problem solving skills for math

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6 Tips for Teaching Math Problem-Solving Skills

Solving word problems is tougher than computing with numbers, but elementary teachers can guide students to do the deep thinking involved.

A growing concern with students is the ability to problem-solve, especially with complex, multistep problems. Data shows that students struggle more when solving word problems than they do with computation , and so problem-solving should be considered separately from computation. Why?

Consider this. When we’re on the way to a new destination and we plug in our location to a map on our phone, it tells us what lane to be in and takes us around any detours or collisions, sometimes even buzzing our watch to remind us to turn. When I experience this as a driver, I don’t have to do the thinking. I can think about what I’m going to cook for dinner, not paying much attention to my surroundings other than to follow those directions. If I were to be asked to go there again, I wouldn’t be able to remember, and I would again seek help.

If we can switch to giving students strategies that require them to think instead of giving them too much support throughout the journey to the answer, we may be able to give them the ability to learn the skills to read a map and have several ways to get there.

Here are six ways we can start letting students do this thinking so that they can go through rigorous problem-solving again and again, paving their own way to the solution. 

1. Link problem-solving to reading

When we can remind students that they already have many comprehension skills and strategies they can easily use in math problem-solving, it can ease the anxiety surrounding the math problem. For example, providing them with strategies to practice, such as visualizing, acting out the problem with math tools like counters or base 10 blocks, drawing a quick sketch of the problem, retelling the story in their own words, etc., can really help them to utilize the skills they already have to make the task less daunting.

We can break these skills into specific short lessons so students have a bank of strategies to try on their own. Here's an example of an anchor chart that they can use for visualizing . Breaking up comprehension into specific skills can increase student independence and help teachers to be much more targeted in their problem-solving instruction. This allows students to build confidence and break down the barriers between reading and math to see they already have so many strengths that are transferable to all problems.

2. Avoid boxing students into choosing a specific operation

It can be so tempting to tell students to look for certain words that might mean a certain operation. This might even be thoroughly successful in kindergarten and first grade, but just like when our map tells us where to go, that limits students from becoming deep thinkers. It also expires once they get into the upper grades, where those words could be in a problem multiple times, creating more confusion when students are trying to follow a rule that may not exist in every problem.

We can encourage a variety of ways to solve problems instead of choosing the operation first. In first grade, a problem might say, “Joceline has 13 stuffed animals and Jordan has 17. How many more does Jordan have?” Some students might choose to subtract, but a lot of students might just count to find the amount in between. If we tell them that “how many more” means to subtract, we’re taking the thinking out of the problem altogether, allowing them to go on autopilot without truly solving the problem or using their comprehension skills to visualize it. 

3. Revisit ‘representation’

The word “representation” can be misleading. It seems like something to do after the process of solving. When students think they have to go straight to solving, they may not realize that they need a step in between to be able to support their understanding of what’s actually happening in the problem first.

Using an anchor chart like one of these ( lower grade , upper grade ) can help students to choose a representation that most closely matches what they’re visualizing in their mind. Once they sketch it out, it can give them a clearer picture of different ways they could solve the problem.

Think about this problem: “Varush went on a trip with his family to his grandmother’s house. It was 710 miles away. On the way there, three people took turns driving. His mom drove 214 miles. His dad drove 358 miles. His older sister drove the rest. How many miles did his sister drive?”

If we were to show this student the anchor chart, they would probably choose a number line or a strip diagram to help them understand what’s happening.

If we tell students they must always draw base 10 blocks in a place value chart, that doesn’t necessarily match the concept of this problem. When we ask students to match our way of thinking, we rob them of critical thinking practice and sometimes confuse them in the process. 

4. Give time to process

Sometimes as educators, we can feel rushed to get to everyone and everything that’s required. When solving a complex problem, students need time to just sit with a problem and wrestle with it, maybe even leaving it and coming back to it after a period of time.

This might mean we need to give them fewer problems but go deeper with those problems we give them. We can also speed up processing time when we allow for collaboration and talk time with peers on problem-solving tasks. 

5. Ask questions that let Students do the thinking

Questions or prompts during problem-solving should be very open-ended to promote thinking. Telling a student to reread the problem or to think about what tools or resources would help them solve it is a way to get them to try something new but not take over their thinking.

These skills are also transferable across content, and students will be reminded, “Good readers and mathematicians reread.” 

6. Spiral concepts so students frequently use problem-solving skills

When students don’t have to switch gears in between concepts, they’re not truly using deep problem-solving skills. They already kind of know what operation it might be or that it’s something they have at the forefront of their mind from recent learning. Being intentional within their learning stations and assessments about having a variety of rigorous problem-solving skills will refine their critical thinking abilities while building more and more resilience throughout the school year as they retain content learning in the process. 

Problem-solving skills are so abstract, and it can be tough to pinpoint exactly what students need. Sometimes we have to go slow to go fast. Slowing down and helping students have tools when they get stuck and enabling them to be critical thinkers will prepare them for life and allow them multiple ways to get to their own destination.

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

posted on September 20, 2023

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

Math Learning

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

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

Benefits of Math Learning

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

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

How to Know What Math You Need to Learn

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

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

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

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How to learn math.

Learning at School

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

Learning at Home

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

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

Learning Math with the Help of Artificial Intelligence (AI) 

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

Using Khan Academy’s AI Tutor, Khanmigo

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

Get Math Help with Khanmigo Right Now

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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How to Improve Problem-Solving Skills: Mathematics and Critical Thinking

In today’s rapidly changing world, problem-solving has become a quintessential skill. When we discuss the topic, it’s natural to ask, “What is problem-solving?” and “How can we enhance this skill, particularly in children?” The discipline of mathematics offers a rich platform to explore these questions. Through math, not only do we delve into numbers and equations, but we also explore how to improve problem-solving skills and how to develop critical thinking skills in math. Let’s embark on this enlightening journey together.

What is Problem-Solving?

At its core, problem-solving involves identifying a challenge and finding a solution. But it’s not always as straightforward as it sounds. So, what is problem-solving? True problem-solving requires a combination of creative thinking and logical reasoning. Mathematics, in many ways, embodies this blend. When a student approaches a math problem, they must discern the issue at hand, consider various methods to tackle it, and then systematically execute their chosen strategy.

But what is problem-solving in a broader context? It’s a life skill. Whether we’re deciding the best route to a destination, determining how to save for a big purchase, or even figuring out how to fix a broken appliance, we’re using problem-solving.

How to Develop Critical Thinking Skills in Math

Critical thinking goes hand in hand with problem-solving. But exactly how to develop critical thinking skills in math might not be immediately obvious. Here are a few strategies:

• Contextual Learning: Teaching math within a story or real-life scenario makes it relevant. When students see math as a tool to navigate the world around them, they naturally begin to think critically about solutions.
• Open-ended Questions: Instead of merely seeking the “right” answer, encourage students to explain their thought processes. This nudges them to think deeply about their approach.
• Group Discussions: Collaborative learning can foster different perspectives, prompting students to consider multiple ways to solve a problem.
• Challenging Problems: Occasionally introducing problems that are a bit beyond a student’s current skill level can stimulate critical thinking. They will have to stretch their understanding and think outside the box.

What are the Six Basic Steps of the Problem-Solving Process?

Understanding how to improve problem-solving skills often comes down to familiarizing oneself with the systematic approach to challenges. So, what are the six basic steps of the problem-solving process?

• Identification: Recognize and define the problem.
• Analysis: Understand the problem’s intricacies and nuances.
• Generation of Alternatives: Think of different ways to approach the challenge.
• Decision Making: Choose the most suitable method to address the problem.
• Implementation: Put the chosen solution into action.
• Evaluation: Reflect on the solution’s effectiveness and learn from the outcome.

By embedding these steps into mathematical education, we provide students with a structured framework. When they wonder about how to improve problem-solving skills or how to develop critical thinking skills in math, they can revert to this process, refining their approach with each new challenge.

Making Math Fun and Relevant

At Wonder Math, we believe that the key to developing robust problem-solving skills lies in making math enjoyable and pertinent. When students see math not just as numbers on a page but as a captivating story or a real-world problem to be solved, their engagement skyrockets. And with heightened engagement comes enhanced understanding.

As educators and parents, it’s crucial to continuously ask ourselves: how can we demonstrate to our children what problem-solving is? How can we best teach them how to develop critical thinking skills in math? And how can we instill in them an understanding of the six basic steps of the problem-solving process?

The answer, we believe, lies in active learning, contextual teaching, and a genuine passion for the beauty of mathematics.

The Underlying Beauty of Mathematics

Often, people perceive mathematics as a rigid discipline confined to numbers and formulas. However, this is a limited view. Math, in essence, is a language that describes patterns, relationships, and structures. It’s a medium through which we can communicate complex ideas, describe our universe, and solve intricate problems. Understanding this deeper beauty of math can further emphasize how to develop critical thinking skills in math.

Why Mathematics is the Ideal Playground for Problem-Solving

Math provides endless opportunities for problem-solving. From basic arithmetic puzzles to advanced calculus challenges, every math problem offers a chance to hone our problem-solving skills. But why is mathematics so effective in this regard?

• Structured Challenges: Mathematics presents problems in a structured manner, allowing learners to systematically break them down. This format mimics real-world scenarios where understanding the structure of a challenge can be half the battle.
• Multiple Approaches: Most math problems can be approached in various ways . This teaches learners flexibility in thinking and the ability to view a single issue from multiple angles.
• Immediate Feedback: Unlike many real-world problems where solutions might take time to show results, in math, students often get immediate feedback. They can quickly gauge if their approach works or if they need to rethink their strategy.

Enhancing the Learning Environment

To genuinely harness the power of mathematics in developing problem-solving skills, the learning environment plays a crucial role. A student who is afraid of making mistakes will hesitate to try out different approaches, stunting their critical thinking growth.

However, in a nurturing, supportive environment where mistakes are seen as learning opportunities, students thrive. They become more willing to take risks, try unconventional solutions, and learn from missteps. This mindset, where failure is not feared but embraced as a part of the learning journey, is pivotal for developing robust problem-solving skills.

Incorporating Technology

In our digital age, technology offers innovative ways to explore math. Interactive apps and online platforms can provide dynamic problem-solving scenarios, making the process even more engaging. These tools can simulate real-world challenges, allowing students to apply their math skills in diverse contexts, further answering the question of how to improve problem-solving skills.

