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20 Effective Math Strategies To Approach Problem-Solving 

Katie Keeton

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

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

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

What are problem-solving strategies?

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

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

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

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

The ultimate guide to problem solving techniques

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

20 Math Strategies For Problem-Solving

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

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

Strategies to understand the problem

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

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

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

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

1. Read the problem aloud

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

2. Highlight keywords 

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

3. Summarize the information

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

4. Determine the unknown

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

5. Make a plan

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

Strategies for solving the problem 

1. draw a model or diagram.

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

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

2. Act it out

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

3. Work backwards

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

For example,

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

4. Write a number sentence

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

5. Use a formula

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

Strategies for checking the solution 

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

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

Here are five strategies to help students check their solutions. 

1. Use the Inverse Operation

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

2. Estimate to check for reasonableness

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

3. Plug-In Method

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

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

4. Peer Review

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

5. Use a Calculator

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

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

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

Polya’s 4 steps include:

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

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

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

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

How Third Space Learning improves problem-solving 

Resources .

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

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

One-on-one tutoring 

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

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

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

Problem-solving

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

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

READ MORE : 8 Common Core math examples

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

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

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

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

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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 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!

Need more help developing problem solving skills?

Read up on  how to set a problem solving and reasoning activity  or explore Mathseeds and Mathletics, our award winning online math programs. They’ve got over 900 teacher tested problem solving activities between them!

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10 Strategies for Problem Solving in Math

Created on May 19, 2022

Updated on January 6, 2024

When faced with problem-solving, children often get stuck. Word puzzles and math questions with an unknown variable, like x, usually confuse them. Therefore, this article discusses math strategies and how your students may use them since instructors often have to lead students through this problem-solving maze.

What Are Problem Solving Strategies in Math?

If you want to fix a problem, you need a solid plan. Math strategies for problem solving are ways of tackling math in a way that guarantees better outcomes. These strategies simplify math for kids so that less time is spent figuring out the problem. Both those new to mathematics and those more knowledgeable about the subject may benefit from these methods.

There are several methods to apply problem-solving procedures in math, and each strategy is different. While none of these methods failsafe, they may help your student become a better problem solver, particularly when paired with practice and examples. The more math problems kids tackle, the more math problem solving skills they acquire, and practice is the key.

Strategies for Problem-solving in Math

Even if a student is not a math wiz, a suitable solution to mathematical problems in math may help them discover answers. There is no one best method for helping students solve arithmetic problems, but the following ten approaches have shown to be very effective.

Understand the Problem

Understanding the nature of math problems is a prerequisite to solving them. They need to specify what kind of issue it is ( fraction problem , word problem, quadratic equation, etc.). Searching for keywords in the math problem, revisiting similar questions, or consulting the internet are all great ways to strengthen their grasp of the material. This step keeps the pupil on track.

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Guess and check.

One of the time-intensive strategies for resolving mathematical problems is the guess and check method. In this approach, students keep guessing until they get the answer right.

After assuming how to solve a math issue, students should reintroduce that assumption to check for correctness. While the approach may appear cumbersome, it is typically successful in revealing patterns in a child’s thought process.

Work It Out

Encourage pupils to record their thinking process as they go through a math problem. Since this technique requires an initial comprehension of the topic, it serves as a self-monitoring method for mathematics students. If they immediately start solving the problem, they risk making mistakes.

Students may keep track of their ideas and fix their math problems as they go along using this method. A youngster may still need you to explain their methods of solving the arithmetic questions on the extra page. This confirmation stage etches the steps they took to solve the problem in their minds.

Work Backwards

In mathematics, a fresh perspective is sometimes the key to a successful solution. Young people need to know that the ability to recreate math problems is valuable in many professional fields, including project management and engineering.

Students may better prepare for difficulties in real-world circumstances by using the “Work Backwards” technique. The end product may be used as a start-off point to identify the underlying issue.

In most cases, a visual representation of a math problem may help youngsters understand it better. Some of the most helpful math tactics for kids include having them play out the issue and picture how to solve it.

One way to visualize a workout is to use a blank piece of paper to draw a picture or make tally marks. Students might also use a marker and a whiteboard to draw as they demonstrate the technique before writing it down.

Find a Pattern

Kids who use pattern recognition techniques can better grasp math concepts and retain formulae. The most remarkable technique for problem solving in mathematics is to help students see patterns in math problems by instructing them how to extract and list relevant details. This method may be used by students when learning shapes and other topics that need repetition.

Students may use this strategy to spot patterns and fill in the blanks. Over time, this strategy will help kids answer math problems quickly.

When faced with a math word problem, it might be helpful to ask, “What are some possible solutions to this issue?” It encourages you to give the problem more thought, develop creative solutions, and prevent you from being stuck in a rut. So, tell the pupils to think about the math problems and not just go with the first solution that comes to mind.

Draw a Picture or Diagram

Drawing a picture of a math problem can help kids understand how to solve it, just like picturing it can help them see it. Shapes or numbers could be used to show the forms to keep things easy. Kids might learn how to use dots or letters to show the parts of a pattern or graph if you teach them.

Charts and graphs can be useful even when math isn’t involved. Kids can draw pictures of the ideas they read about to help them remember them after they’ve learned them. The plan for how to solve the mathematical problem will help kids understand what the problem is and how to solve it.

Trial and Error Method

The trial and error method may be one of the most common problem solving strategies for kids to figure out how to solve problems. But how well this strategy is used will determine how well it works. Students have a hard time figuring out math questions if they don’t have clear formulas or instructions.

They have a better chance of getting the correct answer, though, if they first make a list of possible answers based on rules they already know and then try each one. Don’t be too quick to tell kids they shouldn’t learn by making mistakes.

Review Answers with Peers

It’s fun to work on your math skills with friends by reviewing the answers to math questions together. If different students have different ideas about how to solve the same problem, get them to share their thoughts with the class.

During class time, kids’ ways of working might be compared. Then, students can make their points stronger by fixing these problems.

