

WHY ENCOURAGE PROBLEM SOLVING?

THE VALUES OF TEACHING WITH PROBLEMS

Places the focus of attention on ideas and sense making rather than on following the directions of the teacher.

Develops the belief in students that: they are capable of doing mathematics. mathematics makes sense.

Provides ongoing assessment data so that: instructional decisions can be made. students can be helped to succeed. parents can be informed.

Develops “mathematical power.” All five + 1 process standards are likely to be employed.

Allows entry points for a wide range of students. Good problems have multiple entry points.

Engages students so that there are fewer discipline problems.

It’s just a lot of fun . . . for both teachers and students!!!!!

AN IDEA WORTH CONSIDERING: Most mathematical ideas can be taught via problem

solving.

That is—

Tasks or problems are posed to engage students in thinking about and developing the intended mathematical ideas.

WHAT IS GEORGE POLYA’S PROBLEM SOLVING FRAMEWORK?

1. Understand the Problem

Involves not only applying the skills necessary for literary reading, but determining what is being asked.

2. Devise a Plan

Involves finding or devising a strategy to help in solving the problem and answering the question.

3. Carry Out the Plan

Involves attempting to solve the problem with some chosen strategy, recording thinking to keep an accurate record of work.

4. Look Back

Involves interpreting the solution in terms of the original problem . . . Does it make sense? Is it reasonable?

5. Look Forward

Involves asking the “next question,” generalizing the solution, or determining another method of finding the solution.

HOW IS TEACHING MATHEMATICS LIKE A “PROBLEM-SOLVING ENDEAVOR?”

Understand goals for teaching—be clear about what you want to accomplish.

Key question: What mathematics do I want the children to understand and develop?

Plan the tasks—select problems, explorations, or activities that will most likely require children to wrestle meaningfully and thoughtfully with the math you identified as important.

Key question: How can this problem (or activity) be used to help promote reflective thinking about the math I want children to learn?

Implement your plan—facilitate activities and focus appropriately on the needs and strengths of your individual students.

Key question: What “teacher moves” and “math talk” might I use to help develop understanding?

Look back—through active listening in a student-centered classroom, you can learn what students know and how they understand the ideas being discussed.

Key question: Based on what I learn about what my students understand, what direction do I need to go in my future planning for instruction?

PROBLEM-SOLVING STRATEGIES

Guess and checkLook for a patternMake a listSolve a simpler versionDraw a picture

Use symmetryMake a table or chartDo a simulationLook for a formulaConsider cases

Model the situationWork backwardWrite an open sentenceUse logical reasoningConsider all possibilitiesChange your point of view

SO . . .

. . . HOW DO “TRADITIONAL PROBLEMS” DIFFER FROM THOSE THAT ARE TYPICALLY MORE AUTHENTIC—MORE CLOSELY RELATED TO REAL LIFE?

DIFFERENCES BETWEEN TRADITIONAL WORD PROBLEMS AND MANY REAL-LIFE PROBLEMS

Typical TextbookProblems

The problem is given.

All the information you need to solve the problem is given.

There is always enough information to solve the problem.

Typical Authentic Problems

Often, you have to figure out what the problem really is.

You have to determine the information needed to solve the problem.

Sometimes you will find that there is not enough info to solve the problem.

There is typically no extraneous information.

The answer is in the back of the book . . .

There is usually a right or best way to solve the problem.

Sometimes there is too much information--must decide what is needed.

You, or your team, decides whether your answer is valid.

There are usually many different ways to solve the problem.

Typical Textbook Word Problems Typical Authentic Problems

GOALS FOR PROBLEM SOLVINGStrategy and Process Goals

Develop problem analysis skills

Develop and be able to select strategies

Justify solutions

Extend and generalize problems

Metacognitive Goals

Monitor and regulate actions

Determine when to try a new strategy

Determine when a solution leads to a correct solution

Attitudinal Goals

Gain confidence and belief in abilities

Be willing to try and to persevere

Enjoy doing mathematics