More than Numbers 

In summary, mathematics is more than just numbers and formulas—it’s a world filled with challenges, patterns, and beauty. By understanding its depth and leveraging its structured nature, we can provide learners with the perfect platform to develop critical thinking and problem-solving skills. The key lies in blending traditional techniques with modern tools, creating a holistic learning environment that fosters growth, curiosity, and a lifelong love for learning.

Join us on this transformative journey at Wonder Math. Let’s make math an adventure, teaching our children not just numbers and equations, but also how to improve problem-solving skills and navigate the world with confidence. Enroll your child today and witness the magic of mathematics unfold before your eyes!

FAQ: Mathematics and Critical Thinking

1. what is problem-solving in the context of mathematics.

Problem-solving in mathematics refers to the process of identifying a mathematical challenge and systematically working through methods and strategies to find a solution.

2. Why is math considered a good avenue for developing problem-solving skills?

Mathematics provides structured challenges and allows for multiple approaches to find solutions. This promotes flexibility in thinking and encourages learners to view problems from various angles.

3. How does contextual learning enhance problem-solving abilities?

By teaching math within a story or real-life scenario, it becomes more relevant for the learner. This helps them see math as a tool to navigate real-world challenges , thereby promoting critical thinking.

4. What are the six basic steps of the problem-solving process in math?

The six steps are: Identification, Analysis, Generation of Alternatives, Decision Making, Implementation, and Evaluation.

5. How can parents support their children in developing mathematical problem-solving skills?

Parents can provide real-life contexts for math problems , encourage open discussions about different methods, and ensure a supportive environment where mistakes are seen as learning opportunities.

6. Are there any tools or apps that can help in enhancing problem-solving skills in math?

Yes, there are various interactive apps and online platforms designed specifically for math learning. These tools provide dynamic problem-solving scenarios and simulate real-world challenges, making the learning process engaging.

7. How does group discussion foster critical thinking in math?

Group discussions allow students to hear different perspectives and approaches to a problem. This can challenge their own understanding and push them to think about alternative methods.

8. Is it necessary to always follow the six steps of the problem-solving process sequentially?

While the six steps provide a structured approach, real-life problem-solving can sometimes be more fluid. It’s beneficial to know the steps, but adaptability and responsiveness to the situation are also crucial.

9. How does Wonder Math incorporate active learning in teaching mathematics?

Wonder Math integrates mathematics within engaging stories and real-world scenarios, making it fun and relevant. This active learning approach ensures that students are not just passive recipients but active participants in the learning process.

10. What if my child finds a math problem too challenging and becomes demotivated?

It’s essential to create a supportive environment where challenges are seen as growth opportunities. Remind them that every problem is a chance to learn, and it’s okay to seek help or approach it differently.

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5 Teaching Mathematics Through Problem Solving

Janet Stramel

In his book “How to Solve It,” George Pólya (1945) said, “One of the most important tasks of the teacher is to help his students. This task is not quite easy; it demands time, practice, devotion, and sound principles. The student should acquire as much experience of independent work as possible. But if he is left alone with his problem without any help, he may make no progress at all. If the teacher helps too much, nothing is left to the student. The teacher should help, but not too much and not too little, so that the student shall have a reasonable share of the work.” (page 1)

What is a problem  in mathematics? A problem is “any task or activity for which the students have no prescribed or memorized rules or methods, nor is there a perception by students that there is a specific ‘correct’ solution method” (Hiebert, et. al., 1997). Problem solving in mathematics is one of the most important topics to teach; learning to problem solve helps students develop a sense of solving real-life problems and apply mathematics to real world situations. It is also used for a deeper understanding of mathematical concepts. Learning “math facts” is not enough; students must also learn how to use these facts to develop their thinking skills.

According to NCTM (2010), the term “problem solving” refers to mathematical tasks that have the potential to provide intellectual challenges for enhancing students’ mathematical understanding and development. When you first hear “problem solving,” what do you think about? Story problems or word problems? Story problems may be limited to and not “problematic” enough. For example, you may ask students to find the area of a rectangle, given the length and width. This type of problem is an exercise in computation and can be completed mindlessly without understanding the concept of area. Worthwhile problems  includes problems that are truly problematic and have the potential to provide contexts for students’ mathematical development.

There are three ways to solve problems: teaching for problem solving, teaching about problem solving, and teaching through problem solving.

Teaching for problem solving begins with learning a skill. For example, students are learning how to multiply a two-digit number by a one-digit number, and the story problems you select are multiplication problems. Be sure when you are teaching for problem solving, you select or develop tasks that can promote the development of mathematical understanding.

Teaching about problem solving begins with suggested strategies to solve a problem. For example, “draw a picture,” “make a table,” etc. You may see posters in teachers’ classrooms of the “Problem Solving Method” such as: 1) Read the problem, 2) Devise a plan, 3) Solve the problem, and 4) Check your work. There is little or no evidence that students’ problem-solving abilities are improved when teaching about problem solving. Students will see a word problem as a separate endeavor and focus on the steps to follow rather than the mathematics. In addition, students will tend to use trial and error instead of focusing on sense making.

Teaching through problem solving  focuses students’ attention on ideas and sense making and develops mathematical practices. Teaching through problem solving also develops a student’s confidence and builds on their strengths. It allows for collaboration among students and engages students in their own learning.

Consider the following worthwhile-problem criteria developed by Lappan and Phillips (1998):

• The problem has important, useful mathematics embedded in it.
• The problem requires high-level thinking and problem solving.
• The problem contributes to the conceptual development of students.
• The problem creates an opportunity for the teacher to assess what his or her students are learning and where they are experiencing difficulty.
• The problem can be approached by students in multiple ways using different solution strategies.
• The problem has various solutions or allows different decisions or positions to be taken and defended.
• The problem encourages student engagement and discourse.
• The problem connects to other important mathematical ideas.
• The problem promotes the skillful use of mathematics.
• The problem provides an opportunity to practice important skills.

Of course, not every problem will include all of the above. Sometimes, you will choose a problem because your students need an opportunity to practice a certain skill.

Key features of a good mathematics problem includes:

• It must begin where the students are mathematically.
• The feature of the problem must be the mathematics that students are to learn.
• It must require justifications and explanations for both answers and methods of solving.

Problem solving is not a  neat and orderly process. Think about needlework. On the front side, it is neat and perfect and pretty.

But look at the b ack.

It is messy and full of knots and loops. Problem solving in mathematics is also like this and we need to help our students be “messy” with problem solving; they need to go through those knots and loops and learn how to solve problems with the teacher’s guidance.

When you teach through problem solving , your students are focused on ideas and sense-making and they develop confidence in mathematics!

Mathematics Tasks and Activities that Promote Teaching through Problem Solving

Choosing the Right Task

Selecting activities and/or tasks is the most significant decision teachers make that will affect students’ learning. Consider the following questions:

• Teachers must do the activity first. What is problematic about the activity? What will you need to do BEFORE the activity and AFTER the activity? Additionally, think how your students would do the activity.
• What mathematical ideas will the activity develop? Are there connections to other related mathematics topics, or other content areas?
• Can the activity accomplish your learning objective/goals?

Low Floor High Ceiling Tasks

By definition, a “ low floor/high ceiling task ” is a mathematical activity where everyone in the group can begin and then work on at their own level of engagement. Low Floor High Ceiling Tasks are activities that everyone can begin and work on based on their own level, and have many possibilities for students to do more challenging mathematics. One gauge of knowing whether an activity is a Low Floor High Ceiling Task is when the work on the problems becomes more important than the answer itself, and leads to rich mathematical discourse [Hover: ways of representing, thinking, talking, agreeing, and disagreeing; the way ideas are exchanged and what the ideas entail; and as being shaped by the tasks in which students engage as well as by the nature of the learning environment].

The strengths of using Low Floor High Ceiling Tasks:

• Allows students to show what they can do, not what they can’t.
• Provides differentiation to all students.
• Promotes a positive classroom environment.
• Advances a growth mindset in students
• Aligns with the Standards for Mathematical Practice

Examples of some Low Floor High Ceiling Tasks can be found at the following sites:

• YouCubed – under grades choose Low Floor High Ceiling
• NRICH Creating a Low Threshold High Ceiling Classroom
• Inside Mathematics Problems of the Month

Math in 3-Acts

Math in 3-Acts was developed by Dan Meyer to spark an interest in and engage students in thought-provoking mathematical inquiry. Math in 3-Acts is a whole-group mathematics task consisting of three distinct parts:

Act One is about noticing and wondering. The teacher shares with students an image, video, or other situation that is engaging and perplexing. Students then generate questions about the situation.

In Act Two , the teacher offers some information for the students to use as they find the solutions to the problem.

Act Three is the “reveal.” Students share their thinking as well as their solutions.

“Math in 3 Acts” is a fun way to engage your students, there is a low entry point that gives students confidence, there are multiple paths to a solution, and it encourages students to work in groups to solve the problem. Some examples of Math in 3-Acts can be found at the following websites:

• Dan Meyer’s Three-Act Math Tasks
• Graham Fletcher3-Act Tasks ]
• Math in 3-Acts: Real World Math Problems to Make Math Contextual, Visual and Concrete

Number Talks

Number talks are brief, 5-15 minute discussions that focus on student solutions for a mental math computation problem. Students share their different mental math processes aloud while the teacher records their thinking visually on a chart or board. In addition, students learn from each other’s strategies as they question, critique, or build on the strategies that are shared.. To use a “number talk,” you would include the following steps:

• The teacher presents a problem for students to solve mentally.
• Provide adequate “ wait time .”
• The teacher calls on a students and asks, “What were you thinking?” and “Explain your thinking.”
• For each student who volunteers to share their strategy, write their thinking on the board. Make sure to accurately record their thinking; do not correct their responses.
• Invite students to question each other about their strategies, compare and contrast the strategies, and ask for clarification about strategies that are confusing.

“Number Talks” can be used as an introduction, a warm up to a lesson, or an extension. Some examples of Number Talks can be found at the following websites:

• Inside Mathematics Number Talks
• Number Talks Build Numerical Reasoning

Saying “This is Easy”

“This is easy.” Three little words that can have a big impact on students. What may be “easy” for one person, may be more “difficult” for someone else. And saying “this is easy” defeats the purpose of a growth mindset classroom, where students are comfortable making mistakes.

When the teacher says, “this is easy,” students may think,

• “Everyone else understands and I don’t. I can’t do this!”
• Students may just give up and surrender the mathematics to their classmates.
• Students may shut down.

Instead, you and your students could say the following:

• “I think I can do this.”
• “I have an idea I want to try.”
• “I’ve seen this kind of problem before.”

Tracy Zager wrote a short article, “This is easy”: The Little Phrase That Causes Big Problems” that can give you more information. Read Tracy Zager’s article here.

