steps in problem solving in math for elementary

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20 Effective Math Strategies To Approach Problem-Solving 

Katie Keeton

Math strategies for problem-solving help students use a range of approaches to solve many different types of problems. It involves identifying the problem and carrying out a plan of action to find the answer to mathematical problems.  

Problem-solving skills are essential to math in the general classroom and real-life. They require logical reasoning and critical thinking skills. Students must be equipped with strategies to help them find solutions to problems.

This article explores mathematical problem solving strategies, logical reasoning and critical thinking skills to help learners with solving math word problems independently in real-life situations. 

What are problem-solving strategies?

Problem-solving strategies in math are methods students can use to figure out solutions to math problems. Some problem-solving strategies: 

• Draw a model
• Use different approaches
• Check the inverse to make sure the answer is correct

Students need to have a toolkit of math problem-solving strategies at their disposal to provide different ways to approach math problems. This makes it easier to find solutions and understand math better. 

Strategies can help guide students to the solution when it is difficult ot know when to start.

The ultimate guide to problem solving techniques

Download these ready-to-go problem solving techniques that every student should know. Includes printable tasks for students including challenges, short explanations for teachers with questioning prompts.

20 Math Strategies For Problem-Solving

Different problem-solving math strategies are required for different parts of the problem. It is unlikely that students will use the same strategy to understand and solve the problem. 

Here are 20 strategies to help students develop their problem-solving skills. 

Strategies to understand the problem

Strategies that help students understand the problem before solving it helps ensure they understand: 

• The context
• What the key information is
• How to form a plan to solve it

Following these steps leads students to the correct solution and makes the math word problem easier .

Here are five strategies to help students understand the content of the problem and identify key information. 

1. Read the problem aloud

Read a word problem aloud to help understand it. Hearing the words engages auditory processing. This can make it easier to process and comprehend the context of the situation.

2. Highlight keywords 

When keywords are highlighted in a word problem, it helps the student focus on the essential information needed to solve it. Some important keywords help determine which operation is needed.  For example, if the word problem asks how many are left, the problem likely requires subtraction.  Ensure students highlight the keywords carefully and do not highlight every number or keyword. There is likely irrelevant information in the word problem.

3. Summarize the information

Read the problem aloud, highlight the key information and then summarize the information. Students can do this in their heads or write down a quick summary.  Summaries should include only the important information and be in simple terms that help contextualize the problem.

4. Determine the unknown

A common problem that students have when solving a word problem is misunderstanding what they are solving. Determine what the unknown information is before finding the answer.  Often, a word problem contains a question where you can find the unknown information you need to solve. For example, in the question ‘How many apples are left?’ students need to find the number of apples left over.

5. Make a plan

Once students understand the context of the word problem, have dentified the important information and determined the unknown, they can make a plan to solve it.  The plan will depend on the type of problem. Some problems involve more than one step to solve them as some require more than one answer.  Encourage students to make a list of each step they need to take to solve the problem before getting started.

Strategies for solving the problem 

1. draw a model or diagram.

Students may find it useful to draw a model, picture, diagram, or other visual aid to help with the problem solving process.  It can help to visualize the problem to understand the relationships between the numbers in the problem. In turn, this helps students see the solution.

Similarly, you could draw a model to represent the objects in the problem:

2. Act it out

This particular strategy is applicable at any grade level but is especially helpful in math investigation in elementary school . It involves a physical demonstration or students acting out the problem using movements, concrete resources and math manipulatives .  When students act out a problem, they can visualize and contectualize the word problem in another way and secure an understanding of the math concepts.  The examples below show how 1st-grade students could “act out” an addition and subtraction problem:

3. Work backwards

Working backwards is a popular problem-solving strategy. It involves starting with a possible solution and deciding what steps to take to arrive at that solution.  This strategy can be particularly helpful when students solve math word problems involving multiple steps. They can start at the end and think carefully about each step taken as opposed to jumping to the end of the problem and missing steps in between.

For example,

To solve this problem working backwards, start with the final condition, which is Sam’s grandmother’s age (71) and work backwards to find Sam’s age. Subtract 20 from the grandmother’s age, which is 71.  Then, divide the result by 3 to get Sam’s age. 71 – 20 = 51 51 ÷ 3 = 17 Sam is 17 years old.

4. Write a number sentence

When faced with a word problem, encourage students to write a number sentence based on the information. This helps translate the information in the word problem into a math equation or expression, which is more easily solved.  It is important to fully understand the context of the word problem and what students need to solve before writing an equation to represent it.

5. Use a formula

Specific formulas help solve many math problems. For example, if a problem asks students to find the area of a rug, they would use the area formula (area = length × width) to solve.   Make sure students know the important mathematical formulas they will need in tests and real-life. It can help to display these around the classroom or, for those who need more support, on students’ desks.

Strategies for checking the solution 

Once the problem is solved using an appropriate strategy, it is equally important to check the solution to ensure it is correct and makes sense. 

There are many strategies to check the solution. The strategy for a specific problem is dependent on the problem type and math content involved.

Here are five strategies to help students check their solutions. 

1. Use the Inverse Operation

For simpler problems, a quick and easy problem solving strategy is to use the inverse operation. For example, if the operation to solve a word problem is 56 ÷ 8 = 7 students can check the answer is correct by multiplying 8 × 7. As good practice, encourage students to use the inverse operation routinely to check their work. 

2. Estimate to check for reasonableness

Once students reach an answer, they can use estimation or rounding to see if the answer is reasonable.  Round each number in the equation to a number that’s close and easy to work with, usually a multiple of ten.  For example, if the question was 216 ÷ 18 and the quotient was 12, students might round 216 to 200 and round 18 to 20. Then use mental math to solve 200 ÷ 20, which is 10.  When the estimate is clear the two numbers are close. This means your answer is reasonable. 

3. Plug-In Method

This method is particularly useful for algebraic equations. Specifically when working with variables.  To use the plug-in method, students solve the problem as asked and arrive at an answer. They can then plug the answer into the original equation to see if it works. If it does, the answer is correct.

If students use the equation 20m+80=300 to solve this problem and find that m = 11, they can plug that value back into the equation to see if it is correct. 20m + 80 = 300 20 (11) + 80 = 300 220 + 80 = 300 300 = 300 ✓

4. Peer Review

Peer review is a great tool to use at any grade level as it promotes critical thinking and collaboration between students. The reviewers can look at the problem from a different view as they check to see if the problem was solved correctly.   Problem solvers receive immediate feedback and the opportunity to discuss their thinking with their peers. This strategy is effective with mixed-ability partners or similar-ability partners. In mixed-ability groups, the partner with stronger skills provides guidance and support to the partner with weaker skills, while reinforcing their own understanding of the content and communication skills.  If partners have comparable ability levels and problem-solving skills, they may find that they approach problems differently or have unique insights to offer each other about the problem-solving process.

5. Use a Calculator

A calculator can be introduced at any grade level but may be best for older students who already have a foundational understanding of basic math operations. Provide students with a calculator to allow them to check their solutions independently, accurately, and quickly. Since calculators are so readily available on smartphones and tablets, they allow students to develop practical skills that apply to real-world situations.  

Step-by-step problem-solving processes for your classroom

In his book, How to Solve It , published in 1945, mathematician George Polya introduced a 4-step process to solve problems. 

