strategies for problem solving in math

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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
• Tier 3 Interventions: A School Leaders Guide
• Tier 2 Interventions: A School Leaders Guide
• Tier 1 Interventions: A School Leaders Guide

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 for 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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10 Strategies for Problem-Solving in Math

reviewed by Jo-ann Caballes

Updated on August 21, 2024

It’s not surprising that kids who lack problem-solving skills feel stuck in math class. Students who are behind in problem-solving may have difficulties identifying and carrying out a plan of action to solve a problem. Math strategies for problem-solving allow children to use a range of approaches to work out math problems productively and with ease. This article explores math problem-solving strategies and how kids can use them both in traditional classes and in a virtual classroom. 

What are problem-solving strategies in math?

Problem-solving strategies for math make it easier to tackle math and work up an effective solution. When we face any kind of problem, it’s usually impossible to solve it without carrying out a good plan.In other words, these strategies were designed to make math for kids easier and more manageable. Another great benefit of these strategies is that kids can spend less time cracking math problems. 

Here are some problem-solving methods:

• Drawing a picture or diagram (helps visualize the problem)
• Breaking the problem into smaller parts (to keep track of what has been done)
• Making a table or a list (helps students to organize information)

When children have a toolkit of math problem-solving strategies at hand, it makes it easier for them to excel in math and progress faster. 

How to solve math problems?

To solve math problems, it’s worth having strategies for math problem-solving that include several steps, but it doesn’t necessarily mean they are failproof. They serve as a guide to the solution when it’s difficult to decide where and how to start. Research suggests that breaking down complex problems into smaller stages can reduce cognitive load and make it easier for students to solve problems. Essentially, a suitable strategy can help kids to find the right answers fast. 

Here are 5 math problem-solving strategies for kids:

• Recognize the Problem  
• Work up a Plan  
• Carry Out the Plan  
• Review the Work  
• Reflect and Analyze  

Understanding the Problem 

Understanding the problem is the first step in the journey of solving it. Without doing this, kids won’t be able to address it in any way. In the beginning, it’s important to read the problem carefully and make sure to understand every part of it. Next, when kids know what they are asked to do, they have to write down the information they have and determine what essentially they need to solve. 

Work Out a Plan 

Working out a plan is one of the most important steps to solving math problems. Here, the kid has to choose a good strategy that will help them with a specific math problem. Outline these steps either in mind or on paper.

Carry Out the Plan 

Being methodical at that stage is key. It involves following the plan and performing calculations with the correct operations and rules. Finally, when the work is done, the child can review and show their work to a teacher or tutor.

Review the Work

This is where checking if the answer is correct takes place. If time allows, children and the teacher can choose other methods and try to solve the same problem again with a different approach. 

Reflect and Analyze

This stage is a great opportunity to think about how the problem was solved: did any part cause confusion? Was there a more efficient method? It’s important to let the child know that they can use the insights gained for future reference. 

Ways to solve math problems

The ways to solve math problems for kids are numerous, but it doesn’t mean they all work the same for everybody. For example, some children may find visual strategies work best for them; some prefer acting out the problem using movements. Finding what kind of method or strategy works best for your kid will be extremely beneficial both for school performance and in real-life scenarios where they can apply problem-solving. 

Online tutoring platforms like Brighterly offer personalized assistance, interactive tools, and access to resources that help to determine which strategies are best for your child. Expertise-driven tutors know how to guide your kid so they won’t be stuck with the same fallacies that interfere with effective problem-solving. For example, tutors can assist kids with drawing a diagram, acting a problem out with movement, or working backward. All of these ways are highly effective, especially with a trusted supporter by your side. 

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What are 10 strategies for solving math problems?

There are plenty of different problem-solving strategies for mathematical problems to help kids discover answers. Let’s explore 10 popular problem-solving strategies:

Understand the Problem

Figuring out the nature of math problems is the key to solving them. Kids need to identify what kind of issue this is (fraction problem, word problem, quadratic equation, etc.) and work up a plan to solve it. 

Guess and Check

With this approach, kids simply need to keep guessing until they get the answer right. While this approach may seem irrelevant, it illustrates what the kid’s thinking process is. 

Work It Out

This method encourages students to write down or say their problem-solving process instead of going straight to solving it without preparation. This minimizes the probability of mistakes. 

Work Backwords 

Working backward is a great problem-solving strategy to acquire a fresh perspective. It requires one to come up with a probable solution and decide which step to take to come to that solution. 

A visual representation of a math problem may help kids to understand it in full. One way to visualize a problem is to use a blank piece of paper and draw a picture, including all of the aspects of the issue. 

Find a Pattern 

By helping students see patterns in math problems, we help them to extract and list relevant details. This method is very effective in learning shapes and other topics that need repetition. 

It may be self-explanatory, but it’s quite helpful to ask, “What are some possible solutions to this issue?”. By giving kids time to think and reflect, we help them to develop creative and critical thinking.

Draw a Picture or Diagram

Instead of drawing the math problem yourself, ask the kid to draw it themselves. They can draw pictures of the ideas they have been taught to help them remember the concepts better.

Trial and Error Method

Not knowing clear formulas or instructions, kids won’t be able to solve anything. Ask them to make a list of possible answers based on rules they already know. Let them learn by making mistakes and trying to find a better solution. 

Review Answers with Peers

It’s so fun to solve problems alongside your peers. Kids can review their answers together and share ideas on how each problem can be solved. 

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Math problem-solving strategies for elementary students

5 problem-solving strategies for elementary students include:

Using Simple Language 

Ask students to explain the problem in their own words to make sure they understand the problem correctly.

Using Visuals and Manipulatives 

Using drawing and manipulatives like counters, blocks, or beads can help students grasp the issue faster. 

Simplifying the Problem

Breaking the problem into a step-by-step process and smaller, manageable steps will allow students to find the solution faster. 

Looking for Patterns 

Identifying patterns in numbers and operations is a great strategy to help students gain more confidence along the way. 

Using Stories

Turning math problems into stories will surely engage youngsters and make them participate more actively. 

To recap, students need to have effective math problem-solving strategies up their sleeves. Not only does it help them in the classroom, but it’s also an essential skill for real-life situations. Productive problem-solving strategies for math vary depending on the grade. But what they have in common is that kids have to know how to break the issue into smaller parts and apply critical and creative thinking to solve it. 

If you want your kid to learn how to thrive in STEM and apply problem-solving strategies to both math and real life, book a free demo lesson with Brighterly today! Make your child excited about math!

Jessica is a a seasoned math tutor with over a decade of experience in the field. With a BSc and Master’s degree in Mathematics, she enjoys nurturing math geniuses, regardless of their age, grade, and skills. Apart from tutoring, Jessica blogs at Brighterly. She also has experience in child psychology, homeschooling and curriculum consultation for schools and EdTech websites.

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Math Problem Solving Strategies That Make Students Say “I Get It!”

Even students who are quick with math facts can get stuck when it comes to problem solving.

As soon as a concept is translated to a word problem, or a simple mathematical sentence contains an unknown, they’re stumped.

That’s because problem solving requires us to  consciously choose the strategies most appropriate for the problem   at hand . And not all students have this metacognitive ability.

But you can teach these strategies for problem solving.  You just need to know what they are.

We’ve compiled them here divided into four categories:

Strategies for understanding a problem

Strategies for solving the problem, strategies for working out, strategies for checking the solution.

Get to know these strategies and then model them explicitly to your students. Next time they dive into a rich problem, they’ll be filling up their working out paper faster than ever!

Before students can solve a problem, they need to know what it’s asking them. This is often the first hurdle with word problems that don’t specify a particular mathematical operation.

Encourage your students to:

Read and reread the question

They say they’ve read it, but have they  really ? Sometimes students will skip ahead as soon as they’ve noticed one familiar piece of information or give up trying to understand it if the problem doesn’t make sense at first glance.

Teach students to interpret a question by using self-monitoring strategies such as:

• Rereading a question more slowly if it doesn’t make sense the first time
• Asking for help
• Highlighting or underlining important pieces of information.

Identify important and extraneous information

John is collecting money for his friend Ari’s birthday. He starts with $5 of his own, then Marcus gives him another $5. How much does he have now?

As adults looking at the above problem, we can instantly look past the names and the birthday scenario to see a simple addition problem. Students, however, can struggle to determine what’s relevant in the information that’s been given to them.

Teach students to sort and sift the information in a problem to find what’s relevant. A good way to do this is to have them swap out pieces of information to see if the solution changes. If changing names, items or scenarios has no impact on the end result, they’ll realize that it doesn’t need to be a point of focus while solving the problem.

Schema approach

This is a math intervention strategy that can make problem solving easier for all students, regardless of ability.

Compare different word problems of the same type and construct a formula, or mathematical sentence stem, that applies to them all. For example, a simple subtraction problems could be expressed as:

[Number/Quantity A] with [Number/Quantity B] removed becomes [end result].

This is the underlying procedure or  schema  students are being asked to use. Once they have a list of schema for different mathematical operations (addition, multiplication and so on), they can take turns to apply them to an unfamiliar word problem and see which one fits.

Struggling students often believe math is something you either do automatically or don’t do at all. But that’s not true. Help your students understand that they have a choice of problem-solving strategies to use, and if one doesn’t work, they can try another.

Here are four common strategies students can use for problem solving.

Visualizing

Visualizing an abstract problem often makes it easier to solve. Students could draw a picture or simply draw tally marks on a piece of working out paper.

Encourage visualization by modeling it on the whiteboard and providing graphic organizers that have space for students to draw before they write down the final number.

Guess and check

Show students how to make an educated guess and then plug this answer back into the original problem. If it doesn’t work, they can adjust their initial guess higher or lower accordingly.