Using “Worksheets”

Do you want your students to memorize concepts, or do you want them to understand and apply the mathematics for different situations?

What is a “worksheet” in mathematics? It is a paper and pencil assignment when no other materials are used. A worksheet does not allow your students to use hands-on materials/manipulatives [Hover: physical objects that are used as teaching tools to engage students in the hands-on learning of mathematics]; and worksheets are many times “naked number” with no context. And a worksheet should not be used to enhance a hands-on activity.

Students need time to explore and manipulate materials in order to learn the mathematics concept. Worksheets are just a test of rote memory. Students need to develop those higher-order thinking skills, and worksheets will not allow them to do that.

One productive belief from the NCTM publication, Principles to Action (2014), states, “Students at all grade levels can benefit from the use of physical and virtual manipulative materials to provide visual models of a range of mathematical ideas.”

You may need an “activity sheet,” a “graphic organizer,” etc. as you plan your mathematics activities/lessons, but be sure to include hands-on manipulatives. Using manipulatives can

• Provide your students a bridge between the concrete and abstract
• Serve as models that support students’ thinking
• Provide another representation
• Support student engagement
• Give students ownership of their own learning.

Adapted from “ The Top 5 Reasons for Using Manipulatives in the Classroom ”.

any task or activity for which the students have no prescribed or memorized rules or methods, nor is there a perception by students that there is a specific ‘correct’ solution method

should be intriguing and contain a level of challenge that invites speculation and hard work, and directs students to investigate important mathematical ideas and ways of thinking toward the learning

involves teaching a skill so that a student can later solve a story problem

when we teach students how to problem solve

teaching mathematics content through real contexts, problems, situations, and models

a mathematical activity where everyone in the group can begin and then work on at their own level of engagement

20 seconds to 2 minutes for students to make sense of questions

Mathematics Methods for Early Childhood Copyright © 2021 by Janet Stramel is licensed under a Creative Commons Attribution 4.0 International License , except where otherwise noted.

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Problem Solving Activities: 7 Strategies

• Critical Thinking

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

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

I was so excited!

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

It was a proud moment for me!

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

Genius right? 

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

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

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

When I Finally Saw the Light

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

Problem Solving Activities

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

Seasonal Problem Solving

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

Cooperative Problem Solving Tasks

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

Notice and Wonder

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

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

Math Starters

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

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

Calculators

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

Three-Act Math Tasks

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

Getting the Most from Each of the Problem Solving Activities

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

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

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

Shametria Routt Banks

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

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

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10 Strategies for Problem-Solving in Math

reviewed by Jo-ann Caballes

Updated on August 21, 2024

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

What are problem-solving strategies in math?

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

Here are some problem-solving methods:

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

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

How to solve math problems?

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

Here are 5 math problem-solving strategies for kids:

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

Understanding the Problem 

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

Work Out a Plan 

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

Carry Out the Plan 

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

Review the Work

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

Reflect and Analyze

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

Ways to solve math problems

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

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

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What are 10 strategies for solving math problems?

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

Understand the Problem

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

Guess and Check

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

Work It Out

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

Work Backwords 

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

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

Find a Pattern 

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

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

Draw a Picture or Diagram

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

Trial and Error Method

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

Review Answers with Peers

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

Help your kid achieve their full math potential

The best Brighterly tutors are ready to help with that.

Math problem-solving strategies for elementary students

5 problem-solving strategies for elementary students include:

Using Simple Language 

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

Using Visuals and Manipulatives 

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

Simplifying the Problem

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

Looking for Patterns 

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

Using Stories

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

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

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

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

As adults, we take numbers for granted, but preschoolers and kindergartners have no idea what these symbols mean. Yet, we often demand instant understanding and flawless performance when we start teaching numbers to our children. If you don’t have a clue about how to teach numbers for kids, browse no more. You will get four […]

May 19, 2022

Teaching strategies aren’t something that is set in stone and continue to evolve every year. Even though traditional educational strategies like teachers teaching in front of the classroom seemed to work for decades with little to no adjustments, the digital age has brought along numerous challenges. Teaching methods for kids require new strategies, so educators […]

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Teaching Problem Solving in Math

• Freebies , Math , Planning

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

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

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

The Problem Solving Strategies

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

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

There’s also this visuals strategy wheel practice.

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

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

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

The Problem Solving Steps

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

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

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

We did this over and over with example problems.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Act  – Act on a strategy and try it out.

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

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

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

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

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Problem Solving in Mathematics

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The main reason for learning about math is to become a better problem solver in all aspects of life. Many problems are multistep and require some type of systematic approach. There are a couple of things you need to do when solving problems. Ask yourself exactly what type of information is being asked for: Is it one of addition, subtraction, multiplication , or division? Then determine all the information that is being given to you in the question.

Mathematician George Pólya’s book, “ How to Solve It: A New Aspect of Mathematical Method ,” written in 1957, is a great guide to have on hand. The ideas below, which provide you with general steps or strategies to solve math problems, are similar to those expressed in Pólya’s book and should help you untangle even the most complicated math problem.

Use Established Procedures

Learning how to solve problems in mathematics is knowing what to look for. Math problems often require established procedures and knowing what procedure to apply. To create procedures, you have to be familiar with the problem situation and be able to collect the appropriate information, identify a strategy or strategies, and use the strategy appropriately.

Problem-solving requires practice. When deciding on methods or procedures to use to solve problems, the first thing you will do is look for clues, which is one of the most important skills in solving problems in mathematics. If you begin to solve problems by looking for clue words, you will find that these words often indicate an operation.

Look for Clue Words

Think of yourself as a math detective. The first thing to do when you encounter a math problem is to look for clue words. This is one of the most important skills you can develop. If you begin to solve problems by looking for clue words, you will find that those words often indicate an operation.

Common clue words for addition  problems:

Common clue words for  subtraction  problems:

• How much more

Common clue words for multiplication problems:

Common clue words for division problems:

Although clue words will vary a bit from problem to problem, you'll soon learn to recognize which words mean what in order to perform the correct operation.

Read the Problem Carefully

This, of course, means looking for clue words as outlined in the previous section. Once you’ve identified your clue words, highlight or underline them. This will let you know what kind of problem you’re dealing with. Then do the following:

• Ask yourself if you've seen a problem similar to this one. If so, what is similar about it?
• What did you need to do in that instance?
• What facts are you given about this problem?
• What facts do you still need to find out about this problem?

Develop a Plan and Review Your Work

Based on what you discovered by reading the problem carefully and identifying similar problems you’ve encountered before, you can then:

• Define your problem-solving strategy or strategies. This might mean identifying patterns, using known formulas, using sketches, and even guessing and checking.
• If your strategy doesn't work, it may lead you to an ah-ha moment and to a strategy that does work.

If it seems like you’ve solved the problem, ask yourself the following:

• Does your solution seem probable?
• Does it answer the initial question?
• Did you answer using the language in the question?
• Did you answer using the same units?

If you feel confident that the answer is “yes” to all questions, consider your problem solved.

Tips and Hints

Some key questions to consider as you approach the problem may be:

• What are the keywords in the problem?
• Do I need a data visual, such as a diagram, list, table, chart, or graph?
• Is there a formula or equation that I'll need? If so, which one?
• Will I need to use a calculator? Is there a pattern I can use or follow?

Read the problem carefully, and decide on a method to solve the problem. Once you've finished working the problem, check your work and ensure that your answer makes sense and that you've used the same terms and or units in your answer.

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5 Ways to Build Math Problem Solving Skills (based on brain research)

Whether talking about state tests or meeting with your team to plan the next math unit, the conversation inevitably turns to word problems. But knowing how to build math problem-solving skills without resorting to pages of boring story problem practice can be hard.

These days word problems aren’t the basic one-step wonders that many of us dealt with as students. Instead, multi-step story problems that require students to apply multiple concepts and skills are incorporated into instruction and state assessments.

Understanding brain research can help simply the process of teaching this challenging format of math problem-solving to students, including those who struggle.

What research says about building master problem solvers in math

Have you seen how many math skills we must teach these days? No teacher has enough time to build critical math skills AND effectively teach problem-solving…or do they?

Research would argue we are going about these tasks all wrong. They say there are many reasons students struggle with math word problems , but one big one is that we aren’t doing what’s best for the brain. Instead, here’s what the brain research says about the must-have elements for building step-by-step math problem-solving mastery.

Finding #1: Becoming a master problem solver requires repetition.

Duh, right? Any good teacher knows this…but what’s the best recipe for repetition if you want students to master math word problems? How much practice? How often?

Let’s start with the concept of mastery.

How do you develop math problem solving skills?

In the 1990’s, Anders Ericsson studied experts to explore what made some people excel. Findings showed a positive correlation between the amount of deliberate practice (activities that require a high level of concentration and aren’t necessarily inherently fun) and skill level.

In other words, the more practice someone gets, the more they improve. This became the basis of Malcolm Gladwell’s 10,000-hour rule, which stated that it takes 10,000 hours to make you an expert in a field.

But what should that practice look like for students who struggle with word problems? Is it better to have a deep dive into story problems, or do short bursts of practice do more for long-term understanding?

Designing Better Word Problem Activities: Building Step-by-step Math Problem-Solving Practice

We can look at Ebbinghaus’ work on memory & retention to answer that.  He found spacing practice over time decreased the number of exposures needed. In other words, small amounts of practice over several days, weeks, or even months actually means you need LESS practice than if you try to cram it all in at once.

For over 80 years, this finding has stood the test of time. While research has shown that students who engage in mass practice (lots of practice all at once) might do better on an assessment that takes place tomorrow, students who engage in repeated practice over a period of time retain more skills long-term (Bloom & Shuell, 1981; Rea & Modigliani, 1985).

And how long does the research say you should spend reviewing?

How long should problem-solving practice really be?

Shorter is better. As discussed earlier, peak attention required for deliberate practice can only be maintained for so long. And the majority of research supports 8-10 minutes as the ideal lesson length (Robertson, 2010).

This means practice needs to be focused so that during those minutes of discussion, you can dive deep – breaking down the word problem and discussing methods to solve it.

Teacher Tip: Applying this finding to your classroom

Less is actually more as long as you plan to practice regularly. While students who struggle with word problems may need a great deal of practice to master word problems, ideally, this practice should be provided in short, regular intervals with no more than 8-10 minutes spent in whole group discussion.