Polya’s 4 steps include:

• Understand the problem
• Devise a plan
• Carry out the plan

Today, in the style of George Polya, many problem-solving strategies use various acronyms and steps to help students recall. 

Many teachers create posters and anchor charts of their chosen process to display in their classrooms. They can be implemented in any elementary, middle school or high school classroom. 

Here are 5 problem-solving strategies to introduce to students and use in the classroom.

How Third Space Learning improves problem-solving 

Resources .

Third Space Learning offers a free resource library is filled with hundreds of high-quality resources. A team of experienced math experts carefully created each resource to develop students mental arithmetic, problem solving and critical thinking. 

Explore the range of problem solving resources for 2nd to 8th grade students. 

One-on-one tutoring 

Third Space Learning offers one-on-one math tutoring to help students improve their math skills. Highly qualified tutors deliver high-quality lessons aligned to state standards. 

Former teachers and math experts write all of Third Space Learning’s tutoring lessons. Expertly designed lessons follow a “my turn, follow me, your turn” pedagogy to help students move from guided instruction and problem-solving to independent practice. 

Throughout each lesson, tutors ask higher-level thinking questions to promote critical thinking and ensure students are developing a deep understanding of the content and problem-solving skills.

Problem-solving

Educators can use many different strategies to teach problem-solving and help students develop and carry out a plan when solving math problems. Incorporate these math strategies into any math program and use them with a variety of math concepts, from whole numbers and fractions to algebra. 

Teaching students how to choose and implement problem-solving strategies helps them develop mathematical reasoning skills and critical thinking they can apply to real-life problem-solving.

READ MORE : 8 Common Core math examples

There are many different strategies for problem-solving; Here are 5 problem-solving strategies: • draw a model  • act it out  • work backwards  • write a number sentence • use a formula

Here are 10 strategies of problem-solving: • Read the problem aloud • Highlight keywords • Summarize the information • Determine the unknown • Make a plan • Draw a model  • Act it out  • Work backwards  • Write a number sentence • Use a formula

1. Understand the problem 2. Devise a plan 3. Carry out the plan 4. Look back

Some strategies you can use to solve challenging math problems are: breaking the problem into smaller parts, using diagrams or models, applying logical reasoning, and trying different approaches.

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13 Effective Learning Strategies: A Guide to Using them in your Math Classroom

Differentiated Instruction: 9 Differentiated Curriculum And Instruction Strategies For Teachers 

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Ultimate Guide to Metacognition [FREE]

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3.1 Use a Problem-Solving Strategy

Learning objectives.

By the end of this section, you will be able to:

• Approach word problems with a positive attitude
• Use a problem-solving strategy for word problems
• Solve number problems

Be Prepared 3.1

Before you get started, take this readiness quiz.

• Translate “6 less than twice x ” into an algebraic expression. If you missed this problem, review Example 1.26 .
• Solve: 2 3 x = 24 . 2 3 x = 24 . If you missed this problem, review Example 2.16 .
• Solve: 3 x + 8 = 14 . 3 x + 8 = 14 . If you missed this problem, review Example 2.27 .

Approach Word Problems with a Positive Attitude

“If you think you can… or think you can’t… you’re right.”—Henry Ford

The world is full of word problems! Will my income qualify me to rent that apartment? How much punch do I need to make for the party? What size diamond can I afford to buy my girlfriend? Should I fly or drive to my family reunion?

How much money do I need to fill the car with gas? How much tip should I leave at a restaurant? How many socks should I pack for vacation? What size turkey do I need to buy for Thanksgiving dinner, and then what time do I need to put it in the oven? If my sister and I buy our mother a present, how much does each of us pay?

Now that we can solve equations, we are ready to apply our new skills to word problems. Do you know anyone who has had negative experiences in the past with word problems? Have you ever had thoughts like the student below?

When we feel we have no control, and continue repeating negative thoughts, we set up barriers to success. We need to calm our fears and change our negative feelings.

Start with a fresh slate and begin to think positive thoughts. If we take control and believe we can be successful, we will be able to master word problems! Read the positive thoughts in Figure 3.3 and say them out loud.

Think of something, outside of school, that you can do now but couldn’t do 3 years ago. Is it driving a car? Snowboarding? Cooking a gourmet meal? Speaking a new language? Your past experiences with word problems happened when you were younger—now you’re older and ready to succeed!

Use a Problem-Solving Strategy for Word Problems

We have reviewed translating English phrases into algebraic expressions, using some basic mathematical vocabulary and symbols. We have also translated English sentences into algebraic equations and solved some word problems. The word problems applied math to everyday situations. We restated the situation in one sentence, assigned a variable, and then wrote an equation to solve the problem. This method works as long as the situation is familiar and the math is not too complicated.

Now, we’ll expand our strategy so we can use it to successfully solve any word problem. We’ll list the strategy here, and then we’ll use it to solve some problems. We summarize below an effective strategy for problem solving.

Use a Problem-Solving Strategy to Solve Word Problems.

• Step 1. Read the problem. Make sure all the words and ideas are understood.
• Step 2. Identify what we are looking for.
• Step 3. Name what we are looking for. Choose a variable to represent that quantity.
• Step 4. Translate into an equation. It may be helpful to restate the problem in one sentence with all the important information. Then, translate the English sentence into an algebraic equation.
• Step 5. Solve the equation using good algebra techniques.
• Step 6. Check the answer in the problem and make sure it makes sense.
• Step 7. Answer the question with a complete sentence.

Example 3.1

Pilar bought a purse on sale for $18, which is one-half of the original price. What was the original price of the purse?

Step 1. Read the problem. Read the problem two or more times if necessary. Look up any unfamiliar words in a dictionary or on the internet.

• In this problem, is it clear what is being discussed? Is every word familiar?

Step 2. Identify what you are looking for. Did you ever go into your bedroom to get something and then forget what you were looking for? It’s hard to find something if you are not sure what it is! Read the problem again and look for words that tell you what you are looking for!

• In this problem, the words “what was the original price of the purse” tell us what we need to find.

Step 3. Name what we are looking for. Choose a variable to represent that quantity. We can use any letter for the variable, but choose one that makes it easy to remember what it represents.

• Let p = p = the original price of the purse.

Step 4. Translate into an equation. It may be helpful to restate the problem in one sentence with all the important information. Translate the English sentence into an algebraic equation.

Reread the problem carefully to see how the given information is related. Often, there is one sentence that gives this information, or it may help to write one sentence with all the important information. Look for clue words to help translate the sentence into algebra. Translate the sentence into an equation.

Step 5. Solve the equation using good algebraic techniques. Even if you know the solution right away, using good algebraic techniques here will better prepare you to solve problems that do not have obvious answers.

Step 6. Check the answer in the problem to make sure it makes sense. We solved the equation and found that p = 36 , p = 36 , which means “the original price” was $36.

• Does $36 make sense in the problem? Yes, because 18 is one-half of 36, and the purse was on sale at half the original price.

Step 7. Answer the question with a complete sentence. The problem asked “What was the original price of the purse?”

• The answer to the question is: “The original price of the purse was $36.”

If this were a homework exercise, our work might look like this:

Pilar bought a purse on sale for $18, which is one-half the original price. What was the original price of the purse?