Check out the Printable Math Worksheets for Your Kids!

There are different ways to solve problems that can affect how fast and well students do on math tests. That’s why they need to learn the best ways to do things. If students follow the steps in this piece, they will have better experiences with solving math questions.

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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Khan Academy Blog

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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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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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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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:

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.

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

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5 Essential Problem Solving Techniques

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In the first post in this series, I talked about the difference between solving problems and problem solving. This week, I will continue my series on problem solving and share five essential problem solving techniques for your problem solving routines.

A strong problem-solving routine is essential for helping students develop their problem-solving strategy toolboxes. Over the years, I have used a variety of routines that have helped my students develop problem-solving strategies and critical thinking skills. (Read more about my favorite routine here !) Through a lot of trial and error, I found several routines that worked well for my students. (I will share more about them next week!) Today, I want to share some trade secrets with you to help you get the most from your problem-solving routines with five essential problem-solving techniques.

Five Essential Problem Solving Techniques

1. share student thinking and strategies..

This is essential! I can’t tell you how many times I have seen teachers give a great problem solving or critical thinking task and then never allow students to share their responses. Sometimes, our students are the best teachers and they can get a message across when we struggle to do so. Also, providing an opportunity for students to talk to other students about their thinking increases math vocabulary and builds communication skills.

After students have had an opportunity to share their thinking with a group member or partner, I encourage you to discuss the task as a class. This gives the teacher an opportunity to reiterate correct thinking, modify incorrect thinking, ask questions, build math vocabulary, and increase students’ communication skills.

Read more about getting started with math talk in the classroom here .

2. Solve non-routine problems.

In an earlier blog post, I emphasized the importance of using non-routine problems with students. Not only are students typically more engaged, but students have the opportunity to use strategies beyond writing an equation/number sentence or drawing a picture. If you’re interested in some fun, non-routine tasks, please check out my Solve It! Friday page.

One of the things many people say they love about math is the fact that there is a right and wrong answer. While there certainly are wrong answers, sometimes, there can be more than one right answer. These types of tasks really stretch some kids’ thinking. They also provide a natural venue for discussion. Students can debate the answers only to discover that more than one works!

3. Discuss efficiency.

During problem-solving experiences, students will often use beautiful and complicated solution strategies to solve problems. While we want to encourage outside-of-the-box thinking, we also want students to attend to efficiency. One way to do this is to have several students share their solutions. They can then discuss what strategies are best for specific types of problems. When discussing difficulty becomes a regular part of your routine, students will begin to utilize their problem-solving strategies in a way that not only gets them to the correct answer but also using an efficient method.

4. Make connections.

Recently, I wrote about making connections as part of my Summer PD series. Read it here ! When students make connects, it deepens their understanding of other content and skills. One way to do this is to connect the problem-solving task to grade-level content and skills. Another way is to have students represent problems in a variety of ways, i.e. pictures, numbers, words, or equations. Each representation is crafted in a specific way, so being able to translate words into an equation or numbers into a picture is a big skill that has many benefits.

5. Use “high ceiling, low floor tasks.”

The term “high ceiling, low floor” refers to a task having multiple entry points to allow all students a way to access the task; however, it also includes ways to extend the tasks for those students who are ready for more of a challenge. These types of tasks increase participation because students can participate at a level that is comfortable for them. Students are also able to showcase what they can do instead of what they are unable to do. Even better, these tasks provide instant opportunities for differentiation because all students can participate in a way that allows them to be most successful.

Using a regular problem-solving routine can help students develop the tools necessary to be powerful thinkers of mathematics; however, in order to get the most from the routines, certain problem-solving techniques must be included. While you may not want to add all of the above techniques to your routine, I encourage you to commit to adding one or two of them this year. I highly recommended starting with “sharing student thinking and strategies.” It’s probably the most important technique of all of the problem-solving routines. It will get you the most bang for your buck!

Sound Off! How do get the most from your problem-solving routine? Which problem-solving techniques do you think are most important?

Shametria Routt Banks

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3 Responses

Hi, can you provide an example of a high ceiling, low floor task? Thank you!

Hi Jen! Great question! The high ceiling, low floor tasks give all students a chance to engage in the task but have places to go to extend the learning for students. One problem that comes to mind is a task where students are asked to find combinations of numbers to achieve a goal, like the following problem: Farmer Brown’s niece Angie is in charge of her uncle’s farm while he is on vacation. He gave her strict instructions to make sure none of the animals ran away. When Angie counted the pigs and chickens, she counted 32 legs. How many pigs and chickens did she count? All students should be able to determine a combination of pigs and chickens; however, what if I added a new condition to say: Angie counted a total of 12 animals. This changes the level of rigor because students are now looking for a specific combination. Some students will struggle with this but others may be ready to tackle it; so, using tasks that have a high-ceiling allow for this flexibility. Check out more high ceiling, low floor tasks here: https://www.youcubed.org/task-grades/low-floor-high-ceiling/ .

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Problem-Solving Strategies

January 26, 2021 Brad Hoffman Leave a Comment

Certainly, many students find that it is possible to solve a given word problem with minimal consideration of how to approach it. People have varying degrees of “math sense.” Some find most math problems mysterious. Some, however, can very easily see what to do to find solutions; it almost seems obvious to them. But even for students with strong “math sense,” there come those situations when they don’t intuitively know what to do. For all learners, the recognition of specific problem-solving strategies to solve math problems is useful. Thinking about our own thinking (aka metacognition) is important in developing flexibility so that we can see more than one way to solve a particular problem. Math journaling supports this thinking and development.

Below you will find a list of some very useful problem-solving strategies . One thing that is particularly beneficial about this set of strategies is that they are, in fact, universal. In other words, they will work regardless of the math program a student might be using. Whether it’s Singapore Math or Everyday Math or something else entirely , these problem-solving strategies can provide a clear path toward solutions. Interestingly, they can even extend to problem-solving outside the area of math! Becoming familiar with them and comfortable using them can be a big help to students as they wend their way through problems, be they less or more complex.