Here are a few simple steps to apply these findings to your math classroom:

• Find 8-12 minutes in your daily schedule to focus on problem-solving – consider this time sacred & only for problem-solving.
• Select only 1-2 word problems per day. Target step-by-step math problem-solving to build math problem-solving skills through a less-is-more approach using Problem of the Day .

Finding #2: Students who are challenged & supported have better outcomes.

Productive struggle, as it is called in the research, focuses on the effortful practice that builds long-term understanding.

Important to this process are opportunities for choice, collaboration, and the use of materials or topics of interest (which will be discussed later).

This productive struggle also helps students build flexible thinking so that they can apply previously learned skills to new or unfamiliar tasks (Bransford, Brown, & Cocking, 2000).

“Meaningful learning tasks need to challenge ever student in some way. It is crucial that no student be able to coast to success time after time; this experience can create the belief that you are smart only if you can succeed without effort.” -Carol Dweck

It is also critical to provide support and feedback during the challenging task (Cimpian, Arce, Markman, & Dweck, 2007). This prevents frustration and fear of failure when the goal seems out of reach or when a particularly challenging task arises.

Simple ways to build productive struggle into your math classroom

Giving students who struggle with word problems a chance to struggle with challenging word problems is critical to building confidence and skills. However, this challenge must be reasonable, or the learner’s self-esteem will falter, and students need support and regular feedback to achieve their potential.

Here are a few simple things to try:

• Select problems that are just at the edge of students’ Zone of Proximal Development.
• Scaffold or model with more challenging problems to support risk-taking.
• Give regular feedback & support – go over the work and discuss daily.

Finding #3: Novelty & variation are keys to engagement.

When it comes to standardized testing (and life in general), problems that arise aren’t labeled with the skills and strategies required to solve them.

This makes it important to provide mixed practice opportunities so students are focused on asking themselves questions about what the problem is asking and what they are trying to find.

This type of variation not only supports a deeper level of engagement, it also supports the metacognitive strategies needed to analyze and develop a strategy to solve (Rohrer & Taylor, 2014).

The benefits of novelty in learning

A 2013 study also supports the importance of novelty in supporting reinforcement learning (aka review). The findings suggested that when task variation was provided for an already familiar skill, it offered the following benefits:

• reduced errors due to lack of focus
• helped learners maintain attention to task
• motivated and engaged student

Using variety to build connections & deepen understanding

In addition, by providing variations in practice, we can also help learners understand the skills and strategies they are using on a deeper level.  

When students who struggle with word problems are forced to apply their toolbox of strategies to novel problem formats, they begin to analyze and observe patterns in how problems are structured and the meaning they bring.

This requires much more engagement than being handed a sheet full of multiplication story problems, where students can pull the numbers and compute with little focus on understanding.

Designing word problems that incorporate variety & novelty

Don’t be afraid to shake things up!

Giving students practice opportunities with different skills or problem formats mixed in is a great way to boost engagement and develop meta-cognitive skills.

Here are a few tips for trying it out in the classroom:

• Change it up! Word problem practice doesn’t have to match the day’s math lesson.
• Give opportunities to practice the same skill or strategy in via different formats.
• Adjust the wording and/or topic in word problems to help students generalize skills.

Finding #4: Interest and emotion increase retention and skill development.

Attention and emotion are huge for learning. We’ve all seen it in our classroom.

Those magical lessons that hook learners are the ones that stick with them for years to come, but what does the research say?

The Science Behind Emotion & Learning

Neuroscientists have shown that emotions create connections among different sections of the brain (Immordino-Yang, 2016) . This supports long-term retrieval of the skills taught and a deeper connection to the learning.

This means if you can connect problem-solving with a scenario or a feeling, your students will be more likely to internalize the skills being practiced. Whether this is by “wowing” them with a little-known fact or solving real-world problems, the emotional trigger can be huge for learning.

What about incorporating student interests?

As for student interests, a long line of research supports the benefits of using these to increase educational outcomes and student motivation, including for students who struggle with word problems (Chen, 2001; Chen & Ennis, 2004; Solomon, 1996).

Connecting classwork with student interests has increased students’ intentions to participate in future learning endeavors (Chen, 2001).

And interests don’t just mean that love of Pokemon!

It means allowing social butterflies to work collaboratively. Providing students with opportunities to manipulate real objects or create models. Allowing kids to be authentic while digging in and developing the skills they need to master their learning objectives.

What this looks like in a math class

Evoke emotion and use student interests to engage the brain in deep, long-lasting learning whenever possible.

This will help with today’s learning and promote long-term engagement, even when later practice might not be as interesting for students who struggle with word problems.

Here’s how to start applying this research today:

• Find word problems that match student interests.
• Connect real-life situations and emotions to story problem practice.
• Consider a weekly theme to connect practice throughout the week.

Finding #5: Student autonomy builds confidence & independence.

Autonomy is a student’s ability to be in control of their learning. In other words, it is their ability to take ownership over the learning process and how they demonstrate mastery.

Why students need to control their learning

Research shows that providing students a sense of control and supporting their choices is way to help engage learners and build independent thinking. It also increased intrinsic motivation (Reeve, Nix, & Hamm, 2003).

However, this doesn’t mean we just let kids learn independently. Clearly, some things require repeated guidance and modeling. Finding small ways that students can take control of the learning process is much better in these instances.

We know that giving at least partial autonomy has been linked to numerous positive student learning outcomes (Wielenga-Meijer, Taris, Widboldus, & Kompier, 2011).

But how can we foster this independence and autonomy, especially with those students who struggle to self-regulate behavior?

Fostering independence in students who struggle to stay on task

Well, the research says several conditions support building toward independence.

The first (and often neglected) is to explain unappealing choices and why they are one of the options.

When it comes to word problems, this might include explaining the rationale behind one of the strategies that appears to be a lot more work than the others.

It is also important to acknowledge students’ negative feelings about a task or their ability to complete it. While we want them to be able to build independence, we don’t want them to drown in overwhelm.

By providing emotional support, we can help determine whether a student is stuck with the learning or with the emotions from the cognitive challenge.

Finally, giving choices is recommended. Identifying choices you and your students who struggle with word problems can live with is an important step.

Whether this is working in partners, trying an alternative method, or skipping a problem and coming back, students need to feel like they have some ownership over the challenge they are working through.

By building in opportunities for autonomy, and choice, teachers help students build a sense of self-efficacy and confidence in their ability to be successful learners across various contexts (McCombs, 2002,2006).

We know this leads to numerous positive outcomes and has even been linked to drop-out prevention (Christenson & Thurlow, 2004).

Fostering autonomy in your classroom

You’re not going to be able to hold their hands forever.

Giving opportunities to work through challenges independently and to feel ownership for their choices will help build both confidence and skills.

Here’s how to get started letting go:

• Give students time to tackle the problem independently (or in partners).
• Don’t get hyper-focused on a single method to solve – give opportunities to share & learn together.
• Provide appropriate support (where needed) to build autonomy for all learners – like reading the problem orally.

Finding #6: Students need to be taught how to fail & recover from it.

Despite Ericsson’s findings discussed early on in this post, talent does matter, and it is important to teach students to recover from failure because those are the moments when they learn the most.

A 2014 study by Brooke Macnamara analyzed 88 studies to determine how talent factored into deliberate practice.

Her findings show what we (as teachers) already know, students may require different amounts of practice to reach the same skill level…but how do we keep those struggling students from keeping up?

Growth mindset research gives us insight into ways to support students who struggle with word problems, encourage all students in math problem-solving, and harness the power of failure through “yet.”

You might not be able to do something yet, but if you keep trying, you will. This opens the door for multiple practice opportunities where students learn from each other.

And what about the advanced students?

Many of these students have not experienced failure, but they may have met their match when it comes to complex word problems.

To support these students, who may be experiencing their first true challenge, we need to have high standards and provide constructive, supportive feedback on how to grow.

Then we need to give them space to try again.

There is great power in allowing students to revise and try again, but our grading system often discourages being comfortable with failure.

Building the confidence to fail in your classroom

Many students feel the pressure always to have the right answer. Allowing students to fail safely means you can help them learn from these failures so they don’t make the same mistake twice.

Here’s how you can safely foster growth and build math problem solving skills through failure in your classroom:

• Build in time to analyze errors & reflect.
• Reward effort & growth as much as, if not more than, accuracy.
• ​At least initially, skip the grading so students aren’t afraid to be wrong.

Getting started with brain-based problem solving

The brain research is clear.

Spending 45 minutes focused on a sheet of word problems following the same format isn’t the answer.

By implementing this research, you can save yourself time and the frustration from a disengaged class.

Based on this research, I’ve created Daily Problem Solving bundles to save you time and build math problem-solving skills. You can get each month separately or buy the full-year bundle at a major discount.

Currently, I offer these bundles for several grade levels, including:

Try Daily Problem Solving with your Learners

Of course, you do! Start working to build step-by-step math problem-solving skills today by clicking the button below to sign up for a free set of Daily Problem Solving.

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Study Smarter

17 maths problem solving strategies to boost your learning.

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

Friday, 3rd June 2022

• What are strategies?

Understand the problem

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

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

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

What are maths problem-solving strategies?

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

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

Polya's 4-step process for solving problems

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

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

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

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

Do I understand all the words used in the problem?

What am I asked to find or show?

Can I restate the problem in my own words?

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

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

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

What information is extra or irrelevant?

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

1. Read the problem aloud

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

2. Summarise the information

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

3. Create a picture or diagram

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

4. Act it out

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

5. Use keyword analysis

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

How many more?

How many left?

Equal parts

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

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

6. Look for a pattern

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

7. Guess and check

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

8. Working backwards

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

9. Use a formula

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

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

10. Eliminate the possibilities

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

11. Use direct reasoning

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

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

12. Solve a simpler problem

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

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

13. Solve an equation

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

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

14. Be systematic

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

15. Check your work

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

16. Be flexible

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

17. Don't give up

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

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

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

What was the most challenging part of the problem?

Was one method more effective than another?

Would you do something differently next time?

What have you learned from this experience?

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

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

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

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

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

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

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Fluency, Reasoning and Problem Solving: What This Looks Like In Every Math Lesson

Neil Almond

Fluency, reasoning and problem solving are central strands of mathematical competency, as recognized by the National Council of Teachers of Mathematics (NCTM) and the National Research Council’s report ‘Adding It Up’.

They are key components to the Standards of Mathematical Practice, standards that are interwoven into every mathematics lesson. Here we look at how these three approaches or elements of math can be interwoven in a child’s math education through elementary and middle school.

We look at what fluency, reasoning and problem solving are, how to teach them, and how to know how a child is progressing in each – as well as what to do when they’re not, and what to avoid.