Joaquin bought a bookcase on sale for $120, which was two-thirds of the original price. What was the original price of the bookcase?

Two-fifths of the songs in Mariel’s playlist are country. If there are 16 country songs, what is the total number of songs in the playlist?

Let’s try this approach with another example.

Example 3.2

Ginny and her classmates formed a study group. The number of girls in the study group was three more than twice the number of boys. There were 11 girls in the study group. How many boys were in the study group?

Guillermo bought textbooks and notebooks at the bookstore. The number of textbooks was 3 more than twice the number of notebooks. He bought 7 textbooks. How many notebooks did he buy?

Gerry worked Sudoku puzzles and crossword puzzles this week. The number of Sudoku puzzles he completed is eight more than twice the number of crossword puzzles. He completed 22 Sudoku puzzles. How many crossword puzzles did he do?

Solve Number Problems

Now that we have a problem solving strategy, we will use it on several different types of word problems. The first type we will work on is “number problems.” Number problems give some clues about one or more numbers. We use these clues to write an equation. Number problems don’t usually arise on an everyday basis, but they provide a good introduction to practicing the problem solving strategy outlined above.

Example 3.3

The difference of a number and six is 13. Find the number.

The difference of a number and eight is 17. Find the number.

The difference of a number and eleven is −7 . −7 . Find the number.

Example 3.4

The sum of twice a number and seven is 15. Find the number.

Did you notice that we left out some of the steps as we solved this equation? If you’re not yet ready to leave out these steps, write down as many as you need.

The sum of four times a number and two is 14. Find the number.

The sum of three times a number and seven is 25. Find the number.

Some number word problems ask us to find two or more numbers. It may be tempting to name them all with different variables, but so far we have only solved equations with one variable. In order to avoid using more than one variable, we will define the numbers in terms of the same variable. Be sure to read the problem carefully to discover how all the numbers relate to each other.

Example 3.5

One number is five more than another. The sum of the numbers is 21. Find the numbers.

One number is six more than another. The sum of the numbers is twenty-four. Find the numbers.

Try It 3.10

The sum of two numbers is fifty-eight. One number is four more than the other. Find the numbers.

Example 3.6

The sum of two numbers is negative fourteen. One number is four less than the other. Find the numbers.

Try It 3.11

The sum of two numbers is negative twenty-three. One number is seven less than the other. Find the numbers.

Try It 3.12

The sum of two numbers is −18 . −18 . One number is 40 more than the other. Find the numbers.

Example 3.7

One number is ten more than twice another. Their sum is one. Find the numbers.

Try It 3.13

One number is eight more than twice another. Their sum is negative four. Find the numbers.

Try It 3.14

One number is three more than three times another. Their sum is −5 . −5 . Find the numbers.

Some number problems involve consecutive integers. Consecutive integers are integers that immediately follow each other. Examples of consecutive integers are:

Notice that each number is one more than the number preceding it. So if we define the first integer as n , the next consecutive integer is n + 1 . n + 1 . The one after that is one more than n + 1 , n + 1 , so it is n + 1 + 1 , n + 1 + 1 , which is n + 2 . n + 2 .

Example 3.8

The sum of two consecutive integers is 47. Find the numbers.

Try It 3.15

The sum of two consecutive integers is 95 . 95 . Find the numbers.

Try It 3.16

The sum of two consecutive integers is −31 . −31 . Find the numbers.

Example 3.9

Find three consecutive integers whose sum is −42 . −42 .

Try It 3.17

Find three consecutive integers whose sum is −96 . −96 .

Try It 3.18

Find three consecutive integers whose sum is −36 . −36 .

Now that we have worked with consecutive integers, we will expand our work to include consecutive even integers and consecutive odd integers. Consecutive even integers are even integers that immediately follow one another. Examples of consecutive even integers are:

Notice each integer is 2 more than the number preceding it. If we call the first one n , then the next one is n + 2 . n + 2 . The next one would be n + 2 + 2 n + 2 + 2 or n + 4 . n + 4 .

Consecutive odd integers are odd integers that immediately follow one another. Consider the consecutive odd integers 77, 79, and 81.

Does it seem strange to add 2 (an even number) to get from one odd integer to the next? Do you get an odd number or an even number when we add 2 to 3? to 11? to 47?

Whether the problem asks for consecutive even numbers or odd numbers, you don’t have to do anything different. The pattern is still the same—to get from one odd or one even integer to the next, add 2.

Example 3.10

Find three consecutive even integers whose sum is 84.

Try It 3.19

Find three consecutive even integers whose sum is 102.

Try It 3.20

Find three consecutive even integers whose sum is −24 . −24 .

Example 3.11

A married couple together earns $110,000 a year. The wife earns $16,000 less than twice what her husband earns. What does the husband earn?

Try It 3.21

According to the National Automobile Dealers Association, the average cost of a car in 2014 was $28,500. This was $1,500 less than 6 times the cost in 1975. What was the average cost of a car in 1975?

Try It 3.22

U.S. Census data shows that the median price of new home in the United States in November 2014 was $280,900. This was $10,700 more than 14 times the price in November 1964. What was the median price of a new home in November 1964?

Section 3.1 Exercises

Practice makes perfect.

Use the Approach Word Problems with a Positive Attitude

In the following exercises, prepare the lists described.

List five positive thoughts you can say to yourself that will help you approach word problems with a positive attitude. You may want to copy them on a sheet of paper and put it in the front of your notebook, where you can read them often.

List five negative thoughts that you have said to yourself in the past that will hinder your progress on word problems. You may want to write each one on a small piece of paper and rip it up to symbolically destroy the negative thoughts.

In the following exercises, solve using the problem solving strategy for word problems. Remember to write a complete sentence to answer each question.

Two-thirds of the children in the fourth-grade class are girls. If there are 20 girls, what is the total number of children in the class?

Three-fifths of the members of the school choir are women. If there are 24 women, what is the total number of choir members?

Zachary has 25 country music CDs, which is one-fifth of his CD collection. How many CDs does Zachary have?

One-fourth of the candies in a bag of M&M’s are red. If there are 23 red candies, how many candies are in the bag?

There are 16 girls in a school club. The number of girls is four more than twice the number of boys. Find the number of boys.

There are 18 Cub Scouts in Pack 645. The number of scouts is three more than five times the number of adult leaders. Find the number of adult leaders.

Huong is organizing paperback and hardback books for her club’s used book sale. The number of paperbacks is 12 less than three times the number of hardbacks. Huong had 162 paperbacks. How many hardback books were there?

Jeff is lining up children’s and adult bicycles at the bike shop where he works. The number of children’s bicycles is nine less than three times the number of adult bicycles. There are 42 adult bicycles. How many children’s bicycles are there?

Philip pays $1,620 in rent every month. This amount is $120 more than twice what his brother Paul pays for rent. How much does Paul pay for rent?

Marc just bought an SUV for $54,000. This is $7,400 less than twice what his wife paid for her car last year. How much did his wife pay for her car?

Laurie has $46,000 invested in stocks and bonds. The amount invested in stocks is $8,000 less than three times the amount invested in bonds. How much does Laurie have invested in bonds?

Erica earned a total of $50,450 last year from her two jobs. The amount she earned from her job at the store was $1,250 more than three times the amount she earned from her job at the college. How much did she earn from her job at the college?