10 Problem-Solving Strategies

• Make a model/Act out
• Draw a diagram or picture
• Look for a pattern
• Make an organized list
• Make a table
• Guess & Check
• Make it simpler
• Work backwards
• Use logical reasoning

Here are some examples of problems and how to use these strategies.

“How many complete turns does the hour hand on a clock make in one day?”

From the list of problem-solving strategies above, “make a model or act it out” is an excellent choice for this problem. A student could use a model or a real analog clock and turn the hands and count. Distinguishing between the minute and the hour hand and recognizing that the clock only shows 12 of the 24 hours in a day lets the student see that the hour hand makes two complete turns. A physical clock that a student can actually turn provides an important concrete experience that may prove helpful for finding the solution.

“Using each of the digits 0, 1, 2, 3, 4 only once, make a two-digit number times a three-digit number multiplication problem with the greatest product.”

Students can “ draw a diagram or picture” of an “empty” multiplication problem with a box for each digit. Consider which two digits give the largest product and put them in the highest place value spots. Then, if it’s not immediately evident to the student, use one of the other problem-solving strategies — “ guess and check” — to place the remaining digits in the remaining spots. Check by multiplying the results to identify which is actually the largest (e.g. Is it 430 x 21 or 320 x 41?)

“How many even numbers are there between 201 and 351?”

In this instance, “ look for a pattern” would be especially helpful from the list of problem-solving strategies. Either write all numbers from 201 through 351 and notice the pattern that there are 5 in every set of 10 numbers (e.g. 201-210), and then count how many sets of 10 numbers there are and multiply that by 5, or simply write one set of 10 numbers and identify the 5 in 10 pattern without writing out all of them. Either way is valid.

“You have two noses and three hats. How many different nose-hat disguises can you wear?”

For this problem, “ make an organized list ” from the problem-solving strategies listed above works well. The list will start with Hat A and match with each nose (2), then Hat B with each nose (2), then Hat C with each nose (2). This gives a total of 6 disguises.

“How many numbers between 10 and 30 give a remainder of 2 when divided by 3?” You could “ make a table” to find the solution.

As the Table continues, a pattern becomes evident (“ look for a pattern ” — overlapping strategy!) in which every third number gives a remainder of 2. Count them for a solution.

“If 25 Glinks equal a Glonk, and 15 Glonks equal a Glooie, how many Glinks equal 2 Glooies?”

Please, “ make it simpler”! That strategy is an especially good choice from the list of problem solving-strategies. Let’s look at a simpler, but similar, problem. It’s simpler because the numbers are smaller, and you could even draw a picture to prove it’s correct.

If 3 Glinks equal a Glonk. And 2 Glonks equal a Glooie. How many Glinks equal a Glooie? Multiply 3×2, which equals 6.

So, if 6 Glinks equal a Glooie, then how many Glinks equal 2 Glooies? Multiply 6×2, which equals 12. So, 12 Glinks equal 2 Glooies.

Now with the larger numbers:

If 25 Glinks equal a Glonk. And 15 Glonks equal a Glooie. How many Glinks equal a Glooie? Multiply 15×25, which equals 750. So, 750 Glinks equal a Glooie.

Then, how many Glinks equal 2 Glooies? Multiply 750×2, which equals 1500. So, there are 1500 Glinks in 2 Glooies.

It’s the same process, with bigger numbers! Much simpler!

“If I add 10 to my age and double it, I am 90. How old am I?”

From the list of problem-solving strategies, this problem begs for the student to “ work backwards”. Simply un-double the 90 and subtract ten. 90 divided by 2 = 45 and 45-10=35. Voilà! The answer is 35 years old! Then reverse again to confirm that the answer is correct.

“Arrange these digits and symbols to make a true number sentence (equation.) 3,1,4,9,+,/,= (Note: the forward slash  [/] signifies “divided by”.)

“ Use logical reasoning ” to realize that any order is possible, but a larger number needs to be divided by a smaller number with no remainder (9/3=3) Then 3+1=4, so the sentence 9/3+1=4 is the solution.

For the problems that seem absolutely impossible to solve, your best option is to “ brainstorm” , and that’s on the above list of problem-solving strategies! Try various ideas; work with a partner; explore to see what might work; try everything you can think of! It’s amazing how good ideas will sometimes just pop into one’s head!

As a student works with these problem-solving strategies, it becomes clear that they often overlap (as in the “ draw a picture” / “guess and check” example above, problem #2). Or a student becomes especially attached to a few particular strategies that often work well. Some problems seem to be especially suitable for a particular strategy, while others can be approached from several directions. Having the flexibility to move from one strategy to another helps avoid the serious “I’m STUCK!” situation. Also, using more than one strategy on the same problem allows students to check solutions more efficiently before moving on. Again, however, THINKING about how we are THINKING is very beneficial in developing skills in this area. We call this metacognition .

Solving word problems can be fun, like being a detective who has unusual insight. There are solutions! Enjoy finding them! And make effective use of problem-solving strategies!

By Jean Snyder and Brad Hoffman , Elementary Math Specialists

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Best 13 Strategies for solving math problems

Mathematics can often be a challenging subject for many students. The key to conquering math lies in understanding the concepts and developing effective problem-solving strategies. Let us explore a range of strategies that can help students solve math problems confidently and easily. By implementing these techniques, students can enhance their mathematical skills and achieve success in this critical subject.

Table of Contents

1. Understand the Problem

2. simplify and visualize the problem.

Complex math problems can often be overwhelming. It’s helpful to simplify and visualize the problem to make them more manageable. Break down the problem into smaller, more understandable parts. Draw diagrams or use visual aids to represent the information, relationships, or geometric figures. This approach can provide valuable insights and facilitate a better understanding of the problem.

3. Identify Relevant Concepts and Formulas

4. work backwards.