The hope is that this blog will help elementary and middle school teachers think carefully about their practice and the pedagogical choices they make around the teaching of what the common core refers to as ‘mathematical practices’, and reasoning and problem solving in particular.

Before we can think about what this would look like in Common Core math examples and other state-specific math frameworks, we need to understand the background to these terms.

What is fluency in math?

What is reasoning in math, what is problem solving in math, mathematical problem solving is a learned skill, performance vs learning: what to avoid when teaching fluency, reasoning, and problem solving.

• What IS ‘performance vs learning’?
• Teaching to “cover the curriculum” hinders development of strong problem solving skills.
• Fluency and reasoning – Best practice in a lesson, a unit, and a semester

Best practice for problem solving in a lesson, a unit, and a semester 

Fluency, reasoning and problem solving should not be taught by rote .

The Ultimate Guide to Problem Solving Techniques

Develop problem solving skills in the classroom with this free, downloadable worksheet

Fluency in math is a fairly broad concept. The basics of mathematical fluency – as defined by the Common Core State Standards for math – involve knowing key mathematical skills and being able to carry them out flexibly, accurately and efficiently.

But true fluency in math (at least up to middle school) means being able to apply the same skill to multiple contexts, and being able to choose the most appropriate method for a particular task.

Fluency in math lessons means we teach the content using a range of representations, to ensure that all students understand and have sufficient time to practice what is taught.

Read more: How the best schools develop math fluency

Reasoning in math is the process of applying logical thinking to a situation to derive the correct math strategy for problem solving  for a question, and using this method to develop and describe a solution.

Put more simply, mathematical reasoning is the bridge between fluency and problem solving. It allows students to use the former to accurately carry out the latter.

Read more: Developing math reasoning: the mathematical skills required and how to teach them .

It’s sometimes easier to start off with what problem solving is not. Problem solving is not necessarily just about answering word problems in math. If a child already has a readily available method to solve this sort of problem, problem solving has not occurred. Problem solving in math is finding a way to apply knowledge and skills you have to answer unfamiliar types of problems.

Read more: Math problem solving: strategies and resources for primary school teachers .

We are all problem solvers

First off, problem solving should not be seen as something that some students can do and some cannot. Every single person is born with an innate level of problem-solving ability.

Early on as a species on this planet, we solved problems like recognizing faces we know, protecting ourselves against other species, and as babies the problem of getting food (by crying relentlessly until we were fed).

All these scenarios are a form of what the evolutionary psychologist David Geary (1995) calls biologically primary knowledge. We have been solving these problems for millennia and they are so ingrained in our DNA that we learn them without any specific instruction.

Why then, if we have this innate ability, does actually teaching problem solving seem so hard?

As you might have guessed, the domain of mathematics is far from innate. Math doesn’t just happen to us; we need to learn it. It needs to be passed down from experts that have the knowledge to novices who do not.

This is what Geary calls biologically secondary knowledge. Solving problems (within the domain of math) is a mixture of both primary and secondary knowledge.

The issue is that problem solving in domains that are classified as biologically secondary knowledge (like math) can only be improved by practicing elements of that domain.

So there is no generic problem-solving skill that can be taught in isolation and transferred to other areas.

This will have important ramifications for pedagogical choices, which I will go into more detail about later on in this blog.

The educationalist Dylan Wiliam had this to say on the matter: ‘for…problem solving, the idea that students can learn these skills in one context and apply them in another is essentially wrong.’ (Wiliam, 2018) So what is the best method of teaching problem solving to elementary and middle school math students?

The answer is that we teach them plenty of domain specific biological secondary knowledge – in this case, math. Our ability to successfully problem solve requires us to have a deep understanding of content and fluency of facts and mathematical procedures.

Here is what cognitive psychologist Daniel Willingham (2010) has to say:

‘Data from the last thirty years leads to a conclusion that is not scientifically challengeable: thinking well requires knowing facts, and that’s true not simply because you need something to think about.

The very processes that teachers care about most—critical thinking processes such as reasoning and problem solving—are intimately intertwined with factual knowledge that is stored in long-term memory (not just found in the environment).’

Colin Foster (2019), a reader in Mathematics Education in the Mathematics Education Center at Loughborough University, UK, says, ‘I think of fluency and mathematical reasoning, not as ends in themselves, but as means to support students in the most important goal of all: solving problems.’

In that paper he produces this pyramid:

This is important for two reasons:

1)    It splits up reasoning skills and problem solving into two different entities

2)    It demonstrates that fluency is not something to be rushed through to get to the ‘problem solving’ stage but is rather the foundation of problem solving.

In my own work I adapt this model and turn it into a cone shape, as education seems to have a problem with pyramids and gross misinterpretation of them (think Bloom’s taxonomy).

Notice how we need plenty of fluency of facts, concepts, procedures and mathematical language.

Having this fluency will help with improving logical reasoning skills, which will then lend themselves to solving mathematical problems – but only if it is truly learnt and there is systematic retrieval of this information carefully planned across the curriculum.

I mean to make no sweeping generalization here; this was my experience both at university when training and from working in schools.

At some point, schools become obsessed with the ridiculous notion of moving students through content at an accelerated rate. I have heard it used in all manner of educational contexts while training and being a teacher. ‘You will need to show ‘accelerated progress in math’ in this lesson,’ ‘School officials will be looking for ‘accelerated progress’ etc.

I have no doubt that all of this came from a good place and from those wanting the best possible outcomes – but it is misguided.

I remember being told that we needed to get students onto the problem solving questions as soon as possible to demonstrate this mystical ‘accelerated progress’.

This makes sense; you have a group of students and you have taken them from not knowing something to working out pretty sophisticated 2-step or multi-step word problems within an hour. How is that not ‘accelerated progress?’

This was a frequent feature of my lessons up until last academic year: teach a mathematical procedure; get the students to do about 10 of them in their books; mark these and if the majority were correct, model some reasoning/problem solving questions from the same content as the fluency content; give the students some reasoning and word problem questions and that was it.

I wondered if I was the only one who had been taught this while at university so I did a quick poll on Twitter and found that was not the case.

I know these numbers won’t be big enough for a representative sample but it still shows that others are familiar with this approach.

The issue with the lesson framework I mentioned above is that it does not take into account ‘performance vs learning.’

What IS ‘performance vs learning’?

The premise is that performance in a lesson is not a good proxy for learning.

Yes, those students were performing well after I had modeled a mathematical procedure for them, and managed to get questions correct.

But if problem solving depends on a deep knowledge of mathematics, this approach to lesson structure is going to be very ineffective.

As mentioned earlier, the reasoning and problem solving questions were based on the same math content as the fluency exercises, making it more likely that students would solve problems correctly whether they fully understood them or not.

Chances are that all they’d need to do is find the numbers in the questions and use the same method they used in the fluency section to get their answers (a process referred to as “number plucking”) – not exactly high level problem solving skills.

Teaching to “cover the curriculum” hinders development of strong problem solving skills.

This is one of my worries with ‘math mastery schemes’ that block content so that, in some circumstances, it is not looked at again until the following year (and with new objectives).

The pressure for teachers to ‘get through the curriculum’ results in many opportunities to revisit content being missed in the classroom.

Students are unintentionally forced to skip ahead in the fluency, reasoning, problem solving chain without proper consolidation of the earlier processes.

As David Didau (2019) puts it, ‘When novices face a problem for which they do not have a conveniently stored solution, they have to rely on the costlier means-end analysis.

This is likely to lead to cognitive overload because it involves trying to work through and hold in mind multiple possible solutions.

It’s a bit like trying to juggle five objects at once without previous practice. Solving problems is an inefficient way to get better at problem solving.’

Fluency and reasoning – Best practice in a lesson, a unit, and a semester

By now I hope you have realized that when it comes to problem solving, fluency is king. As such we should look to mastery math based teaching to ensure that the fluency that students need is there.

The answer to what fluency looks like will obviously depend on many factors, including the content being taught and the grade you find yourself teaching.

But we should not consider rushing them on to problem solving or logical reasoning in the early stages of this new content as it has not been learnt, only performed.

I would say that in the early stages of learning, content that requires the end goal of being fluent should take up the majority of lesson time – approximately 60%. The rest of the time should be spent rehearsing and retrieving other knowledge that is at risk of being forgotten about.

This blog on mental math strategies students should learn at each grade level is a good place to start when thinking about the core aspects of fluency that students should achieve.

Little and often is a good mantra when we think about fluency, particularly when revisiting the key mathematical skills of number bond fluency or multiplication fluency. So when it comes to what fluency could look like throughout the day, consider all the opportunities to get students practicing.

They could chant multiplication facts when transitioning. If a lesson in another subject has finished earlier than expected, use that time to quiz students on number bonds. Have fluency exercises as part of the morning work.

Read more: How to teach multiplication for instant recall

What about best practice over a longer period?

Thinking about what fluency could look like across a unit of work would again depend on the unit itself.

Look at this unit below from a popular scheme of work.

They recommend 20 days to cover 9 objectives. One of these specifically mentions problem solving so I will forget about that one at the moment – so that gives 8 objectives.

I would recommend that the fluency of this unit look something like this:

This type of structure is heavily borrowed from Mark McCourt’s phased learning idea from his book ‘Teaching for Mastery.’

This should not be seen as something set in stone; it would greatly depend on the needs of the class in front of you. But it gives an idea of what fluency could look like across a unit of lessons – though not necessarily all math lessons.

When we think about a semester, we can draw on similar ideas to the one above except that your lessons could also pull on content from previous units from that semester.

So lesson one may focus 60% on the new unit and 40% on what was learnt in the previous unit.

The structure could then follow a similar pattern to the one above.

When an adult first learns something new, we cannot solve a problem with it straight away. We need to become familiar with the idea and practice before we can make connections, reason and problem solve with it.

The same is true for students. Indeed, it could take up to two years ‘between the mathematics a student can use in imitative exercises and that they have sufficiently absorbed and connected to use autonomously in non-routine problem solving.’ (Burkhardt, 2017).

Practice with facts that are secure

So when we plan for reasoning and problem solving, we need to be looking at content from 2 years ago to base these questions on.

You could get students in 3rd grade to solve complicated place value problems with the numbers they should know from 1st or 2nd grade. This would lessen the cognitive load , freeing up valuable working memory so they can actually focus on solving the problems using content they are familiar with.

Increase complexity gradually

Once they practice solving these types of problems, they can draw on this knowledge later when solving problems with more difficult numbers.