In the following exercises, solve each number word problem.

The sum of a number and eight is 12. Find the number.

The sum of a number and nine is 17. Find the number.

The difference of a number and 12 is three. Find the number.

The difference of a number and eight is four. Find the number.

The sum of three times a number and eight is 23. Find the number.

The sum of twice a number and six is 14. Find the number.

The difference of twice a number and seven is 17. Find the number.

The difference of four times a number and seven is 21. Find the number.

Three times the sum of a number and nine is 12. Find the number.

Six times the sum of a number and eight is 30. Find the number.

One number is six more than the other. Their sum is 42. Find the numbers.

One number is five more than the other. Their sum is 33. Find the numbers.

The sum of two numbers is 20. One number is four less than the other. Find the numbers.

The sum of two numbers is 27. One number is seven less than the other. Find the numbers.

The sum of two numbers is −45 . −45 . One number is nine more than the other. Find the numbers.

The sum of two numbers is −61 . −61 . One number is 35 more than the other. Find the numbers.

The sum of two numbers is −316 . −316 . One number is 94 less than the other. Find the numbers.

The sum of two numbers is −284 . −284 . One number is 62 less than the other. Find the numbers.

One number is 14 less than another. If their sum is increased by seven, the result is 85. Find the numbers.

One number is 11 less than another. If their sum is increased by eight, the result is 71. Find the numbers.

One number is five more than another. If their sum is increased by nine, the result is 60. Find the numbers.

One number is eight more than another. If their sum is increased by 17, the result is 95. Find the numbers.

One number is one more than twice another. Their sum is −5 . −5 . Find the numbers.

One number is six more than five times another. Their sum is six. Find the numbers.

The sum of two numbers is 14. One number is two less than three times the other. Find the numbers.

The sum of two numbers is zero. One number is nine less than twice the other. Find the numbers.

The sum of two consecutive integers is 77. Find the integers.

The sum of two consecutive integers is 89. Find the integers.

The sum of two consecutive integers is −23 . −23 . Find the integers.

The sum of two consecutive integers is −37 . −37 . Find the integers.

The sum of three consecutive integers is 78. Find the integers.

The sum of three consecutive integers is 60. Find the integers.

Find three consecutive integers whose sum is −3 . −3 .

Find three consecutive even integers whose sum is 258.

Find three consecutive even integers whose sum is 222.

Find three consecutive odd integers whose sum is 171.

Find three consecutive odd integers whose sum is 291.

Find three consecutive even integers whose sum is −36 . −36 .

Find three consecutive even integers whose sum is −84 . −84 .

Find three consecutive odd integers whose sum is −213 . −213 .

Find three consecutive odd integers whose sum is −267 . −267 .

Everyday Math

Sale Price Patty paid $35 for a purse on sale for $10 off the original price. What was the original price of the purse?

Sale Price Travis bought a pair of boots on sale for $25 off the original price. He paid $60 for the boots. What was the original price of the boots?

Buying in Bulk Minh spent $6.25 on five sticker books to give his nephews. Find the cost of each sticker book.

Buying in Bulk Alicia bought a package of eight peaches for $3.20. Find the cost of each peach.

Price before Sales Tax Tom paid $1,166.40 for a new refrigerator, including $86.40 tax. What was the price of the refrigerator?

Price before Sales Tax Kenji paid $2,279 for a new living room set, including $129 tax. What was the price of the living room set?

Writing Exercises

What has been your past experience solving word problems?

When you start to solve a word problem, how do you decide what to let the variable represent?

What are consecutive odd integers? Name three consecutive odd integers between 50 and 60.

What are consecutive even integers? Name three consecutive even integers between −50 −50 and −40 . −40 .

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

ⓑ If most of your checks were:

…confidently. Congratulations! You have achieved your goals in this section! Reflect on the study skills you used so that you can continue to use them. What did you do to become confident of your ability to do these things? Be specific!

…with some help. This must be addressed quickly as topics you do not master become potholes in your road to success. Math is sequential—every topic builds upon previous work. It is important to make sure you have a strong foundation before you move on. Who can you ask for help? Your fellow classmates and instructor are good resources. Is there a place on campus where math tutors are available? Can your study skills be improved?

…no—I don’t get it! This is critical and you must not ignore it. You need to get help immediately or you will quickly be overwhelmed. See your instructor as soon as possible to discuss your situation. Together you can come up with a plan to get you the help you need.

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Teaching Problem Solving in Math

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Every year my students can be fantastic at math…until they start to see math with words. For some reason, once math gets translated into reading, even my best readers start to panic. There is just something about word problems, or problem-solving, that causes children to think they don’t know how to complete them.

Every year in math, I start off by teaching my students problem-solving skills and strategies. Every year they moan and groan that they know them. Every year – paragraph one above. It was a vicious cycle. I needed something new.

I put together a problem-solving unit that would focus a bit more on strategies and steps in hopes that that would create problem-solving stars.

The Problem Solving Strategies

First, I wanted to make sure my students all learned the different strategies to solve problems, such as guess-and-check, using visuals (draw a picture, act it out, and modeling it), working backward, and organizational methods (tables, charts, and lists). In the past, I had used worksheet pages that would introduce one and provide the students with plenty of problems practicing that one strategy. I did like that because students could focus more on practicing the strategy itself, but I also wanted students to know when to use it, too, so I made sure they had both to practice.

I provided students with plenty of practice of the strategies, such as in this guess-and-check game.

There’s also this visuals strategy wheel practice.

I also provided them with paper dolls and a variety of clothing to create an organized list to determine just how many outfits their “friend” would have.

Then, as I said above, we practiced in a variety of ways to make sure we knew exactly when to use them. I really wanted to make sure they had this down!

Anyway, after I knew they had down the various strategies and when to use them, then we went into the actual problem-solving steps.

The Problem Solving Steps

I wanted students to understand that when they see a story problem, it isn’t scary. Really, it’s just the equation written out in words in a real-life situation. Then, I provided them with the “keys to success.”

S tep 1 – Understand the Problem.   To help students understand the problem, I provided them with sample problems, and together we did five important things:

• read the problem carefully
• restated the problem in our own words
• crossed out unimportant information
• circled any important information
• stated the goal or question to be solved

We did this over and over with example problems.

Once I felt the students had it down, we practiced it in a game of problem-solving relay. Students raced one another to see how quickly they could get down to the nitty-gritty of the word problems. We weren’t solving the problems – yet.

Then, we were on to Step 2 – Make a Plan . We talked about how this was where we were going to choose which strategy we were going to use. We also discussed how this was where we were going to figure out what operation to use. I taught the students Sheila Melton’s operation concept map.

We talked about how if you know the total and know if it is equal or not, that will determine what operation you are doing. So, we took an example problem, such as:

Sheldon wants to make a cupcake for each of his 28 classmates. He can make 7 cupcakes with one box of cupcake mix. How many boxes will he need to buy?

We started off by asking ourselves, “Do we know the total?” We know there are a total of 28 classmates. So, yes, we are separating. Then, we ask, “Is it equal?” Yes, he wants to make a cupcake for EACH of his classmates. So, we are dividing: 28 divided by 7 = 4. He will need to buy 4 boxes. (I actually went ahead and solved it here – which is the next step, too.)