Find a pattern

To find patterns, show students how to extract and list all the relevant facts in a problem so they can be easily compared. If they find a pattern, they’ll be able to locate the missing piece of information.

Work backward

Working backward is useful if students are tasked with finding an unknown number in a problem or mathematical sentence. For example, if the problem is 8 + x = 12, students can find x by:

• Starting with 12
• Taking the 8 from the 12
• Being left with 4
• Checking that 4 works when used instead of x

Now students have understood the problem and formulated a strategy, it’s time to put it into practice. But if they just launch in and do it, they might make it harder for themselves. Show them how to work through a problem effectively by:

Documenting working out

Model the process of writing down every step you take to complete a math problem and provide working out paper when students are solving a problem. This will allow students to keep track of their thoughts and pick up errors before they reach a final solution.

Check along the way

Checking work as you go is another crucial self-monitoring strategy for math learners. Model it to them with think aloud questions such as:

• Does that last step look right?
• Does this follow on from the step I took before?
• Have I done any ‘smaller’ sums within the bigger problem that need checking?

Students often make the mistake of thinking that speed is everything in math — so they’ll rush to get an answer down and move on without checking.

But checking is important too. It allows them to pinpoint areas of difficulty as they come up, and it enables them to tackle more complex problems that require multiple checks  before  arriving at a final answer.

Here are some checking strategies you can promote:

Check with a partner

Comparing answers with a peer leads is a more reflective process than just receiving a tick from the teacher. If students have two different answers, encourage them to talk about how they arrived at them and compare working out methods. They’ll figure out exactly where they went wrong, and what they got right.

Reread the problem with your solution

Most of the time, students will be able to tell whether or not their answer is correct by putting it back into the initial problem. If it doesn’t work or it just ‘looks wrong’, it’s time to go back and fix it up.

Fixing mistakes

Show students how to backtrack through their working out to find the exact point where they made a mistake. Emphasize that they can’t do this if they haven’t written down everything in the first place — so a single answer with no working out isn’t as impressive as they might think!

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Problem-Solving Strategies

October 16, 2019

There are many different ways to solve a math problem, and equipping students with problem-solving strategies is just as important as teaching computation and algorithms. Problem-solving strategies help students visualize the problem or present the given information in a way that can lead them to the solution. Solving word problems using strategies works great as a number talks activity and helps to revise many skills.

Problem-solving strategies

1. create a diagram/picture, 2. guess and check., 3. make a table or a list., 4. logical reasoning., 5. find a pattern, 6. work backward, 1. create a diagram/draw a picture.

Creating a diagram helps students visualize the problem and reach the solution. A diagram can be a picture with labels, or a representation of the problem with objects that can be manipulated. Role-playing and acting out the problem like a story can help get to the solution.

Alice spent 3/4 of her babysitting money on comic books. She is left with $6. How much money did she make from babysitting?

2. Guess and check

Teach students the same strategy research mathematicians use.

With this strategy, students solve problems by making a reasonable guess depending on the information given. Then they check to see if the answer is correct and they improve it accordingly.  By repeating this process, a student can arrive at a correct answer that has been checked. It is recommended that the students keep a record of their guesses by making a chart, a table or a list. This is a flexible strategy that works for many types of problems. When students are stuck, guessing and checking helps them start and explore the problem. However, there is a trap. Exactly because it is such a simple strategy to use, some students find it difficult to consider other strategies. As problems get more complicated, other strategies become more important and more effective.

Find two numbers that have sum 11 and product 24.

Try/guess  5 and 6  the product is 30 too high

  adjust  to 4 and 7 with product 28 still high

  adjust  again 3 and 8 product 24

3. Make a table or a list

Carefully organize the information on a table or list according to the problem information. It might be a table of numbers, a table with ticks and crosses to solve a logic problem or a list of possible answers. Seeing the given information sorted out on a table or a list will help find patterns and lead to the correct solution.

To make sure you are listing all the information correctly read the problem carefully.

Find the common factors of 24, 30 and 18

Logical reasoning is the process of using logical, systemic steps to arrive at a conclusion based on given facts and mathematic principles. Read and understand the problem. Then find the information that helps you start solving the problem. Continue with each piece of information and write possible answers.

Thomas, Helen, Bill, and Mary have cats that are black, brown, white, or gray. The cats’ names are Buddy, Lucky, Fifi, and Moo. Buddy is brown. Thoma’s cat, Lucky, is not gray. Helen’s cat is white but is not named Moo. The gray cat belongs to Bill. Which cat belongs to each student, and what is its color?

A table or list is useful in solving logic problems.

Since Lucky is not gray it can be black or brown. However, Buddy is brown so Lucky has to be black.

Buddy is brown so it cannot be Helen’s cat. Helen’s cat cannot be Moo, Buddy or Lucky, so it is Fifi.

Therefore, Moo is Bill’s cat and Buddy is Mary’s cat.

5. Find a pattern.

Finding a pattern is a strategy in which students look for patterns in the given information in order to solve the problem. When the problem consists of data like numbers or events that are repeated then it can be solved using the “find a pattern” problem-solving strategy. Data can be organized in a table or a list to reveal the pattern and help discover the “rule” of the pattern.

 The “rule” can then be used to find the answer to the question and complete the table/list.

Shannon’s Pizzeria made 5 pizzas on Sunday, 10 pizzas on Monday, 20 pizzas on Tuesday, and 40 pizzas on Wednesday. If this pattern continues, how many pizzas will the pizzeria make on Saturday?

6. Working backward

Problems that can be solved with this strategy are the ones that  list a series of events or a sequence of steps .

In this strategy, the students must start with the solution and work back to the beginning. Each operation must be reversed to get back to the beginning. So if working forwards requires addition, when students work backward they will need to subtract. And if they multiply working forwards, they must divide when working backward.

Mom bought a box of candy. Mary took 5 of them, Nick took 4 of them and 31 were given out on Halloween night. The next morning they found 8 pieces of candy in the box. How many candy pieces were in the box when mom bought it.

For this problem, we know that the final number of candy was 8, so if we work backward to “put back” the candy that was taken from the box we can reach the number of candy pieces that were in the box, to begin with.

The candy was taken away so we will normally subtract them. However, to get back to the original number of candy we need to work backward and do the opposite, which is to add them.

8 candy pieces were left + the 31 given out + plus the ones Mary took + the ones Nick took

8+31+5+4= 48   Answer: The box came with 48 pieces of candy.

Selecting the best strategy for a problem comes with practice and often problems will require the use of more than one strategies.

Print and digital activities

I have created a collection of print and digital activity cards and worksheets with word problems (print and google slides) to solve using the strategies above. The collection includes 70 problems (5 challenge ones) and their solution s and explanations.

sample below

How to use the activity cards

Allow the students to use manipulatives to solve the problems. (counters, shapes, lego blocks, Cuisenaire blocks, base 10 blocks, clocks) They can use manipulatives to create a picture and visualize the problem. They can use counters for the guess and check strategy. Discuss which strategy/strategies are better for solving each problem. Discuss the different ways. Use the activities as warm-ups, number talks, initiate discussions, group work, challenge, escape rooms, and more.

Ask your students to write their own problems using the problems in this resource, and more, as examples. Start with a simple type. Students learn a lot when trying to compose a problem. They can share the problem with their partner or the whole class. Make a collection of problems to share with another class.

For the google slides the students can use text boxes to explain their thinking with words, add shapes and lines to create diagrams, and add (insert) tables and diagrams.

Many of the problems can be solved faster by using algebraic expressions. However, since I created this resource for grades 4 and up I chose to show simple conceptual ways of solving the problems using the strategies above. You can suggest different ways of solving the problems based on the grade level.

Find the free and premium versions of the resource below. The premium version includes 70 problems (challenge problems included) and their solutions

There are 2 versions of the resource

70 google slides with explanations + 70 printable task cards

70 google slides with explanations + 11 worksheets

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Math and Special Education Blog

8 problem solving strategies for the math classroom.

Posted by Colleen Uscianowski · February 25, 2014

Would you draw a picture, make a list  possible number pairs that have the ratio 5:3, or guess and check? 

Explicit strategy instruction should be an integral part of your math classroom, whether you're teaching kindergarten or 12th grade.

Teach students that they can choose from a list of strategies to solve a problem, and often there isn't one correct way of finding a solution.

Demonstrate how you solve a word problem by thinking aloud as you choose and execute a strategy.

Ask students if they would solve the problem differently and praise students for coming up with unique ways of arriving at an answer.

Here are some problem-solving strategies I've taught my students:

Below is a helpful chart to remind students of the many problem-solving strategies they can use when solving word problems. This useful handout is a great addition to students' strategy binders, math notebooks, or math journals.  

How do you teach problem-solving in your classroom? Feel free to share advice and tips below!    

Sign up to receive a FREE copy of our problem-solving poster.

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Unlocking the Power of Math Learning: Strategies and Tools for Success

posted on September 20, 2023

Mathematics, the foundation of all sciences and technology, plays a fundamental role in our everyday lives. Yet many students find the subject challenging, causing them to shy away from it altogether. This reluctance is often due to a lack of confidence, a misunderstanding of unclear concepts, a move ahead to more advanced skills before they are ready, and ineffective learning methods. However, with the right approach, math learning can be both rewarding and empowering. This post will explore different approaches to learning math, strategies for success, and cutting-edge tools to help you achieve your goals.

Math Learning

Math learning can take many forms, including traditional classroom instruction, online courses, and self-directed learning. A multifaceted approach to math learning can improve understanding, engage students, and promote subject mastery. A 2014 study by the National Council of Teachers of Mathematics found that the use of multiple representations, such as visual aids, graphs, and real-world examples, supports the development of mathematical connections, reasoning, and problem-solving skills.