Working backward is a powerful strategy that can be employed to solve certain types of math problems. Start with the given answer and consider the steps leading to that solution. By reversing the problem, you can uncover hidden relationships, identify missing information, or even discover alternative solutions.

5. Utilize Logical Reasoning

6. practice problem-solving regularly.

Practice makes perfect, and this holds for math problem-solving as well. Regular practice strengthens your problem-solving skills and builds confidence in tackling different types of math problems. Solve various math problems from textbooks, workbooks, or online resources. The more you practice, the more familiar you become with different problem-solving techniques, enhancing your ability to effectively approach and solve math problems.

7. Seek Help and Collaborate

8. break down complex problems, 9. review and learn from mistakes.

Mistakes are an integral part of the learning process. When you encounter challenges or make errors while solving math problems, take the time to review and learn from them. Analyze your mistakes, identify where you went wrong, and understand the correct approach. This

10. Practice Mental Math

11. use problem-solving strategies, 12. develop time management skills.

Time management is essential when solving math problems, especially during exams or timed assessments. Practice working on math problems under timed conditions to improve speed and accuracy. Learn to prioritize and allocate time to different parts of a problem, ensuring you complete all necessary steps within the given timeframe.

13. Review and Reinforce Fundamentals

Conclusion – strategies for solving math problems.

Regular practice, collaboration, breaking down complex problems, and learning from mistakes are key to improving mathematical problem-solving abilities. With dedication and perseverance, students can unlock the secrets to solving math problems easily and confidently.

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Developing Logical Mathematical Intelligence

How to unlock your problem-solving potential

• Characteristics and Examples
• Develop Logical Mathematical Intelligence
• Tips and Strategies

Logical mathematical intelligence is one of eight intelligences that Howard Gardner, a professor of cognition and education at Harvard University, proposed in his theory of multiple intelligences , which he outlined in his book Frames of Mind . Gardner's theory posited these types of intelligence:

• Visual-spatial
• Linguistic-verbal
• Logical-mathematical
• Body-kinesthetic
• Interpersonal
• Intrapersonal
• Naturalistic

According to Rebecca Mannis, PhD and founder and learning specialist at Ivy Prep, for a long time American psychologists viewed intelligence as a single factor, typified by Lewis Terman in the early to mid-1900s, who referred to intelligence as “g.” Similarly,  Kimberly Nix Berens, PhD , founder of Fit Learning and author of "Blind Spots: Why Students Fail and the Science That Can Save Them," explains that intelligence is often viewed as an innate ability, but there is a lack of scientific evidence to support this.

Instead, Gardner theorized that there are a number of kinds of intelligence that we each may have, that we may vary in the degree we have of each one, and that we may have more or less of each of these throughout our lifetimes. Logical mathematical intelligence is one of the intelligences he described, and it represents the ability to use numbers effectively, to reason well, and to recognize and solve problems using logical patterns.

In this article, we’ll review the characteristics of logical mathematical intelligence, discover how to develop it, explore strategies to enhance it, and look at the benefits of this kind of intelligence.

Anchiy / E+ / Getty

Characteristics and Examples of Logical Mathematical Intelligence

People who have logical mathematical intelligence solve problems using logic, can quickly calculate math problems , and like when things are categorized in a rational way. They’re also good at understanding patterns, the relationships between things, and understanding complex ideas.

Thus, logical-mathematical intelligence encompasses the following:

• Being good with numbers
• Understanding logical concepts
• Having good reasoning skills
• Enjoying experiments
• Enjoying solving puzzles and mysteries
• Being good at manipulating numbers and operations
• Being good at understanding and applying scientific principles

According to Berens, examples of logical mathematical intelligence include “fluently solving multi-step equations, [solving] language-based math problems , interpreting and analyzing scientific findings, and designing experiments.”

Mannis explains that mastery of basic facts and concepts that leads to the ability to adapt those facts and concepts into more complex tasks and to new settings is a hallmark of logical mathematical intelligence.

For instance, Mannis gives the example of a third grader with strong logical mathematical intelligence who not only knows how to distinguish between perimeter and area but can also use those formulas for other things. The child can apply “that information to a math challenge to design a playground… where they are tasked with determining what the area of a complex shape would be if there were sections of semicircles or squares within that figure ‘cut out.’ They would also be able to then identify the cost of paving the entire play area given a particular per foot fee.”

For those of us who are more inclined toward verbal linguistics or another type of intelligence, this may seem beyond our reach, but it is possible to build your strength in the more mathematical areas of intelligence.

How to Develop Logical Mathematical Intelligence

Decades of research indicate that higher-level skills, such as those seen in logical mathematical intelligence, require the mastery of foundational skills, says Berens. As a result, “Young learners require explicit instruction and repeated practice to fluency in core math skills such as numeracy, basic computation, fractions, decimals, percents, and ratios," she says. "They also require fluent reading skills and fluent comprehension, problem-solving, and critical thinking skills.”

Mannis explains that some children are better at memorizing facts and drawing connections between concepts than others, but there are ways to develop these skills.

These methods include:

• Teacher education: According to Mannis, teachers must have a “strong understanding of how children develop these skills and methods [by] first teaching this content systematically and then gradually ‘scaffolding’ or adding complexity and integration."
• Focusing on basic math: As Berens indicates students must attain fluency in the basics before they can begin applying these core skills for mastery of highly complex skills, including algebra, geometry, and calculus.
• Engagement beyond the classroom: Encouraging children to see how mathematical concepts can be part of their real lives can help develop logical mathematical intelligence skills.

Mannis provides an example of the last point: “A child usually reads a book in four days and would like to borrow the series to read during the three weeks between the end of the school year and sleep away camp. How can they estimate how many books they will get through? How does being free of homework shift their estimate? That is an example of living math that offers a chance to systematize, use concepts such as ratios and estimating, and also encourage them to engage through creating a system to make their estimate.”