Step 3 – Solving the problem . We talked about how solving the problem involves the following:

• taking our time
• working the problem out
• showing all our work
• estimating the answer
• using thinking strategies

We talked specifically about thinking strategies. Just like in reading, there are thinking strategies in math. I wanted students to be aware that sometimes when we are working on a problem, a particular strategy may not be working, and we may need to switch strategies. We also discussed that sometimes we may need to rethink the problem, to think of related content, or to even start over. We discussed these thinking strategies:

• switch strategies or try a different one
• rethink the problem
• think of related content
• decide if you need to make changes
• check your work
• but most important…don’t give up!

To make sure they were getting in practice utilizing these thinking strategies, I gave each group chart paper with a letter from a fellow “student” (not a real student), and they had to give advice on how to help them solve their problem using the thinking strategies above.

Finally, Step 4 – Check It.   This is the step that students often miss. I wanted to emphasize just how important it is! I went over it with them, discussing that when they check their problems, they should always look for these things:

• compare your answer to your estimate
• check for reasonableness
• check your calculations
• add the units
• restate the question in the answer
• explain how you solved the problem

Then, I gave students practice cards. I provided them with example cards of “students” who had completed their assignments already, and I wanted them to be the teacher. They needed to check the work and make sure it was completed correctly. If it wasn’t, then they needed to tell what they missed and correct it.

To demonstrate their understanding of the entire unit, we completed an adorable lap book (my first time ever putting together one or even creating one – I was surprised how well it turned out, actually). It was a great way to put everything we discussed in there.

Once we were all done, students were officially Problem Solving S.T.A.R.S. I just reminded students frequently of this acronym.

Stop – Don’t rush with any solution; just take your time and look everything over.

Think – Take your time to think about the problem and solution.

Act  – Act on a strategy and try it out.

Review – Look it over and see if you got all the parts.

Wow, you are a true trooper sticking it out in this lengthy post! To sum up the majority of what I have written here, I have some problem-solving bookmarks FREE to help you remember and to help your students!

You can grab these problem-solving bookmarks for FREE by clicking here .

You can do any of these ideas without having to purchase anything. However, if you are looking to save some time and energy, then they are all found in my Math Workshop Problem Solving Unit . The unit is for grade three, but it  may work for other grade levels. The practice problems are all for the early third-grade level.

• freebie , Math Workshop , Problem Solving

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Problem Solving Activities: 7 Strategies

• Critical Thinking

Problem solving can be a daunting aspect of effective mathematics teaching, but it does not have to be! In this post, I share seven strategic ways to integrate problem solving into your everyday math program.

In the middle of our problem solving lesson, my district math coordinator stopped by for a surprise walkthrough. 

I was so excited!

We were in the middle of what I thought was the most brilliant math lesson– teaching my students how to solve problem solving tasks using specific problem solving strategies. 

It was a proud moment for me!

Each week, I presented a new problem solving strategy and the students completed problems that emphasized the strategy. 

Genius right? 

After observing my class, my district coordinator pulled me aside to chat. I was excited to talk to her about my brilliant plan, but she told me I should provide the tasks and let my students come up with ways to solve the problems. Then, as students shared their work, I could revoice the student’s strategies and give them an official name. 

What a crushing blow! Just when I thought I did something special, I find out I did it all wrong. 

I took some time to consider her advice. Once I acknowledged she was right, I was able to make BIG changes to the way I taught problem solving in the classroom. 

When I Finally Saw the Light

To give my students an opportunity to engage in more authentic problem solving which would lead them to use a larger variety of problem solving strategies, I decided to vary the activities and the way I approached problem solving with my students. 

Problem Solving Activities

Here are seven ways to strategically reinforce problem solving skills in your classroom. 

Seasonal Problem Solving

Many teachers use word problems as problem solving tasks. Instead, try engaging your students with non-routine tasks that look like word problems but require more than the use of addition, subtraction, multiplication, and division to complete. Seasonal problem solving tasks and daily challenges are a perfect way to celebrate the season and have a little fun too!

Cooperative Problem Solving Tasks

Go cooperative! If you’ve got a few extra minutes, have students work on problem solving tasks in small groups. After working through the task, students create a poster to help explain their solution process and then post their poster around the classroom. Students then complete a gallery walk of the posters in the classroom and provide feedback via sticky notes or during a math talk session.

Notice and Wonder

Before beginning a problem solving task, such as a seasonal problem solving task, conduct a Notice and Wonder session. To do this, ask students what they notice about the problem. Then, ask them what they wonder about the problem. This will give students an opportunity to highlight the unique characteristics and conditions of the problem as they try to make sense of it. 

Want a better experience? Remove the stimulus, or question, and allow students to wonder about the problem. Try it! You’ll gain some great insight into how your students think about a problem.

Math Starters

Start your math block with a math starter, critical thinking activities designed to get your students thinking about math and provide opportunities to “sneak” in grade-level content and skills in a fun and engaging way. These tasks are quick, designed to take no more than five minutes, and provide a great way to turn-on your students’ brains. Read more about math starters here ! 

Create your own puzzle box! The puzzle box is a set of puzzles and math challenges I use as fast finisher tasks for my students when they finish an assignment or need an extra challenge. The box can be a file box, file crate, or even a wall chart. It includes a variety of activities so all students can find a challenge that suits their interests and ability level.

Calculators

Use calculators! For some reason, this tool is not one many students get to use frequently; however, it’s important students have a chance to practice using it in the classroom. After all, almost everyone has access to a calculator on their cell phones. There are also some standardized tests that allow students to use them, so it’s important for us to practice using calculators in the classroom. Plus, calculators can be fun learning tools all by themselves!

Three-Act Math Tasks

Use a three-act math task to engage students with a content-focused, real-world problem! These math tasks were created with math modeling in mind– students are presented with a scenario and then given clues and hints to help them solve the problem. There are several sites where you can find these awesome math tasks, including Dan Meyer’s Three-Act Math Tasks and Graham Fletcher’s 3-Acts Lessons . 

Getting the Most from Each of the Problem Solving Activities

When students participate in problem solving activities, it is important to ask guiding, not leading, questions. This provides students with the support necessary to move forward in their thinking and it provides teachers with a more in-depth understanding of student thinking. Selecting an initial question and then analyzing a student’s response tells teachers where to go next. 

Ready to jump in? Grab a free set of problem solving challenges like the ones pictured using the form below. 

Which of the problem solving activities will you try first? Respond in the comments below.

Shametria Routt Banks

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2 responses.

This is a very cool site. I hope it takes off and is well received by teachers. I work in mathematical problem solving and help prepare pre-service teachers in mathematics.

Thank you, Scott! Best wishes to you and your pre-service teachers this year!

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How to Ace Math Problem Solving

When your kids struggle with their math, it’s time to take a step back and take a deep breath. They need to slow down and take their time. Here’s a step by step guide that will help your kids get through those tough math problems.

We’ll use a grade 3 addition word problem as an example to clarify:

Pinky the Pig bought 36 apples while Danny the Duck bought 73 apples and 14 bananas. How many apples do they have altogether?

Read the problem

Carefully read through the problem to make sure you understand what is being asked.

Pinky the pig and Danny the duck bought apples and bananas. The question is how many apples they have together.

Re-read the problem

Read through the problem again and as you read through it, make notes.