Moreover, the importance of math learning goes beyond solving equations and formulas. Advanced math skills are essential for success in many fields, including science, engineering, finance, health care, and technology. In fact, a report by Burning Glass Technologies found that 71% of high-salary, entry-level positions require advanced math skills.

Benefits of Math Learning

In today’s 21st-century world, having a broad knowledge base and strong reading and math skills is essential. Mathematical literacy plays a crucial role in this success. It empowers individuals to comprehend the world around them and make well-informed decisions based on data-driven understanding. More than just earning good grades in math, mathematical literacy is a vital life skill that can open doors to economic opportunities, improve financial management, and foster critical thinking. We’re not the only ones who say so:

• Math learning enhances problem-solving skills, critical thinking, and logical reasoning abilities. (Source: National Council of Teachers of Mathematics )
• It improves analytical skills that can be applied in various real-life situations, such as budgeting or analyzing data. (Source: Southern New Hampshire University )
• Math learning promotes creativity and innovation by fostering a deep understanding of patterns and relationships. (Source: Purdue University )
• It provides a strong foundation for careers in fields such as engineering, finance, computer science, and more. These careers generally correlate to high wages. (Source: U.S. Bureau of Labor Statistics )
• Math skills are transferable and can be applied across different academic disciplines. (Source: Sydney School of Education and Social Work )

How to Know What Math You Need to Learn

Often students will find gaps in their math knowledge; this can occur at any age or skill level. As math learning is generally iterative, a solid foundation and understanding of the math skills that preceded current learning are key to success. The solution to these gaps is called mastery learning, the philosophy that underpins Khan Academy’s approach to education .

Mastery learning is an educational philosophy that emphasizes the importance of a student fully understanding a concept before moving on to the next one. Rather than rushing students through a curriculum, mastery learning asks educators to ensure that learners have “mastered” a topic or skill, showing a high level of proficiency and understanding, before progressing. This approach is rooted in the belief that all students can learn given the appropriate learning conditions and enough time, making it a markedly student-centered method. It promotes thoroughness over speed and encourages individualized learning paths, thus catering to the unique learning needs of each student.

Students will encounter mastery learning passively as they go through Khan Academy coursework, as our platform identifies gaps and systematically adjusts to support student learning outcomes. More details can be found in our Educators Hub . 

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How to learn math.

Learning at School

One of the most common methods of math instruction is classroom learning. In-class instruction provides students with real-time feedback, practical application, and a peer-learning environment. Teachers can personalize instruction by assessing students’ strengths and weaknesses, providing remediation when necessary, and offering advanced instruction to students who need it.

Learning at Home

Supplemental learning at home can complement traditional classroom instruction. For example, using online resources that provide additional practice opportunities, interactive games, and demonstrations, can help students consolidate learning outside of class. E-learning has become increasingly popular, with a wealth of online resources available to learners of all ages. The benefits of online learning include flexibility, customization, and the ability to work at one’s own pace. One excellent online learning platform is Khan Academy, which offers free video tutorials, interactive practice exercises, and a wealth of resources across a range of mathematical topics.

Moreover, parents can encourage and monitor progress, answer questions, and demonstrate practical applications of math in everyday life. For example, when at the grocery store, parents can ask their children to help calculate the price per ounce of two items to discover which one is the better deal. Cooking and baking with your children also provides a lot of opportunities to use math skills, like dividing a recipe in half or doubling the ingredients. 

Learning Math with the Help of Artificial Intelligence (AI) 

AI-powered tools are changing the way students learn math. Personalized feedback and adaptive practice help target individual needs. Virtual tutors offer real-time help with math concepts while AI algorithms identify areas for improvement. Custom math problems provide tailored practice, and natural language processing allows for instant question-and-answer sessions. 

Using Khan Academy’s AI Tutor, Khanmigo

Transform your child’s grasp of mathematics with Khanmigo , the 24/7 AI-powered tutor that specializes in tailored, one-on-one math instruction. Available at any time, Khanmigo provides personalized support that goes beyond mere answers to nurture genuine mathematical understanding and critical thinking. Khanmigo can track progress, identify strengths and weaknesses, and offer real-time feedback to help students stay on the right track. Within a secure and ethical AI framework, your child can tackle everything from basic arithmetic to complex calculus, all while you maintain oversight using robust parental controls.

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You can learn anything .

Math learning is essential for success in the modern world, and with the right approach, it can also be enjoyable and rewarding. Learning math requires curiosity, diligence, and the ability to connect abstract concepts with real-world applications. Strategies for effective math learning include a multifaceted approach, including classroom instruction, online courses, homework, tutoring, and personalized AI support. 

So, don’t let math anxiety hold you back; take advantage of available resources and technology to enhance your knowledge base and enjoy the benefits of math learning.

National Council of Teachers of Mathematics, “Principles to Actions: Ensuring Mathematical Success for All” , April 2014

Project Lead The Way Research Report, “The Power of Transportable Skills: Assessing the Demand and Value of the Skills of the Future” , 2020

Page. M, “Why Develop Quantitative and Qualitative Data Analysis Skills?” , 2016

Mann. EL, Creativity: The Essence of Mathematics, Journal for the Education of the Gifted. Vol. 30, No. 2, 2006, pp. 236–260, http://www.prufrock.com ’

Nakakoji Y, Wilson R.” Interdisciplinary Learning in Mathematics and Science: Transfer of Learning for 21st Century Problem Solving at University ”. J Intell. 2020 Sep 1;8(3):32. doi: 10.3390/jintelligence8030032. PMID: 32882908; PMCID: PMC7555771.

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17 maths problem solving strategies to boost your learning.

Worded problems getting the best of you? With this list of maths problem-solving strategies , you'll overcome any maths hurdle that comes your way.

Friday, 3rd June 2022

• What are strategies?

Understand the problem

Devise a plan, carry out the plan, look back and reflect, practise makes progress.

Problem-solving is a critical life skill that everyone needs. Whether you're dealing with everyday issues or complex challenges, being able to solve problems effectively can make a big difference to your quality of life.

While there is no one 'right' way to solve a problem, having a toolkit of different techniques that you can draw upon will give you the best chance of success. In this article, we'll explore 17 different math problem-solving strategies you can start using immediately to deepen your learning and improve your skills.

What are maths problem-solving strategies?

Before we get into the strategies themselves, let's take a step back and answer the question: what are these strategies? In simple terms, these are methods we use to solve mathematical problems—essential for anyone learning how to study maths . These can be anything from asking open-ended questions to more complex concepts like the use of algebraic equations.

The beauty of these techniques is they go beyond strictly mathematical application. It's more about understanding a given problem, thinking critically about it and using a variety of methods to find a solution.

Polya's 4-step process for solving problems

We're going to use Polya's 4-step model as the framework for our discussion of problem-solving activities . This was developed by Hungarian mathematician George Polya and outlined in his 1945 book How to Solve It. The steps are as follows:

We'll go into more detail on each of these steps as well as take a look at some specific problem-solving strategies that can be used at each stage.

This may seem like an obvious one, but it's crucial that you take the time to understand what the problem is asking before trying to solve it. Especially with a math word problem , in which the question is often disguised in language, it's easy for children to misinterpret what's being asked.

Here are some questions you can ask to help you understand the problem:

Do I understand all the words used in the problem?

What am I asked to find or show?

Can I restate the problem in my own words?

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

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

Is there anything I need to find out first in order to find the answer?

What information is extra or irrelevant?

Once you've gone through these questions, you should have a good understanding of what the problem is asking. Now let's take a look at some specific strategies that can be used at this stage.

1. Read the problem aloud

This is a great strategy for younger students who are still learning to read. By reading the problem aloud, they can help to clarify any confusion and better understand what's being asked. Teaching older students to read aloud slowly is also beneficial as it encourages them to internalise each word carefully.

2. Summarise the information

Using dot points or a short sentence, list out all the information given in the problem. You can even underline the keywords to focus on the important information. This will help to organise your thoughts and make it easier to see what's given, what's missing, what's relevant and what isn't.

3. Create a picture or diagram

This is a no-brainer for visual learners. By drawing a picture, let's say with division problems, you can better understand what's being asked and identify any information that's missing. It could be a simple sketch or a more detailed picture, depending on the problem.

4. Act it out

Visualising a scenario can also be helpful. It can enable students to see the problem in a different way and develop a more intuitive understanding of it. This is especially useful for math word problems that are set in a particular context. For example, if a problem is about two friends sharing candy, kids can act out the problem with real candy to help them understand what's happening.

5. Use keyword analysis

What does this word tell me? Which operations do I need to use? Keyword analysis involves asking questions about the words in a problem in order to work out what needs to be done. There are certain key words that can hint at what operation you need to use.

How many more?

How many left?

Equal parts

Once you understand the problem, it's time to start thinking about how you're going to solve it. This is where having a plan is vital. By taking the time to think about your approach, you can save yourself a lot of time and frustration later on.

There are many methods that can be used to figure out a pathway forward, but the key is choosing an appropriate one that will work for the specific problem you're trying to solve. Not all students understand what it means to plan a problem so we've outlined some popular problem-solving techniques during this stage.

6. Look for a pattern

Sometimes, the best way to solve a problem is to look for a pattern. This could be a number, a shape pattern or even just a general trend that you can see in the information given. Once you've found it, you can use it to help you solve the problem.

7. Guess and check

While not the most efficient method, guess and check can be helpful when you're struggling to think of an answer or when you're dealing with multiple possible solutions. To do this, you simply make a guess at the answer and then check to see if it works. If it doesn't, you make another systematic guess and keep going until you find a solution that works.