This is what Mark McCourt calls the ‘Behave’ phase. In his book he writes:

‘Many teachers find it an uncomfortable – perhaps even illogical – process to plan the ‘Behave’ phase as one that relates to much earlier learning rather than the new idea, but it is crucial to do so if we want to bring about optimal gains in learning, understanding and long term recall.’  (Mark McCourt, 2019)

This just shows the fallacy of ‘accelerated progress’; in the space of 20 minutes some teachers are taught to move students from fluency through to non-routine problem solving, or we are somehow not catering to the needs of the child.

When considering what problem solving lessons could look like, here’s an example structure based on the objectives above.

It is important to reiterate that this is not something that should be set in stone. Key to getting the most out of this teaching for mastery approach is ensuring your students (across abilities) are interested and engaged in their work.

Depending on the previous attainment and abilities of the children in your class, you may find that a few have come across some of the mathematical ideas you have been teaching, and so they are able to problem solve effectively with these ideas.

Equally likely is encountering students on the opposite side of the spectrum, who may not have fully grasped the concept of place value and will need to go further back than 2 years and solve even simpler problems.

In order to have the greatest impact on class performance, you will have to account for these varying experiences in your lessons.

Read more: 

• Math Mastery Toolkit : A Practical Guide To Mastery Teaching And Learning
• Problem Solving and Reasoning Questions and Answers
• Get to Grips with Math Problem Solving For Elementary Students
• Mixed Ability Teaching for Mastery: Classroom How To
• 21 Math Challenges To Really Stretch Your More Able Students
• Why You Should Be Incorporating Stem Sentences Into Your Elementary Math Teaching

Do you have students who need extra support in math? Give your students more opportunities to consolidate learning and practice skills through personalized math tutoring with their own dedicated online math tutor. Each student receives differentiated instruction designed to close their individual learning gaps, and scaffolded learning ensures every student learns at the right pace. Lessons are aligned with your state’s standards and assessments, plus you’ll receive regular reports every step of the way. Personalized one-on-one math tutoring programs are available for: – 2nd grade tutoring – 3rd grade tutoring – 4th grade tutoring – 5th grade tutoring – 6th grade tutoring – 7th grade tutoring – 8th grade tutoring Why not learn more about how it works ?

The content in this article was originally written by primary school lead teacher Neil Almond and has since been revised and adapted for US schools by elementary math teacher Jaclyn Wassell.

Related articles

20 Effective Math Strategies To Approach Problem-Solving 

Why Student Centered Learning Is Important: A Guide For Educators

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

Differentiated Instruction: 9 Differentiated Curriculum And Instruction Strategies For Teachers 

Ultimate Guide to Metacognition [FREE]

Looking for a summary on metacognition in relation to math teaching and learning?

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Problem Solving Skills: Meaning, Examples & Techniques

Table of Contents

26 January 2021

Reading Time: 2 minutes

Do your children have trouble solving their Maths homework?

Or, do they struggle to maintain friendships at school?

If your answer is ‘Yes,’ the issue might be related to your child’s problem-solving abilities. Whether your child often forgets his/her lunch at school or is lagging in his/her class, good problem-solving skills can be a major tool to help them manage their lives better.

Children need to learn to solve problems on their own. Whether it is about dealing with academic difficulties or peer issues when children are equipped with necessary problem-solving skills they gain confidence and learn to make healthy decisions for themselves. So let us look at what is problem-solving, its benefits, and how to encourage your child to inculcate problem-solving abilities

Problem-solving skills can be defined as the ability to identify a problem, determine its cause, and figure out all possible solutions to solve the problem.

• Trigonometric Problems

What is problem-solving, then? Problem-solving is the ability to use appropriate methods to tackle unexpected challenges in an organized manner. The ability to solve problems is considered a soft skill, meaning that it’s more of a personality trait than a skill you’ve learned at school, on-the-job, or through technical training. While your natural ability to tackle problems and solve them is something you were born with or began to hone early on, it doesn’t mean that you can’t work on it. This is a skill that can be cultivated and nurtured so you can become better at dealing with problems over time.

Problem Solving Skills: Meaning, Examples & Techniques are mentioned below in the Downloadable PDF. 

Benefits of learning problem-solving skills  

Promotes creative thinking and thinking outside the box.

Improves decision-making abilities.

Builds solid communication skills.

Develop the ability to learn from mistakes and avoid the repetition of mistakes.

Problem Solving as an ability is a life skill desired by everyone, as it is essential to manage our day-to-day lives. Whether you are at home, school, or work, life throws us curve balls at every single step of the way. And how do we resolve those? You guessed it right – Problem Solving.

Strengthening and nurturing problem-solving skills helps children cope with challenges and obstacles as they come. They can face and resolve a wide variety of problems efficiently and effectively without having a breakdown. Nurturing good problem-solving skills develop your child’s independence, allowing them to grow into confident, responsible adults. 

Children enjoy experimenting with a wide variety of situations as they develop their problem-solving skills through trial and error. A child’s action of sprinkling and pouring sand on their hands while playing in the ground, then finally mixing it all to eliminate the stickiness shows how fast their little minds work.

Sometimes children become frustrated when an idea doesn't work according to their expectations, they may even walk away from their project. They often become focused on one particular solution, which may or may not work.

However, they can be encouraged to try other methods of problem-solving when given support by an adult. The adult may give hints or ask questions in ways that help the kids to formulate their solutions. 

Encouraging Problem-Solving Skills in Kids

Practice problem solving through games.

Exposing kids to various riddles, mysteries, and treasure hunts, puzzles, and games not only enhances their critical thinking but is also an excellent bonding experience to create a lifetime of memories.

Create a safe environment for brainstorming

Welcome, all the ideas your child brings up to you. Children learn how to work together to solve a problem collectively when given the freedom and flexibility to come up with their solutions. This bout of encouragement instills in them the confidence to face obstacles bravely.

Invite children to expand their Learning capabilities

 Whenever children experiment with an idea or problem, they test out their solutions in different settings. They apply their teachings to new situations and effectively receive and communicate ideas. They learn the ability to think abstractly and can learn to tackle any obstacle whether it is finding solutions to a math problem or navigating social interactions.

Problem-solving is the act of finding answers and solutions to complicated problems. 

Developing problem-solving skills from an early age helps kids to navigate their life problems, whether academic or social more effectively and avoid mental and emotional turmoil.

Children learn to develop a future-oriented approach and view problems as challenges that can be easily overcome by exploring solutions. 

About Cuemath

Cuemath, a student-friendly mathematics and coding platform, conducts regular  Online Classes  for academics and skill-development, and their Mental Math App, on both  iOS  and  Android , is a one-stop solution for kids to develop multiple skills. Understand the Cuemath Fee structure and sign up for a free trial.

Frequently Asked Questions (FAQs)

How do you teach problem-solving skills.

Model a useful problem-solving method. Problem solving can be difficult and sometimes tedious. ... 1. Teach within a specific context. ... 2. Help students understand the problem. ... 3. Take enough time. ... 4. Ask questions and make suggestions. ... 5. Link errors to misconceptions.

What makes a good problem solver?

Excellent problem solvers build networks and know how to collaborate with other people and teams. They are skilled in bringing people together and sharing knowledge and information. A key skill for great problem solvers is that they are trusted by others.

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How to Improve Problem Solving Skills

Last Updated: July 24, 2024 Fact Checked

This article was co-authored by Erin Conlon, PCC, JD . Erin Conlon is an Executive Life Coach, the Founder of Erin Conlon Coaching, and the host of the podcast "This is Not Advice." She specializes in aiding leaders and executives to thrive in their career and personal lives. In addition to her private coaching practice, she teaches and trains coaches and develops and revises training materials to be more diverse, equitable, and inclusive. She holds a BA in Communications and History and a JD from The University of Michigan. Erin is a Professional Certified Coach with The International Coaching Federation. There are 11 references cited in this article, which can be found at the bottom of the page. This article has been fact-checked, ensuring the accuracy of any cited facts and confirming the authority of its sources. This article has been viewed 239,973 times.

The ability to solve problems applies to more than just mathematics homework. Analytical thinking and problem-solving skills are a part of many jobs, ranging from accounting and computer programming to detective work and even creative occupations like art, acting, and writing. While individual problems vary, there are certain general approaches to problem-solving like the one first proposed by mathematician George Polya in 1945. By following his principles of understanding the problem, devising a plan, carrying out the plan, and looking back, you can improve your problem-solving and tackle any issue systematically.

Define the problem clearly.

• Try to formulate questions. Say that as a student you have very little money and want to find an effective solution. What is at issue? Is it one of income – are you not making enough money? Is it one of over-spending? Or perhaps you have run into unexpected expenses or your financial situation has changed?

State your objective.

• Say that your problem is still money. What is your goal? Perhaps you never have enough to go out on the weekend and have fun at the movies or a club. You decide that your goal is to have more spending cash. Good! With a clear goal, you have better defined the problem.

Gather information systematically.

• To solve your money shortage, for example, you would want to get as detailed a picture of your financial situation as possible. Collect data through your latest bank statements and to talk to a bank teller. Track your earnings and spending habits in a notebook, and then create a spreadsheet or chart to show your income alongside your expenditures.

Analyze information.

• Say you have now collected all your bank statements. Look at them. When, how, and from where is your money coming? Where, when, and how are you spending it? What is the overall pattern of your finances? Do you have a net surplus or deficit? Are there any unexplained items?

Generate possible solutions.

• Your problem is a lack of money. Your goal is to have more spending cash. What are your options? Without evaluating them, come up with possible options. Perhaps you can acquire more money by getting a part-time job or by taking out a student loan. On the other hand, you might try to save by cutting your spending or by lowering other costs.
• Divide and conquer. Break the problem into smaller problems and brainstorm solutions for them separately, one by one.
• Use analogies and similarities. Try to find a resemblance with a previously solved or common problem. If you can find commonalities between your situation and one you've dealt with before, you may be able to adapt some of the solutions for use now.

Evaluate the solutions and choose.

• How can you raise money? Look at expenditures – you aren’t spending much outside of basic needs like tuition, food, and housing. Can you cut costs in other ways like finding a roommate to split rent? Can you afford to take a student loan just to have fun on the weekend? Can you spare time from your studies to work part-time?
• Each solution will produce its own set of circumstances that need evaluation. Run projections. Your money problem will require you to draw up budgets. But it will also take personal consideration. For example, can you cut back on basic things like food or housing? Are you willing to prioritize money over school or to take on debt?

Implement a solution.

• You decide to cut costs, because you were unwilling to take on debt, to divert time away from school, or to live with a roommate. You draw up a detailed budget, cutting a few dollars here and there, and commit to a month-long trial.

