# «Research-Based Strategies for Problem-Solving in Mathematics K-12 Florida Department of Education, Bureau of Exceptional Education and Student ...»

Classroom Cognitive

and Meta-Cognitive

Strategies for

Teachers

Research-Based Strategies for

Problem-Solving in Mathematics

K-12

Florida Department of Education,

Bureau of Exceptional Education and Student Services

2010

This is one of the many publications available through the Bureau of Exceptional

Education and Student Services, Florida Department of Education, designed to assist

school districts, state agencies that support educational programs, and parents in the

provision of special programs. For additional information on this publication, or a list of available publications, contact the Clearinghouse Information Center, Bureau of Exceptional Education and Student Services, Florida Department of Education, Room 628, Turlington Building, Tallahassee, Florida 32399-0400.

Telephone: (850) 245-0477 FAX: (850) 245-0987 Suncom: 205-0477 E-mail: cicbiscs@mail.doe.state.fl.us Web site: http://www.fldoe.org/ese Classroom Cognitive and Meta-Cognitive Strategies for Teachers Research-Based Strategies for Problem-Solving in Mathematics K-12 Florida Department of Education, Division of Public Schools and Community Education, Bureau of Exceptional Education and Student Services 2010 This product was developed for PS/RtI, a special project funded by the State of Florida, Department of Education, Bureau of Exceptional Education and Student Services, through federal assistance under the Individuals with Disabilities Education Act (IDEA), Part B.

Copyright State of Florida Department of State 2010 Authorization for reproduction is hereby granted to the State System of Public Education as defined in Section 1006.39 (2), Florida Statutes. No authorization is granted for distribution or reproduction outside the State System of Public Education without prior approval in writing.

Table of Contents Introduction

Step 1: Understanding the Problem

Survey, Question, Read (SQR)

Frayer Vocabulary Model

Mnemonic Devices

Graphic Organizers

Paraphrase

Visualization

Cooperative Learning Groups

Analyze Information

Step 2: Devising a Plan to Understand the Problem............. 27 Hypothesize

Estimating

Disuss/Share Strategies

Guess and Check

Make an Organized List

Look for a Pattern

Eliminating Possibilities

Logical Reasoning

Draw a Picture

Using a Formula

Work Backwards

Explain the Plan

Step 3: Implementing a Solution Plan

Implement Your Own Solution Plan

Step 4: Reflecting on the Problem

Reflect on Plan

Appendices

Resources

References

**According to Polya (1957):**

"One of the first and foremost duties of the teacher is not to give his students the impression that mathematical problems have little connection with each other, and no connection at all with anything else....The teacher should encourage the students to imagine cases in which they could utilize again the procedure used, or apply the result obtained" (p. 15-16).

**The Problem-Solving Process:**

Students can learn to become better problem solvers. Polya’s (1957) “How to Solve It” book presented four phases or areas of problem-solving, which have become the framework often

**recommended for teaching and assessing problem-solving skills. The four steps are:**

1. understanding the problem,

2. devising a plan to solve the problem,

3. implementing the plan, and

4. reflecting on the problem.

facilitate work with problem-solving. This process should be seen as a dynamic, non-linear and flexible approach. Learning these and other problem-solving strategies will enable students to deal more effectively and successfully with most types of mathematical problems.

However, many other strategies could be added. These problem-solving processes could be very useful in mathematics, science, social sciences and other subjects. Students should be encouraged to develop and discover their own problem-solving strategies and become adept at using them for problem-solving. This will help them with their confidence in tackling problem-solving tasks in any situation, and enhance their reasoning skills. As soon as the students develop and refine their own repertoire of problem-solving strategies, teachers can highlight or concentrate on a particular strategy, and discuss aspects and applications of the strategy. As necessary, the students should develop flexibility to choose from the variety of strategies they have learned. We have provided some examples later in this document.

is often the most overlooked step in the problem-solving process. This may seem like an obvious step that doesn’t need mentioning, but in order for a problem-solver to find a solution, they must first understand what they are being asked to find out.

**Polya suggested that teachers should ask students questions similar to the ones listed below:**

Do you understand all the words used in stating the problem?

What are you asked to find or show?

Can you restate the problem in your own words?

Can you think of a picture or a diagram that might help you understand the problem?

Is there enough information to enable you to find a solution?

Teachers should decide which strategies to use based on students’ answers to these questions.

For example, if Jason understands the meaning of all of the words in the problem, he does not need a vocabulary strategy, but if he cannot restate the problem, teaching him to paraphrase would be beneficial.

The following pages contain examples of strategies that teachers can use to support students through the first step in problem-solving.

What is SQR?

The SQR strategy involves expansion and discussion between teacher and students. These discussions often lead to a better student understanding of the problem. This strategy was developed to help students arrive at their own solutions through rich discussion.

How do I use SQR?

Survey Read the problem Paraphrase in your own words Question Question the purpose of the problem

Read Reread the question Determine the exact information you are looking for Eliminate unnecessary information Prepare to devise a plan for solving the problem

What is the Frayer Vocabulary Model?

The Frayer model is a concept map which enables students to make relational connections with vocabulary words.

How do you use it?

1. Identify concept/vocabulary word.

2. Define the word in your own words.

3. List characteristics of the word.

4. List or draw pictures of examples and non-examples of the word.

What are mnemonic devices?

Mnemonic devices are strategies that students and teachers can create to help students remember content. They are memory aids in which specific words are used to remember a concept or a list. The verbal information promotes recall of unfamiliar information and content (Nagel, Schumaker, & Deshler, 1986). Letter strategies include acronyms and acrostics (or sentence mnemonics). For example, “PEMDAS” is commonly used to help students remember the order of operations in mathematics.