8. Working backwards

Regressive reasoning, or working backwards, involves starting with a potential answer and working your way back to figure out how you would get there. This is often used when trying to solve problems that have multiple steps. By starting with the end in mind, you can work out what each previous step would need to be in order to arrive at the answer.

9. Use a formula

There will be some problems where a specific formula needs to be used in order to solve it. Let's say we're calculating the cost of flooring panels in a rectangular room (6m x 9m) and we know that the panels cost $15 per sq. metre.

There is no mention of the word 'area', and yet that is exactly what we need to calculate. The problem requires us to use the formula for the area of a rectangle (A = l x w) in order to find the total cost of the flooring panels.

10. Eliminate the possibilities

When there are a lot of possibilities, one approach could be to start by eliminating the answers that don't work. This can be done by using a process of elimination or by plugging in different values to see what works and what doesn't.

11. Use direct reasoning

Direct reasoning, also known as top-down or forward reasoning, involves starting with what you know and then using that information to try and solve the problem . This is often used when there is a lot of information given in the problem.

By breaking the problem down into smaller chunks, you can start to see how the different pieces fit together and eventually work out a solution.

12. Solve a simpler problem

One of the most effective methods for solving a difficult problem is to start by solving a simpler version of it. For example, in order to solve a 4-step linear equation with variables on both sides, you could start by solving a 2-step one. Or if you're struggling with the addition of algebraic fractions, go back to solving regular fraction addition first.

Once you've mastered the easier problem, you can then apply the same knowledge to the challenging one and see if it works.

13. Solve an equation

Another common problem-solving technique is setting up and solving an equation. For instance, let's say we need to find a number. We know that after it was doubled, subtracted from 32, and then divided by 4, it gave us an answer of 6. One method could be to assign this number a variable, set up an equation, and solve the equation by 'backtracking and balancing the equation'.

Now that you have a plan, it's time to implement it. This is where you'll put your problem-solving skills to the test and see if your solution actually works. There are a few things to keep in mind as you execute your plan:

14. Be systematic

When trying different methods or strategies, it's important to be systematic in your approach. This means trying one problem-solving strategy at a time and not moving on until you've exhausted all possibilities with that particular approach.

15. Check your work

Once you think you've found a solution, it's important to check your work to make sure that it actually works. This could involve plugging in different values or doing a test run to see if your solution works in all cases.

16. Be flexible

If your initial plan isn't working, don't be afraid to change it. There is no one 'right' way to solve a problem, so feel free to try different things, seek help from different resources and continue until you find a more efficient strategy or one that works.

17. Don't give up

It's important to persevere when trying to solve a difficult problem. Just because you can't see a solution right away doesn't mean that there isn't one. If you get stuck, take a break and come back to the problem later with fresh eyes. You might be surprised at what you're able to see after taking some time away from it.

Once you've solved the problem, take a step back and reflect on the process that you went through. Most middle school students forget this fundamental step. This will help you to understand what worked well and what could be improved upon next time.

Whether you do this after a math test or after an individual problem, here are some questions to ask yourself:

What was the most challenging part of the problem?

Was one method more effective than another?

Would you do something differently next time?

What have you learned from this experience?

By taking the time to reflect on your process you'll be able to improve upon it in future and become an even better problem solver. Make sure you write down any insights so that you can refer back to them later.

There is never only one way to solve math problems. But the best way to become a better problem solver is to practise, practise, practise! The more you do it, the better you'll become at identifying different strategies, and the more confident you'll feel when faced with a challenging problem.

The list we've covered is by no means exhaustive, but it's a good starting point for you to begin your journey. When you get stuck, remember to keep an open mind. Experiment with different approaches. Different word problems. Be prepared to go back and try something new. And most importantly, don't forget to have fun!

The essence and beauty of mathematics lies in its freedom. So while these strategies provide nice frameworks, the best work is done by those who are comfortable with exploration outside the rules, and of course, failure! So go forth, make mistakes and learn from them. After all, that's how we improve our problem-solving skills and ability.

Lastly, don't be afraid to ask for help. If you're struggling to solve math word problems, there's no shame in seeking assistance from a certified Melbourne maths tutor . In every lesson at Math Minds, our expert teachers encourage students to think creatively, confidently and courageously.

If you're looking for a mentor who can guide you through these methods, introduce you to other problem-solving activities and help you to understand Mathematics in a deeper way - get in touch with our team today. Sign up for your free online maths assessment and discover a world of new possibilities.

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How to Solve Math Problems Faster: 15 Techniques to Show Students

Written by Marcus Guido

• Teaching Strategies

“Test time. No calculators.”

You’ll intimidate many students by saying this, but teaching techniques to solve math problems with ease and speed can make it less daunting.

This can also  make math more rewarding . Instead of relying on calculators, students learn strategies that can improve their concentration and estimation skills while building number sense. And, while there are educators who  oppose math “tricks”  for valid reasons, proponents point to benefits such as increased confidence to handle difficult problems.

Here are 15 techniques to show students,  helping them solve math problems faster:

Addition and Subtraction

1. two-step addition.

Many students struggle when learning to add integers of three digits or higher together, but changing the process’s steps can make it easier.

The first step is to  add what’s easy.  The second step is to  add the rest.

Let’s say students must find the sum of 393 and 89. They should quickly see that adding 7 onto 393 will equal 400 — an easier number to work with. To balance the equation, they can then subtract 7 from 89.

Broken down, the process is:

• (393 + 7) + (89 – 7)

With this fast technique, big numbers won’t look as scary now.

2. Two-Step Subtraction

There’s a similar method for subtraction.

Remove what’s easy. Then remove what’s left.

Suppose students must find the difference of 567 and 153. Most will feel that 500 is a simpler number than 567. So, they just have to take away 67 from the minuend — 567 — and the subtrahend — 153 — before solving the equation.

Here’s the process:

• (567 – 67) – (153 – 67)

Instead of two complex numbers, students will only have to tackle one.

3. Subtracting from 1,000

You can  give students confidence  to handle four-digit integers with this fast technique.

To subtract a number from 1,000, subtract that number’s first two digits from 9. Then, subtract the final digit from 10.

Let’s say students must solve 1,000 – 438.  Here are the steps:

This also applies to 10,000, 100,000 and other integers that follow this pattern.

Multiplication and Division

4. doubling and halving.

When students have to multiply two integers, they can speed up the process when one is an even number. They just need to  halve the even number and double the other number.

Students can stop the process when they can no longer halve the even integer, or when the equation becomes manageable.

Using 33 x 48 as an example,  here’s the process:

The only prerequisite is understanding the 2 times table.

5. Multiplying by Powers of 2

This tactic is a speedy variation of doubling and halving.

It simplifies multiplication if a number in the equation is a power of 2, meaning it works for 2, 4, 8, 16 and so on.

Here’s what to do:  For each power of 2 that makes up that number, double the other number.

For example, 9 x 16 is the same thing as 9 x (2 x 2 x 2 x 2) or 9 x 24. Students can therefore double 9 four times to reach the answer:

Unlike doubling and halving, this technique demands an understanding of exponents along with a strong command of the 2 times table.

6. Multiplying by 9

For most students, multiplying by 9 — or 99, 999 and any number that follows this pattern — is difficult compared with multiplying by a power of 10.

But there’s an easy tactic to solve this issue, and  it has two parts.

First, students round up the 9  to 10. Second, after solving the new equation, they subtract the number they just multiplied by 10 from the answer.

For example, 67 x 9 will lead to the same answer as 67 x 10 – 67. Following the order of operations will give a result of 603. Similarly, 67 x 99 is the same as 67 x 100 – 67.

Despite more steps, altering the equation this way is usually faster.

7. Multiplying by 11

There’s an easier way for multiplying two-digit integers by 11.

Let’s say students must find the product of 11 x 34.

The idea is to put a space between the digits, making it 3_4. Then, add the two digits together and put the sum in the space.

The answer is 374.

What happens if the sum is two digits? Students would put the second digit in the space and add 1 to the digit to the left of the space.  For example:

It’s multiplication without having to multiply.

8. Multiplying Even Numbers by 5

This technique only requires basic division skills.

There are two steps,  and 5 x 6 serves as an example. First, divide the number being multiplied by 5 — which is 6 — in half. Second, add 0 to the right of number.

The result is 30, which is the correct answer.

It’s an ideal, easy technique for students mastering the 5 times table.

9. Multiplying Odd Numbers by 5

This is another time-saving tactic that works well when teaching students the 5 times table.

This one has three steps,  which 5 x 7 exemplifies.

First, subtract 1 from the number being multiplied by 5, making it an even number. Second, cut that number in half — from 6 to 3 in this instance. Third, add 5 to the right of the number.

The answer is 35.

Who needs a calculator?

10. Squaring a Two-Digit Number that Ends with 1

Squaring a high two-digit number can be tedious, but there’s a shortcut if 1 is the second digit.

There are four steps to this shortcut,  which 812 exemplifies:

• Subtract 1 from the integer: 81 – 1 = 80
• Square the integer, which is now an easier number: 80 x 80 = 6,400
• Add the integer with the resulting square twice: 6,400 + 80 + 80 = 6,560
• Add 1: 6,560 + 1 = 6,561

This work-around eliminates the difficulty surrounding the second digit, allowing students to work with multiples of 10.

11. Squaring a Two-Digit Numbers that Ends with 5

Squaring numbers ending in 5 is easier, as there are  only two parts of the process.

First, students will always make 25 the product’s last digits.

Second, to determine the product’s first digits, students must multiply the number’s first digit — 9, for example — by the integer that’s one higher — 10, in this case.

So, students would solve 952 by designating 25 as the last two digits. They would then multiply 9 x 10 to receive 90. Putting these numbers together, the  result is 9,025.

Just like that, a hard problem becomes easy multiplication for many students.