Review and evaluate the outcome.

• The results of your trial are mixed. On one hand, you have saved enough during the month for fun weekend activities. But there are new problems. You find that you must choose between spending cash and buying basics like food. You also need a new pair of shoes but can’t afford it, according to your budget. You may need to a different solution.

Adjust if necessary.

• After a month, you decide to abandon your first budget and to look for part-time work. You find a work-study job on campus. Making a new budget, you now have extra money without taking too much time away from your studies. You may have an effective solution.

Do regular mental exercises.

• Word games work great. In a game like “Split Words,” for example, you have to match word fragments to form words under a given theme like “philosophy.” In the game, “Tower of Babel,” you will need to memorize and then match words in a foreign language to the proper picture.
• Mathematical games will also put your problem solving to the test. Whether it be number or word problems, you will have to activate the parts of your brain that analyze information. For instance: “James is half as old now as he will be when he is 60 years older than he was six years before he was half as old as he is now. How old will James be when his age is twice what it was 10 years after he was half his current age?”

Play video games.

• Play something that will force you to think strategically or analytically. Try a puzzle game like Tetris. Or, perhaps you would rather prefer a role-playing or strategy game. In that case, something like “Civilization” or “Sim-City” might suit you better.

Take up a hobby.

• Web design, software programming, jigsaw puzzles, Sudoku, and chess are also hobbies that will force you to think strategically and systematically. Any of these will help you improve your overall problem solving.

Expert Q&A

You Might Also Like

• ↑ https://www.healthywa.wa.gov.au/Articles/N_R/Problem-solving
• ↑ https://asq.org/quality-resources/problem-solving
• ↑ https://ctb.ku.edu/en/table-of-contents/evaluate/evaluate-community-interventions/collect-analyze-data/main
• ↑ https://www.mindtools.com/pages/article/newCT_96.htm
• ↑ https://www.skillsyouneed.com/ips/problem-solving.html
• ↑ Erin Conlon, PCC, JD. Executive Life Coach. Expert Interview. 31 August 2021.
• ↑ https://www.ncbi.nlm.nih.gov/pmc/articles/PMC5930973/
• ↑ https://www.theguardian.com/lifeandstyle/2018/oct/13/mental-exercises-to-keep-your-brain-sharp
• ↑ https://www.apa.org/monitor/2014/02/video-game
• ↑ https://www.nature.com/articles/d41586-018-05449-7

About This Article

To improve your problem-solving skills, start by clearly defining the problem and your objective or goal. Next, gather as much information as you can about the problem and organize the data by rewording, condensing, or summarizing it. Then, analyze the information you've gathered, looking for important links, patterns, and relationships in the data. Finally, brainstorm possible solutions, evaluate the solutions, and choose one to implement. For tips on implementing solutions successfully, read on! Did this summary help you? Yes No

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Problem solving

At its roots, problem solving is exactly what it sounds like, the process of solving problems. However, problem solving methods permeate the studies of mathematics , science , and technology . The human processes involved in problem solving are often studied by cognitive scientists .

What is Problem Solving?

by Richard Rusczyk

I was invited to the Math Olympiad Summer Program (MOP) in the 10th grade. I went to MOP certain that I must really be good at math… In my five weeks at MOP, I encountered over sixty problems on various tests. I didn’t solve a single one. That’s right – I was 0-for-60+. I came away no longer confident that I was good at math. I assumed that most of the other kids did better at MOP because they knew more tricks than I did. My formula sheets were pretty thorough, but perhaps they were missing something. By the end of MOP, I had learned a somewhat unsettling truth. The others knew fewer tricks than I did, not more. They didn’t even have formula sheets!

At another contest later that summer, a younger student, Alex, from another school asked me for my formula sheets. In my local and state circles, students’ formula sheets were the source of knowledge, the source of power that fueled the top students and the top schools. They were studied, memorized, revered. But most of all, they were not shared. But when Alex asked for my formula sheets I remembered my experience at MOP and I realized that formula sheets are not really math. Memorizing formulas is no more mathematics than memorizing dates is history or memorizing spelling words is literature. I gave him the formula sheets. (Alex must later have learned also that the formula sheets were fool’s gold – he became a Rhodes scholar.)

The difference between MOP and many of these state and local contests I participated in was the difference between problem solving and what many people call mathematics. For these people, math is a series of tricks to use on a series of specific problems. Trick A is for Problem A, Trick B for Problem B, and so on. In this vein, school can become a routine of ‘learn tricks for a week – use tricks on a test – forget most tricks quickly.’ The tricks get forgotten quickly primarily because there are so many of them, and also because the students don’t see how these ‘tricks’ are just extensions of a few basic principles.

I had painfully learned at MOP that true mathematics is not a process of memorizing formulas and applying them to problems tailor-made for those formulas. Instead, the successful mathematician possesses fewer tools, but knows how to apply them to a much broader range of problems. We use the term “problem solving” to distinguish this approach to mathematics from the ‘memorize-use-forget’ approach.

After MOP I relearned math throughout high school. I was unaware that I was learning much more. When I got to Princeton I enrolled in organic chemistry. There were over 200 students in the course, and we quickly separated into two groups. One group understood that all we would be taught could largely be derived from a very small number of basic principles. We loved the class – it was a year long exploration of where these fundamental concepts could take us. The other, much larger, group saw each new destination not as the result of a path from the building blocks, but as yet another place whose coordinates had to be memorized if ever they were to visit again. Almost to a student, the difference between those in the happy group and those in the struggling group was how they learned mathematics. The class seemingly involved no math at all, but those who took a memorization approach to math were doomed to do it again in chemistry. The skills the problem solvers developed in math transferred, and these students flourished.

We use math to teach problem solving because it is the most fundamental logical discipline. Not only is it the foundation upon which sciences are built, it is the clearest way to learn and understand how to develop a rigorous logical argument. There are no loopholes, there are no half-truths. The language of mathematics is precise, as is ‘right’ and ‘wrong’ (or ‘proven’ and ‘unproven’). Success and failure are immediate and indisputable; there isn’t room for subjectivity. This is not to say that those who cannot do math cannot solve problems. There are many paths to strong problem solving skills. Mathematics is the shortest.

Problem solving is crucial in mathematics education because it transcends mathematics. By developing problem solving skills, we learn not only how to tackle math problems, but also how to logically work our way through any problems we may face. The memorizer can only solve problems he has encountered already, but the problem solver can solve problems she’s never seen before. The problem solver is flexible; she can diversify. Above all, she can create.

There are four main steps in successful problem solving — not just in math or even computer science, but also in everyday life (even figuring out how to beat your rival in a video game).

• Identify the Problem : The very first step in problem solving is to identify the problem. Problem solving is harder if one doesn’t know what “problem” to solve! This is often seen in engineering and math by reading the problem statement.
• Make a Game Plan : The next step is to have a plan on what to do. In other words, just rushing in won’t usually go through. Know what skills, properties, and information would be useful in aiding your decisions. Don’t forget about backups in case Plan A fails.
• Solve the Problem : As Rusczyk often says, sometimes, you just have to do something. A problem can’t be solved by just staring at it!
• Reflect on Actions : Skilled problem solvers look back at their work. In math, it usually means figuring out if an answer is reasonable. In engineering (and other areas), problem solvers can look back and see what went right and what went wrong, so they can tackle a similar problem better next time.

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Mathematics for Teaching

This site is NOT about making mathematics easy because it isn't. It is about making it make sense because it does.

Levels of Problem Solving Skills

Level 1 – Recognition

Students at this level have the ability to recognize characteristics of a previously solved problem in a new situation and believe that one can do again what one did before. Solvers operating at this level would not be able to anticipate sources of difficulty and would be surprised by complications that might occur as they attempted their solution. A student operating at this level would not be able to mentally run-through a solution method in order to confirm or reject its usefulness.

Level 2 – Re-presentation

Students at this level are able to run through a problem mentally and are able to anticipate potential sources of difficulty and promise. Solvers who operate at this level are more flexible in their thinking and are not only able to recognize similarities between problems, they are also able to notice the differences that might cause them difficulty if they tried to repeat a previously used method of solution. Such solvers could imagine using the methods and could even imagine some of the problems they might encounter but could not take the results as a given. At this level, the subject would be unable to think about potential methods of solution and the anticipated results of such activity.

Level 3 – Structural abstraction

Students at this level evaluates solution prospects based on mental run-throughs of potential methods as well as methods that have been used before. They are able to discern the characteristics that are necessary to solve the problem and are able to evaluate the merits of a solution method based on these characteristics. This level evidences considerable flexibility of thought.

Level 4 – Structural awareness

A solver operating at this level is able to anticipate the results of potential activity without having to complete a mental run-through of the solution activity. The problem structure created by the solver has become an object of reflection. The student is able to consider such structures as objects and is able to make judgments about them without resorting to physically or mentally representing methods of solution.

The levels of problem solving skills described above indicate that as solvers attain the higher levels they become increasingly flexible in their thinking. This framework is from the dissertation of Cifarelli but I read it from the paper  The roles of reification and reflective abstraction in the development of abstract thought: Transitions from arithmetic to algebra by Tracy Goodson-Espy. Educational Studies in Mathematics 36: 219–245, 1998.  © 1998 Kluwer Academic Publishers. Printed in the Netherlands.

You may also be interested on  Levels of understanding of function in equation form  based on my own research on understanding function.

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One thought on “ Levels of Problem Solving Skills ”

Great! Interesting trial to rank ‘The levels of appreciating, handling, problematics, then methods.’.

Too much weight on ‘methods’ at my sense (because – so often – the main difficulty is not there downstream, but upstream, at the analyse and comparative pace, in the capacity to analyse the datas, their linkage, the system as named in Physics.).

Eventhough this slight remark, THANK YOU, congratulations, and encouragements to continue!

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Manipulatives in Maths - A Classroom Guide for Teachers

Mathematical manipulatives are touted as essential tools for learning, but let's be honest—we've all experienced that moment of dread when we hand them out. Suddenly, your carefully planned lesson turns into chaos: One pupil starts building a fortress with the base ten blocks while another is hiding all the shiny counters.

Yet, despite these challenges, manipulatives play an important role in maths education. They bridge the gap between abstract concepts and tangible understanding, helping pupils grasp basic number sense. In fact, the National Curriculum emphasises their importance across all key stages, recognising that hands-on learning is vital for developing maths fluency, reasoning, and problem-solving skills.

So, how can we take advantage of these tools without losing control of the classroom? Let's explore the world of maths manipulatives—what they are, why they matter, and how to use them effectively in your primary school lessons.