Pinky the pig –36 apples. Danny the duck –73 apples and 14 bananas. How many apples together?

What is the problem asking

In your own words, say or write down exactly what the question is asking you to solve.

The question is asking how many apples the pig and the duck bought together.

Write it down in detail

Go through the problem and write out the information in an organized fashion. A diagram or table might help.

Turn it into math

Figure out what math operation(s) or formula(s) you need to use in order to solve this problem.

The problem wants us to add the number of apples Pinky the Pig and Danny the Duck have together. That means we need to make use of addition to add the apples.

Find an example

Are you still struggling? Sometimes it’s hard to work out the solution, especially if the math problem involves several steps. It’s time to present the problem in an easier way. As teachers and parents we can often help our kids simplify the problem from our own math knowledge. If the problem is a bit harder, there are lots of resources online that you can look up for similar problems that have been worked out on paper or a video tutorial to watch.

In our example, let’s say the double-digit numbers are intimidating our student, so we’re going to simplify the equation for the sake of helping our student understand the operation needed.

Let’s say Pinky the Pig bought 3 apples and Danny the Duck 7 apples and 1 banana. Now, how many apples have they bought together? With 3 apples and 7 apples bought, the total number of apples is 10.

Work out the problem

Now that we have got to the bottom of what is being asked and know what operation to use, it’s time to work out the problem.

Pinky the Pig bought 36 apples. Danny the Duck bought 73 apples. (The 14 bananas do not matter) We need to add up the apples. 36 + 73 = 109

Check and review your answer

Check that your answer is correct. Always ask: does this answer make sense?  You can use estimation using mental math, for example.

Let’s round the numbers: 30 + 70 = 100. That is close to the exact number so it’s in the correct range.

The beauty of the basic operations is that addition and subtraction can be used to check answers too.

If we use the sum and take away one of the numbers, it should equal the other number.

109 – 73 = 36 109 – 36 = 73

If our student did not work out the sum correctly, we would not come to these sums.

(By the way, the same can be done with multiplication and division.)

Finally, go back and review the problem one last time. By going over the concepts, operations and formulas, it will help your kids to internalize the process and help them tackle harder math problems in the future.

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Polya’s problem-solving process: finding unknowns elementary & middle school, by: jeff todd.

In this article, we'll explore how a focus on finding “unknowns” in math will lead to active problem-solving strategies for Kindergarten to Grade 8 classrooms. Through the lens of George Polya and his four-step problem-solving heuristic, I will discuss how you can apply the concept of finding unknowns to your classroom. Plus, download my Finding Unknowns in Elementary and Middle School Math Classes Tip Sheet .

It is unfortunate that in the United States mathematics has a reputation for being dry and uninteresting. I hear this more from adults than I do from children—in fact, I find that children are naturally curious about how math works and how it relates to the world around them. It is from adults that they get the idea that math is dry, boring, and unrelated to their lives. Despite what children may or may not hear about math, I focus on making instruction exciting and showing my students that math applicable to their lives.

Problem solving is a fundamental means of developing students' mathematical knowledge and it also shows them that math concepts apply to real-world concepts.

Problem solving is one way I show my students that math relates to their lives! Problem solving is a fundamental means of developing students' mathematical knowledge and it also shows them that math concepts apply to real-world concepts.

Who Is George Polya?

George Polya was a European-born scholar and mathematician who moved to the U.S in 1940, to work at Stanford University. When considering the his classroom experience of teaching mathematics, he noticed that students were not presented with a view of mathematics that excited and energized them. I know that I have felt this way many times in my teaching career and have often asked: How can I make this more engaging and yet still maintain rigor?

Polya suggested that math should be presented in the light of being able to solve problems. His 1944 book,  How to Solve It  contains his famous four-step problem solving heuristic. Polya suggests that by presenting mathematical thinking as a way to find “unknowns,” it becomes more engaging for students.

He even goes as far as to say that his general four-step problem-solving heuristic can be applied to any field of human endeavor—to any opportunity where a problem exists.

Polya suggested that math should be presented in the light of being able to solve problems...that by presenting mathematical thinking as a way to find “unknowns,” it becomes more engaging for students.

Polya specifically wrote about problem-solving at the high school mathematics level. For those of us teaching students in the elementary and middle school levels, finding ways to apply Polya’s problem-solving process as he intended forces us to rethink the way we teach.

Particularly in the lower grade levels, finding “unknowns” can be relegated to prealgebra and algebra courses in the later grades. Nonetheless, today’s standards call for algebra and algebraic thinking at early grade levels. The  download  for today’s post presents one way you can find unknowns at each grade level.

Presenting Mathematics  As A Way To Find "Unknowns" In Real-Life Situations

I would like to share a conversation I had recently with my friend Stu. I have been spending my summers volunteering for a charitable organization in Central America that provides medical services for the poor, runs ESL classes, and operates a Pre-K to Grade 6 school. We were talking about the kind of professional development that I might provide the teachers, and he was intrigued by the thought that we could connect mathematical topics to real life. We specifically talked about the fact that he remembers little or nothing about how to find the area of a figure and never learned in school why it might be important to know about area. Math was presented to him as a set of rules and procedures rather than as a way to find unknowns in real-life situations.

That’s what I am talking about here, and it’s what I believe Polya was talking about. How can we create classrooms where students are able to use their mathematical knowledge to solve problems, whether real-life or purely mathematical?

As Polya noted, there are two ways that mathematics can be presented, either as deductive system of rules and procedures or as an inductive method of making mathematics. Both ways of thinking about mathematics have endured through the centuries, but at least in American education, there has been an emphasis on a procedural approach to math. Polya noticed this in the 1940s, and I think that although we have made progress, there is still an over-emphasis on skill and procedure at the expense of problem-solving and application.

I recently reread Polya’s book. I can’t say that it is an “easy” read, but I would say that it was valuable for me to revisit his own words in order to be sure I understood what he was advocating. As a result, I made the following outline of his problem-solving process and the questions he suggests we use with students.

Polya's Problem-Solving Process

1. understand the problem, and desiring the solution .

• Restate the problem
• Identify the principal parts of the problem
• Essential questions
• What is unknown?
• What data are available?
• What is the condition?

2. Devising a Problem-Solving Plan 

• Look at the unknown and try to think of a familiar problem having the same or similar unknown
• Here is a problem related to yours and solved before. Can you use it?
• Can you restate the problem?
• Did you use all the data?
• Did you use the whole condition?

3. Carrying Out the Problem-Solving Plan 

• Can you see that each step is correct?
• Can you prove that each step is correct?

4. Looking Back

• Can you check the result?
• Can you check the argument?
• Can you derive the result differently?
• Can you see the result in a glance?
• Can you use the result, or the method, for some other problem?

Polya's Suggestions For Helping Students Solve Problems

I also found four suggestions from Polya about what teachers can do to help students solve problems:

Suggestion One In order for students to understand the problem, the teacher must focus on fostering in students the desire to find a solution. Absent this motivation, it will always be a fight to get students to solve problems when they are not sure what to do.

Suggestion Two A second key feature of this first phase of problem-solving is giving students strategies forgetting acquainted with problems.

Suggestion Three Another suggestion is that teachers should help students learn strategies to be able to work toward a better understanding of any problem through experimentation.