12. Calculating Percentages

Cross-multiplication is an  important skill  to develop, but there’s an easier way to calculate percentages.

For example, if students want to know what 65% of 175 is, they can multiply the numbers together and move the decimal place two digits to the left.

The result is 113.75, which is indeed the correct answer.

This shortcut is a useful timesaver on tests and quizzes.

13. Balancing Averages

To determine the average among a set of numbers, students can balance them instead of using a complex formula.

Suppose a student wants to volunteer for an average of 10 hours a week over a period of four weeks. In the first three weeks, the student worked for 10, 12 and 14 hours.

To determine the number of hours required in the fourth week, the student must  add how much he or she surpassed or missed the target average  in the other weeks:

• 14 hours – 10 hours = 4 hours
• 12 – 10 = 2
• 10 – 10 = 0
• 4 hours + 2 hours + 0 hours = 6 hours

To learn the number of hours for the final week, the student must  subtract the sum from the target average:

• 10 hours – 6 hours = 4 hours

With practice, this method may not even require pencil and paper. That’s how easy it is. 

Word Problems

14. identifying buzzwords.

Students who struggle to translate  word problems  into equations will benefit from learning how to spot buzzwords — phrases that indicate specific actions.

This isn’t a trick. It’s a tactic.

Teach students to look for these buzzwords,  and what skill they align with in most contexts:

Be sure to include buzzwords that typically appear in their textbooks (or other classroom  math books ), as well as ones you use on tests and assignments.

As a result, they should have an  easier time processing word problems .

15. Creating Sub-Questions

For complex word problems, show students how to dissect the question by answering three specific sub-questions.

Each student should ask him or herself:

• What am I looking for?  — Students should read the question over and over, looking for buzzwords and identifying important details.
• What information do I need?  — Students should determine which facts, figures and variables they need to solve the question. For example, if they determine the question is rooted in subtraction, they need the minuend and subtrahend.
• What information do I have?  — Students should be able to create the core equation using the information in the word problem, after determining which details are important.

These sub-questions help students avoid overload.

Instead of writing and analyzing each detail of the question, they’ll be able to identify key information. If you identify students who are struggling with these, you can use  peer learning  as needed.  

For more fresh approaches to teaching math in your classroom, consider treating your students to a range of  fun math activities .

Final Thoughts About these Ways to Solve Math Problems Faster

Showing these 15 techniques to students can give them the  confidence to tackle tough questions .

They’re also  mental math  exercises, helping them build skills related to focus, logic and critical thinking.

A rewarding class equals an  engaging class . That’s an easy equation to remember.

> Create or log into your teacher account on Prodigy  — a free, adaptive math game that adjusts content to accommodate player trouble spots and learning speeds. Aligned to US and Canadian curricula, it’s loved by more than 500,000 teachers and 15 million students.

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Think back to the first problem in this chapter, the ABC Problem . What did you do to solve it? Even if you did not figure it out completely by yourself, you probably worked towards a solution and figured out some things that did not work.

Unlike exercises, there is never a simple recipe for solving a problem. You can get better and better at solving problems, both by building up your background knowledge and by simply practicing. As you solve more problems (and learn how other people solved them), you learn strategies and techniques that can be useful. But no single strategy works every time.

Pólya’s How to Solve It

George Pólya was a great champion in the field of teaching effective problem solving skills.  He was born in Hungary in 1887, received his Ph.D. at the University of Budapest, and was a professor at Stanford University (among other universities).  He wrote many mathematical papers along with three books, most famously, “How to Solve it.” Pólya died at the age 98 in 1985. [1]

 In 1945, Pólya published the short book How to Solve It , which gave a four-step method for solving mathematical problems:

• Understand the problem.
• Devise a plan.
• Carry out the plan.
• Looking back.

This is all well and good, but how do you actually do these steps?!?! Steps 1. and 2. are particularly mysterious! How do you “make a plan?” That is where you need some tools in your toolbox, and some experience to draw upon.

Much has been written since 1945 to explain these steps in more detail, but the truth is that they are more art than science. This is where math becomes a creative endeavor (and where it becomes so much fun). We will articulate some useful problem solving strategies, but no such list will ever be complete. This is really just a start to help you on your way. The best way to become a skilled problem solver is to learn the background material well, and then to solve a lot of problems!

We have already seen one problem solving strategy, which we call “Wishful Thinking.” Do not be afraid to change the problem! Ask yourself “what if” questions:

• What if the picture was different?
• What if the numbers were simpler?
• What if I just made up some numbers?

You need to be sure to go back to the original problem at the end, but wishful thinking can be a powerful strategy for getting started.

This brings us to the most important problem solving strategy of all:

Problem Solving Strategy 2 (Try Something!). If you are really trying to solve a problem, the whole point is that you do not know what to do right out of the starting gate. You need to just try something! Put pencil to paper (or stylus to screen or chalk to board or whatever!) and try something. This is often an important step in understanding the problem; just mess around with it a bit to understand the situation and figure out what is going on.

And equally important: If what you tried first does not work, try something else! Play around with the problem until you have a feel for what is going on.

Problem 2 (Payback)

Last week, Alex borrowed money from several of his friends. He finally got paid at work, so he brought cash to school to pay back his debts. First he saw Brianna, and he gave her 1/4 of the money he had brought to school. Then Alex saw Chris and gave him 1/3 of what he had left after paying Brianna. Finally, Alex saw David and gave him 1/2 of what he had remaining. Who got the most money from Alex?

Think/Pair/Share

After you have worked on the problem on your own for a while, talk through your ideas with a partner (even if you have not solved it). What did you try? What did you figure out about the problem?

This problem lends itself to two particular strategies. Did you try either of these as you worked on the problem? If not, read about the strategy and then try it out before watching the solution.

Problem Solving Strategy 3 (Draw a Picture). Some problems are obviously about a geometric situation, and it is clear you want to draw a picture and mark down all of the given information before you try to solve it. But even for a problem that is not geometric, like this one, thinking visually can help! Can you represent something in the situation by a picture?

Draw a square to represent all of Alex’s money. Then shade 1/4 of the square — that’s what he gave away to Brianna. How can the picture help you finish the problem?

After you have worked on the problem yourself using this strategy (or if you are completely stuck), you can watch someone else’s solution.

Problem Solving Strategy 4 (Make Up Numbers). Part of what makes this problem difficult is that it is about money, but there are no numbers given. That means the numbers must not be important. So just make them up!

You can work forwards: Assume Alex had some specific amount of money when he showed up at school, say $100. Then figure out how much he gives to each person. Or you can work backwards: suppose he has some specific amount left at the end, like $10. Since he gave Chris half of what he had left, that means he had $20 before running into Chris. Now, work backwards and figure out how much each person got.

Watch the solution only after you tried this strategy for yourself.

If you use the “Make Up Numbers” strategy, it is really important to remember what the original problem was asking! You do not want to answer something like “Everyone got $10.” That is not true in the original problem; that is an artifact of the numbers you made up. So after you work everything out, be sure to re-read the problem and answer what was asked!

Problem 3 (Squares on a Chess Board)

How many squares, of any possible size, are on a 8 × 8 chess board? (The answer is not 64… It’s a lot bigger!)

Remember Pólya’s first step is to understand the problem. If you are not sure what is being asked, or why the answer is not just 64, be sure to ask someone!

Think / Pair / Share

After you have worked on the problem on your own for a while, talk through your ideas with a partner (even if you have not solved it). What did you try? What did you figure out about the problem, even if you have not solved it completely?

It is clear that you want to draw a picture for this problem, but even with the picture it can be hard to know if you have found the correct answer. The numbers get big, and it can be hard to keep track of your work. Your goal at the end is to be absolutely positive that you found the right answer. You should never ask the teacher, “Is this right?” Instead, you should declare, “Here’s my answer, and here is why I know it is correct!”

Problem Solving Strategy 5 (Try a Simpler Problem). Pólya suggested this strategy: “If you can’t solve a problem, then there is an easier problem you can solve: find it.” He also said: “If you cannot solve the proposed problem, try to solve first some related problem. Could you imagine a more accessible related problem?” In this case, an 8 × 8 chess board is pretty big. Can you solve the problem for smaller boards? Like 1 × 1? 2 × 2? 3 × 3?

Of course the ultimate goal is to solve the original problem. But working with smaller boards might give you some insight and help you devise your plan (that is Pólya’s step (2)).

Problem Solving Strategy 6 (Work Systematically). If you are working on simpler problems, it is useful to keep track of what you have figured out and what changes as the problem gets more complicated.

For example, in this problem you might keep track of how many 1 × 1 squares are on each board, how many 2 × 2 squares on are each board, how many 3 × 3 squares are on each board, and so on. You could keep track of the information in a table:

Problem Solving Strategy 7 (Use Manipulatives to Help You Investigate). Sometimes even drawing a picture may not be enough to help you investigate a problem. Having actual materials that you move around can sometimes help a lot!

For example, in this problem it can be difficult to keep track of which squares you have already counted. You might want to cut out 1 × 1 squares, 2 × 2 squares, 3 × 3 squares, and so on. You can actually move the smaller squares across the chess board in a systematic way, making sure that you count everything once and do not count anything twice.

Problem Solving Strategy 8 (Look for and Explain Patterns). Sometimes the numbers in a problem are so big, there is no way you will actually count everything up by hand. For example, if the problem in this section were about a 100 × 100 chess board, you would not want to go through counting all the squares by hand! It would be much more appealing to find a pattern in the smaller boards and then extend that pattern to solve the problem for a 100 × 100 chess board just with a calculation.