What are manipulatives?

It can sound complicated, but manipulatives are simply hands-on tools that make abstract mathematical concepts concrete and visual . They're the building blocks, quite literally in some cases, that help pupils wrap their heads around tricky number ideas through good old-fashioned play, exploration, and modelling.

These learning aids come in all shapes and sizes, from the humble counter to the more elaborate Cuisenaire rods . Their key purpose? To give pupils something tangible to manipulate as they grapple with mathematical concepts. Whether it's using multilink cubes to understand place value or fraction circles to visualise parts of a whole, manipulatives help bridge the gap between 'maths on paper' and 'maths in real life'.

Common manipulatives you'll find in primary classrooms include:

Multilink cubes

Cuisenaire rods, base ten blocks, bead strings.

• Balance scales

Clock faces

Digit cards, hundred squares.

These tools align perfectly with the National Curriculum's aims of developing mathematical fluency, reasoning, and problem-solving skills. By allowing pupils to physically interact with mathematical ideas, manipulatives help build a strong foundation for more complex concepts down the line. They're not just toys or distractions—they're powerful learning tools that can transform how your pupils understand and engage with maths.

Why are they important?

Over the past two decades, research has consistently shown the positive impact of using manipulatives in the classroom. A 2013 report published in the Journal of Educational Psychology identified "statistically significant results" when teachers used manipulatives compared with when they only used abstract maths symbols. This highlights the role that manipulatives play in supporting conceptual understanding and facilitating the progression from concrete to abstract thinking.

Alignment with CPA approach

The NCETM agrees that physical manipulatives should play a central role in maths teaching. "Manipulatives are not just for young pupils, and also not just for those who can't understand something. They can always be of help to build or deepen understanding of a mathematical concept."

This approach aligns perfectly with the concrete-pictorial-abstract (CPA) progression. Once children are confident using manipulatives or 'concrete' resources, they can then move onto pictorial representations or the 'seeing' stage. Here, visual representations of concrete objects are used to model problems. This stage encourages children to make a mental connection between the physical object they just handled and the abstract pictures , diagrams or models that represent the objects from the maths problem.

Enhance problem solving

But manipulatives do more than just support understanding—they're powerful tools for enhancing problem-solving skills. By allowing pupils to physically manipulate and visualise mathematical concepts, they can more easily devise strategies to tackle complex problems. This hands-on approach often leads to those 'aha!' moments we all love to see in our classrooms.

Support engagement

Moreover, manipulatives play an important role in fostering engagement and motivation. Let's face it—maths can sometimes seem dry and abstract to young learners. But introduce some colourful counters or interlocking cubes, and suddenly you've got a room full of eager mathematicians. This increased engagement is key to developing a positive attitude towards maths, which in turn supports long-term learning.

This deep understanding allows pupils to move beyond mere memorisation of facts and procedures, towards true mathematical fluency—where they can apply their knowledge flexibly and efficiently across a range of contexts.

In essence, manipulatives are not just helpful additions to our maths teaching toolkit—they're essential components in building a comprehensive, engaging, and effective mathematics education.

Types of manipulatives in primary mathematics

In this section, we'll break it common types of manipulatives into bite-sized pieces, just like we do for our pupils.

Physical manipulatives: the classics

These are the tangible, grab-them-with-your-hands resources that have been the backbone of maths classrooms for years. They're the ones that inevitably end up stuck between classroom seats and occasionally in someone's shoe.

Below is a list of common physical manipulatives in the classroom:

Ideal for teaching place value, addition, and subtraction with regrouping.

Fraction tiles

Excellent for comparing fractions and understanding equivalence.

Great for exploring 2D shapes, symmetry, and area.

Images: Wikipedia.org

Versatile tools for counting, measuring, and understanding volume.

Fantastic for developing number sense and exploring number relationships.

Essential for basic counting, sorting, and introducing simple addition and subtraction.

Useful for teaching multiplication, division, and fractions.

Image: Pinterest

Helpful for developing number sense and practicing skip counting.

Useful for probability exercises and generating random numbers for various activities.

Great for pattern recognition, matching, and basic addition facts.

Essential for teaching time-telling and understanding intervals.

Images: Pinterest & Pinterest

Useful for place value activities and forming large numbers.

Excellent for identifying number patterns and supporting multiplication and division.

Virtual manipulatives: a new kind of tool

Manipulatives have gone digital! These are interactive, online versions of our physical favourites. Think of them as the maths equivalent of e-books.

Some popular virtual manipulatives include:

Online number lines

These number lines are zoomable, clickable, and free of the uneven lines that are often result of our hand-drawn versions.

Digital base ten blocks

All the functionality without the risk of losing pieces under desks.

Interactive fraction tools

Slice and dice up pieces in any way imaginable.

Whether physical or virtual, the best manipulative is the one that helps your pupils understand the concept at hand. Whether that's a handful of multilink cubes or a fancy online simulator, if it's making those mathematical lightbulbs flicker on, you're on the right track!

Implementing manipulatives in the classroom - let them play!

Whether you have a bumper pack of manipulatives, a shared bank of resources or your very own DIY versions, it's important to teach children how to use them independently. Here are some best practices for integrating manipulatives effectively into your lessons:

• Introduce gradually : Bring in manipulatives one at a time. If you don't have enough for each child, set up a 'maths table' where pupils can take turns exploring. This works particularly well with younger years where 'choosing tables' are common.
• Allow for exploration : Give children a chance to play with and explore the manipulatives before using them for instruction. Through this exploration, they can start to imagine how the resource might be useful.
• How could you use this?
• How might this help you when adding or subtracting?
• Why do you think they're different sizes - what could that represent?
• Model usage : Once children are familiar with a resource, introduce a simple maths problem and ask them to use the manipulatives to solve it. Model the problem-solving process step-by-step, then guide children through it.
• Scaffold learning : Start with highly structured activities, then gradually reduce support as pupils gain confidence. For instance, begin with direct instruction on how to use base ten blocks for place value, then move to guided practice, and finally independent problem-solving.
• Year 1: Using counters or number lines to support addition and subtraction within 20.
• Year 2: Use fraction tiles to help pupils recognise, find, name and write fractions of a length, shape, set of objects or quantity.
• Year 3: Utilising place value charts (physical or digital) so pupils can recognise 3-digit numbers (100s, 10s and 1s).
• Integrate into lesson plans : Don't treat manipulatives as an add-on. Instead, weave them into your lessons as essential tools for understanding. Plan specific points in your lessons where manipulatives will be most beneficial.
• Support diverse learners : Manipulatives can be particularly helpful for English Language Learners (ELLs) and pupils with learning disabilities. They provide a universal language of mathematics that transcends verbal communication barriers.

Images: The Average Teacher

Manipulatives across Key Stages 1 and 2

Next, let's breakdown more examples of manipulatives in the classroom by Key Stage.

Key Stage 1 (Years 1-2): Laying the foundations

In these early years, it's all about getting hands-on with numbers and shapes.

• Number and Place Value : Introduce counters, number lines, and base ten blocks. Pupils can observe how 10 ones form a 'ten stick', helping them grasp place value concepts.
• Addition and Subtraction : Utilise multilink cubes for hands-on learning. Pupils can physically join or separate cubes to represent addition and subtraction operations.
• Fractions : Fraction tiles can be effective tools for teaching fractions. They provide a visual and tactile representation of concepts like 'half' and 'quarter'.
• Geometry : Employ geoboards for creating 2-D shapes. Pupils can then be asked to match these shapes on a 3-D surface to enhance spatial understanding.

Key Stage 2 (Years 3-6): Progressing with Purpose

As our mathematicians-in-training grow, so does the sophistication of our manipulatives. We're not ditching the basics, just building on them.

• Multiplication and Division : Array cards and Cuisenaire rods are useful for these operations. For multiplying by 6, pupils can line up 6 rods of 4 to visualise the concept.
• Fractions, Decimals, and Percentages : Fraction circles can be used alongside decimal place value charts. The 100 square is effective for teaching percentages.
• Geometry : The geoboard is a helpful tool for teaching perimeter, area, and symmetry concepts in a hands-on manner.
• Statistics : Data can be represented using multilink cube bar charts or human pictograms, making statistics more engaging for pupils.

CPA Journey: From Concrete to Pictorial to Abstract

Remember, our end goal is for pupils to solve problems without relying on physical props. Here's how we might progress:

• Concrete : Pupils physically manipulate objects to solve a problem. For example, using counters to work out 5 + 3.
• Pictorial : They draw a picture or diagram to represent the problem. Our 5 + 3 might become five circles and three circles.
• Abstract : Finally, they use mathematical symbols and numbers alone. "5 + 3 = 8."

The beauty of this approach? Pupils can always 'go back' a stage if they're struggling with a new concept. Stuck on an abstract problem? Draw a picture! Need more practise? Grab those counters!

Remember, every child's journey through these stages is unique. Some might race through, others might linger longer at certain points. The key is to ensure they have a solid understanding at each stage before moving on.

An example of moving from the concrete, to pictorial, to abstract stages.

Manipulative manners

Once you have introduced your resources, speak as a class and explain that they should come up with a set of rules for how they are treated and used. Giving children ownership over the manipulatives as well as the respect to make their own rules will make them feel accountable and lessen the likelihood of negative behaviours when using manipulatives. Write the rules up as a class and display them so they can be referred to.

Storing manipulatives

NRICH recommends children having access to manipulatives “Give open access to all the resources and allow the children free reign in choosing what to use to model any problem they may be tackling. I would make sure that children of all ages had this access from 3 to 11 years old and beyond.” While this is exactly what teachers would like to replicate in their classrooms, not all classes learn in the same way and this isn’t always achievable due to space, budgets and children’s prior experiences of manipulatives.

Once you have introduced a manipulative, decide as a class where you should store it . You know what works best for your class, so consider different options such as communal drawers, a maths table, individual packs or a collection of manipulatives for each table. Set clear rules around using and treating manipulatives to ensure they are not broken or lost. Additionally, you could create a monitor for each resource so the children can take ownership and make sure they stay tidy and accounted for.

Creating a classroom culture that uses manipulatives will aid children’s fluency and help develop their ability to solve problems, reason mathematically and share! If manipulatives are introduced in a considered and gradual way, with clear boundaries from an early age, children should see them as part of everyday learning and they will not be a novelty. They will be seen as tools instead of toys — and hopefully no more multilink towers!

For the community, by the community

Maths — No Problem! Community Event Conference, 28 November at The Royal Society, London.

Lily Lanigan

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