How do you use letter strategy mnemonics?

1. Decide on the idea or ideas that the student needs to remember.

2. Show the student the mnemonic that you want them to use.

3. Explain what each letter stands for.

4. Give the students an opportunity to practice using the mnemonic.

Example 1: FIRST is a mnemonic device for creating mnemonics (Mercer & Mercer, 1998).

F - Form a word (from your concepts or ideas). Decide if you can create a word using the first letter of each word. Example: PEMDAS I - Insert extra letters to form a mnemonic (only insert extra letters if you need them to create a word).

R - Rearrange the first letters to form a mnemonic word.

S - Shape a sentence to form a mnemonic (If you cannot form a word from the letters, use them to create a sentence). Example: Please Excuse My Dear Aunt Sally.

Example 2: Ride is for problem-solving (Mercer & Mercer, 1993).

R - Read the problem correctly.

I - Identify the relevant information.

D - Determine the operation and unit for expressing the answer.

E - Enter the correct numbers and calculate.

Mnemonics are placed throughout this document to support students with different steps and strategies. Look for this symbol to mark each mnemonic.

What are graphic organizers?

Graphic organizers are diagrammatic illustrations designed to assist students in representing patterns, interpreting data, and analyzing information relevant to problem-solving (Lovitt, 1994, Ellis, & Sabornie, 1990).

How do you use them?

1. Decide on the appropriate graphic organizer.

2. Model for the students using a familiar concept.

3. Allow the students to practice using the graphic organizer independently.

**Examples:**

Hierarchical Diagramming These graphic organizers begin with a main topic or idea. All information related to the main idea is connected by branches, much like those found in a tree.

The Paraphrasing Strategy is designed to help students restate the math problem in their own words, therefore strengthening their comprehension of the problem (Montague, 2005).

How do I use it?

1. Read the problem.

2. Underline or highlight key terms.

3. Restate the problem in your own words.

4. Write a numerical sentence.

**Example #1:**

Step 1 (Read the problem).

The middle school has 560 lockers available for the beginning of the school year. They have 729 students starting school. How many lockers will they be short on the first day of school?

Step 2 (Underline or highlight key terms).

The middle school has 560 lockers available for the beginning of the school year. They have 729 students starting school. How many lockers will they be short on the first day of school?

Step 3 (Restate the problem in your own words).

** If there are 729 students and only 560 lockers, I need to know how much more 729 is than 560, therefore:**

729 - 560 = 169 lockers are still needed.

**Example #2:**

Step 1 (Read the problem).

A survey shows that 28% of 1,250 people surveyed prefer vanilla ice cream over chocolate or strawberry. How many of the people surveyed prefer vanilla ice cream?

Step 2 (Underline or highlight key terms).

A survey shows that 28% of 1,250 people surveyed prefer vanilla ice cream over chocolate or strawberry. How many of the people surveyed prefer vanilla ice cream?

Step 3 (Restate the problem in your own words).

If there are 1,250 students and 28% of them prefer vanilla ice cream, I need to know what 28% of 1,250 is. I also need to know that the word “of” means multiply. I can change 28%

**into a decimal or into a fraction. Therefore:**

Step 4 (Write a numerical sentence).

1,250 x.28 = 350 people prefer vanilla ice cream.

OR 1,250 x 28 = 350 people prefer vanilla ice cream.

What is visualization?

Visualization in mathematics is the practice of creating pictorial representations of mathematical problems. Students are asked to visualize and then draw the problem, allowing them to obtain a clearer understanding of what the problem is asking.

How do I teach visualization?

1. Read the problem.

2. Have the students underline important images in the problem.

3. Ask the students to draw a visual representation of the problem.

4. Write a numerical sentence.

** Be sure the students are drawing a representation of the problem, not just pictures of the items mentioned in the problem.

Example #1 Step 1 (Read the problem).

There are 5 rabbits, 2 goats, and 6 ducks at the petting zoo. How many animals are at the petting zoo?

Step 2 (Have the students underline important images in the problem).

There are 5 rabbits, 2 goats, and 5 ducks at the petting zoo. How many animals are at the petting zoo?

+ + Step 4 (Write a numerical sentence).

5 + 2 + 5 = 12; Answer: 12 animals are at the petting zoo.

Example #2 Step 1 (Read the problem).

Dave was hiking on a trail that took him to an altitude that was 15 miles below sea level.

Susan hiked to an altitude that was 8 miles above Dave. What was the final altitude for Susan’s hike?

Step 2 (Have the students underline important images in the problem).

Dave was hiking on a trail that took him to an altitude that was 15 miles below sea level.

Susan hiked to an altitude that was 8 miles above Dave. What was the final altitude for Susan’s hike?

Step 3 (Ask the students to draw a visual representation of the problem).

Dave is 15 miles below sea level and Susan is 8 miles above Dave.

The final altitude for Susan can be represented by the expression –15 + 8.

-7 was left over after taking out zero pairs, so the final altitude for Susan’s hike is 7 miles below sea level.

Example #3 Karli has $12 to spend at the grocery store. She must buy 1 gallon of milk and some bags of snacks. The gallon of milk costs $4. How many bags of snacks can she buy if each bag costs $2?

Step 1 (Read the problem).

Karli has $12 to spend at the grocery store. She must buy 1 gallon of milk and some bags of snacks. The gallon of milk costs $4. How many bags of snacks can she buy if each bag costs $2?

Step 2 (Have the students underline important images in the problem).