Suggestion Four Finally, when students are not sure how to solve a problem, they need strategies to “hunt for the helpful idea.”

Whether you are thinking of problem-solving in a traditional sense (solving computational problems and geometric proofs, as illustrated in Polya’s book) or you are thinking of the kind of problem-solving students can do through STEAM activities, I can’t help but hear echoes of Polya in Standard for Math Practice 1: Make sense of problems and persevere in solving them.

Mathematically proficient students start by explaining to themselves the meaning of a problem and looking for entry points to its solution. They analyze givens, constraints, relationships, and goals. They make conjectures about the form and meaning of the solution and plan a solution pathway rather than simply jumping into a solution attempt. They consider analogous problems, and try special cases and simpler forms of the original problem in order to gain insight into its solution. They monitor and evaluate their progress and change course if necessary.

In Conclusion

We all know we should be fostering students’ problem-solving ability in our math classes. Polya’s focus on “finding unknowns” in math has wide applicability to problems whether they are purely mathematical or more general.

Grab my  download  and start  applying Polya’s Four-Step Problem-Solving Process in the lower grades!

Four-Step Math Problem Solving Strategies & Techniques

• Harlan Bengtson
• Categories : Help with math homework
• Tags : Homework help & study guides

Four Steps to Success

There are many possible strategies and techniques you can use to solve math problems. A useful starting point is a four step approach to math problem solving. These four steps can be summarized as follows:

• Carefully read the problem. In this careful reading, you should especially seek to clearly identify the question that is to be answered. Also, a good, general understanding of what the problem means should be sought.
• Choose a strategy to solve the problem. Some of the possible strategies will be discussed in the rest of this article.
• Carry out the problem solving strategy. If the first problem solving technique you try doesn’t work, try another.
• Check the solution. This check should make sure that you have indeed answered the question that was posed and that the answer makes sense.

Step One - Understanding the Problem

As you carefully read the problem, trying to clearly understand the meaning of the problem and the question that you must answer, here are some techniques to help.

Identify given information - Highlighting or underlining facts that are given helps to visualize what is known or given.

Identify information asked for - Highlighting the unknowns in a different color helps to keep the known information visually separate from the unknowns to be determined. Ideally this will lead to a clear identification of the question to be answered.

Look for keywords or clue words - One example of clue words is those that indicate what type of mathematical operation is needed, as follows:

Clue words indicating addition: sum, total, in all, perimeter.

Clue words indicating subtraction: difference, how much more, exceed.

Clue words for multiplication: product, total, area, times.

Clue words for division: share, distribute, quotient, average.

Draw a picture - This might also be considered part of solving the problem, but a good sketch showing given information and unknowns can be very helpful in understanding the problem.

Step Two - Choose the Right Strategy

It step one has been done well, it should ease the job of choosing among the strategies presented here for approaching the problem solving step. Here are some of the many possible math problem solving strategies.

• Look for a pattern - This might be part of understanding the problem or it might be the first part of solving the problem.
• Make an organized list - This is another means of organizing the information as part of understanding it or beginning the solution.
• Make a table - In some cases the problem information may be more suitable for putting in a table rather than in a list.
• Try to remember if you’ve done a similar problem before - If you have done a similar problem before, try to use the same approach that worked in the past for the solution.
• Guess the answer - This may seem like a haphazard approach, but if you then check whether your guess was correct, and repeat as many times as necessary until you find the right answer, it works very well. Often information from checking on whether the answer was correct helps lead you to a good next guess.
• Work backwards - Sometimes making the calculations in the reverse order works better.

Steps Three and Four - Solving the Problem and Checking the Solution

If the first two steps have been done well, then the last two steps should be easy. If the selected problem solving strategy doesn’t seem to work when you actually try it, go back to the list and try something else. Your check on the solution should show that you have actually answered the question that was asked in the problem, and to the extent possible, you should check on whether the answer makes common sense.

K-5 Math Centers

K-5 math ideas, 3rd grade math, need help organizing your k-5 math block, 5 ways to include math problem solving activities in your classroom.

Are you looking for math problem solving activities that are not too easy and not too hard, but juuust right? I’ve got something just for you and your students.

Solve and Explain Problem Solving Tasks are open-ended math tasks that provide just the right amount of challenge for your kids. Here’s a little more about them.

Open-ended math problem solving tasks:

• promote multiple solution paths and/or multiple solutions
• boost critical thinking and math reasoning skills
• increase opportunities for developing perseverance
• provide opportunities to justify answer choices
• strengthen kids written and oral communication skills

What Makes These So Great?

• All Common Core Standards are covered for your grade level
• 180+ Quality questions that are rigorous yet engaging
• They are SUPER easy to assemble
• Provide opportunities for meaningful math discussions
• Perfect for developing a growth mindset
• Easily identify student misconceptions so you can provide assistance
• Very versatile (check out the different ways to use them below)

You can find out more details for your grade level by clicking on the buttons below.

I’m sure you really want to know how can you use these with your kids. Check out the top 5 ideas on how to use Solve and Explain Problem Solving Tasks in your classroom.

How and When Can I Use Them?

Solve and Explain Tasks Cards are very versatile. You can use them for:

• Math Centers  – This is my favorite way to use these! Depending on your grade level, there are at least two (Kinder – 2nd) or three (3rd-5th) tasks types per Common Core standard. And each task type has 6 different questions. Print out each of the different tasks types on different color paper. Then, let students choose which one question from each task type they want to solve.

• Problem of the Day  – Use them as a daily math journal prompt. Print out the recording sheet and project one of the problems on your white board or wall.  Students solve the problem and then glue it in their spiral or composition notebooks.

• Early Finisher Activities  -No more wondering what to do next!Create an early finishers notebook where students can grab a task and a recording sheet. Place the cards in sheet protectors and make copies of the Early Finisher Activity Check-Off card for your kids to fill out BEFORE they pull a card out to work on. We want to make sure kids are not rushing through there first assignment before moving on to an early finisher activity.

• Weekly Math Challenges  – Kids LOVE challenges! Give students copies of one of the problems for homework. Then give them a week to complete it. Since many of the questions have multiple solutions and students have to explain how they got their answers, you can have a rich whole group discussion at the end of the week (even with your kindergarten and 1st grade students).

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• Formative Assessments  – Give your students a problem to solve. Then use the Teacher Scoring Rubric to see how your kids are doing with each standard. Since they have to explain their thinking, this is a great way to catch any misconceptions and give feedback to individual students.

So this wraps up the top 5 ways that you can use problem solving tasks in your classroom.  Click your grade level below to get Solve and Explain problem solving tasks for your classroom.

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5 Math Mini-Lesson Ideas that Keep Students Engaged

A Rigorous Elementary Math Curriculum for Busy Teachers

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Problem-Solving in Elementary School

Elementary students practice problem-solving and self-questioning techniques to improve reading and social and emotional learning skills.

In a school district in New Jersey, beginning in kindergarten each child is seen as a future problem solver with creative ideas that can help the world. Vince Caputo, superintendent of the Metuchen School District, explained that what drew him to the position was “a shared value for whole child education.”

Caputo’s first hire as superintendent was Rick Cohen, who works as both the district’s K–12 director of curriculum and principal of Moss Elementary School . Cohen is committed to integrating social and emotional learning (SEL) into academic curriculum and instruction by linking cognitive processes and guided self-talk.

