Problem Solving

The process of applying previously acquired knowledge to new and

unfamiliar situations

NCTM Problem Solving Standard

Instructional programs from pre-K-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

The Handshake ProblemThe evening before classes begin at Knox College, a "Pumphandle" is held. Suppose a thousand faculty and students participate. If everyone shakes hands with everyone else once, how many handshakes occur during the Pumphandle?

If you were the 500th person to enter the Pumphandle line, how many hands would you shake by the end of the evening?

3-5 Minutes

Try it by yourself…

3-5 Minutes

Discuss with a partner…

How did you approach the problem?

Act it out

Guess and check

Compute

Simplify

Use a formula

Write an equation

Make a table

Consider a simpler case

Look for patterns

Use drawings

These are called…

Act it out

Guess and check

Compute

Simplify

Use a formula

Write an equation

Make a table

Consider a simpler case

Look for patterns

Use drawings

Heuristics

Act it out

Guess and check

Compute

Simplify

Use a formula

Write an equation

Make a table

Consider a simpler case

Look for patterns

Use drawings

Why are heuristics important?

How do heuristics help problem solvers?

Problems

Exercises

VS

Problems

Exercises

VS What do you think makes a problem different than an exercise?

Let’s try to identify a few…

Is it a problem or exercise?

How many squares are in the picture below?

Is it a problem or exercise?

Tonya traveled 175 miles in 3.25 hours to go on vacation. What is the average speed she was traveling?

Is it a problem or exercise?

Find 30% of 92.

Is it a problem or exercise?A Tale of the Cats

The lucky cats on Stratton Street Had seven mice apiece to eat. The rest made do With only two; The total score Being twenty-four. How many cats ate mousie meat?

This is a poem from the book Fun With Figures by J. A. H. Hunter (1965, Dover

Publishing).

Problems VS Exercises

•Uses words and/or symbols

•Require prior knowledge of the topic

•Can use calculators to solve

•Challenging

•Solution is not immediately obvious

•Multiple ways to approach/solve

•Use higher level thinking skills

•Apply a strategy/ heuristic

•Math in a context

•Uses algorithms

•Not in a context

•Drill and practice

•Knowledge/ recall level reasoning and thinking

Problem Solving:A Head Scratching Dilemma

• What do you do when you first read a problem and have NO IDEA what it said?

• How do you make sense of an unfamiliar situation?

• How do you help your students to focus on thinking about the problem?

Problems That Emphasize Understanding

and Logical Thinking

• How can two fathers and two sons divide three automobiles among themselves with each receiving one?

Problems That Emphasize Understanding

and Logical Thinking

• Some months have 30 days, some have 31. How many have 28 days?

Problems That Emphasize Understanding

and Logical Thinking

• I have two U.S. coins in my hand which total fifty-five cents. One is not a nickel. What are the coins?

It helps to have an organized process.

Polya’s Problem Solving Process

1. Understand the problem

2. Devise a plan

3. Carry out the plan

4. Look back

Understand the Problem

• Reread the problem

• What are you trying to find out?

• What facts do you have?

• What is the problem asking?

• What do you know that is not stated in the problem?

• What questions do you have about the situation?

Understand the Problem

• Draw a picture to help you understand the problem.

• Try a specific case to help you learn about it.

• Is it possible to solve the problem?• Is it a reasonable problem? Why or why

not?• Restate the problem in your own words

Devise a Plan

• Do you know a related problem?• Think of a familiar problem where you

have to answer a similar question.• Here is another problem related to yours.

Could you use it to help you?• Try to solve a simpler problem.• What strategies (heuristics) do you

know?

Devise a Plan

• How can you organize the information given?

• What plan of attack will you use?

• Can you develop a pattern?

• Guess a solution or a first step.

• Can you write an equation?

• Display examples or cases in a chart or table.

Carry Out the Plan

• Use the strategy/heuristic you identified in the “Devise a Plan” phase

• Sometimes the original strategy you begin you leads to another

• Work the problem/solve

Look Back

• Did you answer the question(s) asked?

• Is your answer in the correct unit?

• Does your answer seem reasonable?

• Describe your solution process.

• Are there extensions/generalizations you can make?

• Could you have solved the same problem another way?

Look Back

• Could you have solved the same problem another way?

• Could you use the same method for another problem?

• What problems are similar to this one?

• Compare your solution to your guess or estimate.

Look Back

• What are the advantages & disadvantages of your method?

• What were the pitfalls in this problem?

• Verbalize the stumbling blocks you overcame to solve this problem.

• What solution strategies were most helpful in solving the problem?

Making Problem Solving the Focus of School Mathematics

• Going beyond the textbook to create meaningful problem-solving situations

• Teaching problem-solving strategies and skills

• Helping students experience success in problem solving to motivate them to do more

• Modeling good problem solving behaviors for the students

Making Problem Solving the Focus of School Mathematics

• Teaching computational skills in the context of problem solving

• Using a problem-solving approach to teaching computation, helping students understand the algorithms they use

• Weaving problem-solving instruction into the entire curriculum

Problem solving in my classroom may look like…

• How could you introduce this process to your students?

• How can you develop their understanding of the process?

• How can you organize this process?

• What are the roadblocks you anticipate?