If you have not done so already, extend the table above all the way to an 8 × 8 chess board, filling in all the rows and columns. Use your table to find the total number of squares in an 8 × 8 chess board. Then:

• Describe all of the patterns you see in the table.
• Can you explain and justify any of the patterns you see? How can you be sure they will continue?
• What calculation would you do to find the total number of squares on a 100 × 100 chess board?

(We will come back to this question soon. So if you are not sure right now how to explain and justify the patterns you found, that is OK.)

Problem 4 (Broken Clock)

This clock has been broken into three pieces. If you add the numbers in each piece, the sums are consecutive numbers. ( Consecutive numbers are whole numbers that appear one after the other, such as 1, 2, 3, 4 or 13, 14, 15.)

Can you break another clock into a different number of pieces so that the sums are consecutive numbers? Assume that each piece has at least two numbers and that no number is damaged (e.g. 12 isn’t split into two digits 1 and 2.)

Remember that your first step is to understand the problem. Work out what is going on here. What are the sums of the numbers on each piece? Are they consecutive?

After you have worked on the problem on your own for a while, talk through your ideas with a partner (even if you have not solved it). What did you try? What progress have you made?

Problem Solving Strategy 9 (Find the Math, Remove the Context). Sometimes the problem has a lot of details in it that are unimportant, or at least unimportant for getting started. The goal is to find the underlying math problem, then come back to the original question and see if you can solve it using the math.

In this case, worrying about the clock and exactly how the pieces break is less important than worrying about finding consecutive numbers that sum to the correct total. Ask yourself:

• What is the sum of all the numbers on the clock’s face?
• Can I find two consecutive numbers that give the correct sum? Or four consecutive numbers? Or some other amount?
• How do I know when I am done? When should I stop looking?

Of course, solving the question about consecutive numbers is not the same as solving the original problem. You have to go back and see if the clock can actually break apart so that each piece gives you one of those consecutive numbers. Maybe you can solve the math problem, but it does not translate into solving the clock problem.

Problem Solving Strategy 10 (Check Your Assumptions). When solving problems, it is easy to limit your thinking by adding extra assumptions that are not in the problem. Be sure you ask yourself: Am I constraining my thinking too much?

In the clock problem, because the first solution has the clock broken radially (all three pieces meet at the center, so it looks like slicing a pie), many people assume that is how the clock must break. But the problem does not require the clock to break radially. It might break into pieces like this:

Were you assuming the clock would break in a specific way? Try to solve the problem now, if you have not already.

• Image of Pólya by Thane Plambeck from Palo Alto, California (Flickr) [CC BY 2.0 (http://creativecommons.org/licenses/by/2.0)], via Wikimedia Commons ↵

Mathematics for Elementary Teachers Copyright © 2018 by Michelle Manes is licensed under a Creative Commons Attribution-ShareAlike 4.0 International License , except where otherwise noted.

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5 Strategies for Successful Problem Solving

• Powerful teaching strategies
• December 26, 2023
• Michaela Epstein

Blog > 5 Strategies for Successful Problem Solving

Problem solving can change the way students see maths – and how they see themselves as maths learners.

But, it's tough to help all students get the most out of a task.

To help, here are  5 Strategies for Problem Solving Success.

These are 5 valuable lessons I've learned from working with teachers across the globe .  You can use these strategies with all your students, no matter their level.

5 Strategies for Problem Solving Success

Your own enthusiasm is quickly picked up by your students. So, choose a problem, puzzle or game that you’re excited and curious about.

How do you know what will spark your curiosity? Do the task yourself!

(That’s why,  in the workshops I run , we spend a lot of time  actually  exploring problems. It’s a chance to step into students' shoes and experience maths from their perspective.)

Often, curriculum content becomes the goal of problem solving. For example, adding fractions, calculating areas or solving quadratic equations.

But, this is a mistake! Here's why-

Low floor, high ceiling tasks give students choices. Choices about what strategies to use, tools to draw on – and even what end-points to get to.

The most valuable goals focus on building confidence and capability in problem solving. For example:

• To make and break conjectures
• To use and evaluate different strategies
• To organise data in meaningful ways
• To explain and justify their conclusions.

The start of a task is what will get your students curious and hungry to get underway.

Consider: What's the least information your students will need?

At  ​our Members' online PL sessions​ , we look at one of four possibilities for launching a problem:

• Present a mystery to explore
• Present an example and non-example
• Run a demonstration game
• Show how to use a tool.

Keep the launch short – under 5 minutes. This is just enough to keep students’ attention AND share essential information.

Let’s face it, problem solving is hard, no matter your age or mathematical skill set.

Students aren’t afraid of hard work – they’re afraid of feeling or looking stupid. And, when those tricky maths moments do come, you can help.

Using questions, tools and other prompts can bring clarity and boost confidence.

(Here's a  free question catalogue  you might find handy to have in your back pocket.)

This careful support will help your students find problem solving far less daunting. Instead, it can become a chance for wonderous mathematical exploration.

Picture this: Your students are elbows deep in a problem, there’s a buzz in the air – oh, and only a minute until the bell.

The  most important  stage of a problem solving task – right at the end – is often the one that gets dropped off.

Why does ‘wrapping up’ matter?

In the last 10 minutes of a problem, students can share conjectures, strategies and solutions. It's also a chance to consider new questions that may open up further exploration.

In wrapping up, important learning will happen. Your students will observe patterns, make connections and clarify conjectures. You might even notice ‘aha’ moments.

Five strategies for problem solving success:

• Choose a task that YOU'RE keen on,
• Set a goal for strengthening problem solving skills,
• Plan a short launch to make the task widely accessible,
• Use questions, tools and prompts to support productive exploration, and
• Wrap up to create space for pivotal learning.

Join the Conversation

Dear Michaela, Greetings !! Thank you for sharing the strategies for problem solving task. These strategies will definitely enhance the skill in the mindset of young learners. In India ,Students of Grade 9 and Grade 10 have to learn and solve lot of theorems of triangle, Quadrilateral, Circle etc. Being an educator I have noticed that most of the students learn the theorems and it’s derivation by heart as a result they lack in understanding the application of these theorems.

I will appreciate if you can share your insights as how to make these topics interesting and easy to grasp.

Once again thanks for sharing such informative ideas.

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Four Principles for Effective Math Intervention

The Institute for Education Sciences, an authority on Response to Intervention (RtI) for math published a groundbreaking report on RtI that outlines a series of intervention recommendations. The report Assisting Students Struggling with Mathematics: Response to Intervention (RtI) for Elementary and Middle schools  covers recommendations that were judged to have strong or moderate evidence to support RtI and provide the foundation for effective math intervention. Here are specific strategies that fall under these four key recommendations:

1. Instruction during the intervention should be explicit and systematic.

This recommendation from the report includes providing models of proficient problem solving, verbalization of thought processes, guided practice, corrective feedback, and frequent cumulative review.

Students who have been classified as Tier 2 or 3 in RtI need their instruction to be organized and scaffolded. They lack the numeracy skills and background knowledge to engage in theoretical exercises with math

Nothing is more systematic than the four-step approach to problem solving, first outlined by educator George Polly in 1945:

•  Understand the problem. Restate the problem, and then identify the information given and the information that needs to be determined.
• Make a plan. Relate the problem to similar problems solved in the past. Consider possible strategies, and then select a strategy or a combination of strategies
• Carry out the plan. Execute the chosen strategy and perform the necessary calculations. Revise and apply different strategies as necessary. 
• Look back at the solution. Evaluate the strategy/strategies used for problem solving and then assess if there is a better way to approach the problem.

2. Interventions should include instruction on solving word problems that is based on common underlying structures.

A stripped-down version of the Gradual Release Model—the “I do, We do, You do” strategy—is effective in all levels of education. As RtI students require as much structure as possible, the strategy gives them an effective way to know what to expect from a lesson. In word problems, not only can “I do, We do, You do” be used to solve problems, it can also be used to have students create their own word problems, reaching synthesis, a higher level of taxonomy. Students should start with sentences that involve a specific math operation and build from there. (McCarney, S. B., Cummins Wunderlich, K., Bauer, A. The Teacher’s Resource Guide.)

3. Intervention materials should include opportunities for students to work with visual representations of mathematical ideas.

This recommendation from the RtI report goes on to say that interventionists should be proficient in the use of visual representations of mathematical ideas. Graphics are important in math instruction, especially as the curriculum becomes more data=based under the Common Core State Standards. RtI students are not excluded from having to be able to read charts, graphs, and other math graphics. Intervention Central , which provides educators with free RtI resources, has a great intervention that uses the Question-Answer Relationship (QAR) to help students break down math graphics. In short, QARs come in four types:

•  “Right There” questions are found explicitly in the graphic.
•  “Think and Search” questions are not quite as explicit, but still can be found in the graphic with some close analysis.
• “Author and You” questions ask students to compare the data with their own life experiences and opinions.
• “On My Own” questions require only the student’s own knowledge and experiences to answer.

4. Interventions at all grade levels should devote about 10 minutes in each session to building fluent retrieval of basic arithmetic facts.

Rehearsal of math facts is a key step in the RtI process because many of the students who fall into Tier 2 or 3 status are missing key pieces of background information, usually in the form of math facts.  

Taking ten minutes each day to review facts is typically done with flash cards, which can be tedious and, therefore, does not always prove to be effective. Another strategy, described by Intervention Central, uses flash-card practice that balances “known” facts with “unknown” facts. Unknown facts are modeled by a teacher or tutor and then presented with known facts—those already mastered by the student—in a sequence. Not only does this systematically build recall, it also builds confidence in the student because they are consistently getting cards correct throughout the process.  

Using Technology to Boost RtI for Math

Digital curriculum, at its best, incorporates the scaffolds and formative assessment that define successful RtI for math. Personalized learning and ongoing, robust data collection provide a meaningful feedback loop for both the student and teacher that leads to deeper learning and higher achievement. Real-time feedback—while learning is happening—is critical so that students don’t practice new math skills, again and again , incorrectly.  