Strategies for Enhancing Logical Mathematical Intelligence

To enhance logical mathematical intelligence you first have to learn the basics, so explicit instruction and repeated practice in math skills is essential, says Berens.

Moreover, per Mannis, to enhance skills in logical mathematical intelligence further:

• Provide opportunities to create systems and patterns, and solve logic problems
• Encourage ‘metacognitive awareness,’ or being aware of how you think, and talk through this approach
• Balance learning facts and math operations with real-world problem-solving

For example, Mannis speaks of a middle school class she consulted with that timed its geometry unit so it was right before the school carnival. “After completing the basics of the course, the students were put in charge of designing, creating, and manning some of the carnival stations using” what they learned. This allowed them to use the skills they got from the course and enhance their logical mathematical intelligence.

Benefits of Logical Mathematical Intelligence

People with logical mathematical intelligence are good at rational thinking, analyzing problems logically, and thinking about issues scientifically. “Not only does mastery of high-level math skills produce long-term academic success," says Berens, "but it also gives learners access to careers in science, technology, and engineering.”

We have a lot of complex problems in these fields to solve, such as climate change, and we need people with logical mathematical intelligence to solve them, she says.

While some people may have more innate ability with logical mathematical intelligence, anyone can enhance their abilities. Developing math skills, engaging in strategy games and logic problems, explaining your thinking, and using your skills in the real world can help develop your logical mathematical intelligence and have you on your way to unlocking your problem-solving potential.

Gardner H.  Frames of Mind: The Theory of Multiple Intelligences . 10th anniversary ed. BasicBooks; 1993.

Arani HK, Mobarakeh SD. Metacognitive strategies and logical/mathematical intelligence in EFL context: Investigating possible relationships. TPLS . 2012;2(2):304-313. doi:10.4304/tpls.2.2.304-313

Šafranj J. Logical/mathematical intelligence in teaching English as a second language .  Procedia - Social and Behavioral Sciences . 2016;232:75-82. doi: 10.1016/j.sbspro.2016.10.019

By Cynthia Vinney, PhD Cynthia Vinney, PhD is an expert in media psychology and a published scholar whose work has been published in peer-reviewed psychology journals.

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5 Research-Based Math Interventions for Your MTSS/RTI

Leah Allen-Manning

While overhauls to reading curricula headline the news across the country, its academic counterpart—math—has been somewhat forgotten, despite equally shocking data from the National Assessment of Educational Progress (NAEP) Report . In fact, the most recent data revealed the sharpest decline in math scores since the assessment's inception in 1990.

To counter declining math scores, educators need effective strategies to incorporate into math instruction. When combined with a solid foundation of core instruction , research-based mathematics interventions enable districts to strengthen their tiered systems of support to help all students learn.

Download our toolkit of 12 research-based math interventions.

How Math Interventions Fit Into an MTSS

A core component of an MTSS (multi-tiered system of supports) or RTI (Response to Intervention) framework is interventions: research-based strategies to support students across three levels, called tiers. Math intervention activities teach mathematics topics at a student’s individual level to meet their needs.

• Tier 1 math interventions are included in universal, core instruction provided to all students. These strategies are effective best practices to establish a solid foundation for math success.
• Tier 2 math interventions offer targeted support in small-group instruction for students who need additional resources on top of core instruction.
• Tier 3 math interventions involve intensive, one-on-one instruction and highly individualized support for students who have received both Tier 1 instruction and Tier 2 interventions.

Below are 5 research-based and evidence-based math interventions to share with school teams and add to your district’s intervention menu . These interventions have been selected from Playbook , Panorama’s professional learning library for K-12 educators.

RTI/MTSS Math Interventions:

Math with pals, draw a picture, explore before explain, concrete representational abstract (cra) approach.

Math Songs use rhyme, rhythm, patterns, and repetition to help students remember math concepts, skills, or basic facts. The use of songs helps students retain information in long-term memory.

• Focus Area: Fluency
• Developmental Stage: Lower Elementary, Upper Elementary, Middle School
• MTSS/RTI Tiers: Tier 1

Preparation:

• Consider which topics would benefit from the inclusion of a math song.
• Find math songs about the relevant topic that your students would enjoy. Websites like YouTube, Teachers Pay Teachers, and  GoNoodle offer a variety of math songs. Teachers can search for songs on specific topics such as multiplication tables, fractions, or geometry.
• Print out lyrics or display lyrics for students to read along as they sing (as appropriate for the grade level).

Instructions:

• Play or sing math songs at any point during the learning of a math topic, whether as an introduction or throughout the course of study.
• Consider asking students to create their own math song or write their own lyrics to a familiar tune. This will provide an engaging way for students to interact with the topic and take authentic ownership of their understanding.

Math with PALS (Peer-Assisted Learning Strategy) is an intervention that supports student growth through collaboration, discussion, and practice. Students work in pairs of tutor and tutee to support one another’ s understanding of a previously taught math skill.

• Focus Area: Problem-Solving
• MTSS/RTI Tiers: Tier 1, Tier 2

Preparation :

• Choose stu dents who know how to use the skill or strategy in a problem set.
• Pair students ahead of time in heterogeneous pairings. One partner should be stronger in the math skill (tutor), and the other can be a student who needs some support to improve (tutee). 
• Create a practice worksheet for students to complete while working in pairs. For younger students, consider adding these coaching questions to the worksheet.

Instructions :

• Pair students and provide worksheets or work materials. 
• Allow 20–40 minutes for pairs to work based on grade level and age group.
• The “tutor” partner will support the other partner’s work through questions and modeling. Prompts or questions can be provided for the tutor partner to use.
• If the “tutee” makes a mistake, the tutor should address it and explain their thinking in a respectful way.
• After the pairs complete half of the worksheet, the students switch roles to complete the rest of the worksheet.
• Once the partner work time is complete, students should work individually to complete additional problems, then exchange worksheets and review their partner’s work. 