Cohen’s first focus was kindergarten students. “I recommended Moss teachers teach just one problem-solving process to our 6-year-olds across all academic content areas and challenge students to use the same process for social problem-solving,” he explained.  

Reading and Social Problem-Solving

Moss Elementary classrooms use a specific process to develop problem-solving skills focused on tending to social and interpersonal relationships. The process also concentrates on building reading skills—specifically, decoding and comprehension.

Stop, Look, and Think.  Students define the problem. As they read, they look at the pictures and text for clues, searching for information and asking, “What is important and what is not?” Social problem-solving aspect: Students look for signs of feelings in others’ faces, postures, and tone of voice.

Gather Information . Next, students explore what feelings they’re having and what feelings others may be having. As they read, they look at the beginning sound of a word and ask, “What else sounds like this?” Social problem-solving aspect: Students reflect on questions such as, “What word or words describe the feeling you see or hear in others? What word describes your feeling? How do you know, and how sure are you?”

Brainstorming . Then students seek different solutions. As they read, they wonder, “Does it sound right? Does it make sense? How else could it sound to make more sense? What other sounds do those letters make?” Social problem-solving aspect: Students reflect on questions such as, “How can you solve the problem or make the situation better? What else can you think of? What else can you try? What other ideas do you have?”  

Pick the Best One.  Next, students evaluate the solution. While reading, they scan for smaller words they know within larger, more difficult words. They read the difficult words the way they think they sound while asking, “Will it make sense to other people?” Social problem-solving aspect: Students reflect on prompts such as, “Pick the solution that you think will be best to solve the problem. Ask yourself, ‘What will happen if I do this—for me, and for others involved?’”

Go . In the next step, students make a plan and act. They do this by rereading the text. Social problem-solving aspect: Students are asked to try out what they will say and how they will say it. They’re asked to pick a good time to do this, when they’re willing to try it.

Check . Finally, students reflect and revise. After they have read, they ponder what exactly was challenging about what they read and, based on this, decide what to do next. Social problem-solving aspect: Students reflect on questions such as, “How did it work out? Did you solve the problem? How did others feel about what happened? What did you learn? What would you do if the same thing happened again?”

You can watch the Moss Elementary Problem Solvers video and see aspects of this process in action.

The Process of Self-Questioning 

Moss Elementary students and other students in the district are also taught structured self-questioning. Cohen notes, “We realized that many of our elementary students would struggle to generalize the same steps and thinking skills they previously used to figure out an unknown word in a text or resolve social conflicts to think through complex inquiries and research projects.” The solution? Teach students how to self-question, knowing they can also apply this effective strategy across contexts. The self-questioning process students use looks like this:

Stop and Think. “What’s the question?”

Gather Information. “How do I gather information? What are different sides of the issue?”

Brainstorm and Choose. “How do I select, organize, and choose the information? What are some ways to solve the problem? What’s the best choice?”

Plan and Try. “What does the plan look like? When and how can it happen? Who needs to be involved?”

Check & Revise. “How can I present the information? What did I do well? How can I improve?”

The Benefits

Since using the problem-solving and self-questioning processes, the students at Moss Elementary have had growth in their scores for the last two years on the fifth-grade English language arts PARCC tests . However, as Cohen shares, “More important than preparing our students for the tests on state standards, there is evidence that we are also preparing them for the tests of life.”

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12.4: The 5-Step Method of Solving Applied Problems

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• Page ID 49420

• Denny Burzynski & Wade Ellis, Jr.
• College of Southern Nevada via OpenStax CNX
• Let \(x\) (or some other letter) represent the unknown quantity.
• Translate the English to mathematics and form an equation.
• Solve this equation.
• Check this result by substituting it into the original statement of the problem.
• Write a conclusion.

3 Ways to Strengthen Math Instruction

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Students’ math scores have plummeted, national assessments show , and educators are working hard to turn math outcomes around.

But it’s a challenge, made harder by factors like math anxiety , students’ feelings of deep ambivalence about how math is taught, and learning gaps that were exacerbated by the pandemic’s disruption of schools.

This week, three educators offered solutions on how districts can turn around poor math scores in a conversation moderated by Peter DeWitt, an opinion blogger for Education Week.

Here are three takeaways from the discussion. For more, watch the recording on demand .

1. Intervention is key

Research shows that early math skills are a key predictor of later academic success.

“Children who know more do better, and math is cumulative—so if you don’t grasp some of the earlier concepts, math gets increasingly harder,” said Nancy Jordan, a professor of education at the University of Delaware.

For example, many students struggle with the concept of fractions, she said. Her research has found that by 6th grade, some students still don’t really understand what a fraction is, which makes it harder for them to master more advanced concepts, like adding or subtracting fractions with unlike denominators.

At that point, though, teachers don’t always have the time in class to re-teach those basic or fundamental concepts, she said, which is why targeted intervention is so important.

Still, Jordan’s research revealed that in some middle schools, intervention time is not a priority: “If there’s an assembly, or if there is a special event or whatever, it takes place during intervention time,” she said. “Or ... the children might sit on computers, and they’re not getting any really explicit instruction.”

2. ‘Gamify’ math class

Students today need new modes of instruction that meet them where they are, said Gerilyn Williams, a math teacher at Pinelands Regional Junior High School in Little Egg Harbor Township, N.J.

“Most of them learn through things like TikTok or YouTube videos,” she said. “They like to play games, they like to interact. So how can I bring those same attributes into my lesson?”

Part of her solution is gamifying instruction. Williams avoids worksheets. Instead, she provides opportunities for students to practice skills that incorporate elements of game design.

That includes digital tools, which provide students with the instant feedback they crave, she said.

But not all the games are digital. Williams’ students sometimes play “trashketball,” a game in which they work in teams to answer math questions. If they get the question right, they can crumble the piece of paper and throw it into a trash can from across the room.

“The kids love this,” she said.

Williams also incorporates game-based vocabulary into her instruction, drawing on terms from video games.

For example, “instead of calling them quizzes and tests, I call them boss battles,” she said. “It’s less frightening. It reduces that math anxiety, and it makes them more engaging.

“We normalize things like failure, because when they play video games, think about what they’re doing,” Williams continued. “They fail—they try again and again and again and again until they achieve success.”

3. Strengthen teacher expertise

To turn around math outcomes, districts need to invest in teacher professional development and curriculum support, said Chaunté Garrett, the CEO of ELLE Education, which partners with schools and districts to support student learning.

“You’re not going to be able to replace the value of a well-supported and well-equipped mathematics teacher,” she said. “We also want to make sure that that teacher has a math curriculum that’s grounded in the standards and conceptually based.”

Students will develop more critical thinking skills and better understand math concepts if teachers are able to relate instruction to real life, Garrett said—so that “kids have relationships that they can pull on, and math has some type of meaning and context to them outside of just numbers and procedures.”

It’s important for math curriculum to be both culturally responsive and relevant, she added. And teachers might need training on how to offer opportunities for students to analyze and solve real-world problems.

“So often, [in math problems], we want to go back to soccer and basketball and all of those things that we lived through, and it’s not that [current students] don’t enjoy those, but our students live social media—they literally live it,” Garrett said. “Those are the things that have to live out in classrooms right now, and if we’re not doing those things, we are doing a disservice.”

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