Technological interventions to boost math achievement can be an important part of a successful math intervention , but a commitment to the process is also a necessary part of helping students succeed in math and sharpen their math skills .

Read More About Math Instruction on the DE Blog!

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7 Real-World Math Strategies

We asked our audience how they’re using the real world to teach math and compiled their most intriguing responses.

Math used to be all rote memorization and pencil-to-paper equations disconnected from the real world, but more and more teachers are realizing the importance of making practical, relevant connections in math.

We asked our audience of educators: How do you use the real world to teach math? We’ve collected some of the most interesting answers, ways teachers are connecting math to the everyday lives of their students.

The Real-World Math Wall

To get her fifth-grade students thinking about the math in their lives—and to head off the inevitable “we’ll never use this in the real world” complaint—Samantha Baumgardner, a teacher at Woodrow Petty Elementary School in Taft, Texas, has them bring in a common item and write three ways it relates to math on a notecard. These objects form the class’s real-world math wall.

An interesting rule: Once something is placed on the wall, the same item can’t be brought in again—pushing students to think outside the box to make real-world math connections. Students bring in objects like playing cards, cake pans, softball score sheets, and cookie recipes. One student brought in a medicine dropper and by way of explanation wrote, “The medicine dropper helps you measure water, and it also helps you with cooking steak.”

A School Water Audit

At North Agincourt Jr. Public School in Scarborough, Canada, Krista Dunlop-Sheppard, a bilingual resource and home school teacher for grades 1–6, has her students conduct a water audit at home and at their school. Her students have a wide range of math ability: In a single class, she has students with learning disabilities, students who are gifted, and students who have no diagnoses but need extra support. Modifying a pilot project created by the Toronto Zoo Education Department , students add, subtract, find averages, and measure liquids—like the flow rate of all the water fountains, toilets, and urinals—to measure the amount of water their school uses in a day.

They also interview custodians to discover their daily water usage while mopping floors, and do online research to find out how much water the cafeteria dishwasher uses. When finished, students suggest ways for the school to conserve—like collecting rainwater in a barrel to water plants, and cleaning paint brushes in a bucket instead of using running water. In June, students repeat the water audit and see if the changes they implemented made a difference.

Acting Out Restaurant Scenarios

There’s nothing wrong with using money to teach negative and positive numbers, or pizza to introduce fractions, but Justin Ouellette—a third- to fifth-grade International Baccalaureate educator at Suzhou Singapore International School in China—takes these exercises a memorable step further, bringing in menus so students can act out true-to-life restaurant scenarios. Going dutch on dinner and tipping reinforces addition, subtraction, decimals, and percents, Ouellette says.

If you need a good resource: Ouellette has used this free lesson plan about a fictional eatery called the Safari Restaurant.

Integrating Math Into English and History

Making real-world math connections can happen outside of math class, too.

While reading Elie Wiesel’s Night , 12th-grade students at Kittatinny Regional High School in Hampton Township, New Jersey, calculate the volume of 11 million pennies to help them imagine the impact of the lives lost during the Holocaust. Ashley Swords, a resource center English teacher for grades nine to 12, uses pennies because they are small and plentiful and allow her to recontextualize a familiar, everyday object.

Students perform other calculations to amplify the impact of the lessons—at the school’s football field, for example, they determine the volume of Swords herself and then calculate how many football fields would be needed to bury the 11 million Holocaust victims if they were each Swords’s size and were buried in graves 10 feet deep. Guesses ranged from two to 20 football fields, with students finally concluding that it would take about 343 football fields.

Swords knew this lesson was a success when a group of six seniors got emotional after completing it and realizing the magnitude of the deaths in World War II.

Math Recipes

Recipes were perhaps the most popular idea among the elementary teachers who responded to our request for real-world math examples. Fifth-grade teacher Gabi Sanfilippo of Meadow Ridge in Orland Park, Illinois, for example, asked her students to write down one to two ways they used math outside of the classroom during spring break—and more than half of her students wrote that they practiced using measurements and fractions while baking and cooking with their families.

In class, students practice halving, tripling, or quadrupling recipes based on how many people they’d cook or bake for. Most teachers don’t actually cook in class, but often students cook at home to practice their new skills.

Another educator, Elizabeth Eagan of Bastrop Independent School District in Texas, brings in a toaster oven to bake in class. She teaches the visually impaired, from newborns to 22-year-olds, using recipes in large print, braille, or audio to show the real-world application of adding and subtracting fractions.

Eagan prints out recipes at 129 percent for her low-vision students or converts the text to braille using  braille transcription software , and then prints them using a braille printer . She purchased Stir It Up , a cookbook in both braille and print that makes it easier for families and teachers to help their students if they haven’t mastered the tactile language. Students may use electronic magnifying glasses like Pebble and Ruby. For audio, Eagan records herself reading the recipes, has a peer or parent help, or uses the app Seeing AI , which can scan and read recipes aloud.

Grocery Store Field Trip

Many teachers make real-world math connections to grocery shopping. Leanna Agcaoili’s second-grade students at Joseph J. Fern Elementary School in Honolulu, Hawaii, are tasked with creating a healthy meal for their family on a $20 budget. On a grocery store field trip, students practice adding and subtracting one- and two-digit whole numbers—and learn about money and budgeting in the process.

Agcaoili says she’ll do a practice run in class next year, noting that the first time through some students had difficulty finding their ingredients.

Graphing Favorite Halloween Candy Wrappers

After Halloween, Dottie Wright Berzins, a retired public school teacher, had her students bring in wrappers from their favorite Halloween candy. Depending on the age, the students then created graphs showing their favorite candies.

Younger ages built life-size candy bar graphs, marking the x- and y-axis with masking tape on the floor and using the wrappers to represent the bars. Older students constructed tally charts and paper graphs, and followed ads, tracked which brands advertised more, interviewed peers about their favorite candy, made predictions, and created what-if scenarios—like what if the price of chocolate increased—how would that variable impact the graph?

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“Teachers should teach math in a way that encourages students to engage in sense-making and not merely to memorize or internalize exactly what the teacher says or does,” says Jon R. Star.

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One way is the wrong way to do math. Here’s the right way.

Harvard Staff Writer

Research by Ed School psychologist reinforces case for stressing multiple problem-solving paths over memorization

There’s never just one way to solve a math problem, says Jon R. Star , a psychologist and professor of education at the Harvard Graduate School of Education. With researchers from Vanderbilt University, Star found that teaching students multiple ways to solve math problems instead of using a single method improves teaching and learning. In an interview with the Gazette, Star, a former math teacher, outlined the research and explained how anyone, with the right instruction, can develop a knack for numbers.

Jon R. Star

GAZETTE: What is the most common misconception about math learning?

STAR: That you’re either a math person or you’re not a math person — that some people are just born with math smarts, and they can do math, and other people are just not, and there’s not much you can do about it.

GAZETTE: What does science say about the process of learning math?

STAR: One thing we know from psychology about the learning process is that the act of reaching into your brain, grabbing some knowledge, pulling it out, chewing on it, talking about it, and putting it back helps you learn. Psychologists call this elaborative encoding. The more times you can do that process — putting knowledge in, getting it out, elaborating on it, putting it back in — the more you will have learned, remembered, and understood the material. We’re trying to get math teachers to help students engage in that process of elaborative encoding.

GAZETTE: How did you learn math yourself?

STAR: Learning math should involve some sense-making. It’s necessary that we listen to what our teacher tells us about the math and try to make sense of it in our minds. Math learning is not about pouring the words directly from the teacher’s mouth into the students’ ears and brains. That’s not the way it works. I think that’s how I learned math. But that’s not how I hope students learn math and that’s not how I hope teachers think about the teaching of math. Teachers should teach math in a way that encourages students to engage in sense-making and not merely to memorize or internalize exactly what the teacher says or does.

GAZETTE: Tell us about the teaching method described in the research.

STAR: One of the strategies that some teachers may use when teaching math is to show students how to solve problems and expect that the student is going to end up using the same method that the teacher showed. But there are many ways to solve math problems; there’s never just one way.

The strategy we developed asks that teachers compare two ways for solving a problem, side by side, and that they follow an instructional routine to lead a discussion to help students understand the difference between the two methods. That discussion is really the heart of this routine because it is fundamentally about sharing reasoning: Teachers ask students to explain why a strategy works, and students must dig into their heads and try to say what they understand. And listening to other people’s reasoning reinforces the process of learning.

GAZETTE: Why is this strategy an improvement over just learning a single method?

STAR: We think that learning multiple strategies for solving problems deepens students’ understanding of the content. There is a direct benefit to learning through comparing multiple methods, but there are also other types of benefits to students’ motivation. In this process, students come to see math a little differently — not just as a set of problems, each of which has exactly one way to solve it that you must memorize, but rather, as a terrain where there are always decisions to be made and multiple strategies that one might need to justify or debate. Because that is what math is.

For teachers, this can also be empowering because they are interested in increasing their students’ understanding, and we’ve given them a set of tools that can help them do that and potentially make the class more interesting as well. It’s important to note, too, that this approach is not something that we invented. In this case, what we’re asking teachers to do is something that they do a little bit of already. Every high school math teacher, for certain topics, is teaching students multiple strategies. It’s built into the curriculum. All that we’re saying is, first, you should do it more because it’s a good thing, and second, when you do it, this is a certain way that we found to be especially effective, both in terms of the visual materials and the pedagogy. It’s not a big stretch for most teachers. Conversations around ways to teach math for the past 30 or 40 years, and perhaps longer, have been emphasizing the use of multiple strategies.