This strategy asks students to make a drawing of an applied problem. Drawing a picture can support student understanding of the problem itself as well as how to solve the problem. Additionally, this strategy can pinpoint misconceptions students may have about how to set up or solve the problem.

• Focus Area: Comprehension
• Developmental Stage: Lower Elementary, Upper Elementary, Middle School, High School
• Analyze the word problems that students will be asked to solve. 
• Choose a word problem to model how to draw a picture to help solve it. 
• Choose one or two to use as guided practice. Students can then work independently using a drawing.

Explicit Instruction:

• Choose a word problem to display and ensure students have a copy.
• Model reading the problem aloud carefully. 
• Talk through the thought process for how to draw a picture to represent the problem. This may include what you notice, what values you are starting with, and what you will do. 
• Draw the picture as you explain, and ask students to do it along with you.

Guided Practice:

• Provide students with a selected word problem.
• Ask students what information is provided in the problem.
• Guide students to draw a picture to represent the problem. Facilitate student thinking with your own thinking aloud and with guiding questions.

Explore Before Explain is a learning environment where students explore real-life situations as they relate to a concept before an explanation of the concept occurs. Activities to practice a concept are not introduced until students have had time to gain meaning through exploration.

Explore Before Explain helps avoid the question “When will I ever have to use this?” Making connections between math concepts and the real world will help students be more engaged and remember what they’ve learned

• Focus Area: Comprehension   
• MTSS/RTI Tiers: Tier 1, Tier 2, Tier 3
• Think about student experiences that can relate to the math concept in consideration. Consider how students could explore an experience to discover useful information about the concept. This might include a demonstration, using the classroom environment describing a relatable scenario, or watching a video
• In elementary school, this might be a great time to bring in a guest expert—a parent or other relative of one of the students who uses this concept in their work or daily lives.
• Engage students in the context. Provide expectations of the task at hand. Do not formally introduce the math concept at this stage.
• Allow students to explore a common experience related to the concept.
• Explain how the exploration connects to the formal concept.
• Have students elaborate on the concept by applying the concept to a new scenario or using follow-up practice questions.
• Evaluate student understanding of the formal math concept.

Students begin the CRA process by using concrete models to represent math concepts. Next, students transition to the use of drawings to represent the same concepts. Finally, students translate model representations to abstract representations using numbers and symbols.

• Focus Area: Computation  
• Gather concrete manipulatives (counters, blocks, cups, algebra tiles, etc.) that can be used to model the math concept being discussed.
• Choose problems that avoid complicated calculations.
• Download the CRA worksheet .

Concrete : Students work with actual objects to solve the math problem.

• Hand out the manipulatives to each student or pair. Give the students a few minutes to explore the manipulatives.
• Demonstrate how the concrete materials connect to the problem. Allow students to see you “think through” some sample problems. 
• Using a whole-class approach, work through a few more problems together. Check for understanding. If needed, work through additional examples.
• Finally, give students a few problems to work through individually or in pairs.

Representational : Students draw or sketch representations of the objects to model problems.

• Using the same math problems from the concrete examples, transition students to pictorial models.
• Ask for suggestions from students of different ways they can represent the concept on paper instead of with objects.
• Have students practice using pictorial models individually or in pairs.

Abstract : Students write symbolic representations using numbers and symbols.

• Continue with the same math problems. Review previous methods using concrete objects and pictorial representations. Transition students from representational models to abstract displays using math notations.
• Have students practice solving problems using only symbolic notations. Provide additional support, if needed. 

Identify Students & Administer Math Interventions With Panorama

District and school users can identify students who score below benchmark on math assessments—such as iReady, FastBridge, STAR, and NWEA MAP—and create support plans in Panorama Student Success . With all student data in one platform, educators can easily take action with intervention planning workflows, collaboration tools, and evidence-based strategies .

These interventions not only enhance comprehension and problem-solving skills, but they also foster a deeper understanding of mathematical concepts across all tiers of support.

By incorporating research-based and evidence-based math interventions into an MTSS or RTI framework, school and district leaders ensure all students receive the support they need. 

Download these and other research-based strategies in our Math Interventions Toolkit.

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Setting Small Goals to Achieve High-Impact Math Gains

Prioritizing key concepts can help teachers design instructional strategies that accelerate learning.

 A well-designed Multi-Tiered System of Support (MTSS) for elementary math supports teachers in effectively meeting students’ diverse needs by prioritizing high-quality, universal instruction that is inclusive and engaging for all students. 

In an MTSS, students have access to intervention focused on high-impact math concepts that support grade-level learning. In order to accelerate learning, we need to prioritize math concepts that have the most impact across a grade level. As elementary educators, we are often unsure which concepts to prioritize. When designing instruction for intervention, teachers should start by asking the following questions: What is the current high-priority grade-level math content? What is the most critical content that students need in this grade level in order to access next year’s grade-level learning? These questions can guide the design of high-impact instruction.

designing high-quality math intervention

High-leverage concepts are the key understandings identified at each grade level that help focus math instruction by identifying the big ideas at each grade level that will support success in the next grade. We know that this growth is not necessarily linear. However, this resource allows teachers to understand the key prerequisite understandings along the way toward mastery. When designing for Tier 2 and Tier 3 layers of support in an MTSS, it is critical to prioritize the most impactful math content if the goal is to accelerate learning.

Once teachers have identified the highest-impact math concepts at their grade level, they need to consider the small goals related to that math concept that will indicate progress toward understanding. Teachers should ask themselves these questions: What evidence will I see to indicate understanding? What models do I expect students to use? What strategies do I expect students will apply? 

These learning progressions are a resource to break down these larger concepts into small goals along the way. The idea is to identify and monitor the developing stages along the continuum of critical math concepts, including models and strategies that students should explore. When designing for intervention, teachers intentionally plan lessons based on what a student already knows and understands to help move them forward in their understanding. This process highlights an asset-based approach for instruction. It may be necessary to start on the concepts from one or two grade levels prior that largely impact the current grade-level math content.