GAZETTE: What are the potential challenges for math teachers to put this in practice?

STAR: If we want teachers to introduce students to multiple ways to solve problems, we must recognize that that is a lot of information for students and teachers. There is a concern that there could be information overload, and that’s very legitimate. Also, a well-intentioned teacher might take our strategy too far. A teacher might say something like, “Well, if comparing two strategies is good, then why don’t I compare three or four or five?” Not that that’s impossible to do well. But the visual materials you would have to design to help students manage that information overload are quite challenging. We don’t recommend that.

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10 Best Strategies for Solving Math Word Problems

1. Understand the Problem by Paraphrasing

2. identify key information and variables, 3. translate words into mathematical symbols, 4. break down the problem into manageable parts, 5. draw diagrams or visual representations, 6. use estimation to predict answers, 7. apply logical reasoning for unknown variables, 8. leverage similar problems as templates, 9. check answers in the context of the problem, 10. reflect and learn from mistakes.

Have you ever observed the look of confusion on a student’s face when they encounter a math word problem ? It’s a common sight in classrooms worldwide, underscoring the need for effective strategies for solving math word problems . The main hurdle in solving math word problems is not just the math itself but understanding how to translate the words into mathematical equations that can be solved.

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Generic advice like “read the problem carefully” or “practice more” often falls short in addressing students’ specific difficulties with word problems. Students need targeted math word problem strategies that address the root of their struggles head-on. 

A Guide on Steps to Solving Word Problems: 10 Strategies 

One of the first steps in tackling a math word problem is to make sure your students understand what the problem is asking. Encourage them to paraphrase the problem in their own words. This means they rewrite the problem using simpler language or break it down into more digestible parts. Paraphrasing helps students grasp the concept and focus on the problem’s core elements without getting lost in the complex wording.

Original Problem: “If a farmer has 15 apples and gives away 8, how many does he have left?”

Paraphrased: “A farmer had some apples. He gave some away. Now, how many apples does he have?”

This paraphrasing helps students identify the main action (giving away apples) and what they need to find out (how many apples are left).

Play these subtraction word problem games in the classroom for free:

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Students often get overwhelmed by the details in word problems. Teach them to identify key information and variables essential for solving the problem. This includes numbers , operations ( addition , subtraction , multiplication , division ), and what the question is asking them to find. Highlighting or underlining can be very effective here. This visual differentiation can help students focus on what’s important, ignoring irrelevant details.

• Encourage students to underline numbers and circle keywords that indicate operations (like ‘total’ for addition and ‘left’ for subtraction).
• Teach them to write down what they’re solving for, such as “Find: Total apples left.”

Problem: “A classroom has 24 students. If 6 more students joined the class, how many students are there in total?”

Key Information:

• Original number of students (24)
• Students joined (6)
• Looking for the total number of students

Here are some fun addition word problems that your students can play for free:

The transition from the language of word problems to the language of mathematics is a critical skill. Teach your students to convert words into mathematical symbols and equations. This step is about recognizing keywords and phrases corresponding to mathematical operations and expressions .

Common Translations:

• “Total,” “sum,” “combined” → Addition (+)
• “Difference,” “less than,” “remain” → Subtraction (−)
• “Times,” “product of” → Multiplication (×)
• “Divided by,” “quotient of” → Division (÷)
• “Equals” → Equals sign (=)

Problem: “If one book costs $5, how much would 4 books cost?”

Translation: The word “costs” indicates a multiplication operation because we find the total cost of multiple items. Therefore, the equation is 4 × 5 = $20

Complex math word problems can often overwhelm students. Incorporating math strategies for problem solving, such as teaching them to break down the problem into smaller, more manageable parts, is a powerful approach to overcome this challenge. This means looking at the problem step by step rather than simultaneously trying to solve it. Breaking it down helps students focus on one aspect of the problem at a time, making finding the solution more straightforward.

Problem: “John has twice as many apples as Sarah. If Sarah has 5 apples, how many apples do they have together?”

Steps to Break Down the Problem:

Find out how many apples John has: Since John has twice as many apples as Sarah, and Sarah has 5, John has 5 × 2 = 10

Calculate the total number of apples: Add Sarah’s apples to John’s to find the total,  5 + 10 = 15

By splitting the problem into two parts, students can solve it without getting confused by all the details at once.

Explore these fun multiplication word problem games:

Diagrams and visual representations can be incredibly helpful for students, especially when dealing with spatial or quantity relationships in word problems. Encourage students to draw simple sketches or diagrams to represent the problem visually. This can include drawing bars for comparison, shapes for geometry problems, or even a simple distribution to better understand division or multiplication problems .

Problem: “A garden is 3 times as long as it is wide. If the width is 4 meters, how long is the garden?”

Visual Representation: Draw a rectangle and label the width as 4 meters. Then, sketch the length to represent it as three times the width visually, helping students see that the length is 4 × 3 = 12

Estimation is a valuable skill in solving math word problems, as it allows students to predict the answer’s ballpark figure before solving it precisely. Teaching students to use estimation can help them check their answers for reasonableness and avoid common mistakes.

Problem: “If a book costs $4.95 and you buy 3 books, approximately how much will you spend?”

Estimation Strategy: Round $4.95 to the nearest dollar ($5) and multiply by the number of books (3), so 5 × 3 = 15. Hence, the estimated total cost is about $15.

Estimation helps students understand whether their final answer is plausible, providing a quick way to check their work against a rough calculation.

Check out these fun estimation and prediction word problem worksheets that can be of great help:

When students encounter problems with unknown variables, it’s crucial to introduce them to logical reasoning. This strategy involves using the information in the problem to deduce the value of unknown variables logically. One of the most effective strategies for solving math word problems is working backward from the desired outcome. This means starting with the result and thinking about the steps leading to that result, which can be particularly useful in algebraic problems.

Problem: “A number added to three times itself equals 32. What is the number?”

Working Backward:

Let the unknown number be x.

The equation based on the problem is  x + 3x = 32

Solve for x by simplifying the equation to 4x=32, then dividing by 4 to find x=8.

By working backward, students can more easily connect the dots between the unknown variable and the information provided.

Practicing problems of similar structure can help students recognize patterns and apply known strategies to new situations. Encourage them to leverage similar problems as templates, analyzing how a solved problem’s strategy can apply to a new one. Creating a personal “problem bank”—a collection of solved problems—can be a valuable reference tool, helping students see the commonalities between different problems and reinforcing the strategies that work.

Suppose students have solved a problem about dividing a set of items among a group of people. In that case, they can use that strategy when encountering a similar problem, even if it’s about dividing money or sharing work equally.

It’s essential for students to learn the habit of checking their answers within the context of the problem to ensure their solutions make sense. This step involves going back to the original problem statement after solving it to verify that the answer fits logically with the given information. Providing a checklist for this process can help students systematically review their answers.

Checklist for Reviewing Answers:

• Re-read the problem: Ensure the question was understood correctly.
• Compare with the original problem: Does the answer make sense given the scenario?
• Use estimation: Does the precise answer align with an earlier estimation?
• Substitute back: If applicable, plug the answer into the problem to see if it works.

Problem: “If you divide 24 apples among 4 children, how many apples does each child get?”

After solving, students should check that they understood the problem (dividing apples equally).

Their answer (6 apples per child) fits logically with the number of apples and children.

Their estimation aligns with the actual calculation.

Substituting back 4×6=24 confirms the answer is correct.

Teaching students to apply logical reasoning, leverage solved problems as templates, and check their answers in context equips them with a robust toolkit for tackling math word problems efficiently and effectively.

One of the most effective ways for students to improve their problem-solving skills is by reflecting on their errors, especially with math word problems. Using word problem worksheets is one of the most effective strategies for solving word problems, and practicing word problems as it fosters a more thoughtful and reflective approach to problem-solving

These worksheets can provide a variety of problems that challenge students in different ways, allowing them to encounter and work through common pitfalls in a controlled setting. After completing a worksheet, students can review their answers, identify any mistakes, and then reflect on them in their mistake journal. This practice reinforces mathematical concepts and improves their math problem solving strategies over time.

3 Additional Tips for Enhancing Word Problem-Solving Skills

Before we dive into the importance of reflecting on mistakes, here are a few impactful tips to enhance students’ word problem-solving skills further:

1. Utilize Online Word Problem Games

Incorporate online games that focus on math word problems into your teaching. These interactive platforms make learning fun and engaging, allowing students to practice in a dynamic environment. Games can offer instant feedback and adaptive challenges, catering to individual learning speeds and styles.

Here are some word problem games that you can use for free:

2. Practice Regularly with Diverse Problems

Consistent practice with a wide range of word problems helps students become familiar with different questions and mathematical concepts. This exposure is crucial for building confidence and proficiency.

Start Practicing Word Problems with these Printable Word Problem Worksheets:

3. Encourage Group Work

Solving word problems in groups allows students to share strategies and learn from each other. A collaborative approach is one of the best strategies for solving math word problems that can unveil multiple methods for tackling the same problem, enriching students’ problem-solving toolkit.

Conclusion 

Mastering math word problems is a journey of small steps. Encourage your students to practice regularly, stay curious, and learn from their mistakes. These strategies for solving math word problems are stepping stones to turning challenges into achievements. Keep it simple, and watch your students grow their confidence and skills, one problem at a time.

Frequently Asked Questions (FAQs)

How can i help my students stay motivated when solving math word problems.

Encourage small victories and use engaging tools like online games to make practice fun and rewarding.

What's the best way to teach beginners word problems?

Begin with simple problems that integrate everyday scenarios to make the connection between math and real-life clear and relatable.

How often should students practice math word problems?

Regular, daily practice with various problems helps build confidence and problem-solving skills over time.

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