Finally, teachers must consider the learning opportunity that includes both teaching and assessing students’ understanding. They need to consider the high-cognitive-demand tasks that students can engage with during intervention that will allow them to show their understanding of these critical concepts. Beyond straight calculation, we need to know the models and strategies that students are using and whether or not those strategies are increasing in complexity. When we look for small gains in strategy use, we are able to increase our impact on student learning while feeling a sense of efficacy in the implementation of our instructional design. 

High-quality Instruction in Practice 

I recently worked with a group of special educators in Illinois to hone their skill set in designing intervention that promotes the acceleration of math learning. Here are the steps we followed.

1. Identify the high-priority grade-level math content to prioritize. The small group of students we planned for were in second grade. The high-priority grade-level math content in second grade is addition and subtraction The Common Core State Standard (CCSS) states the following: Use addition and subtraction within 100 to solve one- and two-step word problems involving situations of adding to, taking from, putting together, taking apart, and comparing, with unknowns in all positions. This group of students was still working on understanding number values and sequences to 20. The concept is critical for addition and subtraction, as students must be able to identify both the name and the total value of a given number in order to add and subtract whole numbers within 100. As a group, we lined up the kindergarten and grades one and two high-leverage progressions documents to identify where instruction should start based on this goal. We identified counting, comparing, and composing/decomposing within 10 on the kindergarten document.

2. Identify the critical strategies and small goals to achieve as an outcome of this intervention. Using the kindergarten and first-grade high-leverage concepts, we discussed our expectation for students to be able to count within 20 both forward and backward and compare quantities within 10. We identified the critical strategy of composing and decomposing quantities using models, specifically a tens frame. We also expected to see evidence of counting by ones, subitizing, organizing, and tracking. 

3. Identify the strategy for teaching and assessing student understanding. Students are much more relaxed and able to show their thinking when playing. We decided to play a simple dice game involving counters or bingo chips, blank printed tens frames, and six-sided dice. In this game, each student rolls a die and answers these questions: How many dots are there? Who has more dots? How do you know? How many chips will you need? How close are you to 10? Students then place that many chips on the tens frame. When all chips are placed, we discuss answers to these questions: How many chips do we have altogether? How do you know? How are your strategies for counting the same, how are they different? 

As we play this game, we are looking for the following:

• Evidence of subitizing
• One-to-one correspondence
• Strategies for counting and cardinality
• Sums to 10 using the tens as a thinking tool and strategies beyond 10 if the situation arises

Following this process, we were able to focus on high-impact math content at the current grade level. We were able to deconstruct the small goals along the way that help us understand student growth toward this concept. Finally, we could design a learning opportunity that was focused on a specific, small learning goal. 

With a small learning goal, we are able to monitor progress toward understanding because we are crystal-clear about what we are looking for. The goal here is for us to set small goals so we know exactly when students know and understand a concept and are ready for the next instructional step on the progression. 

Every small goal we move toward is focused on math concepts that are critical for understanding current grade-level concepts, so that we begin to see impact on grade-level learning at a much faster rate. Through high-quality Tier 1 instruction and high-impact Tier 2 and Tier 3 layers of instruction, we are more readily able to accelerate student learning of key mathematical concepts.

ChatGPT: Disruptive or Constructive?

Thursday, Jul 18, 2024 • Jeremiah Valentine : [email protected]

What is Chat GPT?

ChatGPT is a popular emerging technology using Artificial Intelligence. GPT stands for Generative Pre-trained Transformer, which describes an AI program that looks for patterns in language and data learning to predict the next word in a sentence or the next paragraph in an essay. The website has a friendly interface that allows users to interact with AI in a n efficient conversational tone . ChatGPT provides another opportunity for students, instructors, researchers, workers, and others to find practical solutions to everyday and complicated problems.

At the root of this conversation is Artificial Intelligence. I plan to explore applicable uses of AI and ChatGPT in the classroom , entrepreneurial potential uses, and applications in industry .

Everyday Uses of Artificial Intelligence

The use of Artificial I ntelligence varies based on the user and their end goal. While many individuals will use certain programs or websites to meet specific objectives , many companies and apps have begun to utilize this emerging technology to better meet their customer's needs.

Duolingo is a popular foreign language learning application that I use to supplement my Spanish studies . The app uses Artificial Intelligence to assess users' knowledge and understanding as they interact with the program , thus streamlining users learning outcomes.

As another example, Khan Academy is a free online resource that helps teachers and students learn any level of math or other grade school topics for free. They have created Khanmigo , using AI. The model acts as a tutor that helps work through a problem while not directly providing the answer. It can assist in writing an essay or solving a complex math problem step by step.

These everyday applications continue a trend of companies implementing this new technolog y into students and teachers' lives . . This new AI technology also allows business professionals to enhance aspects of their processes.

Entrepreneurs, A.I. and the Advantages

While AI already provides companies and organizations with new ways to interact with and better support their customers, AI could also provide emerging industries and entrepreneurs with new paths to business success. 

According to Entrpreneur.com, most businesses currently use AI for customer service purposes , however , AI could also help entrepreneurs create effective spreadsheets cataloging useful data with accuracy that can be incredibly specific or broad. Specifically with customer service, AI can quickly find what a customer needs and solve their problems efficiently. It could also analyze how effective marketing campaigns are influencing customers’ purchases.

As I researched for more information about this topic, I found an article in The Journal of Business Venturing Insights published in March 2023, sharing different techniques business students can use ChatGPT as an asset to generate entrepreneurial business pitches. The article titled “ The Artificially Intelligent Entrepreneur” written by Cole Short, an Assistant Professor of Strategy at Pepperdine University, and Jeremy C. Short, a UTA alumni and Professor at the University of North Texas at Denton, showcased different elevator pitch scenarios.

Students and entrepreneurs study CEOs who have impacted an industry dynamically; the CEO's mentality is an asset . I had the opportunity to question Dr. Jeremy Short on how he arrived at the initial question of using AI as a CEO archetype business consultant. An archetype is a symbol, term, or pattern of behavior which others have replicated or emulated.

He responded, “ We used this existing framework and selected a CEO from each archetype and used ChatGPT to create elevator pitches, social media pitches, and crowdfunding pitches. The strength of ChatGPT is based largely on the creativity of the prompt, which is where we aim as authors.”

CEO Archetypes and Prompt Engineering

ChatGPT allows the user to understand the archetypes of successful CEOs and collaborate with entrepreneurial styles. These archetypes are accessible options to consult with AI. Let ’ s break down different CEO archetypes students used during this study:

Creator CEOs are typically serial entrepreneurs and serve during the growth stages of developing new businesses. These individuals are risk takers recognizing opportunities that others don ’ t see. Elon Musk, CEO of Tesla, SpaceX, and Twitter is the creator archetype.

Transformer CEOs are created by climbing the ladder of a successful business and adding new ideas . They have a firm understanding of the company's culture and work to dramatically change the company, separating it from missteps in the past. Indra Nooyi CEO of PepsiCo is the transformer archetype.

Savior CEOs rescue businesses on the verge of failure with disciplined actions, unique experience and insights they forge a successful path forward for declining businesses. Lisa Su, CEO of AMD is the savior archetype.

ChatGPT was prompted to write an elevator pitch in the style of the previously listed CEOs. 

The response for Elon Musk included language about “ building” a product with “ cutting-edge technology.” 

Indra Nooyi ’s response included phrases like “ the world is changing” and making “ a positive impact in the world.” 

Lisa Su's response produced a pitch speaking about being “ accountable, tough and disciplined” with an emphasis on “ a strong focus on efficiency and performance.”

However, I believe these positions can help entrepreneurs develop their own successful business practices; creating a product your former employer could use to gain an advantage over the competition is disruptive. B uying a company on the brink of bankruptcy that has been mismanaged is a scenario entrepreneurs have explored and practiced .

Prompt engineering is the description of a task AI can accomplish , with instructions embedded in the input. Using prompt engineering, users can fine-tune their input to achieve a desired output incorporating a task description to guide the AI model. 

Conversation around ChatGPT and Artificial Intelligence

I asked Dr. Short about how students could use this technology as an asset that guides their learning and, additionally, how instructors can use this as well. He spoke about an assignment he is currently using in his classes. “ Chat GPT might be valuable in helping create a recipe for material that students can then refine. For example, in my social entrepreneurship class students create crowdfunding campaigns for either DonorsChoose , a platform that caters to public school teachers or GoFundMe , a service which allows a variety of project types to a larger userbase . I plan on students using ChatGPT to create a ‘rough draft’ to show me so I can see how they refine their responses for their particular campaigns this upcoming fall.” Th is approach allows students to take advantage of popular technology in a constructive way.

The journal article provided some notable conclusions about ChatGPT , i ncluding “ quality control is essential when using automated tools; a hallmark of success for large language models is their vast associative memory, this strength can also be a weakness. Specifically, models such as OpenAI’s GPT-3.5 and GPT-4 are capable of confidently generating “ hallucinated” output that appears correct but, it is incorrect or completely fabricated. ChatGPT serves as an emerging tool that can efficiently and flexibly produce a range of narrative content for entrepreneurs and serve to inspire future research at the intersection of entrepreneurship and AI.” ChatGPT ’s limitations and potential applications are continually being explored.

Industry Application

After researching various applications of AI, I spoke with Dr. George Benson, Professor and Department Chair of the Department of Management at The University of Texas at Arlington, about AI and ChatGPT from an industry perspective. His research focuses on Artificial Intelligence with Human Resource Management .

Dr. Benson told me that Artificial Intelligence is being invested heavily by human resource departments who are looking to automate hiring practices. Specifically, he mentioned “ HR is using this as a market opportunity. AI is a useful tool to sift through potential applicants by scanning their resumes for qualifications and experiences. Allowing professionals to hire applicants faster.”

This application allows the technology to handle low-level tasks, but the results generated are being handed to a human to review and act on. He spoke about the potential of A.I. “ There are a lot of unknowns, but the technology is new and getting better.” Looking towards the future, technology is already being applied in different ways . These applications are being explored in the classrooms of UTA as well.

Exploration of AI at UTA

The College of Business conduct ed a survey to understand the faculty’s attitude towards A I in the classroom. It was a part of the “Teaching with Chat GPT” workshop on Friday February 9 th , which focus ed on how to integrate Chat GPT and other AI platforms into teaching . 

Dr. Kevin Carr, a Clinical Assistant Professor of Marketing at UTA, was a part of the workshop ; he currently teaches Advanced Business Communication . I talked to him about the purpose of the workshop and what he hopes to gain from the group's sessions. 

Dr. Carr explained "The point of the workshop is designed to give faculty ideas for instruction and to develop classroom activities to work with students . Our goal for th e workshop is to introduce Artificial Intelligence as a teaching tool for faculty, including showing what AI can do potentially in the classroom. We are going to be very open to faculty’s direction, in terms of ongoing discu ssions and meetings.”

Personal Take

Artificial Intelligence or Chat GPT , in my view, is another useful tool in the toolbox of technology. It will take the air out of certain industries, and it will change jobs, yet every major technological advancement has the potential to do so. The automobile was considered radical, the use of plastic, computers in the workplace, and alternative energy have been impactful on society. 

Alternative energy was headlined as the end of oil use. The automobile changed the way cities were formed and led to the creation of a national highway system. Society has always found a way to adapt and overcome major technological innovations, artificial intelligence is not any different.

AI is the technology of tomorrow. It reminds me of something Dr. George Benson said , “ It's cool software that is a sophisticated search engine.” Google, one of the most popular search engines, reshaped the internet, as you search for resources, it is a natural starting point. AI and ChatGPT are an evolution, for students it is a tremendous resource consulting a CEO archetype, creating business pitches, and most importantly shaping the future .

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